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Design Sensitivity and Optimization User`s Guide

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1. This single DRESP2 entry defines a maximum shear for every element in the group Because the underlying first level responses are element level the DRESP2 is as well This also applies to DRESP3 type responses 2 You can multiply nest DRESP2 entries In other words each DRESP2 entry can reference one or more other DRESP2 entries However you can only nest integrated response DRESP1 entries which are referenced as DRESP2 entries a single time i e you can only reference it from a DRESP2 entry that isn t further nested You cannot nest DRESP3 entries inside each other Limitations in Writing Second Level Responses A few restrictions apply to the formulation of second level responses Unless you use a DRSPAN Case Control command you can only reference subcase dependent responses from a DRESP2 or DRESP3 entry with other responses from the same subcase If a DRESP2 or DRESPS entry contains one or more DRESP1 entries that are assigned with a DRSPAN command to particular subcases then all DRESP1 entries in the same DRESP2 or DRESP3 entries must be assigned with the DRSPAN command You can also use a DRSPAN command to assign any integrated response DRESP1 entries to a specific subcase When combining responses of different types e g stress strain etc the responses must be scalar quantities That is the corresponding DRESP1 entries must define a single response only This implies that for displacements only a single grid ma
2. PARAM AUTOSPC YES PARAM POST 0 7 34 Design Sensitivity and Optimization User s Guide Example Problems PARAM GRDPNT 0 PARAM MAXRATIO1 0E 8 CORD2S 2 0 0 0 0 0 1A 2 1A 2 0 0 0 0 0 0 1000 000 1B 2 1B 2 1000 000 0 0 0 0 1C 2 1C 2 CORD2C 1 0 0 0 0 0 1A 1 1A 1 0 0 0 0 0 0 1000 000 1B 1 1B d 1000 000 0 0 0 0 1C 1 1C 1 GRID 1 0 0 0 0 0 0 0 0 GRID 165 0 10 000 2 000 4 000 0 GRDSET 456 CHEXA 1 1 1 2 13 12 34 35 EA 1 EA 1 46 45 CHEXA 80 1 120 121 132 131 153 154 EA 80 EA 80 165 164 MAT1 1 2 0680E 05 0 28999999166 MA 1 MA 1 1 00000000 1 169999996E 05 MB 1 MB 1 1500000 00 1500000 00 68000 00 MC 1 MC 1 PSOLID 1 1 0 0 0 0 SPC 1 1 123456 0 0 SPC 1 12 123456 0 0 SPC 1 23 123456 0 0 SPC 1 34 123456 0 0 SPC 1 45 123456 0 0 SPC 1 56 123456 0 0 SPC 1 67 123456 0 0 SPC 1 78 123456 0 0 SPC 1 89 123456 0 0 SPC 1 100 123456 0 0 SPC 1 111 123456 0 0 SPC 1 122 123456 0 0 SPC BE 133 123456 0 0 SPC 1 144 123456 0 0 SPC 155 123456 0 0 SPC1 1 456 1 THRU 165 FORCE 1 143 0 0 5 0 0 0 0 50 0 FORCE 1 154 0 1 0 0 0 0 0 50 0 FORCE n 165 0 0 5 0 0 0 0 50 0 PARAM DESPCH 1 PARAM NASPRT 1 DESVAR ID LABEL XINIT XLIB XUB DELXV DESVAR 1 UPPER 1 0 00 7 00 0 4 DESVAR 2 LOWER 1 0 00 7 00 0 4 DVBSHAP DVID AUXMID COL1 SF1 COL2 SF2 DVBSHAP 1 1 1 1 0 DVBSHAP 2 1 2 1 0 DLINK ID DDVID CO CMULT IDV1 et IDV2 02 IDV3 c3 DLINK 12 1 0 1 1 0 BOUND
3. 3 44 Approximate Optimization Control Parameters An Overview sn 3 45 Move Limits and Approximate Optimization llle 3 46 Numerically Identifying the Active and Violated Constraints 3 52 Optimizer Convergence Parameters oooooooocanean ee 3 53 Convergence Ar Bmmw 3 54 Convergence of Design Cycles Hard and Soft Convergence o o o o o oo ooo o 3 54 Soft Convergence Decision Logic liliis 3 56 Hard Convergence Decision Logic o o oooooooernoerno rosa 3 57 Input Data ccs iw eh nez ea OR ERREUR a URS we Pe a ew RR wee Ro eR n 4 1 File Management Section Executive Control n a aaa aeaa 4 2 Solution 200 see suc et cet n etit e oe OE oh ER RC ol ERR C C RS DR REESE RR DR 4 2 Shape Optimization and the File Management Section 20205 4 2 Case Control Section aa 4 3 Analysis Discipline Definition lere 4 3 Design Model Definition lt lt oooooooooorooenr nn 4 4 Design Response Characterization oo oooooooooooooerarea 4 5 Shape Basis Vector Computation ollis 4 5 Bulk Data Entries 2 0 0 0 0 ns 4 6 DCONADD Ta Irem 4 6 pues cip nc 4 7 DEQATN 232053040042 dele See Sat aras xc um tado f adiu D fed 4 8 DESVAR eee e ed Sn ed re nae anes Ao ae a a ee we hee Me eg 4 9 DR m 20 4 24 bas Oe Sohne Ce hans hat eet ete ee eee 4 10 4 Design Sensitivity and
4. ATT2 Sait DRESP1 20 W WEIGHT DRESP1 21 U4 DISP DRESP1 22 V4 DISP DRESP1 23 sis STRESS PROD DRESP1 24 S2 STRESS PROD DRESP1 25 S3 STRESS PROD CONSTRAINTS DCONSTR DCID LALLOW UALLOW DCONSTR 21 0 20 0 20 DCONSTR 21 0 20 0 20 DCONSTR 21 15000 20000 DCONSTR 21 15000 20000 DCONSTR 21 15000 20000 OPTIMIZATION CONTROL DSCREEN DISP DSCREEN STRESS DOPTPRM IPRINT Listing 7 13 RESTART VERSION LAST NOKEEP ASSIGN MASTER d200xlc MASTER ID EDS D200X1 TIME 10 SOL 200 OPTIMIZATION CEND TITLE SYMMETRIC THREE BAR TRUSS DESIGN OPTIMIZATION D200X1 SUBTITLE BASELINE 2 CROSS SECTIONAL AREAS AS DESIGN VARIABLES ECHO SORT SPC 100 DISP ALL STRESS ALL DESOBJ MIN 20 DESIGN OBJECTIVE DRESP ID DESSUB 21 DEFINE CONSTRAINT SET FOR BOTH SUBCASES ANALYSIS STATICS SUBCASE 1 LABEL LOAD CONDITION 1 LOAD 300 SUBCASE 2 LABEL LOAD CONDITION 2 LOAD 310 BEGIN BULK 34 ENDDATA Listing 7 14 7 70 Design Sensitivity and Optimization User s Guide Appendix A Glossary of Terms Active Constraint a constraint whose numerical value is near zero In NX Nastran the constant CT default 2 0 03 is used to measure whether a constraint is active or inactive A constraint whose value is less than CT is deemed inactive and if its value is greater than CT it is considered active This information is used at th
5. DCONSTR 10 201 7 99 8 01 doptprm desmax 40 pl 1 p2 8 convi 0 01 S Locus Di aa Vi Mii Di ONES HUNE NE Bios esie D revu 0 param optexit 2 Listing 7 8 7 8 Twenty Five Bar Truss Superelement Optimization This problem often seen in the design optimization literature calls for a minimum weight structure subject to member stress Euler buckling and joint displacement constraints The structure is shown in Figure 7 20 The formulation of the buckling constraints is a good example of constructing normalized constraints based on user defined structural responses In addition this problem will be substructured in order to illustrate superelement optimization 7 44 Design Sensitivity and Optimization User s Guide Example Problems Superelement 1 Superelement 0 Figure 7 20 Twenty Five Bar Truss Analysis Model Description Three dimensional truss tower Symmetric with respect to the x z plane and y z plane Weight density 0 1 Ibs in Materials E 1 0E7 psi Two distinct loading conditions Design Model Description Minimization of structural weight Design variables Cross sectional areas linked to eight independent design variables Constraints Allowable stress Tensile 40 000 psi Compressive 40 000 psi Displacement constraints 0 35 inches at grid 1 and 2 for all translational degrees of freedom Euler buckling constraints for compressive members assuming tubular section diameter to thickne
6. A l FEA __ Finite Element Analysis DESCON Constraint Evaluation and Screening DOMD 1 Hard Convergence l Convergence Check Based on PSLGDV the Data of Two Consecutive Designs FEA Sensitivity Analysis RESPSEN and Optimization with Approximate Models DOM9 E DOM10 Print Output DOMI2 Ea Soft Convergence 3 Check If the Optimizer UPDATE zi Update Analysis Made Progress Model Figure 3 12 Convergence Testing and Program Flow If soft convergence has not terminated the design process a finite element evaluation of the new proposed design is performed followed by constraint evaluation and screening If the analysis results are not appreciably different from those of the prior cycle hard convergence is achieved and the design cycle process is terminated If the hard convergence criteria are not satisfied the design cycle process continues Maximum Number of Iterations Design Cycles DESMAX If neither hard nor soft convergence is achieved the design process will continue until convergence is indicated or until the maximum allowable number of iterations design cycles is met This default maximum number of iterations is five but can be changed on the DOPTPRM entry using DESMAX The following paragraphs flowchart and discuss the soft and hard convergence decision logic Design Sensitivity and Optimization User s Guide 3 55 Chapter 3 Design Sensitivity and Optimization in NX Nastran Soft Convergence
7. 2 LA dx 0 x F x Ax f x Equation 1 25 Since all terms of power Ax and higher have been omitted the error in the approximation is on the order of Ax2 This situation is shown in Figure 1 16 where the error increases with increasing values of Ax Note that the approximation would be exact if the original function were linear Design Sensitivity and Optimization User s Guide 1 29 Chapter 1 Getting Started x IORA Figure 1 16 Errors in Approximating Functions In design optimization we are concerned not just with a single independent variable but rather with a vector of design variables t Under this condition the approximations for the objective and constraint functions become 0 E F x 42 F x VE AX x 1 0 gt D epa x IB 2 PER gt be 4 Il Equation 1 26 where a gradient term replaces the first derivative term of Eq 1 25 We have not yet addressed the issue of how to determine the first derivative or gradient information This information comes from the design sensitivity analysis We introduce design sensitivity for linear static analysis here but reserve the discussion of the other analysis disciplines for Design Modeling for Sensitivity and Optimization Many of our constraints and perhaps the objective function as well are based on responses that depend on the solution of the static equilibrium equations K u P Equation 1 27 For examp
8. DV 3 THETA 60 60 VPREL1 ID TYPE PID FID PMIN PMAX co i 35 DVID1 COEF1 DVID2 COEF2 DVPREL1 11 PCOMP 1 14 DT1 DT1 2 1 0 DVPREL1 12 PCOMP 1 18 90 0 DT2 DT2 2 1 0 DVPREL1 13 PCOMP 1 24 DT3 DT3 3 1 0 DVPREL1 14 PCOMP 1 28 90 0 DT4 DT4 3 1 0 DVPREL1 15 PCOMP 1 34 DT5 DT5 3 1 0 DVPREL1 16 PCOMP 1 38 90 0 DT6 DT6 3 1 0 DVPREL1 17 PCOMP 1 44 DT7 DT7 2 1 0 DVPREL1 18 PCOMP 1 48 90 0 DT8 DT8 2 1 0 DVPREL1 100 PCOMP 1 3 100 DZ1 DZ1 1 0 04 RESP1 ID LABEL RTYPE PTYPE REGION ATTA ATTB ATT1 ATT2 DRESP1 10 W WEIGHT DESOBJ 10 W MIN DRESP1 1 FP CFAILUREELEM 5 1 64 DRESP1 2 FP CFAILUREELEM 5 2 64 DRESP1 3 FP CFAILUREELEM 5 3 64 DRESP1 4 FP CFAILUREELEM 5 4 64 DRESP1 5 FP CFAILUREELEM 5 5 64 DRESP1 6 FP CFAILUREELEM 5 6 64 DRESP1 7 FP CFAILUREELEM 5 7 64 DRESP1 8 FP CFAILUREELEM 5 8 64 CONSTR DCID RID LALLOW UALLOW DCONSTR 20 1 001 0 9 DCONSTR 20 2 001 0 9 DCONSTR 20 J 001 0 9 DCONSTR 20 4 001 0 9 DCONSTR 20 5 001 0 9 DCONSTR 20 6 001 0 9 DCONSTR 20 Ty 001 0 9 DCONSTR 20 8 001 0 9 SDOPTPRM1 gt 20 0 3 DOPTPRM IPRINT 7 DESMAX 20 DELP Da CONV2 1 0E 3 ENDDATA Listing 7 10 Design Sensitivity and Optimization User s Guide 7 55 Chapter 7 Example Problems 7 10 Acoustic Optimization Acoustic Optimization introduces acoustic pressures as a design response These are computed from a
9. 13 14 S21 22 23 24 S31 32 33 S34 S41 942 43 844 S51 852 53 S54 S61 S62 S63 S64 S71 872 73 S74 S81 S82 S83 S84 Di D2 Wo 0 015625 G 9 9 PID FID PMIN DVID2 COEF2 1 4 2 4 5 4 4 4 5 4 6 4 j 7 4 8 4 r responses RTYPE PTYPE REGION STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL STRESS PSHELL DISP 4 DISP i to be used as the WEIGHT Define the design constraints SDCONSTR DCID DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 DCONSTR 10 RID 15 16 22 LALLOW UALLOW 50000 50000 29000 29000 50000 50000 29000 29000 50000 50000 29000 29000 50000 50000 29000 2
10. CONM2 10 2 P 15 0 CONM2 11 3 15 0 CONM2 12 4 gt 15 0 CONM2 14 6 15 0 CONM2 15 7 15 0 CONM2 16 8 A 15 0 MATI 1 1 03E7 0 3 0 1 PARAM WTMASS 0 002588 PARAM GRDPNT 1 Define the design variables SDESVAR ID LABEL XINIT XLB XUB DELXV DESVAR 1 Al 1 0 0 01 100 0 DESVAR 2 A2 1 0 0 01 100 0 DESVAR 3 A3 poo 0 01 100 0 DESVAR 4 Th 0 2 0 001 10 0 DESVAR 5 T2 0 2 0 001 10 0 DESVAR 6 T3 Dek 0 001 10 0 Relate the design variables to analysis model properties SDVPREL1 1D TYPE PID FID PMIN PMAX CO 6 DVIDD1 COEF1 DVID2 COEF2 DVPREL1 1 PROD 201 4 i A DP1 DP1 T 1 0 DVPREL1 2 PROD 202 4 DP2 DP2 2 1 0 DVPREL1 3 PROD 203 4 4 A DP3 DP3 dz 1 0 DVPREL1 4 PSHELL 204 4 7 i n 7 DP4 DP4 4 1 0 DVPREL1 5 PSHELL 205 4 r r 7 DP5 DP5 5 1 0 DVPREL1 6 PSHELL 206 4 r m P DP6 DP6 6 1 0 Identify the analysis responses to be used in the design model SDRESP1 ID LABEL RTYPE PTYPE REGION ATTA ATTB ATT1 ATT2 DRESP1 1 W WEIGHT DRESP1 2 F1 EIGN 2 Ha Use these responses to define the objective in case control and the constraints DCONSTR DCID RID LALLOW UALLOW DCONSTR 10 2 15791 200000 0 lower bound 20 Hz Optional override of design optimization parameters DOPTPRM IPRINT 2 DESMAX 10 D
11. ELEMENT GROUP 1 CQUADA 101 1 100 101 201 200 100 100 100 100 100 8 CQUAD4 102 i 101 102 202 201 1 1 1 1 1 7 PSHELL 1 150 08 150 ELEMENT GROUP 2 CQUAD4 202 2 201 202 302 301 100 100 100 100 100 7 CQUADA 203 2 202 203 303 302 1 1 1 1 1 6 PSHELL 2 150 08 150 ELEMENT GROUP 3 CQUAD4 303 3 302 303 403 402 100 100 100 100 100 6 CQUAD4 304 3 303 304 404 403 1 1 1 1 1 5 PSHELL 3 150 08 150 ELEMENT GROUP 4 CQUAD4 404 4 403 404 504 503 100 100 100 100 100 5 CQUAD4 405 4 404 405 505 504 1 1 1 1 1 4 PSHELL 4 150 08 150 ELEMENT GROUP 5 CQUADA 505 2 504 505 605 604 100 100 100 100 100 4 CQUAD4 506 9 505 506 606 605 1 1 1 1 1 3 PSHELL 5 150 08 150 ELEMENT GROUP 6 CQUAD4 606 6 605 606 706 705 100 100 100 100 100 3 CQUAD4 607 6 606 607 707 706 1 1 1 1 1 2 PSHELL 6 150 08 150 ELEMENT GROUP 7 CQUAD4 707 7 706 707 807 806 100 100 100 100 100 2 CQUAD4 708 75 707 708 808 807 1 1 1 1 1 7 42 Design Sensitivity and Opti
12. G X J J J X XXXXXXXX TRS H 1 0 lil i Px Region 1 Region 2 Region 3 Retained Responses 1 0 2 Assume NSTR 2 Figure 3 5 Constraint Regionalization Constraint Screening and Sensitivity Analysis Constraint screening precedes sensitivity analysis as can be seen from the flowchart in Figure 3 1 and in the detailed Solution 200 flowchart in Design Modeling for Sensitivity and Optimization The default values of TRS and NSTR are usually fine for most design optimization problems however for design sensitivity analysis these default values may not be satisfactory It may be that some of the responses for which you had requested that sensitivities be computed are screened out To ensure that all responses are retained you may need to set TRS to a large negative number perhaps 10 or 100 Also depending on the number of responses per region it may be necessary to increase NSTR This combination ensures that all constraints pass both levels of screening and all responses are retained for sensitivity analysis This procedure is the only way to turn off the screening Formal Approximations NX Nastran uses formal approximation to avoid the high cost associated with repeated finite element analyses during design optimization Two basic types of approximation are used 3 14 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran Direct variable approximations e Reci
13. LOWER SURFACE CQUADA 1000 2 1 2 13 12 0 0 CQUADA 1001 2 2 3 14 13 0 0 CQUADA 1002 2 3 4 15 14 0 0 CQUADA 1003 2 4 5 16 15 0 0 CQUADA 1004 2 5 6 17 16 0 0 CQUADA 1005 2 6 7 18 17 0 0 CQUADA 1006 2 7 8 19 18 0 0 CQUADA 1007 2 8 9 20 19 0 0 CQUADA 1008 2 9 10 21 20 0 0 CQUADA 1009 2 10 11 22 21 0 0 CQUADA 1010 2 12 13 24 23 0 0 CQUADA 1011 2 13 14 25 24 0 0 CQUADA 1012 2 14 15 26 25 0 0 CQUADA 1013 2 15 16 27 26 0 0 CQUADA 1014 2 16 17 28 27 0 0 CQUADA 1015 2 17 18 29 28 0 0 CQUADA 1016 2 18 19 30 29 0 0 CQUADA 1017 2 19 20 31 30 0 0 CQUADA 1018 2 20 21 32 31 0 0 CQUADA 1019 2 21 22 33 32 0 0 UPPER SURFACE CQUADA 950 2 133 144 145 134 0 0 CQUADA 951 2 134 145 146 135 0 0 CQUADA 952 2 135 146 147 136 0 0 CQUADA 953 2 136 147 148 137 0 0 CQUADA 954 2 137 148 149 138 0 0 CQUADA 955 2 138 149 150 139 0 0 CQUADA 956 2 139 150 151 140 0 0 CQUADA 957 2 140 151 152 141 0 0 CQUADA 958 2 141 152 153 142 0 0 CQUADA 959 2 142 153 154 143 0 0 CQUADA 960 2 144 155 156 145 0 0 CQUADA 961 2 145 156 157 146 0 0 CQUADA 962 2 146 157 158 147 0 0 CQUADA 963 2 147 158 159 148 0 0 CQUADA 964 2 148 159 160 149 0 0 CQUADA 965 2 149 160 161 150 0 0 CQUADA 966 2 150 161 162 151 0 0 CQUADA 967 2 151 162 163 152 0 0 CQUADA 968 2 152 163 164 153 0 0 CQUADA 969 2 153 164 165 154 0 0 MAT1 11 2 1E 5 0 8E 5 0 3 0 00 PSHELL 2 11 0 20 11 0 0 SPC1 200 123456 1 12 23 SPC1 200 12 11 22 33 SPC1 200 123456 34 THRU 165 SPCD 220 11 3 1 0 22 3 1 0
14. THIS SECTION CONTAINS THE PROPERTY AND MATERIAL BULK DATA ENTRIES PSHELL 3 1 100 1 PSHELL 1 1 02493 1 PSHELL 2 1 01953 1 pshell 4 1 02047 1 pshell 5 1 02596 1 pshell 6 1 02175 1 pshell 7 1 02426 1 THIS SECTION CONTAINS BULK DATA FOR SUPERELEMENT 0 GRID 1 0 0 0 0 0 0 GRID 2 Zi 0 0 0 0 GRID 3 2 0 0 T GRID 1293 1 8 a T GRID 1294 1 6 9 Jd GRID 1295 ls 5 Li CQUAD4 1 1 1 9 29 28 CQUAD4 2 Hh 9 10 30 29 CQUADA 999 3 1139 1140 275 296 7 62 Design Sensitivity and Optimization User s Guide Example Problems CQUADA 1000 3 1140 897 5 275 THIS SECTION CONTAINS THE LOADS CONSTRAINTS AND CONTROL BULK DATA ENTRIES SPC 1 1 123 0 0 SPC 1 2 123 0 0 SPC 1 986 4 SPC 1 987 4 SPC 1 975 4 MAT1 1 2 411 3 7600 a JR doom e Gumcoccs desec ecee oS a Ya Lee SSSSSSSSSSSSSSSSSSSSSSSSSSSSsssssssss acoustic model SSSSSSSSSSSSS SSSSSSSSSsSsSssssssssss THIS SECTION CONTAINS BULK DATA FOR SUPERELEMENT 0 Sce 2 ee c e d DASS Loses g Dicere 10 GRID 10001 0 0 0 0 0 0 GRID 10002 2 0 0 0 0 1 GRID 10003 2 0 0 Es L GRID 12540 EXA Is 2 1 GRID 12541 2 1 ale 1 CHEXA 10001100 10004 10126 10127 10009 10018 10137 10138 10019 CHEXA 10002100 10009 10127 10128 10010 10019 10138 10139 10020 CHEXA 11999100 12411 12530 12531 12412 12422 12540 12541 12423 CHEXA 12000100 12412 12531 12532 12413 12423 12541 10006 12424 p
15. This section discusses each of these tasks individually Analysis Discipline Definition Design optimization in NX Nastran is multidisciplinary a number of different analyses may be performed in Solution 200 and the results used simultaneously in optimization The analysis types are defined on a subcase basis using the ANALYSIS Case Control command It can be set to any of the following values STATICS MODES BUCK DFREO MFREO MTRANS SAERO FLUTTER ANALYSIS The following are a few Case Control issues to be aware of when using the ANALYSIS command If ANALYSIS is specified above the subcase level all subsequent subcases apply to that analysis type until redefined in a later subcase e The buckling subcase defined by ANALYSIS BUCK refers to the first STATICS subcase e Multiple boundary conditions may be used in Solution 200 for ANALYSIS STATICS Design Sensitivity and Optimization User s Guide 4 3 Chapter 4 Input Data Limited multiple boundary condition support exists for ANALYSIS MODES A normal modes subcase can have its own unique set of boundary conditions but only one normal modes subcase can exist in a Solution 200 run Design Model Definition The design model definition process in Case Control includes identification of the design objective function and the design constraint sets Design Objective Identification The design objective is identified in Case Control with the command pEsosj
16. 0 of modifying a current best design instead of using a completely new design An initial design E must be provided but it need not satisfy all of the constraints Indeed one of the most powerful uses of optimization is to find a feasible solution to a complicated problem To determine a search direction that improves the design gradients of the objective and critical constraints must be supplied Ideally these are computed analytically or semi analytically as is done in NX Nastran This process dramatically increases the size of the problem that can be efficiently solved Finally a one dimensional search is performed by trying several values of a and interpolating for the one that gives a minimum objective while satisfying the constraints During this process the objective and the constraints must be repeatedly evaluated Here the use of approximation methods plays a major role because the evaluation of these constraints would otherwise require a full finite element analysis In NX Nastran we only evaluate the approximate functions which is a relatively inexpensive process In the following sections we will outline the overall optimization process and identify the steps taken to reach the optimum The presentation here will be necessarily brief and the reader is referred to Vanderplaats Numerical Optimization Techniques for Engineering Design with Applications for more details D 2 The Modified Feasible Direction Algorithm
17. AXMPR2 UPDATE End Design Cycle Loop Design Sensitivity and Optimization User s Guide 5 11 OPTEXIT 7 includes computation of sensitivities for the current design cycle all operations up to DOMS9 5 12 Design Sensitivity and Optimization User s Guide Chapter 6 Output Features and Interpretation Output Controlling Parameters Design Optimization Output Design Sensitivity Output Design Sensitivity and Optimization User s Guide 6 1 Chapter 6 Output Features and Interpretation Since design optimization is an iterative process a significant amount of output may be generated depending on the quantity of data requested for each design cycle and the number of cycles performed In design sensitivity and optimization output may be generated from the following four sources 1 NX Nastran analysis 2 Design sensitivity analysis 3 Design optimization including optimizer output 4 Convergence tests and design model updates There are a few parameters that affect the level of detail and the frequency of output for design cycle dependent data While covered elsewhere in this Guide these parameters are collected and summarized in this chapter in Output Controlling Parameters Following this summary in Design Optimization Output there is an output example from a design optimization problem in the test problem library which is supplied with your NX Nastran installation software This
18. Computes the design sensitivity coefficient matrix DSCM2 for output purposes This BEN is invoked only if OPTEXIT is 4 Otherwise design sensitivity information is available to the optimizer via matrix DSCM DOMS Performs the optimization task using the previously created approximating functions DOM10 Provides printed output of the results of the approximate optimization This output includes the values of the proposed new design variables and the estimated values of the objective and constraint functions The amount of output is determined by the parameters P1 and P2 on the DOPTPRM Bulk Data entry DOM11 Updates the element property table EPT grid coordinates COORDO and en Been data blocks GEOM1N in accordance with the new vector of design variables Performs both hard and soft convergence testing outputs the results of these tests updates the design optimization history table and prints out the summary of design cycle history It is called for every normal exit condition in Solution 200 Sets up nonrepetetive tables for design variables design variable linking DTABLE constants designed property attributes initial design variable and property values constraint screening data and optimization parameters 5 2 Design Sensitivity and Optimization User s Guide Solution Sequences Performs pre processing operations for the shape basis vectors sensitivity and optimization It is called in PREDOM for direct input of ba
19. DOPR2 DOPR3 DOPR4 DOPR5 DSABO DSAD DSAE DSAF DSAH DSAJ DSAL DSAP DSAR DSAW Design Sensitivity and Optimization User s Guide 5 3 Chapter 5 Solution Sequences Outputs parameters used for DMAP flow control Depending on the types of retained responses listed in data block DRSTBL parameters are output to indicate whether or not response sensitivities are to be recovered for a particular superelement Generates a correlation table describing the design response column order for the DSCM DSCM2 matrices Design Sensitivity Vector Generator 1 computes terms in the sensitivity analysis pseudo load vectors that depend on AK AB and AM Design Sensitivity Vector Generator 2 This module computes the thermal load contribution to the pseudo load vector Design Sensitivity Vector Generator 4 Forms u Au finite difference change in displacements for design variables that affect mass and or stiffness terms Partitions the original input case control onto individual Case Control records based on analysis type It also outputs parameters that are used at the DMAP level to direct analysis for the various disciplines Partitions the design model entries table EDOM on superelement The result EDOMS is an analysis discipline and superelement partition of the original EDOM See also the DMPR module description Builds the superelement design sensitivity processing data block DSLIST that is used to direct the pseudo load and respons
20. DV ID LABEL INITIAL 1 2 3 4 5 i 1 Al 1 0000E 00 7 6865E 01 2 2 A2 2 0000E 00 7 1480E 01 3 3 A3 1 0000E 00 7 6865E 01 USER INFORMATION MESSAGE 6464 DOM12E RUN TERMINATED DUE TO MAXIMUM NUMBER OF DESIGN CYCLES Lis Figure 7 28 Single Design Cycle Turning now to the pseudo restart the punch file from the previous design cycle is included in the restart deck listed in Listing 7 13 Only the design model portion of the deck has been included in Listing 7 13 since the analysis model portion is identical to Listing 7 12 The optimization now begins from the point in the design space defined by the new DESVAR entries The DOPTPRM and DSCREEN entries are updated in this deck as well to allow more design iterations DESMAX reduce the move limits to 50 DELP and set the truncation threshold back to 0 5 TRS Leaving out the DSCREEN entries altogether accomplishes the same thing since this is the TRS default The optimization results from this input are listed in Figure 7 29 which are seen to be slightly better than the original D200X1 If this job were much larger and more complex the ability to pause examine the results and readjust some of the optimization parameters could be indispensable Design Sensitivity and Optimization User s Guide 7 65 Chapter 7 Example Problems ARK RARA RARA ko ke ek ko ke ok koe ok koe ok TR oe
21. Design Modeling for Sensitivity and Optimization DRESP1 501 selects the x component of displacement at Grid 100 and DRESP1 502 selects the corresponding y component These displacements may also be used in a synthetic relation For example to express the total x y plane displacement at Grid 100 as the square root of the sum of displacement squares we could use DEFINITION OF EQUATION SDEQATN EQUID F DEQATN 510 U UX UY SQRT UX 2 UY 2 SYNTHETIC RESPONSE DRESP2 ID EQID REGION DESVAR 1 DVID2 DTABLE LABEL2 f DRESP1 NR2 Se DNODE DIR1 NID2 D 520 510 DRESP1 502 The equation data is supplied on the DEQATN entry while the DRESP2 entry defines the arguments to this equation in this case the two first level responses DRESP1 501 and 502 DRESP2 520 can now be used either as the objective function or as a constraint Efficiencies in First Level Response Identification 1 Since a large number of responses are often used in design the DRESP1 entry must be able to efficiently define many responses using few entries For example a single DRESP1 entry can identify the z component of displacement at grid points 100 101 and 102 as follows SDRESP1 1D LABEL RTYPE PTYPE REGION ATTA ATTB ATT1 4 ATT2 DRESP1 100 UZ DISP 101 102 This list can be extended as necessary to any number of grids Item codes used on DRESP1 entri
22. LOAD 300 CC FOR OPTIMIZATION ANALYSIS STATICS DESOBJ MIN 8 OBJECTIVE FUNCTION DEFINITION DESSUB 10 CONSTRAINT SET SELECTION BEGIN BULK SPARAM OPTIM NO PARAM OPTEXIT 4 S o ANALYSIS MODEL S Ei EEE EE aa oia aa Soi a aso NEE E NEEE NEE S CEE EREE E E E NEER MAT1 110 10 0E6 0 33 0 1 M1 M1 50000 50000 29000 GRDSET 4 GRID Ji 0 0 0 0 0 0 GRID 2 20 0 0 050 CBEAM di 100 1 2 0 0 1 0 00 PBEAM ENTRY INPUT WITH SLIGHT ERROR IN TERMS THIS HELPS VALIDATE THE DE SIGN MODEL BECAUSE USER WARNING MESSAGE WILL BE ISSUED CONFIRMING OVERRIDE PBEAM 100 110 2 01 167 667 P11 P11 0 0 1 01 0 0 1 04 P12 P12 YES 1 0 0 51 0104 042 P13 P13 0 0 0 51 0 0 0 51 SPC1 100 123456 1 FORCE 300 2 20000 0 0 0 0 0 100 Ss P ee DESIGN MODEL S A asse ED Ce it Ee Eimer m SS A SS a ea E END A DATA A I1 12 C2 D2 SDESVAR ID LABEL XINIT XLB XUB DELXV DESVAR 1 Bl 1 0 Oud 10 0 DESVAR 2 H1 2 0 0 2 20 0 SDVPREL2 ID TYPE PID FID PMIN PMAX EQID P St DESVAR DVID1 DVID2 A F 4 DTABLE CID1 CID2 T DVPREL2 1
23. Move limits on properties are applied to every property referenced by the design model There is no provision to make these limits property dependent that is to apply a different DELP for different properties Eq 3 101 and Eq 3 103 effectively form a box around the current design This effect is shown in Figure 3 8 where these move limits are shown for successive design cycles in a two design variable space For the first cycle the approximate optimum is found to lie at a corner of the box where the objective is minimized and there are no active constraints As a result of the second cycle one of the constraints is slightly violated due to errors in the approximation By the third cycle a near optimal design has been found Of course the situation is usually more complex than in this simple design space The intent of Figure 3 8 is merely to suggest some general features of the overall process Even though the optimizer deals with an approximate model note that the properties are always known precisely That is p j p j x is explicitly given by the design variable to property relations DVPREL1 and DVPREL2 and these relations are made available to the optimizer Design Sensitivity and Optimization User s Guide 3 47 Chapter 3 Design Sensitivity and Optimization in NX Nastran X E Constraint lt Boundaries X1 Figure 3 8 Sequence of Approximations Avoiding Small Moves with DPMIN The D
24. Two methods are used to test for convergence with respect to overall design cycles These methods are denoted as soft convergence and hard convergence Soft convergence is based on the results of the approximate optimization while hard convergence is based on finite element analysis results Recall that at the outset of each design cycle a finite element analysis and a sensitivity analysis are performed The approximate model is then constructed followed by an optimization with respect to this approximation Once a corresponding approximate optimum is found the proposed design is submitted for another finite element analysis to compute the updated responses thus marking the beginning of the next design cycle Hard convergence compares the results of this most recent finite element analysis with those from the previous design cycle Since this test compares the exact results within the limits of the finite element approximations from two consecutive analyses the conclusions are said to be based on hard evidence Since this test is conclusive this is the default test for determining whether or not to terminate the design cycle process Soft convergence compares the design responses from the approximate optimization with those obtained from the previous finite element analysis This test although not as conclusive as hard convergence is often an acceptable criterion to indicate convergence For example if a particular analysis model has a high
25. p f x e Equation 2 9 where pjis the j th property expressed as a function of a collection of design variables and constants The same design variables may appear in both Type 1 and Type 2 relations simultaneously This is depicted in Figure 2 2 where the design variable x appears in both DVPREL1 and DVPREL2 bulk entries DESVAR DEQATN DTABLE mE Analysis DVPREL1 gt DVPREL2 Model Pit Cos FTC xs fp As C5 Properties A eee RN Figure 2 2 Design Variable to Property Relations Design Sensitivity and Optimization User s Guide 2 9 Chapter 2 Design Modeling for Sensitivity and Optimization The DVPREL1 bulk entry defines linear property variations and requires only design variable input from DESVAR bulk entries All necessary constants are supplied directly on the DVPREL1 bulk entry Thus Figure 2 2 shows the DVPREL1 block with a single input and a single output The DVPREL2 bulk entry relies on an equation to describe the generally nonlinear property variations The input to this equation can include both design variable input from DESVAR bulk entries and table constant input from DTABLE bulk entries Thus Figure 2 2 shows the DEQATN block with two possible inputs and a single output and the DVPREL2 block with a single input and a single output Selective Modification of Design Variable Bounds Many practical problems have properties that are linearly related to a single design var
26. 15 0 T 37 1 30 DLINK 22 12 P 01 l 15 05 2 0 9 y 8 0 81 DLINK 23 I3 F 015 1 1 405 2 0 8 35 0 64 DLINK 24 14 F Ody Ly 1 0 2 0 7 t 3 0 49 DLINK 25 155 r Oly 1j 1205 2 0 6 t 3 0 36 DLINK 26 16 7 01 1 AsO 25 0 5 3 0 25 DLINK 27 155 7 20d ty 1 29 25 0 4 3 0 16 DLINK 28 18 OL T 1 01 2j Di 35 0 09 DLINK 29 19 f 01 d 1 05 2 0 2 y 37 0 04 DLINK 30 20 lt 01 1 1 0 2 0 1 3 0 01 DESIGN EQUATION DEQATN 40 I2 H 0 005 H 3 12 TYPE 2 DESIGN VARIABLE TO PROPERTY RELATIONS SDVPREL2 ID TYPE PID FID PMIN PMAX EQID S DESVAR DVID1 DVID2 a 4 DTABLE CID1 CID2 3 10 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran DVPREL2 41 PBAR 101 5 1 0E 12 40 n t DESVAR 11 DVPREL2 42 PBAR 102 5 1 0E 12 40 I t DESVAR 12 DVPREL2 43 PBAR 103 5 1 0E 12 40 t DESVAR 13 DVPREL2 44 PBAR 104 5 1 0E 12 40 t DESVAR 14 DVPREL2 45 PBAR 105 5 1 0E 12 40 Listing 3 2 The basis function multipliers are x1 x2 and x3 Their initial values of 1 0 0 0 and 0 0 correspond to a constant height beam section Negative lower bounds allow a negative contribution from any of the basis functions The dependent variables are defined using DESVAR entries 11 through 20 Since the dependent variables de
27. 2 F ag t aix a x Equation D 46 where aq ay a are constants and x is the variable In this case x corresponds to the one dimensional search parameter a Next assume that three values of F exist for three values of x as listed in the table below Substituting the values of F and x into Eq D 46 results in three equations with three unknowns Gy 0 5 a 0 25 a 125 Equation D 47 ag 1 0 e ai 1 00 e a 3 0 Equation D 48 ag 2 0 e a 4 00 e a 2 0 Equation D 49 Solving for the values of ag a and a results in dg 2 0 dy 8 0 a 3 0 Equation D 50 Substituting this into Eq D 46 results in F 208004300 Equation D 51 Eq D 51 can be used in different ways depending on whether the function F is the objective or a constraint First assume that this is the objective function and we wish to find the value of x that minimizes it The function F has a minimum or maximum value when the derivative of F with respect to x is zero Differentiating Eq D 51 with respect to x yields Design Sensitivity and Optimization User s Guide D 21 Appendix D Numerical Optimization dF 6 0 6 x dx 8 0 6 0 x Equation D 52 Setting this to zero and solving for x gives a proposed solution x 1 33 Substituting this into Eq D 51 gives an estimated value of the function of F 3 33 However this is not sufficient to indicate that F is a minimum The seco
28. CONSTRAINT VALUES G VECTOR 1 4 32761E O01 4 04487E O1 4 32761E 01 6 4 54694E 01 4 67287E 01 4 54694E 01 11 3 13682bE 01 4 46650E 01 3 10988E 01 16 9 27064E 02 1 54449E 01 2 05535E 01 21 4 46650E 01 3 10988E 01 4 18596E 01 26 1 54449E 01 8 45290E 02 1 00000E 00 31 1 71800E 01 1 37363E 00 1 34228E 00 36 3 33333E 01 3 33333E 01 1 00000E 00 41 1 00000E 00 3 33333E 01 3 33333E 01 46 3 66569E 01 3 64299E 01 1 00000E 00 51 1 00000E 00 3 33333E 01 3 33333E 01 OPTIMIZATION IS COMPLETE NUMBER OF ITERATIONS 2d CONSTRAINT TOLERANCE CT 3 00000E 03 THERE ARE 0 ACTIVE CONSTRAINTS AND CONSTRAINT NUMBERS 10 19 20 42 43 THERE ARE 2 ACTIVE SIDE CONSTRAINTS VARIABLE NUMBERS MINUS INDICATES LOWER BOUND 1 2 INATION CRITERIA 21 ITERATIONS FAILED TO PRODUCE A FEASIBLE DESIGN OPTIMIZATION TERMINATED RELATIVE CONVERGENCE CRITERION WAS MET FOR ABSOLUTE CONVERGENCE CRITERION WAS MET FOR OPTIMIZATION RESULTS OBJECTIVE F X 1 12227E 01 DECISION VARIABLES X ade XL X XU ak 1 00000E 03 3 00000E 01 3 00000E 01 2 1 00000E 03 4 00000E 01 4 00000E 01 3 1 00000E 03 1 65920E 01 2 88000E 01 CONSTRAINTS G X 1 9 40960E 01 9 19968E 01 9 40960E 01 6 9 36438E 01 9 51816E 01 9 36438E 01 11 5 17144E 01 7 18986E 01 6 65892E 01 16 4 02165E 01 4 26015E 01 4 39304E 01 21 7 18986E 01 6 65892E O1 2 71000E 01 26 4 26015E 01 3 94653E 01 3 00000E 00 31 2 28411bE 01 2 15125E 00 2
29. Default 0 001 PMAX Maximum value allowed for this property Real Default 1 0E20 CO Constant term of relation Real Default 0 0 DVIDi DESVAR entry identification number Integer gt 0 COEFi Coefficient of linear relation Real Associated Entries Design variables are referenced on DVPREL1 entries by their DESVAR defined IDs ue xw xs x sw 9 Discussion An analysis model property can be expressed as a linear combination of design variables as Py Co t 4X4 9X5 T t e X Equation 4 8 Both independent as well as dependent design variables can appear in this relation see the DLINK entry This form is especially useful in the sense that it can be used to express not only a simple one to one correspondence between a design variable and a property but more complex reduced basis formulations as well Design Sensitivity and Optimization User s Guide 4 21 Chapter 4 DVPREL2 Purpose Input Data Defines a structural property by reference to an equation defined on a DEQATN entry Entry Description Field D TYPE ID FID v PMIN v MAX EQID DESVAR DVIDi DTABLE LABi Contents Unique identification number Integer gt 0 Name of a property entry such as PBAR PBEAM etc Character Property entry identification number Integer gt 0 Field position of the property in the analysis model entry Integer 0 0 Minimum value allowed
30. Design Sensitivity and Optimization in NX Nastran Ki T5 AE Equation 3 66 These forces can be used to compute a strain energy per grid as ETE 18 Equation 3 67 Eq 3 67 is an open product resulting in a vector of strain energies at each grid Locating the maximum term in the vector and taking its square root yields E max E19 L Equation 3 68 We get one such E for each design variable shape basis vector pair This has been found to yield a consistent set of finite difference move parameters for shape optimization Dynamic Response Sensitivities Dynamic response sensitivities are computed in Solution 200 in connection with design sensitivity analysis and optimization The following disciplines are available Direct frequency Modal frequency Modal transient See the DRESP1 Bulk Data entry description for a list of the response types supported for the above analysis disciplines General Considerations The complete stiffness matrix for dynamic response analysis consists of a superposition from damping as well as direct matrix input K K1 ig K1 iV g K K5 Equation 3 69 where K4 structural stiffness g uniform structural damping coefficient PARAM G ge structural damping coefficient on a MAT entry Design Sensitivity and Optimization User s Guide 3 35 Chapter 3 Design Sensitivity and Optimization in NX Nastran Ko direct matrix input at grids DMIG Current
31. Initial value of vector of design variables this characterizes the initial design Vector of design variables at the optimal design Vector of dependent design variables Vector of independent design variables Transpose of a vector Transpose of a matrix One dimensional search parameter Optimal value of one dimensional search parameter for the current search direction gt S Partial derivatives with respect to the i th design variable Finite difference move delta Lagrange multiplier used in the Kuhn Tucker conditions Sensitivity coefficient for the i th design variable and j th response Gradient operator n th eigenvector Modal transformation matrix Mass density Frequency of oscillation of a dynamic system rad s Design Sensitivity and Optimization User s Guide B 3 9 Vector of modal responses B 4 Design Sensitivity and Optimization User s Guide Appendix C Commonly Used Commands for Design Optimization C 1 Case Control Command This section lists commonly used Case Control commands for optimization See the NX Nastran Quick Reference Guide for complete descriptions of these entries C 2 Bulk Data Entries This section lists commonly used Bulk Data entries for optimization See the NX Nastran Quick DRSPAN Reference Guide for complete descriptions of these entries DCONADD DCONSTR DDVAL DEQATN DESVAR DLINK DOPTPRM DRESP1 DRESP2 DRESP3 DTI DFRFNC DSCREEN DTABLE DVBSHAP DVCRE
32. LOAD CONDITION 1 LOAD 300 DESSUB 10 SUBCASE 2 ANALYSIS STATICS LABEL LOAD CONDITION 2 LOAD 310 DESSUB 10 BEGIN BULK ANALYSIS MODEL MAT1 51 1 0E 7 0 33 Dido i M2 M2 50000 50000 29000 SPCl 100 123456 1 11 21 GRID I j Oep 5 0 Design Sensitivity and Optimization User s Guide 7 13 Chapter 7 Example Problems IDV2 10 10 10 10 10 10 8750 7500 6250 5000 3750 10 DL1 DL2 DL3 DL4 DL5 DL6 GRID 2 5 5 5 0 GRID d 10 Bey 0 GRID 4 f 15 5 0 GRID NG 20 Dis 0 GRID by P 25 255 0 GRID Te sg 30 Di 0 GRID 8 n 35 5 0 GRID 9 r 40 255 0 GRID 11 n 0 Oe 0 GRID 12 E Der Duy 0 GRID 13 10 0 0 GRID 14 i54 0 0 GRID 15 P 20 Ow 0 GRID 16 25 05 7 0 GRID 17 30 0 0 GRID 18 E 35 05 0 GRID 19 40 0 0 GRID 21 r 0 Se 0 GRID 22 Diez 555 0 GRID 23 10 Say 0 GRID 24 f 15 5 0 GRID 25 20 5s 0 GRID 26 25 Doy 0 GRID 27 P 30 5 Bey 0 GRID 28 35 55 0 GRID 29 40 Dy 0 CQUAD4 1 iy i 2 19 11 CQUADA 2 2 2 37 13 12 CQUAD4
33. MAT8 100 1 0701E75 4375E50 4 2 523E52 523E52 523E5 0 04 M1 Al A2 TREF XT XC YT YC S M1 2 426E5 3 3205E43 0305E31 9864E41 2025E4 GE F12 STRNS THIS SECTION CONTAINS THE LOADS CONSTRAINTS AND CONTROL BULK DATA ENTRIES PLOAD4 1 1 100 0 PLOAD4 1 2 100 0 PLOAD4 1 3 100 0 PLOAD4 1 158 100 AE 0 PLOAD4 1 159 100 0 PLOAD4 1 160 100 0 SPC 1 1 123456 0 0 SPC 1 2 123456 0 0 SPC 1 3 123456 0 0 SPC 1 4 123456 0 0 SPC 1 5 123456 0 0 SPC 1 57 123456 0 0 SPC 1 58 123456 0 0 SPC 1 59 123456 0 0 SPC 1 60 123456 0 0 SPC 1 112 123456 0 0 SPC 1 113 123456 0 0 SPC 1 114 123456 0 0 SPC 1 115 123456 0 0 7 54 Design Sensitivity and Optimization User s Guide Example Problems SPC 1 167 123456 0 0 SPC 1 168 123456 0 0 SPC 1 169 123456 0 0 PARAM AUTOSPC YES DESIGN MODEL ESVAR ID LABEL XINIT XLB XUB DELXV DESVAR 1 TPLY 1 0 0 001 10 0 DESVAR 2 THETA 85 0 90 0 90 0 DESVAR 3 THETA 60 0 90 0 90 0 DV1 PLY THICKNESS VPREL1 ID TYPE PID FID PMIN PMAX co j DVID1 COEF1 DVID2 COEF2 DVPREL1 1 PCOMP 1 13 DV1 DV1 1 0 01 DVPREL1 2 PCOMP 1 17 DV2 DV2 1 0 01 DVPREL1 3 PCOMP 1 23 DV3 DV3 1 0 01 DVPREL1 4 PCOMP 1 27 DV4 DV4 1 0 01 DVPREL1 5 PCOMP 1 33 DV5 DV5 1 0 01 DVPREL1 6 PCOMP 1 37 DV6 DV6 1 0 01 DVPREL1 7 PCOMP 1 43 DV7 DV7 1 0 01 DVPREL1 8 PCOMP 1 47 DV8 DV8 1 0 01 DV 2 THETA 85 85
34. SORT SPC 100 DISP ALL STRESS ALL DESOBJ MIN 20 DESIGN OBJECTIVE DRESP ID DESSUB 21 DEFINE CONSTRAINT SET FOR BOTH SUBCASES ANALYSIS STATICS SUBCASE 1 LABEL LOAD CONDITION 1 LOAD 300 SUBCASE 2 LABEL LOAD CONDITION 2 LOAD 310 BEGIN BULK on yoga tea Se A A a la i a a eto en es i ee ak Sk ANALYSIS MODEL as A D K ee GRID DATA 2 3 4 5 6 7 8 9 10 GRID T 7 10 0 0 0 0 0 GRID 27 A 0 0 0 0 0 0 GRID 3r F 10 0 0 0 0 0 GRID 4 0 0 10 0 0 0 SUPPORT DATA SPC 100 Ty 123456 2 123456 SPC 100 35 123456 4 3456 ELEMENT DATA CROD T 11 1 4 7 6 Design Sensitivity and Optimization User s Guide Example Problems CROD 2 12 2 4 CROD 3 13 3 4 PROPERTY DATA PROD bip 1 1 0 PROD 12 i 2 0 PROD 137 1 1 0 MAT1 ls 1 0E 7 0 33 0 1 EXTERNAL LOADS DATA FORCE 300 4 20000 0 8 0 6 FORCE 310 4 20000 0 8 0 6 M DESIGN MODEL Ss TE kc en pa EN A DESIGN VARIABLE DEFINITION DESVAR ID LABEL XINIT XLB XUB DELXV OPTIONAL DESVAR 1 Al 1 0 0d 100 0 DESVAR 2 A2 2 0 0 1 100 0 DESVAR 3 A3 1 0 0 1 100 0 IMPOSE X3 X1 LEADS TO A3 A1 DLINK
35. each design variable This equation is valid only for the case of distinct eigenvalues Buckling Load Factor Sensitivities Requesting Buckling Load Factor Sensitivities To obtain buckling load factor sensitivities in Solution 200 you need to Define a design model see Design Modeling for Sensitivity and Optimization e Define a buckling subcase in Solution 200 with the Case Control command ANALYSIS BUCK 3 32 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran Buckling load factor sensitivities are computed automatically in connection with design optimization If you just want to compute and output the sensitivity coefficients you may do so with the DSAPRT Case Control command or the Bulk Data parameter assignment PARAM OPTEXIT 4 Theory The derivation of buckling sensitivity similar is similar to that of eigenvalue sensitivities The equation for determining of the buckling load factor A is K tons 4 K7 6 10j Equation 3 59 where Kg is the differential stiffness also known as the geometric stiffness matrix Differentiating this expression and solving for the buckling load factor sensitivities yields T O K o kK te 20 a DD L n L X f JT f 1 0 3 K4 t f Equation 3 60 As before this equation is solved semi analytically by approximating the structural property matrix derivatives using finite differences Differential Stiffness Approximat
36. g term is assumed to be negligible With the derivatives of the modal coordinates computed from Eq 3 77 the displacement derivatives are available by direct substitution into Eq 3 76 Modal Transient Response Sensitivities The governing differential equation for transient dynamic response is Design Sensitivity and Optimization User s Guide 3 37 Chapter 3 Design Sensitivity and Optimization in NX Nastran M u BI tu K w Pj Equation 3 78 The modal transformation used to reduce problem size is given by uj O Equation 3 79 where p is a matrix whose columns are composed of a sufficient number of eigenvectors such that the reduced problem yields accurate results and is the vector of modal coordinates or transformed displacements which are solved for in the reduced problem then yielding the actual displacements u from Eq 3 79 Introducing this modal transformation into Eq 3 78 and differentiating with respect to a design variable yields the approximate also see Eq 3 82 expression for the modal transient displacement sensitivities 0 M E9 AE P LB 0 A5 0 K 0 As P QAP AM AB 9 AK C Equation 3 80 In Eq 3 80 the assumption is made that a sufficient number of modes are included such that Equation 3 81 is a good approximation This assumption implies that Ab i 20 Equation 3 82 Once the solution for the modal coordinat
37. llle 2 2 Defining the Analysis Disciplines llle 2 3 Case Control Output Requests in Design Optimization lesen 2 5 Defining the Design Variables o o o oo ooo 2 5 Design Variables and the Basic Optimization Problem Statement 2 7 Relating Design Variables to Properties o o oooocooooooee ee 2 8 Special Design Modeling Considerations BEAM Elements lun 2 15 Relating Design Variables to Shape Changes llle 2 21 Basis Vectors in Shape Optimization ooo 2 22 Auxiliary Models in Shape Optimization llle 2 24 Modeling Methods Shape Basis Vector Definition o o o o o ooo 2 26 Example Shape Basis Vectors o o o o e eee 2 30 Identifying the Design Responses o oo oooerooenr II Ies 2 41 Defining Design Responses with External Programs eee ee 2 48 Design Responses and Case Control llli 2 50 Identifying Dynamic Responses 4 224 y Re E Ew d b xk md yox ERE RC 2 52 Defining the Objective Function 0 000 ce ee 2 53 Defining the Constraints 5 3 ce ho RI RSEN uos Registro LR I emer UR edo E Ra 2 54 Normalized Constraints a 2 55 Equality Constraints in NX Nastran lee eh 2 58 Design Sensitivity and Optimization User s Guide 3 Contents Superelement Design Modeling illl 2 59 Supported Su
38. well as an example highlighting some of the differences among these various approaches Other examples can be found in Example Problems Example Problems Interested readers may also refer to Example Problems for details regarding shape sensitivity analysis Basis Vectors in Shape Optimization Shape basis vectors are used in NX Nastran to describe the properties of characteristic allowable shape changes The optimizer then determines the best linear combination of these vectors given the design criteria established by the engineer Although the mathematics of shape basis vectors are quite simple it is probably best to introduce the idea with a simple example The simple angle bracket redesign of Figure 2 9 can be used to illustrate the properties of shape basis vectors Figure 2 9 Angle Bracket Suppose the design goal is to vary the lengths of the x and y axis legs of the bracket while keeping the outer edge straight The location of all grid points along this edge can easily be described in terms of the bracket leg lengths For simplicity assuming only four grids along this edge we can write Say 0 1 3 G _ 1273 0 Giy Gx 0 2 3 G4 G 1 3 0 3y Equation 2 19 2 22 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization A more realistic model certainly would include more grid points along this edge in addition to a number of grids in the structure s interior Adding t
39. 00 5 114175E 01 EE 1 118684E 00 1 118690E 00 5 114954E 06 8 037044E 02 2 7 193502E 01 7 191593E 01 2 653850E 04 1 896803E 01 3 5 404640E 01 5 403725E 01 1 694252E 04 1 283122E 01 4 4 362053E 01 4 362369E 01 7 255247E 05 8 706033E 02 5 3 744896E 01 3 744358E 01 1 438239E 04 5 622089E 02 6 3 359877E 01 3 359930E 01 1 596586E 05 3 742423E 02 7 3 119118E 01 3 119270E 01 4 891782E 05 2 289958E 02 8 2 961263E 01 2 961254E 01 3 119868E 06 1 413279E 02 9 2 855329E 01 2 855373E 01 1 534280E 05 8 562075E 03 10 2 767122E 01 2 767129E 01 2 692531E 06 9 941631E 02 11 2 260737E 01 2 260700E 01 1 634666E 05 9 497630E 02 12 1 608659E 01 1 608551E 01 6 771779E 05 1 364352E 01 13 1 608551E 01 1 608551E 01 0 000000E 00 1 364352E 01 DESIGN VARIABLE HISTORY A A i mmm nien CR ei emi minimi eem INTERNAL EXTERNAL DV ID DV ID LABEL INITIAL jJ 2 3 4 5 1 d 1 0000E 00 7 0000E 01 4 5000E 01 3 3813E 01 2 7297E 01 2 3430E 01 2 2 THETA 8 5000E 01 8 6324E 01 8 6703E 01 9 0000E 01 9 0000E 01 8 9947E 01 3 3 THETA 6 0000E 01 4 7320E 01 4 1473E 01 3 5252E 01 2 9964E 01 2 5470E 01 INTERNAL EXTERNAL DV ID DV ID LABEL 6 7 8 9 10 11 1 2 1024E 01 1 9518E 01 1 8529E 01 1 7867E 01 1 7315E 01 1 4146E 01 2 THETA 8 994 7E 01 8 9825E 01 8 9708E 01 8 9229E 01 7 5845E 01 6 4468E 01 THETA 2 1649E 01 1 8402E 01 1 5642E 01 1 3295E 01 1 1301E 01 9 6059E 00 INTERNAL EXTERNAL DV ID DV ID LABEL 12 i 13 1 TPLY 1 0065E 01 1 0065E 01 2 THETA 5 4798E 01 5 4798E 01
40. 01 21339E 01 4 02165E 01 O0000E 00 1 30444E 00 84390E 01 2 84390E 01 14683E 00 1 14683E 00 31853E 01 9 77449E 01 14683E 00 1 14683E 00 84390E 01 2 84390E 01 Output Features and Interpretation This lowest level of optimizer diagnostic output reports on the initial and final design objective and on the design variable and design constraint values Since constraints are violated at both the initial and final designs that is input to and output from the optimizer respectively for this design cycle it is clear the optimizer was unable to locate a feasible design Note however that the objective function has increased from 5 7844 to 11 2227 in the process What the optimizer is apparently attempting to do is to minimize the constraint violation by adding weight to the structure This can be confirmed by observing that all design variables which are related to sizing features of the structure have increased in value However this may not always be the case Also there may be cases where the objective function decreases with improved satisfaction of constraints All of these are problem dependent issues These conclusions can be confirmed by the summary output for the first design cycle The frequency and extent of the following output is governed by the parameters P1 and P2 In this example P1 1 and P2 15 This summary begins with a listing of the design objective and the design variables KKK Ck k k k OK Ck Ck k k k K
41. 02 2 0460E 02 1 8660E 02 1 8185E 02 1 8185E 02 3 4 P4 3 8862E 02 4 1434E 02 4 1168E 02 4 1655E 02 4 1157E 02 4 1157E 02 4 5 P5 3 3466E 02 3 4754E 02 3 5574E 02 3 7825E 02 3 8932E 02 3 8932E 02 5 6 P6 3 5802E 02 3 4492E 02 3 3876E 02 3 3222E 02 3 2961E 02 3 2961E 02 6 7 P7 6 6110E 03 5 3870E 03 4 9474E 03 4 4700E 03 4 2716E 03 4 2716E 03 T 8 BETA 9 8287E 02 9 8271E 02 9 8721E 02 9 2664E 02 9 3475E 02 9 3475E 02 USER INFORMATION MESSAGE 6464 DOM12E RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT CYCLE NUMBER 11 Figure 7 26 Design Cycle History for Acoustic Optimization Example 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2 28H03 Figure 7 27 Acoustic Optimization Sound Pressure Levels Initial and Final Distributions Table 7 1 lists the first eight initial and final structural eigenvalues as well as the first nine invariant fluid eigenvalues Note that the fluid has three repeated roots at 100 4 Hz and it is the coupling between these fluid resonances and the nearby structural resonances that creates the peak response It is also notable that the final design has two structural frequencies near 60 Hz that are showing significant response 7 60 Design Sensitivity and Optimization User s Guide Table 7 1 Structural and Fluid Eigenfrequencies in Hz Structural Eigenfrequencie Mode No ID UGS TIME 100 SOL 200 modal frequency response CEND SUB
42. 1 5832E 01 1 6455E 01 1 6824E 01 1 6902E 01 10 10 T10 1 3812E 01 1 4050E 01 1 4416E 01 1 4660E 01 1 4734E 01 USER INFORMATION MESSAGE 6464 DOM12E RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT CYCLE NUMBER 10 Figure 7 17 Design Cycle History z disp grid 1110 amp 3 622 58 Hz 3 5 1 2 931 51 Hz A a wee tee ee we eee tee eee cee eee Figure 7 18 Frequency Dependent Displacements Figure 7 18 shows the initial and final response data for the displacements used to formulate the objective Note that the integral of the final curve is indeed less than that of the original curve In addition the response peak has shifted somewhat from a value of 3 622 at 58 Hz to a value of 2 931 at 51 Hz Figure 7 19 shows the final thickness distribution 7 40 Design Sensitivity and Optimization User s Guide Example Problems thicknoss 5 6 Designa Variable Figure 7 19 Final Thickness Distribution Unit Test Data Deck d200t4af SOL200 V68 ID EDS D200t4af TIME 100 SOL 200 CEND TITLE Synthesis of Responses across Different Frequencies d200t4af SET 10 1110 DISPL PHASE SORT1 10 MAGNITUDE PHASE REPRESENTATION FOR RESPONSE ANALYSIS AS WELL AS SENSITIVITY ANALYSIS desglb 10 subcase 1 SPC 100 LOADSET 720 DLOAD 700 FREQ 740 METHOD 500 ANALYSIS MFREQ sdamping 2000 DESOBJ 1
43. 144 0 06 f 10000 10100 10001 10401 10204 10200 20100 10101 10301 LABEL XINIT T PLATE 0 15 T WEB 0 20 A BAR 0 144 20204 1 1 1 728 4 0 6 0 33 2000 2000 2000 2000 2000 10000 THRU ooooo os ss 10300 10002 10402 20101 10102 10302 XLB 0 001 0 001 0 001 0 0 1 0 0 0 1 728 2 1 745 2 0 06 0 6 0 06 0 6 0 283 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 16 10400 10003 10004 10104 10403 10404 10304 20102 20103 20104 10103 10104 10303 10304 XUB DELXV 10 0 10 0 10 0 Relate the design variables to analysis model properties linear relations Express shell thicknesses as functions of xl x2 SDVPREL1 ID TYPE PID FID PMIN PMAX co DVIDD1 COEF1 DVID2 COEF2 DVPREL1 1 PSHELL 1 4 0 01 DP1 1 1 0 DVPREL1 2 PSHELL 2 4 0 01 P P P DP2 2 1 0 Express bar cross sectional area as a function of x3 DVPREL1 3 PBAR 3 4 0 01 A DP3 3 1 0 Proportionally relate bar s 11 12 112 to changes in area x3 nonlinear relations require use of DVPREL2 DEQATN SDVPREL2 1D TYPE PID FID PMIN PMAX EQID DESVAR DVID1 DVID2 A DTABLE CID1 CID2 DVPREL2 11 PBAR 35 5 1 0 6 101 m DESVAR 3 r r DTABLE X3INIT I1INIT DEQATN 101 I1 X3 X3INIT I1INIT I1INIT 7 DELTA SQRT
44. 16 STRESS 12 10 23 STRESS 13 10 24 STRESS 14 10 25 STRESS 15 10 26 STRESS 16 10 27 STRESS LT 10 28 STRESS 18 10 TL STRESS 19 10 12 STRESS 20 10 13 STRESS 21 10 17 STRESS 22 10 18 STRESS 23 10 19 STRESS 24 10 20 STRESS 25 10 21 STRESS 26 10 22 STRESS 27 30 10 DISP INTERNAL PROPERTY FIELD ID ID FLAG 28 1 4 29 2 4 30 3 4 31 3 5 32 3 6 33 3 7 34 3 18 35 3 17 36 3 16 37 3 15 38 3 14 39 3 13 40 3 12 41 3 19 42 1 4 43 2 4 44 3 4 45 3 5i 46 3 6 47 3 7 48 3 18 49 3 17 50 3 16 51 3 15 52 3 14 E LOWER LOWER LOWER LOWER LOWER LOWER LOWER LOWER LOWER LOWER LOWER LOWER LOWER LOWER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER UPPER INTERNAL REGION BO B NO wh ww W i so ab al QD www wn TD Q LB HB HB WW U ww U P P BD N IN OM NON OM MON mW LO UL ww m 5000E 02 0000E 01 2000E 02 0000E 06 2800E 03 4500E 03 0000E 02 0000E 01 0000E 02 0000E 01 0000E 02 0000E 01 0000E 02 0000E 01 2500E 01 0000E 01 1600E 01 0173E 02 7280E 02 7450E 02 0000E 02 0000E 01 0000E 02 0000E 01 0000E 02 SUBCASE ID INPUT VALUE 4 3276E 01 Ds 4 0449E 01 4 3276E 01 a 4 0449E 01 9 4 6729E 01 9 4 5469E 01 9 4 6729E 01 E 4 5469E 01 Y 1 8451E 01 e
45. 2 max Equation 4 1 where n is the set identification of a design response on either a DRESP1 or DRESP2 Bulk Data entry This response must be a single scalar quantity A DESOBJ entry appearing above the subcase level identifies a global response Weight and volume are typical examples of global responses DESOBJ can also be used at the subcase level if the design goal is to minimize or maximize a subcase dependent response Design Constraint Identification Design constraint sets are identified in Case Control by the following commands DESGLB DESSUB where n is the set identification number of a DCONSTR or a DCONADD Bulk Data entry n n DESGLB is used above the subcase level to define a subcase independent constraint set Weight and volume are subcase independent responses as are DRESP2 responses that are not functions of DRESP1 responses e g DRESP2s that are functions of design variables table constants and grid coordinates only DESSUB defines subcase dependent constraint sets at the subcase level A DESSUB command remains in effect until replaced with a new DESSUB in a subsequent subcase The overall design constraint definition process is 1 Identify design responses in the Bulk Data using DRESP1 and or DRESP2 entries 2 Impose limits on these responses using DCONSTR Bulk Data entries 3 Combine these DCONSTR sets if desired using DCONADD Bulk Data entries 4 Identify the DCONSTR DCONADD set identifica
46. 2 10 Basis Vectors for the Angle Bracket Once a shape basis vector description is available the optimizer can vary the shape of the structure by changing the corresponding design variables Reduced Basis Formulations We can generalize the relation given in Eq 2 22 as AG mxl T mxn AX nx1 Equation 2 23 where AG 6 t G l i AX Qn Lo xy pu Equation 2 24 and design cycle number Eq 2 23 expresses a vector of grid point changes as a function of changes in design variables The linear combination of design variable changes times each shape basis vector results in the total change in shape Typically there are more grid points than design variables or m n For this reason Eq 2 23 is often called a reduced basis formulation Auxiliary Models in Shape Optimization An auxiliary model is simply an additional finite element model used to generate shape basis vectors Rather than tediously specifying shape changes on an individual grid by grid basis auxiliary models are a tool that can be used to simplify this process An auxiliary model usually shares 2 24 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization the same geometry element connectivity and possibly material type as the original structure However boundary conditions usually differ Applying loads to this structure will result in sets of displacement vectors U When we think o
47. 3 3 3 4 14 13 CQUAD4 4 4 4 5 15 14 CQUAD4 5 5 5 6 16 15 CQUAD4 6 6 6 Ta iy 16 CQUAD4 7 ce pa 8 18 17 CQUADA 8 8 8 9 I9 18 CQUAD4 11 Ly 11 195 22 21 CQUADA 12 2 12 13 23 22 CQUAD4 13 3 137 14 24 23 CQUAD4 14 4 14 15 25 24 CQUAD4 15 5 15 16 26 25 CQUAD4 16 6 16 i7 2 26 CQUAD4 17 Ty 17 18 28 27 CQUAD4 18 8 18 19 29 28 PSHELL 1 51 3 0 51 E 51 PSHELL 2 51 2 640625 51 PSHELL 3 51 2 3125 51 PSHELL 4 51 2 015625 51 PSHELL 5 51 1 75 51 PSHELL 6 51 1 515625 51 PSHELL 7 51 1 3125 51 PSHELL 8 51 1 140625 51 2 3 4 5 6 7 FORCE 300 9 50000 0 0 0 0 FORCE 300 29 50000 0 0 0 0 PLOAD2 310 60 44 1 THRU 8 PLOAD2 310 60 44 11 THRU 18 M ns DESIGN MODEL Define the design variables SDESVAR ID LABEL XINIT XLB XUB DELXV This group will be the dependent design variables DESVAR 1 T 3 0 0 001 100 0 DESVAR 2 T2 2 640625 0 001 100 0 DESVAR 3 T3 2 3125 0 001 100 0 DESVAR 4 T4 2 015625 0 001 100 0 DESVAR 5 T5 1 75 0 001 100 0 DESVAR 6 T6 1 515625 0 001 100 0 DESVAR 7 T7 1 3125 0 001 100 0 DESVAR 8 T8 1 140625 0 001 100 0 This group will be the independent design variables DESVAR 9 AL
48. 3 13 Since both approximations are readily available both use the same gradient information the choice of which to use can be made just by looking at the sign of the difference between the two 0 2 0 1 23 a g xx Sjp X Ax SiR x Ax pm mm 3 Lj 1 s 0 e Equation 3 14 Since the squared term in the expression is always positive the choice depends on the sign of the product Og 1 if 2 Pa gt 0 Use the direct approximation for x since it yields a larger estimate d 3 Ud of constraint value x Equation 3 15 ee X if al 3x 0 Use the reciprocal approximation for x since it yields a larger d 35 estimate of constraint value Equation 3 16 This criterion is applied on an individual design variable basis Thus a combination of direct and reciprocal approximations can be made for a single constraint For convex linearization the objective function is always linearized regardless of its response type Modeling Guidelines There are several situations where the default mixed method APRCOD 2 is not applicable and therefore not recommended The first is when basis vectors for either property or shape optimization are used In this case the physical justification for the use of reciprocal approximations does not apply see Reciprocal Approximations earlier in this chapter and either direct approximations or perhaps even convex linearization should be used instead 3 18 Design Sensitivity and Optimiza
49. 3 3 THETA 9 5154E 00 9 5154E 00 0 USER INFORMATION MESSAGE 6464 DOM12E RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT CYCLE NUMBER 13 Figure 7 23 Design History for the Composite Tube Design Sensitivity and Optimization User s Guide 7 53 Chapter 7 Example Problems ID COMPOSITE TUBE SOL 200 TIME 200 CEND ANALYSIS CASE CONTROL TITLE TUBE UNDER INTERNAL PRESSURE DIRECT APPROXIMATION SUBTITLE ANTI SYMMETRIC ANGLE PLY 85 85 60 60 60 60 85 85 SPC 1 LOAD 1 SET 1 32 64 STRESS 1 FORCE 1 OPTIMIZATION CASE CONTROL ANALYSIS STATICS DESOBJ MIN 10 DESSUB 20 BEGIN BULK GRID 1 0 4 0 0 0 0 GRID 2 0 3 71915 1 472380 0 GRID 3 0 2 82843 2 828430 0 GRID 217 0 1 472383 71915 20 GRID 218 0 2 828432 82843 20 GRID 219 0 3 719151 47238 20 CQUADA 1 1 1 2 7 6 CQUADA 2 1 2 3 8 7 CQUAD4 3 1 3 4 9 8 CQUAD4 158 1 212 213 218 217 CQUAD4 159 1 213 214 219 218 CQUAD4 160 1 214 46 51 219 LAMINATE DEFINITION ANTISYMMETRIC ANGLE PLY ALL PLIES ARE OF THE SAME MATERIAL KFRP KEV 49 EPOXY AND THICKNESS PID ZO NSM SB FT TREF GE LAM PCOMP 1 1 3E4 HILL PC1 MID1 T1 THETA1 SOUT1 MID2 T2 THETA2 SOUT2 PC1 100 0 01 85 0 YES 85 0 YES PC2 4PCZ 60 0 YES 60 0 YES PC3 PC3 60 0 YES 60 0 YES PC4 PC4 85 0 YES 85 0 YES PLY MATERIAL DATA ALL ALLOWABLES MUST BE DEFINED AS POSITIVE MID El E2 NU12 G12 G1Z G2Z RHO
50. 3 38 and Eq 3 39 show the advantage of generalizing the basis to include not only design variables but also design properties Static Response Sensitivities Requesting Static Response Sensitivities To compute the sensitivities of static responses you need to e Define a design model see Design Modeling for Sensitivity and Optimization Define a static analysis subcase in Solution 200 using the Case Control command ANALYSIS STATICS Either request sensitivity coefficient output with the DSAPRT Case Control command or the Bulk Data parameter assignment PARAM OPTEXIT 4 or PARAM OPTEXIT 7 e On invoke Solution 200 for design optimization Static response sensitivities are computed automatically in design optimization if static responses are present in the design model Theory For a given structure with specified geometry material properties and boundary conditions a displacement based linear static analysis computes the displacement responses due to the applied loads All other static responses such as stresses and strains are determined from the displacement solution For an arbitrary response r the functional dependency on displacement is written as gt r r u Equation 3 40 In the design context the displacement solution is an implicit function of the design variables That is gt i gt uU Ux X Equation 3 41 Since all other design responses may be functions of these displaceme
51. 7 36 Design Sensitivity and Optimization User s Guide Example Problems SPCD 220 33 3 1 0 SPC1 200 3 11 22 33 SPC1 300 123456 133 144 155 SPC1 300 12 143 154 165 SPC1 300 123456 1 THRU 132 SPCD 330 143 3 1 0 154 3 1 0 SPCD 330 165 3 1 0 SPC1 300 3 143 154 165 Listing 7 7 7 7 Dynamic Response Optimization This example demonstrates structural optimization when the structural loads are frequency dependent The system considered is a flat rectangular plate clamped on three edges and free along the fourth as shown in Figure 7 15 The problem investigates minimization of the mean square response of the transverse displacement at the midpoint of the free edge while constraining the volume of the structure and hence weight to be equal to that of the initial design A pressure loading with an amplitude of 1 0 Ib in is applied across a frequency range of 20 0 to 200 0 Hz A small amount of frequency dependent modal damping has also been included Grid 1110 Uniform Pressure Figure 7 15 Pressure Loaded Flat Plate Figure 7 16 shows the finite element representation Due to symmetry conditions only half of the structure needs to be modeled Ten design variables are related to ten plate element property group thicknesses The first such group is shown in the figure as the shaded ring of elements Subsequent design variables control the thicknesses of subsequent rings of elements up to the tenth design variable that cont
52. APRCOD Selects the type of approximation s to be used in the construction of the approximate model The DOPTPRM entry is used to choose from one of the available methods by setting APRCOD to 1 2 or 3 where 1 direct linearization 2 mixed method default and 3 convex linearization See Approximation Concepts in Design Optimization for a theoretical overview CT CTMIN CT and CTMIN state the criteria that the optimizer uses to identify active and violated constraints CT and CTMIN are specified on the DOPTPRM entry DELP DPMIN Move limits on properties for the approximate optimization Since the approximations are only locally valid these quantities which are specified on the DOPTPRM entry are used to limit the region of search during the approximate optimization Move limits are discussed in this section DELX Fractional change in any design variable during approximate optimization This provides move limits on the design variables similar to the move limits on properties supplied by DELP and DPMIN It can be changed using the DOPTPRM entry DELXV Move limit for a particular design variable during approximate optimization It is assigned on an individual design variable basis using the DESVAR Bulk Data entry If not supplied the default is provided using DELX on the DOPTPRM entry DESMAX Maximum allowable number of approximate optimizations that may be performed This is the same as the maximum allowable number of design cycles S
53. B H 3 12 H 2 H 2 RTYPE PTYPE DISP WEIGHT LALLOW UALLOW 3 0 3 0 END B DATA PMIN REGION Design Modeling for Sensitivity and Optimization PMAX EQID ATTA ATTB 101 102 103 104 105 101 102 103 104 105 A I1 12 C2 D2 ATTI Listing 2 1 2 5 Relating Design Variables to Shape Changes To use shape sensitivity and optimization in NX Nastran you must define design variables and relate them to allowable shape variations The amount the design variable is changed during optimization results in a corresponding shape change The allowable shapes are defined using shape basis vectors The engineer uses these to describe how the structure is allowed to change The optimizer then determines how much the structure can change by modifying the design variables There are four methods are available to describe these shape basis vectors Manual grid variation Direct input of shapes Design Sensitivity and Optimization User s Guide 2 21 Chapter 2 Design Modeling for Sensitivity and Optimization e Geometric boundary shapes e Analytic boundary shapes This section begins with a brief theoretical discussion of shape basis vectors followed by an introduction to auxiliary models which you can use in NX Nastran as an aid to shape basis vector generation Following these discussions is an overview of the various modeling methods available as
54. Character Design Sensitivity and Optimization User s Guide 4 11 Chapter 4 Input Data Field Contents VALI Value of the parameter Real or Integer see the DOPTPRM listing in Bulk Data Entries Associated Entries There are no directly associated entries Discussion There are numerous parameters that control various aspects of the optimization process itself While all of these parameters have defaults the DOPTPRM entry is optional the defaults can be changed using the DOPTPRM entry See also the Bulk Data listing in Bulk Data Entries as it contains the complete list of all optimization parameters that may be changed Following is an overview of the various parameters grouped according to their functionality Choice of Approximation Method There are three types of approximation methods to choose from direct linearization mixed method and convex linearization The mixed method is the default Direct linearization APRCOD 1 is based on the simple first order Taylor series expansion directly in terms of the design variables The method is often useful for dynamic response optimization shape optimization and optimization tasks that use basis vector formulations The mixed method APRCOD 2 uses a combination of direct and reciprocal approximations depending on the response type being approximated Volume weight internal force and buckling load responses are directly approximated while all other response types u
55. Conditions The Kuhn Tucker conditions necessary for optimality are stated in Eq D 14 through Eq D 17 In the uncommon case where there are no constraints these conditions degenerate to the familiar case of the vanishing gradient of the objective In practice this is assumed to be true if all components of the gradient are less than 0 001 The usual optimization task in NX Nastran has many constraints imposed In the process of finding a usable feasible search direction it is possible to detect if the Kuhn Tucker conditions are satisfied If they are the optimization process should be terminated However in practice other considerations may affect the results Remember that a constraint is defined as active if it is numerically greater than the parameter CT The optimization starts with CT 0 03 Any constraint within three percent of being satisfied is called critical Since a constraint with this value is considered in the Kuhn Tucker calculations these conditions may be satisfied even if the search is up to three percent away from the constraint boundary Therefore if CT is greater than CTMIN in magnitude the CT value is reduced by some fraction typically 5096 and another attempt is made to find a usable feasible direction If CT is reduced in magnitude to a value of CTMIN and can still satisfy the Kuhn Tucker conditions the optimization process is terminated D 24 Design Sensitivity and Optimization User s Guide Numerical Opti
56. Contents ID Unique entry identifier Integer gt 0 4 10 Design Sensitivity and Optimization User s Guide Input Data Field Contents DDVID Dependent design variable identification number Integer gt 0 CO Constant term Real default 0 0 CMULT Constant multiplier Real default 1 0 IDVi Independent design variable identification number Integer gt 0 Ci Coefficient i corresponding to IDVi Real Associated Entries Design variables are identified in DLINK relations by their IDs both independent and dependent variables Discussion A DLINK entry specifies a design variable relationship of the form e x XD Co Y cx i Equation 4 2 where Xp dependent design variable Co constant cj constant multiplying coefficient X independent design variables Design variable linking can be used to ensure structural symmetry unify sizing changes across property groups etc The efficiency of the design process is usually improved if the number of independent design variables can be kept to a minimum Placing a design variable in the dependent set using a DLINK entry removes it from the independent set DOPTPRM Purpose Overrides default values of parameters used in design optimization Entry Description DOPTPRM PARAM1 VAL 1 PARAM2 VAL2 PARAM3 VAL3 PARAM4 VAL4 Field Contents PARAM Name of the design optimization parameter For allowable names see the DOPTPRM listing in Bulk Data Entries
57. D3 D4 D5 D6 W LABEL DVID1 LABEL1 NR1 NID1 scl 1 Til 1 Sc2 RTYPE STRESS STRESS STRESS STRESS STRESS STRESS STRESS STRESS DISP DISP DISP DISP DISP DISP WEIGHT EQID DVID2 LABEL2 NR2 DIR1 L r r L r r L r r Y r PTYPE PROD PROD PROD PROD PROD PROD PROD PROD r REGION here DIR2 WNHFWNHENNNNNNNN simple Euler buckling ATTB ATT1 NNUNFPFRPOIAOBWNE ALL Example Problems DP1 DP2 DP3 DP4 DP5 DP6 DP7 DP8 DR11 DR12 DR13 DR21 Design Sensitivity and Optimization User s Guide 7 49 Chapter 7 Example Problems DR21 DESVAR 2 i DR22 DR22 DTABLE L2 E E E DR23 DR23 DRESP1 2 DRESP2 18 SC3 d i X i DR31 DR31 DESVAR 3 j i j DR32 DR32 DTABLE L3 i i A E r DR33 DR33 DRESP1 3 DRESP2 19 SC4 i j i j DR41 DR41 DESVAR 4 i i P P F DR42 DR42 DTABLE L4 i gt 7 DR43 DR43 DRESP1 4 DRESP2 20 SC5 15 P A A DR51 DR51 DESVAR 5 i A DR52 DR52 DTABLE L5 E y E DR53 DR53 DRESP1 5 DRESP2 21 SC6 ds i A DR61 DR61 DESVAR 6 4 DR62 DR62 DTABLE L6 d i P DR63 DR63 DRESP1 6 DRESP2 22 SC7 i 5 DR71 DR71 DESVAR 7 i P y DR72 DR72 DTABLE L7 7 i A A
58. Eq 3 103 Recall that lower and upper limits exist for each design variable as defined by the XLB and XUB fields on the DESVAR Bulk Data entry Regardless of the values computed in Eq 3 101 and Eq 3 104 move limits cannot be outside of these values Considering all the effects we have for the lower and upper bounds x max nnt Y DELX DELXV x DXMIN XLB x min nos x DELX DELXV x DXMIN XUB Equation 3 106 For every designed property we have a minimum lower bound PMIN and a maximum upper bound PMAX as defined on either a DVPREL1 or DVPREL2 entry Considering these limits on properties leads to 0 p max4min p DELP p DPMIN PMIN 0 pP min max p DELP p DPMIN PMAX Equation 3 107 _L U Figure 3 9 illustrates the situation for a property that is initially close to zero Pi and are the move limits resulting from Eq 3 103 Since these limits are overly restrictive the limits based on DPMIN or pp and pj are used instead Although not shown in the figure if PMIN is a numerically greater quantity than pp it becomes the lower bound as indicated by Eq 3 107 PMAX may also become the upper bound as well if it is less than pj Design Sensitivity and Optimization User s Guide 3 49 Chapter 3 Design Sensitivity and Optimization in NX Nastran Figure 3 9 Move Limits on Properties Automatic Updates of Move Limits Parameters related to design variables c
59. FIELD ID IS GREATER THAN ZERO IT IDENTIFIES THE FIELD POSITION ON A PROPERTY BULK DATA ENTRY THE DIFFERENCE FLAG IS USED TO CHARACTERIZE DIFFERENCES BETWEEN ANALYSIS AND DESIGN MODEL PROPERTIES THE FLAG IS NONE THEN THERE IS NO SIGNIFICANT DIFFERENCE BETWEEN THE TWO VALUES THE FLAG IS WARNING THEN THE USER IS ADVISED THAT DIFFERENCES EXIST THE FLAG IS FATAL THEN THE DIFFERENCES ARE GREATER THAN 1 00000E 35 AND THE RUN WILL BE TERMINATED If any of the analysis model properties differs from the design model description the design model will be used to override the analysis model This table simply reports on any differences found and provides notification if this override took place The design value column contains the property values computed from the initial design variable values In this example no differences were found Design Sensitivity and Optimization User s Guide 6 3 Chapter 6 Output Features and Interpretation The override of the analysis model property values by the design model affords a convenient way of performing a type of rudimentary restart Simply by changing the initial design variable values you can begin optimization from any point in the design space For shape changes updated GRID entries also need to be included but these are available from the punch file if one or more design cycles have been performed in a previous run Once the analysis and design models are in agreement Solution 200 proceeds wit
60. Now we turn to the actual task of solving the approximate problem The method described here is referred to as the Modified Method of Feasible Directions see 4 At this point it is assumed that we are provided with an objective function FQ and constraints g x lt 0 j 1 2 Ng as well as lower and upper bounds on the design variables Also the gradients of the objective and constraints are available Thus we are solving the general problem defined by Eq 1 1 1 2 and 1 4 which are repeated here for convenience Find the set of design variables x i 1 2 n that gt X minimize F x Equation D 19 subject to g lt 0 j L2 n Equation D 20 TET d i 12 n Equation D 21 At this point the optimizer does not know what kind of a problem it is solving It is simply minimizing a function subject to inequality constraints Given an initial x vector x9 the design will be updated according to Eq D 18 which is also repeated here D 8 Design Sensitivity and Optimization User s Guide Numerical Optimization d _ 4 1 2q x x as Equation D 22 The overall optimization process now proceeds in the following steps gt gt q 0 x x 1 Start 2 q q 1 gt go gt 3 Evaluate and where J 1 2 ng 4 Identify the set of critical and near critical constraints j J gt gt Vgj X 5 Calculate VF and for all j J gt q 6 Determine a usable feasible search dire
61. Optimum From a design perspective convergence to a nonunique design provides useful information In many cases this can indicate that the properties found at the end of the optimization process may be 3 58 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran manually further modified by the engineer with correspondingly little change in the objective function and or constraints In other words the optimal design can be expected to exhibit low sensitivity with respect to changes in the design variables A design space of this type is shown in Figure 3 15 where contours of the objective and active constraint boundary have similar curvatures in the region of the optimum In situations such as these the design may be modified to take advantage of available sheet metal gauges or tube sizes without violating the active constraint s and with correspondingly little change to the optimum objective Of course any proposed changes should be subjected to an analysis and the results checked carefully to ensure that other performance constraints are not violated 2 Nonunique X5 Optimum gt g X F 20 F 15 10 X1 Figure 3 15 Nonunique Optimum Returning to the hard convergence decision logic of Figure 3 14 note that the code only checks the relative changes in properties and design variables if the limit on the maximum constraint value is not satisfied The purpose of this check is
62. PBEAM 100 8 101 DESVAR 1 2 DVPREL2 2 PBEAM 100 9 102 DESVAR 1 2 DVPREL2 3 PBEAM 100 10 103 DESVAR 1 2 DVPREL2 4 PBEAM 100 15 104 t DESVAR 2 DVPREL2 5 PBEAM 100 17 105 DESVAR 2 END B DATA A I1 12 C2 D2 SDESVAR ID LABEL XINIT XLB XUB DELXV DESVAR 3 B2 045 0 05 10 0 DESVAR 4 H2 do RA 20 0 SDVPREL2 1D TYPE PID FID PMIN PMAX EQID St DESVAR DVID1 DVID2 St DTABLE CID1 CID2 2 20 Design Sensitivity and Optimization User s Guide PBEAM 3 PBEAM 3 PBEAM 3 PBEAM 4 PBEAM 4 TYPE DVID1 CID1 PBEAM 3y PBEAM 3 PBEAM 3 PBEAM 4 PBEAM DVPREL2 6 DESVAR DVPREL2 7 DESVAR DVPREL2 8 DESVAR DVPREL2 9 DESVAR DVPREL2 10 DESVAR FIRST INTERMEDIATE S SDVPREL2 ID t DESVAR DTABLE DVPREL2 11 DESVAR DVPREL2 12 DESVAR DVPREL2 13 DESVAR DVPREL2 14 DESVAR DVPREL2 15 DESVAR DEQATN 101 DEQATN 102 DEQATN 103 DEQATN 104 DEQATN 105 SDRESP1 ID ATT2 DRESP1 7 DRESP1 8 SDCONSTR DCID DCONSTR 10 100 168 4 100 169 4 100 170 4 100 175 100 1777 TATION COPY OF PID FID DVI B2 ey CID2 100 24 4 100 225 4 100 26 4 100 31 100 33 B H H B 3 12
63. Purpose Defines in conjunction with a DEQATN entry a second level response to be used either as an objective or as a constraint Entry Description Ceo 3 1 1 1 _4 Field Contents ID Unique identification number Integer gt 0 LABEL User defined label Character EQID DEQATN entry identification number Integer gt 0 REGION Region identifier for constraint screening Integer gt 0 DESVAR Flag indicating DESVAR entry identification numbers Character DVIDi DESVAR entry identification number Integer gt 0 a Flag indicating that the labels for the constants in a DTABLE entry follow DTABLE Characien LABLj Label for a constant in the DTABLE entry Character 4 16 Design Sensitivity and Optimization User s Guide Input Data Field Contents DRESP 1 Flag indicating DRESP1 entry identification numbers Character NRk DRESP1 entry identification number Integer gt 0 DNODE Flag signifying that the following fields are designed grid points Character Gm Grid point identification number Integer gt 0 Cm Degree of freedom number of grid point Gm 1 lt Integer lt 3 Associated Entries The DCONSTR entry can be used to place bounds on a DRESP2 response using the DRESP2 entry ID as a reference The DESOBJ Case Control command can also reference the DRESP2 entry ID DESOB id n max Equation 4 6 Discussion It is often necessary to define st
64. SFi Scaling factor for load sequence identification number Real Default 1 0 Associated Entries In shape optimization shape basis vectors relate the changes in a design variable to changes in grid locations see basis vectors in shape optimization Relating Variables to Shape Changes Design variables must first be defined using DESVAR entries Design Sensitivity and Optimization User s Guide 4 19 Chapter 4 Input Data DVGRID Purpose Defines design variable to grid coordinate relations for shape sensitivity and optimization Entry Description Field Contents DVID DESVAR entry identification number Integer gt 0 GID Grid point identification number Integer gt 0 CID Coordinate system identification number Integer gt 0 Default 0 COEFF Multiplier of the vector defined by Ni Real Default 7 0 0 Ni Components of the vector measured in the coordinate system defined by CID Real at least one Ni 0 0 Associated Entries The DESVAR entry defines a design variable that can be used to describe grid variations as well as other design relations The DNODE fields on the DRESP2 entry allow the coordinates of designed grids to be used in formulating second level responses E DRESPY e e IET 3 EE NODE p p god o qp E Discussion Changes in grid coordinate values are expressed as functions of design variables according to the relation DAT A i Equation 4 7 wher
65. Sensitivity and Optimization in NX Nastran Limitations regarding AP Computation Although the force derivative term AP is shown in Eq 3 73 for completeness this term is assumed to be zero in NX Nastran This is often a good approximation However if gravity thermal or geometry dependent loadings are significant some error will be introduced into the approximation This assumption also applies to the modal frequency and modal transient formulations For all of the dynamic sensitivity solution sequences the limitation of zero force sensitivity contribution should be kept in mind Modal Frequency Response Sensitivities The governing equations for modal frequency response analysis are obtained from the equations for frequency response by again introducing the modal transformation of Eq 3 79 uy P 6 Equation 3 74 to yield T o M 9 iol BID P IKIPE 0 P Equation 3 75 Differentiating Eq 3 75 with respect to a design variable and assuming that enough modes are retained such that Equation 3 76 is a good approximation yields with the assumption of Eq 3 82 NEP T T T Co P M 0 o 0 8 0 6 K IO AS 0 GAP Co AM io AB AX u3 Equation 3 77 As with the case in transient analysis Eq 3 77 assumes that enough modes are retained so that the perturbed space is adequately spanned by the retained set of eigenvectors That is the error incurred by omitting the Ag
66. Series expansions of the objective and retained constraints with respect to the design variables 3 Define move limits on the design variables Typically during one cycle the design variables will be allowed to change by 20 40 but this is adjusted during later cycles 4 Solve the linear approximate optimization problem 5 Check for convergence If satisfied EXIT Otherwise repeat the process from step 1 In step 2 Taylor Series expansions are created in the form 2 eq 1 ER rU yvrRU yc Equation D 57 1 1 G 2gT 4 Vg T ei jeJ J J M Equation D 58 10 Dantzig G B Linear Programming and Extensions Princeton University Press Princeton NJ 1963 Design Sensitivity and Optimization User s Guide D 25 Appendix D Numerical Optimization gt 4 4 1 where 6x x x Equation D 59 and J is the set of retained constraints Note that everything in Eq D 57 and Eq D 58 is constant except the values of the design variables f Therefore Eq D 57 and Eq D 58 can be rewritten as 0 1 FERD F VFR 2 Equation D 60 B 3 1 g B Ve TTD jeJ Equation D 61 2 1 1 where F F T vFRT y 24 Equation D 62 24 1 4 1 DX 0 4 1 andg g x Vg G Equation D 63 The following linear approximate optimization problem is now solved minimize F x Equation D 64 subject to gs 0 jeJ Equation D 65 Equation D 66 D 26 Design Sensitivity and Optimization User
67. The contents of the punch file can be included in the original input deck to restart from this point in the design space Using the venerable Three Bar Truss as an example TPL deck D200X1 DAT suppose that we want to perform a single design cycle with some of the optimization parameters overridden examine these results readjust the parameters and continue with optimization Listing 7 12 shows the input data file for the first design cycle Note The shape optimization capabilities in the software also include output of the updated GRID entries The parameter DESPCH controls the frequency of both updated GRID and DESVAR entry output to the punch file See Parameters for Design Sensitivity and Optimization Since displacement and stress responses vary inversely to the bar cross sectional dimensions APRCOD 2 or the mixed method of approximations is selected on the DOPTPRM entry APRCOD 2is the default anyway but this makes it explicit in the input data Since this is likely to yield a good approximation for truss type structures we have increased the move limits to 100 for this first design cycle DELP 1 0 With these relatively wide move limits we may want to make sure that all of the constraints are included just to be sure DSCREEN entries have been included to reduce the truncation threshold TRS to a large negative number to ensure that none of the displacement or stress constraints are screened out prior to the appr
68. This is a case of extraneous and redundant design data If the constraints can be filtered in a consistent manner a considerable increase in efficiency may be realized A response that is to be constrained must first be defined on either a DRESP1 or DRESP2 entry Response lower and upper bounds are set with a DCONSTR entry For the j th response this definition yields L gt U lt r x lt r jo r X y Equation 3 5 NX Nastran uses these bounds to create a pair of normalized constraints as L Ti l U s ri rj 829 07 U lt 0 li Equation 3 6 where the absolute values of the bounds are used as normalizing factors This normalization provides a convenient method of screening the constraints For example any normalized constraint with a value of 0 5 has violated its bound by 50 while a constraint with a 3 12 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran value of 0 5 is within 5096 of its bound We might argue that it may not be necessary to retain every constraint just those that are greater than some normalized value Constraint Deletion Figure 3 4 represents a group of constraints in bar chart form In constraint deletion any constraint exceeding the truncation threshold value shown as TRS in the figure is retained for the current design cycle Those constraints whose values are less than this threshold are temporarily deleted for the current c
69. a component that is subjected to two static loading conditions The component must also satisfy requirements on natural frequency bounds as determined from a modal Design Sensitivity and Optimization User s Guide 2 3 Chapter 2 Design Modeling for Sensitivity and Optimization analysis Further the structure might also be subjected to transient loading conditions for which peak displacement responses might be of concern We could quite easily introduce these different analysis types in Solution 200 the solution sequence for design optimization with the following input 15 STATICS subcase 1 DESSUB 10 displacement all stress all load 1 subcase 2 DESSUB 20 displacement all strain fiber all load 2 subcase 3 MODES 30 3 40 4 begin bulk Note some of the features in this example SOL 200 Executive command indicating Solution 200 the solution sequence for design sensitivity and optimization is to be invoked ANALYSIS Case Control command indicating the analysis discipline to be used for a particular subcase if appearing above a subcase level all subsequent subcases assume the same ANALYSIS command until changed ANALYSIS may assume any of the following values STATICS Statics MODES Normal Modes BUCK Buckling DFREQ Direct Frequency Response MFREQ Modal Frequency Response MTRAN Modal Transient SAERO Steady Aeroelastic FLUTTER Flutter DESOBJ Selects the design objective from a response de
70. a vanishing gradient at the optimum design x However we must also remember the original optimization problem and the inequality condition of Eq D 2 In addition we note that Eq D 3 for equality constraints is restated in NX Nastran as a set of inequality constraints When all of these conditions are considered they lead to the statement of the Kuhn Tucker necessary conditions for optimality Condition 1 Y is feasible Therefore for all j g x lt 0 Equation D 14 Condition 2 Ag G 7 0 the product of A and g x equals zero Equation D 15 M a aT Condition 3 VF x gt A VgG 0 jc Equation D 16 A 20 j 1 2 M Equation D 17 The physical interpretation of these conditions is that the sum of the gradient of the objective and the scalars Aj times the associated gradients of all active constraints must vectorally add to zero This is much like the statement for static equilibrium where the sum of all internal and external forces at any given point must vectorally add to zero Figure D 2 shows this situation for a simple two variable function space with two active constraints at the optimum Eq D 15 states that the corresponding Lagrange multiplier is zero for all non active constraints D 6 Design Sensitivity and Optimization User s Guide Numerical Optimization Figure D 2 Kuhn Tucker Conditions at a Constrained Optimum These are only necessary conditions and the definition here is actually a bit mor
71. arguments may be design variables defined on DESVAR entries and table constants defined on a DTABLE entry Design Sensitivities and Type 2 Properties Every property defined on a DVPREL2 entry becomes an independently varying property internally in NX Nastran Design sensitivities are first computed with respect to these independently varying properties and then the chain rule applied to relate these sensitivities to changes in the design variables Refer to Design Sensitivity Analysis for further details DVSHAP Purpose Defines a shape basis vector as a linear combination of DBLOCATEd displacement vectors in data block UGD and assigns a design variable multiplier to the result Entry Description Field Contents DVID Design variable identification number on the DESVAR entry Integer gt 0 COLI Column number of the displacement matrix 1 lt Integer maximum column number in the displacement matrix Scaling factor applied to the COLi th column of the displacement matrix Real dl Default 1 0 Associated Entries In shape optimization shape basis vectors relate the changes in a design variable to changes in grid locations see basis vectors in shape optimization in Relating Design Variables to Shape Changes The design variables must first be defined using DESVAR entries 4 4 Parameters for Design Sensitivity and Optimization In addition to the optimization control possible with Bulk Data entries such as
72. aspects of engineering design but is especially so in design optimization To use the current example if we did not specify a maximum allowable beam height to width ratio we might run the risk of introducing twisting or other buckling modes beyond the simple bending stress criteria we had already accounted for Without this constraint the optimizer would be able to reduce the structural volume even further see the design space of Figure 1 7 but would be completely unaware of the introduction of other failure modes possible with narrow beam sections Consequently the engineer not the optimizer must accept ultimate responsibility for the integrity of the final design Summary The intent of this example was to help illustrate the concepts presented in this section as well as to give a general idea of the approach used by the modified method of feasible directions The discussion was simplified by the fact that we had an explicit functional description of the design space beforehand as well as only two design variables In any real structural optimization task each of the data points in the design space can only be determined based on the results of a complete finite element analysis This may be quite expensive Also since a numerical optimizer usually needs a number of these function evaluations throughout the search process the costs associated with this analysis can quickly become enormous These factors combined with tens or even hundre
73. axial force See the NX Nastran Quick Reference Guide for these plot codes Design Sensitivity and Optimization User s Guide 2 47 Chapter 2 Design Modeling for Sensitivity and Optimization Note The order of the DRESP2 continuation lines is not interchangeable See the DRESP2 Bulk Data entry description The DRESP2 entry defines the arguments of the Euler buckling equation DEQATN 600 Note that this information is positional the values are assigned to the argument list of the equation based on the order of specification in the DRESP2 entry Defining Design Responses with External Programs When using the DRESP3 Bulk Data entry capability for type 2 design responses the user can Customize copies of the FORTRAN subroutines that are provided in the installation directory to perform the design response calculations A build script is also provided in the installation directory that is suitable for use with these FORTRAN subroutines Modify copies of the FORTRAN subroutines that are provided in the installation directory to interface with user coded or third party software that performs the design response calculations The step by step procedure to customize copies of the provided FORTRAN subroutines is as follows 1 2 48 Customize the provided FORTRAN subroutines Copy the contents of the install_directory nxnr dr3srv directory to another location Then modify the FORTRAN subroutines r3sgrt F r3svald F for LP a
74. boundary and loading conditions The input data file for this model is given in Listing 7 5 Figure 7 7 Auxiliary Model Outside edges of the culvert are fixed in the auxiliary model to satisfy straight edge requirements Grid points 1 2 3 on the bottom are allowed to move in the x direction and grid points 13 20 27 on the symmetry line can move along the y direction Grids 5 9 14 15 and 16 on the hole boundary are allowed to move as well Along this hole boundary six CBAR elements have been added to help smooth the applied loading effects It is important to allow z rotations along this boundary To generate a set of displacements to be used as basis vectors we can statically load each grid in a direction normal to the boundary as shown in Figure 7 7 Note from Listing 7 5 that this can be performed in Solution 101 or in Solution 200 with OPTEXIT 2 This will result in seven displacement vectors one corresponding to each load case Three of these vectors are also shown in Figure 7 7 Each arrow on a plot indicates that the deformation is due to a concentrated force Design Optimization Input In the optimization run the displacement matrix containing the seven displacement vectors from the Auxiliary Model analysis is retrieved using the DBLOCATE statement The optimization input file is shown in Listing 7 7 Since each shape basis vector is defined in terms of a single displacement vector seven DESVAR and DVSHAP entrie
75. boundary model 1 AUXMID 1 The resultant reduced basis formulation could be written as Il un AG LOU yuymip 1 10U guxmip SUBCASE 10 SUBCASE 20 Ax5 Equation 2 28 where each column is a displacement solution of the auxiliary model Each column is the same size as AG The subscripts on displacement solutions indicate the source of the applied boundary displacements The first shape basis vector is derived from the first auxiliary boundary model subcase while the second shape basis vector is derived from the second auxiliary model subcase 2 6 Identifying the Design Responses Before the objective function and constraints can be defined the analysis responses on which they depend must be identified These responses are called design responses and are specified in the design model using DRESP1 DRESP2 and DRESP3 Bulk Data entries Type 1 Responses DRESP1 Bulk Data entries define type 1 or first level responses These responses are available directly from an NX Nastran analysis Structural weight displacements at grid points element stresses and so on are all examples of type 1 responses For a list of available responses see the DRESP1 Bulk Data entry description For information on how to generate multiple DRESP1 Bulk Data entries for more than one entity such as a grid or an element or frequency and how to generate results of simple mathematical functions such as Sum or Average in frequency response see E
76. cannot draw upon such intuition and experience A basic goal of design optimization is to automate the design process by using a rational mathematical approach to yield improved designs Ways in which this might be put to use include 1 2 Design Sensitivity and Optimization User s Guide Getting Started 1 Producing more efficient designs having maximum margins of safety 2 Performing trade off or feasibility studies 3 Assisting in design sensitivity studies 4 Correlating test data and analysis results model matching In addition to providing a complete description of the optimization tools in NX Nastran part of the aim of this user s guide is to suggest various ways in which design sensitivity and optimization might be used Consider the following examples Example A A complex spacecraft is in a conceptual design stage The total weight of the spacecraft cannot exceed 3 000 pounds The nonstructural equipment including the payload is 2 000 pounds Static loads are prescribed based on the maximum acceleration at launch Also the guidance systems require that the fundamental elastic frequency must be above 12 Hz It is extremely important to reduce the structural weight since it costs several thousand dollars to place one pound of mass in a low earth orbit There are three types of proposed designs truss frame and stiffened shell configurations Currently all of the designs fail to satisfy at least one design requirement and are
77. design optimization If you do not want to perform optimization you can request sensitivity analysis with the DSAPRT Case Control command or the Bulk Data parameter assignment Design Sensitivity and Optimization User s Guide 3 31 Chapter 3 Design Sensitivity and Optimization in NX Nastran PARAM OPTEXIT 4 Theory Normal modes analysis can be requested in Solution 200 by setting the Case Control command ANALYSIS MODES This setting invokes both a normal modes analysis as well as a sensitivity analysis if either sensitivity or optimization is requested The eigenvalue equation is K 2 MD 0 10 Equation 3 56 where A and are the n th eigenvalue and eigenvector respectively K is the structural stiffness M is the structural mass and 0 is the null vector The governing Eq 3 56 can be differentiated with respect to the th design variable x to yield aln ro K 01M On L1 2g MDG OS Equation 3 57 When Eq 3 57 is premultiplied by q the first term is zero Eq 3 57 can then be solved for the eigenvalue derivatives T 0 K epum AA 5 4 gy l L t6 1 EM 16 Equation 3 58 In practice the solution to the above equation is based on a semi analytic approach The derivatives of the mass and stiffness matrices are approximated using finite differences of the form in Eq 3 47 and Eq 3 48 Eq 3 58 is solved for each retained eigenvalue referenced in the design model and for
78. differ somewhat depending on your system In addition the Solution 200 flowchart in Solution Sequences might also be useful Referring to the flowchart will help fix the relationship of the particular output samples to their order in the design optimization process D200X7 Output This example problem is presented in Stiffened Plate If you have not already done so you may want to turn to that section first to briefly review the problem set up Assuming you have run this example on your installation and have the output file available you will notice the following table appearing shortly after the Bulk Data echo PROPERTY PROPERTY DESIGN 440000E 01 440000E 01 000000E 02 000000E 20 728000E 04 728000E 04 000000E 06 000000E 20 728000E 02 728000E 02 000000E 06 000000E 20 745000E 02 745000E 02 000000E 06 000000E 20 000000E 02 000000E 02 000000E 35 000000E 20 000000E 01 000000E 01 000000E 35 000000E 20 000000E 02 000000E 02 000000E 35 000000E 20 000000E 01 000000E 01 000000E 00 000000E 20 000000E 02 000000E 02 000000E 00 000000E 20 000000E 01 000000E 01 000000E 00 000000E 20 000000E 02 000000E 02 000000E 00 000000E 20 000000E 01 000000E 01 000000E 35 000000E 20 500000E 01 500000E 01 000000E 02 000000E 20 000000E 01 000000E 01 000000E 02 000000E 20 3 3 3 3 3 3 3 3 3 3 3 3 1 2 PSHELL IF FIELD ID IS LESS THAN ZERO IT IDENTIFIES THE WORD POSITION OF AN ENTRY IN EPT IF
79. ek ok ke doe dee ok eee SUMMARY OF DESIGN CYCLE HISTORY 3k ek e ke ek de Sek ke e e ke ke Sek ke ek ke dee ke e de e ke e ke ke de ke e ke ee ke de e ee dee hee de e koe dee koe dee dee HARD CONVERGENCE ACHIEVED NUMBER OF FINITE ELEMENT ANALYSES COMPLETED NUMBER OF OPTIMIZATIONS W R T APPROXIMATE MODELS 4 OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY OBJECTIVE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE CYCLE APPROXIMATE EXACT OF OF NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT INITIAL 2 888885E 00 2 564473E 02 1 2 752784E 00 2 752907E 00 4 460214E 05 7 020019E 03 2 2 719121E 00 2 719218E 00 3 551004E 05 4 937890E 03 3 2 697654E 00 2 697639E 00 5 479589E 06 1 350586E 03 4 2 694388E 00 2 694371E 00 6 105647E 06 2 292285E 03 DESIGN VARIABLE HISTORY INTERNAL EXTERNAL DV ID DV ID LABEL INITIAL a z E 2 3 A 4 t 5 1 1 A 7 6865E 01 7 9464E 01 8 2615E 01 8 3462H 01 8 3819E 01 2 2 a2 7 1480E 01 5 0532E 01 3 8250E 01 3 3696E O1 3 2361E 01 3 3 AB 7 6865E 01 7 9464E 01 8 2615E 01 8 3462E 01 8 3819E 01 0 USER INFORMATION MESSAGE 6464 DOM12E RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT CYCLE NUMBER 4 Figure 7 29 Optimization from Updated Design Looking back on the summary of design cycle history for the first run note that two finite element analyses were performed The first provided the baseline analysis while t
80. expected to be overweight We are to determine which configuration s promises the best performance and warrants detailed design study Also the payload manager needs to know how much weight could be saved if the frequency requirement were to be relaxed from 12 Hz to 10 Hz The spacecraft s structure contains about 150 structural parameters which we may want to vary simultaneously Example B One part of a vehicle s frame structure was found to be overstressed Unfortunately it is too expensive to redesign that particular frame component at this stage in the engineering cycle However other structural components nearby can be modified without severe cost increases There are nearly 100 structural design parameters that can be manipulated The design goal is to reduce the magnitude of the stresses by reducing the internal load to the overstressed member Example C A frame structure which supports a set of sensitive instruments must withstand severe in service dynamic loads Modal test results are available from comprehensive tests performed on the prototype structure We need to create a finite element model for dynamic analysis that is much less detailed than the original model created for stress analysis since the costs of dynamic analysis using a complex model would be prohibitive We must ensure however that the first ten modes obtained from our simplified model are in close agreement with those obtained from the test results The goal is
81. for Primary and Auxiliary Models for Shape AXMDRV AXMPR2 PHASEO Design Optimization Data Initialization DESINIT DMPR SDSA SDSB SETSOLAP Begin Design Cycle Loop Database Cleanup From Previous Design Cycle Design Sensitivity and Optimization User s Guide Solution Sequences AMBS Auxiliary Model Boundary Shapes Static Analysis for All Auxiliary Models Loop over Auxiliary Models Boundary Conditions FEAOPT Analysis fAMB or GMBS YES Collect Analysis Results Generate Enforced Boundary Displacements for Auxiliary Structure Analysis AXMDRV DSAJ Finite Element Analysis and Data Recovery for All Analysis Disciplines Superelements and Subcases Basis Vector Solution for Shape Optimization FEA FEA GMBS Geometry Model Boundary Shapes BNDSHP Loop on Design Cycle Partition Out Shape Basis Vector Solutions Generate Basis Vectors If a Design Model Exists and NO PARAM OPTIM YES YES EXIT Begin Design Sensitivity and Optimization Pre Sensitivity Initialization Operations Shape Basis Vector Initialization and Scaling Operations DOM11 DOPR2 DOPR3 DOPR4 DOPR5 PRESENS First Design ll Cycle Only Sis Design Sensitivity and Optimization User s Guide 5 9 Chapter 5 Solution Sequences Design Constraint Evaluati
82. limits are placed both on property as well as design variable changes Move limits are discussed in Optimization with Respect to Approximate Models Direct Approximations A direct variable approximation linearizes the function directly in terms of the design variables For the j th constraint function the direct approximation is written as Design Sensitivity and Optimization User s Guide 3 15 Chapter 3 Design Sensitivity and Optimization in NX Nastran O 20 gt 0 8 oa a i i 50 X Equation 3 8 where the subscript D indicates a direct approximation 0 The quantity Ax represents a total vector move in the design space from the initial design t The partial derivatives dg 0x are available from the design sensitivity analysis An equation similar to Eq 3 8 can be written for the objective function where F replaces g in the expression Reciprocal Approximations The first order Taylor series expansions of Eq 3 8 can alternately be expressed in terms of reciprocal variables This choice turns out to be quite useful for those responses that are inversely proportional to the design variables This simple idea can be easily shown in the case of the axially loaded rod element of Figure 3 6 where the cross sectional area A is taken as the design variable NN IN N TuS Figure 3 6 Axially Loaded Bar Element The axial stress in the bar is equal to P A and the displacement is equal to PL AE If these responses are used
83. linear and quadratic basis functions respectively Each column is a vector and the dimension of each of these vectors is greater than the number of such vectors Relations of this sort are termed reduced basis formulations In general a reduced basis formulation can be expressed as Ding Tl yynt ty Equation 3 3 The most powerful form of Eq 3 3 occurs when the number of rows of 7 is much greater than the number of columns or M gt gt N This concept is used extensively in connection with shape optimization a large set of grid coordinate variations can be governed by a much smaller set of design variables see Relating Design Variables to Shape Changes Listing 3 1 shows the Bulk Data entries that can be used to implement the design formulation of Eq 3 2 Since the reduced basis formulation is just a set of linear equations DVPREL1 Bulk Data entries can be used exclusively Also since each of the plate strips are of constant thickness each element group shares a property Bulk Data entry for a total of ten such entries over 40 plate elements SPSHELL PID MIDI T MID2 121 T3 MID3 PSHELL 101 100 lt 03 100 100 PSHELL 102 100 prom 100 100 PSHELL 103 100 L 0 100 P 100 PSHELL 104 100 LO 100 100 PSHELL 105 100 Ls 100 7 100 PSHELL 106 i00 L 0 100 100 PSHELL 107 100 1 0 100 100 PSHELL 108 100 1 0 100 100 PSHE
84. location and values of the design model override data PROPERTY PROPERTY FIELD ANALYSIS DESIGN LOWER UPPER DIFFERENCE TYPE ID ID VALUE VALUE BOUND BOUND FLAG PBEAM 100 177 5 100000E 0 5 000000E 01 1 000000E 35 1 000000E 20 WARNING PBEAM 100 175 5 100000E 01 5 000000E 01 1 000000E 35 1 000000E 20 WARNING PBEAI 100 170 4 200000E 02 4 166667E 02 1 000000E 03 1 000000E 20 WARNING PBEA 00 169 1 040000E 02 1 041667E 02 1 000000E 03 1 000000E 20 WARNING PBEAI 00 168 5 100000E 0 5 000000E 0 000000E 03 000000E 20 WARNING PBEAM 00 33 5 100000E 0 5 000000E 0 1 000000E 35 000000E 20 WARNING PBEAI 00 31 5 100000E 0 5 000000E 0 1 000000E 35 000000E 20 WARNING PBEAI 00 26 4 200000E 02 4 166667E 02 000000E 03 000000E 20 WARNING PBEAI 00 25 040000E 02 1 041667E 02 000000E 03 000000E 20 WARNING PBEAI 00 24 5 100000E 0 5 000000E 0 000000E 03 000000E 20 WARNING PBEAI 00 17 1 010000E 00 1 000000E 00 1 000000E 35 000000E 20 WARNING PBEAI 00 15 010000E 00 1 000000E 00 1 000000E 35 000000E 20 WARNING PBEAI 00 10 6 670000E 0 6 666667E 0 000000E 03 000000E 20 WARNING PBEAI 00 9 670000E 0 1 666667E 0 000000E 03 000000E 20 WARNING PBEAI 00 8 2 010000E 00 2 000000E 00 000000E 03 000000E 20 WARNING 1 IF FIELD ID IS LESS THAN ZERO IT IDENTIFIES THE WORD POSITION OF AN ENTRY IN EPT 2 IF FIELD ID IS GREATER THAN ZERO IT IDENTIFIES THE FIELD POSITION ON A PROPERTY BULK DATA ENTRY 3 THE DIFFERE
85. matrix is 10096 that is no zero terms are present We can conclude that every response will undergo some finite change for a change in any of the independent design variables 6 14 Design Sensitivity and Optimization User s Guide Output Features and Interpretation If this density were less than 100 a closer inspection of the design model formulation might be warranted A response that is not a function of any of the design variables hence a zero sensitivity coefficient might indicate a design modeling error Quite often these types of errors will show up as a null row or column of DSCM2 A null column indicates a response that is not a function of any of the design variables A null row on the other hand indicates that changing a design variable will have no effect whatsoever on any of the design responses Both of these errors are usually easily addressed and underscore the significance of using design sensitivities as a method for modeling checkout Column order in the design sensitivity coefficient matrix is given by the following abbreviated correlation table IDENTIFICATION OF COLUMNS IN THE DESIGN SENSITIVITY MATRIX THAT ARE ASSOCIATED WITH DRESP1 ENTRIES WEIGHT VOLUME RESPONSES DRESP1 RESPONSE STATICS RESPONSES DRESP1 ENTRY ID STRESS STRESS DISP DISP STRESS STRESS STRESS STRESS STRESS Note that each column is associated with a particular response component for a given subcase For example col
86. of the DRESP1 entry For the BAR element Item 7 is the maximum stress at end A and ltem14 is the maximum stress at end B See Appendix A of the NX Nastran Quick Reference Guide for a list of these plot codes The ATTi fields on these entries identify the property ID here it is 3 DRESP1 number 1 indicates that the maximum end A stress for every BAR element in PBAR group 3 is to be used in the design model for a total of four such responses DRESP1 number 2 identifies similar information for end B of the BAR elements Design Sensitivity and Optimization User s Guide 7 19 Chapter 7 Example Problems Similarly DRESP1 entries 3 6 9 and 12 identify the von Mises stresses at Z1 and Z2 for all elements in the web and in the plate These responses are in turn constrained by the bounds set using a corresponding set of DCONSTR entries numbers 3 6 9 and 12 From the summary of design cycle history note that the initial constraint violation is greater than 100 Even though the initial design is infeasible the optimizer is able to obtain a feasible design after the second iteration However this is at the expense of additional structural mass with a weight increase to 16 4 Ibs from 5 8 Ibs Once a feasible design is achieved the optimizer can begin trying to reduce the weight achieving a final weight of 7 9 lbs Of course this design is still heavier than the original structure by almost 4096 yet that design did not satisfy the performan
87. options exist data recovery in design optimization will use the Case Control Section to determine the output form If no Case Control requests are present the default representations are used These are listed in the NX Nastran Quick Reference Guide For example von Mises stresses are the default form for NX Nastran output This is taken as the default for design optimization as well Thus in the absence of any Case Control requests to the contrary the following will select von Mises stresses for all elements in property group 150 SDRESPI ID LABEL RTYPE PTYPE REGION ATTA ATIB ATTI1 4 ATT2 DRESP1 100 SIC1l STRESS PSHELL 9 L 150 The integer 9 in the ATTA field selects the von Mises stress component as surface Z1 for the shell elements However the following may be used to select stresses in maximum shear rather than von Mises form in case control STRESS SHEAR ALL in bulk data SDRESP1 ID LABEL RTYPE PTYPE REGION ATTA ATTB ATT2 DRESP1 100 SIGl STRESS PSHELL 9 Note that the DRESP1 entry is unchanged with the response form determined solely by Case Control Case Control may also be used to limit output in frequency response and transient response analysis using the OFREQ and OTIME commands This can be used to reduce the cost of design sensitivity analysis as is discussed in the next section Design Sensitivity and Optimization User s Guide 2 51 Chap
88. output disp plot phase 10 output xyout cscale 2 0 ymax 4 0 plotter nastran ytitle displacement at grid 1110 xyplot disp 1110 t3 BEGIN BULK PARAM WTMASS 002588 A Sd ies eS SA i stm ES EE i E Sek EE nn o EE II i SS tes ita E ANALYSIS MODEL Ss 1 GRID AND SPC DATA GRDSET 6 GRID 100 i 0 De 0 1 1 GRID 200 Oy 1 0 0 1 u GRID 300 DV 2 0 0 1 1 z 9 Design Sensitivity and Optimization User s Guide 7 41 Chapter 7 Example Problems GRID 400 Ox 340 0 e d HAD GRID 500 y 4 0 0 e A HAD GRID 600 g 0 5 0 0 FO s HAD GRID 700 y 0 6 0 0 1 dr r 9 GRID 800 y 0 7 0 0 1 E A GRID 900 y 0 8 0 0 ddr GRID 1000 0 9 0 0 dor GRID 1100 0 10 0 0 1 1 z 9 SPC1 100 123456 100 101 102 103 104 105 106 107 108 109 110 200 300 400 p 500 600 700 800 900 1000 1100 SPC1 100 246 1101 1102 1103 1104 1105 1106 1107 1108 1109 SPCl 100 246 1110 ELEMENT DEFINITION AND PROPERTIES ELEMENTS GROUPED BY PID SINCE THICKNESS OF ALL ELEMENTS IN A GROUP ARE TO BE AFFECTED BY A SINGLE DESIGN VARIABLE MATI 150 1 0E7 0 3 0 1
89. proceeds in this direction until it encounters a constraint boundary From Figure 1 8 we see that the structural volume cannot be reduced any further in this search direction without violating the maximum allowable tip deflection requirement The optimizer is now faced with a choice A finite move in the direction of steepest descent would not be admissible since constraints would then be violated yet we know that the objective function can still be reduced 1 18 Design Sensitivity and Optimization User s Guide Getting Started 60 55 Height H cm 50 E 45 40 35 3 4 5 6 7 Width B cm Figure 1 8 Sequence of Iterations Modified Method of Feasible Directions The optimizer in NX Nastran resolves the situation by choosing a search vector that effectively follows the active constraint boundary in the direction of decreasing objective function If the optimizer could not find any direction in which to move an optimum would be at hand since the Kuhn Tucker conditions have implicitly been satisfied The objective can be reduced further and we observe that by the time two such iterations have been completed the true optimum has been reached Default Optimizer in NX Nastran Note that for both of these iterations one or more constraints have been slightly violated in the interim This is a characteristic of the default optimizer used in NX Nastran the modified method of feasible directions which establishes a search dire
90. provided as an upper bound on the DCONSTR entry is used as the normalizing factor or 2 max g lt 0 O max Equation 2 41 The derivative of this constraint with respect to changes in 01 is now 98 _ 1 004 20 max Equation 2 42 As is shown in Figure 2 27 the region over which this constraint is active is now much greater than the previous formulation This change improves the conditioning of the problem since it is less likely that active constraints quickly become either inactive or violated as the search progresses Design Sensitivity and Optimization User s Guide 2 57 Chapter 2 Design Modeling for Sensitivity and Optimization Active Figure 2 27 Well Conditioned Constraint Of course a second level response can be written so that it is already in normalized form The synthetic response in Eq 2 36 could have been written as 04 0 1 2 lt 26 max Equation 2 43 This would give the same results as shown in Figure 2 27 Either normalized formulation is acceptable Equality Constraints in NX Nastran You can define an equality constraint simply by specifying equivalent upper and lower bounds on a DCONSTR entry This will yield a pair of equal and opposite constraints that force the response to be satisfied with equality at the optimum if R lt r x lt R gt R r X 24 _ J J 821 m R J gt gt r X R 85 X i R J lt 0 J Equation 2 44 Often in design we
91. solution cost and the optimizer has not changed the design to any large degree it may be permissible to forego the final finite element analysis The results from the previous analysis are probably of sufficient validity for design purposes therefore soft convergence may be a preferable and cost effective exit point Note Soft convergence does not terminate the design cycle process unless the parameter SOFTEXIT is explicitly set to YES The default for SOFTEXIT is NO These convergence tests are shown in the simplified flowchart of Figure 3 12 On the initial cycle the hard convergence test is skipped and program flow proceeds directly to the sensitivity analysis and approximate optimization Once an approximate optimum is obtained a soft convergence test is performed to compare these results with the prior finite element analysis results However the design process is stopped only if soft convergence is indicated and the parameter SOFTEXIT has been set to YES This test is performed regardless of whether the parameter SOFTEXIT is set to 3 54 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran YES The printout for each design cycle includes a status report on the soft convergence test see Output Features and Interpretation for examples of this output IFP Initialization PREDOM
92. solution of the coupled fluid structure interaction problem An optimal design can thus be found based not only on a consideration of acoustic pressures but structural responses as well This example considers a closed box with fluid elements on the interior An acoustic source is located at one end of the box with a transducer located at the opposite end The design goal is to modify the thicknesses of the box walls such that the peak acoustic pressure at the transducer is minimized This is to be done without increasing the weight of the box The box geometry and property groups of thicknesses to be modified are shown in Figure 7 24 Six design variables are to be related to six of these property groups the third property group in Figure 7 24 remains fixed This model has 1000 structural elements and 2000 fluid elements Figure 7 24 Acoustic Box Showing Portions Designed by Each Design Variable Property 3 is Fixed An interesting feature of this as well as many other dynamic response problems is that a number of response peaks may exist In addition during the course of design optimization these peaks may not only increase or decrease but shift as well This example presents a useful way of addressing this problem Figure 7 25 shows an arbitrary pressure distribution as a function of frequency P f Three frequencies of interest are shown as f4 fo and fz What we would like to do is to minimize the maximum pressure re
93. tapering the cantilever from its root to its tip Basis vectors describing this characteristic shape can be easily generated using the analytic boundary shapes method Design Sensitivity and Optimization User s Guide 7 30 Example Problems Boundary Shape Changes Using Auxiliary Boundary Models Figure 7 11 shows the auxiliary boundary models that can be used to generate these shapes These disjoint models one for the upper surface and one for the lower are built using QUAD4 elements When fixed at the root and loaded at the tip each generates a cubic displacement variation over its length ls Listing 7 8 is an abbreviated input file for the combined analysis and optimization of the primary structure and the auxiliary boundary model specification The complete file can be found in the Test Problem Library as D200AM3 DAT The primary model definition is similar to that of other NX Nastran input files for static analysis shown in this guide so its description will be deferred here The optimization related input for the primary structure will be described shortly Figure 7 11 Auxiliary Boundary Models Turning first to the auxiliary boundary model specification we see that these boundary models are defined in a special Bulk Data Section appearing after the Bulk Data for the primary model The statement BEGIN BULK AUXMODEL 1 is used to indicate the beginning of this section This section essentially defines three component
94. that indicates the levels of time step reduction OTIME t after OTIME reduction NOi t after NOi reduction t TSTEP t for all analysis steps Figure 2 25 Output Time Step Reduction Similarly for frequency response if the value of a frequency in the ATTB field is not equal to any of the frequencies in the relevant frequency set for the applicable subcase NX Nastran replaces the value in the ATTB field with the nearest frequency in the set Design responses can also be computed for a single forcing frequency or for a single time step We may want to define a design response as a displacement average over time Having identified a peak transient response this technique could help us to reduce this overshot Often as a design is modified the transient peak will move around some from one design cycle to the next Computing an average rms or other measure over time or frequency will often yield more useful results The following example illustrates the formation of an average displacement over five time steps This average is based on the z component of displacement for grid 100 2 52 Design Sensitivity and Optimization User s Guide ID LABEL Z051 42052 42053 Z054 Z055 LABEL DVID1 LABEL1 average displacement fy component displacements RTYPE PTYPE TDISP TDISP TDISP TDISP TDISP over time REGION EQID DVID2 LABEL2 NR1 NID1 AVG I DRES
95. the constant k added to avoid a bound of zero on the constraint It should be clear that the constants have been chosen to scale the objective and responses to values that would minimize numerical difficulties and that there are really no rules for defining these factors Figure 7 26 and Figure 7 27 present the optimization results The design cycle history of Figure 7 26 shows the objective has decreased from 10000 0 to 93 475 in eleven iterations Figure 7 27 shows the magnitude of the frequency dependent pressure at the initial and final designs and it is seen that there has been a dramatic reduction in the peak pressures from 140 5 dB to 115 0 dB from 210 7 N m to 11 4 N m 7 58 Design Sensitivity and Optimization User s Guide KKK KK KKK KKK KKK RK KK KK KK ckck ck RARA RK AAA AAA AAA RARA ck KK KKK KKK KE ck ck KKK AA SUMMARY OF DESIGN CYCLE HISTORY KOKI IK koe kk ek ko kk hok koe ek OR KK RI hok ko ke ko ko ko ke kok ok ko k ok ko ko kk ke k kk ek ek HARD CONVERGENCE ACHIEVED SOFT CONVERGENCE ACHIEVED NUMBER OF FINITE ELEMENT ANALYSES COMPLETED 12 NUMBER OF OPTIMIZATIONS W R T APPROXIMATE MODELS li OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY OBJECTIVE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE CYCLE APPROXIMATE EXACT OF OF NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT INITIAL 1 000000E 04 1 106874E 01 1 9 996712E 03 9 996712E 03 0 000000E 00 2 604586E 03 2 4 962984E 03 4 962984E 03 0 000000E 00
96. up to the DVPREL1 and DVPREL2 entries Bounds have also been defined as 0 01 x45 01 0 01 lt x lt 0 08 Equation 2 2 The DELXV field which has been left blank here limits the amount the design variable can change during an optimization cycle by default this amount is 10096 Design Variables and the Dimension of the Design Space The entire set of DESVAR entries can be thought of as defining a vector of design variables or 2 6 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization gt T x x Xo s x l Equation 2 3 Note To date optimization problems that involve over several thousand design variables have been efficiently solved using NX Nastran The quantity n is often referred to as the dimension of the design space The greater the dimension the more complicated the optimization task The optimizer s task is to find the best configuration for these variables Design Variables and the Basic Optimization Problem Statement Recall that the problem we are trying to solve is defined by the basic optimization problem statement Eq 2 1 through Eq 2 5 Note that the design objective and constraint functions are expressed as functions of design variables How is this the case in structural optimization Assume that we are interested in minimizing the total structural mass for a particular mechanical component This will be our objective function We kno
97. used for Design Optimization Executive Control SOL 200 Case Control Analysis Define the design variables Bulk Data DESVAR Relate the design variables to allowable structural variations for properties Bulk Data DVPRELI DVPREL2 for shape Bulk Data DVGRID DVBSHAP DVSHAP BNDGRID Define the design responses Bulk Data DRESP1 DRESP2 Define the objective Case Control DESOBJ and the constraints Bulk Data DCONSTR DCONADD Case Control DESGLB DESSUB Provide any necessary parameter overrides Bulk Data DOPTPRM DSCREEN Figure 2 1 Design Modeling Process The following sections discuss each of these items in greater detail Overriding optimization control parameters with the DOPTPRM and DSCREEN entries is reserved until after approximation concepts have been presented in Example Problems 2 2 Defining the Analysis Disciplines A powerful feature of design sensitivity and optimization in NX Nastran is that you can specify that a structure be subjected to a number of different analysis types for a number of various load conditions The optimizer will consider the results of all of these analyses simultaneously when proposing an improved design This approach is often described as multidisciplinary design optimization and is the only rational way of proposing a useful optimal design For example you may have
98. useful designs then success becomes more certain The description of methods for accomplishing these goals forms the majority of the material of this guide Before moving on to a detailed discussion of NX Nastran design optimization in Design Modeling for Sensitivity and Optimization and Design Sensitivity and Optimization in NX Nastran the next two sections introduce some of the basic concepts of numerical optimizers and how they are used to solve problems in structural optimization 1 2 Numerical Optimization Basics This section introduces some of the basics of numerical optimization in an intuitive manner stressing overall concepts over technical details This section may be a useful introduction to numerical optimization for those entirely new to the subject Once the basics of numerical optimization have been covered Structural Optimization introduces the extension of these methods to the field of structural design These sections in combination should provide adequate preparation for Design Modeling for Sensitivity and Optimization which describes design modeling in NX Nastran How Much Do I Need to Know About Optimizers The safest answer to this question is The more you know about numerical optimizers the better off you are This answer does not imply that you must be an expert in numerical optimization techniques in order to use those implemented in NX Nastran In fact it would be nice if design optimization were such an automatic
99. which allow negative as well as positive contributions from each of the basis functions The thicknesses are never allowed to assume negative values due to the 0 01 value of PMIN or minimum allowable property value in field 6 of each of the DVPREL1 entries Since the design model overrides the analysis model in design optimization the initial analysis model properties will be set equal to the values computed from the design model specification For this reason no effort has been made to calculate the initial thickness distribution on the set of PSHELL entries These thicknesses have arbitrarily been specified as 1 0 since they will just be overridden anyway Note A table is output whenever a design model override of the analysis model occurs A sample of this table is shown in Design Optimization Output Each DVPREL1 entry defines a row in the reduced basis matrix The continuation lines of each entry lists the design variables and coefficients in each of the linear relations Note that each of these continuation lines is simply a row in the reduced basis matrix Limitations in Reduced Basis Vector Formulations In the previous example the design variable to property relations were defined in terms of a reduced basis formulation Essentially the optimizer s task is to find the best combination of these basis vectors This combination might not yield the true optimum and in fact only does so if some combination of the basis ve
100. with respect to each of the design variables This is a relative type of consideration and is somewhat difficult criteria to define in a general sense However the idea is as follows Suppose a weight design objective is on the order of 1 000 kilograms If a structural element in the model is on the order of 1 0E 3 kg or less and its volume is a function of a single design variable then varying this quantity by 10096 or more does not have any appreciable effect on the overall weight It is probably a better idea either to link a number of these types of elements together and relate them all to a single independent variable resulting in a larger weight change or to eliminate these element groups entirely from the redesign process Prior to optimization it is always advisable to perform a design sensitivity analysis first to ensure that the objective function has sufficient sensitivity with respect to the design variables Design Sensitivity and Optimization User s Guide 2 53 Chapter 2 Design Modeling for Sensitivity and Optimization Formulating the Objective Magnitude The second item to consider is the absolute value of the response selected to be the objective function Care should be taken that this value is not too close to zero Here again it is difficult to define what range of values is too small but the absolute value of the objective should be greater than about 1 0 if possible If this requirement is not met there are a couple of way
101. with those from the previous finite element analysis This test while not as conclusive as hard convergence may be a suitable test to indicate convergence especially if the associated analysis cost is high Type 1 Design Variable to Property Relations express analysis model properties as linear functions of design variables They are defined on DVPREL1 Bulk Data entries Type 2 Design Variable to Property Relations express analysis model properties in terms of design variables using NX Nastran s equation input capability They are defined on DVPREL2 Bulk Data entries and use equations defined on DEQATN entries Violated Constraint is strictly speaking any constraint whose value is greater than zero In NX Nastran the constant CTMIN default 0 003 is used to provide a numerical zero Thus up to one half of one percent constraint violation is tolerated before a constraint is considered violated This improves the numerical conditioning of the problem since trying to precisely satisfy an inequality relation is often a waste of computational resources Design Sensitivity and Optimization User s Guide A 5 Appendix B Nomenclature lal B gu EjD Vector dot product Vector cross product Scalar multiplication Absolute value of the quantity a Matrix of damping terms in the finite element analysis Objective function Approximate objective function Lower bound constraint Upper bound constraint Direct variable a
102. 0733E 01 8 8055E 02 1 0595E 01 2 2 T2 8 0000E 02 8 0375E 02 8 8133E 02 8 3091E 02 9 3658E 02 7 7095E 02 3 3 T3 8 0000E 02 6 4726E 02 7 2045E 02 7 0436E 02 7 0832E 02 6 8105E 02 4 4 T4 8 0000E 02 6 5708E 02 5 2567E 02 4 2931E 02 4 4021E 02 4 0016E 02 5 5 T5 8 0000E 02 7 3653E 02 6 0367E 02 4 9695E 02 4 7654E 02 4 9646E 02 6 6 T6 8 0000E 02 7 9358E 02 7 2390E 02 7 0696E 02 7 5820E 02 7 4753E 02 7 4 T7 8 0000E 02 8 5163E 02 8 6551E 02 8 9344E 02 9 6033E 02 9 5843E 02 8 8 T8 8 0000E 02 9 6000E 02 1 1520E 01 1 2241E 01 1 3176E 01 1 3372E 01 9 9 T9 8 0000E 02 9 6000E 02 1 1520E 01 1 2705E 01 1 4099E 01 1 4819E 01 10 10 T10 8 0000E 02 9 6000E 02 1 1520E 01 1 2202E 01 1 2965E 01 1 3417E 01 INTERNAL EXTERNAL DV ID DV ID LABEL 6 E T 8 9 i 10 amp 11 q 1 1 Tl 8 8563E 02 1 0600E 01 8 6447E 02 1 0377E 01 9 1839E 02 2 2 T2 9 2514E 02 7 5489E 02 9 0480E 02 7 3526E 02 8 7834E 02 3 3 T3 6 7852E 02 6 2472E 02 6 6748E 02 5 8343E 02 6 5374E 02 4 4 T4 3 4938E 02 3 2922E 02 2 8585E 02 3 5773E 02 2 7199E 02 5 5 T5 5 5261E 02 5 8574E 02 6 1972E 02 6 0674E 02 6 0484E 02 6 6 T6 7 7450E 02 7 8756E 02 8 1351E 02 8 0363E 02 8 0163E 02 7 7 T7 9 8118E 02 9 8742E 02 1 0202E 01 1 0198E 01 1 0133E 01 8 8 T8 1 3613E 01 1 3606E 01 1 3865E 01 1 3898E 01 1 3793E 01 9 9 T9 1 5483E 01
103. 089278E 01 636887E 01 636941E 01 285838E 05 610547E 03 294620E 01 294636E 01 259646E 05 604820E 02 748927E 00 751337E 00 753791E 04 988992E 02 379158E 00 379156E 00 230973E 07 885500E 02 954716E 00 954452E 00 315011E 05 145312E 04 954473E 00 954452E 00 637622E 06 145312E 04 INTERNAL EXTERNAL DV ID T PLATE 5000E 01 3 0000E 01 4 4321E 01 2 8106E 01 1 4038E 01 1 0879E 01 T WEB 0000E 01 4 0000E 01 6 1265E 01 8 4962E 01 8 2809E 01 7 5169E 01 4400E 01 1 6592E 01 1 2682E 01 6 4814E 02 3 2407E 02 1 6165E 02 INTERNAL EXTERNAL DV ID T PLATE 1 1169E 0 1169E 01 T WEB 8 3887E 01 8 3887E 01 1 6165E 02 1 6165E 02 Design Sensitivity and Optimization User s Guide 6 11 Chapter 6 Output Features and Interpretation From this summary we see that an optimal design was found after 7 design cycles and 8 finite element analyses initial analysis plus one from each design cycle As was also seen in the hard convergence output the final design is feasible The final cycle does not always yield the best design You can identify the best design by examining the objective function values compared with the maximum constraint values Normally when the maximum constraint value indicates that a design is feasible the best design is the one with the minimum or depending on the problem s definition the maximum objective function However if all generated designs are infeasible then the be
104. 1 DVGRI 0045399 1 DVGRI A 0030902 1 DVGRID 1 0015644 1 DVGRID 1 J lt 0 BASIS DVGRID DVGRI DVGRI DVGRI DVGRI DVGRI DVGRI DVGRI DVGRI DVGRI DVGRI CQ QOO OO OC Oo Oo oO Oo NON ON 5 5 5 5 5 92 90 00 SO O 0000 Q QOO Ott t uuu NON OMS GM GN OMS GS GS GS oS oo QOO OC OO OG eo O S oooo coco OO CO Oo D D D D D D D D D D 0 0 0015644 0 0030902 0 0045399 0 0058779 0 0070711 0 0080902 0 0089101 0 0095106 0 0098769 0 01 0 MNONNNNNNNNDN NH Mos NSN NNN 5 5 SN S M C3 2 Q O S OC Q GO QOO D D D D D D D D D D L The first set of DVGRID entries are related to Design Variable 1 DELA Note that the coefficient magnitudes COEFF are the same as the SPCD magnitudes of the previous example We observe that this set describes an elliptical shape variation in the x direction N1 1 0 N2 N3 0 0 dependent on changes in Design Variable 1 The second set of DVGRID entries likewise describes an elliptical shape variation in the y direction which is dependent on changes in Design Variable 2 BNDGRID entries complete the shape boundary definition process and identify two categories of grid components e Grid components that change according to data supplied on the DVGRID entries Grid components that remain fixed The process is similar to the SPC SPCD combination used to impose enforced displacements in static an
105. 1 1 0 1 1 8 DRESP1 201 VOLUME VOLUME DRESP1 20 g1110L FRDISP 3 20 0 1110 1 22 1 0 79 DRESP1 102 G1110H FRDISP 3 102 0 1110 2 e E DEO 48 DRESP2 1 UZ2 1 DRESP1 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 108 110 122 124 1 36 138 150 152 100 102 104 106 112 114 116 118 120 126 128 130 132 134 140 142 144 146 148 154 156 158 160 162 150 g 800 THRU 110 THRU 210 THRU 310 THRU 410 THRU 510 THRU 610 THRU 710 THRU 810 THRU 910 THRU 1010 r Si 10 ENDT 907 100 908 1 1008 100 1009 1109 OSCILLATORY PRESSURE LOAD RELATE LINEARLY TO PLATE THICKNESS 00 1 Example Problems Design Sensitivity and Optimization User s Guide 7 43 Chapter 7 Example Problems 164 166 168 170 172 174 176 178 180 182 184 186 188 190 192 194 196 198 200 DEQATN 1 UZ2 U20 u21 u22 u23 u24 u25 U26 U27 U28 U29 U30 U31 U32 U33 U34 U35 U36 U37 U38 U39 U40 U41 U42 U43 U44 U45 U46 U47 U48 U49 U50 U51 U52 U53 U54 U55 U56 U57 U58 U59 U60 U61 U62 U63 U64 U65 U66 U67 U68 U69 U70 U71 U72 U73 U74 U75 U76 U77 U78 U79 U80 U81 U82 U83 084 U85 U86 U87 U88 U89 U90 U91 U92 U93 0U94 U95 U96 U97 U98 U99 U100 U102 U104 U106 U108 U110 U11
106. 1 1 through Eq 1 5 Design Sensitivity and Optimization User s Guide A 1 Appendix A Glossary of Terms Basis Vector is a collection of constant terms used to relate the changes in a design variable to changes in structural properties They may be used in connection with either property optimization or shape optimization When used for shape optimization they are commonly referred to as Shape Basis Vectors Central Difference Approximations are used to approximate derivatives or sensitivities of continuous functions Central Difference Approximations which are second order yield greatly improved derivative approximations when compared to first order differencing schemes Forward and backward differences are first order Constraint Design Constraint is defined as an inequality that must be satisfied to indicate a feasible design In NX Nastran the constraint may be a function of design variables structural responses and grid coordinates Constraint Deletion is the process that temporarily removes constraints for a given design cycle Once a constraint is removed its sensitivities do not need to be computed and the optimizer does not need to consider the constraint during the approximate optimization Later design cycles may include constraints that had been previously deleted Constraint Regionalization refers to the grouping of constraints by type in preparation for constraint deletion Constraint Screening is a type of appr
107. 1 636941E 01 3 285838E 05 1 610547E 03 2 3 1 294620E 01 1 294636E 01 1 259646E 05 8 604820E 02 1 4 8 748927E 00 8 751337E 00 2 753791E 04 1 988992E 02 10 5 7 379158E 00 7 379156E 00 3 230973E 07 9 885500E 02 21 6 7 948771E 00 7 948517E 00 3 197505E 05 1 863594E 03 19 7 7 948491E 00 7 948517E 00 3 359479E 06 1 863594E 03 19 DESIGN VARIABLE HISTORY INTERNAL EXTERNAL DV ID DV ID LABEL INITIAL 1 2 3 4 5 1 1 T PLATE 1 5000E 01 3 0000E 01 4 4321E 01 2 8106E 01 1 4038E 01 1 0879E 01 2 2 T WEB 2 0000E 01 4 0000E 01 6 1265E 01 8 4962E 01 8 2809E 01 7 5169E 01 3 3 A BAR 1 4400E 01 1 6592E 01 1 2682E 01 6 4814E 02 3 2407E 02 1 6165E 02 INTERNAL EXTERNAL DV ID DV ID LABEL 6 7 8 9 10 11 1 1 T PLATE 1 1180E 01 1180E 01 2 2 T WEB 8 3723E 01 3723E 01 3 3 A BAR 1 6165E 02 6165E 02 USER INFORMATION MESSAGE 6464 DOM12E RUN TERMINATED DUE TO HARD CONVERGENCE TO A UNIQUE OPTIMUM AT CYCLE NUMBER di 7 20 Design Sensitivity and Optimization User s Guide Example Problems ID UGS D200X7 TIME 10 SOL 200 OPTIMIZATION CEND TITLE STATIC ANALYSIS OF A STIFFENED PLATE ECHO UNSORT DISP ALL STRESS ALL SPC 1 ANALYSIS STATICS DESOBJ MIN 15 OBJECTIVE FUNCTION DEFINITION MIN IS THE DEFAULT SUBCASE 1 LABEL LOAD CONDITION 1 LOAD 1 DESSUB 100 CONSTRAINT DEFININITION SUBCASE 2 LABEL LOAD CONDITION 2 LOAD 2 DESSUB 200 CONSTRAINT DEFININITION BEGIN BULK Sa eo ee es ee oi i E S Sn
108. 1 e5 0 5 0 866 0 0 FORCE 103 16 0 1 e5 1 1 0 FORCE 104 9 0 1 e5 0 866 0 5 0 0 FORCE 105 5 0 1 e5 0 9659 0 259 FORCE 106 1 0 1 e5 1 0 0 Boundary conditions satisfy functional and manufacturing requirements SPC1 25 345 1 THRU 40 SPC1 25 6 2 THRU 4 SPC1 25 6 6 THRU 8 SPC1 25 6 10 THRU 12 SPC1 25 6 17 THRU 19 SPC1 25 6 20 THRU 26 SPC1 25 6 27 THRU 33 SPC1 25 6 34 THRU 40 SPC1 25 12 33 THRU 40 SPC1 25 12 4 8 12 19 26 SPC1 25 1 13 20 27 PC1 25 2 1 2 3 ENDDATA 1 T 1 1 oooooo Listing 7 5 Design Sensitivity and Optimization User s Guide 7 27 Chapter 7 Example Problems FMS section for retrieving the auxiliary displacement matrix assign f1_aux culvert1 MASTER dblocate datablk ug ugd geoml geomid geom2 geom2d logical fl aux SOL 200 TIME 100 CEND TITLE CULVERT EXAMPLE USING EXTERNAL AUXILIARY STRUCTURE SUBTITLE THE PRIMARY STRUCTURE ANALYSIS STATICS SPC 25 LOAD 1 DISP ALL STRESS all DESSUB 10 desobj 5 BEGIN BULK PARAM POST 0 PARAM CDIF NO PARAM optexit 4 PARAM NEWSEO 1 GRID 1 3 00000 0 00000 00 GRID 2 4 00000 0 00000 00 GRID 3 5 00000 0 00000 00 GRID 4 6 00000 0 00000 00 GRID 5 2 89464 0 78478 00 GRID 6 3 79369 0 75885 00 GRID 7 4 69274 0 73293 00 GRID 8 5 59178 0 70700 00 GRID 9 2 60164 1 49178 00 GRID 10 3 46229 1 46585 00 GRID 11 4 32293 1 43993 00 GRID 12
109. 10 H 11 1 1 X1 5 8125E 02 2 9063E 02 2 8529E 02 2 2 X2 2 0443E 00 2 0427E 00 2 0427E 00 3 3 X3 3 0237E 00 3 0059E 00 3 0061E 00 4 4 X4 5 3750E 02 2 6875E 02 2 5837E 02 5 5 X5 5 3789E 02 5 4376E 02 5 3374E 02 6 6 X6 6 8365E 01 6 8306E 01 6 8303E 01 gi 7 X7 1 6118E 00 1 6216E 00 1 6215E 00 8 8 X8 2 6649E 00 2 6690E 00 2 6691E 00 USER INFORMATION MESSAGE 6464 DOM12E RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT CYCLE NUMBER 8 ID UGS D200X3 TIME 10 SOL 200 OPTIMIZATION CEND super all ECHO UNSORT OLOAD ALL DISP ALL SPCFORCE ALL ELFORCE ALL STRESS ALL SPC 100 ANALYSIS STATICS DESOBJ MIN 15 OBJECTIVE FUNCTION DEFINITION DESSUB 12 CONSTRAINT DEFININITION SUBCASE 1 LABEL LOAD CONDITION 1 LOAD 300 SUBCASE 2 LABEL LOAD CONDITION 2 LOAD 310 BEGIN BULK SESET 1 1 2 y 10 0E6 a 0 1 F i M1 M1 25000 25000 123 EA THRU 10 200 200 100 100 100 100 Uu y 3 H o o w a 1 n a p o o oooo 1 m o o w Oooouuuocoo 1 n a Oooouuucuo oooooooooo H o o Q ps o iz o MP o dP OO O OY OY O1 On iS e C C C CO SO PO IO IO ID 8 S Q 4 OU dS OY UO OL Q iS IP I IO SN IO l9 I S O0 1 O0 o 1 2 OY iS O1 O OY Q9 U1 dS OY O1 Q0 AN 7 48 Design Sensitivity and Optimization User s Guide CROD 20 CROD 21 CROD 22 CROD 23 FU a o E co 2O0 01 QN A D
110. 100 100 100 100 100 100 00000 OQOGO Oc t X EC COD CB OE RB A boundary condition set II Lh ody LOL FY FR 161 102 FY FR 161 103 FY FR 161 104 FY FR 161 105 FY FR 161 106 FY FR 161 LOT FY FR 161 108 FY FR 161 109 FY FR 161 LLO FY FR 100 100 100 100 100 100 100 100 100 100 XC UD XU X AA AO e 00000000 temperature distribution TEMP 162 1 1 Diy TEMP 162 7 ds 8 162 107 Ls Ad 162 I3 Ls 14 conditions for both load sets 100 1 23456 0 0 100 14 13456 0 0 100 345 Tiy 12 2 38 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization A number of interesting things can be seen in this listing First the CBAR entries define the auxiliary boundary model elements a ring of bar elements along the cutout Note that GRID entries do not appear because grid data is shared with the primary structure and additional grids need not be defined Of course if extra grids are necessary they can be added as required If the location of the hole were to change one might want to define another grid at the center to use as an attachment point for rigid elements Two sets of loads are applied One imposes an elliptical type variation in the x direction and the other applies a similar displacement in the y direction Uniform loading with PLOAD1 s and a uniform temperature load have
111. 10102 10103 10203 10202 CQUAD4 8 1 10103 10104 10204 10203 CQUAD4 9 li 10200 10201 10301 10300 CQUAD4 10 L 10201 10202 10302 10301 CQUAD4 11 1 10202 10203 10303 10302 CQUAD4 12 T 10203 10204 10304 10303 CQUAD4 13 L 10300 10301 10401 10400 CQUAD4 14 17 10301 10302 10402 10401 CQUADA 15 1 10302 10303 10403 10402 CQUAD4 16 Ly 10303 10304 10404 10403 10200 10201 20101 20100 2 CQUAD4 22 2 10201 10202 20102 20101 CQUAD4 23 2 10202 10203 20103 20102 CQUAD4 24 2 10203 10204 20104 20103 CQUAD4 25 2 20100 20101 20201 20200 CQUAD4 26 2 20101 20102 20202 20201 CQUAD4 27 2 20102 20103 20203 20202 CQUAD4 28 gt 20103 20104 20204 20203 CBAR 31 3 20200 20201 0 0 1 0 0 0 CBAR 32 a 20201 20202 0 0 1 0 0 0 CBAR 33 3 20202 20203 0 0 1 0 0 0 Design Sensitivity and Optimization User s Guide 7 21 Chapter 7 CBAR PSHELL 1 PSHELL 2 PBAR 3 PB3 0 MAT1 1 FORCE 1 FORCE 1 FORCE 1 FORCE 1 FORCE 1 FORCE 2 PLOAD2 1 SPCl 1 SPCl 1 SPCl 1 SPCl 1 SPCl 1 SPCl 1 SPCl 1 SPCl 1 SPCl 1 PARAM PARAM PARAM GRDPNT WTMASS AUTOSPC Define the design variables SDESVAR ID DESVAR 1 DESVAR 2 DESVAR 3 om mbn 1 0E 7 10004 10104 10204 10304 10404 10203 50 1236 136 1 0 00259 YES Example Problems 20203 0 15 0 2 0
112. 10964E 00 36 2 84390E 01 2 84390E 01 1 14683E 00 41 1 14683E 00 3 33333E 01 3 33333E 01 46 1 59050E 01 1 56036E 01 1 14683E 00 51 1 14683E 00 2 84390E 01 2 84390E 01 FUNCTION CALLS 49 GRADIENT CALLS 3 Design Sensitivity and Optimization User s Guide m 3 5 ur iT PRINT il il 4 34 lt le 7235 ems eas l eue 3 de 2 6 S36 2 1 2 eds Modified method of feasible directions Set using METHOD on the DOPTPRM entry Level of optimizer output can be set with IPRINT on the DOPTPRM entry d IGRAD 1 indicates that gradients are supplied to the optimizer In this case the user is NX Nastran Initial independent design variable data Initial objective function value 04487E 01 4 67287E 01 84507E 01 1 31188E 00 Ten ol Initial constraints expressed in 86381E 01 7 73185E 01 48565E 01 9 27064E 02 normalized form 00000E 00 1 00000E 00 33333E 01 3 33333E 01 gt 00000E 00 1 00000E 00 g x lt 0 33333E 01 9 83014E 01 00000E 00 1 00000E 00 33333E 01 3 33333E 01 Number of one dimensional searches performed 5 VIOLATED CONSTRAINTS Constraint activity at the optimum 2 CONSECUTIVE ITERATIONS ITRMOP on the DOPTPRM entry 2 CONSECUTIVE ITERATIONS Resultant objective function and constraints based on optimization with 19968E 01 9 51816E 01 t 58553E 01 6 27252E 01 respect to the approximate model 71000B 01 6 21339E 01 96834E 01 2 59612E
113. 18 101 29 30 37 36 CQUAD4 19 101 16 17 24 23 CQUAD4 20 101 17 18 25 24 CQUAD4 21 101 18 19 26 25 CQUAD4 22 101 23 24 31 30 CQUAD4 23 101 24 25 32 31 CQUAD4 24 101 25 26 33 32 CQUAD4 25 101 30 31 38 37 CQUAD4 26 101 31 32 39 38 CQUAD4 27 101 32 33 40 39 FORCE 1 34 0 1250 1 FORCE 1 35 0 2500 1 7 28 Design Sensitivity and Optimization User s Guide Example Problems FORCE 1 36 0 2500 zd FORCE 1 37 0 2500 Sis FORCE 1 38 0 2500 E FORCE 1 39 0 2500 00 zi FORCE 1 40 0 1250 ies PSHELL 101 102 44 MAT1 102 2 7 3 0 731 3 SPC1 25 3456 1 THRU 40 SPC1 25 12 1 THRU 4 SPC1 25 1 13 20 27 34 design model desvar 1 bl 3 1 e6 1 e6 2 desvar 2 b1 3 1 e6 1 e6 2 desvar 3 b3 3 1 e6 1 e6 2 desvar 4 b4 3 1 e6 1 e6 2 desvar 5 b5 3 1 e6 1 e6 2 desvar 6 b6 3 1 e6 1 e6 2 desvar 7 b7 3 1 e6 1 e6 2 A DVSHAP entry defines a shape basis vector by associating one design variable to a dblocated displacement vector dvshap 1 1 66 773 dvshap 2 2 117 35 dvshap 3 3 216 33 dvshap 4 4 443 55 dvshap 5 5 220 89 dvshap 6 6 115 69 dvshap 7 7 65 669 drespl 5 volume volume drespl 2 von mis stress pshell 9 101 DCONSTR 10 2 3 100e43 100e4 doptprm DESMAX 25 APRCOD 1 param nasprt 1 ENDDATA Listing 7 6 7 6 Analytic Boundary Shapes This example illustrates the use of the analytic boundary shapes method in shape optimal design Relating Design Variable
114. 2 4 5 350312E 06 5 350312E 06 0 000000E 00 8 081818E 04 5 350312E 06 5 350312E 06 0 000000E 00 8 081055E 04 AUXILIARY MODEL 1 APRIL 30 2004 NX NASTRAN 04 30 04 PAGE 226 DESIGN VARIABLE HISTORY INTERNAL EXTERNAL DV ID DV ID LABEL INITIAL 1 2 3 4 5 1 1 UPPER 1 0000E 00 1 4000E 00 1 9600E 00 2 7440E 00 2 7700E 00 2 7700E 00 2 2 LOWER 1 0000E 00 1 4000E 00 1 9600E 00 2 7440E 00 2 7700E 00 2 7700E 00 0 USER INFORMATION MESSAGE 6464 DOM12E RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT CYCLE NUMBER Figure 7 12 Summary of Design Cycle History Design Sensitivity and Optimization User s Guide 7 33 Chapter 7 Example Problems my X Figure 7 13 Solid Cantilever Final Stage TITLE CANTILEVERED BEAM HEXA D200AM3 DESOBJ 15 DESSUB 100 SUBCASE 100 ANALYSIS STATICS SPC 1 LOAD 1 DISPLACEMENT ALL OUTPUT PLOT SET 1 ALL VIEW 90 0 0 0 90 0 PLOT SET 1 SET 2 ALL SVIEW 34 0 24 0 0 0 SPLOT SET 2 PLOT STATIC DEFORMATION SET 2 AUXCASE TITLE AUXILIARY MODEL 1 AUXMODEL 1 SUBCASE 200 SPC 200 LOAD 220 LABEL UPPER SUBCASE 300 SPC 300 LOAD 330 LABEL LOWER BEGIN BULK 9 ANALYSIS MODEL
115. 2 099861E 03 3 3 702784E 03 3 702784E 03 0 000000E 00 1 015833E 02 4 3 709424E 03 3 709424E 03 0 000000E 00 7 083985E 03 5 4 914370E 02 4 914370E 02 0 000000E 00 2 863409E 02 6 9 828741E 02 9 828741E 02 0 000000E 00 1 188324E 01 7 9 827139E 02 9 827139E 02 0 000000E 00 2 536414E 02 8 9 872102E 02 9 872102E 02 0 000000E 00 1 190643E 02 9 9 266409E 02 9 266409E 02 0 000000E 00 9 985107E 03 10 9 347542E 02 9 347542E 02 0 000000E 00 2 054260E 03 del 9 347542E 02 9 347542E 02 0 000000E 00 2 054260E 03 Design Sensitivity and Optimization User s Guide 7 59 Chapter 7 Example Problems DESIGN VARIABLE HISTORY INTERNAL EXTERNAL DV ID DV ID LABEL INITIAL A 1 2 3 4 5 HE 1 Pl 2 4930E 02 2 3478E 02 1 7004E 02 2 2451E 02 2 0869E 02 1 7282E 02 2 2 P2 1 9530E 02 1 7818E 02 2 5265E 02 1 9721E 02 2 0533E 02 2 5814E 02 3 4 P4 2 0470E 02 2 5099E 02 2 9327E 02 3 0365E 02 3 0671E 02 3 3864E 02 4 5 P5 2 5960E 02 2 3245E 02 2 5337E 02 2 6928E 02 2 7292E 02 2 8405E 02 5 6 P6 2 1750E 02 2 6218E 02 2 5095E 02 2 8850E 02 2 8434E 02 3 0478E 02 6 7 P7 2 4260E 02 2 1998E 02 2 0115E 02 1 4816E 02 1 5070E 02 8 4526E 03 7 8 BETA 1 0000E 00 9 9967E 01 4 9630E 01 3 7028E 01 3 7094E 01 4 9144E 02 INTERNAL EXTERNAL DV ID DV ID LABEL 6 7 i 8 9 10 i 11 al d Pl 1 2281E 02 1 3758E 02 1 5196E 02 1 7696E 02 1 8326E 02 1 8326E 02 2 2 P2 2 0876E 02 2 1119E
116. 2 U114 U116 U118 U120 U122 U124 U126 U128 U130 U132 U134 U136 U138 U140 U142 U144 U146 U148 U150 U152 U154 U156 U158 U160 U162 U164 U166 U168 U170 U172 U174 U176 U178 U180 U182 U184 U186 U188 U190 U192 U194 U196 0U198 U200 u20 2 u21 2 U22 2 U23 24 U24 2 U25 2 U26 2 U27 2 U28 2 U29 2 4U30 2 U31 2 U32 2 U33 2 4 U34 2 U35 2 U36 2 U37 2 U38 2 4 US39 2 U40 2 k U41 2 U42 2 UA3 2 U44 2 U45 2 U46 2 U47 2 U48 2 U49 2 U50 2 U50 2 U52 2 U53 2 U54 2 U55 2 u56 2 u57 2 u58 2 059 2 u60 2 U61 2 U62 2 U63 2 U64 2 U65 2 U66 2 U67 2 U68 2 U69 2 U70 2 U71 2 U72 2 U73 2 4 U7A4 2 0U75 2 U76 2 U7 2 U78 2 U79 2 U80 2 U80 2 U82 2 U83 2 U84 2 U85 2 u86 2 u87 2 u88 2 u89 2 u90 2 U90 2 U92 2 U93 2 U94 2 U95 2 u96 2 u97 2 198 2 u99 2 u100 2 2 0 u102 2 u104 2 u106 2 u108 2 ull0 2 u112 2 ull4 2 u116 2 u118 2 u120 2 u122 2 u124 2 ul26 2 ul28 2 ul30 2 u132 2 ul34 2 ul36 2 ul38 2 ul40 2 u142 2 ul44 2 u146 2 ul48 2 ul50 2 u152 2 u154 2 ul56 2 ul58 2 ul60 2 u162 2 ul64 2 ul66 2 ul68 2 ul70 2 ul72 2 u174 2 ul76 2 ul78 2 u180 2 u182 2 u184 2 u186 2 ul88 2 ul90 2 u192 2 u194 2 ul96 2 ul98 2 u200 2
117. 21E 00 2 024285E 06 2 369141E 04 5 2 702065E 00 2 702063E 00 7 941219E 07 2 666992E 04 INTERNAL EXTERNAL DV ID DV ID LABEL INITIAL 1 2 3 4 5 1 1 Ai 1 0000E 00 7 1020E 01 7 8436E 01 8 1374E 01 8 1739E 01 8 3569E 01 2 A2 2 0000E 00 9 9976E 01 6 0313E 01 4 3271E 01 3 9699E 01 3 3838E 01 2 3 A3 1 0000E 00 7 1020E 01 7 8436E 01 8 1374E 01 8 1739E 01 8 3569E 01 0 USER INFORMATION MESSAGE 6464 DOM12E RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT CYCLE NUMBER Bs Figure 7 30 Restart Results from OPTEXIT 4 ID UGS D200X1 TIME 10 SOL 200 OPTIMIZATION CEND TITLE SYMMETRIC THREE BAR TRUSS DESIGN OPTIMIZATION D200X1 SUBTITLE BASELINE 2 CROSS SECTIONAL AREAS AS DESIGN VARIABLES ECHO SORT SPC 100 DISP ALL STRESS ALL DESOBJ MIN 20 DESIGN OBJECTIVE DRESP ID DESSUB 21 DEFINE CONSTRAINT SET FOR BOTH SUBCASES ANALYSIS STATICS SUBCASE 1 LABEL LOAD CONDITION 1 LOAD 300 SUBCASE 2 LABEL LOAD CONDITION 2 LOAD 310 BEGIN BULK P P ANALYSIS MODEL GRID DATA 2 3 4 5 6 7 8 9 10 GRID 1 10 0 0 0 0 0 GRID 2 0 0 0 0 0 0 GRID 3 10 0 0 0 0 0 GRID 4 0 0 10 0 0 0
118. 3456 99 cutout edge N D 123456 1 Ii 127 B B B B B You will recognize these as the same set of boundary conditions that were used in the geometric shapes method The auxiliary boundary model solutions combined with the BNDGRID boundary data supply sufficient information to the code to use in interpolating these shape changes to the interior of the structure The result is a family of shape basis vectors Finally the designer ties these basis vectors to changes in the set of design variables using the DVBSHAP entries SDVBSHAP DVID AUXMID COLI sel DVBSHAP 1 Ly iy 1 0 DVBSHAP 2 1 2 1 0 Each entry associates a Design Variable with a basis vector resulting from a particular auxiliary boundary model AUXMID The column number COL indicates the auxiliary model displacement solution number In this example design variable 1 is the multiplier with a 1 0 multiple of the shape basis vector resulting from a solution of auxiliary boundary model 1 AUXMID 1 But recall that this model has two subcases 10 and 20 The COL field on the entry reconciles this COL1 1 implies 2 40 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization the first solution from subcase 10 COL1 2 implies the second solution from subcase 20 Similarly design variable 2 is the second basis vector multiplier computed from the second solution subcase 20 of auxiliary
119. 44E 03 5 333576E 03 3 910047E 03 953615E 03 23 000000E 01 033874E 03 1 339483E 04 341767E 03 501711E 03 792699E 04 722530E 04 000000E 01 033874E 03 1 339483E 04 341767E 03 501711E 03 792699E 04 722530E 04 This is followed by the DMAP information message DMAP INFORMATION MESSAGE 9052 FEA STATIC ANALYSIS COMPLETED DESIGN CYCLE NUMBER 1 If other analyses are performed normal modes dynamic response etc similar DMAP information messages and data recovery output will follow With the finite element analysis complete the code could perform a hard convergence check were this the second or greater design cycle Since this is just the first cycle though DOM12 the module that performs the convergence tests simply reports on the maximum constraint value THIS IS THE FIRST ANALYSIS NO CONVERGENCE CHECK Note See Eq 3 6 for a description of normalized constraints used in NX Nastran 6 4 Design Sensitivity and Optimization User s Guide Output Features and Interpretation Recall that these constraint values refer to the normalized constraints constructed internally in NX Nastran Since the maximum constraint value is positive we can characterize this initial design as infeasible see the basic optimization problem statement Getting Started Furthermore its value of 1 3119 indicates a constraint violation of around 13096 We will see shortly how to determine which constraint is respons
120. 5 18357 1 41400 00 GRID 13 0 00000 3 00000 00 GRID 14 0 78478 2 89464 00 GRID 15 1 49178 2 60164 00 GRID 16 2 12100 2 12100 00 GRID 17 3 00578 2 12100 00 GRID 18 3 89057 2 12100 00 GRID 19 4 77535 2 12100 00 GRID 20 0 00000 3 73200 00 GRID 21 0 68985 3 66176 00 GRID 22 1 32785 3 46643 00 GRID 23 1 91400 3 14600 00 GRID 24 2 67052 3 14600 00 GRID 25 3 42704 3 14600 00 GRID 26 4 18357 3 14600 00 GRID 27 0 00000 4 46400 00 GRID 28 0 59493 4 42888 00 GRID 29 1 16393 4 33122 00 GRID 30 1 70700 4 17100 00 GRID 31 2 33526 4 17100 00 GRID 32 2 96352 4 17100 00 GRID 33 3 59178 4 17100 00 GRID 34 0 00000 5 19600 00 GRID 35 0 50000 5 19600 00 GRID 36 1 00000 5 19600 00 GRID 37 1 50000 5 19600 00 GRID 38 2 00000 5 19600 00 GRID 39 2 50000 5 19600 00 GRID 40 3 00000 5 19600 00 CQUAD4 1 101 1 2 6 5 CQUAD4 2 101 2 3 Ts 6 CQUAD4 3 101 a 4 8 7 CQUAD4 4 101 5 6 10 9 CQUAD4 5 101 6 7 11 10 CQUAD4 6 101 7 8 12 11 CQUAD4 7 101 9 10 17 16 CQUAD4 8 101 10 11 18 17 CQUAD4 9 101 11 12 19 18 CQUAD4 10 101 13 14 21 20 CQUAD4 11 101 14 15 22 21 CQUAD4 12 101 15 16 23 22 CQUAD4 13 101 20 21 28 27 CQUAD4 14 101 21 22 29 28 CQUAD4 15 101 22 23 30 29 CQUAD4 16 101 27 28 35 34 CQUAD4 17 101 28 29 36 35 CQUAD4
121. 836E 02 458149E 02 459160E 02 4186E 00 5778E 00 3324E 00 3288E 00 8193E 00 3514E 00 9252E 00 5406E 00 1977E 00 8964E 00 6367E 00 5696E 01 3964E 01 1894E 00 5311E 00 2363E 00 7481E 00 0664E 00 1914E 00 1229E 00 8610E 00 4057E 00 FRACTIONAL ERROR OF APPROXIMATION 7602E 00 1178E 00 0673E 00 9453E 00 3054E 00 6989E 00 1257E 00 5859E 00 0795E 00 6064E 00 1666E 00 5705E 01 3968E 01 1892E 00 5356E 00 2402E 00 7514E 00 0692E 00 1936E 00 1246E 00 8622E 00 4063E 00 033855E 07 778743E 07 503979E 06 193210E 06 136332E 07 190216E 06 862229E 07 204878E 08 8056E 00 1321E 00 5704E 01 4806E 00 8213E 00 1476E 00 4596E 00 7574E 00 0409E 00 3101E 00 5650E 00 Example Problems MAXIMUM VALUE or CONSTRAINT 072155E 01 562702E 00 796172E 00 925344E 01 399380E 01 652916E 02 164406E 02 082301E 03 481479E 03 0352E 00 8664E 00 3711E 00 5304E 00 8685E 00 1637E 00 4161E 00 6256E 00 7922E 00 9161E 00 9970E 00 3748E 00 1305E 01 1134E 00 5666E 00 1175E 00 5399E 00 8337E 00 9990E 00 0358E 00 9440E 00 7237E 00 ID UGS D200X5 TIME 10 SOL 200 OPTIMIZATION CEND TITLE CANTILEVERED PLATE D200X5 SUBTITLE REDUCED BASIS FORMULATION SPC 100 OLOAD ALL DISP ALL STRESS ALL DESOBJ MIN MAX 35 OBJECTIVE FUNCTION DEFINITION SUBCASE 1 ANALYSIS STATICS LABEL
122. 9 ROWS THRU 30 Ee ROW 2 0435E 05 2 9491E404 5 0688E 03 OCOLUMN 40 ROWS THRU de wm RE epe dei melde SS euer ROW 2 1453E 00 1 4776H 00 1 1223E 00 OCOLUMN 41 ROWS l IRU MEM Lu uL i E LE E ROW 2 1453E 00 1 4776E 00 1 1223E 00 OCOLUMN 42 ROWS THRU 3 SSeS SSS SS SSS eS eS ae aS rne eer SS aim ROW 1 7913E 04 1 7736H 04 1 7589E 04 OCOLUMN 43 ROWS THRU Sei See SS ses SoS soe See SS Se SSS SSS SS SS ROW 1 7913E 04 1 7736H 04 1 7589E 04 OCOLUMN 44 ROWS THRU Be SR SSSR SS Sao cep SASS SSS sear aR aS SS SS aS SSeS ROW 1 7913E 04 1 7736E 04 1 7589E 04 OCOLUMN 45 ROWS THRU a eee SS OS E ROW 1 7913E 04 1 7736H 04 1 7589E 04 OCOLUMN 61 ROWS 1 THRU 3 HGR y ARA uda eut edet eerte tunc eni ROW 1 2 1265E 04 1 0639H 04 5 4152E 03 OTHE NUMBER OF NON ZERO TERMS IN THE DENSEST COLUMN 3 OTHE DENSITY OF THIS MATRIX IS 100 00 PERCENT rows columns zero terms 100 matrix density TONO SDOISRIPE Note that the size of DSCM2 is 3 rows by 61 columns Each row corresponds to a single independent design variable Since design variables 9 10 and 11 are independent rows 1 2 and 3 correspond respectively to these quantities Each column corresponds to a particular response we can note that the sensitivities of 61 responses have been computed The response order is listed in the correlation table discussed shortly Before moving on to the correlation table we should note that the density of this
123. 9 and central differences see Eq 3 28 for example are frequently used in NX Nastran s semianalytic sensitivity analysis to provide low cost derivative approximations First level Response is a response that is directly available from an NX Nastran analysis These responses are identified for use in the design model with DRESP1 entries Formal Approximations are the Taylor series expansions of the implicit responses used in connection with design optimization The resultant explicit approximations are used by the optimizer to find a corresponding approximate optimum Gradient is defined as a vector of partial derivatives See Eq 1 10 Physically the gradient of a function points in the direction of most rapidly increasing function values Gradient based Numerical Optimizer is any optimizer that uses function gradient information to search for an optimal design The optimizer in NX Nastran is of this class Grid Perturbation refers to a small change in a grid point coordinate for a corresponding infinitesimal design variable change These small variations are used in differencing schemes for design sensitivity analysis Grid Variation refers to a finite change in a grid point coordinate for a corresponding change in a design variable This results in a finite change of structural shape Design Sensitivity and Optimization User s Guide A 3 Appendix A Glossary of Terms Hard Convergence refers to the design cycle convergence test that co
124. 9000 50000 50000 29000 29000 50000 50000 29000 29000 50000 50000 29000 29000 50000 50000 29000 29000 50000 50000 29000 29000 50000 50000 29000 29000 50000 50000 29000 29000 analysis model properties linearly in terms PMAX co 7 9 objective 10 10 ATTA ATTB function of design 0 2500 0 1250 ATT1 M Xo 00 00 O0 O0 NNN OY OY MOSS 0 0 CQ 0 B IO NB NEPPE Example Problems DL7 DL8 variables Je DP1 DP2 DP3 DP4 DP5 DP6 DP7 DP8 Design Sensitivity and Optimization User s Guide 7 15 Chapter 7 Example Problems DCONSTR 10 23 50000 50000 DCONSTR 10 24 29000 29000 DCONSTR 10 25 50000 50000 DCONSTR 10 26 29000 29000 DCONSTR 10 27 50000 50000 DCONSTR 10 28 29000 29000 DCONSTR 10 29 50000 50000 DCONSTR 10 30 29000 29000 DCONSTR 10 31 50000 50000 DCONSTR 10 32 29000 29000 DCONSTR 10 33 ck 2 DCONSTR 10 34 ey 2x Override optimization parameter defaults optional DOPTPRM DESMAX 10 DELP 0 5 DPMIN 01 DELX 2 0 DELB 0 01 CONV2 0 1 ENDDATA Listing 7 3 7 4 Stiffened Plate As discussed in Design Modeling for Sensitivity and Optimization an effective way to keep the number of independent design variables to a minimum is by grouping designed elements by property type A smaller set of independent design variable
125. AR 6 P6 0 02175 0 0001 1 DESVAR 7 P7 0 02426 0 0001 1 DESVAR 8 BETA 1 0 0 001 Relate the Design Variables to changes in plate thicknesses VPREL1 ID TYPE PID FID PMIN PMAX co 2 DVID1 COEF1 DVID2 COEF2 DVPREL1 1 PSHELL 1 4 0 0001 1 1 DVPREL1 2 PSHELL 2 4 0 0001 i 2 15 DVPREL1 4 PSHELL 4 4 0 0001 m 4 Ls DVPREL1 5 PSHELL 5 4 0 0001 5 1 DVPREL1 6 PSHELL 6 4 0 0001 E i 6 1 DVPREL1 7 PSHELL 7 4 0 0001 A Te e Define the synthetic Objective as a function of X8 SSRESP2 ID LABEL EQID REGION n z DESVAR DVID1 DVID2 F i DRESP2 100 BETA 100 DESVAR 8 DEQATN 100 OBJ BETA 10000 0 BETA Define the constraint on weight weight budget for optimization RESP1 ID LABEL RTYPE PTYPE REGION ATTA ATTB ATT1 ATT2 DRESP1 2 WEIGHT WEIGHT CONSTR DCID RID LALLOW UALLOW DCONSTR 5 2 2890 2910 Define the constraints on acoustic sound pressure levels one for each forcing frequency DRESP1 1 DRUCK FRDISP 1 11280 DRESP2 11 PRESBET 10 DESVAR 8 DRESP1 1 DEQATN 10 F BETA PRES 100 0 BETA PRES 1000 DCONSTR 10 11 1000 Override miscellaneous Optimization Parameters DOPTPRM DESMAX 20 P1 1 P2 15 CONV1 1E 6 SSSSSSSSSSSSSSSSSSSS SSSSSSSSsssssssss structural model SSSSSSSSSSSSSSSSSSSSSSSSSSSSsssssssss
126. ARY CONDITIONS FOR SHAPE INTERPOLATIONS TOP SURFACE BNDGRID C GP1 GP2 GP3 GP4 GP5 GP6 GP7 GP8 p BNDGRID 123 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 BOTTOM SURFACE BNDGRID 1231234567 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 EXTERIOR SURFACES INTERPOLATION IN X amp Z DIRECTION ONLY BNDGRID 2 34 35 36 37 38 39 40 41 42 43 44 BNDGRID 2 56 57 58 59 60 61 62 63 64 65 66 BNDGRID 2 67 68 69 70 71 72 73 74 75 76 77 Design Sensitivity and Optimization User s Guide 7 35 Chapter 7 Example Problems BNDGRID 2 89 90 91 92 93 94 95 96 97 98 99 BNDGRID 2 100 101 102 103 104 105 106 107 108 109 110 BNDGRID 2 122 123 124 125 126 127 128 129 130 131 132 TIP END BNDGRID 1 11 22 33 44 55 66 BNDGRID 1 77 88 99 110 121 132 BNDGRID 1 143 154 165 FIXED END BNDGRID 123 1 12 23 34 56 67 89 100 122 133 144 155 BNDGRID 1 45 78 111 FORMULATE WEIGHT BASED SYNTHETIC RESPONSE F 1 E5 W DRESP1 1 WEIGHT WEIGHT DRESP2 15 WE1000 1 DRESP1 1 DEQATN 1 F A 100000 A CONSTRAINTS ON VON MISES STRESSES DRESP1 2 STRESS STRESS PSOLID 13 1 DSCREEN STRESS 1 0 10 DCONSTR 100 2 200 OVERRIDE OF OPTIMIZATION PARAMETERS DOPTPRM DESMAX 9 P1 1 P2 15 BEGIN BULK AUXMODEL 1 PARAM PRGPST NO PARAM MAXRATIO1 0E 8 PARAM AUTOSPC YES
127. Concepts Used in Structural Optimization Approximation concepts are actually nothing more than the computational implementation of methods generally used by experienced design engineers In many instances an engineer is handed a stack of analysis data and asked to propose an improved design This raw data usually contains much more information than is necessary to suggest possible design improvements The question becomes one of how to reduce the problem sufficiently so that only the most pertinent information is considered in the process of generating a better design Design Variable Linking A first step may be to narrow the design task to that of determining the best combination of just a few design variables There is virtually no way for a designer to consider fifty or one hundred variables simultaneously and expect to find a suitable combination from the group It is much more efficient to link these together if possible That is it would be advantageous if all the design variables could be varied in a suitably proportional manner according to changes made to a much smaller set of independent variables This way a large number of structural properties might be varied according to a smaller set of well chosen variables Describing a shape defining polynomial surface in terms of just a few characteristic parameters or allowing only linear variations of plate element thicknesses are both examples of types of design variable linking Design Sen
128. DELX and DXMIN are reduced by one half of their current values The reason for the first condition is that frequently the optimizer may violate the constraints somewhat as it makes favorable gains in the objective function in the first few cycles However if this condition continues it may indicate that the problem is becoming ill conditioned as a result of excessive move limits A corresponding User Warning Message is printed as notification that this update has occurred see Output Features and Interpretation If the job is to be restarted an updated DOPTPRM entry with the new move limits should be included in the restart deck Note Under no conditions does an update increase the move limits The only type of update is a reduction by one half The move limits cannot be increased again unless the engineer intervenes and manually changes them 3 50 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran Implementation of Move Limits Move limits on independent design variables are applied directly Since the optimizer will never propose a value for a design variable outside of its bounds the lower and upper move limits computed from Eq 3 101 Eq 3 104 and Eq 3 106 are simply input directly to the optimizer Note Move limits on dependent design variables are imposed as equivalent constraints similar to that of properties in Eq 3 108 below Move limits on properti
129. DOPTPRM and DSCREEN a set of parameters unique to design sensitivity and optimization is also available This section is intended to be a complete quick reference for all such Case Control and Bulk Data parameters Refer to Commonly Used Commands for Design Optimization for further details Controls the selection of finite difference scheme used in sensitivity analysis YES Selects central differences default for shape optimization NO Selects forward differences default for property optimization only Design Sensitivity and Optimization User s Guide 4 23 Chapter 4 Input Data DESPCH Controls the frequency of updated DESVAR and GRID Bulk Data entry output to the PUNCH file N 0 No output N 0 Final design cycle default N gt 0 Every N th as well as final design cycles DSNOKD Indicates if the differential stiffness effect is to be included in buckling sensitivity analysis uM 1 0 Include differential stiffness default 0 0 Do not include differential stiffness NASPRT Controls the frequency of NX Nastran output 1 No output 0 Output on initial and final design cycles default N Output every N th iteration and also prior to exit OPTEXIT Instructs the program to exit at one of six predetermined exit points um No user defined exit default N gt 0 Exit at one of the locations 1 through 7 1 N s 7 SOFTEXIT Determines whether to terminate design cycles if soft convergence is indicated Wl NO Do not s
130. DR73 DR73 DRESP1 7 DRESP2 23 SC8 di i F j DR81 DR81 DESVAR 8 A P A P DR82 DR82 DTABLE L8 i i i DR83 DR83 DRESP1 8 Equations used to define second level responses note fixed field form SDEQATN EQUID F DEQATN 1 F A RL S S RL 2 A 1 E7 39 274 Table constants SDTABLE LABEL1 VALUE1 LABEL2 VALUE2 LABEL3 VALUE3 LABEL4 VALUE4 LABEL5 VALUES DTABLE Ll 75 00 12 130 50 L3 106 80 L4 75 00 L5 75 00 L6 181 14 17 181 14 L8 133 46 Define the design constraints SCONSTR DCID RID LALLOW UALLOW DCONSTR 10 Ag 40000 40000 DCONSTR 10 2 40000 40000 DCONSTR 10 3 40000 40000 DCONSTR 10 4 40000 40000 DCONSTR 10 By 40000 40000 DCONSTR 10 6 40000 40000 DCONSTR 10 Tx 40000 40000 DCONSTR 10 8 40000 40000 DCONSTR 10 OF 0 35 0 35 DCONSTR 10 10 0 35 0 35 DCONSTR 10 11 0 35 0 35 DCONSTR 10 12 0 35 0 35 DCONSTR 10 13 0 35 0 35 DCONSTR 10 14 0 35 0 35 DCONSTR 11 16 1 0E10 1 0 DCONSTR 11 17 1 0E10 1 0 DCONSTR 11 18 1 0E10 1 0 DCONSTR 11 19 1 0E10 1 0 DCONSTR 11 20 1 0E10 1 0 DCONSTR 11 21 1 0E10 1 0 DCONSTR 11 22 1 0E10 1 0 DCONSTR 11 23 1 0E10 1 0 Combine the two constraint sets equivalent to just putting all into the same set to begin with DCONADD 12 10 11 Override opti
131. Decision Logic A schematic of the soft convergence decision logic is shown in Figure 3 13 The result of the test is a true or false value for the logical variable SOFTCV The design cycles terminate only if SOFTCV is TRUE and the parameter SOFTEXIT is YES No other combinations allow soft convergence to halt the design cycle process ME CONV1 NO ACHOBJ x CONV2 XES NO CHGPRP x CONVPR YES CONMAX x GMAX NO Or CHGDV lt CONDV YES SOFTCV TRUE SOFTCV FALSE Figure 3 13 Soft Convergence Decision Logic From Figure 3 13 the first test is a check of the relative and absolute changes in the objective The definitions of these parameters appear in Table 3 1 along with their default values These values may be modified using the DOPTPRM entry and also apply to the hard convergence checks Table 3 1 Convergence Criteria Parameters P EE CHGOBJ LOB CONV1 0 001 OBI 3 56 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran Table 3 1 Convergence Criteria Parameters max CHGPRP SSS CONVPR 0 001 1 is NPROP max Xj MXi CONVDV 0 001 1 lt i lt NDV a max CONMAX If either of the relative or absolute changes in the objective is satisfied the next check is to see whether or not the analysis model properties have changed appreciably If the changes in the objective and analysis model properties are negligible the subsequent ch
132. ELP 0 5 Pl Ls Fi P2 4 5 Listing 7 2 7 10 Design Sensitivity and Optimization User s Guide Example Problems 7 3 Cantilevered Plate Reduced basis formulations were introduced in Approximation Concepts in Design Optimization as an efficient way to reduce the number of independent design variables This example shows how DLINK entries can be used to express these reduced basis formulations Suppose we wish to determine the optimum thickness distribution of the cantilever plate in Figure 7 3 such that the structural mass is minimized Two separate loading conditions exist the first is a tip loading condition as shown in the figure the second is a uniform pressure loading in the Z direction Constraints are placed on maximum allowable tip displacement and von Mises stresses at the upper and lower surfaces of each shell element Y pl p2 yyy ty ZA Es Y SS d OQ 0 u 6 14 9 8 y 40 Figure 7 3 Cantilever Plate Thickness Design Analysis Model Description 2 x 8 array of CQUAD4 elements Material E 1 0E 6 psi Poisson ratio 0 33 Weight density 0 1 Ibs in Two static load conditions Tip loading Two 5 000 Ib loads in z direction Pressure loading Uniform at 6 44 Ib in Design Model Description Objective Structural weight minimization Design Basis functions Constant linear and quadratic in x direction to describe variables plate element thicknesses Explicit design var
133. ELX default of 1 0 and the DELP default of 0 2 perform well in most cases although these values may not always yield the greatest efficiency If any of the design variables or properties are near zero the move limits prescribed by Eq 3 101 and Eq 3 103 may tend to overly restrict design moves and hence the optimizer s progress This can not only waste valuable computational resources but the convergence decision logic may be trapped into believing that the design process has converged when in reality the optimizer had just been unnecessarily restricted To avoid this minimum move limits are defined for both design variables and properties If any design variable is near zero such that x 0 DELX DELXV lt lt 1 the minimum lower and upper limits are defined by x min x DELX DELXV x DXMIN 0 DXMIN I x max x DELX DELXV x L Equation 3 104 where x 4 DELX DELXV and x Y DELX DELXV are the lower and upper bounds from Eq 3 101 3 48 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran Note The notation DELX DELXV indicates that the default provided by DELX can be overridden if a DELXV is present See Eq 3 101 Likewise minimum move limits on properties are defined as p min p DELP p DPMIN p max p DELP p DPMIN Equation 3 105 where p DELP and p 4 DELP are the lower and upper bounds from
134. ERNAL DV ID DV ID 0000E 00 0000E 01 5000E 01 0722E 01 2632E 01 8 2653E 01 0000E 00 0000E 01 5193E 01 4035E 01 4570E 01 4 4566E 01 0000E 00 0000E 01 5000E 01 2494E 01 1281E 01 1 1269E 01 0000E 01 0000E 01 5153E 02 8634E 02 1032E 02 5 0650E 02 0000E 01 0000E 01 0000E 02 6748E 02 3546E 02 4 3211E 02 0000E 01 0000E 01 0000E 02 5000E 02 4701E 02 2 5277E 02 USER INFORMATION MESSAGE 6464 DOM12E RUN TERMINATED DUE TO HARD CONVERGENCE TO A POSSIBLY NON UNIQUE OPTIMUM nastran oldq4k ID UGS D200X6 TIME 10 SOL 200 OPTIMIZATION CEND TITLE VIBRATION OF A BEAM ECHO UNSORT DISP ALL STRESS ALL METHOD 1 ANALYSIS MODES DESOBJ MIN 1 OBJECTIVE FUNCTION DEFINITION DESSUB 10 CONSTRAINT SET SELECTION BEGIN BULK 123456 2456 2456 2456 123456 SN oo oo ooooo ooooo w Design Sensitivity and Optimization User s Guide 7 9 Chapter 7 Example Problems GRID 6 A 20 0 0 0 3 0 2456 GRID 7 I 40 0 0 0 3 0 i 2456 GRID 8 60 0 0 0 3 0 Jj 2456 CROD 1 201 5 6 CROD 2 202 6 7 CROD 3 203 7 8 CROD 7 201 To 2 CROD 8 202 2 3 CROD 9 203 3 4 PROD 201 1 1 0 0 0 PROD 202 i 1 0 0 0 PROD 203 ds 1 0 0 0 CQUAD4 4 204 i 2j 6 5 CQUAD4 5 205 2 3 7 6 CQUAD4 6 206 3 4 8 7 PSHELL 204 1 0 2 PSHELL 205 a 0 2 PSHELL 206 1 0 2
135. Equation D 42 from which a proposed a is zd 04 e F R j Aest 7 5 q 1 ES da Equation D 43 This results in an estimate of a that reduces the objective function by 1096 However since the gradients of some constraints are probably available other possible moves can be calculated as well Remember that Eq D 40 applies to a constraint by simply substituting the constraint gradient for the objective gradient Now assume that some constraint gradients exist for constraints that are not critical and it is desired to estimate how far to move to make one of them critical That is instead of reducing the value of the function by 1096 as was done for the objective it is desired to estimate a move in a that places it near the constraint boundary Thus a linear approximation gt a _ to find gE 99 ig D 18 Design Sensitivity and Optimization User s Guide Numerical Optimization gt q 1 q q 1 dg x Cu gR ae ea 0 0 da Equation D 44 and an estimate for a is q 1 lt 8g 2st 1 l EL i da Equation D 45 Therefore even at the beginning of the one dimensional search a considerable amount of information is available to direct the process Using the estimates for a given by Eq D 43 and Eq D 45and for each noncritical constraint the smallest positive proposed a is taken as the first estimate of how far to move If constraints are currently violated Eq D 45 can still be us
136. Guide Example Problems and then pause to examine these results before restarting The results of the finite element analysis are saved to the database by including SCR NO in the job submittal command The only difference between the original three bar truss and the input here is the Bulk Data parameter OPTEXIT In this case OPTEXIT is set to 4 which instructs the code to perform sensitivity analysis print out the coefficients and stop After the finite element and sensitivity analysis results are examined and verified the decision is made to continue with optimization in the restart run This deck is shown in Listing 7 14 The RESTART command is required and indicates that the most recent version on the database is to be used VERSION LAST and only the latest data is to be retained NOKEEP If we always intend to use the first version on the database for restarts we can instead use RESTART VERSION 1 KEEP This command appends the subsequent results to the database while saving the data from prior runs This might be useful if the restart yields results that we do not want to save and we want to go back and restart from the initial analysis By specifying VERSION 1 KEEP we always have the first version to fall back on The command ASSIGN MASTER d200x1c MASTER assigns the correct DBSET so we do not have to repeatedly assign this with DBS in the job submittal command The lower case filename ensures consistency with t
137. I kk e k k k kk k k ck ck k k k k ck kk ck k k k k k k k I I ck kk k k k ko kk kk ko kk k k ke kk ee 7 12 Design Sensitivity and Optimization User s Guide INTERNAL DV ID FPOWDMIDUYOAWNHE INTERNAL DV ID P Oouo ocdcsuNpP USER I RUN TERMINATED DUE TO HARD CONVERGENCE TO AN OPTIMUM AT Ok KKK KKK k K Ok Ok k k Kk Kk Ck k k k K OK Ok k KKK KR kok KOk Ck k k k Kk Ck k k KK ERK ko k k k k ko k k Kk k ko k kk Kok SUMMARY OF DESIGN CYCLE HISTORY itt eee ee cece OOOO kk kk kk kk kk ko kk ko kk ko ke AR ke kk A A koe HARD CONVERGENCE ACHIEVED SOFT CONVERGENCE ACHIEVED OBJEC APP NUMBER 0o AUDE NNNNNNRR EXTERNAL DV ID ALPHA1 ALPHA2 ALPHA3 T1 T2 T3 T4 T5 T6 T7 EXTERNAL DV ID ALPHA1 ALPHA2 ALPHA3 T1 T2 T3 T4 T5 T6 NFORMATION MESSAGE 6464 NUMBER OF FINITE ELEMENT ANALYSES COMPLETED NUMBER OF OPTIMIZATIONS W R T APPROXIMATE MODELS OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY TIVE FROM ROXIMATE OPTIMIZATION 134805E 02 575682E 02 029126E 02 365482E 02 438008E 02 435833E 02 458150E 02 459160E 02 DOM12E 0000E 00 0000E 00 0000E 00 0000E 00 6406E 00 3125E 00 0156E 00 7500E 00 5156E 00 3125E 00 1406E 00 7168E 01 3916E 01 2996E 00 3884E 00 1253E 00 6654E 00 0086E 00 1549E 00 1044E 00 8570E 00 4128E 00 OBJECTIVE FROM EXACT ANALYSIS 843750E 01 134805E 02 575683E 02 029123E 02 365477E 02 438006E 02 435
138. IC RELATIVE CHANGE IN OBJECTIVE 2 7611E 03 MUST BE LESS THAN 0000E 03 ABSOLUTE CHANGE IN OBJECTIVE 7 4799E 03 MUST BE LESS THAN 0000E 02 AND MAXIMUM CONSTRAINT VALUE 9 2676E 05 MUST BE LESS THAN 0000E 03 CONVERGENCE TO A FEASIBLE DESIGN eas OR aaa MAXIMUM OF RELATIVE PROP CHANGES 1 5165E 01 MUST BE LESS TH 0000E 03 AND MAXIMUM OF RELATIVE D V CHANGES 1 5165E 01 MUST BE LESS 0000E 03 CONVERGENCE TO A BEST COMPROMISE INFEASIBLE DESIGN Tce A A A A A A A A ee eee Ck Ck Ok Ck ck ck ck k ck k RARA kk k kk CK CK CK A K A k k k k ck k k k k k Ok k RR Ok Ck Ok Ok k ck ck ck ck k k kk ok IO SUMMARY OF DESIGN C YC LE HISTORY oko ke Rok OK I III II III III Ro Re RR GR aaa ak HARD CONVERGENCE ACHIEVED NUMBER OF FINITE ELEMENT ANALYSES COMPLETED NUMBER OF OPTIMIZATIONS W R T APPROXIMATE MODELS OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY OBJECTIVE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE APPROXIMATE EXACT OF NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT 828427E 00 234952E 01 007858E 00 008487E 00 093749E 04 737109E 03 820099E 00 819910E 00 696224E 05 902266E 02 735039E 00 735147E 00 957449E 05 482129E 03 709055E 00 709030E 00 152918E 06 097656E 04 701577E 00 701550E 00 884281E 06 267578E 05 ID UGS D200X1 TIME 10 SOL 200 OPTIMIZATION CEND TITLE SYMMETRIC THREE BAR TRUSS DESIGN OPTIMIZATION D200X1 SUBTITLE BASELINE 2 CROSS SECTIONAL AREAS AS DESIGN VARIABLES ECHO
139. ID DDVID CO CMULT IDV1 cl IDV2 c2 IDV3 c3 a DLINK 1 3 0 0 1 05 i 1 00 Listing 7 1 Cont DEFINITION OF DESIGN VARIABLE TO ANALYSIS MODEL PARAMETER RELATIONS DVPREL1 ID TYPE PID FID PMIN PMAX co DVID1 COEF1 DVID2 COEF2 DVPREL1 10 PROD 11 4 j i DP1 DP1 1 1 0 DVPREL1 20 PROD 12 4 j A DP2 DP2 2 1 0 DVPREL1 30 PROD 13 4 F f DP3 DP3 3 1 0 STRUCTURAL RESPONSE IDENTIFICATION DRESP1 ID LABEL RTYPE PTYPE REGION ATTA ATTB ATT1 ATT2 DRESP1 20 W WEIGHT DRESP1 21 U4 DISP 1 i 4 DRESP1 22 v4 DISP R 2 4 DRESP1 23 Si STRESS PROD 2 11 DRESP1 24 S2 STRESS PROD 2 12 DRESP1 25 S3 STRESS PROD 2 13 CONSTRAINTS DCONSTR DCID RID LALLOW UALLOW DCONSTR 21 21 0 20 0 20 DCONSTR 21 22 0 20 0 20 DCONSTR 21 23 15000 20000 DCONSTR 21 24 15000 20000 DCONSTR 21 25 15000 20000 OPTIMIZATION CONTROL DOPTPRM IPRINT 5 DESMAX 10 DELP 0 5 Pl 1 P2 15 Listing 7 1 7 2 Vibration of a Cantilever Beam Turner s Problem This problem was originally published by M J Turner Design of Minimum Mass Structures with Specified Natural Frequencies A AA Journal Vol 5 No 3 March 1967 The problem is to design a minimum weight structure while constraining the fundamental bending frequency at or above 20 Hz The beam is sym
140. L1 DVCREL2 Design Sensitivity and Optimization User s Guide C 1 e DVGEOM e DVGRID DVMREL1 e DVMREL2 e DVPREL1 DVPREL2 e DVSHAP C 3 Parameters This section lists commonly used parameters for optimization See the NX Nastran Quick Reference Guide for complete descriptions of these parameters e AUTOADJ e CDIF e DESPCH DSNOKD e DESPCH1 DPEPS e DSZERO e NASPRT e OPTEXIT SENSUOO e SOFTEXIT e UPDTBSH C 2 Design Sensitivity and Optimization User s Guide Appendix D Numerical Optimization D 1 Introduction The basic features of numerical optimizers were introduced in Getting Started This general overview is probably sufficient in most situations to let you follow the tasks performed by the optimizer to understand how to pose the design problem for effective solution and to interpret most of the optimization related output This section completes the discussion begun in Getting Started and is intended for the intermediate to advanced level reader who wishes to know more about the implementation of the optimizer used in NX Nastran This section has largely been excerpted from the DOT User s Manual In addition Vanderplaats Numerical Optimization Techniques for Engineering Design with Applications is recommended further reading on this topic because it includes theoretical details that are beyond the scope of this section The processes described here are a part of the code and are not subject to
141. L1 Design Variable Connectivity RELation type 1 is used to relate design variables to connectivity properties DVMREL1 Design Variable Material RELation type 1 is used to relate design variables to material properties DVPREL1 Design Variable Property RELation type 1 is used to relate design variables to analysis model properties 2 8 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization Type 1 linear design variable to property relations are of the form Pj C T y Cri i Equation 2 8 where pjis the j th property expressed as a linear combination of the design variables Type 2 Design Variable to Property Relations Type 2 design variable to property relations define a functional relationship between design variables and properties The functional relationships are defined on DEQATN Design EQuATION bulk entries DVCREL2 DVMREL2 and DVPREL2 bulk entries that reference DEQATN bulk entries are used to define a Type 2 design variable to property relations e DVCREL2 Design Variable Connectivity RELation type 2 is used to relate design variables to connectivity properties e DVMREL2 Design Variable Material RELation type 2 is used to relate design variables to material properties DVPREL2 Design Variable Property RELation type 2 is used to relate design variables to analysis model properties Type 2 design variable to property relations are of the form
142. LL 109 i00 L 0 100 F 100 PSHELL 110 100 1 0 100 100 SDESVAR ID LABEL XINIT XLB XUB DELXV DESVAR 1 X1 0 33 1 0 41 0 DESVAR 2 x2 0 33 1 0 41 0 DESVAR 3 X3 0 33 1 0 1 0 SDVPREL1 ID TYPE PID FID PMIN PMAX CO 4 DVID1 COEF1 DVID2 COEF2 DVPREL1 201 PSHELL 101 4 0 01 P 1 LaDy 2 As 3 1 0 DVPREL1 202 PSHELL 102 4 0 01 P ls 1 0 ay Du 9 34 0 81 DVPREL1 203 PSHELL 103 4 0 01 p ls 1 0 Ve 0 8 34 0 64 DVPREL1 204 PSHELL 104 4 0 01 A P Ta ls 1 0 Ay T 34 0 49 DVPREL1 205 PSHELL 105 4 0 01 P Ly I 2 Usb d 0 36 DVPREL1 206 PSHELL 106 4 0 01 P A Ly IU 2 Du 3 0 25 DVPREL1 207 PSHELL 107 4 0 01 g 1 ls 2 0 4 3 0 16 DVPREL1 208 PSHELL 108 4 0 01 g ey 1 Levy ay 0 3 3y 0 09 DVPREL1 209 PSHELL 109 4 0 01 i 1 Ca Uy Ay D 2 3 0 04 Design Sensitivity and Optimization User s Guide 3 7 Chapter 3 Design Sensitivity and Optimization in NX Nastran DVPREL1 210 PSHELL 110 4 0 01 i 1 0 2 0 1 3 0 01 Listing 3 1 The three design variables or basis function multipliers are defined on the three DESVAR entries Each has an initial value of 0 33 therefore the initial design is defined as that which has a one third contribution from each of the basis functions The lower and upper bounds on the design variables are 1 0 and 1 0 respectively
143. NCE FLAG IS USED TO CHARACTERIZE DIFFERENCES BETWEEN ANALYSIS AND DESIGN MODEL PROPERTIES IF THE FLAG IS NONE THEN THERE IS NO SIGNIFICANT DIFFERENCE BETWEEN THE TWO VALUES IF THE FLAG IS WARNING THEN THE USER IS ADVISED THAT DIFFERENCES EXIST IF THE FLAG IS FATAL THEN THE DIFFERENCES ARE GREATER THAN 1 00000E 35 AND THE RUN WILL BE TERMINATED Figure 2 8 Message Indicating an Analysis Model Override by the Design Model III RRA BEAM MODELING TEST VERSION 2 TAPERED BEAM WITH RECTANGULAR SECTION DESIGN SENSITIVITY ANALYSIS SINCE BOTH ENDS A AND B ARE SPECIFIED ON THE PBEAM ENTRY DESIGN CHANGES MUST BE SPECIFIED NOT ONLY FOR THE ENDS BUT ALSO FOR THE FIRST INTERMEDIATE STATION WHICH CONTAINS A COPY OF END B DATA PLANE 2 S 2 k H PLANE 1 S RTS cd ee STRESS RECOVERY LOCATIONS S gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt gt TIME 10 SOL 200 CEND Design Sensitivity and Optimization User s Guide 2 19 Chapter 2 Design Modeling for Sensitivity and Optimization CC FOR ANALYSIS DISP ALL STRESS ALL SPC 100
144. NSTR number 5 which is in turn referenced in Case Control by the DESGLB command The bounds on Design Sensitivity and Optimization User s Guide 7 57 Chapter 7 Example Problems weight allow less than one third of one percent variation from the initial weight Allowing some variation is usually preferred over an equality constraint since it not only yields a better conditioned problem but is often consistent with our design goals as well The objective function is defined as a constant times design variable number 8 see Eq 7 22 the objective function and Eq 7 24 below using the DRESP2 number 100 entry This refers to DEQATN 100 which defines F 10000 x Equation 7 24 This is defined as the objective function by the DESOBJ 100 Case Control command The constraints on response peaks are defined by first identifying the pressure responses themselves with DRESP1 entry number 1 This designates the 1 component of displacement at grid 11280 as the response Since this grid is part of the fluid model the displacement is actually a pressure with units of N m Since the ATTB field is blank on this DRESP1 entry the response is computed for all output frequencies These pressure responses are used as input to DEQATN 10 via DRESP2 number 11 These entries in combination with DCONSTR number 10 identify constraints on pressure response of the form Ey xg PU k Es Equation 7 25 Note that this is similar to Eq 7 23 with
145. Optimization User s Guide Contents o MpCPT wr m 4 11 bmc MCCC 4 15 blico Amp 4 16 dotis MENT rer TT eas ek ee i 4 17 DTABLE aos PEE 4 18 A O A AN 4 19 DYGRID eum 4 20 DVPRELT sex usa pie cd Chee Ree dl ee de Rd a eed ee ERE ca 4 21 DNPRELZ ciar ira e IES TAIANA ee es ee AE ee 4 22 DVSHAP iui a a Ras ia ee AI OS E EA A E a 4 23 Parameters for Design Sensitivity and Optimization clle 4 23 Solution Sequences ed Sane Hoe lee mor a ee 8 RR a a A 5 1 Design Sensitivity and Optimization Modules llle 5 2 Selected Data Blocks llle res 5 4 Design Model Definition ise ke ERR ha Rea Ree AAA 5 4 Shape Optimization 22s 5 5 Constraint Evaluation and Screening 0ooocooococoooo ee 5 6 Design Sensitivity Analysis lile ren 5 6 Design Optimization History sse xk Wea he ee OE A REOR CR ADR RR es 5 6 General uan iia aa a ad iaa aaa 5 6 Solution 200 Program Flow isis RR dei da de he weed ee AR 5 7 Output Features and Interpretation cllc 6 1 Output Controlling Parameters llieleeeee e uhr 6 2 Design Optimization Output a a aaa aaa 6 3 D200X7 QUU eos a A OR eU Ve n BC B in AA 6 3 Design Sensitivity Output esas ew ade ba whe RRA 6 12 Design Sensitivity Output Example aaa 6 13 Example Problems i v rg yx mur m ew RR ok REX ERE EE 7 1 Ihree Bar Truss it tc eR e Fe t S E s Ce ae M
146. Optimization User s Guide 3 25 Chapter 3 Design Sensitivity and Optimization in NX Nastran DVPRELI i 1 NIDV Xj Or DVPREL2 k 2 1 NPR Op Equation 3 35 where NIDV is the number of independent design variables and NIPR is the set of properties defined on all DVPREL2 entries To illustrate the advantage in using a mixed design variable property basis consider the normalized lower bound constraint in Eq 2 35 Defining the Constraints Equation 3 36 This is the form of the lower bound constraint that is automatically generated in NX Nastran from the DCONSTR entry data For a finite move in the design space Ax the approximated constraint is L 4 290 50 E rj FQ AX e ITI s r J Equation 3 37 Since the response is only approximate the constraint is approximate as well For responses that depend on properties that are linear functions of the design variables DVPREL1 the approximation is RE REAL r FAX Ax rx gt Ax Ax i i 50 gt Equation 3 38 while for responses that depend on the properties that are nonlinear functions of the design variables DVPREL2 3 26 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran 2 30 3 30 or 30 2 0 FG Ax rx zo px Ax p k 30 X Equation 3 39 where the properties at the perturbed design in Eq 3 39 can be evaluated precisely from the input equations DEQATN Eq
147. P1 NR2 DIR1 200 12 Design Modeling for Sensitivity and Optimization REGION ATTA using the equation input capability SDEQATN EQUID DEQATN 200 F AVERAGE U1 U2 U3 U4 U5 U1 U2 U3 U4 U5 5 Note If response values change sign an RMS average may yield more useful results DRESP 1 entries 11 through 15 identify the z component of displacement at grid 100 for time steps 051 052 053 054 and 055 Note that each entry defines a single displacement component at a single grid for a single time step The DRESP2 defines the second level response by identifying each of these individual responses as input to a DEQATN entry The resulting second level response can either be used as an objective or constraint 2 7 Defining the Objective Function The objective function is a scalar quantity that is either minimized or maximized by the optimizer You define the design objective using the DESOBJ DESign OBJective Case Control command This command points to a design response defined on a DRESP1 DRESP2 or DRESP3 Bulk Data entry that must define a single scalar response only Formulating the Objective Sensitivity with Respect to Design Variables When formulating the design objective there are a couple of scaling related issues that should be kept in mind since they affect overall performance First the design problem should be posed so that the objective function has sufficient sensitivity
148. PHA1 1 0 1 410 1 410 DESVAR 10 ALPHA2 1 0 1 410 1 410 DESVAR 11 ALPHA3 1 0 1 410 1 410 Explicit design variable linking SDLINK ID DDVID CO CMULT IDV1 C1 IDV3 ES DLINK 1 p 3 a 9 1 0 DL1 11 1 0 DLINK 2 25 A 9 1 20 DL2 11 0 765625 DLINK 3 3 F 9 1 0 DLS y 1l 0 5625 DLINK 4 4 E 9 TsO DL4 11 0 390625 DLINK 5 5 9 1 0 DL5 11 0 2500 DLINK 6 6 i 9 1 0 7 14 Design Sensitivity and Optimization User s Guide DL6 DLINK FET y DLINK DL8 Express 11 7 1i 8 11 DVPREL1 ID DVPREL1 DP1 DVID1 1 1 Listing 7 3 DVPREL1 DP2 DVPREL1 DP3 DVPREL1 DP4 DVPREL1 DP5 DVPREL1 DP6 DVPREL1 DP7 DVPREL1 DP8 Identify the design SDRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 Define the response DRESP1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 ID ATT2 35y 0 140625 7 0 0625 8 TYPE COEF1 PSHELL 1 0 Cont PSHELL 1 0 PSHELL 1 0 PSHELL 1 0 PSHELL 140 PSHELL 1 0 PSHELL 1 0 PSHELL 1 0 LABEL S11 12
149. R 15 maximum shear stresses used for analysis and optimization If left unspecified the defaults will be used which in this case is the von Mises representation See the NX Nastran Quick Reference Guide for defaults Dynamic responses can be computed using either a real imaginary or a magnitude phase form Here again Case Control can be used to define the output representation For example STRAIN IMAG ALL real imaginary form for analysis and optimization STRAIN PHASE ALL magnitude phase form for analysis and optimization Output Frequencies and Output Time Steps In design optimization design responses in dynamic analysis are computed for all output frequencies ANALYSIS DFREQ or MFREQ and time steps ANALYSIS MTRAN This can lead to a huge amount of data Case Control can be used to limit this output and save computational resources by defining output sets with OFREQ OTIME n m where n and m are frequency and time sets respectively Design responses will only be computed for the frequencies and or time steps identified in the above sets Shape Basis Vector Computation Additions to Case Control for shape optimization are only necessary when using the analytic boundary shapes method Design Sensitivity and Optimization User s Guide 4 5 Chapter 4 Input Data Recall from Relating Design Variables to Shape Changes that the analytic boundary shapes approach uses additional Bulk Data Sections to de
150. RELATIVE PROP CHANGES 0 0000E 00 MUST BE LESS THAN 0000E 03 AND MAXIMUM OF RELATIVE D V CHANGES 0 0000E 00 MUST BE LESS THAN 0000E 03 CONVERGENCE TO A BEST COMPROMISE INFEASIBLE DESIGN KARA ke kk kk khe kk kk kk ck Sk kk he kk ARA AAA Kk Sk kk ck ck kc kk ko ko kk kk kk kk ke ke kk ke Note the similarity between the organization of these convergence messages and the logic in the flowcharts of Relating Design Variables to Properties Convergence Tests Since the maximum constraint value is less than 005 we note that we have achieved a feasible optimum solution And since the maximum of relative property and design variable changes is less than 001 this design is unique After final data recovery and output which again is the default for a zero value of NASPRT the following summary of design cycles appears OCK KKK OK RARA OK OK KKK CK CK K k KOK OK Ck k CK CK CK CK k K CK Ok Ok k ROCK CK CK K k OR CK CK Ck k k KCK CK CK k OK K Ok k k KK OK SUMMARY OF DESIGN CYCLE HISTORY kk ke kk kk AE A kk OO OOOO ROO OOOO Kk kk ok kk kk AR ARA kk AR A HARD CONVERGENCE ACHIEVED SOFT CONVERGENCE ACHIEVED NUMBER OF FINITE ELEMENT ANALYSES COMPLETED NUMBER OF OPTIMIZATIONS W R T APPROXIMATE MODELS OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY OBJECTIVE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE APPROXIMATE EXACT OF OF NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT 784520E 00 311878E 00 122267E 01 122355E 01 868297E 05
151. SIEMENS Design Sensitivity and Optimization User s Guide Contents Proprietary amp Restricted Rights Notice ooooooooorocon ooo 7 Getting Started eae eed v ruo a a Fue E i a e a a eee eo eee ee 1 1 Introduction to Design Sensitivity and Optimization lere 1 2 What is Design Sensitivity and Optimization lle 1 2 Why Use Design Sensitivity and Optimization llle 1 2 How Does Design Optimization Differ from Analysis lees 1 3 Optimizer Limitations ooo 1 6 What Do Need to Use Design Optimization Effectively o o o o ooo o 1 6 Numerical Optimization Basics o o o oo ooo 1 7 How Much Do Need to Know About Optimizers lille 1 7 The Basic Optimization Problem Statement llle 1 8 Numerically Searching for an Optimum ooocccco ee 1 12 Structural Optimization us id Ue Foe is 1 20 Overview of Approximation Concepts Used in Structural Optimization 1 21 Design Variable Linking s esa erp A CR Ee a 1 23 Constraint Regionalization and Deletion lt lt lt oooooooroo ooo 1 27 Formal Approximations o o o o 1 29 A Simple Linear Design Space o 1 32 Summary pes a A AA ARA Raub xd AE 1 33 Design Modeling for Sensitivity and Optimization 2 1 Overview of Design Modeling
152. SP1 25 S3 STRESS PROD CONSTRAINTS DCONSTR DCID RID LALLOW UALLOW DCONSTR 21 21 0 20 0 20 DCONSTR 21 22 0 20 0 20 DCONSTR 21 23 15000 20000 DCONSTR 21 24 15000 20000 DCONSTR 21 25 15000 20000 OPTIMIZATION CONTROL DSCREEN DISP 100 10 DSCREEN STRESS 100 10 DOPTPRM APRCOD 2 IPRINT 5 P1 1 P2 15 PMIN REGION DESMAX PMAX ATTA NNNNE co r ATTB ATT1 7 4 4 A 11 12 A 13 Listing 7 12 DP1 DP2 DP3 Design Sensitivity and Optimization User s Guide 7 69 Chapter 7 Example Problems DESIGN VARIABLE DEFINITION DESVAR ID LABEL XINIT XLB XUB DELXV OPTIONAL DESVAR 1A1 7 68653870E 01 1 00000001E 01 D 1V D 1V 1 00000000E 02 1 00000000E 00 DESVAR 2A2 7 14803874E 01 1 00000001E 01 D 2V D 2V 1 00000000E 02 1 00000000E 00 DESVAR 3A3 7 68653870E 01 1 00000001E 01 D 3v D 3V 1 00000000E 02 1 00000000E 00 IMPOSE X3 X1 LEADS TO A3 A1 DLINK ID DDVID CO CMULT IDV1 C1 IDV3 63 id DLINK 1 3 0 0 1 0 1 1 00 DEFINITION OF DESIGN VARIABLE TO ANALYSIS MODEL PARAMETER RELATIONS DVPREL1 ID TYPE PID FID PMIN PMAX co n DVID1 COEF1 DVID2 COEF2 DVPREL1 10 PROD 11 DP1 le 1 0 DVPREL1 20 PROD 12 DP2 2 DVPREL1 30 13 DP3 35 STRUCTURAL RESPONSE IDENTIFICATION DRESP1 ID LABEL RTYPE PTYPE REGION ATTA
153. SUPPORT DATA 7 68 Design Sensitivity and Optimization User s Guide Example Problems SPC 100 1 123456 2 123456 SPC 100 3 123456 4 3456 ELEMENT DATA CROD 1 1i 4 CROD 2 12 2 4 CROD 37 13 3 4 PROPERTY DATA PROD 1i Ty 1 0 PROD 12 Ty 2 0 PROD 13 T5 i30 MAT1 1 1 0E 7 0 33 0 1 EXTERNAL LOADS DATA FORCE 300 4 P 20000 0 8 0 6 FORCE 310 4 20000 0 8 0 6 A EI tn hn a E os OTe ate DESIGN MODEL DESIGN VARIABLE DEFINITION XUB 100 0 100 0 100 0 IDV1 1 DELXV OPTIONAL cl 1 00 IDV2 C2 DEFINITION OF DESIGN VARIABLE TO ANALYSIS MODEL PARAMETER RELATIONS DESVAR ID LABEL XINIT XLB DESVAR 1 Al 10 OL DESVAR 2 A2 2 0 0 1 DESVAR 3 A3 1 0 0 1 IMPOSE X3 X1 LEADS TO A3 A1 DLINK ID DDVID CO CMULT IDV3 C3 DLINK 1 3 0 0 1 0 DVPREL1 ID TYPE PID FID S35 DVID1 COEF1 DVID2 COEF2 DVPREL1 10 PROD 11 4 DP1 1 1 40 DVPREL1 20 PROD 12 4 DP2 2 1 0 DVPREL1 30 PROD 13 4 DP3 ce 1 0 STRUCTURAL RESPONSE IDENTIFICATION DRESP1 ID LABEL RTYPE PTYPE SE ATT2 DEN DRESP1 20 W WEIGHT DRESP1 21 U4 DISP DRESP1 22 v4 DISP DRESP1 23 S1 STRESS PROD DRESP1 24 S2 STRESS PROD DRE
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155. TERIA SATISFIED HARD CONVERGENCE DECISION LOGIC FOI III IIIS OOOO CONVERGENCE ACHIEVED BASED ON THE FOLLOWING CRITERIA HARD CONVERGENCE DECISION LOGIC RELATIVE CHANGE IN OBJECTIVE 2 0467E 04 MUST BE LESS THAN 0000E 03 ABSOLUTE CHANGE IN OBJECTIVE 1 4267E 03 MUST BE LESS THAN 0000E 02 AND MAXIMUM CONSTRAINT VALUE 1 7216E 03 MUST BE LESS THAN 0000E 03 CONVERGENCE TO A FEASIBLE DESIGN OR E MAXIMUM OF RELATIVE PROP CHANGES 2 3337E 02 MUST BE LESS THAN 1 0000E 03 AND MAXIMUM OF RELATIVE D V CHANGES 2 3337E 02 MUST BE LESS THAN 1 0000E 03 CONVERGENCE TO A BEST COMPROMISE INFEASIBLE DESIGN Kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk FIR I RR IRR II II I RI I IIR II II I I ak ak ok SUMMARY OF DESIGN CYCLE HISTORY itt eter OOOO OOOO OOOO kk ke ke kk kk kk ko kk ok ko ko kk ko ko ee AA HARD CONVERGENCE ACHIEVED NUMBER OF FINITE ELEMENT ANALYSES COMPLETED NUMBER OF OPTIMIZATIONS W R T APPROXIMATE MODELS OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY OBJECTIVE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE APPROXIMATE EXACT OF OF MAXIMUM NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT CONSTRAINT 920000E 01 000000E 00 599560E 00 599990E 00 480287E 05 583312E 02 749448E 00 749540E 00 187549E 05 574277E 03 054639E 00 054621E 00 568502E 06 288688E 03 970634E 00 970623E 00 436540E 06 697157E 03 969195E 00 969197E 00 736827E 07 721585E 03 INTERNAL EXT
156. TITLE acoustic and structural elements LABEL boxael dat analysis case control set 20 11280 echo sort param eigc eigrl freq desvar dconstr drespl dresp2 deqatn dvprell spe 1 DISP phase 20 method structure 20 method fluid 30 cmethod 10 frequency 200 dload 100 partn 20 optimization case control ANALYSIS MFREQ DESGLB 5 DESSUB 10 DESOBJ 100 BEGIN BULK EIGRL 20 200 max eigrl 1 307 EA 1553 7 9 0 T0555 max eigc 10 clan max 15 300 4 sound pressure level param rms yes reference pressure for dB and dBA param prefdb 2 5 PARAM AUTOSPC no fluid structure interface acmodl diff p wO 0l structural damping param g 0 02 rload1 100 101 102 darea 101 1288 3 100 tabled1 102 70 2 Ls 10005 1 e amp ndt freq2 200 1 0 200 0 50 Pesos 2 as Qs Dee 2 2 jenem g g 10 freq 200 40 50 60 70 80 90 95 100 105 110 120 130 140 150 160 170 180 190 200 971 5 102 5 2 3 4 Ye 6 eem 8 9 10 Define the Design Variables SESVAR ID LABEL XINIT XLB XUB DELXV DESVAR 1 BI 0 02493 0 0001 1 Example Problems Fluid Eigenfrequencies Design Sensitivity and Optimization User s Guide 7 61 Chapter 7 Example Problems DESVAR 2 P2 0 01953 0 0001 1 DESVAR 4 P4 0 02047 0 0001 1 DESVAR 5 P5 0 02596 0 0001 1 DESV
157. VE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE CYCLE APPROXIMATE EXACT OF OF NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT INITIAL 6 614414E 02 1 102773E 01 1 5 985143E 02 5 985219E 02 1 254311E 05 2 294609E 03 2 5 763787E 02 5 763831E 02 7 130217E 06 8 119856E 04 2 5 605763E 02 5 605792E 02 5 117301E 06 3 697191E 04 4 5 513699E 02 5 513676E 02 4 206515E 06 2 951452E 03 5 5 484974E 02 5 484994E 02 3 560852E 06 3 337860E 05 6 5 465300E 02 5 465295E 02 7 817438E 07 1 285757E 04 7 5 459542E 02 5 459532E 02 1 788729E 06 3 235681E 06 8 5 459244E 02 5 459244E 02 0 000000E 00 3 865957E 04 Design Sensitivity and Optimization User s Guide 7 47 Chapter 7 Example Problems DESIGN VARIABLE HISTORY INTERNAL EXTERNAL DV ID DV ID LABEL INITIAL 1 f 2 H 3 t 4 5 1 1 X1 2 0000E 00 1 0000E 00 9 3000E 01 4 6500E 01 2 3250E 01 1 1625E 01 2 2 X2 2 0000E 00 2 0461E 00 2 0175E 00 2 0560E 00 2 0416E 00 2 0378E 00 3 3 X3 2 0000E 00 3 0000E 00 3 1355E 00 3 0361E 00 3 0550E 00 3 0211E 00 4 4 x4 2 0000E 00 1 0000E 00 8 6000E 01 4 3000E 01 2 1500E 01 1 0750E 01 5 5 X5 2 0000E 00 1 0000E 00 8 6240E 01 4 3120E 01 2 1558bE 01 1 0774E 01 6 6 X6 2 0000E 00 1 0000E 00 6 1363E 01 7 1806E 01 6 8686E 01 6 6251E 01 7 7 X7 2 0000E 00 1 5944E 00 1 5890E 00 1 5927E 00 1 5991E 00 1 6201E 00 8 8 X8 2 0000E 00 2 5863E 00 2 7106E 00 2 6171E 00 2 6308E 00 2 6893E 00 INTERNAL EXTERNAL DV ID DV ID LABEL 6 i EA 8 9 H
158. X3 X3INIT I1NEW I1 DELTA 4 DVPREL2 12 PBAR 3 6 1 0 6 102 DESVAR 3 r DTABLE X3INIT I2INIT DEQATN 102 I2 X3 X3INIT I2INIT I2INIT DELTA SQRT X3 X3INIT I2NEW I2 DELTA 4 DVPREL2 13 PBAR 3 75 1 0 6 103 DESVAR 3 r r DTABLE X3INIT I12INIT DEQATN 103 112 X3 X3INIT I12INIT 112INIT rH DELTA SQRT X3 X3INIT I12NEW I12 DELTA 4 Modify stress recovery points accordingly DVPREL2 14 PBAR 3 12 104 DESVAR 3 r r DTABLE X3INIT CYINIT DVPREL2 15 PBAR 35 13 y 104 DESVAR 3 r r DTABLE X3INIT CZINIT DVPREL2 16 PBAR 35 14 2 y 104 DESVAR 3 r r DTABLE X3INIT DYINIT DVPREL2 17 PBAR 3 15 1 0 104 A 7 22 so use DVPREL1 Design Sensitivity and Optimization User s Guide PB3 DP1 DP2 DP3 Example Problems DESVAR 3 r r DTABLE X3INIT DZINIT DVPREL2 18 PBAR 3 16 1 0 104 DESVAR 3 r DTABLE X3INIT EYINIT DVPREL2 19 PBAR 3 17 1 0 gt 104 5 DESVAR 3 r r of DTABLE X3INIT EZINIT DVPREL2 20 PBAR 3 18 1 0 r 104 DESVAR 3 r r DTABLE X3INIT FYINIT DVPREL2 21 PBAR 3 19 104 DESVAR 3 T P A DTABLE X3INIT FZINIT Equation for stress recovery points DEQATN 104 NEWPOINT X3 X3INIT POINT POINT SQRT X3 X3INIT T
159. a design perspective the actual cross sectional dimensions are of greater interest A design must ultimately be chosen from readily available manufactured I beam sections and a list of optimal cross sectional dimensions rather than properties will allow us to make the proper selection The natural choice is to let the I section dimensions be the design variables In the design model we will specify the manner in which the I section dimensions relate to the cross sectional properties The functional relationships are illustrated in Figure 1 11 Design Sensitivity and Optimization User s Guide 1 23 Chapter 1 Getting Started Design Model Analysis Model CBAR 101 21 tu h lt t Y E Design Variables Analysis Model Parameters h b hy AAA AN A l Lo I5 J eee PBAR 21 6 2 36 A 2bty h E 2tpt mcr y Oe w 3 12 jp p Figure 1 11 Design and Analysis Models for a Symmetric I Beam Section Note These equations can be expressed in the design model using DVPREL2 and DEQATN Bulk Data entries See Design Modeling for Sensitivity and Optimization Depending on the number of bar elements in our analysis model we may not want to allow each to vary independently of the other in our design model We will probably run into cost or manufacturing problems if we try to build a frame structure with 150 elements and 150 different cross sections To impose some order on the design it might b
160. a parameter SOFTEXIT had been set to YES NO is the default A finite element analysis begins the new design cycle as is apparent from the following DMAP information messages DMAP INFORMATION MESSAGE 9051 FEA STATIC ANALYSIS INITIATED DESIGN CYCLE NUMBER 2 0 USER INFORMATION MESSAGE 5293 FOR DATA BLOCK KLL LOAD SEQ NO EPSILON EXTERNAL WORK EPSILONS LARGER THAN 001 ARE FLAGGED WITH ASTERISKS 1 1 0933737E 16 2 1103991E 01 2 8 9942808E 16 8 6491135E 01 DMAP INFORMATION MESSAGE 9052 FEA STATIC ANALYSIS COMPLETED DESIGN CYCLE NUMBER With two successive finite element analyses available one from the current design cycle and one from the previous a hard convergence test can now be performed Recall that this test was skipped on the first design cycle Even though convergence has not yet been achieved the test results are reported nonetheless CONVERGENCE NOT ACHIEVED YET HARD CONVERGENCE DECISION LOGIC RELATIVE CHANGE IN OBJECTIVE 9 4027E 01 MUST BE LESS THAN 0000E 03 ABSOLUTE CHANGE IN OBJECTIVE 5 4390E 00 MUST BE LESS THAN 0000E 02 a AyD MAXIMUM CONSTRAINT VALUE 4 0893E 01 MUST BE LESS THAN 0000E 03 CONVERGENCE TO A FEASIBLE DESIGN sas OR Hs MAXIMUM OF RELATIVE PROP CHANGES 1 0000E 00 MUST BE LESS THAN 0000E 03 MAXIMUM OF RELATIVE D V CHANGES 1 0000E 00 MUST BE LESS THAN 0000E 03 CONVERGENCE TO A BEST COMPROMISE INFEASIBLE DESIGN In the case of test problem D200X7 the des
161. a slightly smaller value than the true objective The linear approximation has allowed the design to become slightly violated when compared to the true design space 1 32 Design Sensitivity and Optimization User s Guide Getting Started H B 12 65r 60 55 Height H cm 50 ds ptimum Approximate Optimum 45 f 40 35 2 5 Width B cm Figure 1 17 Linearly Approximated Cantilever Beam Design Space Though not yet an optimal design the approximate optimum is a useful starting point for the next approximate optimization cycle This next cycle proceeds in exactly the same fashion an analysis is performed to determine the response values along with an evaluation of the response derivatives This information is then used to create another approximate subproblem that should in turn yield an even better approximation of the true optimum After only a few more cycles we may have reached a point sufficiently close to the true optimum that the process converges as measured by the Kuhn Tucker conditions Summary By now you should have a fairly good idea of some of the various applications of design optimization as well as some of the numerical optimization capabilities available to assist you in your design tasks We introduced the basic ideas of numerical optimization and some of the methods that are used to couple it with structural analysis codes Central to this has been the recognition that it is impractical to directly l
162. able constants for all equations DTABLE X3INIT 0 144 I1INIT 1 728 4 12INIT 1 728 2 112INIT 1 745 2 CYINIT 0 06 CZINIT 0 6 DYINIT 0 06 DZINIT 0 6 EYINIT 0 06 EZINIT 0 6 FYINIT 0 06 FZINIT 0 6 Identify the design responses SDRESP1 ID LABEL RTYPE PTYPE REGION ATTA ATTB ATT1 ATT2 E DRESP1 1 SBARA STRESS PBAR s r 3 DRESP1 2 SBARB STRESS PBAR i 14 P 3 DRESP1 3 13 STRESS PSHELL 9 M 1 DRESP1 6 S16 STRESS PSHELL d 1 DRESP1 9 S23 STRESS PSHELL 9 M 2 DRESP1 12 S26 STRESS PSHELL 17 2 DRESP1 13 Di DISP 3 i 10302 DRESP1 14 D2 DISP i 3 i 10203 DRESP1 15 W WEIGHT Place bounds on the responses SDCONSTR DCID RID LALLOW UALLOW DCONSTR 10 1 25000 25000 DCONSTR 10 25 25000 25000 DCONSTR 10 En 25000 25000 DCONSTR 10 6 25000 25000 DCONSTR 10 9 25000 25000 DCONSTR 10 12 25000 25000 DCONSTR 20 13 0 1 0 1 DCONSTR 30 14 0 03 0 03 DCONADD DCID DC1 DC2 DCONADD 100 10 20 summed constraint set for subcase 1 DCONADD 200 10 30 summed constraint set for subcase 2 Optional override of optimization control parameters DOPTPRM IPRINT 1 DESMAX 20 DELP 055 Pl ais P2 15 DELP 0 5 allows larger moves thus overcoming constraint violations quicker ENDDATA Listing 7 4 7 5 Shape Optimization of a Culver
163. adius for weight minimization It gives allowable structural variations hole radius subject to limits on structural responses stress Probably the biggest difference between analysis and design is that analysis leads to the solution within the limits of the analysis model while design optimization leads to a solution In other words in analysis we are usually guaranteed a unique solution while more than one solution may be possible in design optimization Mathematically we can say that our design space may contain relative minima This is analogous to the situation in which we may want to find the low point in Design Sensitivity and Optimization User s Guide 1 5 Chapter 1 Getting Started a valley yet various low points are separated by hills that we cannot see over Simply finding a low point itself represents an acceptable solution but there may be more than one such solution In some instances we may be able to restate the problem and in effect shift the locations and contours of the hills to allow more efficient convergence However the fact remains that our goal has been stated in the context of design improvement and not in determining a unique solution as in the case of analysis Integrity of Analysis and Design Modes One thing the analysis and design models share is that the results are dependent upon the skill and judgement used in their construction A poorly meshed finite element model may lead to inaccurate and m
164. ae di eo ae a ld ek 7 3 Vibration of a Cantilever Beam Turner s Problem selle 7 7 Cantilevered Plate 2 aaa ee 7 11 Stiffened Plate 2 2 0 ee 7 16 Shape Optimization of a Culvert a na saaa aa ee 7 23 Problem Description naaa anaa ee 7 23 Modeling Considerations iussus eom mh d Rd n me og un e deca e whan bee ea 7 24 Creation of Auxiliary Model and Generation of Basis Vectors 7 25 Design Optimization Input aaa rs 7 25 Analytic Boundary Shapes acaso Sch work Rack week Rack ead back e Di RE ce RI dca e edo ACT uicta 7 29 Problem Description 222ll elles 7 29 Boundary Shape Changes Using Auxiliary Boundary Models 7 31 Shape Changes over the Interior oo ooooooooooooono ooo 7 32 Modeling Summary axe sua td ta poo rack esr ol og ache ael re cob bots up Rope Te aoa dc de Dic a 7 32 Shape Basis Vectors lille 7 32 Optimization Results aaa 7 33 Dynamic Response Optimization llle 7 37 Design Sensitivity and Optimization User s Guide 5 Contents Twenty Five Bar Truss Superelement Optimization lll 7 44 Design Optimization with Composite Materials llle 7 51 Acoustic Optimization ae 7 56 Restarts in Design Optimization 0 000 ee ee 7 63 Restarting from a Previous Design Cycle 0 0 0 0 cee ee 7 63 Restarting from an OPTEXIT Poin
165. al Interpretation For Which Responses Are Sensitivities Computed Sensitivities are computed for all of the responses both first and second level that are used to define the objective function and retained constraints Recall from Approximation Concepts in Design Optimization that sensitivity analysis is always performed after the constraint screening phases are completed Figure 3 1 Thus fewer response sensitivities may be computed than there are responses defined in the design model Of course this is the goal since the computational effort associated with sensitivity analysis is reduced If you wish to force the computation of all sensitivities for all responses defined in the design model you can effectively disable the screening process by using a large negative TRS on the DSCREEN Bulk Data entries along with a large NSTR if necessary See the DSCREEN entry description in Input Data for further details Requesting Sensitivity Analysis Output Design sensitivities are output in the form of Eq 3 1 In Solution 200 you request design sensitivity analysis and output with either the DSAPRT Case control command or the following Bulk Data parameter definition PARAM OPTEXIT 4 Note You can also request design sensitivity coefficient output with an OPTEXIT value of 7 See Solution 200 Program Flow Solution 200 Program Flow for details An OPTEXIT of 4 prints the sensitivity coefficients to the standard output fil
166. al analysis code This section discusses the need for and the theory behind the approximations available in NX Nastran You can control the form of these approximations by e Specifying the approximation method using the DOPTPRM entry APRCOD 1 2 or 3 Providing explicit design variable linking with DLINK Bulk Data entries Controlling the regionalization of responses using DRESP1 DRESP2 and DRESP3 entries and the constraint screening criteria using DSCREEN entries Why Are Approximations Necessary in Design Optimization As outlined in Structural Optimization approximation concepts address three fundamental difficulties associated with structural optimization There are often more design variables than are necessary to adequately describe the allowable design variations A typical structural optimization problem often contains hundreds or even thousands of constraints It is likely that many of these constraints will contain redundant or irrelevant design information The structural responses are only implicit functions of the design variables Having to repeatedly invoke a full finite element analysis to evaluate these implicit functions for small design changes is not cost effective Approximations in NX Nastran An Overview The preceding difficulties are effectively resolved with approximation concepts In NX Nastran there are essentially three categories into which these approximations fall These categories and th
167. alysis The DVGRID entry supplies the grid motion on the boundary much like an SPCD enforced displacement The boundary grids are identified using BNDGRID entries much like an SPC entry that identifies the corresponding grids and their components 2 36 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization BOUNDARY CONDITIONS FOR SPCD APPLICATION PROVIDED VIA DVGRIDS OUTER PLATE EDGES DGRID 123456 2 16 17 18 EN 19 20 21 4 X 0 PLANE DGRID 23456 23 295 26 Y 0 PLANE DGRID 13456 99 101 102 CUTOUT EDGE DGRID 123456 1 6 7 11 12 14 The boundary conditions defined by the DVGRID and BNDGRID entries are only used to generate shape basis vectors They are not used in connection with the primary model analysis The first BNDGRID entry identifies all grid components along the plate outer edges Since no DVGRIDs have been furnished for these degrees of freedom their displacements will be fixed to zero For shape optimization the plate outer edges will be invariant The second BNDGRID entry identifies components 2 through 6 for those grids that lie on the x axis This allows the x component to vary during shape optimization consistent with the elliptical changes made to the cutout All other components are considered fixed since no enforced shape changes for these degrees of freedom have been supplied on DVGRID entries The third BNDGRID entry furnis
168. an auxiliary model over the boundaries of the structure that are to be changed This may be a collection of BAR elements along the edges of a two dimensional structure or a skin of plate elements over a three dimensional part We often refer to this model over the boundaries as an auxiliary boundary model The designer then constrains Design Sensitivity and Optimization User s Guide 2 29 Chapter 2 Design Modeling for Sensitivity and Optimization and statically loads the model to produce a shape variation over the boundary which the code then interpolates to the interior of the structure The result is a shape basis vector for optimization Benefits The method is very general and does not require a geometry based pre and postprocessor The user interface is entirely within the NX Nastran environment Shape basis vectors are updated on every design cycle reducing the problems associated with mesh distortion for large shape changes Drawbacks Some creativity is often required in the selection of loads and boundary conditions for the auxiliary boundary models However any combination of static loading available in NX Nastran can be used point loads FORCE enforced displacements SPCD thermal loads TEMP pressure loads PLOAD and so on can all be used to generate the desired shapes Checklist 1 Define auxiliary boundary models using one additional Bulk Data Section for each model These sections are identified with the comman
169. an be changed using the DOPTPRM entry DELX and DXMIN and the DESVAR entry DELXV XLB XUB The designed property limits can be controlled using the DOPTPRM entry DELP DPMIN and the DVPREL1 or DVPREL2 entries PMIN PMAX This set of constants is used to recompute the move limits for each design cycle At times it may be necessary for the code to automatically adjust these move limits if the problem becomes numerically ill conditioned The situation might arise as follows An approximate problem is constructed from which the optimizer determines a corresponding approximate optimum Perhaps some of the approximate constraints are critical for this design The responses are now evaluated by a finite element analysis and it is determined that rather than just critical these constraints are actually violated Errors have thus been detected between the approximate and the true responses If this error continues from one design cycle to the next it can be taken as an indication that the move limits are probably too wide Continued constraint violations have an adverse effect on overall convergence The move limit controlling parameters are updated automatically in NX Nastran if the following criteria are satisfied The current design cycle number is greater than or equal to three There is at least one violated constraint violated by more than 296 and the level of constraint violation is increasing Under these conditions DELP DPMIN
170. any omitted rows at the bottom of the arglis array as zero The corresponding argval array contains the numerical values and character strings associated with the nonzero entries in the arglis array For this example the argval array contains nine entries two attributable to DESVAR entries one attributable to a DRESP1 entry and six attributable to DVPREL 1 entries The nine entries in the argval array would be ordered as follows DESVAR Value 1 DVID2 on the DRESP3 ent NR1 on the DRESP3 ent DVPREL1 Value 1 DPIP2 on the DRESP3 entr DVPREL1 Value 3 DPIP4 on the DRESP3 entr DPIP5 on the DRESP3 entr DVPREL1 Value 6 The actual argval array passed to the executable will contain a numerical value or character string for each entry In this example there are no USRDATA entries However the argchar array and its length NWRDA8 should still be entered into the argument list of either dr3svald F for LP or dr3svals F for ILP This is the case because NX Nastran calls one of these subroutines and r3sgrt F to determine whether the specified response is a valid one Design Responses and Case Control You can reference design responses from the Case Control section in two ways 2 50 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization By using subcase based constraint and or objective function definitions that you specify with DESSUB or DESOBJ Case Control comma
171. arch for a feasible design Frequently the objective function must be increased in this process For example it may be necessary to add structural mass in order to overcome element stress constraints However the optimizer in NX Nastran will search for a feasible region allowing for the smallest possible increases in the objective Having found the feasible domain if one exists it can then proceed with minimizing the objective The question now arises How do we know that we have reached the optimum The answer can be found in what are known as the Kuhn Tucker conditions 4 In the case of an unconstrained problem omitting Eq D 2 through Eq D 4 this is simply the condition where the objective function vanishes i e equals zero In the case of the constrained optimization problem considered here the conditions of optimality are more complex In this case the governing equation is the stationary condition of the Lagrangian function M gt gt gt LA F Y Vg j l Equation D 12 where 3 Vanderplaats G N An Efficient Feasible Direction Algorithm for Design Synthesis A AA Journal Vol 22 No 11 Nov 1984 4 Zangwill W Nonlinear Programming A Unified Approach Prentice Hall Englewood Cliffs NJ 1969 Design Sensitivity and Optimization User s Guide D 5 Appendix D Numerical Optimization M20 Equation D 13 gt gt The Kuhn Tucker conditions dictate that the Lagrangian function L 4 must have
172. are not concerned with satisfying an equality constraint precisely rather any satisfaction within a certain tolerance will do In fact attempting to satisfy a number of equality constraints precisely may be a problem for the optimizer We may want to match frequencies for example to within 1 of some target values This could be expressed using lower and upper bounds of Rj 6 and Rj 5 on the DCONSTR entry where is the allowable tolerance 2 58 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization 2 9 Superelement Design Modeling With few exceptions design modeling for superelement sensitivity and optimization is a largely transparent operation in Solution 200 That is design models spanning multiple superelements can be expressed using the same tools and methods for non superelement models discussed in the previous sections This is a useful tool in any large scale design task This section introduces the differences in superelement versus non superelement design modeling Limitations are summarized at the end of the section Supported Superelements Primary and image superelements can only be used in connection with the design model Primary image and external superelements can all be used in the analysis model The external superelement restriction occurs because external superelements are only known to the code via their structural stiffness mass damping and load matrices Not
173. arity only Actually the initial thickness distribution is only a function of the independent variables ALPHA1 ALPHA2 and ALPHA3 and the constants defined on the DLINK entries If differences were to exist the dependent variables would be overridden accordingly in addition to the corresponding plate element thicknesses The summary of design cycle history indicates that the initial design is infeasible After seven iterations a minimum weight design has been achieved that satisfies all constraints However note that in order to achieve an optimal feasible design the optimizer still found it necessary to add mass to the structure The result is a minimum weight structure that still satisfies all of the constraints NORMAL CONVERGENCE CRITERIA SATISFIED HARD CONVERGENCE DECISION LOGIC EIEE IEE IE AEE AE EAE DE AE FEE EE IEEE KEE E OOOO I I AI I CONVERGENCE ACHIEVED BASED ON THE FOLLOWING CRITERIA HARD CONVERGENCE DECISION LOGIC RELATIVE CHANGE IN OBJECTIVE 4 1124E 04 MUST BE LESS THAN 1 0000E 03 ABSOLUTE CHANGE IN OBJECTIVE 1 0109E 01 MUST BE LESS THAN 1 0000E 01 AND MAXIMUM CONSTRAINT VALUE 2 4815E 03 MUST BE LESS THAN 5 0000E 03 CONVERGENCE TO A FEASIBLE DESIGN Om MAXIMUM OF RELATIVE PROP CHANGES 5 2540E 04 MUST BE LESS THAN 1 0000E 03 AND MAXIMUM OF RELATIVE D V CHANGES 5 2540E 04 MUST BE LESS THAN 1 0000E 03 CONVERGENCE TO A BEST COMPROMISE INFEASIBLE DESIGN kk ek kk kk ek kk khe he kk kk ek ck kk ERI
174. asible Design Optimization Found Nonunique If CHGPRP CONVPR or CHGDV CONVDV Figure 3 14 Hard Convergence Decision Logic As with soft convergence the first check is with respect to the relative and absolute changes in the objective function The reasoning behind the application of an OR test is based on a consideration of the objective function magnitude If the objective is large such as the weight of a heavy machine part in kilograms converging to within a plus or minus range of ten kilograms is probably quite sufficient Forcing convergence to within a fraction of a kilogram may be meaningless as well as expensive On the other hand there are situations where the objective function is a very small number e g minimization of the difference between analysis and test results For such cases minimization of the absolute change may be more meaningful than the relative change If the objective has not changed appreciably in either an absolute or a relative sense a check is performed to ensure that the maximum constraint value is less than its maximum allowable If this criterion is satisfied then hard convergence is achieved However you should check to see if the relative changes in properties CHGPRP or the relative changes in design variables CHGDV are satisfied If they are satisfied then the design process has converged to a unique design If not a nonunique design may have been found Convergence to a Nonunique
175. asis vector data as in the previous approach this method generates the vectors automatically based on geometric data supplied on the boundaries All of the data preparation and input appear in the optimization input file there is no need for a prior auxiliary model analysis In addition the shape basis vectors are updated on each design cycle as the shape changes One problem with basis vectors that are not updated is that they may lead to ill conditioned meshes for large shape changes Updating this data on each design cycle minimizes but not altogether eliminates this difficulty The geometric boundary shapes method begins with a definition of allowable variations over the boundaries These variations can be described with a geometry based preprocessor or with Bulk Data alone Either way the boundaries are defined using BNDGRID entries and the variations defined using DVGRID entries The elliptical variations on the boundaries can be defined as follows Design Sensitivity and Optimization User s Guide 2 35 Chapter 2 Design Modeling for Sensitivity and Optimization DEFINE DESIGN VARIABLES DESVAR 1 DELA 1 0 20 20 DESVAR 2 DELB isO 20 20 AND RELATE THESE TO BOUNDARY VARIATIONS DEFINED USING DVGRIDS SDVGRID DVID GRID CID COEFF N1 N2 N3 BASIS VECTOR 1 DVGRID 1 0 al DVGRID 1 0098769 1 DVGRID 1 0095106 1 DVGRI 0089101 1 DVGRI 0080902 1 DVGRI 0070711 1 DVGRI 0058779
176. at least and for purposes of this example the shape functions of Figure 3 3 are used Suppose the base dimension of this rectangular section is to be a constant 5 mm with the section height as the design variable This choice yields a total of ten variables similar to the thicknesses of the previous example However the only way to describe the cross sectional properties in terms of the cross sectional dimensions is through the use of equations The figure below shows a representative cross section and element orientation and the equations for area 4 and 5 These equations have been evaluated for an initial h of 10 mm N A bh Ah 5 0E 5 m bh 4 167E 10 m 1 4 1042 10 m 12 Since the design variable to property relations for 4 and are not linear in the design variables DVPREL 1 relations cannot be used to express the reduced basis formulation as before The only way this can be implemented is by explicitly linking together the design variables themselves together using DLINK entries Three additional design variables are defined and used as basis function multipliers These design variables form the independent set The dependent design variables linked to the independent set define the element dimensions The corresponding element properties are defined using these dependent variables which are input to DEQATN entries via DVPREL2 entries Design Sensitivity and Optimization User s Guide 3 9 Chapter 3 Design S
177. at over the entire structure boundary plus interior grids The displacement vector subscripts are simply used to indicate the auxiliary boundary model origin of the solution Further a symmetric redesign is enforced by design variable linking with the DLINK entry Optimization Results Figure 7 12 shows the summary of design cycle history from the output file In five design cycles the weight has been reduced from 8 0 E 6 to 5 4 E 6 The final shape is shown in Figure 7 13 and the corresponding stress distribution in Figure 7 14 The maximum stress regions which had been near the root for the initial design are now located near the tip This is somewhat expected since our choice of basis vectors has left the geometry unchanged near the root of the cantilever KORR I SRR ROR OR AEE ER EE RE O KORG EO EGER EER SUMMARY OF DESIGN CYCLE HISTORY HORROR ok ek ICI ek eK KO OOOO OOOO elk dede dele debe TO ele HARD CONVERGENCE ACHIEVED SOFT CONVERGENCE ACHIEVED NUMBER OF FINITE ELEMENT ANALYSES COMPLETED 6 NUMBER OF OPTIMIZATIONS W R T APPROXIMATE MODELS 5 OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY OBJECTIVE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE CYCLE APPROXIMATE EXACT OF OF NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT INITIAL 8 000000E 06 3 827773E 01 Y 7 401214E 06 7 401214E 06 0 000000E 00 3 720914E 01 2 6 562918E 06 6 562912E 06 9 142283E 07 3 570341E 01 3 5 389287E 06 5 389288E 06 2 7832998 07 4 672768E 0
178. ate approach might be to add some very stiff beams to the bottom edge of the auxiliary model Loading the tip of the rightmost beam would then approximate the effect of the SPCD s used here Design Sensitivity and Optimization User s Guide 2 25 Chapter 2 Design Modeling for Sensitivity and Optimization Figure 2 13 Uniform Taper Basis Vector A more arbitrary shape basis vector could be generated by loading one of the individual grids along the lower edge The produces an indent as shown in Figure 2 14 A number of these shapes produced by loading other grids along this edge might comprise a useful set of shape basis vectors depending on the desired effect P Figure 2 14 Point Load Basis Vector Modeling Methods Shape Basis Vector Definition Basic Assumptions Shape optimization in NX Nastran assumes the engineer is starting with a reasonably good design but would like to investigate ways in which the shape of the part or even the entire structure might be modified in order to better meet the design goals Although large shape variations can be investigated in NX Nastran it is important to note that in a practical design environment shape changes are often limited by numerous design considerations Manufacturability of the part interference from neighboring components of the structure aesthetics and so on often limit the degree to which the structural shape can be modified Shape optimization in NX Nastran al
179. ate that the responses of interest are failure criteria The particular failure mode is indicated in part by the information appearing in field 7 the ATTA field According to the DRESP1 entry listing this attribute field contains the failure criterion item code Design Sensitivity and Optimization User s Guide 7 51 Chapter 7 Example Problems These as well as the item codes for all other element types are listed in the NX Nastran Quick Reference Guide See Item Codes for more information Consulting this section for the CQUADA element shows that the only applicable choices are 5 for ply failure or 7 for bonding material failure The ply failure index is determined based on the computed lamina stresses and or strains and the particular failure theory selected on the PCOMP entry With HILL selected on PCOMP 1 the 5 in field 7 of each DRESP1 entry identifies Hill failure indices for use in the Design Model Hard convergence was achieved after thirteen design cycles as seen in Figure 7 22 and Figure 7 23 The results indicate that use of minimum allowable ply thicknesses is feasible if corresponding changes in ply orientation are made The resulting orientations are 54 8 and 9 5 for the outer and inner plies respectively NORMAL CONVERGENCE CRITERIA SATISFIED Figure 7 21 Composite Tube HARD CONVERGENCE DECISION LOGIC kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk ckckck ck ck KK KK KK ck kkk kk kk k
180. ath file For example suppose the evaluator path file you created in the previous step is named nasevals The command line syntax would be Axnr bin nastran exe gmconn network drive user dir dr3srv nasevals input file dat Format of Data Passed to the Executable File The data included on a DRESP3 Bulk Data entry is passed to the executable file in three one dimensional arrays The arglis array contains the number of each type of entry on the DRESP3 Bulk Data entry The argval array contains the numerical value or character string that corresponds with each entry listed on the DRESP3 Bulk Data entry The argchar array contains character strings for the USRDATA entries included on the DRESP3 Bulk Data entry Design Sensitivity and Optimization User s Guide 2 49 Chapter 2 Design Modeling for Sensitivity and Optimization An example is useful for demonstrating how the arglis and argval arrays are formatted Suppose a DRESP3 entry contains two DESVAR entries one DRESP1 entry and six DVPREL1 entries The arglis array would be The zeros in the arglis array designate that there are no DTABLE or DFRFNC entries Likewise there are no DVPREL1 DVCREL1 DVMREL1 DVPREL2 DVCREL2 DVMREL2 and DRESP2 entries However unlike DTABLE and DFRFNC zeros are not required for the DVPREL1 DVCREL1 DVMREL1 DVPREL2 DVCREL2 DVMREL2 and DRESP2 entries The executable the build script creates will automatically interpret
181. been used to generate these boundary deformations The loading and boundary condition sets are selected in the auxiliary boundary model Case Control Section which follows the Case Control for the primary model AUXILIARY MODEL CASE CONTROL AUXCASE AUXMODEL 1 LO TLE AUXILIARY MODEL 1 LOAD CASE 10 100 160 TEMP LOAD 162 DISPLACEMENT ALL SUBCASE 20 SUBTITLE AUXILIARY MODEL 1 LOAD CASE 20 SPC 100 LOAD 161 TEMP LOAD 162 DISPLACEMENT ALL BEGIN BULK The keyword AUXCASE identifies the beginning of the auxiliary model Case Control sections In this example a single auxiliary boundary model AUXMODEL 1 is used to generate two basis vectors in subcases 10 and 20 The resultant boundary displacements are shown in Figure 2 20 and Figure 2 21 Figure 2 20 Boundary Displacement Set 1 Design Sensitivity and Optimization User s Guide 2 39 Chapter 2 Design Modeling for Sensitivity and Optimization Figure 2 21 Boundary Displacement Set 2 Once these boundary displacements have been computed the code interpolates these displacements to the interior grids Once again BNDGRID entries define the degrees of freedom along the boundaries that are allowed to change and those degrees of freedom that are to remain fixed boundary conditions for basis vector generation outer plate edges DGRID 123456 2 15 16 17 18 20 21 22 4 x 0 plane D 23456 23 24 25 26 y 0 plane D 1
182. c response analysis In addition design models can also employ superelements Proficiency in any or all of these disciplines is useful Material Presented in This Guide and Other Resources The topics covered in this user s guide are intended to describe design sensitivity and optimization in NX Nastran No prior knowledge of structural optimization is required This guide attempts to describe all aspects completely so that someone new to the field of design optimization as well as those with more experience have enough information available to use it wisely and effectively Engineering Judgement and Common Sense Probably the most critical requirement in the effective use of design optimization is common sense Any sufficiently general and powerful tool possesses the capacity for misuse This is especially true of numerical optimizers The old adage that engineering is more art than science is probably more true of design optimization than of many other disciplines As you gain experience with numerical optimization you will probably discover a few pitfalls that are particular to your fields of application and in the process you may discover some especially useful and efficient tricks for clearly stating your optimization tasks In any field of application there is no substitute for a well posed problem If your design goals are clear the constraints are meaningful and well conditioned and the design variables are chosen carefully to produce
183. cal function such as of the AVG rather than the function resultant itself the actual AVG Such mathematical function resultants are referred to as integrated responses for brevity Currently you cannot reference these responses from a DRESP3 entry except as a DRESP1 entry Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization Efficiencies in Second Level Response Definition 1 These efficiencies in design response identification extend to type 2 responses as well For example consider the following DRESP1 entries that identify major and minor principal stresses at surface z1 for PSHELL group 10 SDRESPI ID LABEL RTYPE PTYPE REGION ATTA ATIB ATTI1 4 ATT2 Major principal stress at surface zl DRESP1 101 sIGl STRESS PSHELL Minor principal stress at surface zl DRESP1 102 SIG2 STRESS PSHELL This pair of entries defines a pair of stresses for every element in this property group This could easily result in hundreds of design responses Assume we wanted to write the maximum shearing stress as the average of these responses SDRESP2 ID LABEL EQID REGION DESVAR DVID1 DVID2 A DTABLE LABEL1 LABEL2 DRESP1 NR1 NR2 gt j DNODE ID1 DIR1 NID2 DIR2 to equation to compute max shears 201 IAXS 300 DRESP1 101 102 Equation for max shears DEQATN 300 AXS SIG1 SIG2 SIG1 SIG2 2
184. cantilever subject to constraints on von Mises stresses Dynamic Response TPL problem D200RML3 DAT Optimization F E ee eee Uses synthetic responses to express a minimum RMS objective in Modal Frequency Analysis Twenty Five Bar Truss TPL problem D200X3S DAT Superelement Optimization Based on the 25 Bar Truss TPL problem D200X3 this example illustrates superelement optimization of a structure subject to constraints on local Euler buckling Also illustrates the use of synthetic responses and constraint scaling Design Optimization with TPL problem D200C01 DAT Composite Materials P Examines modifying ply thicknesses and orientations to achieve an optimal design based on Hill failure type constraints Acoustic Optimization TPL problem ACCOPT1 DAT Presents a method for minimization of acoustic pressure response peaks over a range of frequencies This approach is also quite useful for other dynamic response applications Restarts in Design Based on TPL problem D200X1 DAT Optimization a f Illustrates basic restart capabilities available in Solution 200 Discusses pseudo restart from a new location in the Design Space and restarts from an OPTEXIT point 7 1 Three Bar Truss A common task in design optimization is to reduce the mass of a structure subjected to not just one but several load conditions Figure 7 1 shows a simple three bar truss that must be built to withstand two separate loading conditions Note that the
185. case except for a small amount of numerical noise Caution should be used when interpreting the data appearing in the fractional error of approximation column One might be tempted to use the fractional error of approximation information as justification for increasing the move limits DELP and DELX on the DOPTPRM entry and or DELXV on the DESVAR entry in order to achieve convergence earlier This might not yield the expected results In this case the linear objective function was approximated linearly which is exact However the stress constraints in this problem are proportional to the inverse of the cross sectional areas and are thus nonlinear in the design variables Too large of a move limit could lead to increased approximation errors as was illustrated in Chapter 3 Here the mixed method of approximation was used which we know to be pretty good at approximating sizing type stress constraints Thus increasing move limits to 5096 is reasonable as it will probably save us the cost of a few additional finite element analyses A recommendation would be to stick with the defaults unless something is known about the problem beforehand that would indicate enhanced efficiency by changing them NORMAL CONVERGENCE CRITERIA SATISFIED HARD CONVERGENCE DECISION LOGIC MARA III III AAA RARA RARA RRA RARA RARA RR RAR RARA RARA RARA RARA RARA RRA RA RAR RRA CONVERGENCE ACHIEVED BASED ON THE FOLLOWING CRITERIA HARD CONVERGENCE DECISION LOG
186. case dependent constraint sets DEQATN Purpose Defines equations for use in synthetic relations These equations can be used to define either second level responses or second level design variable to property relations Entry Description DEQATN EQUATION MEE EQUATION Cont Field Contents EQID Unique equation identification number Integer gt 0 EQUATION Equation s Character Associated Entries The equation ID may be referenced in connection with a second level response on a DRESP2 entry or in the definition of a synthetic property relation on a DVPREL2 entry peser o ue Des JAJAJ wwe wm owe we i Sid O ome e uz 9 L ER wee we owe 9 Tae a b DESVAR DVID1 DVID2 DVID3 EE DTABLE LABL1 LABL2 LABL3 etc Discussion A unique feature of NX Nastran design sensitivity and optimization is that it allows the engineer to create new response quantities and nonlinear design variable to property relations This is 4 8 Design Sensitivity and Optimization User s Guide Input Data accomplished by defining equations much like the function definition procedure of many programming languages The DEQATN entry is used to specify a single equation or a set of nested equations The syntax of the expressions follows the FORTRAN language standard except that all arguments are assumed to be real numbers no integers Intrinsic functions may also be used The equati
187. ce constraints whereas this one does x x x NORMAL CONVERGENCE CRITERIA SATISFIED HARD CONVERGENCE DECISION LOGIC FAO OOOO CONVERGENCE ACHIEVED BASED ON THE FOLLOWING CRITERIA HARD CONVERGENCE DECISION LOGIC RELATIVE CHANGE IN OBJECTIVE 0 0000E 00 MUST BE LESS THAN 1 0000E 03 OR ABSOLUTE CHANGE IN OBJECTIVE 0 0000E 00 MUST BE LESS THAN 1 0000E 02 ANI MAXIMUM CONSTRAINT VALUE 1 8636E 03 MUST BE LESS THAN 5 0000E 03 CONVERGENCE TO A FEASIBLE DESIGN Om MAXIMUM OF RELATIVE PROP CHANGES 0 0000E 00 MUST BE LESS THAN 1 0000E 03 AND MAXIMUM OF RELATIVE D V CHANGES 0 0000E 00 MUST BE LESS THAN 1 0000E 03 CONVERGENCE TO A BEST COMPROMISE INFEASIBLE DESIGN FORA ke kk IRR ek ck kk che kk kk ck ok ke he kk kk check ck AA AAA ke ke kk e ke cock kk khe kk kk IR Kk ke ke ke kk ke he ko kk ke kk kk ke e ko Kk kk ko kk kk ko kk ke ke ke kk ak ok SUMMARY OF DESIGN CYCLE HISTORY PORRO ROO OOOO OOOO OOOO ko kk kk ke kk ko kk ARA ke kk kk e koe HARD CONVERGENCE ACHIEVED SOFT CONVERGENCE ACHIEVED NUMBER OF FINITE ELEMENT ANALYSES COMPLETED 8 NUMBER OF OPTIMIZATIONS W R T APPROXIMATE MODELS 7 OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY OBJECTIVE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE ROW OF CYCLE APPROXIMATE EXACT OF OF MAXIMUM NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT CONSTRAINT INITIAL 5 784520E 00 1 311878E 00 10 1 1 122267E 01 1 122355E 01 7 868297E 05 4 089278E 01 2 2 1 636887E 01
188. ce is given by df dx X RA f x X AX Equation 1 9 where the quantity Ax represents the small step taken in the direction x For most practical design tasks we are usually concerned with a vector of design variables The resultant vector of partial derivatives or gradient of the function can be written as gt OF F x Ax F x Ox Ax gt VF x gt gt p PG Ax y Fix ex mmc Equation 1 10 where each partial derivative is a single component of the dimensional vector Note Details of how NX Nastran computes sensitivities can be found in Design Sensitivity Analysis Direction of Steepest Descent Physically the gradient vector points uphill or in the direction of increasing objective function If we want to minimize the objective function we will actually move in a direction opposite to that of the gradient The steepest descent algorithm searches in the direction defined by the negative of the objective function gradient or gt S VF Equation 1 11 gt since proceeding in this direction reduces the function value most rapidly is referred to as the search vector For now just note that NX Nastran uses the steepest descent direction only when none of the constraints are critical or violated and then only as the starting point for other more efficient search algorithms The difficulty in practice stems from the fact that although the direction of steepest Des
189. cessary to increase the objective function to do so To achieve this we augment our direction finding problem of Eq D 29 through Eq D 31 with a new variable W This process has no direct physical significance to the problem except as a measure of the constraint violation X gx Y grx S 4 SS NS DY N MU pp My d 77 S 0 X1 Figure D 6 Violation of Constraint s The new direction finding problem is now gt Find the search direction S and artificial variable W that will q 1 2d minimize VF R S yW Equation D 33 Design Sensitivity and Optimization User s Guide D 15 Appendix D Numerical Optimization q 1 29 subject to Vg x S OWs 0 jeJ Equation D 34 Equation D 35 For discussion assume that the parameter 0 in Eq D 34 is equal to 1 0 and that the parameter y is a very large positive number Then the second term dominates the minimizing of the function defined by Eq D 33 so any increase in the variable W will drive the objective more and more negative i e reduce the objective of the direction finding problem However for W to increase the first term in Eq D 34 must become more and more negative Since is bounded by Eq D 35 the cosine of A gt 4 the angle between Vg ands must be moved closer and closer to 1 0 For this to happen 24 241 24 5 must point in a direction opposite to Vg Thatis 5 must point straight back toward the feasible region The first term i
190. change by the engineer However many of those methods use numerical constants in connection with convergence detection one dimensional search control and so on Defaults are provided for all of these internal optimizer parameters but these may be overridden using the DOPTPRM Bulk Data entry This section presents the advanced level information necessary to make these override decisions Chapter 1 introduced the Basic Optimization Problem Statement Recall that this provides the definition of the design variables objective and constraints as minimize F x objective Equation D 1 subject to g x s0 j L n inequality constraints Equation D 2 hy 0 k 1 n equality constraints Equation D 3 1 DOT User s Manual Vanderplaats Miura and Associates Inc 1990 2 Vanderplaats G N Numerical Optimization Techniques for Engineering Design with Applications McGraw Hill 1984 Design Sensitivity and Optimization User s Guide D 1 Appendix D Numerical Optimization X amp x amp x i0 l n side constraints Equation D 4 where A 3 EA X m 1 X X ni design variables Equation D 5 It is within the design space defined by the above problem statement that the optimizer searches for a best design Chapter 1 also introduced the analogy of a blindfolded search along the side of a hill for the point of lowest elevation to describe some of the basic processes used by gradient based numerical optimizers At t
191. cks content and format can be found in the NX Nastran Programmer s Manual Design Model Definition CVALO Updated approximate constraint values output from the optimizer DOM9 DESTAB Table containing design variable attributes DOPR1 5 4 Design Sensitivity and Optimization User s Guide Solution Sequences Pe built from DTOS2K containing current values of designed properties This is updated at the beginning of every design cycle DOPR5 DTOS2J Table identifying properties referenced in the design model both DVPREL1 and E This is built directly from the design model data DOPR1 EDOMS Analysis discipline and superelement partitioned EDOM SDSA pdated element property tables based on optimization results DOM11 pdated grid geometry records based on the optimization results DOM11 PROPO Vector of properties output from the optimizer DOM9 First level responses output from the optimizer These are based on the approximate model DOM9 ME ecto responses output from the optimizer These are based on the approximate model DOM9 TTABDEQ Table of unique design variable IDs DOPR4 XINIT Vector of initial design variable values updated at the start of each design cycle DOPRI initialization EQUIVX after DOM9 on subsequent cycles XO Vector of design variables output from the optimizer This becomes XINIT for the next design cycle DOM9 Shape Optimization AMLIST List of auxiliary models that is used to drive a
192. compute this data by hand Either way the following SPCD entries result 2 32 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization LOAD 1 ELLIPTICAL X COMPONENT VARIATION 10MM MAX VALUE D 01 0098769 0095106 0089101 0080902 0070711 0058779 0045399 0030902 0015644 0 0 5 13 11 12 101 13 101 14 102 123456 10 11 103 1 1 D D D D D D D D D D CONS S 555 505 505 wwe No LOAD 2 ELLIPTICAL Y COMPONENT VARIATION 10MM MAX VALUE 104 5 0015644 104 6 0030902 104 7 0045399 104 8 0058779 104 9 0070711 104 10 0080902 104 11 0089101 104 12 0095106 104 13 0098769 104 14 01 105 5 6 10 135 14 106 N M M M M U N NoN NON 5 5 GM GM oS Note that the SPCD enforced displacement values correspond to an elliptical variation along the hole boundary with a maximum value of 10 mm The result of these load applications are shown in Figure 2 18 and Figure 2 19 The deformations have been enlarged for clarity These displacement vectors are used next to generate our shape basis vectors Design Sensitivity and Optimization User s Guide 2 33 Chapter 2 Design Modeling for Sensitivity and Optimization Figure 2 19 Displacement Vector 2 In the optimization run we will DBLOCATE these displacement vectors and use this data to defi
193. connection with advanced applications Following the module and data block descriptions is a top level flowchart of Solution Sequence 200 This flowchart is referenced in Output Features and Interpretation and Example Problems in connection with output interpretation 5 1 Design Sensitivity and Optimization Modules Auxiliary model driver drives the auxiliary model looping for shape optimization using auxiliary boundary models by reading the auxiliary model list and outputting the next sequential auxiliary model ID AUXMID The auxiliary model loop flag AMLPEL is FALSE if this is the last auxiliary model in the list ANE ee the auxiliary model list AMLIST used in the auxiliary model boundary shapes method for shape optimization AXMPR2 Generates complete geometry grid data for the auxiliary boundary models by merging with the geometry of the primary structure It also generates a Case Control data block CASEVEC used later to generate a geometry partitioning vector This is used if additional grids have been defined by the engineer in the auxiliary boundary model Partitions the design model entries table EDOM on analysis type This is the first of two partitioning operations the second step partitioning on superelements is performed later by SDSA Thus for all later data processing operations which usually involve looping over analysis disciplines and superelements we will have on hand only that required subset of design model entries
194. conservative A minimum value of DELP 0 5 is recommended Using DELP 0 5 may improve solution efficiency in other cases as well The DELP design optimization parameter is specified on the DOPTPRM bulk entry Reference by Property Entry In design variable to property relations analysis model properties are referenced by property entry rather than on an element by element basis This relation is shown schematically in Figure 2 3 The property IDs PID1 through PIDm are referenced on the DVPREL1 and DVPREL2 entries Since each of these property groups may contain a large number of elements EID1 through EIDn you can control many elements by using only a small set of DVPREL type entries For example to modify plate thicknesses all that is required is to reference a PSHELL ID on a DVPREL bulk entry All elements in the property group will then change accordingly EID1 EID2 DVPREL1 2 ElDn PIDm Figure 2 3 Design Model Reference by Property Entry Identification of Property Entry Items You must identify a selected property entry item either by its field position on the property Bulk Data entry or by its word position in the element property table EPT that is generated by NX Nastran In general it is far easier to refer to the field position on the Bulk Data entry In some instances BEAM elements in particular reference to the element property table must be made instead None of this is really as difficult as it sounds and is
195. cost of the displacement sensitivity solution can be reduced if the stiffness matrix already available in decomposed form as a result of the finite element solution is used This observation is taken advantage of in NX Nastran Note that the right hand side of Eq 3 46 often referred to as the pseudo load vector includes the partial derivatives 9 PV 0x and a K Ox Since these are generally implicit functions of the design variables these derivatives are approximated using finite differences as follows 3 28 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran For property optimization unless CDIF YES ouk _ KQ Arp KN x AX ee axa LU PE Ox Ax i i Equation 3 47 e For shape optimization or if CDIF YES i e central finite differencing K EG 4 Ax KG Ax Ox E 2Ax 0 0 re II E rd arp OP _ Ons 2Ax Equation 3 48 For property optimization the size of the move Ax is given by AX DELPB e Xj Equation 3 49 where DELB is given on the DOPTPRM Bulk Data entry See Shape Sensitivity later in this chapter for the corresponding grid variation for shape optimization As a result of these finite difference approximated terms in the pseudo load vector the solution of Eq 3 46 is frequently referred to as a semianalytic one A similar approach is used in the other analysis disciplines and is described i
196. ction 7 Perform a one dimensional search to find a q g 24 8 set 1 27 u 9 Check for convergence to the optimum If satisfied exit Otherwise go to Step 2 The critical parts of the optimization task consist of the following gt 1 Finding a usable feasible search direction 5 gt q 1 gt F x x S 2 Finding the scalar parameter a that will minimize subject to the constraints 3 Testing for convergence to the optimum E and terminating if convergence is achieved We will discuss each of these in turn D 3 3 Finding the Search Direction The first step in finding the search direction is to determine which constraints if any are active or violated Here an active constraint is defined as one with a value between CT and CTMIN where CT is a small negative number and CTMIN is a small positive number Remember that constraints must gt be negative to be feasible therefore if gi is less e g more negative than CT it is not considered active Also when we approach a constraint boundary it makes sense to begin pushing away from that fence Therefore we initially choose a relatively large value for CT such as 0 03 i e within three percent of being mathematically active On the other hand any constraint with a positive value is mathematically violated However trying to achieve a precise zero on the computer is not meaningful Also loads material properties etc Design Sensit
197. ction tangent to the critical constraint s If the constraint is nonlinear a finite move in this direction may lead to a small constraint violation Thus continual corrections must be made by stepping back toward the constraint boundary along the current search direction These corrections are performed as part of the search process and are qualitatively represented in Figure 1 8 as small steps back to the constraint boundary Dealing with Initially Infeasible Designs If the initial design is infeasible the optimizer s first task is to return to the feasible region Once this has been achieved the optimizer can then proceed to minimize the objective if possible Often simply finding a design in which none of the constraints are violated is an engineering success If no feasible designs are found this still provides useful information about the original design formulation as one or more of our performance criteria may need to be relaxed somewhat if we hope to produce a Design Sensitivity and Optimization User s Guide 1 19 Chapter 1 Getting Started feasible design By reexamining our design goals we may learn something about the problem that was not evident before Cautionary Notes A critical but related issue involves the identification of various applicable modes of failure In developing a set of design criteria we must ensure that all possible failure modes are adequately addressed by the design specifications This is true in all
198. ctors is able to uniquely define the optimum Including other basis vectors or allowing each property to vary independently probably will yield a different optimum perhaps even one having a lower structural weight Which final design is more useful If an arbitrary variation of plate thicknesses is permissible assuming it can be manufactured of course and it did not require an unreasonable number of design variables to obtain this level of generality then an independent type of relation may be preferable That is each plate element thickness can be a function of just a single design variable However in the preceding example the basis function approach was able to guarantee that the resulting plate section had a smoothly varying thickness in a stepwise fashion that is This may be a more desirable design solution even though the objective function might be reduced further if this condition were relaxed In short the design model formulation represents a statement of design expectations and an implicit acknowledgement of its limitations Linking with User Defined Equations If the number of analysis model properties cannot be changed or reduced and a reduced basis formulation with DVPREL1 type relations is not sufficient or preferred use of the DLINK Design variable LINKing Bulk Data entry should be considered Its purpose is to provide explicit linear relations among the design variables themselves The DLINK entries define relations
199. curacy Synthetic responses defined via DRESP2 and DRESP3 entries cannot have arguments that span superelement boundaries Multiple boundary conditions are not permitted in superelement design modeling This restriction stems from a similar restriction in superelement analysis The DCONSTR and DSCREEN entries have no provisions for specification of superelement IDs If separate response allowables or screening criteria are to be applied the DRESP1 entries should define responses that do not span superelement boundaries Design Sensitivity and Optimization User s Guide 2 61 Chapter 3 Design Sensitivity and Optimization in NX Nastran Approximation Concepts in Design Optimization Design Sensitivity Analysis Optimization with Respect to Approximate Models Convergence Tests Design Sensitivity and Optimization User s Guide 3 1 Chapter 3 Design Sensitivity and Optimization in NX Nastran This chapter presents the theoretical details of design sensitivity and optimization in NX Nastran Intended as a complement to the design modeling topics of the previous chapter each section begins with a general overview of modeling related aspects before moving on to the theoretical details The presentation order of this chapter generally follows the code s order of operations 3 1 Approximation Concepts in Design Optimization Approximation concepts are used in NX Nastran to efficiently couple the numerical optimizer with the structur
200. d BEGIN BULK AUXMODEL n where n is the auxiliary boundary model ID 2 Select loading and boundary conditions in Case Control using AUXCASE and AUXMODEL n to define the auxiliary boundary model Case Control Sections 3 Define the shape boundary conditions on the actual structure using BNDGRID Bulk Data entries The actual structure is often referred to as the primary structure In addition to defining the connection points with the auxiliary boundary model the BNDGRID entries define those boundaries that are invariant during shape optimization 4 Define the shape design variables using DESVAR entries and relate these to the shape basis vectors which the code will compute with DVSHAP entries Example Shape Basis Vectors This example illustrates the process of defining shape basis vectors for each of the previously outlined methods excluding the manual grid variation method for which data preparation can be quite extensive Since DVGRIDs are used in the geometric boundary shapes method anyway their use will be covered here in connection with that method Suppose we want to redesign the shape of a hole cutout in a square plate shown in Figure 2 16 This redesign seeks to minimize the weight subject to constraints on von Mises stresses However rather than allow only circular variations we would like to investigate elliptical shape changes Since the edge traction force T is twice Ty one would expect a 2 1 ratio elliptical hole a
201. d Optimization assigned with a DRSPAN Case Control command then all DRESP1 entries referenced by the DREPS2 or DRESP3 entries must be assigned with a DRSPAN command e Specify the response bounds by using a DCONSTR entry that references the DRESP1 DRESP2 or DREPSS defined response or set of responses for example if it is defined over a frequency set e Select DCONSTR entry sets in the Case Control section using either the DESGLB command for global DRSPAN related therefore subcase dependent or not DRSPAN related therefore subcase independent responses or the DESSUB command for subcase dependent responses Responses or constraints become subcase dependent when any related DRESP1 entries are referenced from a DCONSTR entry that is in turn referenced from a SESUB command belonging to a subcase rather than assigned with a DRSPAN command Therefore you can reference the non DRSPAN related responses from more than one subcase by using differently numbered DCONSTR entries that are referenced from DESUB commands from different subcases The DCONSTR entry bounds are used by NX Nastran to create a pair of normalized constraints one for the lower bound and one for the upper bound For any given response the bounds are a statement of the inequality relation L gt U r lt r x lt r J J J Equation 2 34 where Ge is the lower bound on the j th response and 1 Uis the corresponding upper bound Normalized Constraints The absolute
202. d by its response ID A first level response can also be used as an objective if it defines a single scalar response e g weight an eigenvalue a grid displacement component for a single subcase and so on The objective is defined in Case Control with the command Design Sensitivity and Optimization User s Guide 4 15 Chapter 4 Input Data DESOB id y max Equation 4 5 where n is the DRESP1 or DRESP2 entry ID Discussion Depending on the particular analysis discipline responses such as displacements stresses eigenvalues etc may be computed However not all of these responses may be of interest from a design optimization standpoint The DRESP1 entry is used to identify responses that are to be used in connection with design sensitivity and optimization either as an objective or as a constraint These responses may also be used as input to compute a second level or synthetic response on a DRESP2 entry An internal Case Control for design sensitivity and optimization is built using the set of first level responses Selecting design responses using DRESP1 entries also ensures that the appropriate data recovery is performed regardless of Case Control requests the user may have supplied Output requests if desired still must be specified with the appropriate Case Control commands however this is for postprocessing convenience only and is not a requirement for either design sensitivity or optimization DRESP2
203. d end B specified This case is typical of any number of intermediate stations up to and including 8 Words 6 21 22 37 38 53 and 166 181 must be accounted for on DVPRELi entries Thus every intermediate station plus one internally generated intermediate station must be accounted for on the set of DVPRELi entries End A Station C Intermediate End B Words Words Words Words 6 21 22 37 38 53 166 181 This station is internally generated but must be accounted for on DVPRELi entries Words 38 53 are identical to Words 166 181 End B Figure 2 6 Element Property Table for BEAM Element Item 1 in Figure 2 6 is a diagram of the EPT for the BEAM element when the PBEAM entry furnishes data at end A only End B data is built internally as a copy of end A properties Since this structure is used by subsequent modules to generate the element mass and stiffness matrices the design model must also define equal variations for both sets of data Thus for design sensitivity and optimization you must prescribe EPT word positions 166 181 end B in addition to words 6 21 end A Otherwise the ensuing incompatibility between end A and end B data will invalidate the subsequent analysis results Item 2 in Figure 2 6 shows the structure of the EPT for a tapered beam The data for ends A and B is determined based on the supplied PBEAM input Additionally the first intermediate station Words 22 37 is internally generated and contains a copy o
204. design variable and a solid circular section is assumed then 2 46 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization Equation 2 32 and 4P x xX a _nEA Equation 2 33 The buckling constraint is now a function of the cross sectional area design variable for this ROD element geometry The following Bulk Data entries illustrate one way in which these relations may be implemented E DRESP1 401 DRESP2 501 DESVAR DTABLE DRESP1 DTABLE L DEQATN 600 L1 A L P 4 P L 2 PI 1 0 1 0E7 A 2 note use of PI function see DEQATN entry description SDCONSTR DCID LALLOW UALLOW DCONSTR 500 1 0 The two GRID entries along with the CROD and PROD entries specify ROD Element 100 with a length of 10 units along the x axis and an initial cross sectional area of 0 5 The DESVAR entry defines the area design variable with an initial value equal to the initial cross sectional area of the ROD Lower and upper bounds of 0 25 and 0 75 respectively are set representing 50 move limits on the cross sectional area Since the constraint on Euler buckling is applied here for just a single element the ELEM identifier is used on the DRESP in field 5 along with a 100 in field 9 to specify that the axial load for Element 100 is to be used in the design model The force component is selected by reference to the plot codes the 2 in field 7 indicates
205. dient is considered and this is identically equal to zero i e any finite move in any direction will not decrease the objective function A zero objective function gradient indicates a stationary condition Not only are the Kuhn Tucker conditions useful in determining if we have achieved an optimal design they are also physically intuitive The optimizer in NX Nastran tests the Kuhn Tucker conditions in connection with the search direction determination algorithm The interested reader can find the theoretical details in Appendix D A Simple Structural Example In this section we covered the basic optimization problem statement the concept of a design space gradient based search techniques and the meaning of an optimum in terms of satisfying the Kuhn Tucker conditions We pause here to take a look at a simple example to help fit these pieces together and to introduce some qualitative aspects of the optimizer used in NX Nastran The cantilever beam in Figure 1 6 has a rectangular cross section Suppose we want to minimize the volume and thus the weight of the beam subject to constraints on maximum bending stress and deflection due to the tip loading In addition the beam s cross section must remain below a maximum beam height to width ratio to guard against the introduction of twisting modes of failure P 2250N Section A A A T mu H px A E A L 500 cm a es Figure 1 6 Cantilever Beam The optimization problem sta
206. ditions T T o c d E NS Figure 1 1 Flat Plate with Hole 1 4 Design Sensitivity and Optimization User s Guide Getting Started Analysis Model Design Model R Finite element discretization of Find R such that Structure Mesh Weight is minimized Loads Stresses do not exceed allowables Boundary Conditions 1 4 Plate Representation R is the design variable weight is the design objective and stresses are the design constraints Figure 1 2 Analysis Versus Design Models Design Models In contrast a design model is an idealized statement of changes that might be made to the structure to improve its performance or response In order to accomplish this we need to define what we mean by an improved design It may be the minimum weight or maximum stiffness but whatever our choice is this constitutes a statement of the design objective The design may be varied such that certain bounds on responses are not exceeded Expressions of maximum allowable stresses or minimum permissible frequencies are termed design constraints And in our description of how the design might be changed we use design variables to express what we mean by a suitable variation By convention the mathematical region over which our design variables objective and constraints are defined is called the design space Figure 1 2 also shows the design model corresponding to a redesign of the hole r
207. ds is comprised of both structural as well as fluid forces or Equation 3 94 The mass matrix in Eq 3 92 is defined as 3 42 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran M 0 4 MJ Equation 3 95 where Ms structural mass Mi fluid mass including compressibility effects A fluid structure coupling matrix The stiffness matrix consists of K TIN 0 Kg K Equation 3 96 where structural stiffness effective fluid stiffness Ks K If the loads in Eq 3 92 are frequency dependent a direct formulation can be written just as for the structural case where no fluid effects are present Differentiating the direct frequency equation yields an expression similar to Eq 3 73 which can be solved by exactly the same methods In very large problems it is often desirable to apply the modal decomposition method in order to reduce the cost of the analysis Modal formulations in coupled fluid structure analysis are derived from a separate consideration of both fluid and structural components The structural modes are computed for a structure in a vacuum that is in the absence of any fluid effects The modal transformation for the structural degrees of freedom can then be written as u 0 8 m 09 1M 0 k P LK 1 9 Equation 3 97 where D the modes for structure in a vacuum Design Sensitivity and Optimization Us
208. ds of design variables and thousands of constraints force us to consider methods for efficiently coupling structural analysis routines with numerical optimizers The field of structural optimization is based on the introduction of approximation concepts which reduce the need for repeated finite element function evaluations Approximation concepts are actually quite intuitive and are the topic of the next section 1 3 Structural Optimization In Introduction and Numerical Optimization Basics we introduced some examples of ways in which numerical optimization might be used to solve design problems and gave a brief overview of gradient based numerical optimizers These tools yield an orderly rational approach to solving a minimization problem subject to a set of constraints however a large number of function evaluations may need to be performed These function evaluations may be expensive especially in a finite element structural analysis context This section discusses methods used in NX Nastran to reduce these costs Basic Difficulties in Structural Optimization Historically the first attempts at linking structural analysis with numerical optimization were largely a direct coupling or black box type of approach as seen in Figure 1 9 Whenever the optimizer needed a function evaluation the finite element analysis would be invoked to provide the necessary information The sheer number of analyses quickly tended to make this approach useless
209. e o The evaluation requires the solution of a system of equations to arrive at the value for the design response During the optimization run NX Nastran passes data defined in a DRESP3 Bulk Data entry to the external program The data is passed in one dimensional arrays The external program receives the data and uses it to calculate a value for the design response The design response value is then passed back to NX Nastran for use as the value of the objective function or for use in some constraints Figure 2 22 shows the relationship between DRESP1 and DRESP3 design responses Analysis Results DESVAR DTABLE DNODE etc DRESP1 Nu 4 DRESP3 oN Design Objective VT Constraints Figure 2 22 First and Second Level Responses For additional information on using the DRESP3 Bulk Data entry capability see Defining Design Responses with External Programs To illustrate the difference between DRESP1 and DRESP2 entries suppose we have a case where we want to use the x and y static displacement components at a particular grid point as design responses in our design model These first level responses can be identified using two DRESP1 entries as follows SDRESPI ID LABEL RTYPE PTYPE REGION ATTA ATIB ATTI1 4 ATT2 X DISPLACEMENT AT GRID 100 DRESP1 501 UX100 Y DISPLACEMENT AT 502 UY100 DISP 2 42 Design Sensitivity and Optimization User s Guide
210. e A value of 4 outputs the coefficients in binary form using OUTPUTA formats DSAPRT has priority over the PARAM OPTEXIT 4 or 7 option It does have somewhat more flexibility and options such as formatting 3 24 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran Identifying the Sensitivity Coefficients If requested by a PARAM OPTEXIT 4 or 7 or by the unformatted DSAPRT Case Control command sensitivity coefficients are output in the matrix DSCM2 If you request sensitivity coefficients with the formatted DSAPRT Case Control command then the software prints the coefficients in a table with headers You can use both options with an EXPORT option to write the sensitivities to an external binary file Each column in the matrix DSCM2 corresponds to a particular response quantity while each row corresponds to a design variable The output is formatted in one column after another in sequence such as response by response The component of the th row and j th column is pr DSCM2 T Zr X Equation 3 33 A correlation matrix is also output to help identify the column order of the individual design responses The row order in DSCM2 corresponds to the design variables defined on the set of DESVAR entries sorted by increasing design variable ID See Output Features and Interpretation Design Sensitivity Output for examples illustrating the interpretation of this output I
211. e Sign Changes If the objective function is a response it is important that you know whether the response is expected to change signs for different designs that may be generated during the optimization For example minimizing a positive displacement means driving it in the direction towards zero However if the displacement changed sign during the optimization then the same minimization would try to drive it towards negative infinity In such cases it may be better to use a device that would minimize the absolute value through a DRESP2 entry as applicable such as the square root of the square or of the sum of the squares 2 8 Defining the Constraints Any first or second level response may be constrained simply by referencing it on a DCONSTR Design CONSTRaint Bulk Data entry This entry defines the response to be constrained and its corresponding bounds To define constraints in NX Nastran you need to e Identify a first level response or group of responses using a DRESP1 Bulk Data entry or write a second level response or group of second level responses using a DRESP2 or a DRESP3 Bulk Data entry If the DRESP2 or DRESP3 entry references a DRESP1 entry the DRESP1 entry can be a generic DRESP1 entry or a DREPS1 entry assigned to specific subcases with a DRSPAN Case Control command If the DRESP2 or DRESP3 entry reference a DRESP1 entry that is 2 54 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity an
212. e The volume of the culvert has been reduced by about 2096 while the strength constraint is satisfied Figure 7 8 Final Culvert Design ID AUX1 VT100 FEB 10 1990 TIME 10 TITLE Culvert Example Using External Auxiliary Model SUBTITLE The External Auxiliary Model SPC 25 seven load cases LOAD 102 DISP ALL SUBCASE 4 LOAD 103 7 26 Design Sensitivity and Optimization User s Guide Example Problems DISP ALL SUBCASE 5 LOAD 104 DISP ALL SUBCASE 6 LOAD 105 DISP ALL SUBCASE 7 LOAD 106 DISP ALL BEGIN BULK PARAM POST 0 param newseq 1 The same GRID and CQUAD4 entries as the primary structure GRID 1 3 00000 0 00000 00 GRID 2 4 00000 0 00000 00 GRID 3 5 00000 0 00000 00 see optimization input file GRID 38 2 00000 5 19600 00 GRID 39 2 50000 5 19600 00 GRID 40 3 00000 5 19600 00 CQUAD4 1 101 i 2r 6 5 CQUADA 2 101 2 3 7 6 CQUAD4 3 101 3 4 8 7 see optimization input file CQUADA 25 101 30 31 38 37 CQUADA 26 101 31 32 39 38 CQUADA 27 101 327 33 40 39 PSHELL 101 102 44 MAT1 102 2 7 3 Additional CBAR elements maintain smoothness of the circular boundary CBAR 31 1 13 14 1 CBAR 32 1 14 15 CBAR 33 1 15 16 CBAR 34 1 16 9 CBAR 35 1 9 5 CBAR 36 1 5 1 1 PBAR 1 102 20 0 1 0 1 0 Seven load cases FORCE 100 13 0 1 e5 0 1 0 FORCE 101 14 0 1 e5 0 259 9659 FORCE 102 15 0
213. e This can be used to express the changes in area moments of inertia in terms of A or 7 18 Design Sensitivity and Optimization User s Guide Example Problems 0 9 0 n Ab Ah eL 12 AUD Equation 7 8 where A543 E 1 2 A 4 yo A x Equation 7 9 and so on Since bending stresses in the web cap are to be used in the design model as constraints the stress recovery points must move as the area changes If not then we fail to recover stresses at the outer corners of the cross section where bending stresses are greatest For point C we have in terms of A v Oo Equation 7 10 and so on for the other stress recovery points D E and F Since the above relations are nonlinear in the design variable X3 we need to express these using DVPREL2 Bulk Data entries These entries provide the input arguments to the DEQATN entries on which the preceding relations have been defined In addition the DTABLE Bulk Data entry has been used to define constants that appear in the equations We could have provided these constants directly on the DEQATN Bulk Data entries instead but the loss of generality would have required us to use additional equations In order for a structural response to be used either as an objective or a constraint it first must be identified on a DRESPi Bulk Data entry The DRESP1 entries 1 and 2 for example identify the maximum stress at ends A and B of a bar element Plot codes are used in the ATTA fields
214. e stress constraints for every element in the skin panel since all of these stresses will vary nearly in unison as the panel thickness is varied The constraints from all of the neighboring elements will likely contain redundant information Thus it is probably safe to retain only a few of the largest valued constraints from this region Constraint regionalization is shown in Figure 1 16 which is the same as Figure 1 15 but with the constraints grouped into three regions For now we will assume these regions have been established based on some design model characteristics Of the three regions shown only two have constraints that are numerically greater than the truncation threshold These constraints will pass the first screening test Since the retained constraints within each region are likely to contain redundant information only the largest constraints from each region are retained These constraints are denoted by the check marks in the figure NSTR which in this example is 2 stands for the maximum number of constraints to be retained per region Note NSTR and TRS defaults can be overridden using the DSCREEN Bulk Data entry The number of regions are implicitly defined in connection with the DRESP1 Bulk Data entry G X NSTR 2 J 4 4 a X XXXXXXXX fil iil 1 0 l fi gt E gt Region 1 Region 2 Region 3 Retained Responses 1 0 2 Figure 1 15 Constraint Regionalization 1 28 De
215. e changes in the th grid point are expressed in terms of changes in the j th design variable If the number of design variables is less than the number of grid points this relation is called a reduced basis formulation Reduced basis formulations are discussed in Relating Design Variables to Shape Changes 4 20 Design Sensitivity and Optimization User s Guide Input Data The DVGRID entry defines the particular grid point design variable and components of each T vector Multiple references to the same grid point and design variable pair simply result in vectorial addition of the corresponding T Other shape basis vector definition methods utilize DVBSHAP and DVSHAP Bulk Data entries DVPREL1 Purpose Defines a structural analysis property as a function of a linear combination of design variables Entry Description Field Contents ID Unique identification number Integer gt 0 TYPE Name of a property entry such as PBAR PBEAM etc Character PID Property entry identification number Integer gt 0 FID Field position of the property entry or word position in the element property table of the analysis model Integer 0 0 Minimum value allowed for this property If FID references a stress recovery location then the default value for PMIN is 1 0 35 PMIN must be explicitly set PMIN l to a negative number for properties that may be less than zero for example field ZO on the PCOMP entry Real
216. e control you have over each are first itemized below The remainder of the section discusses each of these items in greater detail Design variable linking Design variable linking allows you to keep the number of independent design variables to a minimum leading to a well formulated design model You link design variables as part of the design modeling process Any combination of three possible methods can be used grouping of the analysis model properties effective use of the DVPREL1 and 2 relations and explicit linking of the design variables via the DLINK entry In addition to reducing the computational overhead both during sensitivity analysis as well as optimization design optimization results interpretation is usually simplified 3 2 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran Note Design variable linking not only reduces the cost of design sensitivity and optimization it also makes results interpretation easier Constraint regionalization and deletion This process temporarily removes non critical redundant constraints from consideration on the assumption that this reduced set still contains adequate information to efficiently guide the design NX Nastran performs this task automatically although you do have some high level control of the process You can use the DSCREEN DRESP1 and DRESP2 and DRESP3 entries to provide this constraint screening contr
217. e linking such that Az Ay Constraints Allowable stress Tensile 20 000 psi Compressive 15 000 psi The input data for this problem is given in Listing 7 1 Grid element and load data are assigned based on the data supplied in the figure The two separate static load cases are defined in Solution 200 Case Control using the parameter ANALYSIS STATICS Turning to the design model description we see three design variables are used one each to control the individual rod element cross sectional areas The set of DESVAR DVPREL1 entries define the relations 7 4 Design Sensitivity and Optimization User s Guide Example Problems 4 910r P2323 1 Equation 7 1 Note that the initial design variable values are equal to the initial rod cross sectional areas With the multiplying coefficients of 1 0 on the DVPREL1 entries the design model and the initial analysis model properties agree Thus no design model override takes place Since the model s geometry and loading are both symmetric we expect the optimizer to yield a symmetric final design with A1 A3 In a more complex structure we may want to enforce some type of continuity ourselves perhaps to address ease of manufacture concerns This can be done using the DLINK entry to explicitly link design variables together Although not strictly necessary DLINK is used to define the relation x 10 Equation 7 2 The value of X3 now depends on X4 Therefore the optimizer only ne
218. e optimizer level in connection with the numerical search process See Violated Constraints Analysis Discipline is used in Solution 200 to refer to the available analysis types Statics Normal Modes Buckling Direct and Modal Frequency Modal Transient Static Aeroelastic Responses and Flutter Analysis Model defines the geometry element connectivity material properties and loads associated with an NX Nastran finite element analysis The analysis model may be varied according to the design model which uses responses computed from the analysis model to guide the design process Analysis Model Properties are for purposes of design sensitivity and optimization in NX Nastran as quantities specified on property Bulk Data entries For example bar area moments of inertia on the PBAR entry are analysis model properties whereas offsets on the CBAR entry are not Approximate Model is an approximate representation of the design space that is used by the optimizer to determine an approximate optimum Since the approximate model contains a reduced number of constraints and is explicit in the design variables it lends itself to efficient solution by a numerical optimizer Approximate Optimization is used to refer to the optimization with respect to approximate design spaces See Approximation Concepts in Design Optimization Approximation Concepts include design variable linking constraint regionalization and deletion and formal approximati
219. e preferable to restrict the design to a set of say ten different sections We can then arrange the elements so that all vary according to just this set of ten Linking by Property Entries One way to link properties in NX Nastran is to use the analysis model property entries directly In the analysis model a property entry is defined and referenced by a number of elements In the design model design variables are related to the analysis model property entries Thus all elements that reference a particular property entry vary in unison as a function of the design model description This is shown in the diagram of Figure 1 12 and the approach is discussed in Overview of Design Modeling Note DVPREL1 and DVPREL2 entries are discussed in Design Modeling for Sensitivity and Optimization and Input Data 1 24 Design Sensitivity and Optimization User s Guide Getting Started EID1 EID2 DESVAR gt DVPREL1 2 PID EIDn Figure 1 12 Property Entry Linking Explicit Design Variable Linking Another design variable linking technique is to express a design variable as a linear function of other design variables This type of relation introduces the concept of an independent design variable set and a dependent design variable set and can be written as Equation 1 21 2 where the dependent design variable set D is a linear function of the independent design variable set T
220. e restrictive than required in some cases However it does provide the essential idea In practice it is difficult to numerically find a design that precisely satisfies the Kuhn Tucker conditions Also numerous designs might actually satisfy these conditions since there may be more than one constrained minimum The importance of the Kuhn Tucker conditions is that an understanding of the necessary conditions for optimality gives us some knowledge of what is needed to achieve an optimum The simple example of descending a hill by the approach outlined above is the physical interpretation of a class of optimization methods known as feasible direction techniques A multitude of other numerical search algorithms is available to solve the general optimization problem see Vanderplaats An Efficient Feasible Direction Algorithm For Design Synthesis 3 Most of the more powerful methods update the design using the aforementioned relation 34 54 1 24 x x a S Equation D 18 where q iteration number vector search direction d scalar move parameter Eq D 18 is the same as Eq D 6 except that we have replaced the superscripts with the iteration gt counter q Just as in the example of searching down the hill the product a S is the design modification at the current step Note that this is very similar to the traditional engineering approach Design Sensitivity and Optimization User s Guide D 7 Appendix D Numerical Optimization
221. e sensitivity superement loops Its structure is similar to SLIST used in superelement analysis Prints the DSCMR correlation table that identifies the column response correlation in the design sensitivity coefficient matrix DSCM2 Generates a new Case Control data block to drive solutions for basis vectors in shape optimization The initial shape data is supplied as enforced displacements on the boundary of the structure and then interpolated to the interior grids via a static solution of the primary structure SHPCAS generates the Case Control data block CASENEW which is used in connection with this solution It also generates a partitioning vector CVEC to extract the appropriate solution vectors from the analysis results Computes the structural weight or volume for use in design optimization Weight or mass is only computed for those elements that have volume The effect of elements such as CONMs are thus neglected 5 2 Selected Data Blocks The data blocks listed in this section are of particular interest in design sensitivity and optimization Although just a subset the data blocks listed here have been chosen based on their usefulness in custom DMAPing results processing etc Should the need arise the brief descriptions in this section may be a useful aid when trying to navigate through various sections of the DMAP The module that creates each data block is noted at the end of each description Full descriptions of these data blo
222. eatures and Interpretation DSCM2 2 Equation 6 1 where is the row order and j is the column order Recall from Design Sensitivity and Optimization in NX Nastran that a sensitivity coefficient is defined as a response rate of change for a corresponding design variable change The resultant partial derivative thus gives the slope of the j th response function for the current design in the th design variable dimension The rows of DSCM2 are sorted on ascending independent design variable IDs given on the DESVAR Bulk Data entries Row 1 thus corresponds to the independent design variable with the lowest ID row 2 the design variable with the next higher ID and so on DSCM2 column order is given in a correlation table For OPTEXIT 4 or 7 the formatted table contents are printed in the output file This output indicates the particular column and response correlation in the DSCM2 matrix For OPTEXIT 4 this correlation information is output as the DSCMCOL table See the NX Nastran Programmer s Manual for details Design Sensitivity Output Example This example is taken from the Test Problem Library problem number D200X5 This cantilever plate is presented as an optimization example in Example Problems and has been modified here by the addition of PARAM OPTEXIT 4 to the Bulk Data Section Thus only sensitivity analysis and not optimization will be performed See Example Problems for details regarding the problem description and se
223. eck determines if the design constraints are satisfied If these constraints are satisfied then soft convergence to a feasible design is achieved If some of the constraints are still not satisfied a check on the change in design variables is performed The conclusion here is that if the design variables properties and objective function are unchanged by the optimizer then convergence is indicated although to an infeasible design Since the optimizer cannot make any further progress an additional finite element analysis is probably unnecessary Hard Convergence Decision Logic Hard convergence is always used to decide whether or not the design cycle process should continue provided that the maximum allowable number of design cycles DESMAX is not yet reached The satisfaction of hard convergence is always sufficient to stop the design process A flowchart of these criteria is shown in Figure 3 14 with the corresponding definitions of the terms appearing previously in Table 3 1 Even though this test may indicate convergence note that one of three possible conclusions may be reached Design Sensitivity and Optimization User s Guide 3 57 Chapter 3 Design Sensitivity and Optimization in NX Nastran CHGOBJ lt CONV1 or ACHOBJ lt CONV2 Yes Yes No CONMAX GMAX CHGPRP CONVPR Yes and No CHGDV CONVDV Convergence to a Best Compromise Continue with Unique Design Infe
224. ed Applying this approximation to all violated constraints the largest proposed value of a is selected as an estimate of how far to move to overcome all constraint violations If the projected move to a constraint that is not currently active is smaller than this then a move to that constraint is made as a first estimate since we do not want to add new constraints to the set of violated constraints Using similar approximations we can also estimate an upper bound on a that forces all design variables to their lower or upper bounds This then provides a maximum allowable value for a While this brief discussion is not detailed enough to cover all of the intricacies of estimating the first step in the one dimensional search it does provide a sense of the complexity of the decision processes Also it should be noted that on subsequent search directions this process is modified to account for knowledge gained on previous iterations For example if an average of 5 improvement is obtained on the last two one dimensional searches then another 596 improvement is attempted on this one The one dimensional search process now proceeds to find the bounds on the a that contain the solution Once the bounds on a are known the constrained minimum is found by interpolation gt q Since 5 has been defined as a direction of improving design the search can be limited to positive values of al D 4 Finding Bounds On a At the beginning
225. eds to consider two independent design variables X4 and X instead of three This not only simplifies the problem but also reduces the cost of the sensitivity analysis This cost savings can be significant in a larger problem An alternate method of linking would have been placing rod elements 1and 3 in the same property group The DLINK entry would then be unnecessary since a single design variable could control the areas of all elements of the group You may want to try this formulation yourself to see that the results are indeed identical DRESP1 entries define the design responses the displacement and stress responses that are to be constrained and the weight response that is to be used as the objective Note that the objective and constraints are called out in Case Control The DESOBJ command points to the weight response defined on DRESP1 20 while the DESSUB command identifies DCONSTR set 21 Since this appears above the subcase level the constraint bounds will be applied to both subcases A DOPTPRM entry is used to override some of the default optimization parameters DESMAX or the maximum number of design cycles to be performed has been increased from the default of 5 to 10 If convergence is indicated fewer than 10 cycles may be performed DELP has also been increased from its default of 0 2 to 0 5 This allows any analysis model parameter to undergo changes of up to 50 on each design cycle This provides move limits on the approxima
226. efine the design variables SDESVAR ID DESVAR DESVAR DESVAR DESVAR DESVAR DESVAR DESVAR DESVAR co JoUu50NPD Relate the design SDVPREL1 ID S DVIDD1 DVPREL1 1 DP1 pd DVPREL1 2 DP2 2 DVPREL1 3 DP3 nay DVPREL1 4 DP4 DVPREL1 5 DP5 E DVPREL1 6 DP6 E DVPREL1 7 DP7 Py DVPREL1 8 DP8 E O0 OOo O0 1 N PRPPPPRPER LABEL XINIT X1 2 0 X2 2 0 X3 2 0 X4 2 0 X5 2 0 X6 2 0 X7 2 0 X8 2 0 variables TYPE PID COEFl DVID2 PROD 1 1 0 PROD 25 1 0 PROD 3 1 0 PROD 4 1 0 PROD 5 1 0 PROD 6 1 0 PROD Ti 1 0 PROD 8 1 0 SU COOOU NNNNNNNN c0oooooo 10 10 oooooooo oooooooo to the analysis FID COEF2 XUB 100 100 100 100 100 100 100 100 oooooooo PMIN 10000 10000 0 0 20000 20000 DELXV model properties PMAX co Identify the responses to be used in the design model REGION ATTA DRESP1 ID ATT2 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 DRESP1 10 DRESP1 11 DRESP1 12 DRESP1 13 DRESP1 14 DRESP1 15 OIHORWNE o Formulate the second level responses SDRESP2 ID DESVAR DTABLE DRESP1 DNODE DRESP2 16 DR11 DESVAR DR12 DTABLE DR13 DRESP1 DRESP2 17 LABEL S1 2 3 S4 S5 S6 S7 s8 D1 D2
227. egative value of a normalized constraint indicates constraint satisfaction while a positive value indicates violation See the Basic Optimization Problem Statement Figure 1 14 illustrates screening based on normalized constraints by representing the current value of each constraint function in a bar chart format If any constraint exceeds the truncation threshold value denoted by TRS we retain it for the ensuing approximate optimization while temporarily deleting all others below TRS Constraint Screening G X X XXXXXXXX TRS 1 0 List of Constraints Figure 1 14 Constraint Deletion Design Sensitivity and Optimization User s Guide 1 27 Chapter 1 Getting Started Note The default value for TRS is 0 5 but may be changed using the DSCREEN entry See Approximation Concepts in Design Optimization for more details If there are a large number of constraints below this truncation threshold as is typical in structural optimization the constraint screening phase will greatly simplify the optimization task However the number of constraints can still be further reduced using regionalization Constraint Regionalization Suppose we have a particular section of an airplane wing skin modeled with a number of quadrilateral elements This wing skin is to be of constant thickness for manufacturing considerations and is found to be overstressed in a number of locations It would not make much sense to retain th
228. ell as example problems see the NX Nastran Aeroelastic Analysis User s Guide Static Aeroelasticity Sensitivity The form of the equations are the same as for sensitivity analysis of static responses 3 40 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran K u P Equation 3 88 where for static aeroelastic analysis a a a Ky Kj KytK MjD M Kn T T K D Mi M D Mi dr M 0 0 T a a T a T a a D Ki u Ku D Kr id Eg m D Kix T Kpy 0 0 0 0 uy u u a u ux P 0 Py i P D P P Equation 3 89 This equation is differentiated with respect to a design variable and solved for the displacement sensitivities using a semianalytic approach as has been described in Eq 3 46 for static sensitivity analysis Flutter Sensitivity Analysis The eigenvalue problem for flutter is given by l 2 T Onn 1 R E Mane fu per eM m j zP on 1 Uys 105 Equation 3 90 where the terms in the above equation are defined in the NX Nastran Aeroelastic Analysis User s Guide Design Sensitivity and Optimization User s Guide 3 41 Chapter 3 Design Sensitivity and Optimization in NX Nastran Flutter sensitivity computes the rates of change of the transient decay rate coefficient y with respect to changes in the design variables y is defined in connection with the complex eigenvalue p p o y 1i DPR P Equation 3 91 As with the other analys
229. emens AG NX is a trademark or registered trademark of Siemens Product Lifecycle Management Software Inc or its subsidiaries in the United States and in other countries NASTRAN is a registered trademark of the National Aeronautics and Space Administration NX Nastran is an enhanced proprietary version developed and maintained by Siemens Product Lifecycle Management Software Inc MSC is a registered trademark of MSC Software Corporation MSC Nastran and MSC Patran are trademarks of MSC Software Corporation All other trademarks are the property of their respective owners TAUCS Copyright and License TAUCS Version 2 0 November 29 2001 Copyright c 2001 2002 2003 by Sivan Toledo Tel Aviv University stoledo tau ac il All Rights Reserved TAUCS License Your use or distribution of TAUCS or any derivative code implies that you agree to this License THIS MATERIAL IS PROVIDED AS IS WITH ABSOLUTELY NO WARRANTY EXPRESSED OR IMPLIED ANY USE IS AT YOUR OWN RISK Permission is hereby granted to use or copy this program provided that the Copyright this License and the Availability of the original version is retained on all copies User documentation of any code that uses this code or any derivative code must cite the Copyright this License the Availability note and Used by permission If this code or any derivative code is accessible from within MATLAB then typing help taucs must cite the Copyright and type taucs must also cite
230. ensional search is complete the B matrix is updated using the BFGS formula T T T T ip Blips ip ini Equation D 72 where py Pu n Of y 1 0 1Blp q q 1 vr Vx Vx o FR Y 4g j 1 1 0 ue 08 10 TBIip if p y 2 0 2 p Bl 0 tpi Bl tp pl y if p y lt 0 2 p B p Now B replaces B and a new iteration is begun D 28 Design Sensitivity and Optimization User s Guide Numerical Optimization D 9 Summary The purpose of this Appendix has been to provide an overview of the computational details of the optimization process The techniques used in the optimizer have been chosen because they are reasonably robust and because they allow for considerable flexibility in formulating the design problem From the variety of decisions that must be made during the optimization process it is clear that anything the user can do to improve the numerical conditioning of the problem will have a strong influence on the efficiency and reliability of the result It is hoped that some understanding of the actual optimization process will assist in this Design Sensitivity and Optimization User s Guide D 29 About Siemens PLM Software Siemens PLM Software a business unit of the Siemens Industry Automation Division is a leading global provider of product lifecycle management PLM software and services with 7 million licensed seats and 71 000 customers worldwide Headquartered in Plano Texas Siemens PLM
231. ensitivity and Optimization in NX Nastran The design model definition is given in Listing 3 2 The analysis model contains ten independent PBAR entries since all ten bar sections in the model are to vary SPBAR PID MID A Il I2 PBAR 101 100 5 E 5 4 167 10 1 042 10 PBAR 102 100 5 E 5 4 167 10 1 042 10 PBAR 103 100 5 E 5 4 167 10 1 042 10 PBAR 104 100 5 E 5 4 167 10 1 042 10 PBAR 105 100 5 E 5 4 167 10 1 042 10 PBAR 106 100 5 E 5 4 167 10 1 042 10 PBAR 107 100 5 E 5 4 167 10 1 042 10 PBAR 108 100 5 E 5 4 167 10 1 042 10 PBAR 109 100 5 E 5 4 167 10 1 042 10 PBAR iTO 100 5 E 5 4 167 10 1 042 10 INDEPENDENT DESIGN VARIABLE SET SDESVAR ID LABEL XINIT XLB XUB DELXV DESVAR 1 Xi 1 0 1 0 1 0 DESVAR 2 X2 0 0 1 0 1 0 DESVAR 3 X3 0 0 1 0 1 0 DEPENDENT DESIGN VARIABLE SET SDESVAR ID LABEL XINIT XLB XUB DELXV DESVAR 11 Hl 01 1 E 4 1 0 DESVAR 12 H2 4045 L E 4 1 0 DESVAR 13 H3 OL L E 4 1 0 DESVAR 14 H4 03 E 4 1 0 DESVAR 15 H5 Di E 4 1 0 DESVAR 16 H6 01 1 E 4 1 0 DESVAR 17 H7 OL 1 E 4 1 0 DESVAR 18 H8 201 1 E 4 1 0 DESVAR 19 H9 01 1 E 4 1 0 DESVAR 20 H10 201 1 E 4 1 0 DESIGN VARIABLE LINKING Hj Hj X1 X2 X3 j 11 20 SDLINK ID DDVID CO CMULT IDV1 CL IDV2 G2 4 IDV3 3 DLINK 21 Ti P 01 d 1405 2
232. er s Guide 3 43 Chapter 3 Design Sensitivity and Optimization in NX Nastran The modes for the fluid are computed under the effects of rigid wall boundary conditions which eliminate the structural coupling effect The resulting modal transformation for the fluid degrees of freedom is then P 0 187 nd PA MyM T kd Ded IK 3104 Equation 3 98 where Ds the modes for structure in a vacuum After modal reduction for both structural and fluid degrees of freedom the equations of motion can be written as m A E la k i o 47 o S _ o E T E D A my Es 0 k E DT Fr Equation 3 99 This modal transient formulation can be differentiated with respect to a design variable to obtain the equations for coupled fluid structure interaction sensitivity The form of the equations is similar to Eq 3 80 for modal transient analysis so it is not repeated here However the solution process is identical If the excitation is frequency dependent rather than time dependent the resultant sensitivity equations assume the form given by Eq 3 77 for structural problems Due to the similarities of their equations the solution process is again identical 3 3 Optimization with Respect to Approximate Models This section describes the process of optimization with respect to approximate models in NX Nastran Many of these operations can be controlled with various parameter and Bulk Data selections To briefly revi
233. erpretation of the Fletcher Reeves Method The advantage of using the conjugate search direction is seen from Figure D 4 and Figure D 5 which show a simple two variable design space In Figure D 4 the search directions are the steepest descent directions whereas in Figure D 5 the conjugate directions are used Note that in Figure D 4 the search direction is always perpendicular to the previous direction On the other hand in Figure D 5 each search direction uses the steepest descent direction plus some fraction of the previous direction This makes physical sense If progress is made in direction S1 it is reasonable to bias direction S a bit in that direction rather than simply using a steepest descent direction This is exactly what Eq D 27 and Eq D 28 accomplish This algorithm is extremely simple but as seen from Figure D 4 and Figure D 5 it dramatically improves the rate of convergence to the optimum As noted earlier there are other methods called variable metric methods that are considered better than the one just described In fact NX Nastran uses the BFGS Broydon Fletcher Goldfarb and Shanno methodf 7 8 9 Also the geometrical interpretation is almost exactly the same as the Fletcher Reeves algorithm Therefore there is no loss in the physical interpretation of the methods However the details of the method are beyond the scope of this discussion A key issue to note is that the steepest descent method which even today is
234. es however are implemented as equivalent constraints For the j th property in the design model the equivalent lower and upper bound constraints are given by L gt P Pj gt gj X WW S Pj gt gt P X pj gp X gt U lt 0 Pj Equation 3 108 where the lower and upper bounds p and pj are given by Eq 3 105 and Eq 3 107 Eq 3 108 is necessary because the optimizer modifies design variables and not properties Of course the property values are known from the DVPREL1 and DVPREL2 relations For any given set of design variables the optimizer can use Eq 3 108 to determine whether or not the corresponding properties are within their allowable limits Relaxation of Move Limits Since a small numerical constraint violation is allowed by the optimizer by default move limits on properties may not be satisfied with equality at their bounds This is usually of little concern except when the design is at PMIN or PMAX in Eq 3 107 Occasionally these bounds might be slightly violated in a numerical sense but usually this is of little consequence A puzzling situation may occur if the current design is infeasible If this is the case the optimizer s primary task is to reduce the level of constraint violation to find a feasible solution if one exists Since the move limits on properties are implemented as equivalent constraints the optimizer does not know the difference between response type and move limit type constraints T
235. es a change in grid location The characteristics of the shape changes depend on the shape basis vectors DESVAR entries 1 and 2 just define the presence of basis vector multipliers that the optimizer is free to vary The bounds on the design variables given here allow up to a 200096 change The basis vector magnitudes will govern the corresponding magnitude of shape change Direct Input of Shapes Direct input of shapes uses an external auxiliary model to generate shape basis vectors The auxiliary model is used to generate a set of displacement vectors that are then DBLOCATEd and identified as shape basis vectors in the optimization run All data for the auxiliary model analysis must be provided for by the designer in a prior NX Nastran run Figure 2 17 shows the auxiliary model and its boundary conditions Its geometry and connectivity are the same as for the primary structure The material types are also the same although this need not be the case Note that the chosen boundary conditions are consistent with our shape redesign goals outer edges fixed symmetry along the x and y axes and a free edge along the hole boundary Figure 2 17 External Auxiliary Model Along the cutout we will apply two sets of enforced displacements one an elliptical variation in the a or x direction the other an elliptical variation in the b or y direction We can use a geometry based preprocessor to give us this data or for simple cases we can just
236. es and their time derivatives is available the displacement derivatives are computed from Eq 3 81 A Special Case of Frequency Response Sensitivities Equivalent Radiated Power Response Sensitivities Equivalent radiated power ERP is an output option used to quantify the noise level that radiates from a vibrating surface ERP output is a measure of the normal frequency response velocity over the face of elements that comprise a panel by summing the elemental responses for all the elements that comprise the panel 3 38 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran The equivalent radiated power attributable to the ith element of a panel at frequency w is given by ERPE o mle c v x v x e ds i Equation 3 83 where C Scaling coefficient that is usually taken to be the product of density and speed of sound in the fluid medium y x 0 Normal velocity as a function of position at frequency w v x 0 Complex conjugate of the normal velocity as a function of position at frequency W Sj Surface area of the ith element Adjoint Load Method for ERP Sensitivity Analysis When equivalent radiated power ERP is used as a design response a single composite adjoint load for the entire panel is created so that the sensitivities derivatives are obtained directly The expression for the total derivative of the element ERP magnitude ERPE w with respect to
237. es can be functions of design variables table constants first level responses and grid coordinates That is Equation 3 24 Chain rule differentiation of Eq 3 24 with respect to the i th design variable gives using to denote partial differentiation with respect to the design variables appearing explicitly in rj 3 20 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran m zd ar a or D ori Ox P 251 G l y 9 dt Equation 3 25 where x is used here to indicate either a design variable or a design property DVPREL2 property see Eq 3 22 or Eq 3 23 In NX Nastran the terms in Eq 3 20 through Eq 3 25 are computed as follows o CO orc J J Ox Op 1 These quantities are available directly from the semi analytic sensitivity analysis of of or o O 2 J J Depending on whether is an objective or constraint these terms are available as objective J esj a 2 or Or J J Equation 3 26 constraint Ee O e lower bound constraint 1 2 L Or Or py J J J Of uncus ou ll 5 0 T upper bound constraint J J J Equation 3 27 2 or Ox 3 This term is approximated using central differences and the equation solving utility in NX Nastran as SO Ar OP Ax Oa Ax os Ja A J bx AX 2Ax Equation 3 28 where Ax FDCH x with FDCH 1 0E 3 currently in the program D
238. es to select response components can be found in the NX Nastran Quick Reference Guide These are also often referred to as plot codes Element level responses can be selected using either element IDs much like the list of grids in the previous example or property IDs By supplying a property ID on a DRESP1 entry we identify the element level response for every element in that property group For example the following selects the axial stress response for all ROD elements in property groups 150 160 170 and 180 SDRESP1 1D LABEL RTYPE PTYPE REGION ATTA ATTB ATT 4 ATT2 DRESP1 250 SIG1 STRESS PROD 2 160 LTO 180 This can generate quite a lot of design data Figure 2 23 is a schematic diagram of this hierarchy If limits are subsequently placed on these responses using a DCONSTR entry discussed in Defining the Constraints the result will be a pair of stress constraints one for the upper bound Design Sensitivity and Optimization User s Guide 2 43 Chapter 2 Design Modeling for Sensitivity and Optimization 2 44 and one for the lower bound for every element in each of these property groups By using just a pair of entries we have been able to define potentially hundreds of stress constraints EID1 EID2 PID1 PID2 EIDn DRESP1 PIDm Figure 2 23 Identification of Element Level Responses Further automatic generation of DRESP1 data is possible in Frequency Response or Transient Response anal
239. ese bounds are determined at the beginning of each design cycle according to xi x DELX DELXV Y x x DELX DELXV 0 X i Equation 3 101 where x is the initial value of the i th design variable The allowable percentage change in the design variable is supplied by DELXV on the DESVAR Bulk Data entry This value is optional and if not supplied defaults to DELX which is provided on the DOPTPRM Bulk Data entry The DELX default is 1 0 or a 10096 change in all design variables 3 46 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran Note If a DELXV is not present the global parameter DELX provides move limits for all design variables If a DELXV is provided then it takes precedence by providing a specific move limit for a given design variable We have a similar situation for designed properties where each is bounded from above and below by Equation 3 102 where p is the lower bound on the j th property and pj is the corresponding upper bound These bounds are based on percentage changes in the property value pp at the outset of the approximate optimization These bounds are computed as pi p py DELP p py p DELP Equation 3 103 where DELP is the maximum allowable percentage change in the property DELP is defined on the DOPTPRM entry and has a default of 0 2 for a 20 maximum property variation Note
240. esign problem itself For example since the goal here is weight minimization the shape basis vectors must clearly be able to reduce the volume of the structure Were they to simply redistribute material the weight would not change and the optimizer would be able to make little progress Furthermore since the redesign is subject to von Mises stresses we can use the initial stress distribution to help us in the selection of appropriate basis vectors This is also shown in Figure 7 6 Here we note that the maximum von Mises stresses occur around the lower portion of the circular hole The stresses are not equally distributed either This tells us that a circular boundary does not provide the optimal design Thus our shape basis vectors must contain other than just radial components We also need to decide how many design variables or shape basis vectors are needed The decision is not unique because the selection can be affected by experience various functional and manufacturing requirements or aesthetic requirements However the shape basis vectors should yield as much generality as possible consistent with our design goals 7 24 Design Sensitivity and Optimization User s Guide Example Problems Creation of Auxiliary Model and Generation of Basis Vectors The top left frame in Figure 7 7 shows the auxiliary model geometry and boundary conditions Note that it has the same geometry as the primary structure Figure 7 6 but with different
241. esign Modeling for Sensitivity and Optimization b 2 C C h 2 Equation 2 17 Stress Recovery Locations Words of Warning Note the error that may have resulted if the location of the stress recovery point had not been allowed to vary Since the stress computations would not have included the direct effects of changes in the stress recovery point the stresses seen by the optimizer would not be those that the engineer had intended This would certainly lead to incorrect results Of course other stress recovery points would probably also need to be defined to ensure recovery of the maximum stresses Special Design Modeling Considerations BEAM Elements In design modeling the BEAM element differs from other element types in that its properties must be referenced according to their word positions in the element property tables The element property tables are internally generated in NX Nastran and used to form the element level structural matrices Note By convention field position identifiers are denoted as positive quantities on DVPRELi entries EPT word positions are denoted as negative quantities The source of this difference is due to the generality of the PBEAM Bulk Data entry Since a single entry can used to specify the BEAM s cross sectional properties for both ends and up to nine intermediate stations a unique correspondence between a BEAM property and its field position on the PBEAM entry is not assured T
242. esign Sensitivity and Optimization User s Guide Output Features and Interpretation Ck Ck ck KKK KKK KKK KK k k k k ck KKK k k k k K Ok k k k Ck k k k k K k ko k k k kk Kok kk TERMINATION OF DESIGN ITERATION SOFT CONVERGENCE kk koh kk kk kk ok kk ke ke ee eee ee kk ke ke ko kk kk ke kk ke ke koe RELATIVE CHANGE IN OBJECTIVE 2 6376E 06 MUST BE LESS THAN 0000E 03 ABSOLUTE CHANGE IN OBJECTIVE 2 0981E 05 MUST BE LESS THAN 0000E 02 MAXIMUM OF RELATIVE PROP CHANGES 0 0000E 00 MUST BE LESS THAN 0000E 03 AND MAXIMUM CONSTRAINT VALUE 4 1453E 04 MUST BE LESS THAN 0000E 03 OR MAXIMUM OF RELATIVE D V CHANGES 0 0000E 00 MUST BE LESS THAN 0000E 03 EXPLANATION THE OPTIMIZATION PROCESS WITH RESPECT TO THE APPROXIMATE MODELS DID NOT CHANGE APPRECIABLY THIS DESIGN MAY NOT WARRANT AN ADDITIONAL COMPLETE FINITE ELEMENT ANALYSIS Since SOFTEXIT is NO this test will not terminate the design cycle process Rather another finite element analysis is performed after which hard convergence is indicated Oc NORMAL CONVERGENCE CRITERIA SATISFIED HARD CONVERGENCE DECISION LOGIC FAO OOOO OOOO CONVERGENCE ACHIEVED BASED ON THE FOLLOWING CRITERIA HARD CONVERGENCE DECISION LOGIC RELATIVE CHANGE IN OBJECTIVE 0 0000E 00 MUST BE LESS 0000 03 ABSOLUTE CHANGE IN OBJECTIVE 0 0000E 00 MUST BE LESS 0000E 02 22 AND MAXIMUM CONSTRAINT VALUE 4 1453E 04 MUST BE LESS 0000E 03 CONVERGENCE TO A FEASIBLE DESIGN ane Risas MAXIMUM OF
243. esign Sensitivity and Optimization User s Guide 3 21 Chapter 3 Design Sensitivity and Optimization in NX Nastran and if FDCH x FDCHM with FDCHM 1 0E 6 currently in the program 2 Or E 4 k This term is also approximated using central differences and the equation utility as 2 1 1 2 f 1 1 of aD UP ean QR an QE od a DO Or Ar 2Ar Equation 3 29 where Arj 1 is determined using FDCH and FDCHM as in item 3 2 a 5 GI This term is approximated using central differences as ar AD HP AG r G AG 01Gj AG AG Equation 3 30 where AG is the variation in the coordinates of the th grid point and GO represents the baseline coordinates of those grids used in the expression for r a AG is determined as AG FDCH G i i sl Equation 3 31 where FDCHM is used for any component that is less than this amount p a This information is available directly from the basis vectors for shape optimization Note that each term of a basis vector defines the constant rate of change of a grid coordinate given a change in a design variable 3 22 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran 3 2 Design Sensitivity Analysis Design sensitivity analysis computes the rates of change of structural response quantities with respect to changes in design variables Since these partial derivatives provide the e
244. establishes a numerical constraint boundary that is not identical to the exact constraint boundary Feasible lt Violated Active g x CTMIN 0 003 0 X lt Numerical Constraint ES Boundary True Constraint Boundary CT 0 03 Figure 3 11 CT and CTMIN Optimizer Convergence Parameters In this section we describe parameters that you can modify to affect convergence although experience indicates that there is seldom a need to vary these parameters All of these may be changed using the DOPTPRM entry see the NX Nastran Quick Reference Guide for more information Parameters related to convergence at the optimizer level include the following Absolute change in the objective to denote convergence at the optimizer level PABOR A DELOBJ Relative change in the objective function to denote convergence at the optimizer BEEN The combination of DABOBJ and DELOBJ is used to identify the diminishing returns associated with convergence at the optimizer level Maximum number of iterations allowed within the optimizer default 40 This ee is provided so that poorly converging problems are caught The number of consecutive iterations for which the DABOBJ DELOBJ LANE s criteria must be satisfied before convergence is indicated The number of consecutive iterations for which the convergence criteria must be satisfied to indicate convergence in the sequential linear programming method This is equi
245. et on the DOPTPRM entry this parameter provides an upper limit on design cycles unless convergence is achieved before then DXMIN Minimum design variable move limit This absolute quantity ensures that earl move limits exist for numerically small design variables This can be changed using the DOPTPRM entry IPRINT Indicates the level of diagnostic printout from the approximate optimization ee pe The default is O no printout and can range in value from O to 7 See Table 1 on the DOPTPRM entry METHOD Indicates whether the modified method of feasible directions METHOD 1 LN linear programming METHOD 2 or sequential quadratic PMAX DVPREL2 entry The set of all PMAX values acts as upper bounds on properties during the approximate optimization Design Sensitivity and Optimization User s Guide 3 45 Chapter 3 Design Sensitivity and Optimization in NX Nastran Minimum allowable property value specified on either a DVPREL1 or DVPREL2 entry The set of all PMIN values acts as lower bounds on properties during the approximate optimization Lower and upper bounds on a single design variable provided on the DESVAR entry The optimizer will never propose a design that is outside of these bounds In addition to the above quantities there are a number of internal optimizer parameters that may be changed as well A complete listing of these parameters is in Table 1 of the DOPTPRM entry Generally there is little reason to modify the
246. ew once the finite element and sensitivity analyses are performed for the current design cycle all the necessary information is available from which to construct the approximate model This approximation is in turn used by the optimizer to search for a corresponding approximate optimum There are a number of parameters that enable you to exercise some control over the optimization phase of the design cycle This section discusses these quantities with an emphasis on their relation to the overall process as well as guidelines for selecting default value overrides 3 44 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran Approximate Optimization Control Parameters An Overview The modification of some of the quantities used to guide the approximate optimization may bring an increase in numerical efficiency an enhanced usability of the final design or a closer look at the optimization process by increasing the diagnostic printout There are a number of such quantities that you may want to override Some appear on the Bulk Data entries for design optimization with the majority of the others available on the DOPTPRM Bulk Data entry Note Since the DOPTPRM entry is optional every value on it has a default Thus the use of this entry is always associated with some type of override The more common quantities their basic function and methods for accessing them are listed below
247. example introduces most of the frequently encountered types of design optimization output within the context of an actual problem Following in Design Sensitivity Output is a similar output example for design sensitivity analysis 6 1 Output Controlling Parameters Briefly those quantities that affect either the frequency or level of detail of the output are as follows Controls the frequency of DESVAR and GRID Bulk Data entry output to the punch file The default is for the final design cycle only but it can be changed to every n th design cycle plus the last with this parameter See Parameters for Design Sensitivity and Optimization Controls the level of printout available directly from the optimizer lts value which may range from 0 to 7 is set on the DOPTPRM entry An increasing value provides levels of detail See the DOPTPRM entry description in Bulk Data Entries Governs the frequency of NX Nastran analysis results output Output may be requested for the first and last designs only the first and every n th design or turned off completely with this Bulk Data parameter See Parameters for Design Sensitivity ptimization When set to a value of 4 or 7 provides output of the design sensitivity coefficient matrix DSCM2 This Bulk Data parameter may assume any value from 1 to 7 indicating exit at any one of seven predefined locations in the solution sequence However only OPTEXIT 4 and 7 generate DSCM2 design sensitivity coefficient out
248. explained in the section on special modeling considerations at the end of this section To illustrate both type 1 and type 2 design variable to property relations consider the stiffened panel section of Figure 2 4 The example is test problem library example number D200X7 presented in Stiffened Plate Three design variables are each related to the plate thickness web thickness and cross sectional area of the web cap respectively By grouping all of the elements in the base plate into one property group their thickness can all be controlled by a single design variable This same grouping is also used for the elements of the web as well as for the cap section elements Design Sensitivity and Optimization User s Guide 2 11 Chapter 2 Design Modeling for Sensitivity and Optimization Web Cap g 10x 4 Plate Figure 2 4 Stiffened Panel Test Problem The base plate thickness is linearly related to a design variable according to Equation 2 16 This linear relation can be defined with a DVPREL 1 Bulk Data entry as follows PSHELL 1 I 0 15 I Define the design variables SDESVAR ID LABEL XINIT XLB XUB DELXV DESVAR 1 T PLATE 0 15 0 001 1000 S Relate the design variables to analysis model properties linear relations so use DVPREL1 SDVPREL1 ID TYPE PID FID PMIN PMAX co COEF1 DVID2 COEF2 PSHELL 1 4 0 01 O The PSHELL 1 entry defines the analysis model property gro
249. f 1 0 Since the lower bound is not of interest e g tensile stresses on the elements will not induce Euler buckling we can just set it to a large negative number on the DCONSTR entry say 1 0E10 Eight independent design variables DESVAR 1 through 8 are used to describe eight ROD cross sectional areas on DVPREL1 entries 1 through 8 Axial stresses and grid displacements are identified on DRESP1 entries 1 through 14 and are constrained using DCONSTR entries with an I D of 10 Additionally the axial rod stresses are used as input in the definition of the Euler buckling responses DRESP2s 16 through 23 which all reference DEQATN 1 Note that element lengths are included in these relations via constants defined on the DTABLE entry This simplifies the input since the element lengths do not need to be hard coded on eight different DEQATN entries Turning to the superelement analysis model in Case Control the SUPER ALL command is recommended rather than an explicit data recovery subcase for each superelement In Bulk Data the SESET has been used to place grids 1 and 2 on the interior of superelement 1 As far as the superelement aspects of the design model are concerned the only necessary addition is in field 9 are of the DRESP1 number 15 entry Here ALL has been used to indicate the total structural weight across all superelements be computed and used as the objective Function The default zero value for this field would have computed the
250. f end B data Your design model must then reference words 6 21 for end A words 166 181 for end B and the first intermediate station The most general case is shown in item 3 where in addition to ends A and B from one to nine intermediate stations are defined As in item 2 the first available intermediate station always contains a copy of end B data This must always be accounted for in the design model definition Note that if nine intermediate stations are defined there is no need to supply an additional copy of end B data since the next station is end B itself To illustrate consider the uniformly tapered beam element in Figure 2 7 The cross sectional dimensions for a rectangular element can be described in terms of design variables b 4 h 4 b 2 and h 2 as shown in the figure An input data file using this element in a single element design sensitivity test is given in Listing 2 1 This example corresponds to a test of Item 2 in Figure 2 6 Design Sensitivity and Optimization User s Guide 2 17 Chapter 2 Design Modeling for Sensitivity and Optimization E E S Figure 2 7 Tapered Rectangular Beam Element The initial design is given by i 1 0 Ll a b 0 5 1 0 h Equation 2 18 This design is input to the DEQATN entries to define the corresponding cross sectional properties for the element In addition to the properties at ends A and B the design model must also include a definition for the first intermediate
251. f these displacement components as representing individual components of grid motion we realize that the U vectors can be quite conveniently used as basis vectors for shape optimization For example consider the cantilever beam of Figure 2 11 The analysis model is clamped along the left hand edge and tip loaded on the right Suppose we would like to redesign the shape of the structure by modifying the profile of the lower edge In order to do so we might like to use an auxiliary model to generate a few characteristic profiles and use these as our shape basis vectors Figure 2 11 Simple 2 D Cantilever Figure 2 12 shows the corresponding auxiliary model and its boundary conditions exclusive of the loading The fixed edges A and B guarantee that the left hand support and the horizontal upper edge will not change during shape optimization The roller supports along edge C will allow the depth of the beam to change without changing its length We can now load edge D any way we want to produce a range of shapes or basis vectors D Figure 2 12 Auxiliary Model for the Simple 2 D Cantilever A uniform taper component might be generated by applying SPCDs as shown in Figure 2 13 Here the displacements have been plotted to show the basis vector characteristics Visualization of these basis vectors prior to their use in connection with shape optimization is highly recommended An altern
252. fficiencies in First Level Response Identification Type 2 Responses DRESP2 and DRESP3 Bulk Data entries define type 2 or second level responses These responses are also referred to as user defined responses because the user utilizes either the built in mathematical operations in NX Nastran or an external program to define design responses from combinations of first level responses design variables grid coordinate values and table constants DRESP2 Bulk Data entries reference mathematical operations that are built into NX Nastran A few built in mathematical operations are available through the FUNC field of a DRESP2 Bulk Data entry A more comprehensive library of built in mathematical operations is available through the EQID field of a DRESP2 Bulk Data entry The EQID field references a DEQATN Design EQuATION Bulk Data entry With a DEQATN Bulk Data entry the user can define a response using algebraic trigonometric hyperbolic and logarithmic functions For example the user can define design responses like error functions and stress averages Design Sensitivity and Optimization User s Guide 2 41 Chapter 2 Design Modeling for Sensitivity and Optimization DRESP3 Bulk Data entries reference external programs With external programs the user might define for example design responses where o The evaluation is not direct but may require an iterative solution as in Newton s method to arrive at the value for the design respons
253. fied on DVPREL1 entries 1 through 8 that define t 1 0x i 1 8 Equation 7 20 Thus all thicknesses are functions of design variable number 1 defined on the DESVAR 1 entry The minimum allowable thickness is defined by the default value of PMIN 0 001 on the DVPREL1 entry This also agrees with the design variable lower bound supplied on the DESVAR entry Changing the individual ply thickness will cause the overall plate thickness to vary Zo or the distance from the reference plane to the bottom surface must also be changed accordingly The relation supplied on DVPREL 1 100 ensures this by defining Zq x1 E 81 04 Equation 7 21 where f is given by Eq 7 20 Ply orientations are modified using DVPREL1 entries 11 through 18 which express the ply angles as functions of design variables 2 and 3 Negative values of PMIN have been used in DVPREL 1 entries 12 14 16 and 18 to allow for ply angles that are less than zero The design constraints on ply failure are given by the DCONSTR entry set 20 that references the ply failure responses defined on DRESP1 entries 1 through 8 This DCONSTR entry set is then identified by the DESSUB in the Case Control section This constrains the failure modes for all plies in element 64 Including other element stress constraints was unnecessary due to the symmetry of the structure and the implicit linking of the design parameters The CFAILURE in field 4 of each of the DRESP1 entries indic
254. fine the bar element heights i e nonnegative physical dimensions their lower bounds are specified as small positive numbers to avoid negative bar cross sectional heights Note The optimizer never proposes a design that is outside the bounds placed on the design variables This often results in a more effective method of applying bounds than is PMIN or PMAX on the DVPRELi Bulk Data entries The dependent variables are related to the independent set on DLINK entries 21 through 30 Each DLINK entry defines a row of the reduced basis matrix similar to the DVPREL1 entries of the previous example This dependent set of design variables is input to equations in DVPREL2 type relations DVPREL2 entries 41 through 50 specify the area moments of inertia about axis 1 for each of the bar element properties as indicated by the 5 in field 5 Note that they all reference the same DEQATN entry 40 containing the functional relation between h and 4 Each of the DVPREL2 continuations describe the equation input arguments consisting of just a single design variable The 1 0E 12 value for PMIN in field 6 ensures that the property does not take on values less than this bound For brevity the equations for areas and and DEVPREL2 for areas and for each property group have not been listed since the method used to define these relations is identical to that shown here Without relating the design variables to changes in bar areas weight minimization
255. fined in Bulk Data with DRESP1 or DRESP2 2 4 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization DESSUB Selects constraints to be applied at a particular subcase level from a DCONSTR set defined in Bulk Data DESOBJ and DESSUB are discussed in greater detail in Glossary of Terms Case Control Output Requests in Design Optimization In addition to the analysis related Case Control commands output requests might also appear such as displacement all strain fiber all Note The frequency of data recovery output e g first design cycle last every n th etc is governed by the Bulk Data parameter NASPRT See Parameters for Design Sensitivity and Optimization for details These are of interest to design optimization in a couple of ways n general case control requests are for data recovery only strictly speaking they are unnecessary for design optimization All necessary data recovery is automatically performed based on the design responses identified in the Bulk Data identifying the responses is discussed in Identifying the Design Responses You do not need to explicitly request this data recovery unless you want to view the results f various output forms are possible e g strain strcur or strain fiber design optimization will use the defaults unless a Case Control request to the contrary is provided To use the preceding example STRAIN STRCUR represen
256. for example will not be possible Also if stresses due to bending are in the design model the dependence of the stress recovery locations on the design variables should also be defined Design Sensitivity and Optimization User s Guide 3 11 Chapter 3 Design Sensitivity and Optimization in NX Nastran Constraint Screening Constraint screening is an automatic process for both design sensitivity and optimization and consists of two parts regionalization and deletion You can override some of the defaults associated with this process by e Providing explicit region IDs on the DRESP1 and DRESP2 entries Using DSCREEN entries to override key screening parameters There is often little need to override the constraint screening defaults since these perform well in the vast majority of cases However computational efficiency can sometimes be gained by modifying these defaults in special cases This section outlines constraint screening so that you can make effective use of it should the occasion arise Basic Concepts The basic assumption behind constraint screening is that most structural optimization problems contain more constraints than are necessary to adequately guide the design For example suppose bending stresses for every beam element are required to be less than some allowable upper limit Not every element in the model may be critically loaded while those that are may undergo more or less similar changes as the design is modified
257. for this property If FID references a stress recovery location field then the default value for PMIN is 1 0 35 PMIN must be explicitly set to a negative number for properties that may be less than zero for example field ZO on the PCOMP entry Real Default 0 001 Maximum value allowed for this property Real Default 1 0E20 DEQATN entry identification number Integer gt 0 DESVAR flag Indicates that the IDs of DESVAR entries follow Character DESVAR entry identification number Integer gt 0 DTABLE flag Indicates that the IDs for the constants in a DTABLE entry follow This field may be omitted if there are no constants involved in this relation Character Label for a constant on the DTABLE entry Integer gt 0 Associated Entries To define a synthetic property relation the DVPREL2 entry identifies a DEQATN entry and declares the design variable DESVAR and any necessary table constant DTABLE arguments EQUATION EQUATION Cont o o 2 I ME A z y gt Ww E m Discussion VALU1 VALU2 VALU3 VALU4 For many applications the linear design variable to property relations provided by the DVPREL1 entry may not be sufficient In these cases equations may be defined to relate design variables to properties in a nonlinear fashion The DVPREL2 entry provides the arguments to an equation defined A 22 Design Sensitivity and Optimization User s Guide Input Data on a DEQATN entry These
258. g sizes and so on In civil engineering we may be interested in how changes in the deflection of a bridge span can be affected by changes in the dimensions of the bridge sections In automotive design we may want to investigate changes in cabin resonant frequencies given changes in panel thicknesses These rates of change what we call partial derivatives in the language of calculus are called design sensitivity coefficients Design optimization refers to the process of generating improved designs In NX Nastran design optimization is performed by an optimizer An optimizer is really nothing more than a formal plan or algorithm that is used to search for a best design Design sensitivity coefficients are used in NX Nastran to assist the optimizer in this search process Once these rates of change are known the optimizer can for example find the optimal set of panel thicknesses that yield the lowest level of cabin resonant frequencies Why Use Design Sensitivity and Optimization Design sensitivity and optimization are used when we seek to modify a design whose level of structural complexity exceeds our ability to make appropriate design changes What is surprising is that an extremely simple design task may easily surpass our decision making abilities Experienced designers those with perhaps decades of experience are sometimes fantastically adept at poring through mounds of data and coming up with improved designs Most of us however
259. gn Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization The EVALUATOR field on the CONNECT statement specifies the alias for the executable file In the following example the group name of TESTGRP on the CONNECT statement is used in the GROUP field of both DRESP3 Bulk Data entries Therefore both DRESP3 entries will use the same executable file However the DRESP3 Bulk Data entry with 31 as the ID uses EULER as the response type and the DRESP3 Bulk Data entry with 32 as the ID uses JOHNSON as the response type These different response types refer to two different response calculations in the executable file The results of both DRESP3 responses in this example are then used in design constraints CONNECT DRESP3 TESTGRP EXTRESP SOL 200 DCONSTR 1 DCONSTR 1 DESVAR 1 0 0 01 DRESP1 STRESS PBAR DRESP3 1 TESTGRP EULER E TESTGRP JOHNSON SIGMAC Define the alias for the executable file Create an evaluator path file that relates the alias name in the EVALUATOR field of the CONNECT statement to the actual path of the executable file For the above example EXTRESP is the alias and dr3serv is the executable file Thus the evaluator path file would include the following line of text EXTRESP network_drive user_dir dr3srv dr3serv Use the GMCONN keyword to reference the evaluator path file You assign the GMCONN keyword to the name of the evaluator p
260. guments are defined via DRESP1 entries and belong to the same superelement If any of these criteria are violated a fatal error message is issued Superelements and Constraint Screening For superelement sensitivity the constraint screening criteria are applied on an individual superelement basis That is if a DRESP1 entry lists several element grid or property IDs that span superelement boundaries the screening criteria will apply individually to all superelements that share responses listed on this DRESP1 entry In other words regions are subsetted by superelement If separate sets of constraint screening parameters TRS truncation threshold and NSTR number of retained constraints per region are required for different superelements then separate sets of DRESP1 and DCONSTR entries should be used to define superelement specific response groups Case Control The simplest way to incorporate superelements is to use the SUPER ALL command in Case Control This automatically satisfies a number of design optimization Case Control requirements You can also use an expanded subcase structure if necessary with each subcase pertaining to a superelement or group of superelements If you choose the latter approach there are some additional requirements that must be satisfied Each superelement subcase must define an analysis type using the ANALYSIS command For ANALYSIS MODES the eigenvalue constraint must be called out from each MODES s
261. h represents the move required to reach the best design possible for this particular direction This value is defined as a The new objective and constraints can now be expressed as 0 1 FL SFG rasy Equation 1 13 0 31 g g x PES T9 1 e y Equation 1 14 From this new point in the design space we can again compute the gradients and establish another search direction based on this information Again we will proceed in this new direction until no further improvement can be made repeating the process if necessary At some point we will not be able to establish a search direction that can yield an improved design We may be at the bottom of the hill or we may have proceeded as far as possible without crossing over a fence In the numerical search algorithm it is necessary to have some formal definition of an optimum Any trial design can then be measured against this criteria to see if it is met and an optimum has been found This required definition is provided by the Kuhn Tucker conditions that are physically quite intuitive 1 14 Design Sensitivity and Optimization User s Guide Getting Started Convergence to an Optimum The Kuhn Tucker Conditions Y g l Figure 1 4 shows a two design variable space with constraints 819 and 82 and objective function EG The constraint boundaries are those curves for which the constraint values are identically zero A few contours of constant objective are shown as well t
262. h an analysis of the model to gather baseline response data A DMAP information message from subDMAP FEA serves notice that a static analysis has been initiated DMAP INFORMATION MESSAGE 9051 FEA STATIC ANALYSIS INITIATED DESIGN CYCLE NUMBER 1 Since this is the first analysis with the parameter NASPRT at its default value of zero full data recovery is performed based on Case Control output requests Since displacements and stresses have been requested we see the familiar NX Nastran static analysis output abbreviated here LOAD CONDITION 1 SUBCASE 1 DISPLACEMENT VECTOR POINT ID TYPE T T2 R1 R2 R3 10000 G 0 0 927221E 02 565913E 03 10001 344270E 03 1 081616E 04 904300E 02 063296E 04 10002 957638E 03 3 128633E 04 E 359672E 02 434650E 05 10003 187130E 03 1 556333E 04 i 907712E 02 084382E 04 10004 248678E 03 1 673541E 03 E 925253E 02 598804E 03 10100 0 3 815551E 04 E 309933E 02 005817E 02 LOAD CONDITION 2 SUBCASE 2 STRESSES IN QUADRILATERAL E TS QUADA ELEMENT FIBRE STRESSES IN ELEMENT COORD SYSTEM PRINCIPAL STRESSES ZERO SHEAR ID DISTANCE NORMAL X NORMAL Y SHEAR XY MAJOR MINOR VON MISES 21 000000E 01 894159E 03 4 092442E 03 7 646278E 03 164022E 04 3 653620E 03 1 383375E 04 000000E 01 894159E 03 4 092442E 03 646278E 03 164022E 04 3 653620E 03 383375E 04 22 000000E 01 656648E 02 2 257865E 03 013844E 03 333576E 03 3 910047E 03 953615E 03 000000E 01 656648E 02 2 257865E 03 0138
263. han 1 on subsequent design cycles Better results were obtained with the default but it required twenty design cycles to achieve an objective of 128 47 in not that much better considering the cost involved FRI TRI IO OOOO RI EC OR EE SUMMARY OF DESIGN CYCLE HISTORY ek deo dede TO IO TOI ok ek eee OOOO HARD CONVERGENCE ACHIEVED NUMBER OF FINITE ELEMENT ANALYSES COMPLETED 11 NUMBER OF OPTIMIZATIONS W R T APPROXIMATE MODELS 10 OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY OBJECTIVE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE CYCLE APPROXIMATE EXACT OF OF NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT INITIAL 2 301266E 02 1 249123E 03 4 2 077541E 02 1 857625E 02 1 183856E 01 1 248111E 03 2 1 750074E 02 1 608924E 02 8 772969E 02 6 922793E 06 3 1 561576E 02 1 551918E 02 6 223301E 03 4 953377E 06 4 1 506372E 02 1 472406E 02 2 306841E 02 2 064902E 05 5 1 401009E 02 1 426366E 02 1 777738E 02 9 372439E 04 6 1 383175E 02 1 392657E 02 6 808980E 03 9 459354E 04 7 1 342377E 02 1 373367E 02 2 256553E 02 9 445066E 04 8 1 324709E 02 1 350924E 02 1 940550E 02 3 131290E 04 9 1 296246E 02 1 331847E 02 2 673070E 02 3 121765E 04 10 1 281848E 02 1 333292E 02 3 858460E 02 3 099144E 04 Design Sensitivity and Optimization User s Guide 7 39 Chapter 7 Example Problems DESIGN VARIABLE HISTORY INTERNAL EXTERNAL DV ID DV ID LABEL INITIAL n 1 2 E 3 4 5 1 1 T1 8 0000E 02 9 6000E 02 9 3134E 02 1
264. hat time we introduced the idea of a one dimensional search and the steepest descent algorithm as a particular method of establishing this search direction However we also noted that this method in addition to not being particularly efficient cannot address those situations in which one or more constraints were active In this section we discuss how these issues are resolved by the optimizer in NX Nastran and we then move on to consider some of the other details of the search process Recall that the design variable vector update can be written as Equation D 6 30 Ru z l where is the initial vector of design variables is the search vector and a is the search 31 parameter Eq D 6 represents a one dimensional search since the update on depends only on the single scalar parameter a a is the value of a that yields the optimal design in the direction defined by gt S In other words to continue with the hill analogy we have either encountered a fence or are at the 31 bottom of a valley in the direction of Finding a completes the first iteration in the design process If we were at the bottom of a valley we could just repeat the process of finding a new steepest descent direction and keep moving In practice there is a better choice of direction called a conjugate direction which we will discuss shortly However regardless of the search direction used the concept is the same We find a search direction down
265. having any of the properties or other modeling data on hand prevents referencing any of this data in connection with the design model Although external superelements may be used in connection with the analysis model their properties are assumed to be constant with respect to the design model Design Variables in Superelement Design Modeling You can relate design variables to properties as in the case with non superelement models by using either linear relationships specified on DVPREL1 Bulk Data entries or user defined equations on DVPREL2 entries Design variable to property relations can be local to a given superelement or can span superelement boundaries For example a design variable can be related to a particular property entry which is in turn used in a number of superelements in the model Shape Basis Vector Restrictions You can also define design variable to grid relationships using DVGRID Bulk Data entries A given design variable can be related to grid sets that span superelement boundaries A DVGRID entry may reference both interior and exterior grids of primary superelements DVGRID entries are the only way to specify shape basis vectors in superelement design models The other methods direct input of shapes geometric boundary shapes and analytic boundary shapes are not supported For secondary superelements the same property and or grid variations prescribed on the primary superelement also apply to the corresponding image s
266. he filenaming on UNIX based machines On some machines quote marks may also be required around the filename 1 38 deletes the OPTEXIT parameter assignment from the Bulk Data which is entry number 38 in the sorted data It is usually a good idea to request a listing of the sorted Bulk Data especially if subsequent restarts are anticipated It is the default anyway but is explicit with the Case Control command ECHO SORT The optimization results from this restart are shown in Figure 7 30 Note that they agree with the results of the original Three Bar Truss The contents of the F04 file not included here indicate the initial analysis was skipped Design Sensitivity and Optimization User s Guide 7 67 Chapter 7 Example Problems KK RK RK KKK KK KK KK KK KK KK KKK KK KK AAA AAA ckck ck RARA AAA KK KK RA KK KKK KKK AAA SUMMARY OF DESIGN CYCLE HISTORY KK KK KR KK RR RR KR RK RR RR KK ok RK RK RK RR KKK HARD CONVERGENCE ACHIEVED NUMBER OF FINITE ELEMENT ANALYSES COMPLETED NUMBER OF OPTIMIZATIONS W R T APPROXIMATE MODELS 5 OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY OBJECTIVE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE CYCLE APPROXIMATE EXACT OF OF NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT INITIAL 4 828427E 00 3 234952E 01 1 3 007897E 00 3 008492E 00 1 977251E 04 3 737402E 03 2 2 821953E 00 2 821638E 00 1 118734E 04 1 967246E 02 3 2 734469E 00 2 734299E 00 6 217039E 05 7 241016E 03 4 2 708915E 00 2 7089
267. he interpolation phase Since we want to allow the tip of the beam to narrow the z displacement component must not be fixed In contrast all must be fixed in the x direction no axial shortening and the corner grids 110 and 132 must be restrained in the y direction no shortening of the beam width This leads to BNDGRID defined components of 12 for grids 110 and 132 and component 1 only for grid 121 Modeling Summary To summarize auxiliary boundary models are created using additional Bulk Data Sections labeled using BEGIN BULK AUXMODEL n We can have as many of these sections as necessary to suit our auxiliary boundary model needs Boundary conditions and loads are applied with AUXCASE labelled Case Control Sections The resultant boundary deformations are then interpolated to the interior of the primary structure using BNDGRID entries to define the boundaries The resulting total displacement vectors can now be combined in any way to yield basis vectors for shape optimization Shape Basis Vectors The shape basis vectors are defined using DVBSHAP Bulk Data entries DVBSHAP entries 1 and 2 relate design variables 1 and 2 respectively to the two displacement solutions by the relation 7 32 Design Sensitivity and Optimization User s Guide Example Problems 7 1 0 Usupcase20 1 Usugca gina X Equation 7 11 Thus each shape basis vector is simply 1 0 times each of the resultant displacement solutions Each solution is th
268. he only relation that is assured is the one between a property item and its word position in the element property table This correspondence requires that the engineer have some understanding of the structure of the EPT for the BEAM element and its relation to the data supplied on the PBEAM entry Note Note that the numbering in Figure 2 6 applies to the new design sensitivity and optimization Solution 200 In general the cross section of a BEAM element may be either constant linear or variable These three possibilities are shown in Figure 2 6 which is a schematic of the EPT configuration for each of these situations Design Sensitivity and Optimization User s Guide 2 15 Chapter 2 Design Modeling for Sensitivity and Optimization 1 Constant section beam only end A specified Words 6 21 and 166 181 must be accounted for on DVPRELi entries End A End B Words Words 6 21 166 181 2 Constant or variable section beam end A and end B specified Words 6 21 22 37 and 166 181 must be accounted for on DVPRELi entries End A Intermediate End B Words Words Words 6 21 22 37 166 181 This station is internally generated but must be accounted for on DVPRELi entries Words 22 37 are identical to Words 166 181 End B 2 16 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization 3 Constant or variable section beam end A intermediate station C an
269. he second updated the set of responses according to the proposed design changes The first task of the subsequent restart repeats this analysis though to provide the new baseline responses If the analysis cost is high this repeated analysis might be troublesome The costs associated with this type of restart must be weighed against its convenience Restarting from an OPTEXIT Point The second class of restarts is based on NX Nastran s automatic restart capability In design optimization this works in conjunction with the predefined Solution 200 exit points OPTEXIT 1 through 6 Recall that setting this parameter in Bulk Data will terminate the program at one of these six exit points Automatic restart for design optimization is somewhat limited since only the analysis restart logic is in place That is only data associated with the analysis results are used on restart and all design optimization related processes are repeated For example restarts from an OPTEXIT point of 4 or greater will result in a repeat of the sensitivity analysis phases This occurs regardless of whether the design model has changed or not Warning the design model must not be changed as unpredictable results may occur To illustrate this procedure suppose we wanted to perform a baseline analysis for the structure compute the sensitivities of the design responses with respect to changes in the design variables 7 66 Design Sensitivity and Optimization User s
270. hes The corresponding initial volume is 8 0 in3 The volume constraint is imposed as follows the volume response is first identified on DRESP1 number 201 bounds are placed on this response by DCONSTR number 10 which is in turn selected as a global constraint in Case Control global since volume is a subcase independent response using the DESGLB Case Control command This results in 7 38 Design Sensitivity and Optimization User s Guide Example Problems 7 99 lt vol lt 8 01 Equation 7 13 The mean square response is to be the objective function The underlying transverse displacements are first identified by DRESP1 entries 20 through 100 and 102 through 200 note the use of Bulk Data replicators These identify the transverse component of displacement at GRID 1110 across the frequency range of interest These first level responses are then used as input to DEQATN 1 via DRESP2 number 1 which defines the equation input The resultant response is then defined as the objective function using the Case Control DESOBJ command Optimization results are shown in Figure 7 17 and Figure 7 18 Figure 7 17 shows the design cycle history where it can be seen that the objective has been reduced from 230 13 to 133 33 in2 This took ten design cycles Note that CONV the relative measure of convergence with respect to the design objective has been set to 0 01 on the DOPTPRM entry This indicates convergence as long as the objective is changing less t
271. hes similar information as the previous entry only with regard to shape changes along the y axis The last BNDGRID entry identifies components 1 through 6 for the cutout edge grids Shape changes are enforced for components 1 and 2 since they have been provided on DVGRID entries All other components are considered fixed Once the free fixed and enforced portions of the shape changes have been defined the code interpolates this information to all of the interior degrees of freedom In this case the resulting shape basis vectors are identical to those shown in Figure 2 18 and Figure 2 19 On subsequent iterations the shape basis vectors may change slightly because the interpolated solutions over the interior of the structure will change due to the modified geometry Recomputing the basis vectors at the beginning of each design cycle helps to minimize the problems associated with mesh distortion Analytic Boundary Shapes This method also uses auxiliary models Here however they are defined over the boundaries of the structure only The so called auxiliary boundary models can be an edge of BAR elements for a two dimensional redesign or a skin of shell elements for a general three dimensional shape optimization task Auxiliary boundary models are used to generate shape variations over the boundaries of the structure providing a shortcut to the geometric boundary shapes method You can define as many auxiliary boundary models as are necessar
272. hese can be thought of as contour lines drawn along constant elevations of the hill The optimum point in this example is the point that lies at the intersection of the two constraints This location is shown as Ve t Figure 1 4 Kuhn Tucker Condition at a Constrained Optimum If we compute the gradients of the objective and the two active constraints at the optimum we see that they all point off roughly in different directions Remember that function gradients point in the direction of increasing function values For this situation a constrained optimum the Kuhn Tucker conditions state that the vector sum of the objective and all active constraints must be equal to zero given an appropriate choice of multiplying factors These factors are called the Lagrange multipliers Constraints that are not active at the proposed optimum are not included in the vector summation Indeed Figure 1 5 shows this to be the case where A and s are the values of the Lagrange multipliers that enable the zero vector sum condition to be met We could probably convince ourselves that this condition could not be met for any other point in this design space Design Sensitivity and Optimization User s Guide 1 15 Chapter 1 Getting Started VF X Figure 1 5 Graphical Interpretation of Kuhn Tucker Conditions The Kuhn Tucker conditions are useful even if there are no active constraints at the optimum In this case only the objective function gra
273. hese grids would simply add more equations to the set in Eq 2 19 leaving its basic form unchanged We can write Eq 2 19 in terms of grid variations as AC 0 1 3 AG 273 0 AG AG 0 2 3 AG AG 1 3 0 3y Equation 2 20 The coefficient matrix is unchanged in Eq 2 20 from that of Eq 2 19 Eq 2 20 suggests that a convenient choice of design variables would be Ax AG Ax AG 2 4x Equation 2 21 2x 0 1 73 ad A 134279 0 Axy AG3 0 2 83 Ax AGs r 3 AGa Equation 2 22 Note that Eq 2 22 expresses a vector of grid coordinate changes in terms of a vector of design variable changes Each column of the matrix on the right hand side of the equation is called a shape basis vector Generating the shape basis vectors is one of the design engineer s primary tasks in shape optimization If we graph each column of the matrix in Eq 2 22 as a set of grid displacements we see that column 1 represents a vertical or y component variation in shape while column 2 represents a horizontal shape variation The column order is arbitrary and just depends on the order of the design variables Design Sensitivity and Optimization User s Guide 2 23 Chapter 2 Design Modeling for Sensitivity and Optimization Note that both shape basis vectors as well as any linear combination of them keep the edge of the bracket as a straight line consistent with our design goals T Column 1 T Column 2 Figure
274. hese relations can be specified using DLINK Design variable LINKing Bulk Data entries This specification reduces the order of the design space since only the design variables in the independent set are considered by the optimizer Apart from numerical efficiency there may be other design related reasons for introducing some type of design variable linking In the design of the transmission tower in Figure 1 13 we may want to enforce sizing symmetry about the vertical axis For shape optimization we may want to enforce symmetry about the y axis in addition to the requirement that the uprights remain straight Design Sensitivity and Optimization User s Guide 1 25 Chapter 1 Getting Started A h4 ho Y Y Figure 1 13 Symmetric Transmission Tower Assume for example that the variables x4 though xg are used to describe the x component variations in the dimensions of the tower We may want to link the x variables such that symmetry hy h straight sides x4 v XS 1 Ix hi hy j Equation 1 22 From these relations note that the independent and dependent variable sets are Equation 1 23 Not only have our structural redesign conditions been met using these simple relations but the order of the design space has also been reduced from six variables to two 1 26 Design Sensitivity and Optimization User s Guide Getting Started Constraint Regionalization and Deletion If the number of constraints can be tempo
275. hus by seeking to reduce the level of constraint violation some of these move limits may be violated For initial designs with highly violated constraints it is not unusual to see analysis model properties vary by 25 or 30 even though the move limits may actually only be 2096 This usually does not pose any practical difficulties but if it does the initial move limits can always be reduced accordingly using the DOPTPRM entry In rare instances this optimization feature may yield a physically meaningless property value e g a negative plate thickness In this case the analysis cannot proceed and a fatal message will be issued The remedy should this occur is to restart with a reduced value of DELP on the DOPTPRM entry Design Sensitivity and Optimization User s Guide 3 51 Chapter 3 Design Sensitivity and Optimization in NX Nastran Using Design Variable Bounds to Enforce Move Limits on Properties In contrast the bounds on design variables are always honored by the optimizer Since the design variables are the quantities that the optimizer varies directly and thus has direct control over their bounds are never exceeded To implement bounds on properties that are critical from a design standpoint it is recommended that a linear design variable to property relation DVPREL1 be prescribed if possible with appropriate bounds on the design variable stated on the DESVAR entry Since the design variable bounds are always enforced the c
276. iable An example is where the thickness of each ply of a composite shell element is proportional to the thickness of all other plies of the same element For such cases if both the properties and the design variable are bounded NX Nastran will automatically remove bounds on the properties by transferring more restrictive bounds to the related design variable For example consider a single property p and a design variable x that are linearly related as follows p 3 0 2 0x Equation 2 10 Solving for x yields x p 3 0y2 0 Equation 2 11 Suppose the bounds for p and x are specified to be 10 0 s p s 30 0 Equation 2 12 and 20sxs9 0 Equation 2 13 Applying the property bounds to the above equation for x gives 3 5 lt x lt 13 5 Equation 2 14 The new lower bound of 3 5 is more restrictive than the specified lower bound of 2 0 Thus the bounds for x are modified such that 3 5 lt x lt 9 0 Equation 2 15 and the bounds on the property p are removed 2 10 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization If an even more restrictive value for the lower bound of the same x is found from another property then the bound is further modified You can optionally disable selective modification of design variable bounds with system cell 539 When design variable bounds are modified as above the current default for property move limit of DELP 0 2 may be too
277. iable linking via DLINK entries Constraints Major principal and von Mises stresses at lower plate surfaces Z1 Minor principal and von Mises stresses at upper plate surfaces Z2 The reduced basis functions used here consist of three basis vectors constant linear and quadratic The thickness distribution can be written as a linear combination of these basis vectors as Design Sensitivity and Optimization User s Guide 7 11 Chapter 7 Example Problems i 1 0 1 0 1 0 f 1 0 0 875 0 7656 ty p 4 10 a 075 a4 0 5625 10 0 125 0 0156 8 Equation 7 3 The eight individual plate thicknesses one for each station along the length of the cantilever are functions of three independent design variables ay az and az One way of implementing Eq 7 3 is with DLINK entries as shown in Listing 7 1 Another way of writing this reduced basis formulation is with DVPREL1 entries alone you may want to try this exercise on your own However by using DLINK entries and a dependent design variable for each plate thickness the final values of the dependent variables correspond exactly to the final plate thickness distribution This is helpful in evaluating the results as is shown in the design variable history summary The dependent design variables defined using DESVAR entries 1 through 8 are assigned initial values that correspond to the initial thickness distribution see the PSHELL entries 1 through 8 This has been done for cl
278. ibed using six design variables three for the web thicknesses and three for the rod cross sectional areas symmetry is imposed on the upper and lower ROD element areas See Listing 7 2 for details The lower bound constraint on the first mode is f 20 Hz Note that the DCONSTR entry places bounds on A 2rrf or 15 791 radians sec The upper bound will not drive the design so it is just set to an arbitrarily large value of 2 0E5 It should be noted that the weight or more appropriately mass objective function is computed in NX Nastran based on a volume density calculation Thus elements that have no volume are not included in the weight response calculation Here the mass contribution of the concentrated masses is ignored as can be verified from the summary of design cycle history In general although 7 8 Design Sensitivity and Optimization User s Guide Example Problems concentrated and distributed masses are not included in the design model responses they are accounted for as expected in the analysis Turning to the optimization results we note that hard convergence has been achieved after five design cycles requiring a total of six finite element analyses We note that a feasible design has been achieved as the maximum constraint value is less than the allowable maximum Again the final weight objective is that due to the plate and rod elements only excluding the concentrated masses k NORMAL CONVERGENCE CRI
279. ible for this maximum value After sensitivity analysis has been performed as indicated by the following DMAP information messages DMAP INFORMATION MESSAGE 9051 FEA STATIC SENSITIVITY ANALYSIS INITIATED DESIGN CYCLE NUMBER DMAP INFORMATION MESSAGE 9052 FEA STATIC SENSITIVITY ANALYSIS COMPLETED DESIGN CYCLE NUMBER all of the information necessary to perform an optimization is available The parameter IPRINT which can be set using the DOPTPRM entry controls the amount of diagnostic output from the optimizer With the default zero value no output is generated during the numerical optimization phase In this example IPRINT has been set to 1 7 is the highest available value leading to the following output Design Sensitivity and Optimization User s Guide 6 5 Chapter 6 Output Features and Interpretation DOT VERSION 4 00 CONTROL PARAMETERS OPTIMIZATION METHOD METHOD NUMBER OF DECISION VARIABLES NDV NUMBER OF CONSTRAINTS NCON PRINT CONTROL PARAMETER GRADIENT PARAMETER GRADIENTS ARE SUPPLIED BY THE USER THE OBJECTIVE FUNCTION WILL BE MINIMIZED IGRAD INITIAL VARIABLES AND BOUNDS LOWER BOUNDS ON THE DECISION VARIABLES 1 1 00000E 03 1 00000E 03 XL VECTOR 1 00000E 03 DECISION VARIABLES 1 X VECTOR 1 50000E 01 2 00000E 01 1 44000E 01 UPPER BOUNDS ON THE DECISION VARIABLES 1 3 00000E 01 4 00000E 01 XU VECTOR 2 88000E 01 INITIAL FUNCTION VALUES OBJ 5 7844
280. ic e si lh tee i ls ui uh See es ek a me ANALYSIS MODEL GRID 10000 0 0 0 0 0 0 GRID 10001 2557 0 0 0 0 GRID 10002 5 0 0 0 0 0 GRID 10003 745 0 0 0 0 GRID 10004 10 0 0 0 0 0 GRID 10100 0 0 2 5 0 0 GRID 10101 255 2 5 0 0 GRID 10102 5 0 25 0 0 GRID 10103 Ted 2 5 0 0 GRID 10104 10 0 2 5 0 0 GRID 10200 0 0 5 0 0 0 GRID 10201 2 5 5 0 0 0 GRID 10202 5 0 5 0 0 0 GRID 10203 745 5 0 0 0 GRID 10204 10 0 5 0 0 0 GRID 10300 0 0 T5 0 0 GRID 10301 2 5 Jp 0 0 GRID 10302 5 0 Taby 0 0 GRID 10303 7 55 TB 0 0 GRID 10304 10 0 7 5 0 0 GRID 10400 0 0 10 0 0 0 GRID 10401 2 5 10 0 0 0 GRID 10402 5 0 10 0 0 0 GRID 10403 7 5 10 0 0 0 GRID 10404 10 0 10 0 0 0 GRID 20100 0 0 5 0 1 0 GRID 20101 215 bsp 1 0 GRID 20102 5 0 550 1 0 GRID 201037 4 7 5 5 0 1 0 GRID 20104 10 0 5 0 1 0 GRID 20200 0 0 5 0 2 0 GRID 20201 2 5 5 0 2 0 GRID 20202 5 0 5 0 2 0 GRID 20203 Tubs 5 0 2 0 GRID 20204 10 0 5 0 2 0 CQUAD4 1 ty 10000 10001 10101 10100 CQUAD4 2 d 10001 10002 10102 10101 CQUAD4 3 T 10002 10003 10103 10102 CQUAD4 4 1 10003 10004 10104 10103 CQUAD4 Dy 1 10100 10101 10201 10200 CQUAD4 6 15 10101 10102 10202 10201 CQUAD4 Ti 1
281. ified This is particularly Design Sensitivity and Optimization User s Guide D 23 Appendix D Numerical Optimization true in NX Nastran because this program is solving an approximate problem that is to be updated on the next design cycle Here two criteria are used The first criterion requires that the relative change in the objective between iterations be less than a specified tolerance DELOBJ Thus the criterion is satisfied if Ir 5 r i cl Equation D 55 The default value for DELOBJ is 0 001 The second criterion is that the absolute change in the objective between the iterations is less than a specified tolerance DABOBJ This criterion is satisfied if lt DELOBJ Ir q 1 pat F 5 F a lt DABOBJ Equation D 56 0 001 e r The default value for DABOBJ is the maximum of and 1 0E 20 The reason for the two criteria is that if the objective function is large the relative change between two successive iterations is an indication of convergence However if F x is a very small number a relative change is not meaningful and thus the absolute change controls the convergence Often no progress is made on one iteration but significant progress is produced on the next Therefore one of these criteria is required to be satisfied on ITRMOP consecutive iterations where the default value of ITRMOP is 2 Recall that this default may be changed on the DOPTPRM Bulk Data entry Satisfaction of the Kuhn Tucker
282. ign Sensitivity and Optimization This chapter introduces some of the basic concepts of numerical optimization with an emphasis on the methods used in NX Nastran Some of the questions answered in this chapter include e What is design optimization and how does it differ from analysis e What is the relationship between design sensitivity and optimization How is an optimization problem formulated How does an optimizer search for an optimum How does an optimizer communicate with the structural analysis If you are interested in seeing some complete example problems in connection with the material covered in this chapter you may want to refer to the first couple of examples from Example Problems You will probably also need to refer to the Bulk Data descriptions in the NX Nastran Quick Reference Guide for the details of the entries used Example Problems should help to give you some idea of what NX Nastran design optimization input and output looks like The details are covered in later chapters What is Design Sensitivity and Optimization Design sensitivity and optimization are two separate though closely related topics For a given design a design sensitivity analysis computes the rates of change of structural responses with respect to changes in design parameters These design parameters are usually referred to as design variables and can be used to represent shell thicknesses beam cross sectional dimensions journal bearin
283. ign Sensitivity and Optimization User s Guide 1 13 Chapter 1 Getting Started descent is usually a very good starting point subsequent search directions often fail to improve the objective function significantly In NX Nastran we use other more efficient methods that can be generalized for the cases of active and or violated constraints We will briefly introduce these methods later in this section The next question to consider is once we have determined a search direction how can this be used to improve our design One Dimensional Search Just as in the hill example once we found a search direction we proceeded downhill until we bumped into a fence or until we reached the lowest point along our current path Note that this requires us to take a number of steps in this given direction which is equivalent to a number of gt function evaluations in numerical optimization For a search direction 5 and a vector of design variables the new design at the conclusion of our search in this direction can be written as gt l 30 o X x aS Equation 1 12 This relation allows us to update a potentially huge number of design variables by varying the single parameter a We have been able to reduce the dimensionality from n to 1 that is from n design variables to a single search parameter a For this reason this process is called a one dimensional search When we can no longer proceed in this search direction we have the value of a whic
284. ign cycle process is repeated without much incident until cycle number 6 when the following message is issued Ok KKK Ck k k Kk k Ck KKK Ck Ck k k k k AR USER WARNING MESSAGE IF YOU WANT TO RESTART AFTER THIS JOB IS COMPLETED YOU MUST INCLUDE A DOPTPRM DATA ENTRY IN THE RESTART BULK DATA WITH THE FOLLOWING ITEMS MODIFIED AS SHOWN 2 5000E 01 5 0000E 03 5 0000E 01 2 5000E 02 DELP DELX DXMIN NOTE THERE MAY BE MORE THAN ONE MESSAGE LIKE THIS THE LAST ONE IN THIS RUN SHOULD BE LOCATED AND USED TP ttc ee fee cee eee eee ee ee ee ee ee ee eo He OROOR OX HH O HH HH F DPMIN The above is related to the automatic update of move limits discussed in Optimization with Respect to Approximate Models Since the design cycle number is greater than two the constraint violation is increasing and the maximum constraint violation is greater than 296 the conclusion is made that the accuracy of the approximate model can be improved if move limits are reduced by 5096 This warning message is just notification that were optimization to be terminated at this point and then restarted these new move limits should be included on a new DOPTPRM entry Note that parameters controlling move limits on properties DELP DPMIN as well as those controlling design variables DELX DXMIN have all been reduced One design cycle later soft convergence is achieved as indicated by the message 6 10 D
285. ign variables to describe changes to a large number of analysis model properties or grid coordinates Essentially each design variable acts as a multiplier of a constant vector The linear superposition of these so called basis vectors represents the total design changes See Eq 2 23 and Eq 3 3 Search Direction Search Vector an n dimensional vector where n is the number of independent design variables that characterizes a particular search path used by the optimizer The optimizer may search along a number of different paths in connection with a given design cycle Each of these A 4 Design Sensitivity and Optimization User s Guide Glossary of Terms searches is termed a design iteration Upon completion of these iterations an approximate optimum is at hand This approximate optimization process is repeated in subsequent design cycles Second Level Responses also known as synthetic responses are formulated by the engineer using NX Nastran s equation input capability These responses are defined using DRESP2 and DEQATN Bulk Data entries Semianalytic Sensitivity Analysis refers to an approximate solution of the analytic sensitivity equations With this approach the appropriate system equations are differentiated and the structural matrices approximated using finite differences The resulting approximate set of equations is then explicitly solved for the response derivatives See Design Sensitivity Analysis to see how this approach is
286. in all but the smallest of problems Recall that the numerical optimizer may not only request response 1 20 Design Sensitivity and Optimization User s Guide Getting Started derivatives with respect to the design variables but a number of function evaluations must also be performed during each of the one dimensional searches This situation could quite easily lead to hundreds of analyses Finite Element Numerical Analysis Optimization A Figure 1 9 Early Structural Optimization Attempts Structural Responses Implicit Functions of the Design Variable The principal complicating factor in structural optimization is that the response quantities of interest are usually implicit functions of the design variables For example a plate element s stress variation with changing thickness can only be determined in the general case by performing a finite element analysis of the structure When we consider that the optimizer may be asked to deal with perhaps hundreds of design variables and thousands of constraints it becomes apparent that we do not have the luxury of invoking a full finite element analysis each time the optimizer proposes an incremental design change To avoid these difficulties certain approximations can be implemented to reduce the computational overhead The remainder of Structural Optimization will introduce each of these methods all of which are available in NX Nastran Overview of Approximation
287. in the design model linearizing with respect to the quantity 7 A will produce approximations that are exact in both cases In the general case of course these approximations are not exact due to the static indeterminacy of the structure However for all element types the proportionality of the stiffness matrices to the inverse of the sizing quantities forms a reasonable basis for arguing the use of reciprocal approximations Reciprocal approximations can be derived from linear approximations if we first substitute the intermediate variables y For the approximate constraint of Eq 3 8 we get 5 gt 0 E 0 Sj 3 Aj gQ SE 2 5 Ay i 150 y 08 y Equation 3 9 Setting the intermediate variables to reciprocals of the design variables 3 16 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran en d e Ud Equation 3 10 and noting that dy 1 x2 x and that y x x x we obtain X Ax 0 i i Xi Og x 20 _ 4 8j gral eds La i 30 x Equation 3 11 where the subscript R indicates a reciprocal approximation Approximation Methods Three different approximation methods are available in NX Nastran You can select any one of them using the APRCOD field on the DOPTPRM entry These methods are as follows APRCOD 1 This specifies that direct approximations Eq 3 8 are to be used for the objective function as well a
288. inite element analysis serves as the new baseline from which to construct another approximate subproblem This cycle may be repeated as necessary until convergence is achieved These loops are referred to as design cycles Design variable linking constraint deletion and formal approximations form the basis for the approximation concepts implemented in NX Nastran The remainder of Structural Optimization takes a closer look at these methods Design Variable Linking Before discussing design variable linking a few words should be said about how design modeling differs from analysis modeling Differences Between Design and Analysis Modeling In analysis our goal is to construct a mathematical idealization of an actual physical system We must determine analysis model properties select appropriate response quantities and determine a suitable finite element mesh so that the desired responses are computed within an acceptable degree of accuracy On the other hand in design modeling we wish to define how our model can change in pursuit of an improved design along with the criteria that we or the optimizer use to judge the effects of these changes Some formulations may be inherently more efficient than others For example suppose we have a structure consisting of symmetric I section beams as shown in Figure 1 11 For analysis purposes we must specify the cross sectional properties A 4 and 2 on a property Bulk Data entry However from
289. ink numerical optimizers and structural analysis codes to create a structural optimizer and that some sort of approximations are required It has also been shown that the set of approximation concepts acts as an interface between the analysis and the optimizer There is really nothing extraordinary about these approximations since in many ways they are similar to the methods experienced designers already use Of course formalizing these methods in structural optimization allows problems of considerably greater size and complexity to be solved than with traditional design methods alone Design Sensitivity and Optimization User s Guide 1 33 Chapter 2 Design Modeling for Sensitivity and Optimization e Overview of Design Modeling e Defining the Analysis Disciplines Defining the Design Variables Relating Design Variables to Properties Relating Design Variables to Shape Changes Identifying the Design Responses e Defining the Objective Function Defining the Constraints e Superelement Design Modeling Design Sensitivity and Optimization User s Guide 2 1 Chapter 2 Design Modeling for Sensitivity and Optimization A design model is required for design sensitivity and optimization in much the same way an analysis model is required for finite element analysis The design model is simply a formal statement of allowable changes that can be made to a structure during the search for an optimal design It also places limits on the
290. ion process Default values of parameters that control the screening process can be overridden using DSCREEN entries Since these defaults are often satisfactory including DSCREEN entries is often unnecessary Constraint Deletion Constraint screening is performed in two stages In the first step deletion the constraint values that fall below a certain threshold level are deleted from the constraint set The default threshold level is 0 5 but this can be overridden by specifying a TRS value on the DSCREEN entry In NX Nastran the normalized constraints are considered violated if they are positive quantities The default TRS states that only those constraints that are within 50 of their critical values are retained Constraint Regionalization The second step regionalization sorts the remaining constraints and retains up to a prescribed maximum per region per load case The default region specifications are given on the DSCREEN Bulk Data entry description NSTR or the number of constraints to be retained per region has a default value of 20 which again can be overridden DTABLE Purpose Defines a table of constants to be used in conjunction with DEQATN equations Entry Description DTABLE VALU1 VALU5 VALU2 VALU3 VALUA vau ee o Field Contents LABLI Label for the constant Character VALUI Value of the constant Real 4 18 Design Sensitivity and Optimization User s Guide Input Data As
291. ions The approximation of the differential stiffness is worth discussing This stiffness is a function not only of geometry but of displacement as well Kj K 2 Equation 3 61 In the finite difference approximation Kg should be approximated for the perturbed configuration as 0 gt 0 O Kj K Ax uAu EK4G i a GS L L Equation 3 62 since the displacements are also implicit functions of the design variables However in NX Nastran this displacement variation is ignored resulting in Design Sensitivity and Optimization User s Guide 3 33 Chapter 3 Design Sensitivity and Optimization in NX Nastran O K j KR Ax K 2 ii Ox Ax 1 L IR Equation 3 63 In most cases Eq 3 63 is a reasonable approximation Eq 3 62 is an often unnecessary costly alternative However if the optimization results suggest that the sensitivities are not of sufficient accuracy one or both of two options might be used The first is to use the DOPTPRM entry to decrease DELB the finite difference move parameter by one or more orders of magnitude This reduction might also be coupled with central difference approximations Setting the Bulk Data parameter CDIF to YES forces central differences The second option might be to reduce the move limits using DELP and DELX on the DOPTPRM entry and DELXV on the DESVAR entries The goal of these changes is to improve the quality of the approximati
292. is disciplines Eq 3 90 is differentiated with respect to the design variables and solved for the quantity dy dx The solution is semi analytic in nature with derivatives approximated using either forward differences which is the default if no shape defining design variables are present or central differences PARAM CDIF YES which is the default if shape variations are allowed Coupled Fluid Structure Interaction Sensitivity Coupled fluid structure analysis is used to solve problems in which the interaction effects between fluid and structure are significant NX Nastran uses a fully coupled formulation for this analysis Although its principal application is in acoustic problems it can also be applied to other general problems such as fluid storage tank sloshing or other situations in which an inviscid irrotational formulation is valid Refer to the NX Nastran User s Guide for details on the analysis formulation The sensitivity equations for fluid structure interaction are identical in form to the dynamic response sensitivity equations and can be solved using the existing dynamic response sensitivity solution algorithms To understand why this is so the general equations of motion for the fluid structure system can be written as MIES K ix5 F Equation 3 92 The solution vector consists of both structural displacements and the fluid pressure 2 fyl i j vh f Pr P Equation 3 93 The vector of external loa
293. is simply a means of relating it to the constraint numbering scheme used in the preceding optimizer diagnostic output The corresponding internal response ID of this maximum constraint is shown as 15 To see which response this is the following data is also output at the same time 6 8 Design Sensitivity and Optimization User s Guide Output Features and Interpretation DESIGN CYCI INTERNAL DRESPl RESPONSE ELEMENT C ID 2 3 2 1 2 4 3 14 1 2 5 3 15 FA 2 6 2 Le 2 7 C 3 l 2 3 6 14 le Lu 9 15 1 5 2 5000E DESIG INTERNAL DRESPl RESPONSE GRID _ CET 73 INTERNAL D ID Il 1 La 2 i 12 1 2 2 4 13 1 3 2 4 14 2 32 2 2 i 15 2 33 4 2 te e zi 7 2 Response corresponding to 13 S 54 5 the Maximum Constraint 20 3 26 2 21 9 27 2 23 12 21 2 24 12 23 2 25 12 24 2 26 12 26 2 27 12 27 2 26 12 28 2 From this output we can see that response number 15 is related to a bar element stress identified on DRESP1 number 2 for element number 33 lts input value as well as its output are indeed greater than the upper bound of 2 5E4 hence the observed constraint violations We can conclude that the optimizer has indeed been able to reduce the constraint violation by reducing this peak stress although at the expense of added weight Interpretation of Approximate Optimization Results It is important to note that the responses in the preceding output are only approximate because they are based
294. is vectors are updated on every design cycle minimizing the problems associated with mesh distortion for large shape changes Benefits Since grid variations need only be specified over the boundaries data preparation is greatly simplified In fact since DVGRID entries are only written for the boundaries many real world problems can be effectively solved without geometry based preprocessors Internal computation of the shape basis vectors means that these can be recomputed updated for every design cycle reducing the problems associated with mesh distortion for large shape changes Drawbacks Describing grid variations over the boundaries may still require a lot of DVGRID information even though orders of magnitude less than the manual grid variation method Checklist 1 Define the shape design variables using DESVAR Bulk Data entries 2 Define corresponding shape variations over the structure s boundaries using DVGRID entries 3 Define the shape boundary conditions using BNDGRID Bulk Data entries The BNDGRID DVGRID entry combination is analogous to the use of SPCDs to impose enforced boundary displacements The DVGRID entry supplies the enforced variation much like an SPCD to the boundary grids specified on the BNDGRID entry similar to an SPC entry Analytic Boundary Shapes This approach is similar to the geometric boundary shapes method but avoids having to use DVGRID entries on the boundary Instead the designer provides
295. isleading results Similarly an ill posed design optimization task may produce unexpected or useless results Since the redesign process is based on analysis results the results of design optimization are strongly dependent on the integrity of the analysis model Optimizer Limitations A numerical optimizer seeks to find an improved design by trying to minimize or maximize a prespecified objective Throughout this process it must adhere to the bounds on responses and design variables given in the design model It does not have the intelligence to modify the objective or relax any of these limits For example suppose you asked a friend to find you a nice apartment on his street Your friend the optimizer may have a somewhat different definition of nice than you do His income might be higher than yours so that the optimal design he proposes may be infeasible in terms of your bank account Even though he is searching just on his street the next block may turn out to have an apartment that you consider a better value The optimizer is not able to go beyond your specifications to search out other possible configurations The optimization problem statement requires an explicit description of the design objective as well as bounds that define the region in which it may search You may ask for a design satisfying a minimum flexibility requirement such as wing tip deflection but without a weight budget the design that the optimizer proposes may tur
296. ition that fences or constraints do not exist or are located so far uphill that they do not affect our search for a minimum This situation is an unconstrained optimization task in contrast to a constrained task Equality Constraints We might also have the condition that we want the design to lie on some prescribed path or curve drawn on the hillside This is an equality constraint Note that if there are as many equality constraints as design variables a unique solution exists as long as the equalities are linearly independent This solution can be found using standard algebraic methods A finite element analysis belongs to this category of problem When the number and type of constraints do not enable a direct unique solution the job becomes complex and numerical optimizers must be used The Basic Optimization Problem Statement We are now in a position to express our optimization task in a quantitative form This mathematical expression of the design problem is called the basic optimization problem statement and can be written as follows minimize F objective Equation 1 1 subject to g x lt 0 TL n inequality constraints Equation 1 2 Design Sensitivity and Optimization User s Guide 1 9 Chapter 1 Getting Started hy 0 k 1l n equality constraints Equation 1 3 I uo s X so x iSl aenn side constraints Equation 1 4 where gt X Xj Xp oss d design variables Equation 1 5 Function Minimi
297. ivity and Optimization User s Guide D 9 Appendix D Numerical Optimization are not really known precisely Furthermore the responses calculated by the finite element analysis are only approximate because of the nature of the method Therefore we allow a small positive constraint value before identifying a constraint as violated This is the value of CTMIN typically taken as 0 003 three tenths of one percent violation Thus the governing definitions are g x lt CT Inactive Equation D 23 CT lt g x lt CTMIN Active Equation D 24 g x gt CTMIN Violated Equation D 25 These tolerances are shown in Figure D 3 for a single constraint in a two design variable space XQ Feasible region eG 0 Infeasible region CT CTMIN Figure D 3 CT and CTMIN The use of Eq D 23 through Eq D 25 underscores the importance of normalizing constraints In NX Nastran this is done automatically using the constraint bounds as normalizing factors However D 10 Design Sensitivity and Optimization User s Guide Numerical Optimization certain precautions should be taken by the engineer when formulating second level responses as discussed in Design Modeling for Sensitivity and Optimization Using the active constraint criteria the algorithm first sorts all the constraints and identifies those that are active or violated Then the gradients of the objective function and all the active and violated constraints are calc
298. ization in NX Nastran K a or Ou Equation 3 52 where K is the stiffness matrix of the finite element model The required sensitivity of the response rj With respect to a particular design variable x is then given by dr dx dr Ox a ap dx a K dx u Equation 3 53 where p Applied load on the finite element model Indicates local partial derivative d Indicates partial derivative over the system As an example of a local derivative for a response quantity consider the case where a stress component Oe is the response quantity The local derivative is given by 00 Ox 0s 0xj u e Equation 3 54 where s is the relevant row of the transformation matrix from the element displacements U e to the Gauss point stresses Thus it is defined by Oc S U e Equation 3 55 Note NX Nastran does not currently use adjoint loads for stress responses Where appropriate the adjoint load method is also used for dynamic response sensitivities as an alternative to the pseudo load method Eigenvalue Response Sensitivities Requesting Eigenvalue Response Sensitivities To compute eigenvalue sensitivities you need to Define a design model see Design Modeling for Sensitivity and Optimization Define a normal modes subcase using the Case Control command ANALYSIS MODES Eigenvalue response sensitivities are computed automatically in connection with
299. j U are less than GSCAL GSCAL is used instead For example consider the synthetic response 04 o 1 2 2 E O max Equation 2 36 and assume the response had been formulated as 04 to 1 72 lt 0 2 O max Equation 2 37 At first glance this seems acceptable since it is expressed in standard form see The Basic Optimization Problem Statement However the zero upper bound is replaced by a nonzero and the resultant normalized upper bound constraint then becomes max La o l d Equation 2 38 Su The derivative of this constraint with respect to changes in one of the stress responses is 98 1 001 2g Equation 2 39 which is a large quantity due to the small e Numerically the optimizer considers a constraint to be active only if it is greater than CT constraint tolerance and less than CTMIN minimum allowable constraint violation If a constraint is greater than CTMIN it is considered violated This suggests that the constraint as expressed by Eq 2 39 is poorly conditioned since as a result of the derivative s magnitude the region over which this constraint is active is relatively small The situation is shown in Figure 2 26 2 56 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization CTMIN Active Figure 2 26 Poorly Conditioned Constraint If the constraint is expressed instead as o o 1 2 2 O max Equation 2 40 then the limit 0 4
300. k k ck k kk Ck k k k k k k k k Ck Ck k ck k k k kk k KKH kk k k k k k k k Kk k k ko k k k ko k ko k kk OPTIMIZATION KARA AKI cock kk koc kk kk kk ko kk ek ko Kok ke ke koe FOIOS DESIGN CYCLE 1 FEI KARA ke heck ok kk IR AAA AA OPTIMIZATION RESULTS BASED ON THE APPROXIMATE MODEL ERA DESIGN OBJECTIVE INTERNAL MINIMIZE RESPONSE OR SUPERELEMENT SUBCASE LABEL MAXIMIZE OUTPUT T PLATE 0000E 03 5000E 01 3 0000E 01 0000E 01 T WEB 0000E 03 0000E 01 4 0000E 01 0000E 01 A BAR 0000E 03 4400E 01 1 6592E 01 0000E 01 This output supports our observations with respect to the preceding optimizer output namely the design objective and design variables have all increased in value on the first design cycle Since weight has been added to the structure during the first design cycle it would be interesting to identify the violated constraints that are driving the design The following constraints table also appears as part of the summary output and helps us to answer just this sort of question Design Sensitivity and Optimization User s Guide 6 7 Chapter 6 Output Features and Interpretation ume DESIGN CONSTRAINTS ON RESPONSES MAXIMUM RESPONSE CONSTRAINTS MARKED WITH INTERNAL INTERNAL DCONSTR RESPONSE RESPONS ID ID ID TYPE 1 10 2 STRESS 2 10 3 STRESS 3 10 4 STRESS 4 10 5 STRESS 5 10 6 STRESS 6 10 7 STRESS 7 10 8 STRESS 8 10 9 STRESS 9 10 14 STRESS 10 10 LS STRESS 11 10
301. kk kk kk kk CONVERGENCE ACHIEVED BASED ON THE FOLLOWING CRITERIA OR AND HARD CONVERGENCE DECISION LOGIC RELATIVE CHANGE IN OBJECTIVE 0 0000E 00 MUST ABSOLUTE CHANGE IN OBJECTIVE 0 0000E 00 MUST AND MAXIMUM CONSTRAINT VALUE 1 3644E 01 MUST CONVERGENCE TO A FEASIBLE DESIGN i i OR MAXIMUM OF RELATIVE PROP CHANGES 0 0000E 00 MUST MAXIMUM OF RELATIVE D V CHANGES 0 0000E 00 MUST BE LESS THAN 1 0000E 03 BE LESS THAN 1 0000E 03 BE LESS THAN 5 0000E 03 BE LESS THAN 1 0000E 03 BE LESS THAN 1 0000E 03 CONVERGENCE TO A BEST COMPROMISE INFEASIBLE DESIGN TK RR KK ARK kk ARA KARA KARA AAA AAA AA AA AAA RAR ARA RARA RK RK RK RK RK RK KKK KKK KKK KK RARA KE ko ke ke ke ke ke ke ke ke Figure 7 22 Hard Convergence Output for the Composite Tube 7 52 Design Sensitivity and Optimization User s Guide Example Problems ek oko RR kk oko koe KK ek RRR RR ROR ROR ke ok ek ek ek ek e ek eee SUMMARY OF DESIGN CYCLE HISTORY e ek e e e kde sek de ke kde kk ke e e ke ke dee ke e de e ke de he ke de ke e ke ee kde e ee dee hee de e ke de ke ek ek ee ee HARD CONVERGENCE ACHIEVED SOFT CONVERGENCE ACHIEVED NUMBER OF FINITE ELEMENT ANALYSES COMPLETED 14 NUMBER OF OPTIMIZATIONS W R T APPROXIMATE MODELS 13 OBJECTIVE FROM OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY FRACTIONAL ERROR OBJECTIVE FROM MAXIMUM VALUE CYCLE APPROXIMATE EXACT OF OF NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT INITIAL 1 598129E
302. l 1 31 198 0 0 6 3 1368E 01 Bis 4 4665E 01 aF 3 1099E O1 v5 4 1860E 01 2 3 4856E 01 6 9 2706E 02 4 1 5445E 01 4 2 0554E 01 4 6 8638E 01 1 7 7319E 01 2 4 4665E 01 7 3 1099E 01 u 4 1860E 01 2 3 4856E 01 6 9 2706E 02 4 1 5445E 01 4 8 4529E 02 IU INPUT VALUE 0000E 00 J 0000E 00 e 0000E 00 aL 7180E 01 224 3736E 00 2 3423E 00 ks 3333E 01 PES 3333E 01 eas 3333E 01 2 3333E 01 Ze 0000E 00 EI 0000E 00 1 0000E 00 1 0000E 00 sus 3333E 01 34 3333E 01 3 3333E 01 e2u 8301E 01 O 6657E 01 1 6430E 01 1 0000E 00 sa 0000E 00 I 0000E 00 pa 0000E 00 Ls 3333E 01 22 OUTPUT VALUE 4096E 01 1997E 01 4096E 01 1997E 01 5182E 01 3644E 01 5182E 01 3644E 01 5855E 01 maximum 2725E 01 normalized 1714E 01 1899E 01 6589E 01 7100E 01 2134E 01 0216E 01 2602E 01 3930E 01 9683E 01 5961E 01 1899E 01 6589E 01 7100E 01 2134E 01 0216E 01 2602E 01 9465E 01 OUTPUT VALUE 0000E 00 0000E 00 3044E 00 2841E 01 1513E 00 1096E 00 8439E 01 8439E 01 8439E 01 8439E 01 1468E 00 1468E 00 1468E 00 1468E 00 3333E 01 3333E 01 3185E 01 7745E 01 5905E 01 5604E 01 1468E 00 1468E 00 1468E 00 1468E 00 8439E 01 constraint move limits on properties From this summary note that the maximum constraint flagged with is a result of an upper bound constraint on a stress response The constraint internal ID is listed as 10 which
303. late thickness web thickness and web cap cross sectional properties Constraints Strength von Mises stress lt 25 000 psi at the centroid of plate elements for Subcase 1 Maximum axial stress lt 25 000 psi at both ends of rod elements for Subcase 2 Displacement Vertical displacement at grid point 10302 for Subcase 1 lt 0 1 inches Vertical displacement at grid point 10203 for Subcase 2 lt 0 03 inches The DVPREL1 entries 1 2 and 3 each express an analysis model property directly in terms of a given design variable folate E 1 0X fweb 1 0X Acap 10X3 Equation 7 4 If the cap section is assumed to have a rectangular cross section then the other cross sectional properties can be expressed directly in terms of the area Design Variable Figure 7 5 shows a rectangular cross section characterized by the dimensions b and h with four stress recovery points C D E and F Design Sensitivity and Optimization User s Guide 7 17 Chapter 7 Example Problems h aa Y 6 e B D b Figure 7 5 The initial area can be written in terms of the original dimensions as Equation 7 5 For a proportional change A in the cross sectional dimensions b and h the corresponding area is A Ab Ak Abh Equation 7 6 or 172 a 4 A Equation 7 7 To look at it another way for a given change in the design variable area we immediately know how the underlying cross sectional dimensions chang
304. le we may have an upper limit constraint on stress or gt max g x 9 6 lt 0 J Equation 1 28 1 30 Design Sensitivity and Optimization User s Guide Getting Started The stress response oj is not only a function of the displacement solution u but also the element geometry and elastic properties or to DILE i3 Equation 1 29 where D B is the stress displacement transformation matrix o is a component of the element stress vector o Partial differentiation of the stress constraint g with respect to the i th design variable and use of the chain rule for differentiation yields Equation 1 30 where do 0xj is referred to as a sensitivity coefficient In general a sensitivity coefficient is defined as the partial derivative of a response with respect to a design variable or r r Ox l Equation 1 31 where r is a general response quantity Now Equation 1 32 The first term of Eq 1 32 is easily determined from relations such as Eq 1 29 The second term can be evaluated if we first differentiate the static equilibrium equation Eq 1 27 with respect to a design variable to obtain Qu _ OP 0 K A Ox i Ox i Ox Equation 1 33 This relation can be solved for d u dx to provide the information necessary to construct the first order approximations for the objective and retained constraints from Eq 1 26 The solution of Eq 1 33 is relatively inexpensive given that we al
305. lerance of CT but are not really critical Therefore we reduce CT and delete any constraints from the active set J that are not within the new tolerance We then solve this direction D 14 Design Sensitivity and Optimization User s Guide Numerical Optimization finding problem again to determine if a more precise satisfaction of the constraints can be achieved If the value of CT is reduced in magnitude to CTMIN CT is negative and no negative objective is found for the direction finding problem then this design is judged to have satisfied the Kuhn Tucker conditions and the optimization process is stopped The next question is How do we solve this sub problem of finding the usable feasible search direction 2 Unfortunately the details are beyond the scope of this discussion For our purposes it is sufficient to note that the problem can be converted to an iterative form of solving a set of simultaneous equations with some conditions on the variables The actual algorithm is similar to that used in traditional linear programming The details of this suboptimization problem are given in Vanderplaats An Efficient Feasible Direction Algorithm for Design Synthesis 3 One or More Violated Constraints Now consider the case where one or more constraints are violated Such a case is shown in Figure D 6 where constraint g4 x is violated and g gt x is active Now we must find a search direction back toward the feasible region even if it is ne
306. licit design variable linking Vibration of a Cantilever TPL problem D200X6 DAT Beam Turner s Problem TET Examines weight minimization of a structure subject to a free vibration frequency constraint Design variables are linearly related to rod areas and shell thickness Discusses interpretation of weight objective function if concentrated masses are used Cantilevered Plate TPL problem D200X5 DAT Illustrates reduced basis formulation using explicit design variable linking DLINK Weight minimization of a statically loaded plate with constraints on principal and von Mises stresses as well as grid displacements Stiffened Plate TPL problem D200X7 DAT Illustrates effective grouping of elements by property type for design modeling purposes and use of the DVPREL2 entry for user defined property relations Weight minimization subject to constraints on static stresses and displacements Design variables are related to plate thicknesses and bar cross sectional properties Shape Optimization of a TPL problems D200CSX DAT D200CS DAT Culvert Illustrates Shape Optimization using the Direct Input of Shapes method The objective is weight minimization subject to von Mises stress constraints 7 2 Design Sensitivity and Optimization User s Guide Example Problems Analytic Boundary Shapes TPL problem D200AM3 DAT Illustrates Shape Optimization using the Analytic Boundary Shapes method to minimize the weight of a solid
307. lid is explicit in the design variables The resultant explicit representation can then be used by the optimizer whenever function evaluations are required instead of the costly implicit finite element structural analysis This coupling is illustrated in Figure 1 10 The finite element analysis forms the basis for creation of the approximate model that is subsequently used by the optimizer The approximate model includes the effects of design variable linking constraint deletion and formal approximations Design variable linking is established by the engineer while constraint deletion and formal approximations are performed automatically in NX Nastran Design Improvements Finite Element Numerical Analysis Optimization TE Approximate Design Model Figure 1 10 Coupling Analysis and Optimization Using Approximations Design Cycles Once a new design has been proposed by the optimizer based on the information supplied by the approximate model the next step would most likely be to perform a detailed analysis of the new configuration to see if it has managed to satisfy the various design constraints and reduce the objective function This reanalysis update of the proposed designs using a complete finite element analysis is represented by the upper segment of the loop in Figure 1 10 If a subsequent approximate 1 22 Design Sensitivity and Optimization User s Guide Getting Started optimization is deemed necessary the f
308. lows you to investigate these real world types of shape variations It will for example let you determine the optimum placement and size of cutouts to lighten a structure What the code will not tell you is how many of these cutouts to start with or whether an alternate method of construction is better In fact the latter questions are still active areas of shape optimization research Goal of Design Modeling for Shape Four different modeling methods are available in NX Nastran The purpose of each method is the same to define a basis vector or a set of basis vectors for shape optimization The methods only 2 26 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization differ in terms of their user interfaces and whether or not the basis vectors are automatically updated on each design cycle The following subsections briefly outline each approach its strong as well as its possible weak points and its requirements for use Manual Grid Variation This is the most general as well as perhaps the most tedious method of generating Shape basis vectors It is listed first since it can be thought of as the lowest level approach somewhat analogous to Assembly Language versus FORTRAN programming A DVGRID Bulk Data entry defines the direction and magnitude of a grid variation for a given change in a design variable This relation is shown in Figure 2 15 and Eq 2 25 A single DVGRID entry is required f
309. ly a functional relation between design variables and g cannot be prescribed The structural damping contributions are assumed constant for design sensitivity They are accounted for in the analysis but do not vary as a function of the design variables Similarly the direct matrix input quantities cannot be expressed as functions of the design variables since these are essentially external quantities That is they are determined based on considerations that are external to the design model These quantities are also assumed to be constant The total damping force is due to a sum of the contributions from viscous elements B4 and direct matrix input at the grids B2 B B4 25 Equation 3 70 For structural mass M M1 EM Equation 3 71 The direct matrix input for damping and mass is assumed constant in the sensitivity analysis as is the case with the direct matrix input for stiffness Direct Frequency Response Sensitivities Displacement sensitivities for direct frequency response are obtained by differentiating the governing equation below o M io B KD u P Equation 3 72 to obtain 2 o M io B K Au AP e AM ie AB AK u Equation 3 73 where A indicates differentiation with respect to a design variable or 0 0x Eq 3 73 is solved once for each forcing frequency load condition and design variable 3 36 Design Sensitivity and Optimization User s Guide Design
310. m however the ability to do so may at times prove to be indispensable Unless you are particularly well acquainted with numerical optimizers though these parameters can usually be left unchanged and still yield quite satisfactory results Move Limits and Approximate Optimization Since the approximations are only locally valid see Approximation Concepts in Design Optimization and Eq 1 6 in Structural Optimization limits must be placed on the amount by which the design is allowed to vary during the approximate optimization In NX Nastran move limits are imposed in terms of both allowable changes in design variables and allowable changes in properties Imposing move limits using design variables is natural since the approximate model is explicit in these quantities However move limits on designed properties must also be considered as well since the analysis model is a function of these properties Moreover these designed properties may be nonlinear functions of the design variables DVPREL2 relations A 1096 change in a given design variable may equate to a 10096 or more change in an analysis model property Since these large design changes can easily invalidate the approximate model move limits on properties must also be included During approximate optimization each design variable is bounded from above and below by Equation 3 100 where x is the lower bound on the i th design variable and x is the corresponding upper bound Th
311. metric and made up of a shear web having top and bottom caps that are modeled with rod elements Turner s original design model consisted of piecewise linear bar cross sectional areas and web thicknesses however we will just approximate this as a step function model with uniform cross sectional rod elements and uniform thickness shear elements within each of three bays Design Sensitivity and Optimization User s Guide 7 7 Chapter 7 Example Problems A2 A3 E E E tl t2 t3 X 77 ZA WY 20 20 a 20 Figure 7 2 Cantilever Beam Vibration Model Analysis Model Description Web is modeled by CQUADA elements Caps are modeled by ROD elements Material E 1 03E7 psi Poisson ratio 0 3 Weight density 0 1 Ibs in Non structural mass Lumped masses at top and bottom nodes 15 lbs each Design Model Description Objective Minimization of structural weight Design variables Cross sectional areas of bar elements symmetry imposed on the top and bottom elements Constraints Fundamental bending frequency at or above 20 Hz Normal modes analysis is requested in Solution 200 by setting the Case Control parameter ANALYSIS MODES In order to determine a minimum weight design subject to a lower bound constraint on fundamental frequency the thicknesses of the three shear panels and the six rod cross sectional areas are allowed to vary These variations are descr
312. mization D 7 Sequential Linear Programming Sequential Linear Programming see reference 10 was briefly introduced in Structural Optimization This section presents the method in greater detail The basic concept is actually quite simple First Taylor Series approximations are created for the objective and constraint functions These approximations are then used for optimization instead of the original nonlinear functions Now when the optimizer requires the values of the objective and constraint functions these are very easily and inexpensively calculated from the linear approximation Also since the approximate problem is linear the gradients of the objective and constraints are available directly from the Taylor Series expansion Traditionally the SIMPLEX algorithm is used for solving the linear approximate problem However the optimizer in NX Nastran uses the Modified Method of Feasible Directions since it solves linear problems nicely and works well for the size of problems normally considered The general algorithm proceeds in the following steps 1 Forthe current values of the design variables sort the constraints and retain the most critical for use during this cycle Typically about 5 NDV constraints are retained The reason that all constraints are not retained is that there may be thousands and most may be far from critical Therefore it would be wasted effort to calculate their gradients 2 Create first order Taylor
313. mization User s Guide 1 PSHELL 7 150 08 150 ELEMENT GROUP 8 CQUAD4 808 8 807 808 908 100 100 100 100 1 CQUAD4 809 8 808 809 909 1 1 1 1 PSHELL 8 150 08 150 ELEMENT GROUP 9 CQUADA 909 9 908 909 1009 100 100 100 100 CQUAD4 910 9 909 910 1010 PSHELL 9 150 08 150 ELEMENT GROUP 10 CQUAD4 1010 10 PSHELL 10 150 EIGENVALUE EXTRACTION INFORMATION 15 RETAINED MODES 5 EIGRL 500 15 0 5 FREQUENCY DEPENDENT LOADING DATA 1009 1010 1110 08 RLOAD1 700 710 LSEQ 720 710 730 PLOAD2 730 1 0 101 PLOAD2 730 E20 201 PLOAD2 730 1505 301 PLOAD2 730 1 0 401 PLOAD2 730 1 0 501 PLOAD2 730 1 0 601 PLOAD2 730 1 0 701 PLOAD2 730 1 0 801 PLOAD2 730 1 0 901 PLOAD2 730 14075 1001 TABLED1 800 t 0 0 1 0 1 0E FREO1 740 20 1 180 tabdmp1 2000 0 0 0 20 1000 0 0 20 endt DESIGN MODEL SPECIFY DESIGN VARIABLES DESVAR 1 Tl 08 001 1 0 DESVAR 2 T2 08 001 1 0 DESVAR 3 T3 08 001 1 0 DESVAR 4 T4 08 001 1 0 DESVAR 5 T5 08 001 1 0 DESVAR 6 T6 08 001 1 0 DESVAR 7 TT 08 001 1 0 DESVAR 8 T8 08 001 1 0 DESVAR 9 T9 08 001 1 0 DESVAR 10 T10 08 001 1 0 RELATE DESIGN VARIABLES TO PLATE THICKNESSES DVPREL1 101 PSHELL 1 4 01 1 1 8 00
314. mization parameter defaults DOPTPRM IPRINT 3 DESMAX 15 DELP 0 5 pl 1 Listing 7 9 7 50 Design Sensitivity and Optimization User s Guide Example Problems 7 9 Design Optimization with Composite Materials In the optimal design of laminated composite structures individual ply thicknesses and orientations are often selected as design quantities This example shows how a simple composites design problem may be solved using NX Nastran Figure 7 21 shows a composite tube that is to be subjected to an internal pressure load of 100 psi The Hill failure theory will be used to ensure a satisfactory design where an index value in excess of 0 9 will be used to indicate failure Both ply thicknesses and orientations are to be modified The tube is modeled with 160 CQUADA elements oriented such that each element x axis is in the hoop direction and the surface normal faces outward Ply angles are thus measured from the hoop direction corresponding to 0 Due to input deck size the file has been abbreviated in Listing 7 10 From the PCOMP entry note that the structure is built up in eight plies The outer layer pairs are initially at x85 orientations to the hoop direction while the four core layers are at 60 orientations All plies are of equal thickness and material type We would like to vary the overall thickness by allowing the individual plies to change uniformly The design variable to ply thickness relations are speci
315. model data 4 14 Design Sensitivity and Optimization User s Guide Input Data DRESP1 Purpose Defines direct or first level analysis responses to be used in design sensitivity and optimization The responses identified here are those that are directly available from the analysis results as opposed to second level responses that are defined using DVPREL2 and DEQATN entries Entry Description DRESP1 LABEL RTYPE PTYPE REGION ATTA ATTB ATT1 E Jm oen cp o n gg o uw d Field Contents ID Unique entry identifier Integer gt 0 LABEL User defined label Character Response type See DRESP1 listing in Commonly Used Commands for RTYPE dade Design Optimization Character Element flag PTYPE ELEM or property entry name Used with PTYPE element type responses stress strain force etc to identify the property type since property entry IDs are not unique across property types Character ELEM PBAR PSHELL etc REGION Region identifier for constraint screening Integer gt 0 Response attributes See DRESP1 listing in Commonly Used Commands PRAET UST for Design Optimization Integer gt O or Real or blank Associated Entries A response identified on a DRESP 1 entry may be used either as a constraint or an objective The response is identified by its ID number on DCONSTR entries A DRESP1 response can also be used as input to synthetic responses defined on DRESP2 entries It is reference
316. mpares the results of two consecutive finite element analyses The terminology is chosen to indicate that the test is based on hard or conclusive evidence Independent Design Variable is a design variable defined on a DESVAR entry and is not expressed as a function of any other design variables DLINK entries are used to define this dependence By default all design variables are independent In design optimization the optimizer varies the independent variables directly Infeasible Design is defined as a design that violates one or more constraints Kuhn Tucker Conditions provide the formal definition of an optimal design see Numerical Optimization These conditions are implicitly evaluated in NX Nastran in connection with convergence detection at the optimizer level Move Limits serve to limit the region in which the optimizer may search during the current approximate optimization Since the approximate subproblem is only valid in the region of the current design see Approximation Concepts in Design Optimization move limits serve to restrict the search in this approximate design space to avoid numerical ill conditioning due to poor approximations Nonunique Optimal Design is an optimum design at which the objective function and all active constraints are relatively insensitive to changes in the design variables See Figure 3 15 in Convergence Tests for a qualitative explanation of the situation Objective Function Design Objecti
317. n Eq D 33 is included simply as a means to reduce the true objective function if possible while seeking a feasible design Now consider the value of 6 in Eq D 34 This is referred to as a push off factor since its value determines how hard to push away from this violated constraint If 0 0 0 even increasing W will not require a move anywhere except tangent to the constraint and this move will probably not bring the design back to the feasible region Therefore some positive value is needed In the optimizer the more the constraint is violated the greater the push off factor value should be This can be accomplished by expressing the push off factor 0 as a quadratic function of the j th 4 1 e D er Thus q 1 2 gr y oia E Equation D 36 constraint such that 0 0 at 8j 9 lt 50 0 Equation D 37 Therefore when a constraint becomes active it is included in the direction finding process However as the constraint becomes more critical particularly as it becomes violated the push off factor increases The value of 6 has been limited to 50 0 as an arbitrary numerical limit This limit is based on experience that shows this value to be large enough to force the design back to the feasible region without causing too much numerical ill conditioning in the algorithm D 16 Design Sensitivity and Optimization User s Guide Numerical Optimization Finally consider the value of y This parameter is initially cho
318. n out to be unrealistic You might get an extremely stiff 200 ton wing out of this process You also might have asked for a minimum weight design but allowed for negative physical dimensions The optimizer may have no trouble minimizing the weight by adding negative mass but this design may produce a true engineering challenge on the factory floor An optimizer is only able to search for your definition of a best design within the region you have defined Configuration or trade off studies cannot be directly addressed by a numerical optimizer although an optimizer can be used to investigate the benefits of one design over another An optimizer cannot weigh the benefits of a cast aluminum versus a welded steel support member but it can tell you the best that each design is capable of and then let you decide which design path to pursue Some of these considerations are in fact active areas of research Even with these limitations design optimization offers an extremely powerful set of design tools What Do Need to Use Design Optimization Effectively Analysis Skills Since design sensitivity and optimization depends on the results of analysis a reasonable level of skill preparing NX Nastran analysis models is required Design optimization can use analysis results from statics normal modes buckling direct and modal frequency modal transient acoustic 1 6 Design Sensitivity and Optimization User s Guide Getting Started and aeroelasti
319. n the design model to propose improved designs Design Optimization refers to the automated redesign process that attempts to minimize a specific quantity subject to limits or constraints on other responses This process is often referred to as Structural Optimization in this guide to indicate the application of these methods to structural redesign tasks Design Sensitivity Analysis is the process that is used to determine rates of change of structural responses with respect to changes in design variables The resulting partial derivatives or design sensitivity coefficients dr dx can be used directly to perform parametric design studies or as input to a numerical optimizer for design optimization A 2 Design Sensitivity and Optimization User s Guide Glossary of Terms Design Sensitivity Coefficient see Design Sensitivity Analysis Design Space is the n dimensional region over which the objective and constraints are defined where n is the number of independent design variables It is within this space that the optimizer searches for an optimal design while simultaneously trying to satisfy all of the inequality constraints an optimal feasible solution Design Variable Linking allows a design variable to be expressed as a linear function of other independent design variables This linking must be explicitly defined by the engineer It is one of the components in the family of approximation concepts Design variable linking can be used to e
320. n the following sections Even though Eq 3 43 through Eq 3 49 are expressed in terms of the design variables x the independently varying properties p may also be used if DVPREL2 relations appear in the design model In practice then the set of design variables in the preceding equations is generalized in the code to include not only the independent design variables but also the independently varying properties Design Sensitivity and Optimization User s Guide 3 29 Chapter 3 Design Sensitivity and Optimization in NX Nastran Use of Adjoint Loads in Design Sensitivity Analysis Design sensitivities can be calculated by solving Equation 3 46 This approach to calculating design sensitivities is often referred to as the pseudo load method where the right hand side of Equation 3 46 represents the so called pseudo load The pseudo load method is useful for calculating design sensitivities in many instances However the pseudo load method may become cumbersome when There are many design variables as compared to the number of response constraints for which gradients need to be calculated The responses in question are functions of more than one of the basic responses such as displacement and velocity For these cases NX Nastran automatically uses the adjoint load method For the case of comparatively many design variables adjoint loads are generated based on the number of responses For the case where the responses are func
321. n we really need since as the panel thickness is changed the stresses will probably vary more or less in unison It is probably reasonable to consider only the few largest stresses in this region and temporarily ignore the rest even though they are greater than TRS Grouping similar constraints together is the idea behind constraint regionalization Design Sensitivity and Optimization User s Guide 3 13 Chapter 3 Design Sensitivity and Optimization in NX Nastran Note The default for NSTR is 20 In a region of high stress modeled using a fine mesh of plate elements reduction to an NSTR number of constraints could easily lead to a hundredfold reduction in the number of design constraints Suppose the first 30 constraints in Figure 3 4 actually belong to three separate regions as shown in Figure 3 5 In Region 3 eight constraints are greater than or equal to the truncation threshold It may be advantageous to consider only a few of the largest retained constraints from this region and disregard the others Suppose that NSTR 2 for Region 3 Then only two out of the eight possible constraints in this region are retained and six are discarded even though their values exceed the truncation threshold In Regions 1 and 2 less than two constraints are retained therefore no further deletion takes place In the end only three constraints out of the original 30 are retained in the approximate problem
322. nd JTMAX Finite Difference Step Sizes Design sensitivity coefficients are computed in NX Nastran using a semianalytic approach as explained in Design Sensitivity Analysis For example static displacement sensitivities are determined from the solution of the differentiated equilibrium equations O u O P K Ox Ox Ox K Equation 4 3 Note Central differences are available in NX Nastran In fact central differences are the default for shape optimization where 0 K Ox is approximated using finite differences as aLK Ke DELB x K Ox DELB Xi 1 Ih Equation 4 4 The value of the finite difference move parameter DELB can be changed with the DOPTPRM entry If the product of DELB and x is small the finite difference approximation may not be very accurate A minimum finite move is then provided by DELBM which is used if DELB x s DELBM Comparison Between the Initial and Calculated Values of Analysis Properties The initial value of an analysis property given on a property Bulk Data entry may not always correspond to the value of the property calculated using the initial values of the design variables If the relative difference between the properties differ by more than PTOL the program terminates with an error condition The default of 1 0E35 implies that nearly all discrepancies are tolerated and the initial analysis property values will be overridden using the design
323. nd derivative must also be considered where EE de 6 0 Equation D 53 Since the result is positive the proposed solution represents a minimum of F instead of a maximum You can verify the results by a rough plot of the original three points given and passing them over a quadratic curve Now assume that the function to be approximated is a constraint In this case the value of x is sought that drives the function to zero i e forces the design to a constraint boundary To achieve this the value of x must be found that makes Eq D 51 equal to zero Setting Eq D 51 to zero and solving for x yields 4 0 J100 0 2792 2 3874 Equation D 54 For each of these values of x the estimated value of F is zero The question is Which one should be chosen In this case note that the initial value F x 0 is positive Therefore x 0 2792 is a proposed lower bound on a since this overcomes the constraint violation to a quadratic approximation Thus this is a possible lower bound on a aj However more information is available The value of the variable x 2 3874 is the point beyond which the constraint again becomes violated Therefore this value of x is a possible upper bound on a ay This can be verified by calculating the gradient of F at a and ay to determine if it is negative at aj i e forcing gj to be more negative and positive at ay i e forcing gj to be violated To verify this simply substitute the
324. nd r3svals F for ILP as necessary to suit your application If necessary add new subroutines or chains of subroutines To include multiple responses in the same routine separate the responses with if else program structures and use the TYPE field on DRESP3 Bulk Data entry to specify which response to calculate The data included on a DRESP3 Bulk Data entry is passed to the executable file in three one dimensional arrays It is necessary to code the external program so that it interprets the information contained in these arrays correctly For details on how the arrays are formatted see Format of Data Passed to the Executable File Create the executable file Once you have completed customizing the provided FORTRAN subroutines update the provided makefile build script with the object file names of any new subroutines When the makefile build script runs it creates an executable file called dr3serv Make sure all files referenced in makefile are in the same directory as dr3serv or point to the necessary directory in makefile Add a CONNECT statement to the input file Add a CONNECT File Management statement to the NX Nastran input file The CONNECT statement creates the relationship between the DRESP3 Bulk Data entries and the executable file The format of the CONNECT statement is CONNECT DRESP3 GROUP EVALUATOR where The GROUP field on the DRESP3 Bulk Data entries should match the GROUP field on the CONNECT statement Desi
325. nds respectively within a subcase o DESSUB refers to the constraint definition DCONSTR bulk data entry or DCONADD bulk data entry which specifies multiple DCONSTR entries which in turn references the appropriate DRESPi DRESP1 DRESP2 or DRESP3 entry or entries o DESOBJ directly references the relevant DRESPi bulk data entry or entries Byusing global and or objective function definitions that you specify with DESGLB and DESOBJ Case Control commands that appear in the Case Control section above any subcases However if such DESGLB or DESOBJ commands indirectly point to a DRESP1 entry assigned to specific subcases with a DRSPAN command the software uses them to combine responses from various subcases For example consider an analysis in which a DESGLB commands points to a DCONSTR constraint entry on a DRESP2 entry that combines two displacement DRESP1 responses For this to be valid the two DRESP1 entries must be assigned to subcases using sets referenced by DRSPAN commands in the relevant subcases for which they are to be computed For more information see DRSPAN DRESP2 and DRESP3 in the NX Nastran Quick Reference Guide Case Control and Design Response Form Structural responses may often be computed in different forms Dynamic responses may either be expressed using complex quantities or in a magnitude phase representation Plate element strains may be output in strain curvature or strain at fiber form Where these types of
326. ne our shape basis vectors This data can be DBLOCATEd using the following FMS Section statements assign fl plate33 aux MASTER logical f1 dblocate datablk ug ugd geom1l geomld geom2 geom2d 2 34 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization where plate33 aux MASTER is the master DBset file corresponding to the auxiliary model analysis results shown in Figure 2 18 and Figure 2 19 Having input this information we need to identify the design variables that are to act as multipliers of these basis vectors This is done using the following DVSHAP Bulk Data entries SDVSHAP DVID COLI SFI COL2 SF2 DVSHAP 1 1 1 0 DVSHAP 2 2 32 0 The first entry defines Design Variable 1 DVID 1 as the multiplier of the first DBLOCATEd displacement vector COL 1 1 using the factor 1 0 SF1 as the constant of multiplication This process defines the first shape basis vector A similar process on the second DVSHAP entry defines the next shape basis vector Note that we now have two basis vectors one for each of the a and b elliptical variations of the hole boundary As the optimizer changes x1 and x2 the grid locations are varied according to Ax AG 1 04 1 0u9 1 AX Equation 2 27 where u4 is the first DBLOCATEd basis vector note the 1 0 multiplying factor and us is the second Geometric Boundary Shapes Rather than externally generating the shape b
327. nforce symmetry ensure equivalent element properties and so on It can also improve the convergence characteristics of the problem by reducing the number of independent design variables that must be considered by the optimizer Designed Coordinates are grid coordinates that are allowed to vary in shape optimization If a designed grid is to only move in the x y plane for example then it will have only two designed coordinates x and y The z coordinate for that grid will not be designed More specifically the designed coordinates correspond to non zero entries in the basis vector matrix for shape optimization Designed Grids are those grids whose locations vary in shape optimization Grids on a moving boundary would thus be designed as would be grids on the interior of the structure that are updated Grids on a fixed boundary one that does not change during shape optimization would not be Designed Grids See also Designed Coordinates Direct Input of Shapes refers to the method of DBLOCATEing displacement vectors that have been generated using an external Auxiliary Model Once input they may be used for shape sensitivity and optimization Feasible Design is a design that satisfies all of the constraints A feasible design may not be optimal though Finite Differences are a numerical method for approximating derivatives The method derives from the Taylor series expansions of arbitrary functions First forward finite differences see Eq 1
328. nstead a lower bound is carried forward to retain the nearest four proposed values of a If fewer than four values are available including a 0 then all are retained For each retained value of a the value of the objective and all constraint functions are also retained These are then used to interpolate for an improved solution to the value of a D 5 Interpolation for a Once bounds on the solution to the one dimensional search problem have been established it is desirable to refine the solution as much as possible To achieve this a polynomial interpolation of the objective and constraint functions is used The basic tool used here is a simple polynomial curve fit which may be linear quadratic or cubic depending on the amount of information available If more than four pieces of information are available then higher order approximations could be used However experience has shown that a cubic fit is adequate to approximate the functions without introducing too much numerical error Whether the objective function or a constraint is to be approximated the basic approach is the same The process is demonstrated using a quadratic fit to three function values The other approximations are similar and are explained in detail in Vanderplaats An Efficient Feasible Direction Algorithm for Design Synthesis 3 A quadratic polynomial is defined as D 20 Design Sensitivity and Optimization User s Guide Numerical Optimization
329. nt set as a union of DCONSTR entry sets Entry Description Field Contents DCID Design constraint set identification number Integer gt 0 DCi DCONSTR entry identification number Integer gt 0 Associated Entries 4 6 Design Sensitivity and Optimization User s Guide Input Data Discussion Constraints must be selected in Case Control The DCONADD entry allows a number of DCONSTR entry sets to be unified into a single set making their selection in Case Control easier DCONSTR Purpose Places limits on a design response When selected in Case Control by either DESGLB or DESSUB the DCONSTR sets define the design constraints Entry Description Field Contents DCID Design constraint set identification number Integer gt 0 RID DRESPi entry identification number Integer gt 0 LALLOW Lower bound on the response quantity Real Default 1 0E20 UALLOW Upper bound on the response quantity Real Default 1 0E20 Associated Entries The DRESP1 and DRESP2 entries identify responses that may be constrained DRESP1 LABEL RTYPE PTYPE REGION ATTA ATTB ATT1 LABEL EQID REGION o DTABLE LABL1 LABL2 The DCONADD entry can be used to form the union of a number of DCONSTR sets Discussion During the optimization process material may be removed from and or added to the structure to achieve the optimum objective As this occurs the response of the structure to the applied loads changes Member st
330. nternal Representation of the Sensitivity Coefficients in NX Nastran For reasons of efficiency as well as accuracy NX Nastran uses a slightly different internal representation of these coefficients than described above depending on how the design variables are related to the analysis model properties For DVPREL 1 type relations the set of independent design variables is taken as the basis for the design sensitivity coefficients see the DLINK entry for a definition of independent and dependent design variables This is an efficient choice since a large number of properties may be a function of a much smaller set of design variables For DVPREL2 relations the basis for sensitivity analysis is taken to be the set of analysis model properties referenced on all of these second level relations Inside the code the sensitivities are then Equation 3 34 Choosing properties as the basis greatly improves the accuracy of the approximate responses used in design optimization Since an explicit relation between the properties and the associated design variables is given by the DVPREL2 DEQATN entry pair this information can be used to evaluate the properties for a given change in the set of design variables Computing the sensitivities with respect to the design variables would have linearized this relation resulting in a less accurate approximation To summarize the sensitivity coefficients used internally in NX Nastran are Design Sensitivity and
331. nts these are implicit in the design variables as well These can include both first as well as second level response types where Design Sensitivity and Optimization User s Guide 3 27 Chapter 3 Design Sensitivity and Optimization in NX Nastran Equation 3 42 In NX Nastran the sensitivities of these responses with respect to changes in the design variables are approximated using first forward finite differences as gt 0 0 0 30 rj r x Ax U Au r x u a M L or A Ox i Ile Equation 3 43 where rj is the j th design response Eq 3 43 accounts for the functional dependency of the response on both the design variables and the displacement solution However since the displacements are implicit functions of the design variables the quantity Au must be computed as gt gt Ou Au Ax Ox Equation 3 44 Eq 3 44 requires that the displacement sensitivity be known The sensitivities can be computed by differentiating the equations of static equilibrium K u P Equation 3 45 to obtain 0t lP o K Pos Uo Ox Ox e Equation 3 46 This equation is solved in NX Nastran to obtain the displacement sensitivities Once these are known the sensitivities of all necessary responses can be found from Eq 3 43 and Eq 3 44 Computationally the solution of Eq 3 46 is identical to the solution for Eq 3 45 except that they differ in the right hand side load vectors In fact the
332. nts are violated at a 0 the minimum proposed a is selected from the a that minimizes the objective and the a that represents the upper bound As indicated by the difficulty in describing the one dimensional search process this is a particularly complex part of the optimization process The description given here is only a simple outline of the process In practice decisions must be made based on how many proposed values of a are available using linear quadratic and cubic curve fits to estimate the best value of the move parameter Since it is impossible to predict all the possible combinations of the situations that may arise this is necessarily a search process that is based more on experience than theory The key is the need to consistently improve the design If the absolute best solution does not appear during this one dimensional search further iterations provide other chances until the convergence criteria are satisfied This explains why more complex methods are not used with their associated number of function evaluations and cost during the one dimensional search In the optimization algorithm used in NX Nastran the one dimensional search process is somewhat more sophisticated but basically follows the same process as outlined here At this point the best search direction to improve the design has been determined and a search conducted in that direction The question now is How can it be determined if the optimal design ha
333. of the type 3 8 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran f X t x E la cj jaj Equation 3 4 Where the items in braces are separately vectors and the items in brackets 0 and C4 are matrices Eq 3 4 indicates that design variable linking effectively partitions the vector of design variables into an independent set x and a dependent set p Each DLINK entry each row in Eq 3 4 defines a single dependent variable as a linear combination of an independent set of design variables The dependent design variable still needs to be defined using a DESVAR entry The DLINK entry just describes the explicit linking relations A dependent design variable can be used in the design model in exactly the same way as an independent design variable It can be used in connection with either property or shape optimization on DVPREL1 and DVPREL2 relations and so on There are no restrictions The optimizer however need only consider those variables that are in the independent set The corresponding dependent design variable changes are determined from the independent set using the linking defined by Eq 3 4 Example Quite often the DLINK entry may be the only way to implement a given design variable linking scheme For example imagine that a rectangular cross sectional bar consisting of ten elements is to be designed This bar should be smoothly tapering in a stepwise fashion
334. of this process the values of the objective and constraints are known at a 0 and a a4 where q4 is the initial estimate for a Following are three cases that must be considered when deciding if a4 is the upper bound ay Case 1 All constraints are initially satisfied all gj a 0 lt CTMIN In this case the search direction is the one that reduces the objective Therefore if at a4 the objective is greater than at ag a 0 Design Sensitivity and Optimization User s Guide D 19 Appendix D Numerical Optimization then a is an upper bound on a Also since the initial constraints are all satisfied if any g 01 gt CTMIN then a4 is an upper bound because the search has moved into the infeasible design space If either the objective function has begun to increase or some constraint has become violated then ay is chosen as the upper bound q Case 2 One or more constraints are violated greater than CTMIN at a 0 and the objective function is increasing dF xa da gt 0 Here a search direction has been chosen that should reduce the constraint violation Thus at a4 if the maximum constraint value is greater than the maximum constraint value at a 0 then ay is the upper bound a This usually occurs when some new constraint becomes violated at a4 If this happens it is probably not possible to overcome the constraint violations in this direction but the maximum constraint violation may at least be red
335. ol Note Constraint deletion reduces the cost of sensitivity analysis by temporarily deleting design constraints and the associated responses The sensitivity of these responses does not need to be computed for the current design cycle Formal approximations allow the optimizer to make design changes without having to invoke a full finite element analysis each time Formal approximations Formal approximations allow the optimizer to make design changes without having to invoke a full finite element analysis each time In the code the objective function and all retained constraints are cast in terms of high quality approximations that are explicit in the design variables The optimizer refers to these approximations when it requires function evaluations instead of the costly and implicit finite element analysis These approximations are constructed automatically in NX Nastran according to one of three possible methods This choice is problem dependent You can choose from direct linearization mixed method and convex linearization The method is selected by setting APRCOD to 1 2 or 3 using the DOPTPRM Bulk Data entry The mixed method APRCOD 2 is the default Approximate Model Figure 3 1 illustrates the connection between the finite element analysis the approximate model and the optimizer Note that the approximate model constructed using the results of the finite element analysis provides the optimizer with the de
336. olation is now quite useful for describing shape changes over the Primary Structure s boundary However these boundary displacements must still be interpolated over the structure s interior in order to form shape basis vectors for the entire structure This is achieved in the primary model section of the Bulk Data Shape Changes over the Interior BNDGRID entries provide the boundary conditions for the shape interpolation steps The relation between BNDGRID entries and the auxiliary boundary model solutions is similar to that of SPC s and SPCD s The auxiliary boundary model solutions are imposed as enforced displacements on the primary structure This is like an SPCD These displacement degrees of freedom must be present on BNDGRID entries as with SPC entries Degrees of freedom listed on the BNDGRID entries are either enforced or fixed depending on whether or not an auxiliary boundary model solution exists for them All other degrees of freedom are considered free and are solved for in the interpolation For example grids 143 154 and 165 are located at the tip of the upper surface see Figure 7 9 Their displacement components are determined from the auxiliary model analysis This is indicated by using a BNDGRID entry with a translational component of 123 given for each of these grids see listing 7 7 In contrast the displacements at grids 110 121 and 132 located directly below the previous set of grids must be solved for during t
337. on and Screening DSAD DSAR DSPRM WEIGHT DESCON SDR2STAT Hard Convergence Check DOM12 L and Design Cycle 2 EXITOPT e g One Pass Through Loop YES Ex EXITOPT 7 Maximum Design Cycles ompleted E y NO O amp 3 JEXITOPT D a g If Hard Y e Convergence 3 Data Recovery and NO Exit FEAOPT FEA DATARECOVERY EXITOPT Generate Pseudo load Vectors for 7 Sensitivity Analysis Across Analysis Disciplines Subcases and Superelements DSABO DSAE DSAM DSAN DSAP DSAR DSVG1 DSVG2 DESAERDR Generate Solutions for PSLGDV Sensitivity Analysis SEDRDR FEAOPT SENSITIVITY ok S A 5 10 Design Sensitivity and Optimization User s Guide Loop on Design Cycles Solution Sequences Compute Necessary Response Sensitivities for All Retained Constraints RESPSEN DSAF DSAH DSAL DSAW DSDVRG SEDRDR DSFLTF DSFLUTE DSPRM DSVG1 DSVG4 SDR2STAT G 4 Exit and Print Sensitivities DSTAP2 DOM6 SDSC EXITOPT Optimization with Respect to Approximate Models DOMO Output Updated Grid Entries DOM11 Design Optimization Output DOM10 Soft Convergence Check DOM12 5 1 If Soft Convergence YES Data Recovery and Exit FEAOPT FE A DATARECOVERY EXITOPT NO 7 Analysis Model Updates Auxiliary Primary
338. on input arguments are formal arguments that are defined at runtime The DRESP2 entry for responses and the DVPREL2 entry for design variable to property relations specify the input or actual arguments to the appropriate DEQATN entry Note If nested equations are used on a DEQATN entry the return value is determined from the last equation on the entry A nested form of equation definition may also be used Multiple equations can be specified on a single DEQATN entry with the input to the latter equations based on the results of the previous equations The return value of the function call is the last equation in the DEQATN entry The argument list however must contain all of the formal arguments used by all equations on the entry For more details see Relating Design Variables to Properties and Identifying the Design Responses DESVAR Purpose Defines the design variables to be used in design sensitivity and optimization Design sensitivity analysis computes the rates of change of design responses with respect to changes in the design variables In design optimization the set of design variables are the quantities modified by the optimizer in the search for an improved design Entry Description Field Contents ID Unique design variable identification number Integer gt 0 LABEL User supplied name for printing purposes Character XINIT Initial value Real XLB lt XINIT x XUB XLB Lower bound Real defa
339. on seen by the optimizer Note The parameter DSNOKD can be used to suppress the computation of the differential stiffness derivative See Parameters for Design Sensitivity and Optimization Shape Sensitivity Shape sensitivities are computed in Solution 200 whenever design variables are used to describe shape changes They are computed automatically in connection with design optimization or if a sensitivity analysis alone DSAPRT Case Control command or PARAM OPTEXIT 4 is requested In shape optimization the design responses are functions of the shape defining design variables Recall the reduced basis formulation used in connection with shape optimization AG T Ax Equation 3 64 In Eq 3 64 changes in grid coordinates AG are expressed in terms of changes in the design variables Ax and the shape basis vectors T For semianalytic shape sensitivities grid coordinates are perturbed a finite amount 8g Er L i Equation 3 65 The subscript in Eq 3 65 characterizes the relation as a variation in shape for the i th design variable E is the maximum strain energy norm computed using the T basis vector STPSCL is a scaling factor default 1 0 that can be changed using the DOPTPRM Bulk Data entry The maximum strain energy norm used in Eq 3 65 is computed by first determining the grid forces due to a shape basis vector displacement 3 34 Design Sensitivity and Optimization User s Guide
340. on the approximate model See Approximation Concepts in Design Optimization The optimizer has been able to achieve a reduction in the peak approximate stresses These and all other necessary structural responses will be updated again on the next design cycle with a full finite element analysis and the process will be repeated again Once the approximate optimization is complete for this cycle a soft convergence test is performed and the results are reported in the following table KERR KK Kk Ok k k CK RRR K K OK Kk Ck k k A KARA AA A AAA K KORR OR Re INSPECTION OF CONVERGENCE DATA FOR THE OPTIMAL DESIGN WITH RESPECT TO APPROXIMATE MODELS SOFT CONVERGENCE DECISION LOGIC AAA ek Kok kk eee ee AR AA he ke kc kk kk kk kk kk kk kk ck kk kk kk kk kk ke ko kk ee ke RELATIVE CHANGE IN OBJECTIVE 9 4012E 01 MUST LESS THAN 0000E 03 ABSOLUTE CHANGE IN OBJECTIVE 5 4381E 00 MUST LESS THAN 0000E 02 MAX OF RELATIVE PROP CHANGES 1 0000E 00 MUST LESS THAN 0000E 03 AND MAXIMUM CONSTRAINT VALUE 6 2725E 01 MUST LESS THAN 0000E 03 OR MAX OF RELATIVE D V CHANGES 1 0000E 00 MUST LESS THAN 0000E 03 St fff fe eee Sk Sk ke e he kk kk ke kk Sk ke kk kk kc koc kk koc ko kk kk koc kk kc ARA kk ke ke kk ke ke ko kk e Design Sensitivity and Optimization User s Guide 6 9 Chapter 6 Output Features and Interpretation Even if soft convergence had been achieved which it was not this test will not terminate the design cycles unless the Bulk Dat
341. one Such a direction would be perpendicular to the previous direction However there are very solid theoretical reasons not to do this and experience attests to the accuracy of the theory Instead we move in what is called a conjugate search direction or more precisely an A conjugate direction where A is the matrix of second partial derivatives of the objective function The A matrix is not actually computed but there are methods for approximating A that offer a guaranteed convergence rate for problems where the objective is a quadratic function In fact the successive use of steepest descent directions may never converge While there are methods that are considered more powerful than the one used here the simple conjugate direction method has proved to be quite reliable The Fletcher Reeves conjugate direction method used here defines a search direction as 5 Fletcher R and Reeves C M Function Minimization by Conjugate Gradients Computer J Vol 7 No 2 pp 149 154 1964 Design Sensitivity and Optimization User s Guide D 11 Appendix D Numerical Optimization gt 1 29 1 vr pS Equation D 27 where B ved p vea Equation D 28 with the double vertical bars indicating the Euclidean norm of each gradient Figure D 4 Geometric Interpretation of the Steepest Descent Method D 12 Design Sensitivity and Optimization User s Guide Numerical Optimization Figure D 5 Geometric Int
342. ons All of these approximations are available in NX Nastran These tools allow numerical optimizers to be coupled effectively with structural analysis codes where the responses are usually implicit functions of the design variables Solving the resultant approximate subproblem yields an approximate optimum Auxiliary Boundary Model is similar to an Auxiliary Model for generation of shape basis vectors though defined just on the boundary of the finite element model It may be for example an assembly of bar elements along the edge of a two dimensional structure or a group of plate elements over the surface of a three dimensional model It is used to generate the boundary portion of the shape basis vector from which the motion of interior points and the remaining portion of the basis vector is interpolated Auxiliary Model or Auxiliary Structure is a finite element model used to generate grid variations for purposes of shape sensitivity or optimization It usually shares geometry and connectivity with the original or primary structure but with different loads boundary conditions and perhaps material types The resultant deformations are used to generate a basis vector for shape sensitivity and optimization Basic Optimization Problem Statement is a formal definition of the optimization task This problem statement is invoked by the engineer when constructing the design model and by the optimizer when searching for an optimal design See Eq
343. ons from changes in K DSVG1 EGM _ Pseudo load contributions from changes in M DSVG1 EGB Pseudo load contributions from changes in B DSVG1 a Optimization History HIS Design optimization history table DOM12 gt APPEND gt EQUIVX optimization history table DOM12 gt APPEND gt EQUIVX SADO Additions to the O Retony tale dus IG he dond optimization history table due to the current design cycle DOM12 NEWPRM Update of design optimization parameters due to the automatic reduction of move limits DOM12 General DSLIST Superelement processing list for design optimization similar to an SLIST for superelement analysis SDSB DTB Constants from the DTABLE Bulk Data entry DOPR1 OPTPRW Optimization parameters DOPR1 5 6 Design Sensitivity and Optimization User s Guide Solution Sequences 5 3 Solution 200 Program Flow Design sensitivity analysis and optimization both are only available in Solution Sequence 200 The following flowchart describes the organization of Solution 200 or main subDMAP DESOPT and lists some of the more significant modules and subDMAPs invoked Each of the boxes in this high level flowchart describes a major program phase as well as the design sensitivity and or optimization modules used to accomplish that particular task These module names are included inside each of the boxes If program flow is directed to a particular subDMAP each name is listed alongside each of the bo
344. or every design variable grid pair AG Figure 2 15 Grid Point Variation Defined by a DVGRID Entry AG N1 AG COEFF No Ax AG N3 z ij Equation 2 25 DVGRID entries alone can be used to describe shape changes although the method is probably more useful when combined with the direct input of shapes approach Its most powerful form however is in connection with the geometric shapes approach These methods are described later in this section Benefits Since this can be considered the lowest level approach its strength lies in its generality Using DVGRIDs alone the designer has direct control over every designed grid point in the model that is every grid point whose location is to change during shape optimization Design Sensitivity and Optimization User s Guide 2 27 Chapter 2 Design Modeling for Sensitivity and Optimization Drawbacks In all but the simplest of problems the data input can be formidable without a preprocessor The resultant basis vectors are treated as constant and not updated with each design cycle so the risks of mesh distortion are increased Checklist 1 Define the shape design variables using DESVAR Bulk Data entries 2 Establish the corresponding grid variations using DVGRID Bulk Data entries one entry for every design variable designed grid pair in the model This entry ties design variable changes to changes in the structure s shape Of course this may lead to a large amoun
345. orresponding property then never exceeds its lower and upper bounds An exception here is with respect to dependent design variable bounds which are always treated as constraints Numerically Identifying the Active and Violated Constraints A numerical optimizer must use numerical criteria to judge when a constraint is active and when it is violated These criteria are established using the values of CT and CTMIN which appear on the DOPTPRM entry These quantities are shown in Figure 3 10 for a single inequality constraint in a simple two design variable space A constraint is considered active if its numerical value exceeds CT Once a constraint is active its gradient is included in the search direction computation See Appendix D on numerical optimization for further details An active constraint may subsequently become inactive if its value falls below CT X Feasible Region g x 0 Infeasible Regio SS 24 _ gj 5 CI gj X gt 0 gj 0 8j 3 CTIMN SS r Violated Constraint Figure 3 10 Active and Violated Constraints The optimizer identifies a constraint as violated if its value is greater than CTMIN The default for CTMIN is 0 003 as is seen in Figure 3 11 Thus some small constraint violation three tenths of one 3 52 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran percent by default is tolerated Note that this
346. oximate optimization The design cycle history is shown in Figure 7 28 Compare these results to the original Three Bar Truss example in Three Bar Truss Note that the 100 move limits allow a greater move in x for the first cycle which results in a somewhat lower value of the objective In fact notice that x actually changed by slightly more than 50 in the original problem As mentioned in Optimization with Respect to Approximate Models this may occur since move limits are imposed as equivalent constraints on the corresponding properties Critical constraints may not be satisfied exactly as is seen to be the case here This is usually inconsequential since this design just serves as a starting point for the next optimization cycle 7 64 Design Sensitivity and Optimization User s Guide Example Problems FORK KK RARA hok ck ke k kc ko kk ER oko ke ek ok RRA KK ARK kc KKK eee ke eek eoe ke ke ke SUMMARY OF DESIGN CYCLE HD ST O E Y oe de sk de ek OR ee ke e e e e e ke ee Sk ee de Sk ee ROO ORO OOOO ee TO He ek dee ek dee dee e NUMBER OF FINITE ELEMENT ANALYSES COMPLETED 2 NUMBER OF OPTIMIZATIONS W R T APPROXIMATE MODELS L OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY OBJECTIVE FROM OBJECTIVE FROM FRACTIONAL ERROR MAXIMUM VALUE CYCLE APPROXIMATE EXACT OF OF NUMBER OPTIMIZATION ANALYSIS APPROXIMATION CONSTRAINT INITIAL 4 828427E 00 3 234952E 01 1 2 888302E 00 2 888885E 00 2 020325E 04 2 564473E 02 INTERNAL EXTERNAL DV ID
347. oximation concept that seeks to reduce the number of constraints in the optimization problem This reduction is based on the assumption that the entire constraint set is likely to contain redundant design information Both regionalization and deletion are used to group and screen constraints for temporary deletion Constraints that are temporarily deleted may be retained on subsequent design cycles Dependent Design Variable is a design variable defined on a DESVAR entry and is linearly dependent on one or more independent design variables This dependence is defined by DLINK Bulk Data entries Design Cycle refers to the process that invokes the optimizer once for each design cycle Within a design cycle a finite element analysis is performed an approximate problem is created the optimizer is invoked and convergence is tested A number of design cycles may need to be completed before overall convergence is achieved Design Iteration is the optimizer level process of determining a search direction from available gradient information and performing a one dimensional search in this direction A number of these iterations may be performed before an approximate optimum is found Design Model defines the design variables objective and constraints In the design model design variables are used to express permissible analysis model variations and design responses are used as the objective and constraint functions The optimizer uses the information i
348. p section of the stiffened plate of Figure 2 4 is a rectangular section bar as shown in Figure 2 5 Suppose the bar is characterized by the cross sectional dimensions b and h and is oriented with respect to principal planes 1 and 2 as shown A stress recovery location such as point c might be defined as well The Bulk Data describing the analysis and design model relations could be written as follows ANALYSIS MODEL PROPERTIES PBAR 120 220 0 12 1 6E 3 9 0E 4 Se LO 22 BAR CROSS SECTIONAL DIMENSIONS SESVAR ID LABEL XINIT XLB DESVAR 10 B 23s Nur 11 H 4 BAR PROPERTY RELATIONS A SDVPREL2 ID TYPE PID DESVAR DVID1 DVID2 DTABLE CID1 CID2 DVPREL2 250 PBAR 120 DESVAR 10 Tu DVPREL2 251 PBAR 120 T DESVAR 10 11 DVPREL2 252 PBAR 120 DESVAR 10 11 EQUATIONS DEQATN 501 AREA B H B H DEQATN 502 I1 B H B H 3 12 DEQATN 503 I2 B H H B 3 12 STRESS RECOVERY POINT LOCATIONS SDVPREL1 ID TYPE PID FID S DVID1 COEF1 DVID2 COEF2 DVPREL1 260 PBAR i20 12 t 10 25 DVPREL1 261 PBAR 120 13 Fy T1 NS Design Sensitivity and Optimization User s Guide 2 13 Chapter 2 Design Modeling for Sensitivity and Optimization PBAR 120 defines the bar properties corresponding to the initial cross sectional dimensions Note that the initial values of the design variables listed on the DESVAR entries agree with the initial properties If any di
349. perelements l lille 2 59 Design Variables in Superelement Design Modeling 00 000 ee 2 59 Design Responses in Superelement Design Modeling o o o ooooooo 2 59 Case Contralor 2 60 Balle a Rr 2 60 Design Sensitivity and Optimization in NX Nastran o o ooooo oo 3 1 Approximation Concepts in Design Optimization e 3 2 Why Are Approximations Necessary in Design Optimization Ls 3 2 Approximations in NX Nastran An Overview llle 3 2 Design Variable Linking iude ue 0022023 00 dee bee Pee bee eee E POR REGE o DR 3 4 Constraint Screening gos ced a Women oak e a all cog ie os atom cbr le are 3 12 Formal Approximations lll 3 14 Advanced Topics in Approximations llle 3 19 Design Sensitivity Analysis lt lt oooooooooooooorenean a 3 23 General Considerations ii uuu ct dee wee a op pack wee bee ae RUE CR CE e e RC De a 3 23 Static Response Sensitivities 2l 3 27 Eigenvalue Response Sensitivities leen 3 31 Buckling Load Factor Sensitivities 0 0 00002 ee 3 32 Shape Sensitivity vc 3 34 Dynamic Response Sensitivities celle 3 35 Aeroelastic Sensitivity Analysis eee 3 40 Coupled Fluid Structure Interaction Sensitivity liess 3 42 Optimization with Respect to Approximate Models ooo
350. pproximation for the j th constriant Reciprocal variable approximation for the j th constraint j th constraint function Approximation of the j th constraint function Variation in the i th grid coordinates for finite difference sensitivity analysis Coordinate variation in the i th grid point for shape optimization k th equality constriant Design Sensitivity and Optimization User s Guide B 1 Appendix B Nomenclature Nw T mxN U v xj Identity matrix Stiffness matrix in the finite element analysis Differential stiffness matrix Mass matrix in the finite element analysis Upper move limit on j th property Lower move limit on j th property Vector of initial property values Load vector in the finite element analysis Lower bound on the j th structural response Upper bound on the j th structural response j th first level DRESP2 response j th second level DRESP1 response Vector of analysis model responses displacements stresses eigenvalue etc Search vector in the optimization s one dimensional search Matrix of constant coefficients in reduced basis formulation where M gt N Vector of displacements Vector of intermediate variables used in formal approximations Lower bound side constraint on i th design variable Upper bound side constraint on i th design variable B 2 Design Sensitivity and Optimization User s Guide Pn 9 Nomenclature Vector of design variables
351. problem is solved using the Modified Method of Feasible Directions The matrix B is a positive definite matrix that is initially the identity matrix On subsequent iterations B is updated to approach the Hessian of the Lagrangian functions Now assume the approximate problem of minimizing Q subject to the linearized constraints has been solved At the optimum for this problem the Lagrange multipliers A 1 m can be calculated With 11 Powell M J D Algorithms for Nonlinear Constraints that use Lagrangian Functions Math Prog Vol 14 No 2 pp 224 248 1978 12 Vanderplaats G N and Sugimoto H Application of Variable Metric Methods to Structural Synthesis Engineering Computations Vol 2 No 2 June 1985 Design Sensitivity and Optimization User s Guide D 27 Appendix D Numerical Optimization the Lagrange multipliers available an approximate Lagrangian function can be constructed and used gt to search in the direction defined by S The task is to find a in order to m minimize o F x tp De u max 0 g X j 4 Equation D 71 where q 1 gt ES ES as u N j 1 m first iteration 1 i F uj max jaj 5 A j 1 m subsequent iterations and uj uj from the previous iteration During the one dimensional search approximations are made to the components of because this function has discontinuous derivatives at the constraint boundaries but the components are smooth After the one dim
352. procal variable approximations These approximations are used in various ways according to three different methods available in NX Nastran e Direct Approximations APRCOD 1 Mixed Approximations APRCOD 2 default e Convex Linearization APRCOD 3 Note A DOPTPRM entry is used to select APRCOD Choosing to override the default approximation is usually problem dependent and often without strict rules or guidelines This subsection presents theoretical details that might help you decide when such an override might be appropriate The formal approximations used in NX Nastran are all based on simple first order Taylor series expansions The general form of this expansion is gt gt 30 Or pix JSN Ax E At r ax Ax i 30 2 Equation 3 7 Approximation Errors The required derivative information is available from the design sensitivity analysis as outlined in 30 A2 Design Sensitivity Analysis Note that the approximate response NN wine in Eq 3 7 is linear in the design variable change A Thus we expect some error when we try to approximate responses that are actually nonlinear in A Figure 1 16 in Structural Optimization shows how the error is a function of Ay Move Limits and Approximations During approximate optimization the code places limits on the maximum allowable moves in the design space in order to minimize the errors associated with these approximations In NX Nastran move
353. procedure that all an engineer had to do was to push the optimization button and an optimal design would result Some knowledge of the basic procedures involved in numerical optimizers will aid in understanding why an optimizer does what it does in interpreting the final results of an optimization run and in understanding what may have happened if the results are unexpected For example it is not necessary to know all of the details of sparse matrix decomposition in order to perform a linear static finite element analysis but if singularities are present in your model and the decomposition procedure fails the results are far less mystifying and a modeling solution to the problem may be much more Design Sensitivity and Optimization User s Guide 1 7 Chapter 1 Getting Started apparent if you know something of the basics of the solution procedure Most of the parameters that control the optimizer in NX Nastran can be changed to improve performance for various classes of problems Understanding the significance of these choices allows you to take full advantage of the tools at your disposal Design Optimization and Operations Research Design optimization in structural redesign is actually an application of operations research a branch of applied mathematics to problems in engineering design Generally these are classes of problems in which the optimum allocation of scarce resources is desired Operations research is frequently used to sol
354. published as a breakthrough in engineering optimization is generally a poor method that should not be used in a modern design environment Active But No Violated Constraints The more common search direction problem in which the initial design is feasible and there is at least one active constraint is to find a search direction that improves the design but moves parallel to or 6 Broyden C G The Convergence of a Class of Double Rank Minimization Algorithms Parts and II J Inst Math Appl Vol 6 pp 76 90 222 231 1970 7 Fletcher R A New Approach to Variable Metric Algorithms Computer J SIAM Vol 13 pp 317 322 1970 8 Goldfarb D A Family of Variable Metric methods Derived by Variational Means Math Comput Vol 24 pp 23 26 1970 9 Shanno D F Conditioning of Quasi Newton Methods for Function Minimization Math Comput Vol 24 pp 647 656 1970 Design Sensitivity and Optimization User s Guide D 13 Appendix D Numerical Optimization gt away from the constraint To solve this problem a search direction S must first be found that reduces the objective function without violating any currently active constraints The following equations state the problem in mathematical terms 21 Find the search direction that a q 1 24 minimize VF Q i S Equation D 29 q 1 24 CX subject to Vg S jeJ Feasibility Condition Equation D 30 gt S 2 lt 1 Bounds on S Equa
355. put See Parameters for Design Sensitivity and Optimization as well as the Solution 200 flowchart in Solution Sequences Controls the frequency of design cycle output By default initial and final optimization summaries are output but it can be changed to every n th cycle using this parameter See the DOPTPRM entry in Bulk Data Entries Controls how much design cycle information is output at the intervals determined by P1 P2is set using the DOPTPRM entry See Bulk Data Entries 6 2 Design Sensitivity and Optimization User s Guide Output Features and Interpretation 6 2 Design Optimization Output This section presents design sensitivity and optimization output in the context of one of the example problems from NX Nastran s test problem library One way to present the optimization output would be to include a listing of all of the various output messages and summaries with a short description of each However this approach tends to separate them from their procedural context Instead this section traces selected design optimization output from the stiffened plate example of Stiffened Plate This problem is chosen because it contains most of the output commonly encountered in a typical optimization example The best approach is for you to locate this problem in the test problem library and run it on your installation The output from the actual problem can then be referenced while reading this section Note Your results may
356. r features including both Property and Shape Optimization Dynamic Response Optimization Superelements Composites Acoustic Responses and Restarts Aeroelastic Optimization examples are presented in the NX Nastran Aeroelastic Analysis User s Guide The results in this chapter were obtained on a workstation which may be different than your workstation Although your results may differ slightly running these examples on your own machine is a good way to familiarize yourself with some of the software s capabilities All of these decks are available in the NX Nastran Test Problem Library delivered with your system If you are unable to find these files you should contact your System Administrator for assistance Another good way to learn is by modifying these decks You might try adding Shape variables to a Property Optimization example adding analysis disciplines modifying the constraints and so on Most of these decks are relatively small and should not incur any great CPU related costs In any event the time spent learning with simple examples can more than pay for itself when faced with larger more complex problems Examples in this chapter include Three Bar Truss TPL problem D200X1 DAT This is a classic getting started example It examines weight minimization of a structure subject to stress and displacement constraints from multiple statics subcases Design Variables are linearly related to rod cross sectional areas It introduces exp
357. rarily reduced we would expect a reduced computational cost at both the numerical optimization level and more importantly at the sensitivity analysis level This is because we reduce the number of structural responses for which gradient information must be computed Normalized Constraints in NX Nastran In NX Nastran constraints are internally generated in a normalized form based on upper and lower response bounds selected by the engineer Constraints may be present that depend on displacement stress eigenvalue responses and so on These response types may vary by orders of magnitude To allow all constraints to be treated equally regardless of the response magnitude constraints in NX Nastran are normalized by the absolute values of the bounds The result is that a constraint having a value of 1 0 for example is violated by 100 while a constraint with a value of 0 5 is only 50 of its critical value 0 0 Upon scanning the values of all the normalized constraints we may find that we only need consider a subset above a particular threshold Constraints below this value might then be temporarily disregarded on the assumption that they are not currently driving the design Note This topic is discussed in Design Sensitivity and Optimization in NX Nastran in greater detail but the general idea can be given here Defining the Constraints discusses constraint generation in NX Nastran in greater detail Note A n
358. ready have the stiffness matrix K available in decomposed form as a result of static analysis Design Sensitivity and Optimization User s Guide 1 31 Chapter 1 Getting Started A Simple Linear Design Space At this point it is probably worthwhile to take a look at a linearly approximated design space just to gain a qualitative understanding of the nature of the approximation For this example we can again refer to the simple cantilever beam of Numerical Optimization Basics Figure 1 6 Recall that the design variables for the cantilever beam are the base B and height H of the beam cross section If we construct linear approximations to the objective and constraint equations according to Eq 1 26 we have WB 2A SRDS Ie HM IL Ae AH B 0 0 oH 0 0 B H B H 0 0 0 0 Oo Qo B AB H AH L o BP H L 2 Ap ES AH 0B H y B H B H BU AB HAE ose a Ap 29 AH OB 0 0 oH 0 0 B H B H Equation 1 34 For an initial design at 6 45 the derivatives in the above equation can be evaluated to yield P B AB H AH L 1 35 x 10 2 25 x 105AB 3 0 x 10 AH G B AB H AH L 555 56 92 593AB 24 601AH amp B AB H AH L 2 0576 0 34294AB 0 13717AH Equation 1 35 The design space resulting from Eq 1 35 is shown in Figure1 17 superimposed on the true design space From the graph note that the optimum resulting from this linear approximation happens to be at
359. resses joint displacements natural frequencies etc are altered by the changes that the optimizer is making It is important then to place bounds on the structural responses so that the optimized structure responds in a manner that is acceptable to the designer For example it is common to place upper and lower bounds on the member stresses so that the maximum allowable stresses are not exceeded Joint displacement buckling load factors and natural frequencies might also be bounded Design Sensitivity and Optimization User s Guide 4 7 Chapter 4 Input Data When a bound is placed on a structural response this condition is called a constraint Both first and second level responses may be constrained To constrain a given response quantity a DCONSTR Design CONSTRaint entry is used This entry in turn points to a specific response ID identified on either a DRESP1 or a DRESP2 Bulk Data entry Note that the DCONSTR entry serves only to place bounds on a response defined elsewhere in the design model data These bounds are used to construct normalized constraints for use by the optimizer as discussed in Defining the Constraints DCONSTR entry sets or the DCONADD union of these sets must be selected in the Case Control Section to complete the constraint definition process The Case Control command DESGLB is used to identify subcase independent constraints global constraints while the DESSUB Case Control command is used to select sub
360. roblem statement is used directly in NX Nastran and influences the nomenclature adopted here For example the design objective is defined in the Case Control Section by the DESOBJ command while the design variables and constraints are defined in the Bulk Data Section using the DESVAR and DCONSTR entries respectively We will see how these are used in Design Modeling for Sensitivity and Optimization In conjunction with problems in structural optimization we will discuss design spaces objectives constraints and design variables It is useful to take a look at a simple problem that is not explicitly related to structural optimization just to become familiar with these concepts Example Consider the following optimization problem 1 10 Design Sensitivity and Optimization User s Guide minimize the objective subject to the constraints Feasible Designs Getting Started F X x4 x Equation 1 6 tud EL en gt g4 X dm X1 X5 Equation 1 7 x 2 04 x 201 Equation 1 8 The objective and constraint functions are dependent on two design variables x and x2 The 2 g objective 9 is linear in the design variables while the constraint gt 1 is nonlinear Figure 1 3 shows the two variable design space where shading is used to denote regions in which the constraint or side constraints on the design variables are violated If no constraints are violated we say the current design is feasible although it i
361. rols the single element connected to GRID1110 Design Sensitivity and Optimization User s Guide 7 37 Chapter 7 Example Problems Symmetric Boundary Conditions a Crid 1110 A LLLA Figure 7 16 Clamped Free Plate The dynamic excitation is applied at 180 equal intervals across the range of frequencies The objective is to minimize the mean sum of the squares of the transverse displacements at GRID 1110 or 100 i 2 100 2i 2 min q y 44110 2 gt 71110 i 20 51 Equation 7 12 The above notation indicates that the square of the displacement responses are summed over 1 0 Hz intervals from 20 0 to 100 0 Hz and over 2 0 Hz intervals from 102 0 to 200 0 Hz This particular form was selected simply for illustrative purposes other mean square measures could serve equally as well Fifteen modes were retained for the modal representation This number was selected based on a convergence study which indicated good sensitivity information was obtained with this amount Note A guideline recommendation is to retain at least twice as many modes for modal sensitivity and optimization studies as would be appropriate for an analysis Of course the resultant sensitivities should be checked for accuracy and the number increased if necessary From the input file Listing 7 9 it can be seen that the thickness distribution of the initial design is a constant 0 08 inc
362. ructural responses that NX Nastran does not calculate directly e g local buckling criteria The response equation is written on a DEQATN entry This type of response is referred to as second level as opposed to first level responses that are directly available from the analysis The DRESP2 entry defines the input or actual arguments to these equations The arguments may be design variables table constants first level structural responses or grid coordinate locations DSCREEN Purpose Defines override information necessary to screen the constraints for temporary deletion Entry Description Field Contents RTYPE Response type for which the screening criteria apply Character TRS Truncation threshold Real Default 0 5 Maximum number of constraints to be retained per region per load case Integer NETE gt 0 Default 20 Design Sensitivity and Optimization User s Guide 4 17 Chapter 4 Input Data Associated Entries The regions for constraint screening are automatically established by default These defaults can be overridden though and new ones can be established by defining new regions on the DRESP1 and DRESP2 entries DRESP1 iD LABEL RTYPE PTYPE ATTA ATTB ATT1 Y Am e 1 Discussion To reduce the costs associated with the sensitivity analysis and to reduce the size of the optimization problem many of the noncritical constraints are dynamically deleted during the optimizat
363. s a conventional NX Nastran analysis except that the auxiliary model geometry and boundary conditions have been established with consideration given to the shape redesign goals 2 DBLOCATE the results for shape optimization using the following FMS section commands assuming the master file is filel MASTER ASSIGN FILE1 filel MASTER DBLOCATE DATABLK UG UGD GEOM1 GEOM1D GEOM2 GEOM2D LOGICAL FILE1 2 28 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization Note The data blocks UG GEOM1 and GEOM2 are DBLOCATEd and renamed to UGD GEOM1D and GEOMZ2D respectively The data block names UGD GEOM1D and GEOM2D cannot be changed 3 Define shape design variables using DESVAR entries and correlate these to the DBLOCATEd basis vectors using DVSHAP Bulk Data entries 4 Preview the resultant shape basis vectors to check for modeling errors Geometric Boundary Shapes This method defines allowable shape variations using only the boundary of the structure These shape variations can be supplied manually or with a geometry based preprocessor Either approach relies on BNDGRD Bulk Data entries to define the structure s boundaries and DVGRID entries to furnish the shape variations over these boundaries Shape basis vectors are automatically generated by the code through a process of interpolation of the boundary shape changes to the interior of the structure Additionally the shape bas
364. s Guide Numerical Optimization m4 where 3 x Dx Equation D 67 QU and 7 x D X Equation D 68 The multiplier D in Eq D 67 and Eq D 68 is initially set to RMVLMZ Depending on the progress of the optimization this parameter will be sequentially reduced Typically the first approximate optimization produces a design that violates one or several constraints However as the optimization proceeds this constraint violation should be reduced If it actually increases then D is reduced as well as the move limits on the design variables The convergence criteria for the SLP method are much the same as for the other methods with the addition that the process will not terminate if there are significant constraint violations In this case the process continues until the constraints are satisfied or the maximum number of iterations is reached D 8 Sequential Quadratic Programming This method is considered to be an excellent method by many theoreticians 12 The basic concept is very similar to Sequential Linear Programming First the objective and constraint functions are approximated using Taylor Series Approximations However a quadratic rather than a linear approximation of the objective function is used Linearized constraints are used with this to create a direction finding problem of the form minimize Q S F VF e S 0 58 BS Equation D 69 subject to ve eS 2 0 j 1m Equation D 70 This sub
365. s a large number of properties in terms of a much smaller set of design variables This is shown in the following example Example Assume an optimal plate thickness distribution is to be determined using a combination of constant linear and quadratic basis functions as shown in the figure The design variables act as multipliers of these basis functions that are in turn used to specify the element plate thicknesses Figure 3 2 Regions of Validity for the Plate Thickness Design Sensitivity and Optimization User s Guide 3 5 Chapter 3 Design Sensitivity and Optimization in NX Nastran F x L x L ES kd AS Six pd I A E C 10 Figure 3 3 Basis Functions Suppose the plate thicknesses are to vary in the x axis direction but are to remain constant in the y direction Evaluating the basis functions for each of these ten thickness stations and writing these design variable to thickness relations in matrix form we get Hh ty 1 0 1 0 1 0 f 1 0 0 9 0 81 1 0 0 8 064 4 10 07 049 y fs _ 1 0 0 6 0 36 X A 1 0 0 5 0 25 t 10 0 4 0 16 43 i 1 0 0 3 0 09 8 1 0 0 2 0 04 to 1 0 0 1 0 01 f10 Equation 3 2 3 6 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran Reduced Basis Formulations The thicknesses of the ten element groups are controlled using only three design variables X4 Xo and Xs Each of the matrix columns correspond to constant
366. s all constraints The resulting approximate optimization is then performed with respect to an entirely linear approximate design space You should use this option if you know that the structural responses are well approximated by linear functions in the design variables Otherwise without carefully chosen move limits a greater number of approximate optimization cycles may be required before convergence is achieved APRCOD 2 Default This choice specifies that mixed approximations or more precisely a mixture of approximations are to be used Direct approximations are used for volume weight element force and buckling load responses while reciprocal approximations are used for all other response types This method works well in a wide variety of problem types Because of its reliability it has been selected as the default method APRCOD 3 This selects the convex linearization method Essentially this method chooses either a direct or reciprocal constraint approximation depending on which one provides the larger estimation of the constraint function In other words this method chooses the more conservative of the two approximations For example the direct approximation for the j th constraint is x 0g 2 A1 e 39 o D Ax j Xi i 30 x Equation 3 12 while the reciprocal approximation is Design Sensitivity and Optimization User s Guide 3 17 Chapter 3 Design Sensitivity and Optimization in NX Nastran Equation
367. s are used to define seven shape basis vectors The seven DVSHAP entry scaling factors have been selected such that the maximum component of each shape basis vector is Design Sensitivity and Optimization User s Guide 7 25 Chapter 7 Example Problems unity For example DVSHAP entry number 1 identifies the first shape basis vector in terms of the first DBLOCATE d displacement vector 1 in field 3 The 1 in field 2 indicates that design variable number 1 is to be the multiplier of this vector Furthermore since the maximum component of this vector is 1 0 66 773 as determined from a manual inspection of the data the scaling factor has been given as 66 773 effectively unit normalizing the vector Each design variable acts as a multiplying coefficient of a shape basis vector Initial values are somewhat arbitrary and have been selected as 3 0 here However when combined with 2096 allowable move limits 0 2 in field 7 of the DESVAR entries it can be seen that part of the reason for the choice is that these initial values provide for reasonable move limits on the initial design Lower and upper limits of 1 0E6 and 1 0E6 indicate that the design variables are to be considered effectively unbounded during optimization The final shape after six design cycles is plotted in Figure 7 8 bottom half with the corresponding stress contour plot of the deformed structure top half Note the favorable distribution of von Mises stresses for the final shap
368. s available except the function values and their derivatives with respect to a In fact at first glance it is not obvious that the derivatives of the objective and constraints will respect to a are available However consider the objective function the same algebra applies to constraints and create a first order Maclaurin series approximation to F a in terms of a Remember that gt 1 1 q FR pd tas Equation D 38 24 Thus a first order approximation to EE is N q 1 i OF a Ox ox 0a i 1 Equation D 39 or 1 1 dF x 4 1 eL X F x r P Ses da Equation D 40 Design Sensitivity and Optimization User s Guide D 17 Appendix D Numerical Optimization Ox l 5s F z Ox vr qa 0a But is just the j th entry of the vector Q Also from Eq D 22 Where sj is the ith entry of the search direction vector Therefore 4 1 dF x gam ed do Equation D 41 Since both terms in Eq D 39 are available we have the slope of the function at a 0 for any function objective or constraint for which the gradient is available Now consider how this information might be used Since this is the first step in the optimization process we may optimistically expect to reduce the objective function of our minimization problem by some fraction for example 1096 which can be stated as 7 1 1 1 dF i 1 zi rd j FG LLL COBRE ea F 3 j 01e Ira d j da
369. s been achieved The answer is based on the decisions that determine when to terminate the optimization process D 6 Convergence to the Optimum Since numerical optimization is an iterative process one of the most critical and difficult tasks is determining when to stop The optimization software uses several criteria to decide when to end the iterative search process and these are described here It is important to remember that the process described in this section relates only to the solution of the approximate optimization problem The number of cycles through the entire design process is controlled by other similar criteria Maximum Iterations As with any iterative process a maximum iteration counter is included The default for this is 40 iterations search directions Usually an optimum is found sooner than this therefore the maximum is mainly intended to avoid excessive computations No Feasible Solution If the initial design is infeasible constraints are violated the first priority is to overcome these violations and find a feasible solution However if there are conflicting constraints a feasible solution may not exist Therefore if a feasible design is not achieved in 20 iterations the optimization process is terminated Point of Diminishing Returns Probably the most common situation is where the optimum is approached asymptotically Therefore while some progress is still being made continued iterations are not just
370. s decreases the cost associated with the sensitivity analysis allows the optimizer to perform more efficiently and makes interpretation of the final results much easier A simple example is shown in Figure 7 4 The design goal is to reduce the weight of the stiffened panel subject to stress and displacement constraints under two separate static load conditions The thickness of the plate and web and the cross sectional area of the web cap are all allowed to vary The boundary conditions are selected so as to model an infinite plate The first load case includes both uniaxial tension in the x direction and a vertical pressure load in the z direction The second load case is a concentrated load applied in the z direction at grid 10203 which is directly under the web Figure 7 4 Stiffened Plate Model Analysis Model Description 16 CQUAD4 element to model uniform thickness base plate 7 16 Design Sensitivity and Optimization User s Guide Example Problems 8 CQUADA elements to model the uniform thickness web 4 CROD elements to model the uniform cross sectional area web cap Material E 1 0E7 psi Poisson ratio 0 33 Weight density 0 283 Ibs in Loadings Subcase 1 In plane tensile load of Nx 1000 Ibs in Uniform pressure load of 50 psi over the plate in the positive z direction Subcase 2 Lumped vertical load of 10 000 Ibs at Grid 10203 Design Model Description Objective Structural weight minimization Design variables P
371. s in which this may pose problems Note Does your design converge after only a single design cycle If so and your initial objective function value is on the order of 1 0E 2 or smaller CONV2 on the DOPTPRM entry should be reduced For very small absolute values of the objective function the default convergence criteria will not be satisfactory The convergence criteria logic is detailed in Convergence Tests but of interest here is the test on absolute change in the objective function This default is 0 01 but can and should be changed using a DOPTPRM Design OPTimization PaRaMeters Bulk Data entry With the default if the objective function changes by less than 0 01 absolute from the previous design cycle and all constraints are satisfied the process has converged This may not be a reasonable test if the original objective function is small The optimizer itself may also run into numerical difficulties for small absolute values of the objective function This situation may affect the optimizer s estimates of an initial value of a for the one dimensional search as well as the convergence criteria at the optimizer level Again the defaults can be changed using the DOPTPRM entry although determining correct values for the override may not be easy It may be far better to rescale the problem multiply the objective by 100 or other such avoidance rather than try to change the optimizer s defaults Formulating the Objectiv
372. s of the model its connectivity its loads and its boundary conditions This is just like any other NX Nastran model for static analysis The auxiliary boundary model connectivity is defined using CQUADA elements The geometry should not be redefined since the primary model geometry is used If necessary additional grids can be included For example an additional grid may be necessary for use as a rigid element connection point This grid should then appear in this Bulk Data Section The structure is constrained using two SPC sets 200 and 300 Each set one for each of the two auxiliary model subcases is coupled with one of two separate enforced displacement type loading conditions an enforced tip displacement in the z direction for the lower surface and an enforced Design Sensitivity and Optimization User s Guide 7 31 Chapter 7 Example Problems tip displacement in the z direction for the upper surface These are defined using SPCD sets 220 and 330 respectively The Case Control for these models appears after the primary model Case Control in a section beginning with the label AUXCASE Each subcase selects one of the two SPCD defined loads using a different boundary condition for each Static loading is always assumed for these auxiliary model analyses The results of these two analyses are sets of displacement vectors which essentially interpolate the tip displacement along the length of each plate This smooth shape interp
373. s probably not optimal A design is infeasible if one or more of the constraints are violated For example the point 2 2 is a feasible design whereas the point 0 05 3 violates not only the inequality constraint but also the lower bound constraint on x4 Design Sensitivity and Optimization User s Guide 1 11 Chapter 1 Getting Started Feasible Region Xx F 4 Optimum DEEL EEE x GH YYyyya EEE LLL 0 1 2 3 4 X Figure 1 3 Two Variable Function Space The optimal design at 1 1 can be found by inspection The explicit description of the design problem has allowed a graphical solution in two dimensions In practice we usually have more than two design variables and non explicit constraints and objective function This complexity requires an efficient searching procedure recalling that the optimizer is essentially blindfolded Numerically Searching for an Optimum Gradient Based Algorithms The optimization algorithms in NX Nastran belong to the family of methods generally referred to as gradient based since in addition to function values they use function gradients to assist in the numerical search for an optimum The numerical search process can be summarized as follows for a given point in the design space we determine the gradients of the objective function and constraints and use this information to determine a direction in which to search We then proceed in this direction for as far a
374. s to Shape Changes includes a checklist for setting up the design model using this method You may want to refer to that section in connection with this example To use this method you need to define auxiliary models over the boundaries of the structure When constrained and loaded these boundary models produce static deformations that can be used to describe shape variations over the boundaries The code then interpolates this information to the interior grids resulting in basis vectors for shape optimization A static analysis is used for this interpolation Problem Description Figure 7 9 shows the initial structure It is a simple cantilever modeled with eighty solid elements fixed at the support and tip loaded at the free end The design goal is to minimize the structure s weight subject to constraints on von Mises stresses We ll investigate minimizing the weight by tapering the cantilever s shape The initial stress distribution is shown in Figure 7 10 Design Sensitivity and Optimization User s Guide 7 29 Example Problems Chapter 7 PE MILLI N Man MMM an Ueda Figure 7 9 Solid Cantilever _ A A MMM A MMMM A JA MM A j ra N Figure 7 10 Solid Cantilever Initial Stress Distribution Since the cantilever is oriented and loaded in an x z plane removing material from the upper and lower surfaces is an effective way to reduce the weight
375. s we can go whereupon we investigate to see if we are at an optimum point If we are not we repeat the process until we can make no further improvement in our objective without violating any of the constraints Essentially this is the procedure used by the optimizer in NX Nastran although the task is complicated by the structural optimization context Here we consider each of the aspects of this process in more detail with an emphasis placed on the intuitive aspects rather than a rigorous mathematical treatment Finite Difference Gradient Approximations The first step in a numerical search procedure is determining the direction to search The situation may be somewhat complicated if the current design is infeasible one or more violated constraints or if one or more constraints are critical For an infeasible design we are outside of one of the fences to use the hill analogy For a critical design we are standing right next to a fence In general we at least need to know the gradient of our objective function and perhaps some of the constraint functions 1 12 Design Sensitivity and Optimization User s Guide Getting Started as well The process of taking small steps in each of the design variable directions suppose we are not restricted by the fences for this step corresponds exactly to the mathematical concept of a first forward finite difference approximation of a derivative For a single independent variable the first forward differen
376. screpancies exist between the initial analysis model properties given on the property entries and those calculated based on the initial values of the design variables the properties based on the design model have precedence The design model data is then used to override the analysis model properties In this example the design and analysis model agree and no overrides take place Since the bar area and moments of inertia are nonlinear in the design variables they must be defined using equations These relations are defined on the DEQATN entries 501 502 and 503 The input arguments are assigned via the DVPREL2 entries 250 251 and 252 The DESVAR identifier on the first continuation of each of these entries indicates design variables 10 and 11 are to be used as input b and h for each of the referenced equations Note Override of analysis model data by design model data provides a convenient method of restarting in design optimization See the restart example in Restarts in Design Optimization Equation Arguments The input arguments defined on the DEQATN entries are formal arguments which are defined at run time by the actual arguments provided on the DVPREL2 entries The rules for defining these equations follow directly from FORTRAN syntax requirements However all arguments are assumed to be real numbers FORTRAN intrinsic functions may also be used Some of these functions such as ABS and MAX are not recommended since
377. scribe the auxiliary boundary models Each of these additional sections must have a corresponding Case Control Each of the auxiliary boundary model Case Control Sections are identified by the delimiter AUXCASE The auxiliary boundary model Case Control Sections must follow the Case Control Section for the primary model Within the auxiliary boundary model Case Control Section individual auxiliary models are identified using AUXMODEL n where n is the auxiliary boundary model ID referenced in the corresponding BEGIN BULK n Bulk Data Section delimiter An example using AUXCASE and AUXMODEL is shown on 70 in Relating Design Variables to Shape Changes 4 3 Bulk Data Entries The design model is defined in the Bulk Data Section This definition includes the design variables the design variable to property relations shape basis vectors design constraints and so on In short all of the quantities related to the basic optimization problem statement Eq 1 1 through Eq 1 5 in Getting Started are defined here This section contains a brief listing and description of all the Bulk Data entries related to design sensitivity and optimization This information has been extracted from the complete Bulk Data listings in Bulk Data Entries For each entry in this section related entries have also been listed with appropriately shaded fields to help clarify the interrelations among the various data DCONADD Purpose Defines a new constrai
378. se allowable changes and defines limits on the structural responses Constructing the design model requires good engineering judgement if the design results are to be of any practical use A design model must Define the design variables that may be modified Describe the relationship between the design variables and the analysis model properties and or grid locations for shape optimal design Define the objective function which provides a scalar measure of design quality e Place bounds constraints limiting the design responses to an acceptable range This chapter describes each of these features in greater detail In connection with this chapter you may want to refer to Input Data as well as Example Problems These examples should all be available on your system Running some of these in connection with the following discussions is a good way to begin to discover the many design modeling options available to you in NX Nastran 2 1 Overview of Design Modeling Given the wide range of problem types that may be addressed using design sensitivity and optimization the process of generating a design model is hardly a rote or mechanical operation However most of the operations are sufficiently similar that one might generate a flowchart outlining the design modeling process as follows 2 2 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization Define the analysis disciplines to be
379. se reciprocal approximations APRCOD 2 is the default Convex linearization APRCOD 3 uses either a direct or reciprocal approximation depending on which one yields the more conservative approximation The choice is made on an individual design variable and individual response type basis For theoretical details see Approximation Concepts in Design Optimization Print Controls Two types of print control are available The first governs output at the design optimization cycle level the second governs output at the optimizer level The first type of output control is provided by the parameters P1 and P2 which control the overall design cycle printout P1 controls the frequency of output while P2 controls the quantity Values for P1 and P2 are shown in the following tables P1 0 output for initial and optimal designs default pn output for every n th design cycle P2 Ono output 1 objective and design variables default esigned properties esign constraints 8 design responses 4 12 Design Sensitivity and Optimization User s Guide Input Data The various P2 options can be summed to produce output combinations For example P2 15 prints out all available data The second type of output control is provided by the parameter IPRINT which controls the amount of optimizer output This parameter is normally turned off default 0 since the output generated upon exit from the optimizer P1 and P2 is usually s
380. se two loads subject the outer truss members to both compressive as well as tensile loads Due to the loading symmetry we expect the design to be symmetric as well As an exercise we ll show how to enforce this symmetry using design variable linking An important but often overlooked consideration is that the optimization capability in NX Nastran is multidisciplinary That is the final optimal design is the result of a simultaneous consideration of all analysis disciplines across all subcases In this case the optimal three bar truss design will satisfy the load requirements for both statics subcases which is to be expected If for example a normal modes subcase were to be added the resultant design would have to not only satisfy the static strength requirements but also constraints on eigenvalues As an exercise you may wish to try adding an eigenvalue constraint Design Sensitivity and Optimization User s Guide 7 3 Chapter 7 Example Problems i X 10 N 20 000 lbs 20 000 Ibs Subcase 2 Subcase 1 Figure 7 1 Three Bar Truss Analysis Model Description Three rod pin jointed structure in the x y plane Symmetric structural configuration with respect to the y axis Two distinct 20 000 Ib load conditions Material E 1 0E7 psi Weight density 0 1 Ibs in Design Model Description Objective Minimization of structural weight Design variable Cross sectional areas A4 A2 and Az Design variabl
381. sen as a relatively small number such q 1 La VER IT V T as 5 0 Also we always normalize and so that the first term in Eq D 33 and Eq D 34 is near unity Thus the second term in Eq D 33 dominates but not too strongly allowing for the possibility of reducing the objective function while overcoming the constraint violation If this iteration does not overcome the violation y is increased by a factor of 10 although limited to an upper bound of about 1000 again to avoid numerical ill conditioning Usually this has the result of bringing the design back to the feasible region in a few iterations If 20 iterations pass without overcoming the constraint violations the optimization process is terminated on the assumption that no feasible solution exists The One Dimensional Search Having determined a usable feasible search direction the problem now becomes one of determining how far the design can be moved in that direction Again a variety of possibilities exist depending on q 1 the starting point BC However in each case polynomial approximations are used for the objective gt and constraint functions in the one dimensional search direction S The basic concept is to try some initial value for a in Eq D 22 and evaluate the corresponding objective and constraint functions Therefore the first question to ask is What is a good first estimate for a At the beginning of the optimization process very little information i
382. sign model The situation is shown in Relating Design Variables to Properties Figure 2 3 A DVPRELi entry i 1 or 2 defines a design variable to property relation for a particular property entry identified by its property ID or PID A large number of elements in turn might reference this property entry Since they all reference the same PID all elements in this group will undergo equal variations as the design variables are changed By considering the design implications when defining property groups you may be able to reduce their numbers A smaller number of designed property groups might then require fewer independent design variables since groups of elements have already been linked by their property groups memberships 3 4 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optimization in NX Nastran Linking with Linear Relations If the number of analysis model property entries cannot be reduced it may be worthwhile to consider explicit methods of reducing the number of independent design variables One way you can do this is by using linear design variable to property relations The DVPREL1 Bulk Data entry is used to expresses an analysis model property as a linear function of design variables or Pi Co t 40X4 CoXy FC X Equation 3 1 In this equation the th property p is a function of the set of design variables x through x One feature of Eq 3 1 is that it can be used to expres
383. sign Sensitivity and Optimization User s Guide Getting Started If NR is equal to the number of regions in our design model and NSTR is the maximum number of constraints to be retained per region then the maximum number of retained constraints is NR NSTR This is often one or two orders of magnitude less than the number of constraints in the original set Formal Approximations By applying constraint regionalization and deletion we can reduce the number of constraints to only that small subset that is necessary to adequately guide the design However we would still like to replace the implicit and costly finite element analyses with explicit approximations for the objective and constraint functions The approximating functions used in NX Nastran are based on Taylor series expansions of the objective and constraints For any function f x an infinite series about a known value f x0 in terms of the change in the independent variable Ax can be written as ava dt AL dy ar el St a9 3 to 0 x X UG Ax foo L Equation 1 24 In addition to the function value f x this series requires that all derivatives at x be known as well Determining these derivatives may present some difficulty so the series is often truncated to a given power in Ax yielding an approximate representation of the original function For example if we only included through the first derivative term in the series we would obtain a linear approximation
384. sign Sensitivity and Optimization User s Guide Getting Started In our hill example we could easily find a search direction even though blindfolded by taking small steps from side to side and then forward and back to test for elevation changes Based on this estimate of a downhill direction we can then proceed until we hit a fence or the hill starts to climb up again What we have done is to find the local value of the gradient of our objective function and then used this information to establish a probable direction in which to search for a minimum Numerical optimization algorithms that rely on gradient information are termed gradient based Once we have done the best we can possibly do in this direction we find another search direction and again proceed as before We continue to repeat this procedure until we cannot reduce the objective function any further Design Variables Constrained and Unconstrained Problems To quantify the location of a point on the hill we might use north south and east west coordinates corresponding to the elevation at a given point In design optimization terms this is a two design variable space since two coordinate values are required to uniquely specify a point in the design space Two design variables are the most we can easily visualize Considering that an optimizer may have to deal with tens or even hundreds of design variables the task becomes understandably more complex We might also have the cond
385. sign information it requires instead of directly from the finite element analysis The approximate model in Figure 3 1 incorporates the effects of the reduced set of design variables the screened set of constraints and the approximated set of structural responses Design Sensitivity and Optimization User s Guide 3 3 Chapter 3 Design Sensitivity and Optimization in NX Nastran P NEL ang Initial Improved Design oe Structural Response Analysis Constraint Approximate Optimization Screening Algorithm Vv Model Sensitivity Analysis Finite Element Analysis Figure 3 1 Approximation Concepts Design Variable Linking The optimizer improves the design by changing the values of the design variables The complexity and computational cost of the overall process tends to increase as the number of design variables is increased One of your design modeling goals should be to describe the permissible design changes using as few design variables as necessary A problem consisting of 200 to 300 independent design variables is currently considered a large design task in NX Nastran On the next few pages is a discussion of the methods that you can use to achieve this goal Linking by Analysis Model Properties One of the more efficient methods you can use to minimize the number of independent design variables is to consolidate the number of analysis model properties that are referenced in the de
386. sis model property changes Relating Design Variables to Shape Changes discusses shape variations Definition of Analysis Model Properties Analysis model properties are quantities that appear on bulk data property entries Plate thicknesses area moments of inertia elastic spring stiffnesses elastic modulii coefficients of thermal expansion grid point locations and so on are all examples of analysis model properties Analysis model properties can be written as functions of design variables Similarly connectivity properties and material properties can be written as functions of design variables For additional information on properties that can be written as a function of design variables see the DVCREL1 DVCREL2 DVGEOM DVGRID DVMREL1 DVMREL2 DVPREL1 and DVPREL2 bulk data entries in the NX Nastran Quick Reference Guide Design Variable to Property Relation Types There are two ways to relate design variables to properties e Type 1 relations define a linear relationship between design variables and properties Type 2 relations use the equation input capability of NX Nastran to define a functional relationship between design variables and properties Type 1 Design Variable to Property Relations Type 1 design variable to property relations define a linear relationship between the design variables and properties DVCREL1 DVMREL1 and DVPREL1 bulk entries are used to define Type 1 design variable to property relations e DVCRE
387. sis vectors and DESOPT for the AMBS and GMBS options Creates the shape basis vectors in basic coordinates as well as the necessary design variable correlation data Generates tables that are used to compute both first and second level responses constraints and the objective function Since these tables are invariant they are formed on the first design cycle Expands tables DTOS2J and DTOS4J to account for DVPREL2 properties which are treated as internal design variables Design variable numbering in the resultant DTOS2K and DTOSAK tables is in terms of the external design variables DESVAR input and the internal DVPREL2 variables the new vector of design variables XINIT In the semianalytic sensitivity analysis approach used in NX Nastran perturbed structural matrices must be generated DSABO constructs tables to which later modules refer when building property related variational terms These tables are primarily in the form of perturbed element summary tables and design sensitivity processing tables Evaluates and screens constraints that are functions of either first or second level responses For purposes of computational efficiency the design variables are split into two sets The first set consists of those design variables that are related to structural properties while the second set consists of variables related to grid coordinates DSAE merges the tables and data blocks associated with these groups of design variables so
388. sitions on the hill Only one out of all points on the hill can be considered an optimum though For simplicity we are neglecting the presence of relative minima Finding the lowest point on the hill while staying inside the fences is no real problem All we really need to do is have a good look about and note from our perspective which point on the hill appears to be the lowest We have scanned the hill analyzed thousands of possible candidates at a glance and made an immediate decision If we were blindfolded though our decision making process would not be as simple and that is exactly the task a numerical optimizer is faced with In a computational sense the elevation of a single point on the hill or the numerical value of our objective function must be determined by an analysis that may take considerable effort Evaluating hundreds or thousands of candidate designs may be prohibitive We need a systematic method of searching for an optimal design There are numerous techniques available to solve such a problem all of which are classified as numerical optimization algorithms Search Directions Based on Gradient Information Generally numerical optimization methods seek to determine a direction of travel or search direction that moves us down the hill as quickly as possible yet allows us to find an optimum that lies within the fences A sequence of search directions is usually employed during the overall search procedure 1 8 De
389. sitivity and Optimization User s Guide 1 21 Chapter 1 Getting Started Constraint Deletion The next step might then be to identify a few constraints that are violated or nearly violated There may be just a few constraints that are currently guiding the design such as stress concentrations due to thermal loads while others such as the first bending mode may be nowhere near critical and can be temporarily disregarded This is called a constraint deletion process Constraint deletion allows the optimizer to consider a reduced set of constraints simplifying the numerical optimization phase This also reduces the computational effort associated with determining the required structural response derivatives i e sensitivity analysis costs are reduced as well Formal Approximations Once the engineer has determined the constraint set that seems to be driving the design the next step might be to perform some sort of parametric analysis in order to determine how these constraints vary as the design is modified The results of just a few analyses might be used to propose a design change based on a compromise among the various trial designs A parametric study of the problem is carried one step further in structural optimization with formal approximations or series expansions of response quantities in terms of the design variables Formal approximations allow us to construct an approximation to the true design space that although only locally va
390. sociated Entries DRESP2 and DVPREL2 entries list the input arguments to equations that define synthetic response and property relations respectively These input arguments may include table constants defined on the DTABLE entry presz w usa sae reso ar ww owoz ewe wee we owe 9 wee far e poet NN m mee Po Fb DESVAR DVID1 DVID2 DVID3 DTABLE Discussion Constants used in equations can either be built into the DEQATN entry when the equation is defined or passed as arguments Building in the values of constants may be inconvenient and can prevent one equation from easily being used in different contexts The DTABLE entry allows these constants to be stored in a table and then used in the equations as necessary On the DTABLE entry a constant is given a name and a value When the equation argument list is defined on a DVPREL2 or DRESP2 entry the constant is referenced by its name Only one DTABLE entry may appear in the Bulk Data DVBSHAP Purpose Defines a shape basis vector as a linear combination of analytic boundary shapes solutions and assigns a design variable to the result Entry Description Field Contents DVID Design variable identification number of a DESVAR entry Integer gt 0 AUXMOD Auxiliary model identification number Integer gt 0 COLI Load sequence identification number from AUXMODEL Case Control command Integer gt 0
391. solid 100 100 1 pfluid mat10 100 1 293 200 ENDDATA Listing 7 11 7 11 Restarts in Design Optimization In design optimization two types of restarts are possible The first should probably be called a pseudo restart since it is simply a cold start using the results of the previous design The second type is more of a true restart in the NX Nastran sense of the word since it allows results saved on the database to be used by a subsequent job This feature is somewhat limited however in design optimization Probably the best way to illustrate these restart features is by example This section describes both types of restarts and discusses some of the limitations of each type Restarting from a Previous Design Cycle If the design and analysis models differ in Solution 200 the design model data is used to override the analysis model This feature can be used to our advantage to perform a simple type of restart For example if the initial design variable values are modified on the DESVAR entries the code uses this as the initial design for the subsequent optimization This set might have been determined based on a prior optimization cycle or may reflect other changes that the engineer might want to make Design Sensitivity and Optimization User s Guide 7 63 Chapter 7 Example Problems For every design cycle completed a punch file is created PCH that contains the DESVAR Bulk Data entries for this latest design
392. sponse over all of these points 7 56 Design Sensitivity and Optimization User s Guide Example Problems P f an f h fs Figure 7 25 Minimization of Response Peaks We can choose a design variable to represent a peak threshold here shown in the figure as ax a is just a constant of proportionality to facilitate scaling the threshold The difference between ax and the pressure distribution must be a positive quantity at all frequencies of interest Thus constraints can be written that require cy c5 and cz to be positive distances The design objective can then be to minimize ax or minimize Ox Equation 7 22 subject to P f1 ax lt 0 P f2 ax lt 0 P f3 ax lt 0 Equation 7 23 As the optimizer decreases the threshold the constraints ensure that the peaks are reduced as well Any number of these constraints can be written to cover all frequency ranges of interest or in the case of transient analysis time steps Due to the size of this problem Listing 7 12 is an excerpt of the input file consisting mainly of the design model entries as well as some the analysis entries Analysis modeling for acoustics is covered in the NX Nastran User s Guide and in the NX Nastran Advanced Dynamic Analysis User s Guide Turning to the listing we see that the weight budget is established as a global constraint The weight response is defined on DRESP1 number 2 bounds are placed on this response with DCO
393. ss ratio of approximately 100 Euler buckling occurs when the magnitude of a member s compressive stress is greater than a critical stress that for the first buckling mode of a pin connected member is Design Sensitivity and Optimization User s Guide 7 45 Chapter 7 Example Problems O Py 1 El 9j A A L Equation 7 14 We will assume thin walled tubular members with a diameter to thickness ratio of approximately 100 Under this condition the cross sectional area A is nearly equal to A nDt where 2 100 Equation 7 15 and the area moment of inertia can be approximated by nDt D Iz 3 Equation 7 16 Since cross sectional areas are chosen as design variables we seek to express the member buckling stresses as functions of member areas as follows 2 _ AE n 100 1 D n E 8 100 where 100 Equation 7 17 For the buckling stress constraints to be satisfied the member stresses must not be more compressive than Op that is 020 Equation 7 18 where Op is a negative quantity representing a compressive stress Since the constraint in the form of Eq 7 18 can t be used directly on a DCONSTR entry the bound is a function of the design variables we can first normalize the expression as S ud Op Equation 7 19 7 46 Design Sensitivity and Optimization User s Guide Example Problems This is now a well posed constraint since the feasible condition has now been normalized to be on the order o
394. ssential gradient information to the optimizer they are always computed in connection with design optimization Sensitivity analysis is always performed automatically in NX Nastran whenever design optimization is requested There may be times when you want the software to compute just the sensitivity coefficients and not perform optimization You can then use this sensitivity information to perform parametric design studies or to link with your own optimizer This section presents the theoretical basis for design sensitivity analysis in NX Nastran It shows how to request sensitivity analysis describes how the sensitivity coefficients are computed and presents modeling guidelines General Considerations A design sensitivity coefficient is defined as the rate of change of a particular response quantity r with respect to a change in a design variable x or in other words as dr 0x These coefficients are gt 0 evaluated at a particular design characterized by the vector of design variables giving C5 F gt Sm Il X L 0 7 Equation 3 32 where subscripts are used to indicate the th design variable and the j th response Eq 3 32 is just the slope of the response with respect to x as is shown in Eq 3 7 Design Sensitivity and Optimization User s Guide 3 23 Chapter 3 Design Sensitivity and Optimization in NX Nastran rx 1 gt x 0 L j Figure 3 7 Design Sensitivity Coefficient Graphic
395. st design is the one in which there is the lowest maximum constraint violation Fractional Error of Approximation Definition The objective function history traces the progress made by the optimizer during successive optimizations with respect to approximate models The fractional error of approximation is a measure of the error in the approximated objective function versus the true objective The true objective is computed from the analysis at the beginning of the next design cycle Here the error is negligible since the weight objective is just a linear function of the design variables For objectives based on nonlinear structural responses virtually any response other than weight or volume the error may be more significant If these approximation errors are large tighter move limits may be warranted Fractional Error of Approximation Interpretation Note also that as a measure of approximation quality the fractional error of approximation is incomplete at best Even though the error in the objective function may be small errors in constraints may be much greater especially if the constraints depend on responses that are more nonlinear in the design variables than is the objective In fact this is often the case For example weight minimization subject to constraints on stresses has an objective function that is nearly linear in the sizing variables while the constraints are nonlinear functions of these variables Increasing move limi
396. statements must be used ssign fl file MASTER dblocate datablk ug ugd geoml geomld geom2 geom2d logical f1 e Connecting with external programs to compute DRESP3 type responses With this approach you use the CONNECT statement to connect a user defined external program that evaluates DRESP3 responses The filename file MASTER is the name of the database for the auxiliary model analysis The filename file is arbitrary The UG GEOM1 and GEOM2 data blocks must be DBLOCATEd and renamed to UGD GEOM1D and GEOMZ2D respectively These new names cannot be changed since the code 4 2 Design Sensitivity and Optimization User s Guide Input Data looks for the data in these locations explicitly See Relating Design Variables to Shape Changes and Shape Optimization of a Culvert for a discussion of and examples using this method 4 2 Case Control Section In NX Nastran analysis the Case Control Section is used to specify the applied loads boundary conditions and subcases and to request the form and type of analysis output The additions to the Case Control for design optimization are few and do not require any modification of the Case Control commands already required for analysis Case Control commands for design sensitivity and optimization are associated with the following four tasks Analysis discipline definition Design model definition Design response characterization Shape basis vector computation
397. station The properties at end A are expressed as functions of Design Variables 1 and 2 via DVPREL2s 1 5 end B properties are expressed in terms of Design Variables 3 and 4 via DVPREL2s 6 10 and the first intermediate station by DVPREL2s 11 15 In order to make identification of the EPT word positions easier the PBEAM entry has been set up to define cross sectional properties that are slightly different from the properties computed using the initial design variable values The design model overrides the analysis model properties and the differences reported in a differences comparison table Since the word positions in the EPT are also output these results can be used to check the validity of the formulation Note The design model should include stress recovery point locations Serious design errors may result if you omit these from your design model formulation 2 18 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization The difference comparison table associated with the design model override is shown in Figure 2 8 From the corresponding field IDs FID it can be seen that not only is the data for ends A and B in the expected word positions but the intermediate station is also present as expected The analysis and design property columns indicate that it is indeed just a copy of the data for end B Furthermore the design model formulation appears to be correct judging from the number
398. structural model it is only performing a mathematical search It is the design engineer who has defined exactly how the structure will change For example a single design variable might have been used to describe a number of bar cross sectional areas several plate element thicknesses as well as a fillet radius As the optimizer changes this single variable the entire structure changes Expressing allowable structural variations with an economy of design variables is part of the challenge of design modeling DESVAR Bulk Data Entry A Design Variable is defined with a DESVAR Bulk Data entry From the Bulk Data listings in Input Data notice that the DESVAR entry provides an ID an initial value for the variable and bounds or side constraints These bounds limit the region of search with respect to each variable as L U Xx SX lt X 4 1 L Equation 2 1 The optimizer will never propose a design that violates these bounds See also the Selective Modification of Design Variable Bounds in the Relating Design Variables to Properties section The following DESVAR entries define a pair of variables x49 and x44 with initial values of 0 05 and 0 03 respectively SDESVAR ID LABEL XINIT XLB 10 AREA1 0 05 0 01 1l THICK1 0 03 0 01 From the label fields we can assume these design variables will be used to define an area and a thickness although the DESVAR entry does not actually provide these relations this is
399. t saaana aaa 7 66 Glossary O TES xa os cede KE Oe a e RE CR A C RR e 0 Je e A A 1 Nomenclature ssa A EU RERO E GCOX NE EORR Re EORR a RORCR EUR RACES RO A B 1 Commonly Used Commands for Design Optimization C 1 Case Control Command 22er C 1 Bulk Data Entries 00 rss C 1 Parameters iii ase bP ce dodge ERR eda Mode ded edt acad de at deiecit ie p ie he RR C 2 Numerical Optimization lee ee D 1 IMtrOdUCTION TT D 1 The Modified Feasible Direction Algorithm lee D 8 Finding the Search Direction uas idc uec o e aa a D 9 Finding Bounds Ona uos cp rel crack exusta I Ce di d bat ex CCS D 19 Interpolation A D 20 Convergence to the Optimum 1 ee em D 23 Maximum Iterations llle rs D 23 No Feasible Solution lille res D 23 Point of Diminishing Returns sss ss comes ska A Baek one eee RA D 23 Satisfaction of the Kuhn Tucker Conditions llle D 24 Sequential Linear Programming llle D 25 Sequential Quadratic Programming llle D 27 cui a RAN D 29 6 Design Sensitivity and Optimization User s Guide Proprietary amp Restricted Rights Notice O 2014 Siemens Product Lifecycle Management Software Inc All Rights Reserved This software and related documentation are proprietary to Siemens Product Lifecycle Management Software Inc Siemens and the Siemens logo are registered trademarks of Si
400. t This example considers shape optimization using the direct input of shapes method In order to use this method sets of displacement vectors are first generated using a separate auxiliary model analysis These displacement vectors are then DBLOCATEd in a subsequent design optimization run and used to generate a set of shape basis vectors The optimizer then seeks to find the best combination of these shape basis vectors Problem Description Figure 7 6 shows the finite element model of one half of a symmetric culvert structure Symmetry exists with respect to the y axis The structure is made of steel and has been modeled using Design Sensitivity and Optimization User s Guide 7 23 Chapter 7 Example Problems CQUADA elements in plane strain This example has been taken from A D Belegundu and S D Rajan Shape Optimal Design Using Isoparametric Elements Proceedings of 29th AIAA ASME ASCE AHS ASC Structures Dynamics and Materials Conference pp 696 701 Williamsburg VA April 18 20 1988 ibTranslationl Figure 7 6 Initial Culvert Design With the bottom surface fixed pressure loads are applied on the top surface The design task is to minimize the volume of the structure by changing the shape of the initially circular hole subject to von Mises stress constraints over the interior Modeling Considerations The process of shape basis vector generation is closely associated with the nature of the d
401. t of data up to G NDV where G is the number of designed grids in the model and NDV is the number of shape design variables A model with only 100 grids and 2 design variables would thus require on the order of 200 DVGRID entries 3 Preview the resultant basis vectors and modify if necessary Direct Input of Shapes This approach greatly simplifies the process of defining shape basis vectors With this method externally generated vectors are DBLOCATEd and used to define shape basis vectors An auxiliary model analysis provides these externally generated vectors Benefits Basis vectors for shape optimization can be generated using an external auxiliary model analysis and DBLOCATEd allowing for easy generation of shape basis vectors In theory any method of external generation is possible as long as the number of degrees of freedom in the auxiliary model is the same as in the primary structure i e G sets must be equivalent Drawbacks The process is not fully automatic since an auxiliary model analysis must be performed beforehand saved on the database and DBLOCATEd for shape optimization Since the basis vectors are externally generated they are not updated for each design cycle This may cause mesh distortion problems for large shape changes Checklist 1 Define an auxiliary model perform an analysis and save the results to the database using the SCR NO option on the NASTRAN job submittal command This process is the same a
402. t the optimum rather than simply a circle Two design variables will be used each one representing an axis length 2 30 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization T 10E4 N m T 2 0E4 N m Material aluminum 7075 T6 sheet E 7 2E10 N m u 0 3 3 p 2 8E3 kg m Figure 2 16 Plate Quarter Model Recall that the equation for an ellipse centered at the origin of the x y plane is given by S eel gt l N S gt nin Il Equation 2 26 where a and b are the x and y axis intercepts respectively of the ellipse We want the optimizer to vary the shape of the ellipse so a natural choice is to define two design variables each representing the changes in a and b These quantities can be defined as define design variables SDESVAR ID LABEL XINIT XLB XUB DELXV DESVAR 1 1 0 z205y 20 DESVAR 2 205 20 These two entries define two design variables each with initial values of 1 0 lower bounds of 20 and upper bounds of 20 The initial design variable values are somewhat arbitrary as are the lower and upper bounds The lower bound must be less than XINIT which in turn must be less than the upper bound or a fatal message will result Design Sensitivity and Optimization User s Guide 2 31 Chapter 2 Design Modeling for Sensitivity and Optimization Note from Eq 2 23 that a change in a design variables value caus
403. t up D200X5 contains eleven design variables three independent and eight dependent The independent design variables are used as multipliers of basis functions which have been implemented using DLINK entries The problem includes weight displacement and stress responses with two separate loading conditions In short enough of the basic components are present to give a fairly complete description of the output format in design sensitivity analysis The following is an abbreviated listing of the DSCM2 matrix output Design Sensitivity and Optimization User s Guide 6 13 Chapter 6 Output Features and Interpretation MATRIX DSCM2 GINO NAME 101 IS A REAL 61 COLUMN X 3 ROW RECTANG MATRIX Matrix OCOLUMN 1 ROWS 1 THRU A i E E E E EE Dimension ROW 1 4 0000E 01 2 2500E 01 1 5938E 01 OCOLUMN 2 ROWS 1 THRU dr ia ROW 1 3 2871B401 1 9345B 01 1 3255E 01 OCOLUMN 3 ROWS 1 THRU Bo semeemenrem ON uN ROW Ors Ou 29 f 1 3 2871E 01 1 9345E 01 1 3255E 01 Ls 3 See the following OCOLUMN 4 ROWS 1 THRU ds Nase Xop ec secer See L f G i ee 3 3 7 SUBCASEI Correlation Table 1 2 5733E 04 2 9702E 04 3 2985E 04 OCOLUMN 5 ROWS 1 THRU 3U uncendclniddgu a a aa ROW 1 2 5733E 04 2 9702E 04 3 2985E 04 COLUMN 6 ROWS 1 THRU A rl a o eee ROW 1 1 5881E405 1 5696E 05 1 5544E 05 OCOLUMN 7 ROWS 1 THRU Ee ROW 1 1 8881E405 1 5696E 05 1 5544E 05 OCOLUMN 3
404. tation is the default If STRAIN FIBER is provided data recovery and design optimization responses are both computed using this form Defaults are listed in the NX Nastran Quick Reference Guide Shape optimization and superelement optimization each have some additional Case Control considerations beyond those related to data recovery Superelement Design Modeling discusses the Case Control structure for superelement design models In addition Example Problems includes examples of both shape and superelement optimization 2 3 Defining the Design Variables In design optimization design variables are the quantities that are modified by the optimizer during the search for an improved design In design sensitivity analysis rates of change of responses are computed with respect to the design variables Defining the design variables is part of the design modeling task of the engineer Once defined the design variables may be used to Describe analysis model property variations Define shape variations Design Sensitivity and Optimization User s Guide 2 5 Chapter 2 Design Modeling for Sensitivity and Optimization Design variables are often thought of as plate thicknesses radii of cutouts etc Of course design variables may be used to describe these quantities but the idea is actually much more general and powerful Design Variable Generality As the optimizer changes the design variables it does not know it is changing a
405. te model And finally IPRINT P1 and P2 have all been increased from their defaults to allow more diagnostic output Turning to the results of the optimization we see from the hard convergence decision logic output that the problem converges to a feasible design since all constraints are satisfied at the optimum However note that the properties and design variables have still changed appreciably from their values on the previous design cycle This suggests that we may have some flexibility in choosing final dimensions for our design without greatly affecting the overall weight The summary of design cycle history and the design variable history tables both indicate that convergence was achieved in five iterations A total of six finite element analyses were performed one for the initial analysis and one in connection with each of the design cycles Also listed in the summary of design cycle history is the fractional error of approximation in the objective function which compares the approximate objective function on exit from the optimizer with the true value computed from the subsequent finite element analysis This is an indication of the quality of the Design Sensitivity and Optimization User s Guide 7 5 Chapter 7 Example Problems objective function approximation Since the weight is linear in the design variables W L4X4 LX L3L3 we expect the linearized weight objective to be an exact approximation this indeed is seen to be the
406. tement could be written as follows 1 16 Design Sensitivity and Optimization User s Guide Getting Started minimize V B H L Equation 1 15 subject Equation 1 16 BIS apr gt lt 2 54 BH SEI Equation 1 17 H E B 12 Equation 1 18 1 lt B lt 20 Equation 1 19 20 lt H lt 50 Equation 1 20 Since the objective and constraints are available explicitly we can graphically display the two variable design space as shown in Figure 1 7 Design Sensitivity and Optimization User s Guide 1 17 Chapter 1 Getting Started H B 12 V 200000 175000 100000 y 150000 SS SS SS H 50 Ky MMMM LLL A AW Optimum Q 77 AN N Height H cm 7 y 45 7 om Lp i NY 75000 Dr 40 50000 rrr N o 700 25 3 4 5 6 7 75 Width B cm Figure 1 7 Cantilever Beam Design Space In Figure 1 7 note that the optimum lies at the vertex formed by the intersections of the beam bending stress constraint and the constraint on maximum allowable ratio of cross sectional height to width This optimum occurs at an objective of approximately 1 000 cm3 Let us examine the path that the optimizer might take as it searches for this constrained minimum Assume that we begin with an initial design of H 44 cm and B 7 cm Since none of the constraints are active the direction of steepest descent is used as an initial search direction and the optimizer
407. ter 2 Design Modeling for Sensitivity and Optimization Identifying Dynamic Responses In Frequency Response analysis a number of forcing frequencies may be used Similarly a number of analysis time steps may be used in a transient analysis Both of these analysis types have the potential to generate a huge amount of design response data This could get quite expensive as sensitivities discussed in Example Problems may need to be computed for all of these design responses Limiting the Number of Design Responses with OTIME and OFREQ To keep this data to a minimum the OFREQ and OTIME Case Control commands can be used to limit the number of design responses For frequency response design responses are only computed for those frequencies referenced by the OFREQ command Otherwise design responses will be computed for all forcing frequencies by default Note Time steps can be stated explicitly in the ATTB field of the DRESP1 entry If this value is not equal to any of the time values shown in Figure 2 25 then the nearest available reduced time step in the figure will be used For transient response design responses are only computed for the output time steps Output time steps are defined by the NOi fields on the TSTEP Bulk Data entry and may be further limited by the OTIME Case Control command In the absence of any NOi or OTIME definitions design responses are computed for all analysis time steps Figure 2 25 is a schematic
408. that later modules can compute the perturbed structural configurations Generates an updated element summary table ESTDCN for recovery of responses in design sensitivity analysis It contains the subset of elements necessary to recover element level responses and incorporates the effects of constraint deletion that are used in the formation of the design response sensitivities Converts grid variations defined on the boundaries of the structure geometry model boundary solution for shape optimization to a shape basis vector This in turn is used to define enforced displacements on the primary structure for purposes of interpolation to the interior grids of the mesh Actually computes the matrix of partial sensitivity coefficient DSCM given the baseline and perturbed responses and corresponding design variable data The form of the terms in DSCM is Ar DELB This form is converted to Ar Ax in modules DOM6 and DOMO generate the pseudo load vectors In direct and modal frequency analysis DSAP sums the variational terms that are functions of AK AB and AM to form the pseudo load vector Splits the modal transient solution containing data for displacement velocity and acceleration onto separate data blocks one data block each for displacement velocity and acceleration for all solution time steps Computes the delta weight and or volume according to the changes in the design variables in connection with the semi analytic sensitivity analysis
409. the element nodal displacements gives the adjoint load for that particular element Therefore this derivative summed over the elements over which the ERP is found as for a panel gives the corresponding adjoint load for the ERP in question The adjoint load for the ERP magnitude on a single element is Pag o ANT 00 ND Iu Equation 3 84 where J The uth Gauss point hj The relevant weighted surface area for the Jth Gauss point N A matrix that converts the nodal displacements to the displacements at the Gauss point 6 A vector that transforms the Gauss point displacements to the surface normal displacement at the Gauss point u The real part of the element nodal displacements vector Ug C A scaling coefficient usually taken to be the product of density and speed of sound for the fluid medium wW The frequency in cycles per unit time at which the ERP magnitude is computed Design Sensitivity and Optimization User s Guide 3 39 Chapter 3 Design Sensitivity and Optimization in NX Nastran The element ERP adjoint load becomes the basis for panel adjoint load for the ERP design response The adjoint load for the panel is then used to find the ERP derivatives with respect to the design variables in the usual manner As for the panel ERP density derivatives given that the ERP density for a panel is defined as D E A panel Equation 3 85 where E is the ERP total magnitude over the panel and A is the normal s
410. the hill and move in that direction as far as possible Now assume instead that we or the optimizer have encountered a fence In order to make any gt further improvement in the design a new search direction must be found that continues to reduce our elevation on the hill but keeps us inside of the fence Here we seek a usable feasible direction in which a usable direction is one that moves us downhill and a feasible direction is one that keeps us inside of the fence This situation is shown in Figure D 1 The mathematical definition of a usable search direction is D 2 Design Sensitivity and Optimization User s Guide Numerical Optimization 2 VF x S lt 0 Equation D 7 Eq D 7 is just the scalar product dot product of the gradient of the objective function with the search direction The dot product is the magnitude of vr times the magnitude of times the cosine of the angle between the two vectors Thus the cosine of the angle determines the sign of the product since the magnitudes are positive numbers For the cosine to be zero or negative the angle between the vectors must be between 90 and 270 degrees If the cosine is zero the search direction is at an angle of 90 or 270 degrees from the gradient of the objective function A move in this direction follows a contour on the hill and for a small move does not increase or decrease the function If the cosine is 1 0 the direction is opposite to the direc
411. the j th constraint these choices result in az a a OE Las B x Ax gj ub es AX DVPREL1 Equation 3 17 E E Og g x 4 g x 3s Ap DVPREL2 i gt 0 Equation 3 18 where 0 0 Ap pix F At px Equation 3 19 Similar expressions can also be written for the objective function The benefit of this generalization is that properties are computed precisely in Eq 3 19 for a given change in design variables The approximation in Eq 3 18 thus retains all of the non linearities of the DVPREL2 relations yielding a more accurate approximation Design Sensitivity and Optimization User s Guide 3 19 Chapter 3 Design Sensitivity and Optimization in NX Nastran Depending on whether the response quantities are first or second level the derivatives are evaluated as follows If we let the quantity f represent either an objective or a constraint function then for first level responses we have w Of Of Ox 250 Oh DVPREL1 yy Equation 3 20 ait Of _ ies OP 20 0 OP j DVPREL2 J Equation 3 21 where the superscript on r 1 denotes a first level DRESP1 response For objectives or constraints that are functions of second level responses we have 2 of f 9 Ox p 25 0 x DVPREL1 J J Equation 3 22 af of apto E EN DVPREL2 OP 2 50 OP Jj Jj Equation 3 23 where r 2 is a second level response In general these second level respons
412. they may yield functions having discontinuous first derivatives The sensitivity analysis as well as the optimization results may be incorrect if any of these discontinuities are encountered Note The available intrinsic functions and their uses are listed with the DEQATN bulk data entry in the NX Nastran Quick Reference Guide Minimum Allowable Property Values Field 6 on the DVPREL2 entries defines PMIN which is the minimum allowable value for the particular property during design optimization Generally its purpose is to prevent properties from taking on values near zero The default is 0 001 which is acceptable for many applications Here l2 is already less than this default and may take on values less than the default as well based on the minimum B and H values Therefore we override the default in both cases and specify a PMIN of 1 0E 5 on DVPREL2 251 and 252 based on engineering judgment A similar situation is seen in the DVPREL1 relations that in this case relate the location of the stress recovery point c to the values of the design variables The y coordinate location is a negative quantity but the default PMIN for stress recovery locations is 1 0E 35 see the DVPREL1 and DVPREL2 Bulk Data entry descriptions Just to be explicit a negative PMIN has been provided on DVPREL1 260 DVPREL1 260 and 261 respectively define the following linear relations 2 14 Design Sensitivity and Optimization User s Guide D
413. this License and the Availability note Permission to modify the code and to distribute modified code is granted provided the Copyright this License and the Availability note are retained and a notice that the code was modified is included This software is provided to you free of charge Availability TAUCS As of version 2 1 we distribute the code in 4 formats zip and tarred gzipped tgz with or without binaries for external libraries The bundled external libraries should allow you to build the test programs on Linux Windows and MacOS X without installing additional software We recommend that you download the full distributions and then perhaps replace the bundled libraries by higher performance ones e g with a BLAS library that is specifically optimized for your machine If you want to conserve bandwidth and you want to install the required libraries yourself download the lean distributions The zip and tgz files are identical except that on Linux Unix and MacOS unpacking the tgz file ensures that the configure script is marked as executable unpack with tar zxvpf otherwise you will have to change its permissions manually Design Sensitivity and Optimization User s Guide 7 Chapter 1 Getting Started Introduction to Design Sensitivity and Optimization Numerical Optimization Basics e Structural Optimization Design Sensitivity and Optimization User s Guide 1 1 Chapter 1 Getting Started 1 1 Introduction to Des
414. timization gt subject to Vg SEO el Equation D 10 gt gt S S lt 1 Equation D 11 where J is the set of constraints whose values are zero within some numerical tolerance This is the set of active constraints The purpose of Eq D 11 is simply to prevent an unbounded solution to the problem defined by Eq D 9 and Eq D 10 In the case of a simple two variable problem finding the appropriate search direction is quite easy and may be done graphically In the more general case where there are numerous design variables as well as several active constraints this becomes a gt subproblem that is solved as part of the optimization This problem is linear in S except for the quadratic constraint of Eq D 11 Details regarding the solution of this subproblem can be found in Numerical Optimization Techniques for Engineering Design with Applications2 Assuming we can find a usable feasible search direction we can now search in this direction until we can make no further improvement The subproblem of finding a new usable feasible search direction is repeated and continues until no search direction can be found that improves the design without violating one or more constraints We call this point the optimum In the present example we assumed the initial design was feasible In practice we often start outside of one or more fences or from an initially infeasible design If this is the case the optimizer s first task is to se
415. tion D 31 This is the same problem as shown in Eq D 9 through Eq D 11 for finding a usable feasible search direction in the physical example of searching for the lowest point on the hill Note that the scalar product is the magnitude of the two vectors times the cosine of the angle between them Thus the objective of this suboptimization problem is ver D leos Equation D 32 Since we are minimizing this function we want the cosine of the angle between the two vectors to be as large a negative number as possible but within the restriction of Eq D 30 Alternatively for any gt angle 8 between 90 and 270 degrees Eq D 32 can be made more negative if the magnitude of S is gt increased Also if S satisfies the requirements of Eq D 30 then any increase in the magnitude of gt gt S also satisfies this equation Therefore it is necessary to bound S which is accomplished using Eq D 31 Assuming the resulting objective function from this subproblem is negative the usability requirement of Eq D 7 is satisfied If the objective function defined by Eq D 29 cannot be forced to be negative then it follows that no direction exists that reduces the objective function while staying inside of the constraints If this is the case the Kuhn Tucker conditions are satisfied and the optimization process can be terminated In practice the decision is not quite this simple because Eq D 30 may include constraints that are within a to
416. tion User s Guide Design Sensitivity and Optimization in NX Nastran The second situation occurs if design variables can pass through zero during the course of optimization This can occur if the lower bound on the DESVAR entry is a negative quantity and the upper bound is positive In this case APRCOD is automatically set to 1 to avoid ill conditioning about the zero value of the design variable Finally experience has also shown that the mixed method is not always the best performer for dynamic response optimization tasks For these highly nonlinear responses direct approximations coupled with more stringent move limits are often more reliable Advanced Topics in Approximations The direct and reciprocally approximated constraints of Eq 3 8 and Eq 3 11 require the evaluation of function derivatives with respect to design variables Computing these function derivatives is complicated somewhat by design variable to property relations which may be either type 1 or type 2 DVPREL1 or DVPREL2 and responses which may be either first or second level DRESP1 DRESP2 or DRESP3 To accommodate type 1 and type 2 design variable to property relations the concept of a design variable is generalized internally in the code For type 1 relations the basis for the constraint and objective function approximations is the set of independent design variables For type 2 relations this basis is the set of properties defined on the DVPREL2 entries For
417. tion numbers in Case Control using DESGLB for globally applied constraints and DESSUB for subcase dependent constraints 4 4 Design Sensitivity and Optimization User s Guide Input Data Design Response Characterization Strictly speaking Case Control output requests are unnecessary in design sensitivity and optimization In a number of instances they can however be quite useful Data Recovery in Solution 200 Solution 200 builds its own internal Case Control for data recovery based on the list of DRESP1 responses identified in the design model This provision ensures that all necessary data recovery is always performed for design sensitivity and optimization However if you want to view NX Nastran results Case Control output requests are necessary The frequency of this output with respect to design cycle number is controlled by the Bulk Data parameter NASPRT Case Control and Design Responses Case Control output commands can often be used to resolve design response ambiguities For example element stresses can be output using either von Mises or maximum shear The ambiguity arises since both share the same plot code ID referenced on the DRESP1 entry See the NX Nastran Quick Reference Guide for a list of these plot codes The form identified in Case Control is taken as the representation for design sensitivity and optimization For example STRESS VONMISES 15 von Mises stresses used for analysis and optimization STRESS SHEA
418. tion of VEO andis ie direcion ef steepest descent Thus we wish to find a search direction that makes the left hand side of Eq D 7 as negative as possible However this direction must remain within a critical constraint fence This is the feasibility requirement that is similar to the usability requirement but now is stated with respect to the constraint gt gt Vgj x S lt 0 Equation D 8 Design Sensitivity and Optimization User s Guide D 3 Appendix D Numerical Optimization lt X Feasible gt Sector SS ISS a Vg4 x d Tr SS v al A F x constant Usable Y Sector SS C on o Ti O o a O 77 go x 0 O m 0 PX Figure D 1 Usable Feasible Search Direction gt Just as for the objective function the angle between the search direction 5 and the gradient of the constraint must be between 90 and 270 degrees If the angle is exactly 90 or 270 degrees the search direction is tangent to the constraint boundary fence To find the search direction that makes the greatest possible improvement in the objective function but still follows or moves inside of the fences we combine the usability and feasibility requirements This combination creates the following gt suboptimization task Find the components of the search direction S in order to gt minimize VF X Equation D 9 D 4 Design Sensitivity and Optimization User s Guide Numerical Op
419. tions of more than one of the basic responses basic adjoint loads can be combined to produce a composite adjoint load that leads to the gradient of the response The adjoint load for a response quantity is given as the derivative of that response quantity with respect to the vector of displacements u If the response quantity is denoted r then the adjoint load for rj is given by or Ku Equation 3 50 As an example suppose the response quantity is the static displacement of a degree of freedom at a grid point That is suppose rj uj Then the adjoint load is 1 0 for that degree of freedom because du du 1 e 0 0 for all other degrees of freedom because du du 0 If rj represents the static displacement of grid point k at an angle of 30 to a local X axis in a two dimensional displacement field the adjoint loads that correspond to the static displacements in the X and Y directions can be combined into a single adjoint load vector that corresponds to 1 as follows or Quy r du cos 30 i sin 30 jj Equation 3 51 where u and uy are the static displacements at grid point k and i and j are unit vectors in the X and Y directions respectively Application of an adjoint load on a finite element model yields a set of adjoint variables a For linear static behavior the governing equation can be represented by 3 30 Design Sensitivity and Optimization User s Guide Design Sensitivity and Optim
420. to determine suitable properties for the lumped quantities in our simplified dynamic model so that the first ten eigenvalues correlate well with the prototype How Does Design Optimization Differ from Analysis Although design optimization and analysis can be viewed as complementary there are some important conceptual differences between the two that must be clear in order to make effective use of both Design Sensitivity and Optimization User s Guide 1 3 Chapter 1 Getting Started Analysis Models When we perform an analysis we create a mathematical idealization of some physical system in order to obtain estimates of certain response quantities The class of responses that we are interested in defines the applicable analysis discipline to be used while the accuracy of these responses is dependent on the quality of the analysis model and our general knowledge of the true system Our choice of finite element types representation of boundary conditions loads and definition of the finite element mesh all play critical roles in determining how well our model is able to predict the responses of the physical structure The goal is to obtain an accurate prediction of the responses that can be expected from the real structure For example consider the plate subjected to uniform tensile loads in Figure 1 1 The corresponding analysis model in Figure 1 2 is a discretized finite element representation of idealized geometry loads and boundary con
421. to determine whether the design is changing for the case of violated constraints If the design is still varying we are justified in continuing optimization in order to try to overcome the constraint violation s However if the design is not changing then we are at a point in the design space that represents a best compromise solution among the violated constraints This situation is shown in Figure 3 16 where the optimal design is the one that minimizes the sum of the constraint violation Design Sensitivity and Optimization User s Guide 3 59 Chapter 3 Design Sensitivity and Optimization in NX Nastran a gG g Xq Figure 3 16 Design Space No Feasible Solutions Given the implications of hard convergence a careful review of the output is advised in order to determine which convergence conditions have been met Just because the process has been terminated does not necessarily imply that a unique feasible design has been found Caution Always check and confirm the conditions under which convergence is achieved 3 60 Design Sensitivity and Optimization User s Guide Chapter 4 Input Data File Management Section Executive Control e Case Control Section e Bulk Data Entries Parameters for Design Sensitivity and Optimization Design Sensitivity and Optimization User s Guide 4 1 Chapter 4 Input Data This chapter describes all Executive Control Case Control Bulk Data and parameter requiremen
422. top if soft convergence is indicated default YES Terminate design cycles if soft convergence is achieved UPDTBSH Controls the update of the boundary shapes in the analytic boundary shapes method for shape optimization NO Do not update the boundary shapes default YES Update the boundary shapes Note Regardless of the value of UPDTBSH shape basis vectors are still updated for every design cycle interpolation to the interioi 4 24 Design Sensitivity and Optimization User s Guide Chapter 5 Solution Sequences Design Sensitivity and Optimization Modules e Selected Data Blocks e Solution 200 Program Flow Design Sensitivity and Optimization User s Guide 5 1 Chapter 5 Solution Sequences For most applications it is unlikely that an extensive understanding of the design optimization modules and data blocks will be required However a general knowledge of the module functions may be of help when evaluating design optimization messages user warning messages or error messages An awareness of the contents of some of the more significant data blocks may also be useful when examining intermediate results debugging input data etc This section briefly outlines the function of those modules that are unique to design optimization Most of the module names begin with the letters DO which stands for design optimization or DS for design sensitivity If necessary the NX Nastran Programmer s Manual should be consulted in
423. ts for design sensitivity and optimization These various input formats enable the following 1 Specification of the applicable analysis discipline s 2 Definition of the design model 3 Overrides of the optimizer s internal parameters 4 Control of the program flow and results output Use of the material in this chapter is covered in Design Modeling for Sensitivity and Optimization and Example Problems 4 1 File Management Section Executive Control Solution 200 In design sensitivity and optimization the only required Executive Control statement is the SOL statement SOL 200 This states that subDMAP DESOPT is to be invoked which is the main subDMAP for design sensitivity and optimization The following section lists the supported analysis types in Solution 200 these are chosen using the Case Control command ANALYSIS Solution Sequences 108 111 and 112 had previously supported dynamic response sensitivities This capability has been moved from these solution sequences to Solution 200 Shape Optimization and the File Management Section File Management Section statements see File Management Statements in the NX Nastran Quick Reference Guide for more information are required in SOL 200 for Shape optimization for the direct input of shapes method This approach DBLOCATEs a set of displacement vector data from a database and uses this data to generate a set of shape basis vectors for design optimization The following
424. ts are imposed with respect to analysis model properties as well as design variables They can be changed from their defaults by modifying DELP and DPMIN for properties and DELX DELXV and DXMIN for design variables Convergence Criteria Convergence at the Design Cycle Level The parameters CONV1 CONV2 GMAX CONVDV and CONVPR are used to test for overall design cycle convergence These parameters are used in connection with tests for both hard and soft convergence Convergence Tests describes the types of convergence testing as well as the convergence decision logic Soft Convergence Design Sensitivity and Optimization User s Guide 4 13 Chapter 4 Input Data Soft convergence compares the results of the approximate optimization with the results of the finite element analysis performed at the beginning of the design cycle Soft convergence is not sufficient to terminate the design cycle iterations unless the parameter SOFTEXIT is set to YES NO is the default Hard Convergence Hard convergence testing compares the analysis results of current design cycle with those of the previous cycle This test is a more conclusive test of convergence since it is based on hard evidence Hard convergence will always terminate the design cycle process Convergence at the Optimizer Level Convergence at the optimizer level can also be controlled using the DOPTPRM entry Parameters that can be changed include DABOBJ DELOBJ ITMAX ITRMOP ITRMST a
425. ts solely on the basis of this information may not bring about the expected increases in efficiency The design variable history as a function of design cycle is listed at the end of the summary The external design variable ID and the label columns are taken directly from the DESVAR Bulk Data entries The internal design variable ID column simply refers to the order of internal sort on the design variables The internal and external design variable sort may differ if DLINK entries are present Independent design variables are first sorted in ascending numerical order followed by the dependent design variables again in ascending order 6 3 Design Sensitivity Output Output for design sensitivity analysis is activated by setting the Bulk Data parameter OPTEXIT to 4 or 7 OPTEXIT 4 or 7 prints the design sensitivity data to the standard output file and is used primarily for manual interpretation and postprocessing OPTEXIT 4 sends the Design sensitivity data to a FORTRAN readable file using the NX Nastran OUTPUT2 and OUTPUTA functions This form is intended for those who would like to couple NX Nastran computed sensitivities with their own postprocessors external optimizers etc Design sensitivity output consists of two parts a matrix of sensitivity coefficients and a table listing the column order of this matrix The DSCM2 matrix contains these sensitivity coefficients as 6 12 Design Sensitivity and Optimization User s Guide Output F
426. ubcase using the DESSUB command The METHOD command included in the residual subcase must also be used in the upstream superelement subcases This requirement also holds for Component Modes Synthesis CMS Limitations The limitations that apply to superelement design modeling are summarized as follows The design model may not reference external superelements External superelements can be part of the analysis model but are considered invariant with respect to changes in the design model Image superelements have the same design variations and design responses as defined on the referenced primary superelement The exception is external grids which are invariant for the image superelements e Shape basis vectors can only be defined using the DVGRID entry 2 60 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization Eigenvalue sensitivity is evaluated for the residual superelement Since the residual contains the boundary degrees of freedom from all superelements the computed eigenvalue sensitivities pertain to the entire structure Eigenvalues computed during upstream superelement reduction are not available as design responses Note In addition Guyan reduction the default is exact for stiffness properties but only approximate for mass Thus you may want to consider using advanced reduction methods such as component modes synthesis to improve the sensitivity analysis ac
427. uced Alternatively a value of a4 might be found where all constraints are satisfied If this happens a feasible design is found and so again ay is an upper bound on a Here there is little concern that the objective function is increasing since the first priority is to overcome the constraint violations If the constraint violations can be overcome it is desirable to do so with a minimum increase in the objective function Case 3 One or more constraints are violated greater than CTMIN at a 0 and the objective function is decreasing dF x4 1 da gt 0 This is the same as Case 2 except that the objective function can be reduced Therefore everything is the same except that the search is not stopped if the constraint violation s are overcome If a feasible design can be found that allows further reduction of the objective function movement continues in this direction until some new constraint becomes active or the objective begins to increase The corresponding a is the upper bound a During the process of finding the upper bound a on a it may be required to move further than the initial step ay If ay does not yield an upper bound another step a 2 a is taken and tried again In practice it may take many steps before a bound is found on the solution If this is the case the best four solutions are retained and called a a4 G2 and ay If more than three steps are required the lower bound a will not equal zero I
428. uences G set size basis vectors in the global coordinate system It contains information for all grids in the model Constraint Evaluation and Screening CASEDS Case Control for the recovery of design responses DOPR3 DRSTBL Table providing the number of retained responses for each subcase for each of the response types DSAD RIVAL Vector of type 1 Vector of type 1 response values DSAD values Vector of type 1 response values DSAD A Vector of type 2 response values DSAD R3VAL Vector of type 3 response values DSAD R1VALR Vector of retained type 1 response values DSAD R2VALR Vector of retained type 2 response values DSAD R3VALR Vector of retained type 3 response values DSAD CVAL Vector of constraint values DSAD CVALR Vector of retained constraint values DSAD Design Sensitivity Analysis DELWS Delta weight DSAW DELVS Delta volume DSAW DSCM Design sensitivity coefficient matrix form 1 Each term is A DELB DSAL sensitivity coefficient matrix form 1 Each term is Ar DELB DSAL Bse EE A OSA GUT EE sensitivity coefficient matrix form 2 generated for output purposes only Each term is Ar Ax DOM6 IDSCMCOL Correlation table for DSCM DSTAP2 Correlation table for DSCM DSTAP2 SDSC Eero Seo ee element summary table used to compute property sensitivities DSABO ae Perturbed element summary table used to compute shape sensitivities DSAM Pseudo load contributi
429. ufficient However sometimes it is useful to follow what is happening within the optimizer itself The levels of optimizer print are as follows 0 No output default 6 Same plus scaling factors and miscellaneous search information Maximum Number of Design Cycles The optimization process is iterative since the optimizer obtains data about the design space from approximations The approximate model constructed based on a detailed finite element analysis is used by the optimizer to find an approximate optimum This design is resubmitted for another finite element analysis followed by another approximate optimization This process is repeated until convergence with respect to these overall design cycles is reached The default maximum number of such cycles is five unless the problem converges before then Usually a near optimum design is found by this point however more cycles are often necessary If the maximum number of design cycles is reached before convergence is achieved the problem can always be restarted from the last design see Restarts in Design Optimization Move Limits on the Approximate Optimization As the optimizer modifies the design variables the structure s properties and or shape will vary depending on the design model description As discussed in Section 3 3 Optimization with Respect to Approximate Models move limits need to be placed on the approximate subproblem for efficiency reasons These move limi
430. ulated Thereafter a usable feasible search direction is found if one exists In this case there are three possibilities 1 There are no active or violated constraints 2 There are active constraints but no violated constraints 3 There are one or more violated constraints Each of these possibilities is handled differently No Active or Violated Constraints Unconstrained Minimization Frequently at the beginning of the optimization process there are no active or violated constraints In this case the feasibility requirement is automatically met because we can move in any direction for at least a short distance without violating any constraints Thus we only need to find a usable direction which is the one that points downhill It does not have to be the steepest descent direction but to start the process this is the preferred choice Therefore the initial search direction is simply gt 1 CAS Equation D 26 The steepest descent direction is only used if this is the beginning of the optimization q 1 or if the results of the last iteration yielded no active or violated constraints Now assume the last search direction is in the steepest descent direction and there are still no active constraints If there were no violated constraints before there will not be any now since the optimizer would have stopped the search before any violation We could again search in the steepest descent direction and this is commonly d
431. ult 1 0E 20 XUB Upper bound Real default 1 0E 20 DELXV Move limit for the design variable during approximate optimization Real gt 0 0 ID of a DDVAL entry that provides a set of allowable discrete values Blank or DDVAL Integer gt 0 Default blank for continuous design variables Associated Entries Design variables may be related to properties on DVPREL1 or DVPREL2 entries related to changes in shape using DVBSHAP DVGRID or DVSHAP entries linked using DLINK entries or input to user defined responses on the DRESP2 entry Design Sensitivity and Optimization User s Guide 4 9 Chapter 4 Input Data DTABLE LABL1 LABL2 LABL3 NE Pete i i Discussion In design optimization structural properties are changed in order to determine the optimal value of the objective function To accomplish this the optimizer varies the design variables Not only must design variables be defined but they also must be functionally related to analysis model properties and or changes in structural shapes The DESVAR entry is used to define an individual design variable Design variables may be defined as members of independent and dependent sets using DLINK Bulk Data entries if desired DLINK Purpose Imposes a linear relationship among the design variables The set of all DLINK entries partitions the design variables into an independent set and a dependent set Entry Description E Lupe x E FRE E Field
432. umn 3 of DSCM2 contains the sensitivities of the z component displacement for grid 29 subcase 1 Column 41 contains the sensitivities of the same displacement component for subcase 2 In general responses are grouped first by superelement then by subcase and finally by DRESP1 or DRESP2 order A pseudo code of the output loop may be written as follows for superelement 1 no of superelements for load case 1 no of load cases for response first last response print sensitivity coefficient The responses are sorted by individual response type This order is given in Table 1 of the DRESP1 Bulk Data entry description in the NX Nastran Quick Reference Guide Design Sensitivity and Optimization User s Guide 6 15 Chapter 7 Example Problems e Three Bar Truss Vibration of a Cantilever Beam Turner s Problem e Cantilevered Plate Stiffened Plate Shape Optimization of a Culvert Analytic Boundary Shapes Dynamic Response Optimization Twenty Five Bar Truss Superelement Optimization Design Optimization with Composite Materials Acoustic Optimization Restarts in Design Optimization Design Sensitivity and Optimization User s Guide 7 1 Chapter 7 Example Problems This chapter includes a wide range of examples intended to highlight and illustrate the Design Sensitivity and Optimization features of the software Though by no means exhaustive these eleven examples cover most majo
433. up to which a number of elements belong The DESVAR 1 entry defines design variable 1 T PLATE with an initial value of 0 15 and lower and upper bounds of 001 and 10 0 respectively Design variable 1 is in turn related to PSHELL property group 1 thickness via DVPREL1 number 1 The 4 in DVPREL1 number 1 field 5 states that it is field 4 on the property entry the thickness that is to be varied The continuation line on the DVPREL1 entry gives the functional relationship defined by Eq 2 16 Field 6 on the DVPREL1 entry defines the minimum allowable value of the property here a minimum thickness of 0 01 This value enforces a lower bound even though greater than the design variable s lower bound See Stiffened Plate for the remainder of the design model Frequently you may want to specify the bar cross sectional dimensions in the design model instead and use this data to compute the resultant cross sectional properties such as areas and moments of 2 12 Design Sensitivity and Optimization User s Guide Design Modeling for Sensitivity and Optimization inertia Since these properties are often nonlinear functions of the design variables this approach requires the use of type 2 design variable to property relations Probably the easiest way to introduce type 2 relations is with a simple example 2 F b 03 o h 04 bh q 2 n 0 477 p b 2 12 b J Figure 2 5 Cap Section Detail Assume that the ca
434. uperelements All references to external grids of secondary superelements are ignored however This stems from the fact that the superelement mapping matrix is used to compute the secondary superelement matrices using the primary superelement Design Responses in Superelement Design Modeling You can select design responses using DRESP1 DRESP2 or DREPS3 Bulk Data entries or using a combination of those entries A single DRESP1 entry can be used to select a number of design responses by referring to multiple grid locations property IDs or elements The responses identified on a single DRESP1 entry can be from multiple superelements For weight and volume type responses the applicable superelement IDs SEIDi or ALL must be specified Weight or volume responses can be computed for the entire model or just a subset Design Sensitivity and Optimization User s Guide 2 59 Chapter 2 Design Modeling for Sensitivity and Optimization defined by superelement ID reference The default is ALL which is the total weight volume of all superelements except external superelements Weight and volume sensitivities can be computed on an individual superelement basis or for ALL superelements DRESP2 and DRESP3 Restrictions The design responses input to DRESP2 equations or to DRESP3 external programs cannot span multiple superelements or load cases unless they are DRSPAN related All synthetic responses are checked to ensure that all response ar
435. urface area the derivative of D with respect to a single design variable is then dD dx 1 AXE dx 1 A E dA dx Equation 3 86 NX Nastran currently disregards changes in the panel normal surface area with changes in the design variables Therefore the second term drops out and the derivative expression is simplified to dD dx 1 A dE dx Equation 3 87 where dE dx is obtained with the adjoint load approach described above Aeroelastic Sensitivity Analysis Aeroelastic response sensitivities are computed in Solution 200 in connection with design optimization Sensitivities alone can be computed if the DSAPRT Case Control command is used or the Bulk Data parameter OPTEXIT is set to 4 The following sensitivities are available e Static aeroelastic responses including not only displacement stresses forces etc but also trim variables This allows the evaluation of a structural change on the trim angle of attack Stability derivative sensitivities This for example allows the determination of the effect that strengthening a wing spar has on the rolling moment produced by a deflection of the aileron Flutter sensitivities These compute the changes in damping responses in a Flutter analysis at specified velocities due to changes in structural parameters This section contains a brief description of the theory for aeroelastic sensitivity analysis For both theoretical and design modeling details as w
436. used for the various analysis disciplines Shape Basis Vector is a basis vector used in connection with shape sensitivity and optimization The vector is a collection of directions along which designed grids can move Each element of the vector represents the magnitude of change in a grid coordinate due to a unit change in a design variable See the definition for Basis Vector Shape Design Variable is a design variable used to describe structural shape variations Theoretically shape design variables act as multipliers of shape basis vectors but in practice can often be thought of for example as a radius a coordinate location defining the center of a cutout the percentage addition of taper to a control arm and so on Shape Optimization defines a design task in which boundaries and connection point locations are allowed to change in search of an improved design Shape Sensitivity computes the rate of change of structural responses with respect to shape defining design variables The grid coordinate changes are expressed in terms of design variable changes using the methods of Relating Design Variables to Shape Changes Side Constraint is either an upper or a lower bound on a design variable Since the optimizer never searches outside of these bounds these side constraints essentially define the limits of the design space Soft Convergence refers to the design cycle convergence test that compares the results of an approximate optimization
437. uxiliary model loops for shape optimization AXMPR1 BASVEC jBasis vectors for shape optimization This form is G set size and is in basic coordinates It is used in the AMBS GMBS and external input of basis vectors methods VECPLOT for AMBS and GMBS SMPYAD for External Input COORDO Old set of coordinate values for designed grids only It is constant over all design sl cycle for AMBS and GMBS DOPR2 DNODE input data for the DRESP2 Bulk Data entries DOM11 pue eRuMES vector used to strip out the shape basis vector solutions for the AMBS and GMBS methods SHPCAS nM vector defining the finite difference moves to be used in shape sensitivity It is computed based on shape basis vector energy norms ADD5 IDESGID List of designed coordinates and identification data DOPR2 BE containing designed grid perturbations Its form is the same as DTOSAK and is used to define element level variations for shape sensitivity analysis DOPR5 DTOS4J Table of shape basis vector data It is in external user defined design variable ID order DOPR2 DTOS4K Differs from DTOS4J only in that it is sorted on internal design variable ID DOPR4 DVIDS Design variables related to shape changes DSAJ Basis vectors for shape optimization G set size expressed in the global coordinate system DESVECP contains one column for each independent design variable It is Design Sensitivity and Optimization User s Guide 5 5 Chapter 5 Solution Seq
438. valent to ITRMOP and defines the maximum number of approximate linear problems that may be solved JTMAX Maximum number of iterations allowed in the sequential linear programming DEM used only if METHOD 2 see DOPTPRM entry Its function is equivalent to ITMAX used for the modified method of feasible directions Design Sensitivity and Optimization User s Guide 3 53 Chapter 3 Design Sensitivity and Optimization in NX Nastran 3 4 Convergence Tests Since structural optimization is an iterative process numerical criteria must be established to determine when the overall process has converged There are two levels at which convergence is tested the first and lower level is at the optimizer level the second and higher level is with respect to the overall design cycles This section is concerned with design cycle level convergence Convergence at the optimizer level is briefly discussed at the end of the previous section and in greater detail in Introduction The numerical values of all of the convergence criteria discussed in this section can be changed using the DOPTPRM Bulk Data entry This entry is optional though since defaults for all of these quantities have been provided However it is recommended that you always check to ensure their validity for the problem at hand The trade off between analysis cost and acceptable design accuracy should always be of prime consideration Convergence of Design Cycles Hard and Soft Convergence
439. values of the response bounds are used as normalizing factors in the resulting inequality constraints L gt 3 Fc r x p E 82 10 H i U R r x r g X ll J 2j E J lt 0 Equation 2 35 that are of the same inequality form as the constraints in the basic optimization problem statement Physical Significance of Normalized Constraints Normalized constraints are especially useful since the dependence on the magnitude of the response quantity has been removed A constraint having a value of 1 represents a 100 violation irrespective of the magnitude of the response or whether it is an upper or lower bound Normalization is used in the constraint deletion phase to temporarily screen out constraints that are not active Design Sensitivity and Optimization User s Guide 2 55 Chapter 2 Design Modeling for Sensitivity and Optimization An overview of constraint deletion is given in Approximation Concepts in Design Optimization in connection with approximation techniques Using GSCAL to Provide Minimum Constraint Normalizing Factors Since response bounds are used as normalizing factors it is important to avoid specifying the bounds as zero To avoid a divide by zero error in the code a small nonzero number is used instead which may or may not be acceptable from a design standpoint is provided by the parameter GSCAL which can be defined on the DOPTPRM entry Its default is 0 001 If either of the normalizing factors Int or I
440. values of x into Eq D 52 or examine the plot of the function The optimization process leads to an estimate for a for the minimum of the objective and the zero of one constraint In this case the constraint is initially violated so we have both a lower and an upper bound on a However we must also estimate values of a for all constraints The maximum lower bound and the minimum upper bound define the range for the proposed solution to the one dimensional search It may happen that the proposed upper bound is less than the proposed lower bound If this happens it usually means that for each proposed a there are one or more violated constraints In this case Eq D 52 is used to minimize the maximum constraint violation and this decision overrides other conditions Let us assume that the polynomial approximation for the objective and all constraints is used and concludes with a lower and upper bound on the constraint values so that the upper bound is greater D 22 Design Sensitivity and Optimization User s Guide Numerical Optimization than the lower bound If no constraints are violated at a 0 this is the usual case Next a choice must be made as to which value of to select If the search starts with violated constraints and the lower bound is less than the upper bound either the lower bound is selected or the value is chosen that minimizes the objective assuming that it is between the lower and upper bounds If no constrai
441. ve is the function that the optimizer seeks to minimize The objective function must be a continuous function of the design variables One Dimensional Search is the search process at the optimizer level in which all changes to the vector of design variables are characterized by changes in a single scalar parameter a See the definition for Search Direction and Eq 1 12 Optimum Design is defined as a point in the design space for which the objective function is minimized and the Kuhn Tucker Conditions are satisfied The optimum might be feasible if all constraints are satisfied or infeasible if one or more constraints are violated If relative minima exist in the design space other optimal designs can exist Primary Model or Primary Structure refers to the finite element model used for analysis The distinction arises when Auxiliary Models are used for shape optimization See the definitions for Auxiliary Models and Auxiliary Boundary Models Property Optimization defines a design task in which the properties describing the analysis model are allowed to vary Also frequently referred to as sizing optimization quantities such as plate thicknesses bar element moments of inertia composite ply orientations and so on are modified in search of an improved design Pseudo Load Vector refers to the right hand side vector in the sensitivity analysis equations See Design Sensitivity Analysis Reduced Basis Formulation uses a small number of des
442. ve scheduling problems such as the routing of airplanes among various airport facilities The allocation of 100 airplanes among 68 airports may not seem to have much to do with engineering design but the optimization methods employed are similar and can be extended to structural problems An efficient engineering solution to a design problem involves the optimum allocation of scarce resources For example tensile stresses cannot be allowed to assume unlimited values and must be restricted to within reasonable limits the distribution of strain energy density must be made in an optimal manner Likewise reducing structural mass leads to a savings of material and possibly maintenance fuel or other indirect costs The Basic Optimization Problem Statement Before introducing any formal relations assume that we are assigned the type of task that a numerical optimizer might be asked to solve We must examine ways to approach the problem From this experience we can build the equations that describe the basic optimization problem statement Objective and Constraint Functions Design Spaces Suppose we are standing on the side of a hill and would like to find the point of lowest elevation this is our objective Suppose also that some fences exist that force us to restrict our search to within the region enclosed by these fences These fences or constraints act as bounds in our design space which is the region that defines all of our possible po
443. w that the mass is a function of the properties used to describe the component as well as its shape This can be expressed as gt gt M M p G Equation 2 4 gt where the vector P is the collection of properties that describe the model and the vector is the collection of grid point coordinates In the next sections we will show how the properties and shape of the structure can be expressed as functions of design variables or gt Pp p E gt gt G G x Equation 2 5 By direct substitution of Eq 2 5 into Eq 2 4 gt gt 2 gt M M p x G x Equation 2 6 or simply Design Sensitivity and Optimization User s Guide 2 7 Chapter 2 Design Modeling for Sensitivity and Optimization M M x Equation 2 7 To summarize as the optimizer changes the design variables the analysis model properties and shape will also change as defined by the design model The modified properties and shape result in changes to the computed responses that we have used to define the objective and constraint functions Based on the modified objective and constraints the optimizer can measure the amount of design improvement 2 4 Relating Design Variables to Properties For the analysis model to vary as the design variables are changed its properties and or its shape must be expressed in terms of the design variables You need to provide these relations as part of the design model specification This section discusses analy
444. weight using just those elements in the residual structure Note that the initial design is infeasible as indicated by the maximum constraint value in the summary of design cycle history After eight design cycles and a total of nine finite element analyses a lighter weight structure is found that satisfies all constraints NORMAL CONVERGENCE CRITERIA SATISFIED HARD CONVERGENCE DECISION LOGIC Ck kk kk ko ke kk kk kk kk ke ke ke kk ke kk kk kk ok ek OOO OK AA A AA ee CONVERGENCE ACHIEVED BASED ON THE FOLLOWING CRITERIA HARD CONVERGENCE DECISION LOGIC RELATIVE CHANGE IN OBJECTIVE 5 2768E 05 MUST BE LESS THAN 1 0000E 03 OR ABSOLUTE CHANGE IN OBJECTIVE 2 8809E 02 MUST BE LESS THAN 1 0000E 02 AND MAXIMUM CONSTRAINT VALUE 3 8660E 04 MUST BE LESS THAN 5 0000E 03 CONVERGENCE TO A FEASIBLE DESIGN aes PR MAXIMUM OF RELATIVE PROP CHANGES 3 8638E 02 MUST BE LESS THAN 1 0000E 03 AND MAXIMUM OF RELATIVE D V CHANGES 3 8638E 02 MUST BE LESS THAN 1 0000E 03 CONVERGENCE TO A BEST COMPROMISE INFEASIBLE DESIGN KK RRA ete eee eee eee ee eee kk ee Kk kc ek ck ck AAA ARA AAA kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk k kkkkkkx SUMMARY OF DESIGN CYCLE HISTORY itt cco kk kk ok kk ke ke kk kk kk kk ko ke ke kk kk kk ko AR ke ke kk kk e koe HARD CONVERGENCE ACHIEVED NUMBER OF FINITE ELEMENT ANALYSES COMPLETED 9 NUMBER OF OPTIMIZATIONS W R T APPROXIMATE MODELS 8 OBJECTIVE AND MAXIMUM CONSTRAINT HISTORY OBJECTI
445. xes Note Finite element analysis and sensitivity analysis are the most computationally intensive phases These CPU intensive phases have been highlighted in the flowchart The circled numbers 1 through 6 appearing in the flowchart indicate the locations of the six user defined exit points Any one of these exits can be taken by setting the parameter OPTEXIT to an integer value from 1 through 6 An OPTEXIT value of 7 provides sensitivity coefficient output upon normal program termination the sensitivities are printed to the f06 output file Normal program termination includes both hard and soft convergence and maximum number of design cycles Sensitivities for the final design may be particularly useful depending on your postprocessing requirements Be aware however that some additional costs will be incurred for the hard convergence exit condition since the sensitivities must be recomputed for the last design cycle Design Sensitivity and Optimization User s Guide 5 7 Chapter 5 5 8 Solution Sequences NX Nastran Input File Processing for Primary IEPL and All Auxiliary Models AXMDRV AXMPR1 Does a Design Model Exist YES Generate Nonrepetitive Tables Used in Design Sensitivity and Analysis Analysis Model Override by Design Model DOPR1 DOPR2 DOPR4 DOM11 PREDOM D EXITOPT Case Control Partitioning Based on Analysis Type MDCASE PHASEO Operations Restart Checking
446. y be listed on the DRESP1 entry for element level responses the ELEM identifier must be used with only a single element ID and so on Design Sensitivity and Optimization User s Guide 2 45 Chapter 2 Design Modeling for Sensitivity and Optimization Designed Grids The DNODE input fields on the DRESP2 entry may be used to input the coordinates for designed grids only A designed grid is one whose coordinates may vary during shape optimization Mathematically a designed grid has a nonzero coordinate component in one or more of the shape basis vectors See the DRESP2 entry for a detailed description of designed grids Example Suppose we would like to include Euler buckling constraints in the design model for some simple pin ended rod elements A representative element is shown in Figure 2 24 A E I P MK gt x SSS 1 2 Figure 2 24 Pin ended Rod Element The critical load that induces the first Euler buckling mode is given by n El I Equation 2 29 Per The design requirements call for the axial load in this ROD element to be greater less compressive than this critical load Pcp or E NT Por Equation 2 30 which can be rewritten as 2 P x xi n EI Equation 2 31 Note that the critical load is a function of the least area moment of inertia yet this quantity is not a part of the analysis model specification for the ROD element If the cross sectional area of the element is the
447. y using any number and types of applied static loading to produce sets of desired boundary variations The code interpolates these boundary changes to the structure s interior thus automatically generating the shape basis vectors Design Sensitivity and Optimization User s Guide 2 37 Chapter 2 Design Modeling for Sensitivity and Optimization Each auxiliary boundary model requires its own Bulk Data Section Each model can be loaded in any number of subcases to produce more than one shape basis vector per auxiliary boundary model Load and boundary condition sets are selected using the auxiliary boundary model Case Control Sections The following Bulk Data Section provides the auxiliary boundary model definition This section must appear after the Bulk Data for the primary model BEGIN BULK AUXMODEL PARAM PRGPST NO auxiliary model element definition geometry from primary structure 101 20 1 5 102 20 5 103 20 6 104 20 T 105 20 8 9 106 20 9 10 L07 20 10 I1 108 20 117 12 109 20 12 I3 110 20 13 14 20 102 1 0E 3 I 0E 12 102 7 2410 0 33 S QOO OO 6 Oo Ur 0 Te 0E 12 1 0E 12 8015 3 1 0 2 NPRO0OO0O000O000oOo0oOo boundary condition set I D1 160 101 FX FR 160 102 FX FR 160 103 FX FR 160 104 FX FR 160 105 FX FR 160 106 FX FR 160 107 FX FR 160 108 FX FR 160 109 FX FR 160 110 FX FR 100 100 100 100
448. ycle These deleted constraints may again become active during subsequent design cycles and thus retained G X X XXXXXXXX TRS 1 0 List of Constraints Figure 3 4 Constraint Deletion The constraints retained for the current design cycle are denoted by an X in the figure You can change the value of TRS from its default of 0 5 using a DSCREEN entry TRS is defined on a per response type basis Constraint Regionalization Each set of structural responses is automatically assigned a unique region identifier All constraints associated with this region for a given subcase comprise a unique group The assumption is that constraints from this group are likely to contain redundant information from this group only a maximum number of constraints is retained as determined by NSTR on the DSCREEN entry Responses are usually grouped into regions based on their DRESP1 entry identification or by their property group association See the DSCREEN entry for default region specifications For example imagine a highly stressed constant thickness panel comprised of a large number of shell elements Suppose we are seeking to redesign the panel thickness subject to element stress constraints Due to the large loads a number of elements in this panel may yield stress constraints that are well above the truncation threshold TRS They will thus pass the first screening test However this still gives us more design information tha
449. yses In a Frequency Response analysis if you leave the frequency value field blank in the DRESP1 entry NX Nastran generates that particular DRESP1 entry for every frequency in the relevant frequency set If you are using an OFREQUENCY Case Control command the software generates that particular DRESP1 frequency for all frequencies specified by the OFREQUUENCY command Similarly in a Transient Response analysis if you leave the time value field blank NX Nastran generates multiple DRESP1 entries In addition if you specify multiple entities grid or element NX Nastran generates multiple DRESP1 entries for each listed entity For Frequency Response or Transient Response analyses you can obtain a simple mathematical function resultant of the multiple responses by entering character input in the field instead of either specifying a frequency or time value or leaving the field blank For example if you specify AVG instead of a frequency value in a DRESP1 entry NX Nastran evaluates DRESP1 at all frequencies in the relevant set It then calculates the average of these responses See Remark 20 for the DRESP1 entry in the NX Nastran Quick Reference Guide for further details If you reference such a DRESP1 entry from a DRESP2 entry you should reference it as a DRESP2 entry If you reference it as a DRESP1 entry then the software generates the referencing DRESP2 entry for multiple frequencies using the frequency based components of the mathemati
450. zation Maximization The objective function is the scalar quantity to be minimized It is a function of the set of design variables Although we stated the problem as a minimization task we can easily maximize a function by minimizing its negative Side constraints are placed on the design variables to limit the region of search for example to plate thicknesses that are nonnegative or tubes whose wall thicknesses are less than one tenth of the outer radii The inequality constraints represent the fences on the hill and are expressed in a less than or equal to zero form by convention We have satisfied a constraint and are thus within the boundary defined by the fence if the constraint s value is negative We have 2w oL C0 0 o e violated the constraint if its value is positive The location of the j th fence lies at constraints if present must be satisfied exactly at the optimal design Equality Linear Versus Nonlinear Problems The objective and constraint functions may either be linear or nonlinear functions of the design variables If all of these functions are linear we may use linear techniques to find an optimal solution if one exists If just one of these functions is nonlinear then search algorithms that can deal with this nonlinearity must be used NX Nastran includes capabilities for solving both linear and nonlinear optimization problems As seen throughout the remainder of this user s guide the basic optimization p

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