Home

FElt: User's Guide and Reference Manual pdfsubject

1. 96 10 1 Set up function for the truss element definition 118 10 2 Stress function for the truss element definition 119 11 1 Solving problems with the finite element intrinsic functions 150 List of Tables 3 1 Symbolic names used to define material properties 19 3 2 Meaning given to the various analysis parameters 23 4 1 Description of the materials required for and local DOF affected by the various element types coronarse 30 5 1 Contents of the stress vector for different element types 45 10 1 Finite element related variables 24 4 545445406 020 400 4 108 10 2 Fields of an element object 52 bee ee ee eee een ds 110 103 Peldsoranode object s soss a Saws So aw So A BA Be RO 111 104 Fields of a material object em aK EW a a 8 112 10 5 Pdo aforceobj tt lt s ses RARA Ee eG 113 10 Fields of a constraint object gt o e be ee eee RS 6 Re eH a naa 114 10 7 Fields of a distributed load object co cane ee Re Hee HR EE ES 114 10 8 Directions of a distributed load 2 2 2 o oo 115 10 9 Fields of an element definition Object o o 115 10 10Fields of a problem definition object o o 115 10 11 Fields of an analysis parameters object o ooo 116 10 12 Arguments to the add_definition function 117 11 1 Enumeration constants in burl
2. accept dismiss Figure 7 3 The configuration dialog box The same is true for the element numbers option A check mark will appear next to the menu entry if the option is enabled 7 4 2 Object coloring Beyond the basic node color and element color controls provided in the canvas configura tion dialog Figure 7 3 velvet also allows you to assign unique colors to individual forces constraints material properties and distributed loads Selecting color control from the canvas menu will pop up the colors dialog shown in Figure 7 4 The four smaller scrolled lists on the left list all of the attributable objects materials distributed loads constraints forces see section 7 5 defined in the current problem The larger list running down the right side lists the available colors By highlighting an entry in one of the four object lists you can assign a color to that object In Figure 7 4 we see from the highlighted entries that the color for the material steel is red Any elements with material property steel will be rendered in red on the main drawing canvas If we wanted to change the color for steel to yellow we could simply click on yellow in the colors list There is no accept button on this dialog color assignments take effect as soon as you select a color from the colors list For any color changes to be reflected in the drawing area however you need to click on recolor on the bottom of the color control dialog Because
3. A distributed load typedef struct distributed char name Direction direction unsigned Pair nvalues value Distributed A force l reaction force E node number a affected degree of freedom x coordinate Ef y coordinate z coordinate computed stress values element symbolic name z7 element initialization function z stress computation function o element dimensional shape number of nodes in element XY number of nodes which define shape number of computed stress values xj number of degrees of freedom degrees of freedom retain element K after assemblage name of distributed load direction of load number of values nodes and magnitudes 162 CHAPTER 12 THE ALGORITHMS BEHIND FELT typedef struct force char name name of force VarExpr force 7 force vector VarExpr spectrum 7 vector of input spectra Force A constraint typedef struct constraint char name char constraint 7 double ix 7 double ax 4 double vx 4 VarExpr dx 7 Constraint A material typedef struct material char name double E double Ix double Iy double Iz double A double J double G double t double rho double nu double kappa double Rk double Rm double Kx double Ky double Kz dou
4. x Y Z Z XopY matrix matrix matrix Vi j Zi j X j op Y matrix scalar matrix Vi j Zij Xij op y scalar matrix matrix Vi j Z xo0pY scalar scalar scalar z x op y scalar comparison string string scalar z x op y lexicographic comparison Table 11 5 Type table for the relational operators The result of the expression is 1 if the expression is true and 0 if the expression is false If two matrices are compared then their dimensions must be identical 1 write 1 2 3 lt 3 2 write EL 2 3 gt 3 2 Li 3 write aardvark lt aardwolf 1 11 4 7 Range operator Z wp The range operator produces a row vector starting at x and ending at z with y used as the step between successive elements veya rxty 5 y if x lt zand y gt 0 or x gt z and y lt 0 z Z null otherwise 11 4 OPERATORS 133 If y is not specified then 1 is used as the step value Each of x y and z must be scalar expressions as shown in Table 11 6 X y Z W W X Y 2Z scalar scalar scalar matrix as defined above Table 11 6 Type table for the range operator 1 write 1 3 2 write 3 1 null 3 write 1 2 10 4 write 10 2 1 10 8 6 4 2 As illustrated above if the range is improperly specified then a null value is returned The null value is illegal in most operations and is used to indicate an uninitialized
5. 3 1 1 General rules The input file for a typical FElt problem will consist of nine sections In general each object within a problem node element force material distributed load constraint has a symbolic name For nodes and elements these names are positive in tegers for everything else the name is a user assigned string e g steel point load roller Each type of object is defined within its own section each section containing a list of definitions for objects of that type As a rule of thumb white space can occur anywhere and the definition sections can be given in any order and repeated even The exceptions to this are the problem description section and the end statement which must come first and last in an input file respectively and cannot be repeated Comments are denoted as in the C programming language anything between and 13 14 CHAPTER 3 STRUCTURE OF A FELT PROBLEM will be ignored as a comment no matter where it appears in the file 3 1 2 Expressions 3 1 2 1 Continuous functions As a convenience wherever a numeric value is required for a material characteristic mag nitude of a force load or displacement boundary condition or a nodal coordinate you can specify an arbitrary mathematical expression including the operators and the standard mathematical library functions sin cos tan sqrt hypot pow exp log log10 floor ceil fabs and fmod Note that argu
6. 3 2 5 Constraints The constraints section s must define all of the named constraints within the problem Each constraint is defined by a name followed by a list of DOF specifications of the form Tx Ty Tz Rx Ry Rz where the can either be c for constrained u for unconstrained or a valid possibly time dependent expression for cases where a dis placement boundary condition is required e g settlement of support or time varying tem perature along a boundary in a transient thermal analysis problem Note that specifiying Tx c is equivalent to Tx 0 0 The T and R refer to translation and rotation respectively and the subscripted axis letter indicates that the specification refers to translation along or rotation about that axis An additional specification of h or hinged is allowed for the rotational DOFs for cases where it is necessary to model a nodal momentless hinge Note that a hinge specification currently has meaning only on beam beam3d and timoshenko elements If a specification is never made for a DOF then the problem is assumed to be uncon strained in that degree of freedom Getting the constraints right is an important part of 20 CHAPTER 3 STRUCTURE OF A FELT PROBLEM getting a reasonable solution out of a finite element problem so you should be aware of what DOF are active in a given problem this will depend on which types of elements are being used even in a 2 d problem the global stiffness ma
7. The nodes section s must define all of the nodes given in the problem Each node must be located with an x y and z coordinate using x y and z assignments Coordinates are taken as 0 0 if they are not otherwise defined If a coordinate is left unspecified for a given node it takes the value for that coordinate from the previous node A node must also have a constraint assigned to it by a constraint statement The default constraint leaves the node completely free in all six degrees of freedom Like coordinates 1f a constraint is left unspecified the node will be assigned the same constraint as the previous node Forces applied point loads and moments on a node are optional and are applied using the force statement if a force is not specified there will be no force applied to that node You can also specify an optional lumped mass at a given node with a mass statement 18 CHAPTER 3 STRUCTURE OF A FELT PROBLEM 3 2 3 Elements An element definition section begins with the keywords xxxxx elements where xxxxx is the symbolic name of a type of element All elements under this section heading will be taken to be the given element type There could be multiple sections that defined beam elements but each must begin with the keywords beam elements If there were truss elements in the same problem they would have to be defined in sections which be gan with the keywords truss elements Currently available types are spring truss beam b
8. beam type elements are a bit of an anomaly because they need their stiffness matrix back and a bunch of local lt gt global transforms E int timoshenkoEltStress element Element element unsigned i loop index el int count count of errors static Vector dlocal NULL local nodal displacements 186 CHAPTER 13 ADDING ELEMENTS TO FELT static Vector di static Vector E Vector equiv Matrix T static Matrix khat static Matrix Tts static Matrix temp global nodal displacements actual internal forces equivalent nodal forces tf transform matrix local stiffness matrix transpose of transform temporary matrix for multiply our usual trick to set up the matrices and vectors that we need memory for but that are really just local to this function if dlocal NULL dlocal CreateVector 4 d CreateVector 6 f CreateVector 4 khat CreateMatrix 4 4 Tt CreateMatrix 6 4 temp CreateMatrix 4 6 if dlocal NullMatrix khat NullMatrix AllocationError element d NullMatrix NullMatrix Tt NullMatrix stresses set the number of points where we will calculate stresses In this case 1t s two 1 element gt ninteg 2 one at each end Fill out a vector with the element s nodal displacements These are in global coordinates of co
9. felt plot transient flt gt transient result would produce a file transient result which looked like the following time Tx 8 0 0 0 05 0 00077749 0 1 0 028323 00 19 0 10137 0 2 0 23788 0 25 0 45923 0 3 0 76125 0435 1 1366 0 4 1 5518 0 45 1 9716 0 5 22392 0 55 2 6652 0 6 2 8977 0 65 3 0442 0 7 3 1074 48 CHAPTER 5 THE FELT APPLICATION 0 75 3 08 0 8 2 9652 5 3 INTERPRETING THE OUTPUT FROM FELT 49 5 3 3 Modal analysis The output from a modal analysis of a simple one story frame is shown below Note that the matrices command line switch was used to generate supplemental output the ma trices prints before the results from the actual analysis and that the eigen switch was not activated so full modal analysis was performed The actual modal analysis results consist of a listing of modal frequencies a table of mode shape vectors the eigenvectors the modal mass damping and stiffness matrices and the damping ratios in each mode M 15 62 0 0 0 0 0 0 15 62 0 0 0 0 0 0 8 558e 04 0 0 0 0 0 0 15 62 0 0 0 0 0 0 15 62 0 0 0 0 0 0 8 558e 04 C 194 8 0 720 184 6 0 0 0 801 7 170 4 0 1 049 170 4 720 170 4 1 123e 05 0 170 4 1 846e 04 184 6 0 0 194 8 0 720 0 1 049 170 4 0 801 7 170 4 0 170 4 1 846e 04 720 170 4 1 123e 05 K 9 711e 04 0 3 6e 05 9 231e 04 0 0 0 4 005e 05 8 521e 04 0 524 4 8 521e 04 3 6e 05 8 521e 04 5 446e 07 0 8 521e 04 9 231e 06 9 231e 04 0 0 9 71
10. problem description but wanted to do some refinement using your favorite CAD program The patchwork command line to do something like that is simply the reverse of above patchwork felt mesh flt dxf mesh dxf 28 CHAPTER 3 STRUCTURE OF A FELT PROBLEM 3 5 2 patchwork details patchwork operates by providing routines to read and write the different file formats based around a single data structure The input formatted file is translated into the uniform database and then the output formatted file is generated from this same database The ability to read or write any given format is dependent on whether or not routines exist to go back and forth between the file syntax and the uniform data structure Currently patchwork can read i e the allowable input formats are DXF files standard FElt files and data files formatted for use in graphing applications like gnuplot patchwork can write 1 e allow able output formats are FElt DXF gnuplot and files formatted for the software that is distributed with Logan s introductory finite element text 13 Capabilities for additional formats will be added as time permits and demand warrants Note that the DXF handling routines are limited to line and polygonal polyline entities When invoking patchwork the input format and file always come first The syntax is patchwork iformat ifile oformat ofile where iformat can currently be one of dxf felt graph and oformat can be one dxf
11. 2 19 6 Mo 2 19 We can fix the damping ratio in two modes i and j simply by substituting C RmMi R Ki 2 20 Cj RmM R Kj 2 21 into equation 2 19 and solving the resulting linear system for Rm and Rg 1 o R 26 Qi mj I 2 22 EEA After solving the n uncoupled single degree of freedom equations either as initial value problems or steady state oscillation problems for the individual q we can form the solu tion in physical coordinates as the superposition of the motion in each mode x t Yugi Ug 2 23 i 1 2 7 Spectral analysis Spectral analysis is intended to give us information about the response of our structural system in the frequency domain In a way we can think of it as a direct way to calculate the results that we would get from estimating a power spectrum of our time domain results using a fast fourier transform If we assume that the force vector in equation 2 8 is harmonic in time and of the form F t Foe 2 24 2 8 NONLINEAR STATIC ANALYSIS 11 then we can also assume the solution x t is of the form x t ioe 2 25 Note that o is a complex constant which incorporates both the magnitude and phase of the output motion do xe 2 26 If we differentiate substitute into equation 2 8 and divide out the time dependent harmonic term then we find that Fo M i C K o 2 27 If this were a single degree of freedom system we would recognize the term in squ
12. double L the element length double phi bending stiffness shear stiffness 13 6 A DETAILED EXAMPLE 189 double factor common factor in stiffness matrix We use a trick to make sure we only allocate this memory once and then use it over and over again each time we need to create an element of this kind if k NULL k CreateMatrix 4 4 if k NullMatrix AllocationError element local stiffness L ElementLength element 2 if L lt TINY error length of element d is zero to machine precision element gt number return NullMatrix phi 12 0 L L element gt material gt E element gt material gt Ix element gt material gt kappa element gt material gt G element gt material gt A We know how the integration works out for the stiffness matrix so we re just going to fill it out an entry at a time For some element types this wouldn t be possible and we would do some integrating right here to fill in k Also because this is a symmetric matrix we ll just fill in everything above the diagonal and then use MirrorMatrix xf atrixData k 1 1 12 0 atrixData k 1 2 6 0 L atrixData k 1 3 12 0 atrixData k 1 4 6 0 L atrixData k 2 2 4 0 phi L L atrixData k 2 3 6 0 L atrixData k 2 4 2 0 phi L L atrixData k 3 3 12 0
13. se if element gt distributed i gt value 1 node 2 wb element gt distributed i gt value 1 magnitude wa element gt distributed i gt value 2 magnitude Again since we know how the integration turns out we ll just go head and plug straight into the entries in the equiv vector The order of entries in equiv is Fy1 Mzl Fy2 Mz2 There are three cases we need to deal with The first is a uniform load The second two are sloped loads which we 11 treat as the superposition of the uniform case and a case in which the load can be treated as q x q0 1 x L i e a load which goes from q0 to 0 wa wb uniform distributed load VectorData equiv 1 wa L 2 0 VectorData equiv 3 wa L 2 0 VectorData equiv 2 wa L L 12 0 VectorData equiv 4 wa L L 12 0 198 CHAPTER 13 ADDING ELEMENTS TO FELT els els e if fabs wa gt fabs wb load sloping node 1 node 2 factor wa wb L 120 0 1 0 phi VectorData equiv 1 wb L 2 0 factor 42 0 40 0 phi VectorData equiv 3 wb L 2 0 factor 18 0 20 0 phi VectorData equiv 2 wb L L 12 0 factor 6 0 5 0 phi L VectorData equiv 4 wo L L 12 0 factor 4 0 5 0 phi L e if fabs wa lt fabs wb load sloping node 2 node 1 factor wb wa L 120 0 1 0 phi VectorData equiv 1 wa L 2 0 factor
14. 0 3E 1 1 on Pit Bit 5000 Ibs sf 30 fi Figure 3 4 Transient analysis of a simple frame structure The input file for this case could look something like the following problem description title Dynamic Frame Analysis nodes 8 elements 9 analysis transient 26 CHAPTER 3 STRUCTURE OF A FELT PROBLEM analysis parameters dt 0 05 duration 0 8 beta 0 25 gamma 0 5 alpha 0 0 nodes 8 dofs Tx mass mode lumped nodes 1 x 0 y 0 constraint fixed 2 x 360 y 0 3 x 0 y 180 constraint free force f1 4 x 360 5 x 0 y 300 force f2 6 x 360 7 x 0 y 420 force f3 8 x 360 beam elements nodes 1 3 material wall_bottom nodes 3 5 material wall_top nodes 5 7 nodes 7 8 material floor_top load top_wt nodes 5 6 nodes 3 4 load bottom_wt nodes 8 6 material wall_top material floor_bottom load bottom_wt w nodes 6 4 0 0 310044 WHY e nodes 4 2 material wall_bottom material properties wall_bottom A 13 2 Ix 249 E 30e6 rho 0 0049 rho is in lb s 2 in 4 wall_top A 6 2 Ix 107 E 30e6 rho 0 0104 floor_top A 12 3 Ix 133 E 30e6 rho 0 01315 floor_bottom A 24 7 Ix 237 E 30e6 rho 0 0136 distributed loads top_wt direction perpendicular values 1 62 5 2 62 5 bottom_wt direction perpendicular values 1 130 2 130 forces f1 Fx 1000 t lt 0 2 25 t 5 f2 Fx 800 t lt 0 2 25 t 5 3 Fx 500 t lt 0 2 25 t 5 constraints fixed Tx c Ty c Rz c 3
15. 2 3 write a b 2 e The values are not swapped as desired since they are passed by value For the swap function to work correctly the data of the formal and actual parameters must be shared function swap shared x shared y t x key yt end 11 7 CONTROL FLOW CONSTRUCTS 153 ll a 1 b 2 2 swap a b 3 write a b 21 This type of parameter passing is also known as call by reference or var parameters Fortran passes all parameters by reference C passes all parameters by value burlap adopts the term shared parameters to illustrate that the space associated with the formal and actual parameters is shared 11 7 Control flow constructs burlap supports the standard control flow constructs found in most programming lan guages The principal difference is that control flow constructs in burlap are expressions not statements so they can be used anywhere 11 7 1 IF expressions if expression then expression list else if expression then expression list else expression list end First the if expression is evaluated If the result is non zero the then expression list is executed Otherwise each successive if expression is evaluated in turn and if its result is non zero the corresponding then expression list is executed If none of the if expressions evaluates to non zero the else expression list is executed Each if expression must evaluate to a scalar The IF expression has
16. 4 write 1 2 3 4 31 11 4 OPERATORS 137 stdin 4 size mismatch in expression 2 x 2 1 x 2 5 write 1 2 3 4 A 5 6 4 4 5 Finally burlap also supports a remainder operation as shown in Table 11 11 In the left division right division and remainder operators a division by zero will cause a mathemat ical exception x mod y x x yly X Y Z Z X Y scalar scalar scalar z x mod y scalar matrix matrix Vi j Z j x mod Y matrix scalar matrix Vi j Zij X mod y matrix matrix matrix Vi j Zi j Xi j mod Y j Table 11 11 Type table for the remainder operator 11 4 10 Exponentiation operator x y xy The exponentiation operator has two equivalent forms as associated from right to left Both x and y must be scalar expressions as shown in Table 11 12 X y Zz Jz x ty scalar scalar scalar z Table 11 12 Type table for the exponentiation operator Additionally x must be non negative or y must be an integer otherwise a mathematical exception will result 1 write 2 3 8 2 write 2 5 1 41421 138 CHAPTER 11 THE BURLAP SYNTAX 3 write 2 5 stdin 3 exception in expression illegal base and exponent 11 4 11 Transposition operator x The transposition operator returns the transpose of its operand and associates from left to right The transpose of a scalar is defined to be itself as sh
17. Computers and Structures 47 473 481 1993 5 J A George Computer implementation of the finite element method Technical Re port Tech Rep STAN CS 71 208 Computer Science Department Stanford Univer sity Stanford CA 1971 6 Norman E Gibbs A hybrid profile reduction algorithm Assoc for Computing Ma chinery Trans on Math Software 2 378 387 1976 7 Norman E Gibbs William G Poole and Paul K Stockmeyer An algorithm for reducing the bandwidth and profile of a sparse matrix SIAM Journal of Numerical Analysis 13 236 249 1976 8 Ernest Hinton and D R J Owen Finite Element Programming Academic Press 1977 9 Thomas J R Hughes The Finite Element Method Linear Static and Dynamic Finite Element Analysis Prentice Hall Englewood Cliffs NJ 1987 10 Thomas J R Hughes Robert L Taylor and Worsak Kanoknukulchai A simple and efficient finite element for plate bending International Journal for Numerical Meth ods in Enginerring 11 1529 1543 1977 213 214 REFERENCES 11 Barry Joe and R B Simpson Triangular meshes for regions of complicated shape International Journal for Numerical Methods in Enginerring 23 751 778 1986 12 Brian W Kernhigan and Dennis M Ritchie The C Programming Language Prentice Hall Englewood Cliffs second edition 1989 13 Daryl L Logan A First Course in the Finite Element Method PWS Kent Boston second edition 1992 14 Larry J Segerlind
18. Magnification 0 2 Z Distance Scaling Figure 6 3 The solution control box For more control over what gets generated whenever you solve a problem you can use the output control dialog box figure 6 3 available by selecting Controls from the Solutions menu The toggle buttons and controls in this dialog box control all of the available solution and output options both text and graphical The toggles in the top left corner determine the basic type of output that you would like to generate tabular available for all problem types line graphs available for transient and spectral problems color contour plots for static problems with planar elements and wireframe drawings for static problem with any type of element If appropriate multiple toggles can be selected for any given problem solution The next grouping of toggles are specific to spectral and modal analysis and control exactly what types of results get generated during those analyses The final grouping of toggle buttons on the left side of figure 6 3 allows for the inclusion of ancillary information in the tabular text based output Not all of these toggle buttons will have an effect on solutions of every analysis type for instance global matrices are not available in the solution of a nonlinear problem and the static analyses currently do not print any details 56 CHAPTER 6 USING WINFELT 6 3 Text output The text output from WinFElt is the most basic form
19. a full featured graphical environment for the FElt system Chapter 8 covers problem solutions and post processing options within velvet Chapter 9 discusses the syntax and usage of corduroy the command line mesh genera tor program for FEIt Chapter 10 introduces the powerful and flexible interactive environment burlap Chapter 11 describes the syntax of burlap in detail Chapter 12 describes some of the algorithms that FElt uses in solving an arbitrary prob lem Chapter 13 is an attempt at teaching you how to add elements to the FElt library Appendix A discusses building installing and administering the FElt system A must for potential administrators Appendix B provides a list of Geompack error codes You ll want to keep this handy if you find yourself doing a lot of mesh generation Appendix C contains a copy of the GNU General Public License the terms under which FElt is distributed Typographical Conventions In writing this guide a number of typographical conventions were employed to mark but tons command names menu options screen interaction etc Bold Font Used to mark buttons and menu options in graphical environments Italics Font Used to indicate an application program name e g felt Typewriter Font Used to represent screen interaction either with the velvet command line or the shell prompt Also used for example input files keywords that belong in input files and code examples Key Repr
20. cz e node 2 z e node 1 z L o T cx cy cz 0 0 0 0 O e K T K T po SOR EY 7 CZ return 0 end Figure 10 1 Set up function for the truss element definition The set up function assigns the stiffness and mass matrices to the element by assigning to the K and M fields of the element structure If the set up function is successful then it should return zero Otherwise it should return a non zero value The stress function is a burlap function that will compute the stress values for any element of the new definition Our sample set up function is shown in Figure 10 2 The stress function assigns the stresses by assigning to the stress field of the element structure If the stress function is successful then it should return zero Otherwise it should return a non zero value Our simple truss element has only one integration point and only a single stress value at that point Since an element may have an arbitrary number of integration points the ninteg field of the element structure must be set before the stress vector can be accessed The number of stresses and thus the length of the values vector is determined by the element definition In the computation of the stress value for the truss element the two displacement row vectors are subtracted and then multiplied by the column vector c to yield a scalar The remaining arguments to add_definition are easily calculated The shape of our truss element is li
21. equivalent forces 13 2 3 Inside the element stress function Like the fooE1t Setup routine the fooE1tStress function must also be defined in a stan dard way int fooEltStress element Element element 13 3 THE FELT MATRIX AND MEMORY ALLOCATION ROUTINES 175 The fooEltStress routine s only responsibility is to fill a Stress structure for that ele ment This will likely involve computations with the nodal displacements accessible via element gt node j gt dx k and element geometry If necessary element stresses or internal forces can be adjusted for equivalent nodal loads There is no pre allocated space for element stresses because the number of stresses and the number of magnitudes for each stress structure vary so greatly from element to ele ment For this reason you as the element writer are responsible for allocating an array of stress structures basically you will need one structure for each point within the element where you will compute stresses you should also set element gt ninteg to this num ber and for the stress magnitude array within each of these structures For example for a three dimensional beam element six stress components are computed at both ends the six possible internal forces In this case then we need to allocate space for two stress structures set element gt ninteg to two and allocate space for six magnitudes in each stress structure 13 3 The FElt matrix and memory allocat
22. graphs es ee caes eh ee heh a he a oa 56 6 4 3 Wireframe drawings ew OR BR RE BR 56 7 The velvet Application 61 7 1 Introduction tothe velvet GUI ccc ect eee ee eee ee 61 7 2 General features of the interface ooo ek A RR eR HER ERS 61 7 3 Working wie 62 as ks e amp SG SS ooe Oe E ae aG 63 7 4 Configuring the drawing area 2 2 2 eee ee ee 65 dhl SOOO oe o Be SP Pw Rew Se WS OM Seo s 65 122 Object coloring 6 oa we eA Ree Re ERA RK ARK REESE 66 T43 Zooming lt lt coi eR RR OR RR EER OER ER ERS 67 7 4 4 Dumping the drawing area o 68 73 Drawing and defining a problem cier css a Ra 68 13 1 Dehning attributable Objects 64 0 Sek Baw Go Dew Gas 69 Telok Th matenaldialot lt c ee ee ewe eA SS 71 7 5 1 2 The constraint dialog lt lt ewe ws 71 AR Th forcedialog gt sas s oce dacen aie ai ae 72 75 14 Theload dialog lt lt e soa cea Send ew San cens 72 7 5 2 Working with nodes and elements ios S46 546 Saws 13 7 6 More on nodes and elements ooo cisco isis a de 75 7 6 1 Editing nodes oca Se RO ROR ERE R ERA 75 7 6 2 Automatic node renumbering osaa o 76 7 6 3 Editing elements lt s sa a sqm 42444408484 AR 77 ETF MVR MOO os te Ge Sew dl Deter Beton Be Bs 78 7 8 Material databases and defaults files 80 7 9 Automated element generation 2 ee ei we eee ee 81 CONTENTS vil 7 9 1 Generating a grid of line or quadril
23. the FEIt syntax also includes a mechanism for specifying a discrete representation of a transient forcing function The basic specification consists of a series of time magnitude pairs of the form t F where F is the value of the function at time t In evaluating the function FElt will linearly interpolate between each adjacent pair for times that fall between two pairs A single time magnitude pair will be interpreted as an impulse response function at the given time Note that the pairs must be given in order of increasing time 3 1 INPUT FILE SYNTAX 15 You can express a periodic discrete function simply by defining one period and then entering a symbol at the end of the expression You could represent a simple sawtooth forcing function having a period of 2 seconds and a maximum magnitude of 1000 with an expression like forces sawtooth Fx 0 0 0 2 1000 0 Figure 3 1 illustrates different ways to define some common types of loading functions with either continuous or discrete representations Fike Fibs F 0 0 coe SO F 10r medt 5 00 F p0 0 5 150014 4 0 Fem firth 1 00 0 5 7 30D mod 05 3000 OS mod 0 510 4 Jo 15 n Ciel J 3 3 rG Fike F p0 SOO AD 2000 Fi ba E OH LOL O F p0 0 E EE O SO A OL O F itat 4 00 207 2000 01 FebkexS Pear Heb 7 SSO whe 12 ESO AZ E Figure 3 1 Example loading functions 16 CHAPTER 3 STRUCTURE OF A FELT PROBLEM 3 1 3 Units There are no set units f
24. the sign convention for loads in the global directions follow the sign convention of the global coordinate axes Each beam element is limited to two applied distributed loads After displacements are found six internal force quantities are computed for each beam element These quantities are the axial force shear force and bending moment at both nodes The sign convention for these forces is shown in Figure 4 1 This convention is based on a coordinate system in which the local x axis points from node 1 to node 2 As z and 2 always coincide for the 2d beam element and we define the positive z axis to point out of the page z x The three beam type elements beam beam3d and timoshenko all are capable of re solving hinged boundary conditions for the rotational degrees of freedom The adjustment is made to the element stiffness matrix in global coordinates according to the following procedure Given a hinged DOF dof then for all entries in k the element stiffness matrix not associated with dof all entries not in row or column dof Kind Mid poe The inherent problem in this method of dealing with hinged conditions is that we cannot k i dof 4 13 calculate any displacements associated with the hinged DOF and thus the internal forces calculated for any element with a hinged node will not be correct Displacements other than at the hinged DOF will be accurate 34 CHAPTER 4 THE FELT ELEMENT LIBRARY 4 2 2 2 A
25. with Sfx then you can also compute the output spectra at the out put DOF due to input at any forced DOF with spectral inputs This is the default behavior when the transfer switch is not specified on the command line If none of the forces in a 52 CHAPTER 5 THE FELT APPLICATION problem have non zero spectral input components then the only solution that is non trivial is with the transfer switch invoked Just like for transient analysis the output for spectral analysis either transfer functions or output spectra can be tabular and or plotted ASCII plots of transfer functions or output spectra whichever are appropriate for the current problem are generated if the plot switch is specified Tabular output is generated by default and can be turned off with the table switch Chapter 6 Using WinF Elt 6 1 Introduction to WinF Elt The primary feature of the main WinFElt window is a standard text editor 6 2 Solving a problem Depending on what you want your solution to include there are a couple of different ways to go about solving a FElt problem from within WinFElt In addition to the standard tabular type FElt output you can also have WinFElt generate two dimensional color shaded stress or displacement plots and or plots of the structure with magnified displacements applied For transient analysis problems graphical time displacement or time temperature plots are available in spectral analysis problems plots of the transfe
26. 00 0 01 0 1 00000 5 and the consistent mass matrix as 140 0 0 7 0 0 0 156 22L 0 54 13L AL a 31 PAL O 22L 4L O 13L 3L 4 11 m 90 70 0 0 140 0 0 0 54 13L 0 156 22L O 13L 312 0 2L 42 The basis for these mass matrices should also be readily available in many textbooks The transformation from local element coordinates into global coordinates is of the same form as equation 4 4 the actual transform matrix for two dimensional beams is given by cos cos 0 0 0 0 cos Oy cos 0 0 0 0 0 0 1 0 0 0 4 12 j 0 0 0 cos cosOy 0 0 0 O cosB cos 0 0 0 0 0 0 1 where the direction cosines cos 8 and cos6 are defined by equations 4 6 and 4 7 A distributed load assigned to a beam element can either be directed perpendicular to the element as in the self weight case or the linearly sloping loads in the second example 4 2 STRUCTURAL ANALYSIS ELEMENTS 33 Figure 4 1 Sign convention for local forces on a beam element above parallel to the element just like a linearly distributed axial load in the truss case or in the GlobalX or GlobalY directions Note that loads given in the LocalY and LocalX directions will be taken as equivalent to the perpendicular and parallel cases respectively For perpendicular loads a positive magnitude indicates that the load points in the direction of positive A positive parallel loads point from node 1 to node 2 as in the truss case Similarly
27. Pro gram you indicate your acceptance of this License to do so and all its terms and conditions for copying distributing or modifying the Program or works based on it 6 Each time you redistribute the Program or any work based on the Program the recipient automatically receives a license from the original licensor to copy distribute or modify the Program subject to these terms and conditions You may not impose any further restrictions on the recipients exercise of the rights granted herein You are not responsible for enforcing compliance by third parties to this License 7 If as a consequence of a court judgment or allegation of patent infringement or for any other reason not limited to patent issues conditions are imposed on you whether by court order agreement or otherwise that contradict the conditions of this License they do not excuse you from the conditions of this License If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations then as a consequence you may not distribute the Program at all For example if a patent license would not permit royalty free redis tribution of the Program by all those who receive copies directly or indirectly through you then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program If any portion of this section is held invalid or unenforceable under any pa
28. THE DRAWING AREA 65 changes Note that when velvet saves a file it will consist of the standard sections of a FElt file plus special sections that describe the appearance of the drawing area These ancillary sections contain information that allows velvet to reconstruct the drawing area including tool figures canvas configuration options and zoom state when the corresponding FElt file is loaded back into velvet You should never have to worry about whether or not such information exists If a file with these sections is run through felt the information simply will be ignored If the information does not exist and you load the file into velvet then velvet simply will make a best guess as to how to configure the main canvas area The advantage to having such a plain text description of the drawing area is that you can change this information outside of velvet if you want to make quick changes to the problem appearance but do not want to load velvet explicitly Such an arrangement also allows you to easily create defaults files which you can use to start up velvet with your preferred configuration 7 4 Configuring the drawing area 7 4 1 Basic controls The primary means for setting up the drawing area is through the configure dialog under the canvas menu Figure 7 3 The left side of this dialog deals primarily with the minimum and maximum coordinates of the drawing area the snap size and the grid size The right side allows you to specify t
29. be a list of symbolic DOF names Tx Ty Tz Rx Ry Rz You will get solution output for each of these DOF at each of the nodes that you specified in the node list Finally you can specify global Rayleigh damping parameters with Rk and Rm If either of these parameters is non zero then the global damping matrix will be formed using these two parameters and the global mass and stiffness matrices as opposed to being formed from elemental Rayleigh parameters and elemental mass and stiffness matrices You should refer to the chapter on algorithmic details section 12 4 7 for a little more insight on the meanings of the integration parameters The individual element descriptions in chapter 4 discuss the different mass matrices for that element The analysis parameters and their meaning are summarized in table 3 2 3 2 9 Load cases 3 3 An illustrated example Figure 3 3 illustrates a slightly more complicated problem a simply supported beam with two triangular loads applied and a spring support at the center We can model the spring as a slender truss element such that EA L k the spring stiffness The input file for this problem would probably look something like this 3 3 AN ILLUSTRATED EXAMPLE 23 parameter description example r o start frequency range start SP load range start LR 0 0 r stop time T TT frequency SP or load LR range end 10 0 r step time T TT frequency SP or load LR
30. defines all of the point loads used in the problem and actually looks a lot like the constraints section The magnitudes of the forces in the six directions are specified by Fx Fy Fz Mx My Mz If the value for a given force component is not given it is assumed to be zero The directions for both forces and the constraints as defined above should be given in the global coordinate system right hand Cartesian If you are doing transient analysis the force definitions can be more complicated than a simple numerical assignment or expression FElt allows you to specify a transient force as either a series of time magnitude pairs a discrete function in time or as an actual con tinuous function of time This latter fact means that you can define a force as Fx sin t rather than having to discretize the sine function These continuous forcing functions are a special case of expressions as discussed above For spectral analysis problems you can explicitly specify input spectra using Sfx Smy etc if you want to to compute the actual power spectrum of the output These 3 2 SECTIONS OF A FELT INPUT FILE 21 2000 Figure 3 2 An example of a complex distributed load spectra can be analytic functions of w or discrete frequency power pairs 3 2 7 Distributed loads The distributed loads section s must contain a definition for each distributed load that was assigned in the element definition section s A valid definit
31. definition easier in cases where the added accuracy that the ad ditional nodes offer is not worth the extra effort Like the generalized case if the fourth node is the same as the third node the element will be degenerated into a triangle This allows for easy mixing and matching of element shapes without changing element types though this too would be allowed Distributed loads on the four node isoparametric elements work exactly as they do in the CST case The stresses computed for each four node isoparametric are also the same as those computed for CSTs Like CST elements only a lumped mass formulation is available for the mass matrices of isoparametric quadrilateral elements The lumped formulation is generated by lumping the total mass of the element equally at the four nodes or the three nodes if the element is being degenerated into a triangle 4 2 6 Plate bending element The plate bending element in FElt is a simple four node isoparametric quadrilateral that uses selective reduced integration to prevent shear locking The classic reference for this 38 CHAPTER 4 THE FELT ELEMENT LIBRARY my mya Immy y ITTY qa Figure 4 3 Sign convention for force resultants on an htk element approach is 10 The effect is achieved simply by using one point Gaussian quadrature to under integrate the shear contribution to the stiffness where two point quadrature is used on the bending contributions htk elements must exist ent
32. down will slow down or speed up the animation The play backward and play forward buttons lt and gt control which direction time moves during the animation You can use the stop button to freeze the structure at a given point during the animation The text indicator in the right bottom corner of the window keeps track of the elapsed time The animation will repeat itself until it is either stopped or the animation window is dismissed 90 CHAPTER 8 POST PROCESSING WITH VELVET Figure 8 3 The mode shape plotting window 265 1 1 44338 e Figure 8 4 An example of textual output from velvet 8 1 SOLVING A PROBLEM WITH VELVET 91 Figure 8 5 An example of a displaced structure plot Figure 8 6 An animation in velvet 92 CHAPTER 8 POST PROCESSING WITH VELVET 8 2 Problem description and analysis parameters The problem description and analysis parameters section of a regular FElt input file are mimicked in velvet by the problem and analysis dialog box Figure 8 7 This dialog 1s available by selecting problem analysis from the solutions menu The problem title text entry defines the header that will be used for textual output and on graphical line plots The analysis mode is defined by engaging one of the toggle buttons just below the problem title at the top of the dialog The mass mode toggles below the analysis toggles allow you to select either consistent or lumped mass formulations for eleme
33. equivalent forms and groups from left to right It returns 1 if both of its operands compare unequal to zero and 0 otherwise The AND operator guarantees left to right evaluation x is first evaluated if it is equal to 0 the value of the expression is 0 Otherwise y is evaluated and if it is equal to 0 the result of the AND expression is 0 otherwise the result is 1 130 CHAPTER 11 THE BURLAP SYNTAX der 1 ifxAOandy 0 vices l O otherwise Both x and y must be scalar expressions as shown in Table 11 3 It makes little sense to compare a matrix or string with zero for example There are however common ways of mapping a matrix to a scalar Two such ways are provided by the any and every functions discussed in Section 11 5 3 x Y Z 2 X and Y scalar scalar scalar z xand y Table 11 3 Type table for the logical AND operator 1 write 1 lt 2 and 3 lt 4 2 write 1 lt 2 and 3 gt 4 3 write 0 and 1 0 4 write 1 and 1 0 stdin 4 exception in expression right division by zero 11 4 5 Equality operators X Y X l Y X lt gt Y The equality and inequality or lt gt operators associate from left to right The result of the expression is 1 if true and 0 if false ae def 1 ifx y O otherwise le ff ifx7y O otherwise Unlike the previously discussed operators these operators like most of the operators in burlap accept a variety of types as
34. felt graph logan An input or output file name of a hyphen indicates that input should be read from standard input and or output should be written to standard output respectively Chapter 4 The FEIt Element Library 4 1 Introduction The FElt element library contains line plane and solid elements Line elements in clude a one dimensional spring spring a bar or truss element truss two and three dimensional Euler Bernoulli beams beam and beam3d and a two dimensional beam based on Timoshenko theory timoshenko The set of planar elements consists of constant strain triangles CSTPlaneStrain and CSTPlaneStress isoparametric quadrilaterals iso2d_PlaneStress iso2d_PlaneStrain quad_PlaneStrain quad_PlaneStress and an HTK plate bending element htk The only solid element in the library is an eight node brick brick There is an axisymmetric triangular element axisymmetric The only elements available for thermal analysis are a simple one dimensional line element rod and a constrant temperature gradient triangular element ctg In the above list the names in parentheses indicate the actual symbolic type names of the elements The number of nodes and the material properties needed for defining each element and the DOF that the element affects are summarized in the table below A brief description for each element follows the table detailed derivations of element stiffness matrices can be found in any number of finite e
35. filename in the list with the first mouse button Clicking okay will finalize your selection You can maneuver through directories by selecting them and clicking okay just like files An alternative to selecting an entry then clicking the okay button is to click on list entries with the middle mouse button This will instantly select that entry or move to that directory Clicking the cancel button will dismiss the file selection dialog without making a selection Additional entries on the file menu should appear familiar to most users of modern GUI applications Choosing new will erase the current problem and allow you to start with a blank slate You ll be prompted if there is a chance that you might lose some unsaved changes to the current problem save does just what it says if the current problem does not have a filename already associated with it choosing save acts like save as otherwise it will silently save the current state of the problem to the associated file The current filename is always displayed as the title in the window manager title bar save as allows you to name the file that the problem will be saved to After saving through a save as action this filename becomes the current filename for this problem The restore option is equivalent to selecting open and choosing the current input file i e it simply reloads the currently named input file exit quits velvet entirely you will be prompted if you have unsaved 7 4 CONFIGURING
36. function exists then it is executed Otherwise the directories named by the environment variable BURLAP _PATH are searched in order for a file named first F and then F b If a file is found then it is processed just as if include were called After the file is processed the global scope is searched again If the function still cannot be found then a type error results This is the only case in which a global variable used inside a function does not need to be explicitly declared in a global declaration 11 5 Intrinsic functions The intrinsic functions are those functions that are burlap provides by default They are however just variables and their values can be reassigned although this is not typically done The functions can be grouped into five basic categories 11 5 INTRINSIC FUNCTIONS 141 mathematical functions sin sqrt floor etc Section 11 5 1 e matrix functions lu gr eig etc Section 11 5 2 predicate functions nul1 any matrix etc Section 11 5 3 finite element functions assemble area etc Section 11 5 4 e miscellaneous functions concat write exit etc Section 11 5 5 11 5 1 Mathematical functions burlap supports most of the functions available in the C math library as shown in Ta ble 11 15 Each of the functions can take either a scalar or a matrix as an argument If a matrix is passed as an argument then the result is also a matrix with the funct
37. in each application directory e g src Velvet src Felt to relink 180 CHAPTER 13 ADDING ELEMENTS TO FELT the applications with the modified libraries Your new element should now be ready for use Writing up a simple test file that uses the new element and running this through the command line application felt is usually the easiest way to check numerical results 13 6 A detailed example If everything above is still a bit unclear fear not it s really not as bad as it looks Let s walk through a step by step description of how the Timoshenko beam element was added to the FElt library This is a very simple element We only consider bending and shear deformation no axial stiffness is taken into account and we don t allow distributed loads applied in the global DOF For some basic mathematical details for this element refer to section 4 2 3 More details are available in 4 10 15 In general all the code for a given element type will be contained in one source file Hopefully the routines in this file will be completely specific to that element type Ina perfect world common functionality would be available as convenience functions This is the basic model that we will follow here All of the following code is taken from the file timoshenko c in the directory lib Elements We start by including the necessary header files prototyping our local functions we make them static to insure that they will be local to this file and settin
38. in expression must be a matrix vector scalar or the null value If it is a matrix then each column of the matrix is assigned to the for expression If it is a vector then each element is assigned If it is scalar then the scalar is assigned and the expression list is evaluated only once If the in expression is null then the expression list is not executed function sum_of_first n s 0 156 CHAPTER 11 THE BURLAP SYNTAX for i in 1 n d end return s 1 write sum_of_first 3 6 2 write sum_of_first 10 55 3 write sum_of_first 1 0 This last write expression works correctly since the range 1 1 produces a null value and thus the body of the FOR expression is not executed function sum x s 0 for v in x do for i in v do s sti end end return s end This example correctly computes the sum of all elements of x for scalars vectors and matrices If x is simply 3 then v is simply 3 and i is also 3 If x is the vector 1 2 3 then v is successively assigned 1 2 and 3 and i is simply assigned each value of v If x is the matrix 1 2 3 4 5 6 then v is successively assigned the column vectors 1 4 2 5 and 3 6 andi is assigned the values 1 4 2 5 3 and 6 in that order 11 7 4 BREAK NEXT and RETURN expressions return expression break expression next The RETURN expression may be used to return a value from a function If the expression is absent then a null v
39. installation of the executable However as a special exception the source code distributed need not include anything that is normally distributed in either source or binary form with the major components compiler kernel and so on of the operating system on which the executable runs unless that component itself accompanies the executable If distribution of executable or object code is made by offering access to copy from a designated place then offering equivalent access to copy the source code from the same place counts as distribution of the source code even though third parties are not compelled to copy the source along with the object code You may not copy modify sublicense or distribute the Program except as expressly provided under this License Any attempt otherwise to copy modify sublicense or B 2 TERMS AND CONDITIONS 209 distribute the Program is void and will automatically terminate your rights under this License However parties who have received copies or rights from you under this License will not have their licenses terminated so long as such parties remain in full compliance 5 You are not required to accept this License since you have not signed it However nothing else grants you permission to modify or distribute the Program or its deriva tive works These actions are prohibited by law if you do not accept this License Therefore by modifying or distributing the Program or any work based on the
40. it has been successfully built doamake install The entire package has been compiled and tested under SunOS 4 x and 5 x and Linux the OS under which it was developed This version or earlier versions compiled under HP UX 8 0 and 9 0 SystemV386 R3 2 2 and various SGI DEC and IBM workstations using Irix OSF 1 and Ultrix and AIX with little or no problem The files felt exe and feltvu exe a graphing application are available as pre compiled executables for the DOS environment The most recent version of the source code for FElt should always be available via anonymous ftp at felt sourceforge net in the directory pub FE1t We also try to make pre compiled binaries available for a wide variety of platforms and operating systems If you want to be made aware of each revision of the FElt system then we strongly encour age you to subscribe to the FElt mailing list To subscribe visit the list information page athttp lists sourceforce net mailman listinfo felt announce Only major revisions and changes will be announced broadly via Usenet 201 202 APPENDIX A INSTALLING AND ADMINISTERING FELT A 2 Translation files The translation files which map non English terms to the English terms which FElt expects should be installed in the library directory which you specified during the build This path will automatically be searched when cpp goes looking for include files which is really what the translation files are If they are
41. latter we like Hughes 9 and Zienkiewicz and Taylor s 16 classic textbooks Logan s 13 book is a good introductory text He approaches the method from the same direction that many of us come to finite elements through classical matrix structural analysis He s a bit lean on the mathematical basis and theory but that s probably why it makes a good introductory text It is probably our ignorance but we are not aware of any definitive texts on the algorithmic implementation of the finite element method Texts that have been recommended but that we have not really worked with in clude Segerlind 14 a very well recommended introductory text by Burnett 1 and a book by Hinton and Owen 8 12 2 Elementary C programming The advantages to coding FElt entirely in C rather than the more traditional at least for finite element analysis choice of Fortran are 1 keeping the package consistent the system and graphical interface stuff required working in C having all of the mathematics in C made interfacing the two aspects a lot easier 2 C s flexibility in working with data structures makes it really easy to implement some of the concepts of finite element analysis 159 160 CHAPTER 12 THE ALGORITHMS BEHIND FELT in the code elements have pointers to materials and loads nodes have pointers to forces and constraints There is no need to keep track of lots of separate arrays of things generally speaking passing th
42. matrix suitable for use in plane strain analysis 1 Elementlength element n Given a dimension n this function returns the length of an element in n dimensions GaussPoints n points weights Given the number of Gaussian integration points n and the addresses of two pointers to double weights and points this will fill these two arrays with the appropriate values for use in Gaussian quadrature Currently this 178 CHAPTER 13 ADDING ELEMENTS TO FELT function only knows the values for 1 2 and 3 point Gaussian quadrature ResolveHingeConditions element If you want your element to deal with the possibility of hinged DOFs then you can use this routine to modify the element stiffness matrix appropri ately Generally you would call this routine only after completely defining element gt K The given element s nodes are checked for hinged DOF if a DOF is hinged the the coefficients of the element stiffness matrix are adjusted according to the following procedure given a hinged DOF dof then for all entries in k the element stiffness matrix not associated with dof all entries not in row or column dof k i j k i j r ae k i dof 13 1 dof dof The inherent problem in this method of dealing with hinged conditions is that we cannot calculate any displacements associated with the hinged DOF In general these displacements will not be zero SetupStressMemory element After settting element gt nint
43. missing then it is assumed to be 0 The DOF related fields of the problem structure must be initialized as by calling find_dofs felt s Reads the FEIt file named by s to define the current FElt problem If s does not name an absolute path i e starts with or then the environment variable FELT_PATH is used to search for the file named by s The variable should be a colon separated list of directories The directories are searched from left to right for a file first named s and then s f1t If the variable FELT_PATH is not set then only the current directory is searched If s is null then an empty problem is defined find_dofs Computes the set of active DOFs for the current problem As a result the DOF related fields of the problem structure are initialized The number of active DOFs is returned This function must be called before most of the other finite element related functions can be called global_dof n d Returns the global DOF corresponding to a local DOF The local DOF is specified by its node n and its DOF d The node n may be specified as either a node object or a node number The DOF d should be one of amp tx 11 5 INTRINSIC FUNCTIONS 147 aty amp tz amp rx amp ry or amp rz The DOF related fields of the problem structure must be initialized as by calling find_dofs integrate_hyperbolic K M C Pp Solve the discrete equation of motion Ma Cv Kd f using Newmark s metho
44. named by the FEIt input syntax as shown in Table 10 4 For example m rho will return the density of the material m and m A will return its cross sectional area 10 3 4 Force objects Force objects allow the forces and moments to be accessed as individual components or as a vector as shown in Table 10 5 For example f force will return the force vector of f and f force 4 will return the moment about the x axis which is equivalent to f force Mx For transient problems a force or moment may vary with time In felt this is described by an expression in terms of t A component of a force may be assigned a string value which will be interpreted by burlap as an expression Note that evaluating the compo nent will evaluate the expression at time t 0 There is currently no way to determine if a component is assigned a time varying expression 10 3 BURLAP AND FELT 111 Field name Description number constraint force eq_force dx X Y Rx Ry node number constraint object force object equivalent force vector displacement vector x coordinate y coordinate z coordinate lumped mass translation along x axis dx 1 translation along y axis dx 2 translation along z axis dx 3 rotation about x axis dx 4 rotation about y axis dx 5 rotation about z axis dx 6 Table 10 3 Fields of a node object 1 Fx 2 write cos t Fx 10 3 5 Constraint objects Con
45. not in some standard place users need to specify a I flag on the felt command line to tell cpp where to look for them The current filenames are in English you ll obviously want to link them or move them to something reasonable on your system and make users aware of what the appropriate file to include is called The following tables list the currently available translations that we have at least You can look at the current german trn and spanish trn files if you want to change something or provide a new one If you do write a new one please send it to us This is only a first cut at internationalization but as we get more translations we will be in a better position to see how we can make it all work a little more fluently Also remember that because this is a first cut all you can really do is type in a FElt file by hand using these translations Anything velvet saves or even the file that felt writes with the debug command will be in English even if the input file was given in another language All output is still in English Lastly if you use these translations you have to use them exactly as they are given here They are not case insensitive and if there are underscores in the place of spaces you must use the underscores Things that are user specifiable to begin with object names problem title can still be whatever you want of course If a keyword is not listed in a table below use the original English keyword the English and non
46. of output that all applications in the FElt system produce That is no matter how you specify and solve a FElt problem either with felt velvet or WinFElt the same kind of text output is available Chapter 5 details the interpretation of this basic output form 6 4 Graphical output 6 4 1 Contour plots The controls for color contour plots are available on the right side of the main controls dia log 6 3 The two radio buttons control whether element stress or nodal displacements are used as the basis for the interpolation For stress plots the interpretation of the component number is element specific and must be a valid index in the stress vector for each element type in the problem Consult Table 5 1 for details on what the stress vector consists of for each element type For displacement plots the component is simply the displacement DOF that you want to see plotted In general this should be one of the active global DOF for the current problem Other contouring controls include toggles for histogram equalization and element boundary overlay Histogram equalization is a standard technique in image processing for enhancing the contrast of images If the overlay elements toggle is checked then the outline of all the elements in the problem will be drawn in black on top of the color im age An example of stress contours rendered here in greyscale for the wrench example supplied with WinFElt is shown in figure 6 4 Note that both histogra
47. of planar four node elements is almost identical to the grid of line elements described above The only difference is that there are no specifications along the z direction for quadrilateral grids An example of a mesh for a long rectangular plate using htk plate bending elements the default element type for quadrilateral grids is quad_PlaneStress might look like quadrilateral grid element type htk start 0 0 end 10 3 x number 20 y number 6 The result would be a rectangular region meshed with 120 equi sized square elements 9 2 5 Generating a grid of solid brick elements The case of a grid of solid eight node elements is also very similar to grids of line and planar elements described in the preceding two sections An example of a mesh for a long rectangular plate using brick elements the default type for grids of solid elements might look like brick grid start 0 0 0 end 8 12 6 x number 4 y number 6 z number 3 The result would be a three dimensional solid region three elements deep four elements wide and six elements tall 9 2 6 Grid spacing rules For all of the line and grid generation examples given above we could also have di rected that corduroy generate the elements with non linear spacing rules Such spacings are common when we know that we want a higher mesh density at one corner of the grid or only at the end of the line The rules available are linear this is the default equi spaced sinu
48. operands as shown in Table 11 4 11 4 OPERATORS 131 X Y Z Z XopY matrix matrix matrix Vi j Z j Xij op Y matrix scalar matrix Vi j Zij Xi j op y scalar matrix matrix Vi j Z j x op Yj scalar scalar scalar z x op y scalar comparison string string scalar z x op y lexicographic comparison object object scalar z x op y object identity comparison Table 11 4 Type table for the equality operators Although the table might look confusing it provides a precise semantics for all possible type operands Essentially burlap does element by element comparison for matrices and scalars string i e lexicographic comparison for strings and identity 1 e pointer com parison on all other type objects When comparing two matrices their dimensions must be identical 1 write 1 2 3 2 0 al 0 2 write 2 1 2 3 1 0 1 3 write 1 2 3 3 2 3 0 1 1 4 write hello hello 5 write hello there 0 6 write nodes 1 nodes 1 07 write nodes 1 elements 1 stdin 7 type error in expression node element 11 4 6 Relational operators 132 CHAPTER 11 THE BURLAP SYNTAX Xx gt Y X lt Y X gt Y The relational operators are similar to the equality operators with the only difference being that they do not allow an arbitrary object type as an operand for comparison Ta ble 11 5
49. plot stresses Plot displacements plot displacements Plot structure plot structure Contouring contour Wireframe wireframe Canvas Configure configure Color control colors Zoom to fit zoom all Ctrl Z Zoom window zoom Ctrl z Save canvas dump Snap snap Ctrl s Grid grid Ctrl g node numbering node numbers Ctrl n element numbering element numbers Ctrl e Apply Materials apply material Ctrl a m Forces apply force Ctrl a f Constraints appply constraint Ctrl a c Loads apply load Ctrl a 1 Tools Circle draw circle Ctrl t c Arc draw arc Ctrl t a Rectangle draw rectangle Ctrl t r Polygon draw polygon Ctrl t p Line draw line Ctrl t 1 Text draw text Ctrl t t Delete delete tool Move move tool Nodes Add add node Edit edit node button 1 on node Delete delete node Move move node button 2 on node Lumped mass lumped mass Renumber perm renumber nodes Renumber temp toggle renumber Elements Add add element Edit edit element button 1 on element Set type set element Generate generate Materials edit material Ctrl d m Constraints edit constraint Ctrl d c Forces edit force Ctrl d f Loads edit load Ctrl d 1 Quit Esc or button 3 Abort Ctrl c 7 11 COMMAND LINE OPTIONS 85 7 11 Command line options Some of the above functionality can be enabled disabled or altered via command line switches when you invoke velvet In addition to the standard X toolkit options display etc you can use any of the following to control the behavior and set the start
50. quotation marks Special enumeration constants begin with an ampersand as in local_x Variable names are a sequence of letters digits and the underscore character Finally comments are enclosed within a pound sign may also be used to indicate a comment until the end of the current line 1 a 3 14159 2 b hello 30 Ly lt 2 8 de Dx 26 4 write a 3 14159 5 write b hello 6 write c The following sections discuss the operators and functions of burlap in detail They are intended to be the definitive word on the semantics of the operators functions matrix formers etc 123 124 CHAPTER 11 THE BURLAP SYNTAX 11 1 Literals It seems appropriate to begin the discussion with the literals since they are the building blocks of the burlap syntax As previously mentioned floating point literals are numbers such as 123 or 3 14159 Specifically a numeric literal consists of a base followed by an optional exponent The base consists of a series of digits with an optional decimal point The exponent consists of the letter E or e followed by the digits of the base 10 exponent The digits in the exponent may be preceded by a or to indicate a positive or negative exponent respectively String literals are enclosed within double quotation marks as in hello The following escapes may be used to introduce special characters within a string literal audible alert la carriage return Ar backspace b horizontal
51. recoloring the drawing area can be computationally intensive it is a good idea to make all your color assignments first and then select recolor only once when you are finished The precedence of coloring from highest to lowest is force constraint default for 7 4 CONFIGURING THE DRAWING AREA 67 Materials Constraints Colors springy fixed black steel free white pin red roller blue green yellou orange purple cyan beige brown erey pink magent help recolor dismiss Figure 7 4 The object coloring control box nodes and distributed loads material default for elements What this means is that if a node has a force applied and that force has a color assigned then the node will be colored according to the force color If the force does not have a color assigned but the constraint for that node does have one then the node will be colored according to the constraint color If neither force nor constraint has a color assigned then the node will be colored according to the standard node color defined via the regular canvas configuration dialog Figure 7 3 Element coloring works the same way 7 4 3 Zooming A model with a dense mesh can often be difficult to work with because it is difficult to position the cursor exactly on top of a given node or element The way around this problem is to use the zoom controls to zoom in or out on a given part of the mesh If you want to zoom in on a selected part of yo
52. section discusses the various operators available in burlap The operators are dis cussed in order of precedence with the lowest precedence first In general uppercase letters denote matrices and lowercase letters denote scalars For many operators a pre cise definition of the operator is given along with a table specifying the legal types of the operands and example burlap output 11 4 1 Expression separators A semicolon is used to separate multiple expressions to be evaluated and to separate the rows of a matrix A comma is used to separate the arguments to a function and to separate the elements within a single row of a matrix Since an expression usually ends at the end of a line typing a semicolon at the end of every line to separate one expression from the next would be extremely tedious Instead 11 4 OPERATORS 127 burlap will automatically insert a semicolon for you at the end of a line under certain conditions When reading from a terminal in interactive mode if the last token on the line may end an expression then a semicolon is inserted When reading from a file if the last token on the line may end an expression and the first token on the next line may begin an expression then a semicolon will be inserted A line can be explicitly continued by ending the line with a backslash character 1 write hello write there hello there 2la 1 DADAS 3 write a 6 4 b 1 Although these semantics may
53. sound complicated they are designed so that 99 9 of the time you don t need to worry about when to add a semicolon However if you find yourself with a nasty bug you might want to review the rules for inserting semicolons or try putting in an explicit semicolon or backslash 11 4 2 Assignment expressions The assignment operator has two equivalent forms and groups from right to left An as signment expression assigns Y to X and returns X X must be a variable name a submatrix a field reference or the result of a function call returning a global variable Generally any expression can be assigned to a variable However in certain cases the expression on the right hand side is restricted For example if X is a submatrix created as 128 CHAPTER 11 THE BURLAP SYNTAX the result of indexing a matrix then the dimensions number of rows and columns of X and Y must agree If X is the result of a field reference of a FElt structure then there may be type constraints on Y For example a matrix cannot be assigned to the component of a force Some fields are also read only such as the name of a constraint or the number of a node or element 1 a 1 2 3 2 write a 1 2 3 3 a b 0 4 write a b 0 0 5 a b 2 1 6 write a b 3 2 7 nodes 1 number 3 stdin 7 type error in expression changing a read only variable 8 nodes 1 force Fx 4 9 nodes 1 force Fy 1 2 3 stdin 9 type error
54. stiffness matrix This step of the process in FElt actually accomplishes three things For each element this routine calls an element setup routine based on element type Besides filling out element gt K most element setup routines will also calculate the equivalent nodal forces on the element s nodes if appropriate i e if there is a distributed load on the element Assembling the stiffness and mass matrices in a transient analysis problem works the same way For transient analysis the element setup routines calculate a mass matrix ac cording to the the mass mode set in the analysis parameters section The global stiffness and mass matrices are formed exactly as in the static case Nodal lumped masses are su perposed after the element mass matrices have been assembled The damping matrix can be formed in either of two ways both of them based on a Rayleigh damping model In the first method the damping matrix is assembled at the same time as the stiffness and mass matrices by creating and assembling element damping matrices based on the Rayleigh damping coefficient for each element s material Ce Rg Ke Rm Me 12 1 12 4 DETAILS OF A FEW GENERAL FELT ROUTINES 165 Where R and Rm are the Rayleigh damping coefficients for the given material Alterna tively if non zero Rz and Rm values are given in the analysis parameters of the FElt file then these coefficients will be applied to form a global damping matrix only aft
55. tab t formfeed AGE vertical tab v newline n quote yn Matrices are delimited by square brackets with rows separated by semicolons and el ements within a row separated by commas An important thing to realize when writing matrices is that burlap may automatically insert a semicolon at the end of line In particu lar a semicolon is inserted at the end of a line in interactive mode when the last literal or operator on the line can end an expression 2 4 5 6 4 a 1 2 3 4 gt 4 5 6 5 write a Note that since an expression cannot end with the next line is taken to be a continu ation of the previous line A matrix element may be an arbitrary expression however all expressions on the same row must have the same height Additionally all rows must have the same number of columns 11 2 VARIABLES 125 1 x 1 2 a 0 x 2 3 write a 4 b a 3 4 5 5 write b 6 write b b 7 write b a stdin 7 inconsistent number of rows 11 2 Variables Variables are named by a sequence of letters digits the underscore character and the question mark symbol A variable must begin with a letter or the underscore character The question mark may only appear as the last character in a variable name There is no limit on the length of a variable name Variables are assigned values either through the assignment operator parameter pass ing or function definitions A variable is either lo
56. that loads and distributed are synonymous as are num_loads and num distributed 10 3 2 Node objects Like elements nodes have some fields that are defined through the FEIt input file with addi tional fields to contain the result of computations as shown in Table 10 3 As an example n constraint will return the constraint assigned to the node n and n constraint name will return the name of that constraint n dx will return the displacement vector and n dx 1 will return the first component of that vector Note that the displacement vector may be accessed either as a vector by using dx or as individual components such as Tx This 110 CHAPTER 10 THE BURLAP APPLICATION Field name Description number element number nodes array of nodes K stiffness matrix M mass matrix material material object definition element type definition loads array of distributed loads num_loads number of loads stresses array of element stresses ninteg number of integration points distributed array of distributed loads loads num_distributed number of distributed loads num_loads Table 10 2 Fields of an element object is simply a shorthand assigning to an individual component is the same as assigning to the result of indexing the displacement vector The equivalent force vector eq_force is used in transforming distributed loads on an element into equivalent forces on its nodes 10 3 3 Material objects Materials have the fields
57. the user to easily add new element definitions that can be quickly tested without 10 4 ADDING NEW ELEMENT TYPES TO BURLAP 117 having to recompile the felt or velvet applications To add a new element to the set of ex isting elements the add_definition function is used Similarly to remove an existing element definition the remove_definition function is used Argument Description Type Example name name of definition string truss set_up set up function K and M function truss_set_up stress stress computation function function truss_stress shape element shape scalar amp linear num_nodes number of nodes in element scalar 2 shape_nodes number of nodes that define shape scalar 2 num_stresses number of stress values scalar 1 dofs vector of DOFs row vector 1 2 3 0 0 0 retainK retain stiffness matrix indicator scalar amp false Table 10 12 Arguments to the add_definition function The add_definition function requires a variety of arguments as shown in Table 10 12 For the reminder of the discussion regarding the arguments to the add_definition function we will use the truss element as an example The name of the element definition is a string by which the element type is referred Once the element definition has been successfully added then new elements of that type can be created In our example the name is truss so a FElt file can now be loaded that contains truss elements If an element defi
58. they expand to 2 and 6 no real magic there it is just little more intuitive to look at El MultiplyMatrices eq_global Tt equiv element gt node 1 gt eq_force Tx VectorData eq_global 1J element gt node 1 gt eq_force Ty VectorData eq_ global 2l element gt node 1 gt eq_force Rz VectorData eq_ global Sule element gt node 2 gt eq_force Tx VectorData eq_global 41 element gt node 2 gt eq_force Ty VectorData eq_global 51 element gt node 2 gt eq_force Rz VectorData eq_global 6 return 0 Now that we have written the source file timoshenko c in the directory lib Elements we need to change the OBJS line in Makefile in this directory to include our new file That line should now read like OBJS beam o beam3d o cst o truss o iso_2d o iso_quad o misc o timoshenko o All we have to do now is type make at the shell prompt and of course because we do such good work there are no errors On the off chance that you re not so lucky and you do have a few errors all you have to do is fix up your source file and keep trying make until they go away Once we have the new element library made we need to update the initialization procedure which will define the default set of elements for a FElt application We do this by editing lib Felt initialize c and adding extern struct definition and AddDefinition lines just like the lines already in that file With that change made d
59. this repetitive information Starting velvet with a FElt file that contains a couple of common constraint definitions means that those definitions will not need to be specified explicitly within velvet they will be there for use on start up Similarly for other types of objects Note that loading a defaults file from within velvet is just like opening a new input file everything in the current problem will be lost You can use the canvas configuration and display list sections of a FElt file to create a custom canvas configuration in a defaults file Material databases are also special cases of FElt input files They are given special treatment within velvet because they are so convenient Whole classes of material types and shapes W shape beams standard diameter bars etc can be stored in a material database and loaded into velvet at any time with the open database command under the file menu Any changes to material properties or additions of new materials can be saved simply by selecting update database from the file menu The current set of materials is the only thing that is retained when a new input file is loaded or a new problem is started within velvet 7 9 AUTOMATED ELEMENT GENERATION 81 Generation Parameters help okay cancel Figure 7 12 The form for defining line and grid generation parameters 7 9 Automated element generation Velvet can generate grids and triangular meshes automatically
60. to the axes and then specified as two separate loads 4 2 5 Two dimensional isoparametric elements 4 2 5 1 General four to nine node element Like CST elements isoparametric elements are available for either plane strain or plane stress analysis Isoparametric elements are 2 d planar quadrilateral elements with nine nodes Any of the last five nodes are optional however and the fourth node can be the third node repeated The numbering convention is shown in Figure 4 2 If any of nodes 5 9 are left out their place must be filled with a zero in the nodes specification for that element i e you must always specify nine numbers it is simply that any or all of the last five might be specified as zero None of the first four nodes can be zero If the fourth and third nodes are the same then the element will be degenerated to a triangle In this case nodes 5 9 must be zero Currently no stresses are computed for the generalized isoparametric element Dis tributed loads on the generalized isoparametric element are ignored and do not affect the problem solution in any way 4 2 STRUCTURAL ANALYSIS ELEMENTS 37 ic Lin Figure 4 2 Node numbering scheme for nine node isoparametric planar element 4 2 5 2 Simple four node element quad_PlaneStress and quad_PlaneStrain elements are just like the generalized isopara metric elements described above but they can only have the first four nodes They are intended to make problem
61. to the number of active DOFs The dofs_num vector corresponding to the dofs_pos vector given above would then be 1 26 10 3 BURLAP AND FELT 109 since the first second and sixth DOFs are active burlap ensures that the two vectors are always kept consistent All arrays start at one so nodes 1 designates the first node in the problem and nodes 1 constraint designates the constraint associated with that node A period is used to access a field of a FEIt object In general the field names are the same as the names used in the FElt input file For example constraint is the field name for accessing the constraint object assigned to a node with the const raint syntax in the FEIt file 2 write nodes array of node 3 write elements array of element 4 write nodes 1 node 1 5 write nodes 1 constraint constraint free 6 write elements 1 material A 0 0004 7 elements 1 material 8 write A m A E m E A 0 0004 E 2 1e 11 10 3 1 Element objects Elements have the same field names as those used in the FElt file with additional fields for holding the computed matrices and stress values as shown in Table 10 2 For example e number will return the element number of the element e and e material t will return the thickness of the element s material e nodes 1 will return the first node of the ele ment and e nodes 1 x will return the x coordinate of that node Note
62. value 11 4 8 Additive operators XY X Y The type semantics of the additive operators are quite similar to the semantics of the relational and equality operators except that only scalars and matrices are allowed As shown in Table 11 7 adding two scalars produces a scalar as expected adding a matrix to a scalar simply adds the scalar to each element of the matrix adding two matrices results in standard matrix addition again as expected Like the equality and relational operators the additive operators associate from left to right Also two matrices must have identical dimensions for addition or subtraction 1 write 1 2 3 134 CHAPTER 11 THE BURLAP SYNTAX X Y Z Zz X op Y scalar scalar scalar z xopy scalar matrix matrix Vi j Zij x op Y j matrix scalar matrix Vi j Zij Xij op y matrix matrix matrix Vi j Zi j Xij op Y Table 11 7 Type table for the additive operators 2 write 1 2 3 4 3 write 3 4 5 1 4 write 1 2 3 4 5 write 1 2 3 4 5 stdin 5 size mismatch in expression 1 x 3 1 x 2 11 4 9 Multiplicative operators x AAA 7 Sa SK KK oe The multiplicative operators which associate from left to right are distinct from the other operators in that the results are not computed on an element by element basis As illustrated in Table 11 8 multiplying two scalars results in scalar mul
63. 1 j lt element gt distributed i gt nvalues j if element gt distributed i gt value j node lt 1 element gt distributed i gt value j node gt 2 error invalid node numbering for elt d distrib load s element gt number element gt distributed i gt name count if element gt distributed i gt value 1 node element gt distributed i gt value 2 node error incorrect node numbering for elt d distributed load s element gt number element gt distributed i gt name count Have we had any errors If so bail out ay if count return count 13 6 A DETAILED EXAMPLE 197 phi for i if el if 12 0 L L element gt material gt E element gt material gt Ix element gt material gt kappa element gt material gt G element gt material gt A loop over all of the applied distributed loads superposing the effects of each i lt element gt numdistributed i First we have to sort out what order the load values were supplied in We need to get it so that wa is the value on element node 1 and wo is the value on element node 2 element gt distributed i gt value 1 node 1 wa element gt distributed i gt value 1 magnitude wb element gt distributed i gt value 2 magnitude
64. 1 3 The force dialog The two toggle buttons at the top of the dialog toggle the display of force information between the time domain forces Fx Mz in a FElt file and frequency domain spectral inputs Sfx Smz in a FElt file Forces and moments or spectra of forces and moments in each of the three directions can be entered in the appropriate text field If an entry is left blank that component of the force will be taken as 0 0 The dialog pictured in Figure 7 7 is from a model of a bicycle in which there are two static forces defined Just like in a FElt file forces can be a time or frequency dependent function using the independent variables t and w 7 5 1 4 The load dialog Figure 7 8 shows what the load dialog box looks like after we have defined both of the distributed loads used in our sample problem The eight toggle buttons define the direction of the load Note that these toggles function as a radio group 1 e only one of them can be selected at any one time The text fields allow you to enter node magnitude pairs much the same way that you would define a load in a standard FElt input file Remember that the node specification refers to the local element number i e 1 or 2 for a beam 1 2 or 3 fora 7 5 DRAWING AND DEFINING A PROBLEM 73 Name Node Magnitude LocalX Globalx 1 10000 side2 LocalY J cibay 2 o ToralZ Globalz perpand J parallel J dismiss delete help accep
65. 10 1 Introduction to the burlap environment burlap is a mathematical environment designed for adding new element types to FElt and extending the current set of finite element analyses burlap can best be described as a mathematical software package similar to matlab or octave but with the data structures of FElt It was designed as an alternative to trying to understand and modify the C code of the FEIt library which was difficult for even us to do much less engineers with little C programming experience burlap has its own syntax for manipulating matrices like that of matlab However burlap also includes all of the FElt data types as well such as nodes elements constraints etc which eliminates the need for having large arrays of numbers where the third number is the material thickness and the fourth number is the cross sectional area etc Any analysis that can be performed with felt can be performed with burlap Although the burlap code is slower than the compiled C code of the FElt library burlap s interac tive environment makes prototyping new element types and new analyses much easier At present however there is no support for executing burlap code from velvet 10 2 Using burlap 10 2 1 Interacting with burlap The syntax of burlap is designed to be intuitive and is best illustrated by examples A detailed discussion of the syntax can be found in Chapter 11 The simplest way to start 105 106 CHAPTER 10 THE BURLAP APP
66. 18 0 20 0 phi VectorData equiv 3 wa L 2 0 factor 42 0 40 0 phi VectorData equiv 2 wa L L 12 0 factor 4 0 5 0 phi L VectorData equiv 4 wa L L 12 0 factor 6 0 5 0 phi L if this is mode 2 we re done just hand the equiv vector back by setting eq_stress if mode 2 egq_stress equiv ret ay SetEqui urn 0 We have the load vector in local coordinates now All of this is taken care of by a convenience routine What it is doing is checking if the eq_force array has been allocated for this element s nodes If it hasn t it will set it up If it has it will do nothing and simply return to us It has to allocate space for six doubles even though we will only ever use two entries for Timoshenko elements because other element types may try to insert something into this array in different locations Also remember that we will access it as a standard array it s not a Vector or Matrix type valentForceMemory element 13 6 A DETAILED EXAMPLE 199 The equiv vector has four things in it We need to transform these to global coordinate and then add them incrementally into the eq_force array on the nodes because some other element may have also already added something onto this node Note the use of Tx Ty and Rz to access the eq_force array These are just enumerated so that
67. 1e 04 0 3 6e 05 0 524 4 8 521e 04 0 4 005e 05 8 521e 04 0 8 521e 04 9 231e 06 3 6e 05 8 521e 04 5 446e 07 Modal Analysis Example Modal frequencies rad sec The current version of felt cannot generate a graphical representation of the mode shapes See the dis cussion of mode shape plotting in chapter 8 for an example of how velvet can generate these kinds of plots 50 CHAPTER 5 THE FELT APPLICATION 1 12 049 1 9177 Hz 2 22 807 3 6299 Hz 3 30 09 4 7889 Hz 4 110 14 17 529 Hz 5 160 25 465 Hz 6 160 21 25 499 Hz Mode shapes Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 1 Al 1 1 1 J 0 003005 7 4866e 16 0 011436 6 6597e 17 1 9981e 16 137 75 0 007032 0 50358 0 025963 0 00036256 0 012534 1 1007 1 1 1 1 3 667 1 0 003005 4 565e 15 0 011436 1 0303e 16 1 9981e 16 13775 0 007032 0 50358 0 025963 0 00036256 0 041164 1 1007 modal 39 71 0 0 0 0 0 0 4 344e 04 0 0 0 0 0 0 146 6 0 0 0 0 0 0 3 21 0 0 0 0 0 0 1 248e 34 0 0 0 0 0 0 5 93e 09 modal K 5766 0 0 0 0 0 0 2 259e 07 0 0 0 0 0 0 1 328e 05 0 0 0 0 0 0 3 794e 05 0 0 0 0 0 0 3 194e 38 0 0 0 0 0 0 1 522e 14 modal C 13 12 0 0 0 0 0 0 4 693e 04 0 0 0 0 0 0 271 4 0 0 0 0 0 0 760 0 0 0 0 0 0 6 393e 35 0 0 0 0 0 0 3 047e 11 5 3 INTERPRETING THE OUTPUT FROM FELT 51 5 3 4 Thermal analysis The result from a static thermal analysis is a simple table showing the node number and corresponding s
68. 2 3 Transient structural analysis Just like static structural analysis is an extension of simple static equilibrium for a spring we can draw an analogy between dynamic structural analysis and a simple spring mass dashpot oscillator For the system shown in figure 2 3 a sum of forces at the mass and Newton s law gives mi cx kx f t 2 7 where the overdot indicates differentiation in time To generalize this to multiple degrees of freedom we take the same steps as in the static case recognizing that we can assemble 8 CHAPTER 2 FELT ANALYSIS TYPES xit Pit Figure 2 3 Dynamic equilibrium of a spring mass dashpot system global mass and damping matrices from elemental constructs just as we did for the stiffness matrix in the static case In that case the matrix equation of motion becomes M amp CX Kx F t 2 8 where M C and K are now global mass damping and stiffness matrices x is a vector of displacements at each DOF just as before and F t is a time dependent vector of forces at each degree of freedom The damping matrix is the most difficult quantity to estimate in equation 2 8 both the stiffness and mass matrices are relatively simple to derive for a wide variety of elements see chapter 4 There is no explicit way to specify dashpot constants in FElt Instead damping is based on a Rayleigh model whereby the damping matrix is simply a linear combination of the mass and stiffness matrices C RmM R
69. 2 ERPIESSIONS a HE Gok Dooa OR OE OH 14 3 1 2 1 Continuous functions ass ein ed wR Ew HO 14 31 22 Discrete NCHS s 2 4444 24 be oe ee ee SS 14 313 TNS 22 coe 63506 Ped wee ees oo 2 434 524 5845 16 31A Asmpleexkampl lt ao piei i Boh RR ORS 16 3 2 Sections of a FElt input file aa 104 OR AR GOR OOS BOR BH 17 32 1 Probl md scfipti lt 2242242444 8 K ad0 8464444 17 Fae MO eae ee ek ra So OER eee Se Oe 17 32 3 Elements occ ope ede ede ee eed peek bs esd iden 18 3 2 4 Material properties lt lt ses eS eR HR RE REE 18 E COSTAE as es Bee Bite es Ba Bee De wm Bee Re wm Bk we 19 EA ee tO eoe RO EO ERE Ee HERE HEROES 20 327 Distributed loads lt lt cec es HR OR ESE a eS eR AR 21 3 2 8 Analysis parameters 00 eee eee eee 22 32 9 LAO CASS lt o a eG op eGR Gh hh ee Bd Be ee Ro ORs 22 29 An TSO ORM s sos She Stn See Pe Sk Boe Rew GS 22 3 4 An example of a transient analysis problem 25 39 POMMAICOMVEIION os ra dea eem ved eed dedu RE Lee Ye OHS 27 30 1 Conversion basis 2 22d e h ee 58 e Gh noiae tatad 21 3 5 2 patchworkd tails lt dad acce moes a ace a e AR AR RRA 28 4 The FElt Element Library 29 A IEC 2 a a eo ae DA aaea SA Oe he e SE 29 4 2 Structural analysis elements o 30 4 2 1 Truss and spring elements 2 222222 eke eke eee eds 30 CONTENTS 4 2 2 Euler Bernoulli beam elements 31 4 2 2 1 Special case two dimen
70. 2 Sie eee PR ER EER ER BEES 130 CONTENTS ix 11 4 6 Relational operators 4064 444048554 eR Ew RE ES 131 11 4 7 Range operator ke hee eho Shee LG eS 132 114 8 Additive persion oo e Bee hee Be Boe ae eR aS Bas 133 11 4 9 Multiplicative operators 2 242244 8 Rew ROH REG RE 134 11 4 10 Exponentiation operator o ee a a 137 11 4 11 Transposition operator lt 6 6 dg 6 ek ESE DRS A ESS 138 11 4 12 Unary operators ourense ER SEES EES SESS 138 11AT o AA 139 11 4 14 Function expressions lt lt enr sir ag KO KOM ROG KE 140 LLS TSE IUNCUONS e 2 ss ra bw as A ee ee di ao a i a 140 11 5 1 Mathematical functions 6 cox occiso corra a 141 11 5 2 Matrix functions gt pb kee heehee he dde 141 1133 PERICO nCUONS berries e 144 11 54 Finite clement functions y lt s oes ss e Gone Sow a SE 145 11 5 5 Miscellaneous functions sn e wt de ee ds 149 11 6 User defined ICONS c sace sacs sace AE SEE SERS EES 152 11 7 Control flow constructs 2 22s eben eee ene hes 153 RUSTY IF eee MR He Ree SR ee a od es 153 11 28 WHILE SXPTESSIONS 3 a Soe a Bo ews Sw NE 154 11 743 PORENDIESSIONS o s seon be ew a e A ee ee iaa 155 11 7 4 BREAK NEXT and RETURN expressions 156 12 The Algorithms Behind FEIt 159 124 Some URE cada LR ek em oe KE SR ee KE 159 12 2 Elementary C programming oo eee bee eee dee ewes 159 12 3 Introduction to the general FElt routines 2 243 44 Med See es 163 124 Details of
71. 3 2 8 Analysis parameters The analysis parameters section is required only if you are doing some type of tran sient modal or spectral analysis e g analyis transient analysis spectral in the problem description section For modal analysis it is simply used to set the type of el ement mass matrices that will be formed but for transient and spectral analyses it contains information that further defines the problem and the parameters for the numerical integra tion in time The variables that you can define in this section include start for the start of the frequency range of interest in spectral analyis st op for the end of the time tran sient analysis or frequency spectral analysis range of interest duration is an alias for stop step for the time or frequency step to be used between the start and stop points dt is an alias for step beta gamma and alpha for integration parameters in the structural and thermal dynamic integration schemes mass mode for the types of element mass matrices that should be formed either lumped or consistent note again that this is the only assignment that is required in this section for a modal analysis problem nodes defines a list of nodes which you are interested in seeing output for dofs defines the list of local DOF that you are interested in for the nodes that you are inter ested in The list of nodes should just be a comma delimited list of node numbers The list of DOF should
72. 300 these are defaults boundary 0 50 20 10 9 2 THE CORDUROY SYNTAX 101 40 0 40 50 55 70 95 70 115 50 115 0 135 10 155 50 155 90 115 160 170 404 122 404 67 160 0 90 end To generate a mesh in a rectangular plate with two side by side rectangular holes in it the input file could look like triangular mesh use all defaults including CSTPlaneStress boundary 0 0 20 0 20 10 102 CHAPTER 9 THE CORDUROY APPLICATION end Note that the hole definitions do not need to come before the boundary but that points must always be specified in a counter clockwise order 9 3 Using corduroy The only options to corduroy are the standard cpp options as in both felt and velvet like those two programs corduroy runs all of its input through the pre processor before actually operating on it and a debug flag which causes corduroy to reproduce as output what it thinks it received for input To generate all you need to do is create an input file with a text editor and then run corduroy with the name of this file as the last argument on the command line Output will be directed to standard out So if you had a generation description in a file foo crd you could turn it into the basics of a FElt file called foo f1t with the following command corduroy foo crd gt foo flt 9 4 Incorporating output into a FElt file Because
73. 5 FORMAT CONVERSION 27 free Tx u Ty u Rz u end 3 5 Format conversion Any problem that you want to solve with FElt must be formatted according to the rules that we have just described However we recognize that there are countless other formats that provide for descriptions of these same kinds of problems at least the geometric aspects of the problem Because of this FElt provides a mechanism for translating back and forth between FElt and alternative syntaxes 3 5 1 Conversion basics A common example of a popular geometric description syntax is DXF drawing inter change format DXF files can be produced by any number of CAD and geometric mod eling programs The FElt conversion tool patchwork can translate DXF files to and from the FElt syntax patchwork is invoked with a command line like patchwork dxf bridge dxf felt bridge flt This example would convert the geometric description given in the file bridge dxf to the geometric basics of a standard FElt file bridge f1t Geometric basics in this case means that only the nodes and elements sections will be translated the DXF file cannot contain any information about material properties forces loads constraints and therefore this information will be left incomplete in the resulting FEIt file patchwork can also translate FElt files into other formats You might want to do this if you had used some of FEIt s mesh generation capabilities for the bulk of your geometric
74. 6 in 164 CHAPTER 12 THE ALGORITHMS BEHIND FELT the third position indicates that the third active DOF for this problem is rotation about the z axis Both of these arrays are available globally through the analysis structure 12 4 2 Node renumbering We all know that the profile and bandwidth of a global stiffness or mass matrix critical factors in determing memory requirements and ultimate solution speed are a function of the global node numbering The contribution from each element is assembled into the global matrices based on the global numbers of the elements nodes Better node number ing schemes take this into account and try to keep the contribution from each element as near to the diagonal of the global matrix as possible Given the compact column storage scheme described below a matrix with most entries very near the diagonal can use signifi cantly less storage than a very full matrix The linear equation solver can also make use of this reduced matrix and operate a lot faster by not having to operate on a lot of zero entries There are numerous algorithms available to try to optimize node numbering with just such a goal in mind The one that we use in FElt is popularly known as Gibbs King 6 the profile reduction variant of the Gibbs Poole Stockmeyer algorithm 7 which primarily tries to minimize bandwidth Other popular algorithms include Cuthill Mckee 3 and Reverse Cuthill McKee 5 12 4 3 Assembling the global
75. A grid is defined as ann x m array of two dimensional line or quadrilateral elements A triangular mesh is an arbitrary polygonal shape possibly with interior holes discretized into two dimensional triangular elements When you select generate from the elements menu the type of generation is automatically determined from the current element type Because both nodes and elements will be generated both a constraint and a material property must be active You should make sure that the active object of each type will be appropriate for the majority of nodes and elements that will be generated This will save you from applying a lot of objects individually once the generation is complete 7 9 1 Generating a grid of line or quadrilateral elements Generating a grid requires that you complete the form pictured in figure 7 12 The entries in the form match parameters in the corduroy input syntax see chapter 9 The x y and z start and end locations define the two extreme corners of the grid The number entries set the number of elements to be generated along each direction and the rule entries set the spacing rule to be used along each axis Rules can either be typed in or entered by giving the focus to one of the entries and selecting a rule from the pull down menu available by clicking on the Rule button 7 9 2 Generating a mesh of triangular elements Generating a mesh of triangles requires a little more work on your part but it is still a lot easi
76. ATION and if the problem has been solved the stress results for this element If an element uses more than six nodes you can use the side to side arrow buttons next to the node list to scroll through them The dialog gives you a simple mechanism for changing the nodes material and loads for a given element Editing the material and loads is just like the constraint and force fields in the node dialogs You can either type into the text field or click on the appropriate label and choose from a menu of currently available objects Like forces in the node dialog changing the information in the element dialog is the only way to completely remove all the loads from an element Once again of course you need to click accept to make any changes effective The other buttons across the bottom and their functions should be very familiar to you by now 7 7 Using tools The drawing tools are intended to be aids for meshing up or annotating a problem For instance you can draw a circle by selecting circle from the tools menu and then place nodes along the circle The actual figure on the screen does not affect the solution of the problem in any way velvet will prompt you in the command window for the necessary coordinates for each figure type As usual you can choose coordinates either with the mouse or by typing in the command window In general however you must enter all of the coordinate locations for a given figure with the same input method what t
77. Applied Finite Element Analysis Wiley New York second edition 1986 15 A Tessler and S B Dong On a hierarchy of conforming Timoshenko beam elements Computers and Structures 14 335 344 1981 16 O C Zienkiewicz and Robert L Taylor The Finite Element Method Basic Formu lations and Linear Problems volume 1 McGraw Hill London fourth edition 1988
78. English were probably equivalent A 2 TRANSLATION FILES 203 For German use Instead of problembeschreibung titel knoten elemente krafte kraefte zwangsbedingung zwangsbedingungen materialeigenschaften richtung senkrecht GlobalesX GlobalesY GlobalesZ LokalesX LokalesY LokalesZ werte verteilte lasten last gelenk ende problem description title nodes elements force forces constraint constraints material properties direction perpendicular GlobalX GlobalY GlobalZ LocalX Local Y LocalZ values distributed loads load hinge end For Spanish use Instead of descripcion_del problema titulo nodos elementos fuerza fuerzas restriccion restricciones propriedades_del_material direccion paralela valores cargas distribuidas carga articulacion_plana final problem title title nodes elements force forces constraint constraints material properties direction parallel values distributed loads load hinge end 204 APPENDIX A INSTALLING AND ADMINISTERING FELT A 3 Defaults and material databases Though the defaults files and material databases are not a necessity for velvet they are convenient and it might be desirable to install them in a location where all users have easy access to them probably the same library directory as the translation files These files too are really nothing more than include files which can
79. K 2 9 where Rm and R are user specified constants of proportionality 2 4 Static thermal analysis 2 5 Transient thermal analysis 2 6 Modal analysis There are two basic levels to modal analysis The first is the eigenvalue problem which determines the natural frequencies and mode shapes of our structure in free F t 0 2 6 MODAL ANALYSIS 9 undamped C 0 vibration If we assume a solution to the undamped unforced form of equation 2 8 of the form x t ue 2 10 where u is a vector of displacement amplitudes and substitute this into the equation of motion we find that K Moy u 0 2 11 Linear algebra tells us that this has a non trivial solution only if det K Mw 0 2 12 Evaluating the determinant leads to a polynomial of order n where n is the number of DOF in the problem in The roots of this polynomial give us the free vibration natural frequencies 0 i 1 n If we substitute each of these into equation 2 11 we can solve for n mode shape vectors yl By itself natural frequency and mode shape information can be very useful the natural frequencies are an immediate indication of where the resonances for this system will be even when we put damping back into the system The second level of modal analysis however uses the fact that the eigenvectors mode shapes form an orthogonal basis set to decouple an arbitrarily complex multiple degree of freedom system into n single degr
80. LICATION burlap is to simply type burlap on the command line or burlap foo b if you wish to execute the burlap file foo b You can also specify command line options that qualify what you want to do help h Display a brief help message which lists all of the available command line options and then exits quiet q Do not print the start up message regarding copyright information alias a Do not define the set of built in aliases interactive 1 Enter interactive mode by reading expressions from the terminal after pro cessing the input files Normally burlap will simply exit after processing any input files no interactive n Do not enter interactive mode This flag tells burlap to simply exit if no files are given It is useful if you have burlap aliased to burlap i and wish to override the effect of the i flag source command file s command file Read commands from command file on start up felt felt file felt file Use felt file to define the current FElt problem Since burlap can load and process a FEIt file it also accepts the standard nocpp cpp I U and D flags common to felt and velvet If you simply type burlap you will be presented with some copyright information and a prompt This is burlap copyright 1995 by Jason I Gobat and Darren C Atkinson This is free software and you are welcome to redistribute it under certain conditions but there is absolutely no warranty Type hel
81. NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PRO GRAM IS WITH YOU SHOULD THE PROGRAM PROVE DEFECTIVE YOU ASSUME THE COST OF ALL NECESSARY SERVICING REPAIR OR COR RECTION B 3 HOW TO APPLY THESE TERMS 211 12 INNO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER OR ANY OTHER PARTY WHO MAY MODIFY AND OR REDISTRIBUTE THE PROGRAM AS PERMIT TED ABOVE BE LIABLE TO YOU FOR DAMAGES INCLUDING ANY GEN ERAL SPECIAL INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCU RATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBIL ITY OF SUCH DAMAGES END OF TERMS AND CONDITIONS B 3 Appendix How to Apply These Terms to Your New Programs If you develop a new program and you want it to be of the greatest possible use to the pub lic the best way to achieve this is to make it free software which everyone can redistribute and change under these terms To do so attach the following notices to the program It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty and each file should have at least the copy
82. User s Guide and Reference Manu The FElt Syst e ystem E Jason I Gobat Darren C Atkinson FEIt User s Guide and Reference Manual Jason I Gobat Applied Physics Laboratory University of Washington Darren C Atkinson Department of Computer Engineering Santa Clara University Computer Science Technical Report CS94 376 University of California San Diego Legal Notice Copyright 1993 2005 Jason Gobat and Darren Atkinson jgobat apl washington edu and datkinson engr scu edu The FElt System User s Guide and Reference Manual may be reproduced and distributed in whole or in part subject to the following conditions 0 The copyright notice above and this permission notice must be preserved complete on all complete or partial copies Any translation or derivative work of The FElt System User s Guide and Reference Man ual must be approved by the authors in writing before distribution If you distribute The FElt System User s Guide and Reference Manual in part instruc tions for obtaining the complete version of The FElt System User s Guide and Reference Manual must be included and a means for obtaining a complete version provided Small portions may be reproduced as illustrations for reviews or quotes in other works without this permission notice if proper citation is given The GNU General Public License referenced below may be reproduced under the condi t
83. a few general FElt routines 2 444524 5 24544504 163 124 1 Finding the active DOF 2 24 2 eee hee eee he ee ee EES 163 2 CONTENTS 1242 Dede renumbenng lt a oso esa pe ee a a eH Se BX 164 12 4 3 Assembling the global stiffness matrix 164 12 4 4 Compact column representation 165 12 4 5 Dealing with boundary conditions 166 12 4 6 Solving for nodal displacements 166 12 4 7 Time integrating in transient structural analysis 166 12 4 8 Time integrating in transient thermal analysis 168 12 4 9 Solving the eigenvalue problem 2 246 5 be wk Ge es 168 13 Adding Elements to FElt 171 13 1 How to oot Stated ge ee ee ee ww RE ES 171 13 2 Necessary definitions and functions 2 2 02 172 132 1 Thedefinition structure gt 2 25 c2kedeetbeecibaadsveaises 172 13 2 2 Inside the element setup functions 22462 See es Ss 174 13 2 3 Inside the element stress function aoao 174 13 3 The FElt matrix and memory allocation routines 175 13 4 Element library convenience functions 177 13 9 Putung itall setter lt c daci deh ech AG AS ARA RA A ROA 179 1360 BAC Sate o s odos ani aoee aena aia dolg a Dle OE Be h a 180 A Installing and Administering FElt 201 A 1 Building the FElt system from source saoao 201 AZ Tons huon DIES soei a ank s e a e a o Sh Be BO Be 202 A 3 Defaults an
84. a storage or distribution medium does not bring the other work under the scope of this License You may copy and distribute the Program or a work based on it under Section 2 in object code or executable form under the terms of Sections 1 and 2 above provided that you also do one of the following a Accompany it with the complete corresponding machine readable source code which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange or b Accompany it with a written offer valid for at least three years to give any third party for a charge no more than your cost of physically performing source dis tribution a complete machine readable copy of the corresponding source code to be distributed under the terms of Sections 1 and 2 above on a medium cus tomarily used for software interchange or c Accompany it with the information you received as to the offer to distribute corresponding source code This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer in accord with Subsection b above The source code for a work means the preferred form of the work for making mod ifications to it For an executable work complete source code means all the source code for all modules it contains plus any associated interface definition files plus the scripts used to control compilation and
85. ach node in each DOF is printed Of course in most problems not all DOF will be active so many of the displacements will simply come up as zero The information in the element stress table will vary depending on the element type Each row contains up to six columns of information if an element calculates more than six stresses or loads the information for that element will take up more than one row beam3d elements for instance have 12 loads calculated and thus the information for each element takes up two rows The output format for each element type is summarized in Table 5 1 Element col 1 col2 col3 col4 col5 col6 truss Ox beam ji iy ml E A m beam3d HIR l m m m P Rf me mim timoshenko F m a m CSTPlaneStress Ox Oy Txy 61 0 6 CSTPlaneStrain Ox Oy Txy 61 02 6 quad_PlaneStress o o va o o o o oF th Of oO 64 quad PlaneStrain o o om o o o o o Gy OF of 84 htk Myx Myy Myy qx dy brick oh ok t Ee th MESE DE eee Table 5 1 Contents of the stress vector for different element types The reaction force section contains the global forces calculated at each constrained DOF The first column indicates the global node number at which this reaction occurs the second column contains the DOF at which this force is applied and the last column gives the magnitude of this reaction force If we were to save the input file from the detailed example of
86. ade in the element stiffness matrix routines all we do here is zero out rows and columns of the stiffness matrix and force vector as in the fixed case 12 4 6 Solving for nodal displacements The FEIt routine to solve for nodal displacements or to solve any linear system of equa tions takes the compact column representation and solves Kd F for the global displace ment vector using a skyline solver This procedure basically involves a Crout factorization and forward and backward substitution on the compact column representation of the global stiffness matrix 12 4 7 Time integrating in transient structural analysis Time stepping in a transient structural analysis problem is done with the Hilbert Hughes Taylor alpha variant of Newmark integration There are three critical parameters in this algorithm The first two B and y are the standard parameters of Newmark integration Depending on the values for these two parameters we can implement several well known 12 4 DETAILS OF A FEW GENERAL FELT ROUTINES 167 algorithms Setting B i and y 5 results in an unconditionally stable average acceler ation trapezoidal rule implementation B 7 and y 5 results in a linear acceleration algorithm The HHT o algorithm adds a third parameter amp to account for the decrease in order of accuracy that results when you introduce numerical damping into the Newmark method Setting amp 0 reduces the problem to a standard Newmark method Cho
87. al element stiffness matrices were placed into the global stiffness matrix Our equation for static equilibrium is k k 0 Xa Fa ki k k ko x Fp 2 5 0 Kko k2 Xe F Of course one dimensional springs are not the most useful element in terms of mod eling complex structural behavior The concepts of static equilibrium superposition and assembly however are identical no matter what types of elements we are using All that changes the is nature of the element stiffness matrices rather than simple linear springs we take into account bending and torsional stiffness three dimensional solid deformations plate bending motion etc In structural analysis there are only six possible degrees of free dom translations along and rotations about the x y and z axes and thus if we know how each element affects each of these degrees of freedom we can even mix different element types into the same global stiffness matrix Given a global stiffness matrix K which repre sents the contributions from an arbitrary number of individual elements and a vector F of the force at each global degree of freedom then the general form of equation 2 5 is simply Kx F 2 6 where x is a vector of the displacements at each global degree of freedom which we solve for using matrix techniques for linear systems of equations Note the direct analogy be tween this matrix equation and the simple linear spring relationship f kx cited earlier
88. alue is returned A RETURN expression may only occur inside a function body and has the lowest possible precedence 11 7 CONTROL FLOW CONSTRUCTS 157 The BREAK expression may be used to explicitly exit and return a value from a FOR or WHILE loop If the expression is absent then a nu11 value is returned A BREAK expression is executed by first evaluating the expression if it is present and then transferring control out of the nearest enclosing FOR or WHILE loop The BREAK expression has the lowest possible precedence function search x a found 0 for v in x do if v a then found 1 break end end write found found end If the value a is found within the vector x then there is no point in continuing with the search after setting the found indicator The NEXT expression may be used to skip to the next iteration of the nearest enclosing FOR or WHILE loop and has the highest possible precedence Chapter 12 The Algorithms Behind FElt 12 1 Some background In general there is a massive base of experience in the numerical analysis community that relates to programming the finite element method Any time you see the method imple mented chances are that the implementation is in some piece of software So why is it that good books on computational techniques are hard to come by and good books on the underlying theory and mathematics abound Go figure As we mentioned in chapter 2 if you need some references to the
89. ap o oo o 126 11 2 Type table for the logical OR operator o o 129 XV xvi LIST OF TABLES 11 3 Type table for the logical AND operator noaoo 130 11 4 Type table for the equality operators s aooaa a 131 11 5 Type table for the relational Operators lt lt wee aw ees 132 11 6 Type table for the range operator 0 02000 133 11 7 Type table for the additive operators 2 42 cece a Sea Sew beds 134 11 8 Type table for the multiplication operator 135 11 9 Type table for the right division operator 135 11 10Type table for the left division Operator 136 11 11 Type table for the remainder operator kc ee ee ee ees 137 11 12 Type table for the exponentiation operator 137 11 13 Type table for the transposition operator 2 138 11 14Type table for the unary operators o o 138 11 15 Intrinsic functions from the math library 141 11 16Argument table for the norm function 143 11 17Predicate functions available in burlap 144 11 18 Arguments to the add_definition function 145 Foreword FElt is intended as a tool for teaching finite element analysis methods There are better tools there are bigger tools there are tools that can do many many things tha
90. are brackets on the right hand side as the inverse of the transfer function In a multiple degree of freedom system this term is known as the impedance matrix Z 0M i0C K 2 28 and its inverse is equivalent to a matrix of single degree of freedom transfer functions H Z 2 29 In essence the transfer function matrix predicts the amount of response per unit force at frequency Given entries from the transfer function matrix Hjj we can calculate the output spectrum at DOF i due to system inputs as se Y Hlo Ps 2 30 k 1 where the input spectra si are simply the power spectra of the applied forces 2 8 Nonlinear static analysis 2 9 Nonlinear dynamic analysis It is a property of linear systems that the output will always be at the same frequency as the input Chapter 3 Structure of a FElt Problem 3 1 Input file syntax A FElt problem is defined by an input file which you must create using a text editor such as vi or from within velvet though in this case you do not actually need to see the resulting file The input file contains a complete description of everything that defines a problem the nodes the elements analysis parameters the constraints and forces on the nodes and the distributed loads and material properties of the elements An informal description of the file format and these objects is given below For a complete formal definition of the syntax you should refer to the felt 4fe manual page
91. argument argument expression list end burlap provides a simple syntax for allowing users to define their own functions A function definition creates a new function with the specified name overwriting any previ ously assigned value Function names are simply variables just as in any of the previous examples Functions can be passed as parameters to other functions or returned from func tions just like ordinary scalar variables The expression list is a semicolon separated list of expressions that comprise the body of the function The return value of the function is the value of the last expression al though usually a RETURN expression is used to explicitly return a value from the function Section 11 7 4 Each argument specifies the name of a formal parameter to the function Sec tion 11 4 14 By default all arguments are passed by value the actual parameter is eval uated and a copy is passed to the function with assignment to the formal parameter having no effect on the caller of the function If the keyword shared is placed before the name of the formal parameter e g shared foo then the data associated with the formal is shared with the data of the actual An assignment to one is implicitly an assignment to both burlap uses this ability in several intrinsic functions e g lu eig to return more than one value from a function function swap x y t x x yy y t end a 1 b 2 swap a b 1
92. as an argument it will contain the unit lower triangular matrix on return Similarly U will contain the upper triangular matrix and P will contain the permutation matrix such 11 5 INTRINSIC FUNCTIONS 143 that PLU X norm X s Returns the norm of X X The type of the norm depends upon the type of X and the value of s which must be a string as shown in Table 11 16 X oll iv 2 rro scalar X X X X vector XI Xi Xll IX lle matrix X r XI illegal X Table 11 16 Argument table for the norm function The notation X indicates the Frobenius norm of X the Frobenius norm of a vector X is equivalent to its 2 norm X 2 Since a scalar value can always be converted to a string value it is legal to use norm X 1 or norm X 2 butnot norm X fro unless fro evaluates to 1 2 or EKO ones m n Returns an m x n matrix whose elements are all 1 If n is not specified then an m x m matrix is returned Both m and n must evaluate to scalars greater than zero qr X Q R Computes the QR factorization of X such that QTX R and QQ I The result is a right upper triangular matrix R The orthogonal matrix O may be retrieved by specifying it as the second parameter R may also be retrieved in this manner if desired The matrix X must be overdetermined i e tall and thin rand m n s Returns an m x n matrix whose elements are randomly dist
93. at vi s include s end Finally it is important to realize that the readline library capabilities and aliases work on lines not on expressions History substitutions apply to the lines entered not to the individual expressions on a line Similarly aliases and history expansion characters are not recognized when burlap is reading from a file 10 6 Common error messages This section attempts to explain some of the common error messages associated with using burlap All error messages are prefixed with the name of the file and the number of the line that was found to be in error An error on a line may not be discovered until several lines later so the line number may not be the line that actually contained the error If an error occurs in a function then the function call is terminated and the error is propagated back to the outermost level In this case the calling sequence is shown to assist in tracking down the cause of the error In the examples given the file name and line number have been omitted type error in expression function call to null This error indicates that the function call or array index operators were ap plied to a null expression such as foo where foo is null This error 122 CHAPTER 10 THE BURLAP APPLICATION often results from misspelling a variable name or from forgetting to declare a global variable using the global declaration in a function body You can generally locate this error by searchin
94. ate name in the list The accept button is simply there in case you want to type in the element name Typing beam3d in the text field and clicking accept is equivalent to simply clicking on the beam3d entry in the list Given this mechanism the logical way to add the elements in our sample problem is to set the current element type to beam and add elements 1 and 2 Elements are added by selecting add from the elements menu and then clicking on nodes in the appropriate order or entering node numbers in the command window velvet will wait until you have selected the appropriate number of nodes for the current type of element before it prompts you to add the next element Click quit to stop adding elements Click abort if you goofed up on an element and just want to start over for that element e g if you entered the first two nodes wrong on a CST element Once elements and 2 are added we can change the element type to truss and add element 3 by clicking on nodes 2 and 4 Before we did this we would probably also change the active material to springy or we could just change it with material from the apply menu later We also still need to apply the distributed loads All we have to do is set sidel active by clicking it in the load dialog and apply it to element 1 from the apply menu Then we do the same thing for side2 and element 2 Deleting nodes and elements via delete under the appropriate menu must proceed in reverse order of adding n
95. ateral elements 81 7 9 2 Generating a mesh of triangular elements 81 740 Keyboard interface mechanisms 4 4 54 riera rra 82 7410 1 Keyboard shortcuts gt gt s ess OO ED RES E oretta 82 TADA Command Names oes sos saou siui seu ER EDS SES EES 83 TAI Command line ptions 2 2 6 e ce coi e ea sorea tahoa set hais 85 8 Post processing with velvet 87 8 1 Solving a problem with velvet 26 ces caw ee yee ReG HOw neg xe 87 8 2 Problem description and analysis parameters 92 8 3 Controlling the post processing ge ce eR ee RR ERR HR Es 92 8 3 1 Controlling contour plots 24 0 cbc cbsae deh RA 92 8 3 2 Controlling structure plots oo 94 8 3 3 Controlling animation oaoa 2 4 RE 94 9 The corduroy Application 97 OI Tatoducton kb e e a a ade Ao a ERS EX 97 92 Thecorduroy Synta or sda RARA REA 97 9 2 1 Specifying basic parameters cs ko hE RARER RE 97 9 2 2 Generating elements along a line a oo lt oosorzssss 98 9 2 3 Generating a grid of lineelements osos ss ea heahea he 98 9 2 4 Generating a grid of quadrilateral planarelements 99 9 2 5 Generating a grid of solid brick elements 99 M20 Grids pacing rules lt lt oes s sem mastega aea ted tets 99 9 2 7 Generating a triangular mesh o o oo o 100 993 USME NI O we a 102 9 4 Incorporating output into a FElt file 102 viii CONTENTS 10 The burlap Applica
96. atrixData k 3 4 6 L 190 CHAPTER 13 ADDING ELEMENTS TO FELT MatrixData k 4 4 4 0 phi L L MirrorMatrix k k the above numbers aren t quite right we ve got a term out front of the matrix that we need to scale the entire matrix by 7 factor element gt material gt E element gt material gt Ix 1 0 phi L L L ScaleMatrix k k factor 0 0 that s all for this part return k a simple little function to compute the transform matrix for a simple 2d beam element with no axial DOF This should be a convenience routine but none of the other elements actually use this one because they are more complicated del static Matrix TransformMatrix element Element element double S C direction cosines static Matrix T NULL transform matrix to return double Ly element length no surprise here we only want to allocate memory for this guy once xl if T NULL T CreateMatrix 4 6 13 6 A DETAILED EXAMPLE 191 if T NullMatrix AllocationError element transform matrix This is a pretty sparse matrix so we ll just zero it out then fill in the few relevant entries ZeroMatrix T L ElementLength element 2 c element gt node 2 gt x element gt node 1 gt x L s element gt node 2 gt y eleme
97. ax for the FElt input file is based on a high level grammar which frees you from comma delimited lists of numbers and hours of debugging due to not having the right number after the right comma both the parser and the mathematical routines do extensive error checking and report what we hope are informative and useful error messages FElt offers several user interface options The basic felt application gives you the ca pability to define an input file in a favorite editor and then solve the problem from the shell command line Graphics for the felt applications can be handled by any number of graphing packages Under Windows WinFElt provides a text editor and encapsulator en vironment with some post processing capability velvet is the full featured graphical user interface to the FElt system velvet which is smoother than felt knows most of what there is to know about the FEIt system and provides a consistent CAD like interface for draw ing defining solving and visualizing everything about a problem Though it s probably not always the best way to set up a problem it s hard to beat vi for quick and dirty prob lems a user working strictly with two dimensional problems three dimensional graphics are only partially supported in velvet need never actually see the internals of a FEIt file or run the felt application from the shell command line velvet provides access to all of the two dimensional functionality of FElt in one completely stan
98. be included as a convenience for the user Appendix B The GNU General Public License Printed below is the GNU General Public License the GPL or copyleft under which FElt is licensed It is reproduced here to clear up some of the confusion about FElt s copyright status FEIlt is not shareware and it is not in the public domain The FEIt system is copyright 1993 1995 by Jason Gobat and Darren Atkinson Thus FElt is copyrighted however you may redistribute it under the terms of the GPL printed below GNU GENERAL PUBLIC LICENSE Version 2 June 1991 Copyright C 1989 1991 Free Software Foundation Inc 675 Mass Ave Cambridge MA 02139 USA Everyone is permitted to copy and distribute verbatim copies of this license document but changing it is not allowed B 1 Preamble The licenses for most software are designed to take away your freedom to share and change it By contrast the GNU General Public License is intended to guarantee your freedom to share and change free software to make sure the software is free for all its users This Gen eral Public License applies to most of the Free Software Foundation s software and to any other program whose authors commit to using it Some other Free Software Foundation software is covered by the GNU Library General Public License instead You can apply it to your programs too When we speak of free software we are referring to freedom not price Our General Public Licenses are desi
99. ble c Material A node typedef struct node name of constraint xi constraint vector initial displacement vector initial acceleration vector initial velocity vector boundary displacement vector name of material Young s modulus moment of inertia about x x axis moment of inertia about y y axis moment of inertia about z z axis cross sectional area polar moment of inertia bulk shear modulus thickness density Poisson s ratio shear force correction Rayleigh stiffness damping coefficient Rayleigh mass damping coefficient conductivity in x direction conductivity in y direction conductivity in z direction heat capacity 7 y 7 A El 2y 7 Ly 7 7 12 3 INTRODUCTION TO THE GENERAL FELT ROUTINES 163 unsigned number node number Constraint constraint constrained degrees of freedom Force force force acting on node double m lumped mass at this node double eq_force equivalent force uw double dx 7 displacement double X x coordinate da double y y coordinate double 23 z coordinate Node Hopefully the organization of the data structures 1s intuitive enough to someone familiar with finite elements that a detailed understanding of the mechanics of the C language will not be neces
100. box will be assigned the active object You set the active object of a given type by selecting it in the appropriate dialog box These dialogs can be popped up by clicking on the materials forces loads or constraints buttons in the main velvet control area A picture of the constraint dialog after defining all the constraints in our sample problem is shown in Figure 7 5 Several of the features that you see here are exactly the same in the other three dialogs Each has a name field and a list of all the defined objects of that type The six buttons help accept dismiss delete new copy across the bottom are also present in all four dialogs The main area to the right of the list and name is used for object specific definition Selecting an entry in the list will display the definition for that object and it will instantly make that object the active definition for that kind of object It would remain active even if you were to dismiss the dialog box You can use the name field to either set the name when you are defining a new object or to rename an already defined object For example if you bring up the material dialog in a brand new problem you will be faced with a completely blank slate You can name your first material aluminum define some material properties and click accept This would register aluminum as an available material and set it as the currently active material Now you want to define another material property say springy You
101. cal or global Local variables are those variables referenced within a function body Global variables are those variables referenced outside a function body To explicitly reference a global variable inside a function body it must be declared as global using the global declaration function foo x global y z return x t y z end The function foo references two global variables y and z If the function did not use the global declaration then y and z would be local variables The only exception to this rule is a function call as discussed in Section 1 1 4 14 126 CHAPTER 11 THE BURLAP SYNTAX 11 3 Constants There are several predefined enumeration constants in burlap Most of the constants corre spond to keywords used in the FElt input file to define properties such as the direction of a load The enumeration constants are listed in Table 11 1 analysis types amp static transient amp modal amp static_thermal amp transient_thermal spectral load directions amp local_x amp local_y local_z global_x amp global_y global_z amp parallel perpendicular degrees of freedom amp tx amp ty amp tz amp rx amp ry amp rz amp fx amp fy amp fz amp mx amp my amp mz element shapes amp linear planar solid miscellaneous amp constrained tunconstrained hinged amp lumped consistent null true amp false Table 11 1 Enumeration constants in burlap 11 4 Operators This
102. ce FEIt knows how an element interacts and looks all that remains is for you to tell it how to define the element stiffness and mass matrices and how to compute element stresses or internal forces given nodal displacements 13 2 2 Inside the element setup functions For our mythical foo element the setup routine would look like this int fooEltSetup element mass_mode Element element char mass_mode The name of the element setup function must match the function name given in the element definition structure The setup function must return an integer the number of errors encountered in performing their respective function zero on success and must take a single element and a flag indicating the mass matrix to compute as input If this is a static problem that flag will be zero If transient analysis is being performed then the flag will either be c or l depending on whether the setup routine should calculate a consistent or lumped mass matrix If no element mass matrix is required then the mass_mode will be Zero Given the element you have access to the material property distributed loads and nodes assigned to that element The fooE1tSetup routine s primary responsibility is to allocate and fill out the element stiffness matrix and mass matrix if necessary for this element However if there are distributed loads on this element then this routine must also compute the affects of those loads on the element s nodes
103. complete it by mirroring MirrorMatrix m and we re done Ef return m m static Matrix LumpedMassMatrix element Element element static Matrix m NULL the local stiffness matrix double factor constant term double I_factor inertia term for rotation double Lie element length if m NULL m CreateMatrix 4 4 if m NullMatrix AllocationError element mass matrix ZeroMatrix m L ElementLength element 2 factor L element gt material gt rho element gt material gt A 2 I_factor factor L L 12 194 CHAPTER 13 ADDING ELEMENTS TO FELT MatrixData m 1 1 factor MatrixData m 2 2 I_factor MatrixData m 3 3 factor MatrixData m 4 4 I_factor return m The last local function is the routine to compute equivalent nodal loads based on dis tributed loads which the user has applied to this element These routines can get kind of tricky because there is a lot of error checking and lots of different possibilities that we have to deal with directions node ordering etc Furthermore this one needs to take care of two different ways in which it might be called The stiffness function calls this routine to set the equivalent forces on its nodes We get these forces by transforming our equivalent force vector to global coordinates and then adding the terms of this global vector onto the eq_force ar
104. con vection acts over the circumferential surface area of the rod If the node values are equal in the two pairs then the convecting surface is taken to be an exposed end of the rod 4 3 2 Constant Temperature Gradient CTG element The ctg element is a thermal analog to the CST triangular elements for mechanical prob lems The temperature is assumed to vary linearly over the element As with rod elements convection on ctg elements is specified using distributed loads Only the three edges are handled The node magnitude pairs of the values definition of the load are used to specify the edge on which the convection is taking place and as for the rod the convection coef ficient and free stream temperature For a ctg element the definition values 1 20 2 50 specifies a convection coefficient of 20 a free stream temperature of 50 and that the convection acts on the edge defined by local element node numbers 1 and 2 Chapter 5 The felt Application 5 1 Using felt The basic way to solve a problem with felt is simply to type felt foo f1t on the com mand line where foo f1t is the name of a FElt input file There are several command line options however which offer additional capabilities version help debug preview renumber table display the current version number of felt and exit display a brief help message which lists all of the available command line options will generate a FElt file that if all
105. corduroy only generates the node and element sections of a FEIt file you still have some editing to do before you can call the problem completely defined and actually try to solve 1t with felt or velvet In general you will have to define forces and constraints to assign to nodes and material properties and possibly distributed loads to assign to elements Remember the material and constraint specification are only a convenience These two names will simply be assigned to the first element and node respectively and all the following elements and nodes by default You still need to actually define those objects and assign any different objects to appropriate nodes and elements To do all this defining and assigning you can either use your favorite text editor or you can load the problem into velvet and do it all from there One thing to keep in mind whenever you use corduroy to generate a problem for you is that corduroy is not very good at node numbering Using the renumbering features of either felt or velvet is highly recommended with corduroy generated problems In general the problems that you will use corduroy for will be quite large and thus the reduction in memory and solution time would be significant even if node numbering had been intelligent 9 4 INCORPORATING OUTPUT INTO A FELT FILE 103 to start with the reductions for large problem with initially very bad numbering can be remarkable Chapter 10 The burlap Application
106. ction body even if the include should be inside a 150 CHAPTER 11 THE BURLAP SYNTAX find_dofs if problem mode amp static then K assemble F construct_forces Kc zero_constrained K Fc zero_constrained F d solve_displacements Kc Fc else if problem mode amp transient then K assemble M C D integrate_hyperbolic K M C else if problem mode amp modal then K assemble M C Ke remove_constrained K Mc remove_constrained M Cc remove_constrained C l compute_modes K M X else if problem mode amp static_thermal then K assemble F construct_forces Kc zero_constrained K Fc zero_constrained F d solve_displacements Kc Fc else if problem mode amp transient_thermal then K assemble M D integrate_parabolic K M end Figure 11 1 Solving problems with the finite element intrinsic functions 11 5 INTRINSIC FUNCTIONS 151 function body If s does not name an absolute path i e starts with or then the environment variable BURLAP PATH is used to search for the file named by s The variable should be a colon separated list of directories The directories are searched from left to right for a file first named s and then s b If the variable BURLAP PATH is not set then only the current directory is searched length X Returns the length of the object X If X is a matrix then it returns the number of elements in X If X
107. d and element definition routines do not calculate a mass matrix Also note that because brick elements are the only solid elements in the FElt library they are also the only elements which are not fully supported by velvet both in terms of problem definition and in terms of post processing 40 CHAPTER 4 THE FELT ELEMENT LIBRARY 4 2 8 Axisymmetric elements The axisymmetric element is a three node triangular element based on linear shape func tions The axis of symmetry revolution is always the y axis Distributed loads can be directed in either the axial y or radial x directions 4 3 Thermal analysis elements Thermal elements only have one degree of freedom per node the nodal temperature Pre scribed and initial temperatures are assigned using constraints Tx and iTx Heat sources are specified using nodal forces Fx For uniform heat sources simply apply the same force to all nodes 4 3 1 Rod element The rod element is a two node straight line element for thermal analysis Convection sur faces are specified using distributed loads The direction of the load does not matter and does not need to be specified The node magnitude pairs of the values definition of the load are used to specify the surface on which the convection is taking place and the con vection coefficient and free stream temperature For example values 1 20 2 50 specifies a convection coefficient of 20 a free stream temperature of 50 and that the
108. d alone application velvet s current post processing capabilities include plotting the displaced shape two dimensional color contours of stress and displacement for planar elements animation of dynamic struc tural simulations line plots of for time and frequency domain results and graphical pre sentation of mode shapes Automated element generation for a FElt problem is provided for both simple grids of line quadrilateral and brick elements and arbitrary meshes of triangular planar elements The latter capability is derived from J R Shewchuk s Triangle mesh generation routine You can interface this functionality either graphically through velvet or through a separate command line application called corduroy that has its own input file syntax much like the regular syntax for FElt problems 1 3 FELT WHAT IT CANNOT DO FOR YOU 3 Support applications are provided for file format conversion and unit conversion and problem scaling patchwork can translate between the standard FElt syntax and several other common graphical description formats yardstick can be used to scale numerical quantities within a FElt file including special options for conversion between different types of units FElt should be able to handle most types of linear static and dynamic problems that you throw at it but there are no guarantees Most elements allow arbitrary oriented dis tributed line loads Displacement e g settlement of support and force e g
109. d material databases 2 2c be oaa HEE ES 204 B The GNU General Public License 205 B Preamble a ee oe ke ee eR A KS ee Rw e Oe Bee oe ee 205 B 2 Terms and Conditions 22222242 44 592 54455485 a 206 B 3 Howto Apply These Terms o 211 CONTENTS xi References 213 List of Figures 2 1 22 2 3 2 1 3 2 ENE 3 4 4 1 4 2 4 3 4 4 6 1 6 2 6 3 6 4 6 5 6 6 TA Static equilibrium of a general linear spring Static equilibrium of two springs in series ooa Dynamic equilibrium of a spring mass dashpot system Example loading functions og ek oaao ee Dew ee BH lt An example of a complex distributed load A mixed element problem with distributed loads Transient analysis of a simple frame Structure Sign convention for local forces on a beam element Node numbering scheme for nine node isoparametric planar element Sign convention for force resultants on an htk element An example of valid node ordering on a brick element Main WinFElt editor with a sample problem Thetextoutput Dox s s n ao ee hee AAA AA A The GOMOD control DEE lt 3 sels se moai AAA BOHR RON A color contour plot showing the principal stress component in a static problem using constant strain triangular elements Main velvet drawing area with a
110. d or new versions of the Gen eral Public License from time to time Such new versions will be similar in spirit to the present version but may differ in detail to address new problems or concerns Each version is given a distinguishing version number If the Program specifies a version number of this License which applies to it and any later version you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation If the Program does not specify a version number of this License you may choose any version ever published by the Free Software Foundation If you wish to incorporate parts of the Program into other free programs whose distri bution conditions are different write to the author to ask for permission For software which is copyrighted by the Free Software Foundation write to the Free Software Foundation we sometimes make exceptions for this Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally NO WARRANTY BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE THERE IS NO WARRANTY FOR THE PROGRAM TO THE EXTENT PERMITTED BY AP PLICABLE LAW EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND OR OTHER PARTIES PROVIDE THE PROGRAM AS IS WITHOUT WARRANTY OF ANY KIND EITHER EXPRESSED OR IM PLIED INCLUDING BUT
111. d we want to solve for the deformations Now however our applied force is a vector our structural stiffness is a matrix and to solve for the deformations we must now solve a linear system of equations The stiffness matrix on the left hand side of equation 2 3 is just the element stiffness matrix in the finite element method When we want to solve a problem of static equilibrium for a structure that is more complex than our simple spring all we have to do is stick a bunch of springs together and then linearly superpose the contributions from each element s stiffness matrix into our global stiffness matrix Consider the case of two springs in series as in figure 2 2 Each individual spring has a stiffness matrix like the one on the left hand side of equation 2 3 The system now has three degrees of freedom xa xp Xc and thus we know that our global stiffness matrix will be 3 x 3 We assemble the global stiffness matrix construct the superposition of all the element stiffness matrices by considering which degrees of freedom each spring affects spring 1 affects the deformations x and xp spring 2 affects x and xe We know then that the stiffness at b in our global stiffness matrix 2 3 TRANSIENT STRUCTURAL ANALYSIS 7 will have contributions from both spring 1 and spring 2 The global stiffness matrix is k k 0 K ky ky k2 ko 2 4 0 Kka k2 The two boxes indicate exactly how the individu
112. d with the Hilbert Hughes Taylor amp correction for improved accuracy with numerical damping The return value is a matrix D of nodal displace ments with each column of D corresponding to a single time step If the nodes of the FElt problem have been renumbered then p should be used to specify the permutation vector as returned by the renumber_nodes func tion The sizes of the matrices must be consistent with the definition of the current problem Compact column matrices are expected but full symmet ric matrices will be accepted by coercing them to compact column matrices The DoF related fields of the problem structure must be initialized as by calling find_dofs integrate_parabolic K M p length e local_dof Solves the discrete parabolic differential equation Mv Kd f using a gen eralized trapezoidal method If the nodes have been renumbered then p should be used to specify the permutation vector The sizes of the matrices must be consistent with the definition of the current problem Compact column matrices are expected but full symmetric matrices will be accepted by coercing them to compact column matrices The DOF related fields of the problem structure must be initialized as by calling find_dofs Returns the length of the element e The element must be a linear element 1 e e definition shape should be amp linear g 1 Returns the number of the node corresponding to the global DOF g If l is spec
113. ding on the type of output you may have the option of saving in one of several different file formats Time displacement wireframe structure and mode shape plots can be saved in either XWD or PostScript format Text output can be saved as an ASCII file Color contours of stress and displacement can be saved in PPM or encapsulated PostScript EPS format 8 1 SOLVING A PROBLEM WITH VELVET 89 Velvet Line Plot dynamic frame analysis dismiss sa Figure 8 2 A time displacement plot in velvet You can leave all output windows up as long as you like If you want to unview one at some point in your work simply click dismiss For a transient analysis problem you can build a special case simulation for animation simply by selecting animate from the solutions menu The solution that is constructred is special in that the displacement of all nodes in the x and y and possibly z translational DOF will be automatically recorded for each time step This is in contrast to the normal mode of solution for a transient analysis problem in which only specially requested nodes and DOF are recorded for output at the end of the simulation Once this complete table is built the animate window Figure 8 6 will pop up and you can use the playback controls to determine the speed and direction of the animation The buttons that look like rewind and fast forward lt lt and gt gt are really the speed controls Holding them
114. e all the regions are defined velvet will automatically begin generating the mesh 7 10 Keyboard interface mechanisms 7 10 1 Keyboard shortcuts As in most graphical environments the pointer can only do so much for you often it will be easier for you to use the keyboard True you could get away with using the mouse for everything but in the long run you will work more efficiently if you learn the keyboard interface mechanisms There are two basic ways that you can use key presses in place of mouse clicks The first are commonly called keyboard shortcuts This means that rather than selecting solve from the problem menu you can simply press ctrl v A complete list of these shortcuts is given in the quick reference table at the end of this section 7 10 KEYBOARD INTERFACE MECHANISMS 83 The second common way to use the keyboard is to move through groups of buttons and text fields with tab to change the input focus and use space to activate the button or change the toggle state This is a very convenient way of working with dialog boxes like the object dialogs forces materials constraints loads the file selection dialog the node and element dialogs the configure dialog the triangular mesh parameter dialog and the solution dialog When one of the dialogs has the window manager focus there are a few special keyboard shortcuts that you can use as well ctrl h is equivalent to help return is accept or oka
115. e as those that would be used in a normal solution for that problem in transient analysis mode except for the nodes and DOF As mentioned above an animation will automatically solve for all nodes at the x and y and z if the problem is 3d translational DOF The magnification of the displacements during the animation can be controlled via the same magnification control that is used in structure plots Likewise for 3 d animations the 3 d drawing parameters will be taken from the controls in the wireframe control dialog box 95 8 3 CONTROLLING THE POST PROCESSING 96 CHAPTER 8 POST PROCESSING WITH VELVET Figure 8 10 The wireframe plotting control dialog box Chapter 9 The corduroy Application 9 1 Introduction corduroy 1s a command line application for automatically generating nodes and elements It takes a text input file and generates FElt format nodes and elements for inclusion into a FElt problem Like felt then it is a command line interface to some of the same functionality that you can get in velvet However unlike felt and velvet which share their file syntax the standard FElt syntax corduroy has its own special syntax to describe the way elements are generated 9 2 The corduroy syntax 9 2 1 Specifying basic parameters corduroy allows you to generate elements in five different ways along a single line as a three dimensional grid of line elements as a two dimensional grid of four node plana
116. e complicated example is the definition structure for a beam struct definition beamDefinition beam beamEltSetup beamEltStress Linear 2 2 3 O ly Di Oy 0 O 3d The difference between this definition and that for the foo element is that while beams affect three DOF the third DOF is Rz position six in the DOF array Also we ll want to keep the element stiffness matrix around after we assemble it into the global stiffness matrix beams use it for internal force calculations so we set element gt retainK to 1 true For the general four to nine node isoparametric element for plane stress the definition structure is struct definition iso2d_PlaneStressDefinition iso2d_PlaneStress iso2d_PlaneStressEltSetup iso2d_PlaneStressEltStress Planar 9 4 2 0 1 2 0 0 0 0 O hi Here the basic type is planar rather than linear the total number of nodes is nine the number of nodes which define the element shape is four i e the first four nodes com pletely define the geometric shape of the element and the number of DOF per node is two Two dimensional isoparametric elements only affect translations in the x and y directions positions one and two in the DOFs array There is no need to retain the element stiffness matrix after assembling so element gt retaink is set to O false 174 CHAPTER 13 ADDING ELEMENTS TO FELT The definition structure is the only tricky part of setting up a new element On
117. e given dialog box The advantage to this mode of working with objects is that these dialogs can be left up throughout your work session You can have them up and easily change the active object for a given type There is no need to be constantly popping things up selecting something or changing something and then popping it down again only to need it again in 30 seconds If you re sure you won t be needing them after some initial setup you can safely dismiss them if you don t like too much screen clutter 7 5 DRAWING AND DEFINING A PROBLEM 71 Materials Name E 2 1e 11 A 4 76e 07 steel Y help accept dismiss delete new copy Figure 7 6 The material dialog box 7 5 1 1 The material dialog The object specific information for a material property is simply entered through the labeled text boxes to the right of the material list Figure 7 6 Additional help regarding each property including a list of which elements require that property to be defined can be gotten by clicking on the label for the given property 1 e click on E for a description of Young s modulus If a field is left blank that property will be taken as 0 0 7 5 1 2 The constraint dialog The six toggle buttons shown in Figure 7 5 allow you to specify which DOFs are con strained by a boundary condition If a toggle button is engaged the corresponding text field will also be used in constructing the boundary condition defin
118. e message which again should look like the syntax for printf the program will automatically exit AllocationError element string This is a simple little convenience routine for fatal errors during mem ory allocation in the element routines The message displayed will be allocation error computing element n string where nis the ele ment number and st ring is some short descriptor of the program location 1 e stiffness matrix 13 5 Putting it all together Once the routines have been written you ll need to update the Makefile for the ele ment library and the initialization routine that tells FElt applications what kinds of el ements are available by default Add the filename of your new element to the OBJS line in lib Elements Makefile You can type make in that directory to make sure the code compiles Once you think you ve got all the bugs worked out and the new element library is made you need to add two lines to the initialization procedures in lib Felt initialize c To the list of extern struct declarations in that file add a line that looks like extern struct definition fooDefinition and to the list of AddDefinition function calls in the function add_all_definitions add a line that looks like AddDefinition amp fooDefinition replace fooDefinition with whatever variable name you gave to the definition structure for your element of course Once those changes are made just do a make in lib Felt and then a make
119. e node and element arrays around is all the information that a routine will need or even just passing the elements around as they contain pointers into the nodes array For those unfamiliar with C here s a quick and dirty lesson that might help you read the code with a little more comprehension Basically in C you can declare new types of variables including entire data structures What this means is that we can define a structure called an Element which has several different members think of each member as a different field or record in a database When we declare a variable to be of type Element that variable then contains all these different fields and we can get at all the information for that element through that one variable For instance in FElt an element structure contains all of the following Element unsigned number this element s global number xy Node node array of pointers to element s nodes EOL Matrix K element stiffness matrix ef Definition definition definition structure type of element Material aterial pointer to material property E Distributed distributed 4 array of distributed loads EJ unsigned numdistributed number of distributed loads assigned Ef Stress stress element stresses hf unsigned ninteg number of integration points xy Several of these members are structures themselves Node Matrix Distributed Definition Material and Stress are other structures us
120. eVector 4 eq_global CreateVector 6 if equiv NullMatrix eq_global NullMatrix AllocationError element equivalent nodal loads ZeroMatrix equiv count 0 Again we want to do as much error checking and descriptive error reporting as possible Seem like overkill It probably is but it s not hurting anybody either xj if element gt numdistributed gt 2 error Timoshenko beam element d has more than 2 distributed loads element gt number count L ElementLength element 2 if L lt TINY error length of element d is zero to machine precision element gt number count for i 1 i lt element gt numdistributed i if element gt distributed i gt nvalues 2 error Timoshenko beam element d must have 2 values for load element gt number count 196 CHAPTER 13 ADDING ELEMENTS TO FELT We only want to deal with loads in the perpendicular LocalY direction this is a very simple instantiation of this element after all ay if element gt distributed i gt direction LocalY amp amp element gt distributed i gt direction Perpendicular error invalid direction for element d distributed load element gt number count make sure that the user isn t try to apply part of this load to a non existent node some local node other than number 1 or 2 Pel for j
121. eam3d timoshenko CSTPlaneStrain CSTPlaneStress iso2d_PlaneStrain iso2d_PlaneStress quad_PlaneStrain quad_PlaneStress htk brick rod and ctg A FElt problem is not limited to one element type the routines for assembling the global stiffness matrix takes care of getting the right parts of the right element stiffness matrices into the global matrix Each element must have a list of nodes to which it is attached The node list is defined with the nodes statement The length of the list inside the square brackets varies with element type but must always contain the full number of nodes which the element type definition requires see chapter 4 for a complete definition of what each type requires A material property must also be assigned to every element with a material statement If the material is never specified an element will take the same material property as the previous element If nothing ever gets assigned to an element the default material property will have zeros for all of its characteristics Chances are this is not what you want Finally an element can have up to three optional distributed loads Each load is assigned with a separate load assignment Each element type may treat a distributed load differently so you should be careful that the name given for the loads on a given element match the names of distributed loads which are defined in a manner conformant with what that element type is expecting 3 2 4 Material pro
122. ed and you are putting the finishing touches on the problem definition 9 2 2 Generating elements along a line The simplest case of element generation is a line which is divided up into an arbitrary number of elements You might use something like this if you were examining the accuracy of a cantilever beam model and wanted to use successively higher numbers of elements from one case to the next The corduroy specification for a line section looks like this line element type beam start 0 0 0 end 5 9 5 number 20 This example would generate twenty beam elements all lying along the line from the origin to a point at x 5 y 5 z 5 in rectangular Cartesian coordinates The default element type for both line and grid generation is truss Valid type names are the same as in a standard FEIt file see chapter 3 9 2 3 Generating a grid of line elements Another relatively simple case is when you want to generate a three dimensional grid of line elements An excellent example of when this might be useful is for a model of a steel frame structure The specification for a grid is grid element type beam 0 0 0 end 100 500 100 x number 4 y number 20 z number 4 start 9 2 THE CORDUROY SYNTAX 99 This example would generate a three dimensional frame structure with four bays along both the x and z axes and 20 bays along y axis 9 2 4 Generating a grid of quadrilateral planar elements The case of a simple grid
123. ed before many of the built in finite element functions may be called burlap ensures that the arrays are kept 114 CHAPTER 10 THE BURLAP APPLICATION Field name Description name name of constraint constraint constraint vector ix initial displacement vector VX initial velocity vector ax initial acceleration vector dx boundary displacement vector iTx initial displacement along x axis ix 1 iTy initial displacement along y axis ix 2 iTz initial displacement along z axis ix 3 iRx initial rotation about x axis ix 4 iRy initial rotation about y axis ix 5 iRz initial rotation about z axis ix 6 Vx initial velocity along x axis vx 1 Vy initial velocity along y axis vx 2 Vz initial velocity along z axis vx 3 Ax initial acceleration along x axis ax 1 Ay initial acceleration along y axis ax 2 Az initial acceleration along z axis ax 3 Tx translation along x axis dx 1 Ty translation along y axis dx 2 Tz translation along z axis dx 3 Rx translation along x axis dx 4 Ry translation along y axis dx 5 RZ translation along z axis dx 6 Table 10 6 Fields of a constraint object Field name Description name name of distributed load direction load direction num_values values number of values array of values node magnitude pairs Table 10 7 Fields of a distributed load object 10 3 BURLAP AND FELT 115 Field na
124. ed in FEIt Because most of the FElt routines access the structures as pointers that is usually when the code references a structure it is actually referencing the address of that structure in memory this is something you shouldn t need to worry about to simply read and un derstand the mathematics the usual way that you will see code that refers to elements is like this element gt node 1 gt x or element gt material gt E The first of these examples gets the value of the x coordinate of the element s first node not neces sarily globally numbered node 1 The second gets the value of the Young s modulus of the material assigned to this element If the array of elements is under consideration you might seem something like this element 3 gt stress 1 gt values 1 This will be the value of the first stress component at the first integration point on element 3 The 12 2 ELEMENTARY C PROGRAMMING 161 other common data structures in FElt are A reaction force typedef struct reaction double unsigned unsigned Reaction force node dof Element stress typedef struct stress double x double y double z double values Stress An element definition typedef struct definition char name setup li stress shape dofs 7 numnodes shapenodes numstresses numdofs l retainK
125. ee of freedom systems If we form a matrix U of all the eigenvectors ul then we can compute the modal mass and stiffness matrices as M UTMU 2 13 UTKU 2 14 The modal matrices have the remarkable property that they are diagonal and thus if we form a transformed system of coordinates q U x 2 15 and a transformed force vector Q U F 2 16 we can write our matrix equation of motion as n uncoupled single degree of freedom equa tions Migi Kigi Qi i 1 n 2 17 We say that M and K are the modal mass and stiffness in the ith mode They are simple the ith entries along the diagonals of the modal mass and stiffness matrices We can further 10 CHAPTER 2 FELT ANALYSIS TYPES simplify the resulting equations of motion by making use of the fact that the eigenvec tors can be arbitrarily normalized they are only fixed to within an arbitary multiplicative constant by equation 2 11 If we choose an appropriate normalization then we can fix it so that M 1 the identity matrix The appropriately normalized mode shapes are called orthonormal modes Because FElt uses a Rayleigh damping model we can also construct a diagonal damp ing matrix C U CU RmM R 2 18 Also because the motion in each mode is now just like a single degree of freedom motion we can use concepts from the theory of single degree of freedom oscillators to help us choose Rg and Rm The damping ratio in the ith mode is simply Ci i
126. eg to determine the number of points in your element for which you want to calculate stresses you can call this routine to allocate all of the necessary memory for stress structures and the arrays of stress values in those structures SetEquivalentForceMemory element This routine will allocate space for the eq force array on an element s nodes if such an allocation has not already been done from a previous ele ment MultiplyAtBA C A B Given matrices A B and a pre allocated destination matrix C this routine will form the matrix product of A transpose B A This function is par ticularly useful for doing the multiplication TtKT without any temporary storage and without ever explicitly forming the transpose Finally in order to insure a consistent error protocol across the different interfaces to FElt there are a few error routines which your routines should call when they encounter trouble either of the recoverable or irrecoverable variety 13 5 PUTTING IT ALL TOGETHER 179 error error_message varl var2 This is the general routine for reporting a non fatal recoverable error The calling syntax is exactly like that for printf There is a format string error_message and a variable length list of variables which are substi tuted into the message Fatal error message varl var2 This is the routine for non recoverable errors Generally we use this routine when memory allocation fails After displaying th
127. en the appropriate action to be taken just as if you had selected solve or animate from the main solutions menu 8 3 Controlling the post processing 8 3 1 Controlling contour plots The controls for color contour plots are available by selecting contouring from the post processing menu On the dialog Figure 8 8 there are identical controls for stress and displacement plots On the stress side the component specifies which stress component 8 3 CONTROLLING THE POST PROCESSING 93 Figure 8 7 The analysis parameters dialog box Figure 8 8 The control dialog box for contour plots will be plotted This number is element specific and must be a valid index in the stress vec tor for each element type in the problem Consult Table 5 1 for details on what the stress vector consists of for each element type For displacement plots the component is simply the displacement DOF that you want to see plotted In general this should be one of the active global DOF for the current problem Other contouring controls include toggles for histogram equalization and element boundary overlay Histogram equalization is a standard technique in image processing for enhancing the contrast of images If the overlay elements toggle is checked then the outline of all the elements in the problem will be drawn in black on top of the color im age An example of stress contours rendered here in greyscale for the wrench example pictured earlier is show
128. ents are identical to those of X provided that X is a symmetric matrix Matrices are normally full matri ces with the size of an m x n full matrix requiring O mn space The space required by a compact column matrix is approximately equal to the num ber of non zero entries Most of the FElt functions expect compact column matrices The compact representation of a scalar is itself Returns the determinant of X X provided that X is a nonsingular matrix The determinant of a scalar is itself Returns a column vector A containing the eigenvalues of X provided that X is a square matrix Therefore Vi 3y Xy Ajy If a variable V is specified and X is asymmetric matrix then V will contain the matrix of eigenvectors on return Each column of V will contain a single eigenvector Therefore Vi XV AjV If V is missing or X is not a symmetric matrix then only the eigenvalues are returned Returns an identity matrix of size m x n Imn If n is omitted then an m x m matrix is returned Both m and n must be scalar expressions that are greater than zero Returns the inverse of X X which must be either a nonsingular matrix or a non zero scalar la Xr by Up P Computes the LU decomposition of X which must be a nonsingular ma trix The return value is a row permuted superposition of L and U with the diagonal of L not being stored since L is unit lower triangular Thus the result is not very useful However if L is specified
129. er than meshing any sort of complicated geometry and most simple geometries even by hand When triangles are being generated a form will pop up Figure 7 13 for you to define the parameters of the generation The parameters are the target number of elements 82 CHAPTER 7 THE VELVET APPLICATION reremen 2 0 o eel tay cance okay ES Figure 7 13 The form for defining triangle generation parameters that you want to generate the amp area constraint factor and the number of holes in your generation region The area constraint factor is defined as OA region AE 7 1 target where Aregion iS the total area of the generation region and Aex is the maximum area of a generated element Once the parameters are defined click okay to begin defining the boundaries of your problem Regions are defined by putting down marker points in the drawing area they must be put down in a counter clockwise order To finish the boundary click on the quit button If the number of holes was greater than zero then each hole must be defined just like the boundary was defined by laying down a series of marker points but in clockwise order Each hole is finished by clicking the quit button The generation process can be aborted by clicking the abort button Typing shift bkspc will delete the last marker point that was laid down thus allowing you to correct mistakes without restarting the entire sequence Onc
130. er the global mass and stiffness matrices have been completely assembled C RK RmM 12 2 12 4 4 Compact column representation All of the global matrices are assembled directly into a compact column representation This representation tries to minimize storage requirements by taking each column of the matrix and only storing the entries from the first non zero entry to the diagonal In this scheme the following 6 x 6 would require a vector of length 14 to store as opposed to a matrix with 36 entries x 110000 eT OTTO x x 1 0 x 01 x X x Assembly is done in two passes through the elements The first pass is used to optimize the storage scheme by finding out just how tall each column needs to be Given this in formation vectors are actually allocated to hold the global matrices This first pass is also used to construct a vector of diagonal addresses For each row of what would be the full matrix this vector contains the position in the compact form of the diagonal of that row For our above example that vector would be Wie es ee IA The second pass of the assembly process actually inserts the appropriate contribution from each element into these vectors Any routine that needs to access the global stiffness or mass matrices can behave just as if they were full two dimensional matrix representations We achieve this transparency through calls to a function which uses the information in the diagonal addresses array
131. esents a key or key combination to press as in press Return to con tinue ACKNOWLEDGEMENTS XIX Acknowledgements We would like to acknowledge the work of the following people or groups Different bits and pieces of their work have either made it possible for us to develop FElt or have contributed to making FElt a more functional and powerful system e Everyone who has ever worked on the Linux GNU X11 and XFree86 projects We worked almost exclusively under Linux using gcc as a compiler The X11 project provided a powerful and flexible graphical environment and the folks at XFree86 made it possible for us to use X11 on our Linux boxes e Barry Joe developed the Geompack code for triangular mesh generation that we used in earlier versions of the program The new triangular mesh generator is Triangle by Jonathan Shewchuk e Some of the ideas for 3d structure plots are based on the way gnuplot by Thomas Williams and Colin Kelley does it e The code to generate PostScript graphics files is based on the code from xmgr by Paul J Turner The basic look of time displacement plots is also based on the way that xmgr would have drawn them because we ve always liked the way results from xmgr looked e XWD dumps are produced using the same code as in the actual xwd application The man page says it was authored by Tony Della Fera and William F Wyatt e Encapsulated PostScript image files are created using code from pn
132. for the use of macro definitions and include files within a FElt input file An example of this capability is the international translation files which allow you to specify a FElt input file in a language other than English These translation files are nothing more than include files which contain define statements which map the non English terms to what FElt actually expects the regular English terms An input file that takes advantage of these powerful features might look like this tinclude german trn define map x x cos x cos x sin x sin x problembeschreibung titel A cantilever beam knoten 3 elemente 2 knoten 1 x 0 y 0 krafte point_load zwangsbedingung pin 2 x map 240 y 0 beam elemente 1 knoten 1 2 material steel materialeigenschaften steel e 30e6 ix 100 a 10 kraefte point_load fy 1000 zwangsbedingungen pin tx c ty c rz u ende The C preprocessor options and capabilities are described more fully in the felt 1fe manual page The international translation capability is discussed more fully including current translation tables in appendix A 44 CHAPTER 5 THE FELT APPLICATION 5 2 Solving a problem As mentioned above the easiest way to solve a problem is to simply do felt foo flt at the shell prompt If you want to save the output it is easy to simply re direct standard output to a file by doing felt foo flt gt foo results Th file foo results could then be printed
133. for their element The static declaration makes them private to this file There is nothing magical about them They could be called anything your element may not use any local functions etc etc It s all a matter of preference and style Matrix LocalK Matrix TransformMatrix Matrix LumpedMassMatrix Matrix ConsistentMassMatrix int EquivNodalForces The next thing we want to do is define the element setup routine which will define element gt K for every Timoshenko beam element All this function really does is set some memory call a few functions and do a few matrix multiplications The brute force filling in of matrices occurs in our own private routines The element setup function the one that the general routines actually call to define element gt K for CHAPTER 13 ADDING ELEMENTS TO FELT Timoshenko beams We ll break it up a little more for our own internal purposes and call some functions of our own to actually fill out the guts of the thing 7 int timoshenkoEltSetup element mass_mode Element element char mass_mode int count a count of errors encountered Matrix T transform matrix Matrix khat local coordinate stiffness matrix Matrix mhat local coordinate mass matrix Since we re nice and we like to do as much error checking as possible we ll also check to make sure that all necessary ma
134. g for parentheses on the offending line type error in expression typehas no such field The field index operator was applied to an object of the specified type but the object has no such field You can generally locate this error by searching for a period on the given line type error in expression typel operator type2 The specified operator does not allow operands of the specified types The way to fix this error is to determine what operand types are allowed as discussed in Section 11 4 and determine why one or more of your operands are of the incorrect type parse error before string At some time at or before string the expression syntax was illegal Often string is the cause of the error You should now be back at the top level prompt in which case you can correct your error and try again size mismatch in expression a x b operator c x d The types of the operands were legal for the specified operator but the sizes of the operands were illegal The legal operand sizes are discussed in Section 11 4 Chapter 11 The burlap Syntax burlap has a rich variety of operators and functions for manipulating matrices scalars strings and FElt objects Matrices are denoted by square brackets with matrix elements separated by commas and matrix rows separated by semicolons Scalars are floating point numbers such as 3 14159 or 123 Any scalar can also be converted to a matrix with one row and one column Strings are enclosed in double
135. g help foo will give you detailed help on topic foo In short burlap works just like most other interactive programs including csh gdb gnuplot matlab octave and gs 10 3 burlap and FElt You can load a FElt file into burlap by either specifying the file on the command line with the felt flag or by using the felt function 1 felt truss f1t If everything goes okay then the function should simply return silently Several vari ables are now automatically defined as shown in Table 10 1 Variable Description nodes array of nodes elements array of elements dofs_pos global DOF positions dofs_num global DOF numbers problem problem definition structure analysis analysis parameters structure Table 10 1 Finite element related variables FElt maintains the global DOFs in two vectors The first vector dofs_pos is an alias for problem dofs_pos and contains the position number for each of the six DOFs A zero entry in the vector indicates that the DOF is not active An example vector for a problem consisting entirely of beam elements would be 120003 since a beam element allows translation along the x axis and y axis and rotation about the z axis The dofs_pos vector can be used to determine if a given DOF is active and which global number it is assigned The second vector dofs_num is an alias for problem dofs_num and contains the num bers of the active DOFs The length of the vector is equal
136. g the Program Whether that is true depends on what the Program does 1 You may copy and distribute verbatim copies of the Program s source code as you receive it in any medium provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty keep intact all the notices that refer to this License and to the absence of any warranty and give any other recipients of the Program a copy of this License along with the Program You may charge a fee for the physical act of transferring a copy and you may at your option offer warranty protection in exchange for a fee 2 You may modify your copy or copies of the Program or any portion of it thus forming a work based on the Program and copy and distribute such modifications or work un der the terms of Section 1 above provided that you also meet all of these conditions a You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change b You must cause any work that you distribute or publish that in whole or in part contains or is derived from the Program or any part thereof to be licensed as a whole at no charge to all third parties under the terms of this License c If the modified program normally reads commands interactively when run you must cause it when started running for such interactive use in the most ordinary way to print or display an announcement incl
137. g up the definition structure BR KIRK KK IK I KR KR kkk RIOR RARA File timoshenko c X Description This file contains the definition structure and is set up functions for a Timoshenko beam element E History V1 4 by Jason Gobat and Darren C Atkinson IRR RR AAR AAI I RR RRA RARA RARA RARA RARA include lt math h gt include allocate h include element h Se HEHE include misc h Here s the definition structure This is a very simple implementation 2 nodes possible effect on 3 global DOF per node We need to prototype the setup and stress functions 13 6 A DETAILED EXAMPLE 181 first thing so we can use them in the definition declaration a int timoshenkoEltSetup int timoshenkoEltStress struct definition timoshenkoDefinition timoshenko the symbolic name used in input files j timoshenkoEltSetup the element setup function timoshenkoEltStress the element stress function Linear The shape of this element E 0 2 nodes per element 2 nodes define the shape it s a line 2 magnitudes in each stress structure 3 global DOF node Lg 25 6 Oy Oy OF DOF 1 is Tx DOF 2 is Ty DOF 3 is Rz 1 retain stiffness after assembling y static static static static static We ll declare these three functions as static because other people might use these same names
138. generalized eigenvalue problem can be written Kx M 12 15 where A and x are the eigenvalue and eigenvector in a given mode We can transform this to standard form A AI x 0 12 16 12 4 DETAILS OF A FEW GENERAL FELT ROUTINES 169 by forming the Cholesky factorization of the mass matrix M QQ and realizing that Q MQ7 I With this factorization we just need to make the substitution x olx and pre multiply both sides of equation 12 15 by Q 7 the problem then becomes QTKO 8 M 12 17 which is now in standard form FElt solves this standard eigenvalue problem by applying the QL method to the tridiagonal form of the transformed coefficient matrix Note that this procedure will not work if there are zeros on the diagonal of the mass matrix the global mass matrix must be positive definite for the Cholesky factorization to be non singular This kind of condition often occurs in problems with lumped mass formulations of the element mass matrices Chapter 13 Adding Elements to FElt 13 1 How to get started Adding additional element types to FElt is meant to be fairly straightforward Coding an element definition is always an excellent way to really understand the ins and outs of finite element analysis A good starting point is probably to look at the source code for the currently available elements not so much because they are mathematical algorithmic wonders but simply because they provide examples of ho
139. gned to make sure that you have the freedom to distribute copies of free software and charge for this service if you wish that you receive source code or 205 206 APPENDIX B THE GNU GENERAL PUBLIC LICENSE can get it if you want it that you can change the software or use pieces of it in new free programs and that you know you can do these things To protect your rights we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights These restrictions translate to certain responsibilities for you if you distribute copies of the software or if you modify it For example if you distribute copies of such a program whether gratis or for a fee you must give the recipients all the rights that you have You must make sure that they too receive or can get the source code And you must show them these terms so they know their rights We protect your rights with two steps 1 copyright the software and 2 offer you this license which gives you legal permission to copy distribute and or modify the software Also for each author s protection and ours we want to make certain that everyone understands that there is no warranty for this free software If the software is modified by someone else and passed on we want its recipients to know that what they have is not the original so that any problems introduced by others will not reflect on the original authors reputations Finally a
140. have two options Clicking new will clear out all of the property fields and the name field and you can name the new material springy and define its properties Alternatively you can click copy and only the name field will be cleared This is simply a convenient way 70 CHAPTER 7 THE VELVET APPLICATION Name free pin roller Init Trans Init Rot Init Accel help accept dismiss delete new copy Figure 7 5 The constraint dialog box to define a new material with properties similar to the previous material All you have to enter are a new name and any changed or additional material properties In either case you need to click accept to register springy as a new material If you still want aluminum as the active material you simply have to select aluminun in the list after you have accepted springy Finally 1f you really meant to refer to aluminum by the name steel you can simply change the name in the field above the list and click accept It is important to note that the basic behavior of these buttons is exactly the same in all four dialogs Only the object specific information discussed in more detail below in the right side of each behaves differently To delete an object you can simply click delete in the appropriate dialog box Note that velvet will not let you delete an object that is currently assigned to an element or node Clicking and holding the help button will display a brief help message for th
141. he backs of our minds Of course all our good intentions are for nought if no one uses FElt If you find something wrong let us know our email addresses are in the legal notice Alterna tively and an even better option you can subscribe to the FElt mailing list by send ing a one line email message that says subscribe felt 1 without the quotes to list serv mecheng fullfeed com If something is not there that you think should be let us know We make no promises but we do try to respond and gear development toward user feedback If you re happily using FElt and have no complaints let us know that too At least then we know that we re on the right track 1 2 CHAPTER 1 INTRODUCTION TO THE FELT SYSTEM 1 2 FElt What it can do for you FElt currently has support for the following element types three dimensional truss or bar one dimensional spring two and three dimensional Euler beam two dimensional Timo shenko beam constant strain triangles CSTs for both plane stress and plane strain anal ysis planar isoparametric elements four to nine node and a separate type for simple four node quadrilaterals again for both plane stress or plane strain analysis a linear axisymmet ric triangular element an HTK plate bending element an isoparametric eight node brick element and a rod and a constant temperature gradient triangle for thermal analysis FElt allows for an arbitrary mixing of element types within a problem The synt
142. he colors and fonts to use for the different objects that appear in the drawing area The label font is used for node and element numbers Also included on the configure dialog are toggle buttons to control the status of snap grid element num bering and node numbering Note that these four toggles are also available by selecting the appropriate entry right on the canvas menu You must click accept to make any of your changes active As mentioned above the snap and grid options act as toggle switches for these common drawing aids For those unfamiliar with these terms grid produces a ruled grid across the drawing area to help you locate yourself as you work and snap insures that mouse selected coordinates will fall on regular fractions i e a snap size of 0 25 insures that all of your selections will be rounded to the nearest quarter unit Note that the grid size the spacing of the rulings on the screen and the snap size the fraction to which your selected coordinates will be automatically rounded are controlled separately by entries in the configure dialog The node numbers option acts as a toggle switch for node numbering in a crowded problem visible node numbers may not be helpful as they can tend to clutter the display 66 CHAPTER 7 THE VELVET APPLICATION min X nede color max X 12 6 elt color min Y 1 6 tool color max Y 11 6 tool font grid 1 2 label font snap 1 2 A node numbers E snap E grid element numbers
143. hear forces along its length The sign convention for the magnitude of the load pairs is positive for loads pointing from local node 1 to local node 2 The spring element included in FEIt is simply the truss element described above without the transformation from local element coordinates to global coordinates What this means is that the element matrices are only 2 x 2 and because of this they must be defined only along the global x axis Distributed loads are not supported by spring elements 4 2 2 Euler Bernoulli beam elements 4 2 2 1 Special case two dimensional element A beam element is a two dimensional two node linear element which can carry in plane axial shear and bending forces The moment of inertia used in stiffness calculations spec ified with Ix is the bending moment of inertia about the cross section x x axis which in 32 CHAPTER 4 THE FELT ELEMENT LIBRARY global coordinates is bending about the z axis Like the truss element described above the beam element in the FElt library is also a standard in many textbooks the following stiffness matrix should also look relatively familiar AE L 0 0 AE L 0 0 0 12E1 L9 G6El L 0 12EI L 6El L f 0 6EI L 4EI L 0 6El L 2ElI L 4 9 AE L 0 0 AE L 0 0 0 12EI1 L 6EI L 0 12E1 L5 G6El L 0 6El L 2E1 L 0 6El L 4El L The lumped mass matrix for the beam element is defined as 10 0 00 0 01 0 00 0 2 PAL O0O 0 4 00 0 AA 4 1 7 joo 0100 oy
144. hematical models we are talking about when we talk about the various analysis types There is of course some very basic information on just how these models work in the context of FEA but we really do suggest that you try some of the excellent textbooks 9 16 13 1 14 that are out there if you want any sort of real background information 2 2 Static structural analysis The basic equation for static structural analysis can be seen as a generalization of Hooke s law for the deformation of a linear spring f kx For a spring if we know the applied force and the spring stiffness k then we can find the deformation as x f k We can generalize this to multiple degrees of freedom by considering the static equilibrium of the general spring in figure 2 1 If the spring is in static equilibrium then at point a we must have a Xp k Fa 2 1 and at point b xp da k Fy 2 2 6 CHAPTER 2 FELT ANALYSIS TYPES 7A 16 a gt aAA NG ee Figure 2 1 Static equilibrium of a general linear spring Hh 16 aC ca _ a Tn ee _ A kl k2 E Le FE Figure 2 2 Static equilibrium of two springs in series These two conditions form a linear system of equations in two unknowns We can rewrite this system in matrix notation as k k a Fy ea l 2 3 k k Xb F The problem is still essentially the same as our simple Hooke s law calculation we know the applied force and the structural stiffness an
145. hey were added Once the nodes are down we need to remember that all four have the constraint free assigned to them it was active when we added them So now we change the active con straint to pin by clicking it in the constraint list select constraint from the apply menu and click on node 1 We make node 3 a roller by selecting roller on the constraint list and applying it to the node via the apply mechanism Similarly for the fixed condition on 74 CHAPTER 7 THE VELVET APPLICATION node 4 We could have gotten around this by simply changing the active constraint before we added each node All we would have needed to do was leave the constraint dialog up and click on the appropriate entry in the constraint list before we added the node that used that constraint Before we can add the elements we need to know how to set the element type Like an active object we have to set a current element type before we can actually add any elements There is a distinction however as we cannot go back and apply a different element type to an element The only way to change the type of a given element is to delete it and add a new element We set the current element by choosing set type from the elements menu This will bring up a dialog similar to the standard file selection mechanism Like the object dialogs the element selection dialog can be left up throughout your work session The current element type can be set simply by clicking on the appropri
146. his means is that if you enter the coordinates of the first point of a line with the keyboard then you will have to enter the second point with keyboard input as well or alternatively if you select the center of a circle with the mouse then you will have to select the radius with the mouse as well Titles and comments can be entered in the drawing area by selecting text from the tools menu option As mentioned earlier the drawing area could then be dumped to an XWD and the result could serve as documentation or a figure in a presentation complete with color annotation Figure 7 11 shows such a dump with examples of the available tools Tool color and font is controlled from the configure option under the canvas menu In addition to text and circles the velvet drawing toolbox currently includes lines arcs polylines and rectangles To draw a line with the mouse you have to click at the start point and hold the mouse button while you drag the line to the end point Rectangles work the same way click and hold while you drag out a box For circles your mouse click will select a center point and as you move the mouse while still holding the button your circle will expand about that center When drawing polylines with the mouse use the first mouse button to select the 7 7 USING TOOLS 79 Types of Tools Available in Velvet circle LES pal LA oe SAET EA ee ME closed poyt ne esas Figure 7 11 Examples of all the to
147. ide a quoted string the default set of completions is not used instead file names are completed 1 include Examples f lt Tab gt fe b find dofs b foo flt foo_d flt 1 include Examples f Using the completion mechanism can save a lot of typing especially when referring to file names Another trick is to define your own aliases for common lines alias v system vi foo b alias print write 1 2 alias vi system vi 3 4 vi foo b 10 6 COMMON ERROR MESSAGES 121 The first alias defines v to be system vi foo b The second is more general and defines vi to be system vi The string will be replaced with the last word on the line A complete description of the history expansion can be found in the documentation for the readlinelibrary and for csh The final alias print is simply a convenient wrapper for the write function If burlap is not compiled with the readline library you still have access to a limited form of aliasing The only expansion string legal in an alias is s which will expand to the entire line except the first word A similar alias to vi above would be system vi s An alternative to writing aliases is to define your own convenience functions If you wish to be able to both edit and include a file then you can write a small function called edit and place it in a file called edit b in your search path as described in Sec tion 11 4 14 function edit s system conc
148. ient from Timoshenko beam theory Cowper 2 provides an approxi mation for based on Poisson s ratio 10 1 v K 4 16 12 11v am if a better estimate is not available This approximation will automatically be assumed if you provide nu rather than kappa in the material property for a Timoshenko beam element Because there is no axial DOF in this formulation you need to be careful about horizontally oriented elements nothing will be assembled into the translational x DOF in these cases so you should be extra careful about constraints The lumped mass matrix for the t imoshenko element looks just like that for the 2 d Euler Bernoulli element eq 4 10 minus the axial DOF of course The definition of the consistent mass matrix varies from one formulation of Timoshenko theory to the next The definition in the formulation that we are using is considerably more complicated than the Euler Bernoulli formulation see 4 for details Distributed loads on t imoshenko elements can only be directed in the perpendicular equivalent to LocalyY direction There are no axial DOF after all The sign conventions for these loads and for internal forces is the same as that for the standard beam element The internal forces calculated will be the shear forces and bending moment at each end 4 2 4 Constant Strain Triangular CST elements Two different CST elements are in the FElt library one for plane stress analysis and one for plane strain analysi
149. ified then it will contain the local DOF on return The number of the node is returned rather than the node object itself since the nodes of the problem may have been renumbered The DOF related fields of the problem structure must be initialized as by calling find_dofs remove_constrained K Removes the rows and columns of K at all DOFs with a fixed boundary condition and returns the new matrix K itself is not modified K should be either a symmetric matrix or a column vector the size of K must be 148 CHAPTER 11 THE BURLAP SYNTAX consistent with the definition of the current problem The DOF related fields of the problem structure must be initialized as by calling find_dofs remove_definition n Removes the element definition named by n If the definition is successfully removed then 0 is returned Otherwise 1 is returned The definition must not be in use by any elements renumber_nodes Renumbers the nodes of the current problem using the Gibbs Poole Stockmeyer and Gibbs King node renumbering algorithms for bandwidth and profile reduction The result is a permutation vector p of the node numbers restore_numbers p Restores the original node numbers of the current problem The permutation vector as returned by the renumber_nodes function is specified by p The return value is null set_up e s Calls the set up function for element e The argument s may be used to specify the mass mode and should be ei
150. in expression scalar matrix 10 a 1 2 3 4 5 6 11 write a 12 a 1 2 0 13 write a 11 4 3 Logical OR operator x or y x y The logical OR operator has two equivalent forms and groups from left to right It returns 1 if either of its operands compares unequal to zero and 0 otherwise The OR operator guarantees left to right evaluation x is first evaluated if it is unequal to 0 the 11 4 OPERATORS 129 value of the expression is 1 Otherwise y is evaluated and if it is unequal to 0 the result of the OR expression is 1 otherwise the result is 0 der f 1 ifx40oryX0 ae a l 0 otherwise Both x and y must be scalar expressions as shown in Table 11 2 It makes little sense to compare a matrix or string with zero for example There are however common ways of mapping a matrix to a scalar Two such ways are provided by the any and every functions discussed in Section 11 5 3 X y Z 2 X or Y scalar scalar scalar Z xory Table 11 2 Type table for the logical OR operator 1 write 1 lt 2 or 5 gt 6 2 write 1 gt 2 or 5 gt 6 3 write 1 or 1 0 4 write 0 or 1 0 stdin 4 exception in expression right division by zero a 1 2 3 write a or 1 oy ul stdin 6 type error in expression matrix in conditional context 7 write any a or 1 11 4 4 Logical AND operator x and y X amp amp y The logical AND operator has two
151. ion applied to each element individually abs x x hypot x y yx2 y ceil x x log x Inx cos x cosx log10 x logx exp x e pow x y x fabs x x sin x sinx floor x x sqrt x yt fmod x y xmody tan x tanx Table 11 15 Intrinsic functions from the math library In the case of the hypot function which requires two arguments the type semantics are the same as that of the addition operator The fmod function is equivalent to the remainder operator the pow function is equivalent to the exponentiation operator 11 5 2 Matrix functions burlap provides a variety of functions for manipulating and factoring matrices All of these functions with the exception of rows and cols functions could be written in burlap itself however the built in versions will execute faster Detailed explanations of the various factorizations can be found in any basic linear algebra text 142 CHAPTER 11 THE BURLAP SYNTAX chol X cols X compact det X eig X eye m inv X X V Returns the cholesky factorization of X The cholesky decomposition of a matrix X is a matrix B such that BB X provided that X is symmetric and positive definite The cholesky decomposition of a scalar is simply its square root Returns the number of columns of the matrix X A scalar is defined to have a single column A synonym for this function is columns Returns a compact column matrix whose elem
152. ion for a distributed load is a symbolic name followed by the keywords direction xxx and values n x m y The direction assignment must be set to one of parallel perpendicular LocalX LocalY LocalZ GlobalX GlobalY GlobalZ radial axial The valid directions for a given element type and what those directions refer to for that ele ment type are described in the individual element descriptions in chapter 4 The values assignment is used to assign a list of load pairs to the named load A load pair is given in the form n x where n is the local node number to which the magnitude given by x applies Generally two pairs will be required after the values token The imaginary line between the two magnitudes at the two nodes defines an arbitrarily sloping linearly distrib uted load This allows you to specify many common load shapes a constant distributed load of magnitude y including cases of self weight a load which slopes from zero at one node to x at a second node or a linear superposition of these two cases in which the load has magnitude y at node 1 and magnitude x y at node 2 This latter case is illustrated in figure 3 2 The definition of this load would be distributed loads load_case_1 direction perpendicular values 1 2000 2 6000 In the future higher order load shapes may be supported by some elements and thus require the specification of more than two load pairs 22 CHAPTER 3 STRUCTURE OF A FELT PROBLEM
153. ion routines When looking through any of the FElt code you will notice that we make use of several convenience functions and macros The most important of these deal with two additional variable types that FElt defines Besides the general finite element type data structures FElt makes available a Matrix and a Vector type Because these types are really structures a reliable way to get at the data is MatrixData a 1 1 or VectorData a 7 These macros simply expand to a gt data 1 1 and a gt data 7 1 The macro method is preferrable simply because the actual definition of the types may change as FElt develops and the macros will make any such changes transparent The following list illustrates how each of the matrix routines can be used In general a b and c represent variables of type Matrix or Vector width and height define di mensions and status is an integer error code Note that the actual operational functions transpose multiply add etc as opposed to create delete can take either matrices or vec tors interchangeably Appropriate dimensions must always match of course All matrices that go into the matrix routines both source and destination matrices must be previously created by one of the matrix creation routines The operational matrix routines as op posed to create and destroy functions all return integer status codes with zero indicating a successful operation non zero return values are defined in include s
154. ions given within it All source code in the FElt system is placed under the GNU General Public License See ap pendix B for a copy of the GNU GPL The authors are not liable for any damages direct or indirect resulting from the use of information provided in this document Contents Foreword xvii MOU this Manual sso c a dua a d a iLa sde db ae xvii Organization of this Manual aoaaa aaa eee eee eee xvii Typographical Conventions 2 aooaa a xviii Acknowledgemenis y as eosin ada AA A deat a a Se ae t a a eh t XIX 1 Introduction to the FElt System 1 Li Intenti ns ci eee eh ee hee eae a LAA G 1 12 PEI Whatitcando Tor yO peo cas ee ok OS ORES ES Hs 2 1 3 FElt What it cannot do for you o e e 3 2 FElt Analysis Types 5 22 Introduction o ea ra ae WEES EERE EE ES 5 2 2 BIE SITGES analysis s sos ccc edaran ig Rew KREG REG KE 5 2 3 Transient structural analysis s s aos 4 406 4 be eee a de we we 7 24 Staticthermal analysis s s sa s sabe sans sanu darra aira aa a 8 2 5 Transient thermal analysis 224 524 4s 45 e 8 26 Mogalanalysis o e es ee ee eek Se Sb SB SSR SSR d ah ira 8 Ok e o a Sem Stave B are Slaw ae Blew Bo ew BO Be EO 10 28 Non nearsta canalysis sosa CER ERS EEE ES 11 2 9 Nonlinear dynamic analysis o 11 111 iv CONTENTS 3 Structure of a FElt Problem 13 31 Inp tfilesyntax cco secreacra enaar e toa eaae kaea 13 SAI General A 13 3
155. iption name name of force force force vector spectrum input spectra Fx force along x axis force 1 Fy force along y axis force 2 Fz force along z axis force 3 Mx moment about x axis force 4 My moment about y axis force 5 Mz moment about z axis force 6 Sfx spectra along x axis spectrum 1 Sfy spectra along y axis spectrum 2 Sfz spectra along z axis spectrum 3 Smx spectra about x axis spectrum 4 Smy spectra about y axis spectrum 5 Smz spectra about z axis spectrum 6 Table 10 5 Fields of a force object tion is simply a scalar value as indicated in Table 10 8 10 3 7 Element definition objects An element definition object is one of the few object types not defined in a FElt file An element definition or simply definition contains all information necessary for a particular element type such as a beam truss etc The fields given in Table 10 9 are discussed at length in Section 10 4 10 3 8 Problem definition The problem definition object contains all information about the current FElt problem with the exception of the analysis parameters Most fields of the problem definition given in Table 10 10 are read only If you want to change the characteristics of a problem you need to edit the input file that defines the problem and load it with the felt function The only exceptions are the DOF arrays dofs pos and dofs_num Since these arrays must be properly initializ
156. irely in the x y plane the three DOF at each node each represent out of plane deformation Rotations are positive in the right hand sense You can generate plate bending triangular elements by using htk elements with the third and fourth nodes being equal just like you can generate triangles from isoparametric quadrilateral elements Mass matrices for htk elements also work the exact same way as the mass matrices for the in plane quad PlaneStress and quad_PlaneStrain elements Figure 4 3 illustrates the sign convention for the internal force resultants calculated for each htk element 4 2 STRUCTURAL ANALYSIS ELEMENTS 39 Figure 4 4 An example of valid node ordering on a brick element 4 2 7 Solid brick element The eight node solid brick element in FElt is based on an isoparametric formulation using linear shape functions The element incorporates all three translational DOF at each node for a total of 24 local degrees of freedom The 24 x 24 element stiffness matrix is evaluated using a fourth order accurate 2 x 2 x 2 Gaussian quadrature rule This formulation results in stress calculations at eight integration points Figure 4 4 shows the local node number ing convention for bricks As the figure illustrates nodes 1 4 and nodes 5 8 must define opposite faces All six independent components of the stress tensor are calculated for brick elements Brick elements are still somewhat limited however in that distributed loads are not handle
157. is a string then it returns the number of characters in X load s Loads a file s of saved variables created with the save function At present this function is not implemented read Reads a line from standard input and returns it as a string Any newline character is discarded The function returns null upon end of file reads Reads a string i e a sequence of characters separated by spaces from stan dard input and returns it as a string A null value is returned upon end of file save s Saves the current set of global variables in the file named by s At present this function is not implemented system s Executes the UNIX command named by s The command is executed in its own subshell The return value of the function is the return status of the command type X Returns a string containing the type of X which may be of any type write Writes its arguments to standard output with no intervening spaces but with a newline at the end of the output The arguments may be of any type A matrix is always written on a series of separate lines The return value is always 0 writes Writes its arguments to standard output with no intervening spaces and with out a newline at the end of the output The arguments may be of any type A matrix is always written on a series of separate lines The return value is always 0 152 CHAPTER 11 THE BURLAP SYNTAX 11 6 User defined functions function name
158. is working well should look exactly like the original input file The generated file represents what the application thinks it was given produces an ASCII rendering of the problem geometry The graphics may not be great but the result is often good enough for a simple sanity check will invoke a Gibbs Poole Stockmeyer Gibbs King automatic node renum bering algorithm for bandwidth profile reduction of the global stiffness ma trix and mass and damping matrices in transient analysis The renumber ing will only affect solution time and memory requirements results will be referenced to the originally assigned node numbers Generally there is no benefit in using this capability for problems with small numbers of nodes disables tabular output for transient and spectral analysis problems 41 42 CHAPTER 5 THE FELT APPLICATION plot generates an ASCII character based plot or plots of results for transient or spectral analysis problems transfer only calculate the transfer functions during spectral analysis The output can be graphical and or tabular depending on the plot and table options eigen in a modal analysis problem only calculate the eigenvalues natural fre quencies and eigenvectors mode shapes before quitting i e do not cal culate modal mass damping and stiffness matrices orthonormal use orthonormal mode shapes to calculate modal mass damping and stiff ness matrices in a modal analysis problem matrice
159. ition If a button is en gaged and the text field is blank that DOF will be fixed if a button is engaged and that field contains the word hinged then that DOF will be treated as a hinge remember hinged DOFs only make sense on certain element types if a button is engaged and the text field contains a number then that DOF will be defined as a displacement boundary condition if a button is engaged and the corresponding text field has a time dependent expression then that DOF will be treated as time varying BC Finally if a button is not engaged then that DOF will remain completely unconstrained Initial conditions for transient analysis problems are defined with the initial displace ment velocity and acceleration entries For initial displacement and velocity conditions empty entries are taken as 0 0 For initial acceleration conditions empty entries are un 72 CHAPTER 7 THE VELVET APPLICATION handlebars seat Plot Force Plot Moment help accept dismiss delete Figure 7 7 The force dialog box defined The distinction is that if all of the initial accelerations are undefined then the mathematical engine will solve for an initial acceleration vector based on the initial dis placement and velocity vectors If any of the initial accelerations are specified in a problem then any others which were left undefined will be taken as 0 0 and the mathematical engine will use this vector as the initial acceleration 7 5
160. l forces for each element A distributed load on a 3 d beam can be directed parallel LocalX is equivalent or in the LocalY LocalZ GlobalX GlobalY or GlobalZ directions A parallel load will be taken as a linearly distributed axial force Like beam elements the sign convention for distributed loads is based on the appropriate coordinate axes system Positive parallel loads point from node 1 to node 2 Positive loads in the local directions point in the direction of of the positive and 2 axes respectively The sign conventions for loads in the global directions follow the global coordinate axes beam3d elements are limited to at most three applied distributed loads 4 2 3 Timoshenko beam element The Timoshenko beam element currently in the library is really intended as a well worked example of how to add an element to the FElt system see Chapter 13 It is limited to in plane behavior and does not support an axial degree of freedom There are lots of approaches to defining an element using Timoshenko beam theory Classic examples can be found in 10 15 The formulation we use is from 4 In all 4 2 STRUCTURAL ANALYSIS ELEMENTS 35 formulations the stiffness matrix is defined as 12 6L 12 6L 2 _ ayy2 y El 6L 4 0 6L 2 6 L a 1 0 13 12 6L 12 6L 6L 2 L 6L 44 6 L In the above equation is defined as the ratio of the bending stiffness to shear stiffness 2 4 15 KGA x is the shear coeffic
161. l worth it for large problems both in terms of memory requirements and in solution speed For both options the same caveat applies as in the discussion of the command line switch in felt for small problems the algorithm probably won t be able to do much if anything in terms of profile reduction in these cases it is probably easiest simply not to worry about either of the renumbering options 7 6 3 Editing elements Editing elements is very similar to editing nodes You can either select edit from the ele ments menu particularly in cases where you will want to choose the element by typing in its number or click on an element or its number with the first mouse button to display the element dialog Figure 7 10 Like the node dialog you can change the displayed elements either with the up and down arrow buttons or by clicking on a different element with the first mouse button Also just like nodes you can click on an element with third mouse button to raise the element dialog and the object dialogs materials and loads with the ob jects assigned to that element already highlighted Again remember that any changes to the assigned objects can affect other elements as well as the element currently displayed in the element dialog The information displayed in this dialog includes the element type the nodes to which this element is attached the material property and distributed loads assigned to this element 78 CHAPTER 7 THE VELVET APPLIC
162. lement gt material gt E element gt material gt Ix element gt material gt kappa element gt material gt G element gt material gt A phi2 phi phi constl element gt material gt rho element gt material gt Ix 30 0 1 0 phi 1 0 phi L const2 element gt material gt rho element gt material gt A L 210 0 1 0 phi 1 0 phi fill out the top half of the mass matrix no need to explicitly integrate of course atrixData 1 1 36 0 constl 70 0 phi2 147 0 phi 78 const2 atrixData 1 1 2 L 15 0 phi 3 0 constl 35 0 phi2 77 0 phi 44 0 L 4 0 const2 atrixData 1 3 36 0 const1 35 0 phi2 63 0 phi 27 0 const2 atrixData m 1 4 L 15 0 phi 3 0 constl 35 0 phi2 63 0 phi 26 0 L 4 0 const2 atrixData m 21 2 10 0 phi2 5 0 phi 4 L L const1 13 6 A DETAILED EXAMPLE 193 7 0 phi2 14 0 phi 8 0 L L 4 0 const2 atrixData m 2 3 MatrixData m 1 4 atrixData m 21 4 5 0 phi2 5 0 phi 1 0 L L constl 7 0 phi2 14 0 phi 6 0 L L 4 0 const2 atrixData 3 3 36 0 const1 70 0 phi2 147 0 phi 78 0 const2 atrixData 3 4 MatrixData m 1 2 atrixData 4 4 10 0 phi2 5 0 phi 4 0 L L constl 7 0 phi2 14 0 phi 8 0 L L 4 0 const2
163. lement textbooks 29 30 CHAPTER 4 THE FELT ELEMENT LIBRARY Element type name nodes Material properties DOFs truss 2 E A rho Tx Ty TZ spring 2 E A rho TX beam 2 E A Ix rho Tx Ty Rz beam3d 2 E A ly 1z J G rho Tx Ty Tz Rx Ry Rz timoshenko 2 E A 1x G kappa rho Tx Ty Rz CSTPlaneStress 3 E t nu rho Tx Ty CSTPlaneStrain 3 E t nu rho Tx Ty quad PlaneStress 4 E t nu rho Tx Ty quad PlaneStrain 4 E t nu rho Tx Ty iso2d_PlaneStress 9 E t nu rho Tx Ty iso2d_PlaneStrain 9 E t nu rho Tx Ty htk 4 E t nu kappa rho Tz Rx Ry brick 8 E nu rho Tx Ty Tz axisymmetric 3 E nu rho Tx Ty rod 2 A Kx c rho T ctg 3 t Kx Ky c rho Tx Table 4 1 Description of the materials required for and local DOF affected by the various element types The material properties in indicate properties that are only required during a non static analysis 4 2 Structural analysis elements 4 2 1 Truss and spring elements Truss or bar elements are the simplest type of element in the library They are two node three dimensional linear elements which are assumed to deform in the axial direction only They might find application in a simple bridge simulation the classic example of truss elements or as springs as in the above example where the spring stiffness k is given by EA L of the truss element The truss element in the FElt library uses the classical linear shape f
164. les mimick command line switches that are available in felt see chapter 5 to control the printing of additional information within the textual output For graphical output the toggles for stress structure and displacement plots simply allow you to automatically invoke these post processing options see below on solution The other two options time displacement plots and mode shape plots are only available during a problem solution i e only by selecting the appropriate toggle in this dialog and then solving the problem The line plot option generates a graphical line plot as opposed to the ASCII plot that you would get with felt output for transient or spectral results A graphical time displacement plot is shown in Figure 8 2 Figure 8 3 is an example of a mode shape plot Note that velvet only draws a single mode at a time and that the regu lar problem geometry is drawn as an underlying dotted line you can cycle forward and backward through the individual modes using the lt and gt buttons Given the settings in our example control box the output for a static model of a bicycle would be the two windows shown in Figures 8 4 and 8 5 The input file for this problem can be found as bicycle boys f1t in the tests directory of the standard FEIt distributions You can save any of the velvet output by clicking save at the bottom of the output window A file dialog will pop up and allow you to name the file which you wish to save to Depen
165. lies all terms by the scalar factor x and then adds y The result is stored in a AddMatrices a b c Adds matrices b and c and stores the result in a SubtractMatrices a b c Subtracts matrix c from matrix b and stores the result in a 13 4 ELEMENT LIBRARY CONVENIENCE FUNCTIONS 177 In addition to the matrix functions we often use several convenient macros to make memory allocation for non matrix types a little easier Examples of these macros are ptr Allocate type size This is equivalent to ptr type malloc sizeof type size 1 e this creates space for ptr to hold size items each of type type UnitOffset ptr The allocation macro creates a matrix or vector that is zero offset Since most FElt arrays are one offset we use this macro to make the change ZeroOffset ptr If ptr has been previously unit offset this macro will reset it zero offset You must use this macro before de allocating a pointer that has previously been unit offset Deallocate ptr This frees any memory associated with ptr 13 4 Element library convenience functions The file 1ib Elements misc c contains several functions which are useful across a wide variety of element types If you want to use these functions you should include misc h in your source file These functions are D PlaneStressD element Returns a constitutive matrix suitable for use in plane stress analysis D PlaneStrainD element Returns a constitutive
166. m equalization and element overlay were enabled for this plot 6 4 2 Line graphs 6 4 3 Wireframe drawings Most of the controls for wireframe drawing are specific to three dimensional visualization The three sliders control the rotation of the drawing about the three spatial axes The magnification determines the multiplicative factor by which nodal displacements will be increased before the nodes are plotted z scaling controls the front to back aspect ratio of the resulting plot 6 4 GRAPHICAL OUTPUT 57 Figure 6 4 A color contour plot showing the principal stress component in a static problem using constant strain triangular elements 58 CHAPTER 6 USING WINFELT Tx 209760 y Tx 709504 Tx 709120 Figure 6 5 A time displacement plot in WinFElt Figure 6 6 An example of a displaced structure plot 6 4 GRAPHICAL OUTPUT 59 Chapter 7 The velvet Application 7 1 Introduction to the velvet GUI velvet was designed as the primary user interface to the FElt system Generally if the FElt system can solve a given problem you will find it easier to specify solve and analyze the results for that problem right within the velvet GUI velvet embodies all of the mathematical features of FElt as well as graphical pre and post processing and element generation The main velvet window consists of three major areas Figure 7 1 The first is the list of menu buttons down the left side of the windo
167. mass matrix If we need to we call a local function just like we did for the local stiffness matrix depending on the mass_mode then use the same multiplication function to transform to global coordinates el if mass_mode if mass_mode c mhat ConsistentMassMatrix element else if mass_mode 1 mhat LumpedMassMatrix element 13 6 A DETAILED EXAMPLE 185 if mhat NullMatrix return 1 element gt M CreateMatrix 6 6 if element gt M NullMatrix AllocationError element mass matrix MultiplyAtBA element gt M T mhat we made it here everything must have worked Al return 0 Not bad right Note our use of ResolveHingeConditions in the above because we want these elements to be able to deal with hinged DOF For many elements hinges are meaningless and this check would not be necessary The next function we need is the element stress routine Again this is pretty simple because we do a lot of the work elsewhere in this case In order to retrieve the four internal forces that we are interested in all we need to do is transform the element stiffness matrix back to local coordinates transform the nodal displacements back to local coordinates multiply and subtract any equivalent nodal forces The element stress function that actually gets called to fill in the element s stress structures I realize that a lot of this seems awfully inefficient
168. matrix we never allocated any memory for these two because the functions that we are calling will do that AT khat Localk element if khat NullMatrix return 1 T TransformMatrix element We can form the element stiffness matrix now through some simple matrix multiplications The special multiply function here just saves us having to allocate some temporary space and actually transposing the transform matrix it will simply carry out k T trans K T 184 CHAPTER 13 ADDING ELEMENTS TO FELT ai element gt K CreateMatrix 6 6 if element gt K NullMatrix AllocationError element stiffness matrix MultiplyAtBA element gt K T khat Things can get a little tricky here we ll check if there are any distributed loads if there are we need to resolve them and modify this element s node s equivalent nodal forces If not we re home free In this case I have relegated all the distributed load handling to a separate little module A if element gt numdistributed gt 0 count EquivNodalForces element Tt NULL 1 if count return count there s also the possibility that some of this element s nodes have a hinged DOF that s easy to deal with because we have a convenience routine to do all the checking and modifying for us Pel ResolveHingeConditions element check to see if we need to form a
169. me Direction Value LocalX amp local_x LocalY amp local_y LocalZ amp local_z GlobalX amp global_x GlobalY amp global_y GlobalZ amp global_z Parallel amp parallel Perpendicular amp perpendicular Table 10 8 Directions of a distributed load Description n setup S n S n d n Y ame tress um_nodes hape_nodes _dofs cS un O um_stresses etainK Table 10 9 Field name name of definition element set up function element stress computation function number of nodes in element number of nodes that define shape number of degrees of freedom degrees of freedom number of stresses flag to retain stiffness matrix after assembly Fields of an element definition object Description mode title nodes elements dofs_pos dofs_num num_dofs num_nodes num_elements analysis mode problem title array of node objects array of element object global DOF positions global DOF numbers number of active DOFs number of nodes number of elements Table 10 10 Fields of a problem definition object 116 CHAPTER 10 THE BURLAP APPLICATION consistent The individual components of the arrays are read only however either array may be assigned a row vector to change the set of active DOFs When one array is changed both arrays and the number of active DOFs are updated We don t know why you would want to do this Usually find_dofs should be used to initiali
170. ments generally nodes come first since elements have to attach between them 7 5 DRAWING AND DEFINING A PROBLEM 69 7 5 1 Defining attributable objects The way that you work with these attributable objects forces loads materials constraints is by setting one of each as the active object of that type This is particularly important for materials and constraints When you add a node it will automatically be assigned the active constraint and when you add an element it will automatically be assigned the active material If you want to set the load on an element or the force on a node or change the constraint on a node or material on an element you can use the apply menu For instance choosing loads from this menu will allow you to apply the currently active distributed load on an element of your choosing To apply an object simply choose the appropriate entry and then click on the nodes or elements which you wish to apply it to velvet will keep prompting you for additional objects until you click on quit or abort If you wish to apply an object to a group of nodes or elements say for instance you wanted to apply a self weight load to all elements in the problem you could simply drag out a bounding box by clicking and holding the second mouse button If you are applying forces or constraints every node in this box will be assigned the active object of that type If you are applying materials or loads every element completely within the
171. ments to the trigonometric functions should be given in terms of radians just as if you were calling them from a C program using the stan dard math library Other than this difference these functions should be used and should behave as they are described in the manual pages for your local mathematics library In a transient analysis problem the symbol t denotes the time variable in expressions for force magnitudes or time varying boundary conditions These expressions will be dy namically evaulated throughout the course of the simulation For other parameters loads nodal coordinates etc and for all parameters in a static problems these expressions will simply be evaluated as if t 0 For spectral inputs the symbol w can be substituted for the independent variable for clarity and to distinguish frequency domain force spectra from time domain forces Expressions can also contain the ternary conditional operator as in the C programming language if a then b else c is symbolized in a FElt input file as a b c where a b and c are all valid expressions The logical operators to use in constructing a are the same as those in C amp amp lt lt gt gt The conditional construct is particularly useful in defining things like discontinuous dynamic forces 3 1 2 2 Discrete functions Because some forcing functions are easier to express in a discretized as opposed to con tinuous form e g earthquake records
172. methods that they want There must be a definition struct defined for the element type This structure contains information that everything else needs to know about elements of a given type symbolic name shape number of nodes number of nodes which define the shape number of degrees of freedom per node the degrees of freedom which this element affects and what functions to use for element setup and stress calculations Secondly there must be a routine to create and define the element stiffness matrix for elements of this type The setup function should also calculate and fill in the equivalent nodal forces if there is a distributed load on an element and create the element mass matrix if necessary for the current analysis Lastly there has to be a routine which calculates the stress or internal loads on an element If no stress information will be computed there must still be a routine which tells the global routines that no stresses will be computed 13 2 1 The definition structure From our previous discussion of the structures in FElt you might remember that each ele ment had a pointer to a definition structure which looked like typedef struct definition char name element name E int setup initialization function nt stress stress computation function hape shape element dimensional shape Ek nsigned numnodes number of nodes in element E ned shapenodes number of nodes which define shape
173. mtops which is part of Jef Poskanzer s fabulous PBMPLUS image format toolkit e The bivariate interpolation routines are hand translations into C of Fortran code orig inally written by Hiroshi Akima The Fortran version is readily available as one of the ACM TOMS algorithms e The routines to do Gibbs Poole Stockmeyer Gibbs King node renumbering are also hand translations of Fortran code that was originally published in ACM TOMS Chapter 1 Introduction to the FElt System 1 1 Intentions FElt is a package for introductory level Finite ELemenT analysis It is centered around a mathematical engine designed to simply effectively accurately and flexibly solve most types of structural mechanical problems that would be encountered in an introductory course in finite element analysis It was developed in an overzealous fit of we can do better than that based on some antiquated Fortran code that we had been using for just such a course Our intention was to design a system capable of doing everything that code could do in the mathematical sense but bring it into the 90 s or at least the late 80 s in terms of input syntax including error checking problem debugging etc flexibility and most importantly a graphical user interface We hope we ve succeeded or at the least are on the right track FEIt is a work in progress and chances are that if a feature isn t there now there are at least pipe dreams of it somewhere in t
174. n in Figure 8 9 Note that both histogram equalization and element 94 CHAPTER 8 POST PROCESSING WITH VELVET overlay were enabled for this plot In addition to the standard buttons for help accept and dismiss you can use the s plot and d plot buttons as an alias for selecting either plot stresses or plot displacements from the main postprocessing menu item Both buttons cause the dialog state to be accepted before the plot is generated 8 3 2 Controlling structure plots Additional control over structure plots is available through the wireframe dialog box Fig ure 8 10 The majority of these controls are specific to three dimensional visualization The three dial widgets control the rotation of the drawing about the three spatial axes The magnification determines the multiplicative factor by which nodal displacements will be increased before the nodes are plotted z scaling controls the front to back aspect ratio of the resulting plot The toggle for hidden line removal is non functional in the current version of velvet The defaults reflected in the dialog pictured in Figure 8 10 were used to generate our earlier example of the structural plot of the bicycle Figure 8 5 The extra button at the bottom of the dialog plot is equivalent to pressing accept and selecting plot structure from the postprocessing menu 8 3 3 Controlling animation The analysis parameters used in constructing a displacement table for an animation are the sam
175. n interesting sample problem xiii xiv LIST OF FIGURES 7 2 The velvet file selection mechanism aoaaa 64 73 The configuration dialog DOOR lt c e a coe acc arradan ee 66 74 The object coloring control box coco mirarnos 67 7 5 The constraint dialog box 2 2 8 eeeoe es beeen ebaeas 70 7O TR ee aee Seo Ot SR ae A Raw Bey 71 LI The force dialog DOX e i aae eoe e AA 72 7 6 Th lgaddialog DOR lt qoc cacra Gh an he AR AR eR 73 7 9 The node information and editing dialog 75 7 10 The element information and editing dialog 77 7 11 Examples of all the tools available in velvet 79 7 12 The form for defining line and grid generation parameters 81 7 13 The form for defining triangle generation parameters 82 8 1 The output control dialog box 244 6444845 84 bo eed eee ee eS 88 8 2 A time displacement plot in velvet o ooo 89 8 3 The mode shape plotting window e 90 8 4 An example of textual output from velvet o o 90 8 5 An example of a displaced structure plot 91 56 E AMA oe eS e oe ASE SS ok 91 8 7 The analysis parameters dialog box o 93 8 8 The control dialog box for contour plots a aoaaa 93 8 9 The stress output window for the wrench example 95 8 10 The wireframe plotting control dialog box
176. nds you use may be called some thing other than show w and show c they could even be mouse clicks or menu items whatever suits your program You should also get your employer if you work as a programmer or your school if any to sign a copyright disclaimer for the program if necessary Here is a sample alter the names Yoyodyne Inc hereby disclaims all copyright interest in the program Gnomo vision which makes passes at compilers written by James Hacker signature of Ty Coon 1 April 1989 Ty Coon President of Vice This General Public License does not permit incorporating your program into propri etary programs If your program is a subroutine library you may consider it more useful to permit linking proprietary applications with the library If this is what you want to do use the GNU Library General Public License instead of this License References 1 David S Burnett Finite Element Analysis From Concepts to Applications Addison Wesley Reading MA 1987 2 G R Cowper The shear coefficient in Timoshenko s beam theory ASME Journal of Applied Mechanics 33 335 340 1966 3 E Cuthill and J McKee Reducing the bandwidth of sparse symmetric matrices In Proceedings of the 24th National Conference of the ACM ACM Publ P 69 pages 157 172 New York 1969 Association for Computing Machinery 4 Z Friedman and John B Kosmatka An improved two node Timoshenko beam finite element
177. ne row and one column it may also be indexed ELTA S Ply li o Ds 67 2 x 1 2 3 BI Y My 2 3 4 write A 1 2 3 4 5 5 write A 2 6 write A 2 3 7 write x 8 write x 2 9 write y 140 CHAPTER 11 THE BURLAP SYNTAX 10 write y 1 1 11 write A 2 3 Thus there is some ambiguity in using X i since if X is a column vector then the result will be its ith element and if X is a matrix then the result will be its ith column Therefore caution should be used octave does not support indexing a matrix with a single index However the single index notation can be useful and is consistent with the semantics of the for expression of Section 11 7 3 11 4 14 Function expressions A function call is an expression F followed by parentheses containing a possibly empty comma separated list of arguments which may be arbitrary expressions The ar gument list is evaluated in order from left to right The result of the expression is the return value of the function named by F If more arguments are specified than required by the function the extra arguments are simply discarded If too few arguments are specified then null values are appended to the argument list Thus it is impossible to distinguish a nu11 value due to an uninitialized variable from a missing argument If F evaluates to null and is an identifier such as write then the global scope is searched for a function called F If a
178. near and has two nodes both of which are required and needed to define the shape of the element The truss element permits displacements along the X Y and Z axes but does not allow for rotation about any of the axes Therefore the first second and third DOFs are active The entire call to add_definition is 10 5 TIPS ON USING INTERACTIVE MODE 119 function truss_stress e L length e EonL e material E L c zeros 6 1 c 1 e node 2 x e node 1 x L c 2 e node 2 y e node 1 y L c 3 e node 2 z e node 1 z L e ninteg 1 e stress 1 x e node 1 x e node 2 x 2 e stress 1 y e node 1 y e node 2 y 2 e stress 1 values 1 EonL e node 2 dx e node 1 dx c return 0 end Figure 10 2 Stress function for the truss element definition add definition truss truss_set_up truss_stress amp linear 2 2 1 1 2 3 0 0 0 We have neglected to pass a value for the last argument which indicates whether the local stiffness matrix should be retained after assembly into the global stiffness matrix The default value for this argument is amp false which is fine for our purposes Only the last ar gument to add_definition may be omitted the remaining argument do not have default values If we wish to remove the truss element definition then the remove_definition function must be used The remove definition function takes a single argumen
179. ng with files If you already have a FElt input file defined the easiest way to start working with that file is by specifying the filename on the shell command line when you invoke velvet For example velvet foo flt Will start velvet and automatically load in the file foo f1t The drawing area will be initially configured to some reasonable values to allow you to view the entire problem If this file was saved from velvet then it will contain special sections that describe all of the figures and the configuration of the drawing area when the file was saved velvet will use this information to reconstruct all of these things exactly as you left them 64 CHAPTER 7 THE VELVET APPLICATION Open Problem Problem name fusersjgobat FElt 3 02 examples eft M README axisymm flt beam f1t beam3d_2d f1t beam3d_a flt beam3d_b f1t beam3d_dist flt beam3d_dist_a flt beam_1c flt okay Figure 7 2 The velvet file selection mechanism Alternatively you can invoke velvet with no filename and then use the standard velvet file selection mechanism to load a file Select open from the file menu This will bring up a dialog box that looks something like the one shown in Figure 7 2 This dialog is the standard way for you to specify filenames either for loading or saving problem files or graphical dumps Within this dialog you can choose a file either by typing into the entry box at the top of the dialog or by clicking on a
180. nition of the same name already exists then add_definition returns 1 otherwise it returns 0 The set_up function is a burlap function that will compute the local stiffness matrix K and local mass matrix M for any element of the new definition Our sample set up function is shown in Figure 10 1 For our simple truss element we are ignoring the mass matrix and the effect of distributed loads The set up function will be called with two arguments The first argument is the element whose stiffness and mass matrices are to be computed The second argument is the mass mode which is one of amp false amp lumped or amp consistent where false indicates that no mass matrix is required In our simple example we have simply omitted the second argument since we will not be considering mass matrices Note that the set up argument is a burlap function itself not the name of the function As discussed in Section 11 6 functions in burlap are simply variables that can be passed as arguments to functions and also returned from functions Our simple function for truss elements computes the length of the element by calling the length function which is discussed in Section 11 5 4 118 CHAPTER 10 THE BURLAP APPLICATION function truss_set_up e L length e AEonL e material A e material E e material L K AEonL AEonL AEonL AEonL cx e node 2 x e node 1 x L cy e node 2 y e node 1 y L
181. nodal hinge boundary conditions are also allowed Time varying force and boundary conditions can be expressed either as continuous or discrete functions 1 3 FElt What it cannot do for you As of this release FElt can only handle linear static and dynamic problems We realize the shortcomings that this presents for some people and we have some vague plans for non linear analysis but nothing is here yet velvet can t really draw in 3 d at least in the problem definition stage and thus isn t a terribly good way to define 3 d problems it will always assume that it should work in the x y plane z 0 This is probably going to stay this way for a long time There are certainly other shortcoming as well depending on just what you would like the package to do What it really comes down to is that FElt was never intended to solve everybody s real world or cutting edge research problems so we re probably never going to incorporate lots of different analysis types etc If you want to take a crack at modifying FElt for your own local needs however then we encourage you to do so we ll even help out where possible Chapter 2 FElt Analysis Types 2 1 Introduction Our intent in writing this chapter was to give some basic details on the kinds of analyses that are intrinsic to FElt It is not at all meant to be an introduction to the finite element method but rather it is meant to provide a background on what types of physical and mat
182. ns will accept any type of object as an argument 1 x 1 2 3 2 y 4 3 write scalar x scalar y 01 4 write matrix x matrix y l 0 5 z x x 6 write symmetric z compact z L 0 7 write compact compact z 11 5 INTRINSIC FUNCTIONS 145 11 5 4 Finite element functions The finite element functions provide an interface to the functions in the FElt library The functions can all be written in burlap itself but the built in functions will execute faster add_definition area e Adds a new element definition After the definition is added elements of that particular type can be loaded using the felt function The types of the arguments are given in order in Table 11 18 Examples and a more in depth discussion of the arguments can be found in Section 10 4 name of definition string set up function K and M function stress computation function function element shape scalar number of nodes in element scalar number of nodes that define shape scalar number of stress values scalar vector of DOFs row vector retain stiffness matrix indicator scalar Table 11 18 Arguments to the add_definition function Returns the area of the element e The element must be a planar element 1 e e definition shape should be planar assemble M C clear_nodes Computes and returns the global stiffness matrix K for the currently de fined FElt problem by c
183. nsigned numstresses number of computed stress values nsigned numdofs number of degrees of freedom nsigned dofs 7 degrees of freedom GCG SE ESE G amp G G amp G ES UN P un fer Q nsigned retaink retain element K after assemblage Definition For an arbitrary element foo the definition structure might be filled out to look some thing like this struct definition fooDefinition foo fooEltSetup fooEltStress Winear yy 2y Dy O a Oi Oi 0 20 13 2 NECESSARY DEFINITIONS AND FUNCTIONS 173 This defines an element for which the symblic name will be foo It is a linear element with two total nodes two nodes which define its shape and three DOF per node The general FElt routines will call the functions fooEltSetup and fooEltStress for element setup and stress calculations The DOF array always starts with a zero in the Oth position The rest of the array tells the general FEIt routines that foo elements affect global DOF 1 2 and 3 Tx Ty and Tz and that Tx is its first DOF Ty its second and Tz its third The last member of the structure is the flag to retain or destroy element gt K after it has been assembled In this case we don t need it around so we set element gt retainK to zero false If you don t quite see how this all works yet it may help you to know that foo elements have the same definition as truss elements in the actual FElt library A slightly mor
184. nt gt node 1 gt y L atrixData T 1 1 s atrixData T 1 2 c atrixData T 21 3 1 0 atrixData T 31 4 s atrixData T 3 5 c atrixData T 4 6 1 0 return T Now we need two local functions to create the two different kinds of mass matrices They look an awful lot like the local stiffness matrix because once again we don t actually need to do any numerical integration we know how it all turns out so we just have to fill in some matrix entries much like the local K function above all we do here is fill in the mass matrix this function fills it out for consistent mass the following function is used if the user wanted a lumped mass El static Matrix ConsistentMassMatrix element Element element static Matrix m NULL the local stiffness matrix double L the element length x double phi bending stiffness shear stiffness 192 CHAPTER 13 ADDING ELEMENTS TO FELT double phi2 phi squared double constl constant term for rotational mass E double const2 constant term for translational mass if m NULL m CreateMatrix 4 4 if m NullMatrix AllocationError element mass matrix the constants that we 11 need including the constant terms in front of the rotational first terms and translational second terms portions of the matrix L ElementLength element 2 phi 12 0 L L e
185. nt mass matrices For a transient structural and thermal analysis problems there are several parameters which control the numerical integration in time The text entries down the left side of the dialog allow you to fill in values for these parameters in the y B and text entries For both transient and spectral analysis the range of time or frequency over which to perform the computations is defined by the start stop and step text entries Note that the value for start is ignored in transient analysis types Values for the global Rayleigh damping proportionality constants are defined with the Rk and Rm text entries Remember that if either of these values is non zero then the Rayleigh damping for this problem will be based on these values and the global stiffness and mass matrices as opposed to elemental material values of Rk and Rm and the element stiffness and mass matrices The DOF toggles and the node entries allow you to define the list of nodes and which DOF at each of those nodes that you would like to have included in the output when the problem is solved Click on the left and right arrow keys to scroll through the complete list of nodes if you are interested in more than six of them The push buttons on the bottom of the dialog box are standard of course The solve and animate buttons are simply there as a convenience pushing either button will cause the current state of the dialog to be registered as if you had pushed accept and th
186. ntation or for inclusion in a report Additional drawing tools see section 7 7 are available which allow you to put a title annotation or additional graphic objects in the drawing area in addition to the standard presentation of nodes and elements Note that because of the nature of XWD s you need to make sure that no window is covering any part of the drawing area whenever you do a dump in XWD format This includes the file selection dialog make sure you move it out of the way before you click okay 7 5 Drawing and defining a problem Starting with a blank slate either by invoking velvet without an input file or selecting new from the file menu you need to do several things to completely define a new problem Say for instance that you want to define the four node three element beam and truss exam ple problem considered earlier Section 3 3 Figure 3 3 This problem uses two different distributed loads four different constraints two element types and two materials All of these things need to be defined in order to properly solve the problem The material prop erties must be defined before elements are added because an element must have a material assigned to it Likewise at least one of the constraints must be defined before nodes can be added because nodes must have a constraint assigned In general the procedure is to setup whatever objects the problem might require loads forces materials constraints be fore adding nodes and ele
187. ntations of mode shapes for modal analysis problems One way to solve a problem is to select the solve option from the solutions menu By default this simply generates the standard FElt output This window will contain either the mathematical solution of your problem or the syntax errors encountered in solving the problem For more control over what gets generated whenever you solve a problem you can use the output control dialog box Figure 8 1 available by selecting results output from the solutions menu The toggle buttons in this dialog box control all of the available solution and output options both text and graphical The three toggles at the top left of the dialog mimick three of the switches available in felt they control the computations that are performed for modal or spectral analysis The felt output toggle controls whether or not you want to see any of the standard textual output that fe t would generate If the felt output toggle is on then the other three textual 87 88 CHAPTER 8 POST PROCESSING WITH VELVET E Omput solution control Results Graphical Output pi eigen analysis i stress contours E orthonormal mode shapes displacement contours Hg transfer functions E wireframe structure Textual Output E line plot felt tabular _ mode shapes material summary a analysis details _ debug lg print matrices accept dismiss Figure 8 1 The output control dialog box output togg
188. ny free program is threatened constantly by software patents We wish to avoid the danger that redistributors of a free program will individually obtain patent licenses in effect making the program proprietary To prevent this we have made it clear that any patent must be licensed for everyone s free use or not licensed at all The precise terms and conditions for copying distribution and modification follow B 2 Terms and Conditions for Copying Distribution and Modifica tion 0 This License applies to any program or other work which contains a notice placed by the copyright holder saying it may be distributed under the terms of this General Public License The Program below refers to any such program or work and a work based on the Program means either the Program or any derivative work under copyright law that is to say a work containing the Program or a portion of it either verbatim or with modifications and or translated into another language Hereinafter translation is included without limitation in the term modification Each licensee is addressed as you Activities other than copying distribution and modification are not covered by this B 2 TERMS AND CONDITIONS 207 License they are outside its scope The act of running the Program is not restricted and the output from the Program is covered only if its contents constitute a work based on the Program independent of having been made by runnin
189. o a make in lib Felt Now that all of the necessary libraries have been built to be aware of the new element we need to re link an application with the updated libraries felt is the easiest because it is so simple Go to the bin Felt directory and do a make Then create a simple test file that uses your new element and run it through felt If the numbers don t come out right you can 200 CHAPTER 13 ADDING ELEMENTS TO FELT modify the source in 11b Elements do a make in that directory and a make in bin Felt to re link your changes Repeat this process until everything comes out to your satisfaction Eventually you should do a make in the directory for each separate application so that your new element is accessible from all of the application in the FEIt system Appendix A Installing and Administering FElt A 1 Building the FElt system from source The FElt package is intended to be easily portable It should build on most reasonable Un x systems without any modifications To start the build cd to the toplevel directory of the FElt source tree From there do a configure This will try to automatically determine where to find relevant include files and libraries on your system and create a new etc Makefile conf file If you feel comfortable with it you can go in and tweak the Makefile conf file by hand if something does not go right After configuring do a make clean followed by a make all To install the package after
190. odes and elements That is a node can not be deleted if an element is still attached to it so the element must be deleted first Elements can be deleted freely Deleting multiple nodes and elements is accomplished by clicking on one after the other quit or abort will finish deleting or by clicking and holding the middle mouse button and dragging out a box containing the nodes or elements to be deleted As in deleting one at a 7 6 MORE ON NODES AND ELEMENTS 75 Translation X 6 o 0 Y 10 0 0115221 Z O 0 1 29173e 20 Location help accept dismiss delete new copy Figure 7 9 The node information and editing dialog time none of the selected nodes which are still attached to elements will be deleted 7 6 More on nodes and elements 7 6 1 Editing nodes In addition to simply adding and deleting nodes velvet provides a powerful mechanism for displaying and editing all of the information about a node The node dialog box is shown in Figure 7 9 You can raise this dialog either by selecting edit from the node menu or by clicking directly on the node which you want to edit with the left mouse button You may want to use the former option even though it seems a bit clumsier in cases where you can t quite click on the right node If you proceed from the menu option you will be prompted to either select the node with the mouse or to type the node number into the command window Once the dial
191. og is up you can change the displayed node either with the arrow keys in the dialog box by clicking on a different node in the drawing area or by selecting a new node through the edit option under the node menu As a third option for editing nodes you can click on a node with the third mouse button this will raise the node editing dialog and the constraint and force 1f appropriate dialogs with the assigned objects for the selected node already highlighted This is an easy way to take a complete look at a given node remember however that any changes you make to the node s objects will also affect any other nodes that have that object assigned to them The information available in the node dialog are the exact nodal coordinates all six nodal displacements valid only after a problem has been solved the lumped mass for the node the name of the constraint applied to the node and the name of the force if any 76 CHAPTER 7 THE VELVET APPLICATION applied to the node The location lumped mass constraint and force can be modified simply by editing the information in this dialog box To change the applied constraint you can enter a new name into the text field or you can click on the constraint label to pop up a menu that will let you choose from a list of the currently defined constraints The same is true for forces by clicking on the force label Note that while you can change the applied force on a node via the force option under the a
192. ols available in velvet 80 CHAPTER 7 THE VELVET APPLICATION starting point by a simple click not a click and hold Additional end points are defined by additional clicks of the first mouse button You can stop adding endpoints by clicking the third mouse button If you want a closed polyline you can simply click the second mouse button and velvet will draw a line from your current position to the starting point of the polyline Tools can be deleted from the screen just like nodes and elements either one at a time or with a window Once in delete mode select delete from the tools menu individual figures can be selected by clicking with the left mouse button or a window can be drawn with the middle mouse button The delete operation proceeds until quit or abort is clicked Moving a tool figure that already is drawn on the screen is also straightforward simply select move from the tools menu click on the figure that you want to move and then click again once you have dragged it to its new location The ancillary sections in the FElt file that are created when you save a problem from velvet record all your tool figures If these sections are available when you reload a problem the tools will automatically be redrawn for you 7 8 Material databases and defaults files Often you will find that certain objects or material properties are being used over and over again in different problems Dummy FEIt files can be used to make easy use of
193. ombined with transposition is used to compute the inverse without introducing numeric instability as shown in Equation 11 1 In all cases the number of rows of X must equal the number of columns of Y 136 CHAPTER 11 THE BURLAP SYNTAX Z X Y Z xy ZY X 11 1 YTZ pe stdin 2 exception in expression right division by zero 1 write 1 2 0 5 2 write 1 0 3 write 1 2 3 10 0 1 0 2 0 3 The left division of X and Y is approximately equivalent to the inverse of X multiplied by Y Left division is most often used to solve equations of the form Ax b X Y Z Z X Y scalar scalar scalar z y x scalar matrix matrix Vi j Zi Y j x matrix scalar matrix Vi j Zij Xi xy matrix matrix matrix Z XZ Y Table 11 10 Type table for the left division operator As shown in Table 11 10 the third case involves the explicit computation of the matrix inverse and thus should be avoided Left division is also known as a backsolve since if performed on two matrices an LU decomposition or Crout factorization for compact matrices is computed followed by a backsolve operation This is easy to remember since the backslash is used to perform a backsolve The number of columns of X must equal the number of rows of Y stdin 2 exception in expression left division by zero 1 write 2 1 05 2 write 0 1 3 write 10 1 2 0 1 0 2
194. omputing the local stiffness matrices and assembling them into the global matrix For transient problems if M is specified then it will contain the global mass matrix on return Similarly C will contain the global damping matrix All matrices are compact column matrices The DOF related fields of the problem structure must be initialized as by calling the find_dofs function Clears the displacements and equivalent nodal force vectors for all nodes defined in the current FEIt problem The return value is null 146 CHAPTER 11 THE BURLAP SYNTAX compute modes K M X Computes the modes for given stiffness matrix K and mass matrix M The result is the vector of eigenvalues A If X is specified then it will contain the matrix of eigenvectors upon return see eig in Section 11 5 2 The DOF related fields of the problem structure must be initialized as by calling find_dofs compute_stresses e Computes the stresses for the element e The return value is the return value of the element s stress function The stress field of the element is initialized by this function This function is merely a shorthand for e definition stress e construct_forces t Constructs and returns the global nodal force vector f for the current FEIt problem The vector is constructed based on all nodal forces and the global DOFs at those nodes For transient problems may be a scalar expression used to specify the current time If is
195. on title Mixed Element Sample nodes 4 elements 3 nodes 1 x 0 y 0 constraint pin 2 x 6 y 0 constraint free 3 x 12 y 0 constraint roller 4 x 6 y 10 constraint fixed beam elements 1 nodes 1 2 material steel load left_side 2 nodes 2 3 load right_side truss elements 3 nodes 2 4 material spring material properties steel A 1 E 210e9 Ix 4e 4 spring A 4 76e 7 E 210e9 k 10 kN m constraints pin Tx c Ty c Tz c Rz u we better constrain Tz because free Tx u Ty u Tz c Rz u there is a truss element roller Tx u Ty c Tz c Rz u fixed Tx c Ty c Tz c Rz c 3 4 AN EXAMPLE OF A TRANSIENT ANALYSIS PROBLEM 25 distributed loads left_side direction perpendicular values 1 10000 2 0 right_side direction perpendicular values 1 0 2 10000 end 3 4 An example of a transient analysis problem Figure 3 4 illustrates a simple frame problem subjected to a time dependent loading Our input file for this problem differs from our above example in only one significant way we must include an analysis parameters section in this case Also we need to remember to define the density of all the materials so we can get a proper mass matrix for each element In this case we have also calculated what the self weight effect of this mass will be assuming bays on 15 ft centers depth of frame into page and assigned it to each element as a distributed load in terms of pounds per inch OSEM lof
196. or the dimensional quantities that you specify in defining a problem for FElt The important thing is to remain consistent in the units that you use numerical results will then be consistent with the input dimensions Some examples of consistent units would be nodal coordinates in meters forces in Newtons elastic moduli in Pascals N m2 moments of inertia would be in m The displacement results for a problem like this would be in meters stresses would be in Pascals Convenient English units are often pounds inches and psi pounds per square inch or kips kilopounds inches and ksi The FEIt application yardstick is intended for simple scaling of the numerical quanti ties in a FElt input file As a special case of this unit conversion of files is given special treatment and made particularly easy To convert an input file that was originally specified in kips and feet to Newtons and meters the yardstick command line would simply look like yardstick if kips il feet of N ol m foo flt gt foo_si flt The yardstick manual page details the list of units that the program recognizes and all of the available options 3 1 4 A simple example An input file for a simple cantilever beam problem with an end point load and considering self weight might look something like this problem description title Cantilever Beam Sample Problem nodes 2 elements 1 analysis static nodes 1 x 0 0 y 0 0 constraint fixed 2 x 10 0 y 0 0 constraint f
197. osing amp 3 0 y 1 20 2 and B 1 4 results in an unconditionally stable second order ac curate algorithm 9 With these three parameters the time discrete equation of motion in the HHT a method is written as Ma 1 1 01 Cvj 1 acyv 1 0 Kd 1 0Kd F li QLAL 12 4 The standard Newmark finite difference formulas are used as approximations for dj 1 and Vi l gt dis dis BAt ai 12 5 Vir Pipi YyAtaj 1 12 6 where the predictor variables di 1 and P 1 are defined as di 1 1 di Atv At 5 5 di 12 7 vi 1 y Atai 12 8 Vi 1 We can form an implicit update equation with d as the only unknown by rearranging equation 12 5 and substituting the result along with equation 12 6 into the equation of motion equation 12 4 If we denote the left hand coefficient matrix as K and the right hand contributions not due to the force vector as F then the update equation is K dja F 41 F t 41 At 12 9 where K M yAtC 1 a BArK 12 10 and Flin M YAtC dis 1 BA C91 OBA Cv Kd 12 11 168 CHAPTER 12 THE ALGORITHMS BEHIND FELT 12 4 8 Time integrating in transient thermal analysis As in transient structural analysis the time stepping in a transient thermal analysis problem is done using an approach similar to Newmark integration In this case however there is only one integration parameter because of
198. our changes become effective This behavior is similar to an okay button in many GUIs the difference is that accept will not cause the dialog box to disappear You will need to click the dismiss button to do that Where there is a help button you can click and hold it to view a brief help message about that dialog In some dialogs additional help is available by clicking and holding over special labels This type of help will be discussed in more detail later A final factor that is common across GUI components is the idea of multiple mecha nisms In general there are three different ways that you can perform an action in velvet The first is by the standard point click and drag operations of the mouse The second is through keyboard shortcuts The third is by entering commands or data into the command window at the bottom of the main drawing area For the most part we will limit our initial discussion to the mouse operations Section 7 10 presents a summary of the equivalencies between the different interface methods for those of you who want to become velvet power users For now it should be enough to know that whenever velvet prompts you for a co ordinate you can either click in the drawing area or type in an x y pair in the command window followed by a return of course Similarly if you are being asked to select an element or node you can either type in the appropriate number or click on the appropriate object 7 3 Worki
199. own in Table 11 13 x Y y x scalar scalar y x matrix matrix Y XT Table 11 13 Type table for the transposition operator 11 4 12 Unary operators X X not X X The prefix unary operators associate from right to left The unary minus operator sim ply returns the negative of X The unary plus operator simply returns X which must be a matrix or scalar expression The logical negation operator not or logically negates each element of X not x del 1 ifx 0 O otherwise The type tables for the unary operators are shown in Table 11 14 x Y X opy scalar scalar x 0p y matrix matrix Vi j Xij op Yi Table 11 14 Type table for the unary operators 11 4 OPERATORS 139 11 4 13 Index expressions X X I J An index expression is used to access the components of matrices Each index must be an expression that is either a scalar a vector whose elements are contiguous and increasing or the special vector which is used to specify all rows or columns of a matrix and should not be confused with the range operator If I or J is a vector then it specifies a range of indices that are to be used Thus is a shorthand for writing 1 rows X for J or 1 cols X forJ If X is a matrix then X i is its ith column and X 1 3 is the value of the element of its ith row and jth column However if X is a vector then X i is its ith element Since a scalar is a matrix with o
200. p copyright for details Use the q option to suppress this message al sz 10 2 USING BURLAP 107 If burlap was compiled with the GNU readline library then you have complete command line editing and history as in bash A complete discussion of the editing ca pabilities can be found in the documentation of the readline library At the prompt you can enter expressions to be evaluated ED ASHE 2 write 1 2 As illustrated above write is used to print results and is the assignment operator Also notice from the first expression that burlap does not print the result of every expression like matlab does But just like matlab matrices are delimited by square brackets with elements separated by commas and rows separated by semicolons If you type alias you should see the following list If burlap is not compiled with the readline library then the list will be slightly different exit exit h history 20 help help ls system ls quit exit burlap has a very simple syntax with all computations being done through either oper ators such as or functions such as write But since exit and help are rather non intuitive you can alias commands to be whatever you want just like csh So you can just type exit or quit to leave burlap Typing help will present you with a list of all the 108 CHAPTER 10 THE BURLAP APPLICATION operators and functions that burlap provides Typin
201. perties The material properties section s is quite simple Each material has a name followed by a list of characteristics Currently available characteristics are listed in table 3 1 Note that no element types require a material property with every one of these charac teristics defined In fact most only use three or four characteristics You should consult the element definitions chapter 4 for a complete list of what each element type requires from a material property Density rho is always necessary for the materials in a transient modal or spectral analysis problem if you actually want your elements to have any iner tia Different element types can certainly use the same material as long as that material 3 2 SECTIONS OF A FELT INPUT FILE 19 property name description E Young s modulus or elastic modulus A cross sectional area t thickness rho density nu Poisson s ratio bulk or shear modulus torsional stiffness Ix Lx moment of inertia Iy I moment of inertia Iz I moment of inertia kappa shear force correction factor Rk Rayleigh stiffness damping coefficient Rm Rayleigh mass damping coefficient Kx thermal conductivity along the x direction Ky thermal conductivity along the y direction Kz thermal conductivity along the y direction c heat capacitance Table 3 1 Symbolic names used to define material properties definition contains the right characteristics for each type of element that uses it
202. ple problem E mx I ngan Sample l rnhipm A 2 1 p AM xx Nudal Displacements Anne E BI DIN Du i wmi a 1 8 8 2 0 00192 i H HHH Y 4 9 0017 0 00221 0 00213 H HHIH 0 00668 0 00585 5 6 I lement SPresses 0 0170 8 0 9 27 18u 05 6 00023 H MMII 6 00032 6 00052 6 66119 118 0 077905 85 845 3 5547 Ale 247 LETETI 48 595 4 2375 57 728 9 1758 40 57 53 437 SABE 7U AN 1 5782 29 782 Vide 22 766 MB 175 65 9914 11n de 118 35 845 Ll 48 595 57 728 40 57 Ah aA 6 67795 3 5547 H 777A 4 9378 90 1758 33 437 2 7a fhH 1 aa 2 9835 34 3 131 H74 65 997 Figure 6 2 The text output box 6 2 SOLVING A PROBLEM 55 Results Controls IX Generate standard tabular output aph plot of cutput F Diaw dispaced shape Analysis Specific Solution Contrels E Only calculate transfer functions Only calculate eigenvalues and vectors F Use orthonormal mode shapes Additional Text Output with Solution I Pint Global Matrices I Pint Material Usage Sumnary F Pint Debugging Cutput Print Detals Output Contour Specific Controls Contour element stress values C Contour nodal displacement values IX Histogram equalize contour inage FX Draw elemert outlines on contour image E Contour conponen Wiraframe Drawing Controls ETA sean o LL IT 71 Y rotation 20 LL T gt 2 rectation 2 FX Underlay unceformec structure 10 0 Displacemen
203. pply menu editing the information in this dialog is the only way to completely remove the applied force on a node Just like the object dialogs you need to click accept to make your changes effective The rest of the buttons on the bottom of the node dialog also behave exactly the same as the buttons on the object dialogs velvet provides two ways for you to change a node s location The first is by changing the location fields in the node dialog The second is by selecting move from the node menu You can then select the node that you want to move by clicking on it move to the new location and click again to put it back down Alternatively just like you can click the first mouse button on a node to edit you can click on a node with the second mouse button to pick it up move to a new location and click again with the second mouse button to put it back down Note that because velvet is basically a package for working with two dimensional problems you cannot change the z coordinate of a node in the coordinate entry fields of the node dialog 7 6 2 Automatic node renumbering velvet gives you two ways to optimize the node numbering in terms of reducing the profile of the global stiffness matrix The first way is to select renumber perm from the nodes menu This will actually rearrange all of the node numbers in the current problem per manently Generally you would choose this option where the node numbering was not particularly important to yo
204. r elements as a three dimensional grid of solid elements and as a two dimensional mesh of triangular elements Each type of generation is described in its own section there can be multiple types and multiple sections of a given type in any given file Besides these descriptive sections there are only a few basic parameters all of which are optional Order of specifications does not matter in a corduroy file except for numbering of generated nodes and elements and the end statement which must always come at the end of a file and is not optional As in regular FElt files comments can be denoted with and as in the C programming language and symbolic expressions may be substituted for any non integer value 97 98 CHAPTER 9 THE CORDUROY APPLICATION You can use start node and start element to specify the number for the first things that get generated Normally corduroy starts the numbering at one and then numbers sequentially as it moves from section to section of your input file These two specifications allow you to override these starting points in situations where you may already have a few elements and nodes defined and you are going to attach the things that get generated after corduroy has done its work constraint and material will assign a constraint or material name to the initial node and element This is simply a convenience you will need to be sure to go back and actually define these objects after everything is generat
205. r functions or ouput spectra are available All of the WinFElt output windows have a File menu from which you can save or print the results in that window Line graphs are saved in Windows Metafile format WMF and color contour plots are saved as Windows bitmaps BMP Text output can be saved as an ASCII file Once an output window is closed there is no way to bring the window back up without resolving the problem One way to solve a problem is to select the solve option from the Solutions menu By default this simply generates the standard FElt tabular output in a text output window figure 6 2 This window will contain either the mathematical solution of your problem or the syntax errors encountered in solving the problem 53 54 CHAPTER 6 USING WINFELT Elle Edit Soluilons problem descript tirte 1ngan Sam nudes 1 x 0 y 0 Y o R 1H yt K 20 y 9 x 30 y 15 x B2 y 11 25 x iH YH bcan cleacnts nmudes 1 2 nodes 2 3 Draw Fonts bout ion pie l rnhipm A 2 1 p PAH noades lt f ePlenents lt z constraint pin 79H constraing trer 2 9 lt 8 Furce handlebars z 0 79H eronstraint n 1 ier Invernp seat Walerigl sleel_1 material steel 2345 nnnps 1 31 hodes 2 4 nudes 3 5 nodes 4 5 nnes A Walerigl properties steel 1 E Se 07 A 0 1 SrpP i vile stccl_67B E 8c O7 N 0 3 1x 0 01 ilPIHZ A H 1 1x H M7 1kK 1 material steel 678 Figure 6 1 Main WinFElt editor with a sam
206. rays of the element s nodes The stress function calls this routine because it needs to adjust the internal force vector to account for equivalent nodal forces It only needs the basic vector in local coordinates We need to compute the equivalent nodal load vector here Just for convenience we are going to call his function in two different ways mode 1 and mode 2 he first way is for the element stiffness function lement s nodes The second is for the stress routine t T which just wants to get the forces applied to the e which actually needs the equiv force vector in local coordinates T here are lots of ways to handle all these cases see the Bernoulli beam elements for example In mode 1 eq_stress can be NULL in mode 2 Tt can be NULL a static int EquivNodalForces element Tt eq_ stress mode Element element Matrix Te passing it in saves a few FLOPs Vector eg_stress vector pointer to set in mode 2 int mode mode of operation static Vector equiv NULL the equiv vector in local coord static Vector eq_global equiv in global coordinates asf double wa wh values of load at nodes Ef double L the element length 13 6 A DETAILED EXAMPLE 195 unsigned 19 47 some loop conuters double factor constant factor for sloped load double phi bending shear stiffness int count error count if equiv NULL equiv Creat
207. rbitrarily oriented three dimensional element Like their 2 d special case cousins three dimensional beam elements are two node linear elements 3 d beams however can carry forces in any of the six DOFs axial local x direction vertical shear local y direction horizontal shear local z direction rotation about the x axis torsion rotation about the y axis out of plane bending and rotation about the z axis in plane bending Both the elastic and shear moduli E and G must be specified for a 3 d beam element The cross sectional area A torsional moment of inertia J Zy moment of inertia Ly and moment of inertia Iz must also be specified beam3d elements can define either a lumped or consistent local mass matrix The lumped formulation includes entries at every DOF i e there is an inertia entry at rotational DOF just like the two dimensional beam The local coordinate system used for the FElt 3 d beam element is based on the element geometry presented in 13 Positive points from node 1 to node 2 Then is defined by the cross product of z and global z and local x i e P zx Z is selected such that it is orthogonal to the plane 2 x Given this geometry clockwise moments and rotations about are positive for moments and rotations about 2 counter clockwise is defined as positive Local internal forces calculated for each 3 d beam element include all six forces at both nodes twelve tota
208. ree force end_load beam elements 1 nodes 1 2 material steel load self_weight material properties steel A 10 0 E 30e6 Ix 357 properties can also be lowercase distributed loads self_weight direction perpendicular values 1 2000 2 2000 3 2 SECTIONS OF A FELT INPUT FILE 17 constraints fixed Tx C Ty C Rz C column alignment is unimportant free Tx U Ty U Rz U I could have used tx ty and rz forces end_load Fy 1000 end 3 2 Sections of a FElt input file 3 2 1 Problem description The problem description section is used to define the problem title and the number of nodes and elements in the problem These numbers will be used for error checking so the specifications given here must match the actual number of nodes and elements given in the definition sections Note that the definitions for nodes and elements do not have to be given in numerical order as long as nodes 1 m and elements 1 n where m is the number of nodes and n is the number of elements all get defined in one of the element and node definition sections in the file The analysis statement defines the type of problem that you wish to solve Currently it can either be static transient static substitution modal static thermal transient thermal or spectral If you do not specify any thing static analysis will be assumed The problem description section is the only section which you cannot repeat within a given input file 3 2 2 Nodes
209. ributed in 0 1 If n is not specified then an m x m matrix is returned If both m and n are not specified then a scalar is returned The third argument s will be used to seed the random number generator if it is specified and is a non zero scalar burlap initially seeds the generator with the current time rows X Returns the number of rows of the matrix X A scalar is defined to have a single row 144 CHAPTER 11 THE BURLAP SYNTAX zeros m n Returns an m x n matrix whose elements are all 0 If n is not specified then an m x m matrix is returned A synonym for this function is zeroes The eye ones rand and zeros functions provide alternative ways of creat ing a matrix without using the matrix formers 11 5 3 Predicate functions Predicate functions Table 11 17 return 1 if the predicate is true and 0 otherwise The names of all predicate functions in burlap end with a question mark but this is simply a convention and is not required should you write your own functions any X Is any element of X non zero compact X every X null X matrix X scalar X symmetric X Is X a compact column matrix Is every element of X non zero Is X null Is X a matrix Is X a scalar Is X a symmetric matrix Table 11 17 Predicate functions available in burlap The any compact every and symmetric functions require X to be either a scalar or a matrix The remaining functio
210. right line and a pointer to where the full notice is found one line to give the program s name and a brief idea of what it does Copyright 19yy name of author This program is free software you can redistribute it and or modify it under the terms of the GNU General Public License as published by the Free Software Foundation either version 2 of the License or at your option any later version This program is distributed in the hope that it will be useful but WITHOUT ANY WARRANTY without even the implied warranty of MERCHANTABIL ITY or FITNESS FOR A PARTICULAR PURPOSE See the GNU General Public License for more details You should have received a copy of the GNU General Public License along with this program if not write to the Free Software Foundation Inc 675 Mass Ave Cambridge MA 02139 USA 212 APPENDIX B THE GNU GENERAL PUBLIC LICENSE Also add information on how to contact you by electronic and paper mail If the program is interactive make it output a short notice like this when it starts in an interactive mode Gnomovision version 69 Copyright C 19yy name of author Gnomovision comes with ABSOLUTELY NO WARRANTY for details type show w This is free software and you are welcome to redistribute it under certain conditions type show c for details The hypothetical commands show w and show c should show the appropriate parts of the General Public License Of course the comma
211. rticular circumstance the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances It is not the purpose of this section to induce you to infringe any patents or other prop erty right claims or to contest validity of any such claims this section has the sole purpose of protecting the integrity of the free software distribution system which is implemented by public license practices Many people have made generous contri butions to the wide range of software distributed through that system in reliance on consistent application of that system it is up to the author donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice 210 APPENDIX B THE GNU GENERAL PUBLIC LICENSE 10 11 This section is intended to make thoroughly clear what is believed to be a conse quence of the rest of this License If the distribution and or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries so that distribution is permitted only in or among countries not thus excluded In such case this License incorporates the limitation as if written in the body of this License The Free Software Foundation may publish revised an
212. s This means that the only difference between the two is that the constitutive matrix D used in the element stiffness formulation is different for the two cases A CST element is a three node two dimensional planar element Each CST element should exist completely in a single x y plane The node numbers must be assigned to a CST element in counter clockwise order to avoid the element having a negative area 36 CHAPTER 4 THE FELT ELEMENT LIBRARY The lumped mass matrix for a CST element is formed simply by dividing the mass of the element equally between the three nodes i e 10000 0 01 0 0 0 0 pario 0 100 0 m z lo 0010 0 Crp 00001 0 000001 where A is the computed planar area of the element not the area as defined by the material property A There is no consistent mass formulation available for the current set of CST elements Six stress quantities are computed for each CST element Oy Oy Try 01 02 O where Ox Oy Try are the stresses in the global coordinate system O O2 are the principal streses and 6 is the orientation of the principal stress axis A distributed load on a CST element is taken as an in plane surface traction Valid directions for loads are GlobalX and GlobalY The sign convention for load magnitude follows the orientation of the global axes Loads which are perpendicular or parallel to element sides which are not parallel to one of the global axes must be broken down into components which are parallel
213. s will print the global matrices that are appropriate to this problem In static analysis this would simply be the stiffness matrix in transient analysis this is the stiffness damping and mass matrices in modal analysis this is the condensed stiffness damping and mass matrices summary will generate a summary of all the material properties that were used in the problem The number of elements using a material total length surface area and mass of that material if masses are desired then the material properties definition must include a density and the total mass of the structure are given graphics fool will create a graphics file called foo1 so the user can visualize the structure with a standard graphing package to make sure that everything is connected where it should be The format of this file will be a list of coordinate triplets connecting each element with blank lines delimiting the end of an element cpp filename nocpp Idirectory Uname substitute filename for the pre processor to run on the input file do not run the input file through the pre processor add directory to the standard search path for include files in the pre processor undefine the macro name in the pre processor 5 1 USING FELT 43 Dname value define name to be the macro value in the pre processor Additionally every FElt file that is run through felt is pre processed by the C preproces sor except in DOS This allows
214. sary 12 3 Introduction to the general FElt routines Throughout FElt simplicity and readability have generally won out over speed and effi ciency at least where the mathematics are concerned the system and GUI code are another story entirely What this means is that FElt does not make use of many of the al gorithmic tricks that have been developed over the years for working with finite elements The most glaring example of this is probably the fact that the FElt matrix manipulation library is fairly simple and does not make use of symmetry in most cases 12 4 Details of a few general FElt routines 12 4 1 Finding the active DOF The first step in the solution of a FElt problem is to find all the degrees of freedom that the different types of elements in the problem use This information gets stored in two different six element arrays In the dofs_pos array a non zero value in the ith position indicates that the ith DOF is active The actual number in the ith position indicates which DOF this is for the current problem i e in a problem consisting solely of beams a 3 in the sixth position indicates that rotation about the z axis is the third DOF in the current problem The second array is dofs_num In this array the entries from one through the number of active DOF are all non zero A value of j in the ith position indicates that the standard DOF i is the jth DOF for this problem i e for our example problem consisting of beams only a
215. section 3 3 to a file mixed flt then the command o felt summary mixed flt gt mixed result 46 CHAPTER 5 THE FELT APPLICATION would produce a file mixed result which looked like the following Mixed Element Sample Nodal Displacements Node Dor 1 DOF 2 DOF 3 DOF 4 DOF 5 DOF 6 1 0 0 0 0 0 0 0032019 2 0 0 011522 0 0 0 0 3 0 0 0 0 0 0 0032019 4 0 0 0 0 0 0 13 0 29942 0 0 57 587 Ze 0 57 587 59654 0 29942 Node DOF Reaction Force 1 0 1 29942 1 Z 0 2 Z 0 3 29942 3 Zz 0 4 x 0 4 y 115 17 4 Z 0 4 Mz 0 Material Usage Summary Number 2 Length 12 0000 Weight 0 0000 Material spring 5 3 INTERPRETING THE OUTPUT FROM FELT 47 Number al Length 10 0000 Mass 0 0000 Total mass 0 0000 5 3 2 Transient analysis The output for transient analysis is much simpler It can consist of a table listing times and nodal displacements for all the nodes and DOF which were specified in the analysis parameters section or multiple tables if everything will not comfortably fit across one screen width and or a time displacement ASCII character plot relating the same information in a graphical sense or mulitple plots if there are more than ten nodes DOF Tabular output is the default You can disable it with the command line switch table To see graphical output use plot If we were to save the input file from the detailed example of section 3 4 to a file transient flt then the command o
216. sional element 31 4 2 2 2 Arbitrarily oriented three dimensional element 34 4 2 3 Timoshenko beam element o o 34 4 2 4 Constant Strain Triangular CST elements 35 4 2 5 Two dimensional isoparametric elements 36 4 2 5 1 General four to nine node element 36 4 2 5 2 Simple four node element cocoa 37 4 2 0 Plate bending element 44 ov Ge See Sed a a 37 42 7 Solid brickeleme nt lt oe Be Re Re tare ROR YS 39 4 2 8 Axisymmetricelements o a h eee bbe eaea 40 4 3 Thermal analysis elements 2 2 144 63 eee estare EO a 40 43d OO GICMIGNE yor A a g i 40 4 3 2 Constant Temperature Gradient CTG element 40 5 The felt Application 41 sl SmE l aoe gocs coce AR Be we Owe BA KR OG K OM K GG KG 41 5 2 Solvimg APPODIGM airada ha eee are 44 5 3 Interpreting the output from felt 26 cin ene aa 44 IAL Stale BOVIS o Boge Baw Bae IN 44 J32 Transient analysis 6 6 8 kw Be Be Reh tek unre eR eS 47 5 3 3 Modal analysis bib eed kth gtd a 49 So Thernal analysis ss a Se meus wR eS ESE ER OEE 51 Died SPeEcMalANAlySIS lt lt ss RAR AAA A ae a es aa 51 6 Using WinFElt 53 6 1 Introduction to WAPEE o haa ed how esse 53 62 Solving a proble ss soe ca ee ee ee KO KOM KREG KS 53 Go Textovipol erroe cerea drid a ieg 56 vi CONTENTS 64 Graphical outp t lt dra ses bed ees EERE SE EEE OHS 56 641 Contour plots 2 62 8 e oo norris 56 042 Lin
217. soidal cosinusoidal logarithmic parabolic reverse parabolic 100 CHAPTER 9 THE CORDUROY APPLICATION and reverser logarithmic If we are placing the sides of n elements along a line then we need to locate n 1 nodes on the line The coordinate x of each node is defined as follows for each of the spacing rules linear x LB 9 1 cosinusoidal x L Lcos 58 9 2 sinusoidal x Lsin 58 9 3 logarithmic xj L Llog p 1 9B 9 4 parabolic x LB 9 5 reverse logarithmic x L Llog 10 9B 9 6 reverse parabolic x L B 9 7 9 8 where B i 1 n 9 2 7 Generating a triangular mesh The triangular mesh generation capabilities of corduroy use Jonathan Shewchuk s Triangle routine In addition to specifications that describe the boundary and the holes of your two dimensional region you must also define the approximate number of elements to generate and a constraint on the maximum area of any one generated element The number of elements is specified with target The area constraint is given using alpha where is defined such that 9 9 The entire corduroy input file to generate the nodes and elements for the wrench model pictured in Figure 7 1 would look like this start node 1 unnecessary as this is the default start element 1 triangular mesh element type CSTPlaneStress angtol 20 dmin 0 5 min 100 except for min and max angspc 30 kappa 0 25 max
218. step 0 05 r alpha o in HHT o and transient thermal integration T TT 0 5 o gamma yin HHT Q integration T 0 25 o beta B in HHT a integration T 0 5 o mass mode element mass matrices to use T TT SP M lumped r nodes list of nodes for which you want results T TT SP LR LC 1 4 5 r dofs list of DOF at each result node T TT SP LR LC Tx Rz r Rk global Rayleigh damping constant for stiffness T M SP 3 0 o Rm global Rayleigh damping constant for mass T M SP 1 4 o input node node to be parametrically excited LR 4 T input dof DOF at the input node to be excited LR Ty T tolerance convergence tolerance SUB le 3 r iterations maximum permitted number of iterations SUB 100 r relaxation under over relaxation factor SUB 0 8 r load steps number of incremental steps to full load SUB 10 r Table 3 2 Meaning given to the various analysis parameters T indicates transient analysis TT is transient thermal SP is spectral M is modal LR is static analysis linear or nonlinear with load ranges LC is static analysis linear or nonlinear with load cases SUB is non linear static substitution analysis The r o column indicates whether the parameter is r equired for the given analyses or if the o ptional default value of 0 0 will be used if nothing is specified 24 CHAPTER 3 STRUCTURE OF A FELT PROBLEM Figure 3 3 A mixed element problem with distributed loads problem descripti
219. straint objects also allow the various information to be accessed as vectors or as in dividual components as shown in Table 10 6 If c is a constraint then c ix is its initial displacement vector and c ix 1 or c iTx is its initial translation along the x axis Com ponents may be assigned time varying expressions as in the case of forces 10 3 6 Distributed load objects Distributed loads or simply loads have the fields named by the FElt file syntax as shown in Table 10 7 The individual values can be accessed using node and magnitude For example 1 values 1 node returns the node object of the first load point and 1 values 1 magnitude returns its magnitude The direction field is used to indicate the direction of the distributed load The direc 112 CHAPTER 10 THE BURLAP APPLICATION Field name Description name m F pat Ix Iy Iz rho nu kappa Rk Rm Kx Ky name of material Young s modulus moment of inertia about x x axis moment of inertia about y y axis moment of inertia about z z axis cross sectional area polar moment of inertia bulk shear modulus thickness density Poisson s ratio shear force correction Rayleigh damping coefficient K Rayleigh damping coefficient M conductivity along x axis conductivity along y axis conductivity along z axis heat capacity Table 10 4 Fields of a material object 10 3 BURLAP AND FELT 113 Field name Descr
220. t Figure 7 8 The load dialog box CST etc Also it is always your responsibility to make sure that the loads assigned to an element have a valid direction and node ordering for that element type 7 5 2 Working with nodes and elements The first step in actually laying out the geometry of a problem is usually to lay the nodes out elements have to be attached to nodes after all Let s assume that we went through everything above and defined all the materials constraints and loads in our sample problem there are no forces We made steel the active material and free the active constraint we ll deal with the loads later Now to add a node all we have to is select add from the node menu You choose the location of the node simply by clicking in the drawing area A node will be created at that location or the nearest snapped location if snap is on Alternatively you can specify the exact coordinates within the command window You do not need to select add for each individual node Once selected you are effectively in add node mode and can simply add additional nodes by repeatedly clicking or typing coordinates To stop adding nodes click on the quit or abort button in the main control area Most other command functions available from the main control area will be greyed out while you add nodes In our sample problem all we need to do is click on the location of our four nodes Nodes will be numbered consecutively in the order in which t
221. t FElt cannot do Those tools are for the most part not free and if they are they re usually 20 years old FElt is a new continually evolving system that tries to provide lots of modern workstation type features and of course it s completely free About this Manual This manual documents version 3 05 of FEIt It is part of a futile attempt to provide com prehensive accurate documentation for the FElt system We do our best to try to keep it up to date with the latest version of the software we make no guarantees however and chances are good that there are things wrong in here If you find something that behaves differently than the way this document says it should behave then please let us know Organization of this Manual This manual is organized in the following way Chapter 1 gives you an introduction to the types of problems that FEIt can solve Chapter 2 details some of the underlying mathematics for each of the analysis types supported by FEIt Chapter 3 discusses the basic structure of a FElt input file and how a general finite element problem is translated into the FElt language Chapter 4 describes the currently available elements within FEIt Chapter 5 introduces you to the simplest user interface to the FElt system the command line application felt xvii xviii TYPOGRAPHICAL CONVENTIONS Chapter 6 covers the WinFElt graphical encapsulator for the MS Windows based envi ronments Chapter 7 introduces velvet
222. t that is the name of the element to be removed In our case the call would be remove_definition truss 10 5 Tips on using interactive mode Most of the time burlap will be used in interactive mode expressions will be typed in at the prompt and evaluated Functions are best created with an editor as a file which is then included 1 system vi foo b 120 CHAPTER 10 THE BURLAP APPLICATION 2 include foo b 3 system vi foo b 4 include foo b It soon becomes tedious to retype the same expressions If burlap is compiled with the readline library then you can use the built in history mechanism system vi foo b include foo b ls li For those not familiar with csh or bash s executes the last line that begins with s Similarly i executes the last line that begins with i You can save yourself more typing by using the word completion abilities of the readline library Pressing the tab key will complete the current word If there is not a unique completion then you will hear a bell and pressing the tab key again will list the completions 1 sy lt Tab gt symmetric system 1 system vi foo b 2 in lt Tab gt in integrate_hyperbolic inv include integrate_parabolic 2 include foo b The default set of built in completions includes all keywords such as function and return intrinsic functions such as include and sin and enumeration constants such as amp linear and null Ins
223. tatus h Finally Vector and Mat rix types are unit offset The create and destroy routines should make this transparent to the developer 176 CHAPTER 13 ADDING ELEMENTS TO FELT a CreateMatrix width height w Creates matrix of the given width and height This function allocates space and initializes fields within the matrix structure CreateVector height Creates a vector of length height DestroyMatrix a This will free the memory associated with a previously created matrix Once destroyed a matrix variable should not be used again until it is recreated DestroyVector a status status status status status status status This will free the memory associated with a previously created vector MultiplyMatrices a b c Multiplies matrices b and c and stores the result in a This is one of the few functions in which your destination matrix a cannot be one of your source matrices b or c TransposeMatrix a b Transposes matrix b and puts the result into a Source and destination ma trices cannot be the same MirrorMatrix a b This function is useful for filling in symmetric matrices Given a symmetric matrix b with only the diagonal and above diagonal terms filled in this function will complete all entries below the diagonal The result a will be the complete symmetric matrix ZeroMatrix a Fills all entries in matrix a with zeros ScaleMatrix a b x y Given a matrix b multip
224. teady state temperature value thermal stresses are not computed Output for transient thermal analysis can be tabular and or graphical The tabulated results simply show the temperature as a function of time for whatever nodes were listed in the nodes specification of the analysis parameters no dofs specification is necessary in transient thermal analysis If you specify the plot option on the command line then felt will generate a simple ASCII plot of time versus temperature for these same nodes of interest Just as with transient structural analysis tabular output can be turned off witht the table command line switch 5 3 5 Spectral analysis Depending on how felt is invoked and the nature of the problem the output for a spec tral analysis problem can either consist of transfer functions or of output spectra If the transfer switch is given on the command line then all output will be for trans fer functions only The transfer functions computed will be for all combinations of out put at the nodes and DOF defined by the nodes and dofs statements in the analysis parameters section of the input file and input at all DOF with a force applied For exam ple 1f you define nodes 2 and 4 and DOF Tx and Ty as the relevant DOF in the analysis parameters and there is force object with Fx and Fy non zero applied to both nodes 3 and 7 then you will get sixteen transfer functions as results If the input forcing is defined in a spectral sense e g
225. ted The expression list is a semicolon separated list of expressions If the while expression evaluates to zero then WHILE construct terminates and the result is a null value The BREAK expression of Section 11 7 4 may be used to alter the result of the WHILE The while expression must evaluate to a scalar function sum_of_first n i 0 11 7 CONTROL FLOW CONSTRUCTS 155 s 0 while i lt n do s sti i i3 1 end return s 1 write sum_of_first 3 2 write sum_of first 10 3 write while 0 do end null This example uses a WHILE expression to compute the sum of the first n integers The last write function illustrates the result of the WHILE loop The WHILE expression has the highest precedence although this rarely matters 11 7 3 FOR expressions for expression in expression do expression list end The FOR construct first evaluates the for expression and then the in expression For each element in the in expression the element is assigned to the for expression and the expression list is executed The expression list is a semicolon separated list of expres sions The result of the FOR expression is a null value unless a BREAK expression Sec tion 11 7 4 is used to explicitly produce a result The FOR expression has the highest precedence although this rarely matters The for expression must be an expression that can be assigned a value i e can appear on the left hand side of an assignment expression The
226. terial properties are set for this element E count 0 if element gt material gt E 0 0 error timoshenko element d has 0 0 for element gt number count if element gt material gt Ix 0 0 error timoshenko element d has 0 0 for element gt number count if element gt material gt G 0 0 error timoshenko element d has 0 0 for element gt number count if element gt material gt A 0 0 error timoshenko element d has 0 0 for element gt number count Young s modulus E moment of inertia Ix bulk modulus G bulk modulus G nu and kappa are somewhat special because we have to have 13 6 A DETAILED EXAMPLE 183 at least one If we have nu we ll use it to estimate element gt kappa according to Cowper s 1966 approximation If we have kappa we will of course always use it If we have neither it s an error iS if element gt material gt kappa 0 0 if element gt material gt nu 0 0 error timoshenko element d has 0 0 for Poisson s ratio nu element gt number count else element gt material gt kappa 10 0 1 0 element gt material gt nu 12 0 11 0 element gt material gt nu if we ve had any errors there is no point in continuing Ay if count return count get the local stiffness matrix and the transform
227. the highest precedence sox if y lt z then y else z works as expected The expression list is a semicolon separated list of expressions The result of the expression list is the result of the last expression in the list The else if and else con structs are optional The else if syntax is combined into one keyword that allows the entire IF expression to end with a single end keyword To have else if treated as two 154 CHAPTER 11 THE BURLAP SYNTAX keywords thus requiring multiple end keywords place a semicolon or other expression between the else and the if If an expression list is missing then it evaluates to null Similarly if the final else is missing and none of the if expressions evaluated to non zero the result of the entire IF expression will be null function max x y if x gt y then x else y end end 1 write max 1 2 2 In this example the IF expression returns x if x gt y and returns y if x lt y Since the return value of a function is the value of the last expression the function max returns the result of the IF expression function max x y if x gt y then return x else return y end end This function illustrates a more conventional way of writing the max function The result is the same regardless of which function is used 11 7 2 WHILE expressions while expression do expression list end As long as the while expression evaluates to non zero the expression list is evalua
228. the simpler nature of the governing ODE in the thermal analysis case Using a generalized trapezoidal rule we express the temperature vector update equation as Tar T At 1 a afi 12 12 Through some algebraic manipulations of the governing equation at time steps i and i and subsitution of equation 12 12 to eliminate time derivative terms we can write the implicit update equation for T as K Tay M At 1 a K T 1 a Ath 0At F41 12 13 where K M aAtK 12 14 and everything on the right hand side is known Logan 13 gives the following summary of the methods that result from various choices of a amp 0 simple forward difference scheme which is only conditionally stable 5 Crank Nicolson or trapezoidal rule which is unconditionally stable 3 Galerkin which is also unconditionally stable amp 1 backward difference which is unconditionally stable 12 4 9 Solving the eigenvalue problem FEIt uses a relatively unsophisticated procedure to solve for the eigenvalues and eigenvec tors in a modal analysis problem The first step is to actually remove constrained DOF from the global stiffness and mass matrices This procedure is less efficient than the simple zeroing out of constrained DOF used in transient and static analysis but it is necessary for the actual numerical algorithm used to solve the eigenvalue problem In terms of the reduced mass M and stiffness K matrices the
229. ther amp lumped or consistent If s is not specified then no mass matrix will be computed for e This function is merely a shorthand fore definition set_up e s solve_displacements K f Solves the linear system Kd f for the vector of global nodal displace ments d The sizes of the inputs must be consistent with the definition of the problem Additionally K and f should both be condensed K is ex pected to be compact The DOF related fields of the problem structure must be initialized as by calling find_dofs volume e Returns the volume of the element e The element must be a solid element 1 e e definition shape should be amp solid zero_constrained K Zeroes the rows and columns of K at all DOFs with a fixed boundary con dition and returns the new matrix Keona K itself is not modified K should be either a symmetric matrix or a column vector If K is a symmetric matrix then a 1 is placed on the diagonal of each zeroed row column The size of K 11 5 INTRINSIC FUNCTIONS 149 must be consistent with the definition of the problem The DOF related fields of the problem structure must be initialized as by calling find_dofs These functions alone are sufficient to solve all static and transient problems that fe t or velvet can solve As shown in Figure 11 1 it is relatively simple to solve a FElt problem using the finite element intrinsic functions of burlap 11 5 5 Miscellaneous functions The remaining in
230. tion 105 10 1 Introduction to the burlap environment 105 10 2 Using burlap lt c Sg Heed Lake mani ioina AA ARA RA A 105 10 2 1 Interacting with burlap ossaa aaae 105 10 3 burlap and FEIE s o sos coem e ade e om a A AAA AAA RA 108 1031 Element objects e e ee ae ek YR EMSS OEE OE a a Ge 109 10 3 2 Node objects 2 222 2 e845 ee be eS nee ed 109 103 3 Maternal Objects ciaoe RR SSS eR Bk SRR GOR a 110 IOSA Force OSC ee cce e gorr doa RRA SHOR p Goe BW ROR BS 110 113 72 Constant ObjeciS occiso Re HER EDM RED dS 111 10 3 6 Distributed load objects 4 62a ven beat eeeeweees 111 10 3 7 Element definition objects o o 113 10 3 8 Problem definition 6 air e RR ak oa 113 G32 A alysispaf meteis 200 atea don dt e a Ba es 116 10 4 Adding new element types to burlap o o o o o ooo o 116 10 5 Tips on using interactive mode 22 42 45 2i4 ee ee bee ewe ee 119 10 6 Common error messages ocio he eee Re REE ee 121 11 The burlap Syntax 123 ULA LISIS sae oe AAA A AR A 124 LE SAO e E A A Dl Be Se Date a i 125 ULA Constants a Se eae eee a tee ee eee dla e a oe 126 118 Operators 2 1 52 See Ge Pe Ree eii A SERRE a AT 126 11 4 1 Expression Separators ee ee eR RS eR ES 126 11 4 2 Assignment expressions 2c ee ee ee 4 5 127 LIAS Logical OR operator 2 0 24 0 oa Dee oom Dre Bee Se Ss 128 11 4 4 Logical AND operator 129 11 4 5 Equality operators 2 222
231. tiplication multiplying a matrix by a scalar is equivalent to scaling the matrix by the scalar and multi plying two matrices results in standard matrix multiplication In matrix multiplication the inner dimensions must agree 1 write 2 3 6 11 4 OPERATORS 135 X Y Z Z X Y scalar scalar scalar z xxy scalar matrix matrix Vi j Zij x x Y j matrix scalar matrix Vi j Zij Xij Xy matrix matrix matrix Vi j Zij Di Xix X Yj Table 11 8 Type table for the multiplication operator 2 write 2 1 31 2 6 3 write 1 2 3 1 2 31 14 4 write 1 2 3 1 31 1 3 2 6 3 9 5 write 1 2 3 4 stdin 5 size mismatch in expression 1 x 2 1 x 2 burlap supports two forms of division left division and right division The right divi sion of X and Y is approximately equivalent to multiplying X by the inverse of Y X X Z Z X Y scalar scalar scalar z x y scalar matrix matrix Vi j Zij x x ve matrix scalar matrix Vi j Zij Xi j y matrix matrix matrix Z ZY X Table 11 9 Type table for the right division operator The first three cases in Table 11 9 are as expected Note that in the second case where Y is a matrix its inverse must be explicitly computed which may result in numeric instability and thus should be avoided In the fourth case where two matrices are divided then an LU factorization c
232. trinsic functions are concerned with a variety of topics including file access input output and the user interface concat s t Returns the concatenation of s and both of which must be strings eval s Evaluates the string s as a burlap expression The return value of the func tion is the return value of the evaluated expression exit n Exits burlap with exit code n Normally an exit code of 0 indicates success and a non zero exit code indicates an error If n is missing then 0 is used Note that the built in aliases define exit as exit so from the prompt typing exit will exit burlap with exit code 0 There is no return value from the function help s Requests help on topic s If s is not specified or is the empty string then a list of help topics is printed If s is the name of an intrinsic function then the help topic for that function is printed Note that the built in aliases define help as help so from the prompt you need to type help foo rather than help foo to retrieve information on topic foo in fact help foo will not work since it will expand to help history n Prints the command history list If n is given then only the last n events are printed The built in set of aliases define h as history 20 include s Includes the file named by s The file is interpreted just as if it it were specified on the command line Therefore it is executed in the global scope and not within a fun
233. trix will take displacement in the z direction into account if there are 3 d elements in use consequently those displace ments should be constrained In order to account for initial conditions in transient analysis problems a constraint specification may also include the initial at time t 0 displacements and velocity and acceleration in the translational DOFs Initial displacements are given with ITx IRz etc You can specify initial accelerations with Ax Ay and Az Initial velocities are given by Vx Vy and Vz Unspecified velocities and displacements will be taken as 0 0 If there are no initial accelerations specified 1 e none of the nodes have a constraint with an acceleration assigned then the initial acceleration vector will be solved for by the math ematical routines based on the initial force and velocity vectors If any of the nodes have a constraint which has an acceleration component defined even if that component is assigned to 0 0 then the mathematical routines will not solve for an initial acceleration vector they will build one based on the constraint information assigning 0 0 to any component that was not specified What this means is that you cannot specify the initial acceleration for only a few nodes and expect the mathematical routines to simply solve for the rest of them If you specify any of the initial accelerations then you are effectively specifying all of them 3 2 6 Forces The forces section s
234. ts the font used for node and element numbers The default font is 5x8 86 CHAPTER 7 THE VELVET APPLICATION Associated resource labelFont toolFont font Sets the font used for text drawn using the tools The default font is g 22 Associated resource toolFont cpp filename substitute filename for the pre processor to run on the input file nocpp do not run the input file through the pre processor Idirectory add directory to the standard search path for include files in the pre processor Uname undefine the macro name in the pre processor Dname value define name to be the macro value in the pre processor Chapter 8 Post processing with velvet 8 1 Solving a problem with velvet Depending on what you want your solution to include there are a couple of different ways to go about solving a FElt problem from within velvet In addition to the standard tabu lar type FElt output see section 5 3 you can also have velvet generate two dimensional color shaded stress or displacement plots and or plots of the structure with magnified dis placements applied For transient analysis problems graphical time displacement or time temperature plots replace the ASCII versions that felt produces Transient structural analy sis can also include an animation of the simulation For spectral analysis graphical plots of the transfer functions or ouput spectra replace felt s ASCII versions velvet can also draw graphical represe
235. uding an appropriate copyright notice and a notice that there is no warranty or else saying that you provide a warranty and that users may redistribute the program under these conditions and telling the user how to view a copy of this License Exception if the Program itself is interactive but does not normally print such an announcement your work based on the Program is not required to print an announcement These requirements apply to the modified work as a whole If identifiable sections of that work are not derived from the Program and can be reasonably considered independent and separate works in themselves then this License and its terms do not apply to those sections when you distribute them as separate works But when you distribute the same sections as part of a whole which is a work based on the Program the distribution of the whole must be on the terms of this License whose permissions for other licensees extend to the entire whole and thus to each and every part regardless of who wrote it 208 APPENDIX B THE GNU GENERAL PUBLIC LICENSE Thus it is not the intent of this section to claim rights or contest your rights to work written entirely by you rather the intent is to exercise the right to control the distri bution of derivative or collective works based on the Program In addition mere aggregation of another work not based on the Program with the Pro gram or with a work based on the Program on a volume of
236. unctions Nj x L and Np 1 x L Any introductory text on finite element analysis for mechanics at least probably starts out by deriving the stiffness matrix for just such an element see Chapter 3 of 13 for instance The stiffness matrix for this type of element in the local element two dimensional reference frame is the familiar A EA L EA L k 4 1 EA L EA L The lumped mass formulation of the local mass matrix is given by pAL 1 0 4 2 m oal 4 2 4 2 STRUCTURAL ANALYSIS ELEMENTS 31 and the consistent mass formulation by n PL p l Me 6 1 2 4 3 The 2 x 2 matrices in local element coordinates are transformed into the global coordi nate system and into 6 x 6 form according to k TYkT 4 4 where the transformation matrix is given by cos cos0 cos0 0 0 0 T 0 0 0 cos cos cos 4 5 The direction cosines are simply the projections of the local coodinate axes onto the global coordinate axes X1 X2 p 5 4 cos 7 4 6 Y1 7 y2 8 4 7 cos 0 EE 4 7 21 22 4 cos 8 7 4 8 The stress calculated for truss elements is the axial stress not load A positive quantity indicates tension negative indicates compression If a distributed load is assigned to a truss element it is assumed to be a linearly distributed axial loading condition 1 e it must be directed parallel Nothing else makes much sense since a truss element cannot carry bending or s
237. up condi tion of velvet Each of these options can also be set through your Xresources using the associated resource name sensitive Enable sensitive insensitive menu switching velvet has two major modes of operation normal mode and edit mode Each mode allows only certain operations to be performed This option allows disabling of the menus con trolling the inactive operations This is the default behavior but may be inappropriate on a slow server Associated resource sensitiveMenus sensitive Disable sensitive insensitive menu switching Associated resource sensitiveMenus numbers Enable element and node numbering on start up This is the default behav ior Associated resource numbers numbers Disable the drawing of element and node numbering on start up On some servers the node and element numbers may take some extra time to draw on big problems and since you ll probably turn them off anyways to reduce clutter you can use this switch to disable them from the start Associate resource numbers nodeColor color Sets the color of nodes and node numbers The default color is blue Asso ciated resource nodeColor elementColor color Sets the color of elements and element numbers The default color is black Associated resource elementColor toolColor color Sets the color of tools rectangles lines circles and interactive windows The default color is red Associated resource toolColor labelFont font Se
238. ur drawing just use zoom window from the canvas menu After selecting zoom window you can can define a bounding box by typing coordinate pairs in the command window or clicking and dragging out a box with the mouse Once you release the mouse button or enter the second x y pair velvet will rescale the drawing such that the portion you selected will fill the entire drawing area Selecting the zoom to fit option under the canvas menu will rescale the drawing area such that the full extent of the problem can be viewed in the current window If you want to move around a zoomed drawing you can either zoom out and zoom in on a new section 68 CHAPTER 7 THE VELVET APPLICATION and repeat this as needed or you can use the scroll bars on the left and bottom sides of the drawing area to move around a zoomed canvas 7 4 4 Dumping the drawing area Use the save canvas option under the canvas menu when you want to save a snapshot of the drawing area Note that this will not be a dump of the entire velvet window as was shown in Figure 7 1 The drawing area only will be saved either as an XWD or a standard PostScript file Figure 7 11 illustrating the types of drawing tools available in velvet was dumped with the save canvas option You select the filename through the standard file dialog Extra toggle buttons near the bottom of the dialog control whether the saved file will be in XWD or PostScript format This file could then be rendered for a prese
239. ur own understanding of the problem This might be the case if the elements had been automatically generated If on the other hand you have a paper sketch of the problem which you used to setup the problem then you may not want to lose that particular node numbering scheme In this latter case what you can do is toggle temporary renumbering by selecting renum ber temp from the nodes menu This toggle is equivalent to the renumber command line switch in the command line application felt If this switch is on the nodes will be renum bered internally during the computations and then restored to their original numbers before you actually see any output i e the output will reflect the original numbering scheme Both renumbering methods have their advantages The permanent scheme reduces over 7 6 MORE ON NODES AND ELEMENTS 77 Number 1 Type beam 4 b Nodes 1 2 Material steel Load 2 Load 1 sidel Load 3 Stresses No stresses available help accept dismiss delete new copy Figure 7 10 The element information and editing dialog head if you will be solving the problem numerous times do it once and then why waste the time to figure out that it can t get any better every time you try to solve it The temporary internal only scheme keeps the problem and the results referenced the way that you had originally defined it The additional overhead of doing the renumbering each time can be wel
240. urse We need to do a transformation to get them into local coordinates VectorData d 1 element gt VectorData d 2 element gt VectorData d 3 element gt VectorData d 4 element gt node 1 gt dx Tx node 1 gt dx Ty node 1 gt dx Rz node 1 gt dx Tx 13 6 A DETAILED EXAMPLE 187 VectorData d 5 element gt node 2 gt dx Ty VectorData d 6 element gt node 2 gt dx Rz T TransformMatrix element MultiplyMatrices dlocal T d We already have the element stiffness matrix because we set element gt retainK 1 in the definition structure This means that the global stiffness assembly routine didn t trash element gt K after it was done with it and we can use it again We will have to transform it back to local coordinates however Ay TransposeMatrix Tt T MultiplyMatrices temp T element gt K MultiplyMatrices khat temp Tt we can get the internal force vector through a simple matrix multiplication Ay MultiplyMatrices f khat dlocal Of course we may need to modify that for equiv nodal forces Ef if element gt numdistributed gt 0 count EquivNodalForces element NULL amp equiv 2 if count return count for i 1 i lt 4 i VectorData f i VectorData equiv i set up some memory for the stress structure and for the
241. using whatever printer facilities exist at your site lp lpr print etc If you want to plot the structure the command line might look like this felt graphics foo graph foo flt gt foo results 9 6 9 E myplotter foo graph The file foo graph could then be used with a program like gnuplot xmgr or xgraph here we used a mystical program called myplotter to visualize the structure and make sure that all the elements were connected properly 5 3 Interpreting the output from felt The output from felt is the most basic form of output that all applications in the FElt system produce That is no matter how you specify and solve a FElt problem either with felt or velvet the basic output will look the same The main output will vary depending on the analysis type defined in the input file In addition to the results from the specific analysis supplemental kinds of output which are available include material usage statistics debugging information and print outs of the global stiffness and mass matrices Supplemental output from felt is controlled via command line switches see section 5 1 5 3 1 Static analysis For a static analysis problem there are three sections of output nodal displacements ele ment stresses and reaction forces The nodal displacements are given in a table with the six 5 3 INTERPRETING THE OUTPUT FROM FELT 45 global DOF running across the top and the list of nodes going down The displacement of e
242. values in the stress structure We ll just use a quicky little convenience routine to do it for us It s important to 188 CHAPTER 13 ADDING ELEMENTS TO FELT set element gt ninteg before we call this function SetupStressMemory element establish the location of the stresses and the magnitudes of the stresses at each point This particular loop only works because there are two stress points and two stress values at each point a for i 1 i lt 2 itt element gt stress i gt x element gt node i gt x element gt stress i gt y element gt node i gt y element gt stress 1 gt values i VectorData f i element gt stress 2 gt values i VectorData f i 2 return 0 Now we get into the few routines that are local to this file 1 e private routines which we would only call when we were defining Timoshenko beam elements The first of these simply fills in the 4 x 4 local stiffness matrix There is no fancy integration here because we know how it all turns out so we save ourselves some computations by plugging straight into the entries in the matrix The second of these two routines will compute a transformation matrix for these elements Our own function to define the stiffness matrix in local coordinates 7 static Matrix LocalK element Element element static Matrix k NULL the local stiffness matrix
243. w This is the main control area for all of the operations in velvet The second is the command status line across the bottom of the window This line is used to display messages and for keyboard based input The third is the drawing area which occupies most of the window This is the area in which you can dynamically define and interact with the FElt problem The menus are used to define everything about a problem nodes elements forces constraints material properties distributed loads to configure velvet to use the drawing tools to configure the drawing area and to save open and solve problems 7 2 General features of the interface Many of the interface modules that you will see in velvet look very similar to one another Becoming familiar with a few basic operations that are common across all of these modules will allow you to use velvet much more easily and flexibly In general clicking an accept button will register the current state of a dialog with the 61 62 CHAPTER 7 THE VELVET APPLICATION F 2uelpruzess 193 0 159 5 A Cammare Figure 7 1 Main velvet drawing area with an interesting sample problem 7 3 WORKING WITH FILES 63 rest of the application Say you have a dialog with a few toggle buttons and a couple of entry blanks You could change the button states and type things into the entry blanks all you wanted without actually affecting the rest of velvet Once you click accept however all of y
244. w the analyst has to interface the mathematical code with the system code elements take up memory so memory will need to be allocated the generalized analysis routines will expect that the element routines take certain parameters and return certain values or perhaps initialize certain arrays Besides simply defining an el ement stiffness matrix it is the responsibility of the element routines to calculate equivalent nodal forces due to distributed loads and to calculate their own stresses once the general routines have computed the nodal displacements If you want to be able to do transient analysis the same element set up routine that initializes the element stiffness matrix must also build the element mass matrix If you re unfamiliar with the C programming language you should probably start there You might be able to get away with just hacking something together based on existing ele ments but efficiency and elegance will probably suffer The material in 12 is considered by many to be the definitive word on C you may want to start there Other good books on C undoubtedly abound Another simple starting point for C novices would be section 12 2 on understanding the data structures used in FEIt 171 172 CHAPTER 13 ADDING ELEMENTS TO FELT 13 2 Necessary definitions and functions The general routines will expect three things to be defined Aside from these three things the element routines can use whatever local functions and
245. which is attached as a member of the standard matrix data structure to convert from a row column location to a single address in the global vector representations 166 CHAPTER 12 THE ALGORITHMS BEHIND FELT 12 4 5 Dealing with boundary conditions FElt can handle several types of boundary conditions The simplest is a fixed DOF at a given node In this case the row and column of the global stiffness matrix corresponding to the fixed DOF are simply filled with zeros and a one is placed on the diagonal at that location a zero is placed in the appropriate DOF of the global force vector just for clarity The second type of condition is a displacement condition such as might be found in a settlement of support problem In this case before we eliminate the rows and columns of the stiffness matrix we need to adjust the force vector to account for the displacement Given a globally numbered DOF n which has a displacement condition dx global force vector F and global stiffness matrix K then for each DOF i in the problem F i F i K i n dx 12 3 After this adjustment the rows and columns of K associated with DOF n can be zeroed out as in the ordinary fixed case The magnitude of the displacement condition dx is placed in the F n to insure that dx will re appear exactly in the final solution for nodal displacements The final type of condition is a hinge and because the primary adjustments that this condition requires were m
246. y esc is dismiss or cancel ctrl d is delete ctrl n is new and ctrl c is copy 7 10 2 Command names All of the commands that are available through menu options in the main control area are also available by typing a text command into the command window at the bottom of the main window To use a text command simply type it into the command window and hit return If you just type return with no command entered velvet will repeat the last command that you entered The following table lists the equivalencies for each command The first column lists the name of the menu button in the control area the second lists the applicable entry under that menu if any and the third lists the command name s that perform the same action The fourth column provides additional ways that you can perform the same function either through a keyboard shortcut or through mouse interaction with the drawing area 84 CHAPTER 7 THE VELVET APPLICATION Menu Command Text command Other alternatives File New new Ctrl f n Open open Ctrl f o Save save Ctrl f s Save as save as Ctrl f w Restore restore Ctrl f r Open database open database Update database update database Exit exit Ctrl f x Solutions Solve solve Ctrl v Animate animate Results Output define output Problem Analysis define problem Postprocess Plot stresses
247. ze the arrays Changing the DOFs yourself will probably just lead to non singular matrices 10 3 9 Analysis parameters The analysis parameters structure contains information related to a specific analysis type Field name Description gamma parameter in Newmark integration beta parameter in Newmark integration alpha parameter in H H T mass_mode mass mode nodes array of node numbers of interest dofs array of dofs of interest start starting time or frequency stop ending time or frequency step time or frequency increment num_dofs number of dofs of interest num_nodes number of nodes of interest Table 10 11 Fields of an analysis parameters object Unlike the problem definition structure most of the fields in the analysis parameters structure are changeable However the dofs and nodes fields are similar to the dofs_num and dofs_pos fields of the problem definition The individual components are read only but they may be assigned a row vector If analysis dofs is assigned a valid row vec tor then analysis num_dofs is updated and is equivalent to length analysis dofs Similarly if analysis nodes is assigned a valid row vector then analysis num_nodes is updated and is equivalent to length analysis nodes The length function returns the number of items in an array matrix or string 10 4 Adding new element types to burlap Adding a new element type to FElt can be very time consuming and error prone burlap allows

FElt: User's Guide and Reference Manual pdfsubject

Contents

Download Pdf Manuals

Related Search

Related Contents