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1. ok lt ak OK K DOUBLE PRECISION FUNCTION PARX T1 T2 T3 IBCT IMPLICIT REAL 8 A H 0 Z PARX T1 END DOUBLE PRECISION FUNCTION PARY T1 T2 T3 IBCT IMPLICIT REAL 8 A H 0 Z PARY T2 END DOUBLE PRECISION FUNCTION PARZ T1 T2 T3 IBCT IMPLICIT REAL 8 A H 0 Z PARZ T3 GOTO 99999 END KKK K K ook ook k k oko kK kK kK okok oko oko 2 ok ok ok ook oko oko ook ok ook ook ook ook 2 K 2K K K kK OK K SUBROUTINE DCORVG KNPR KVEL NVT NVEL IMPLICIT DOUBLE PRECISION A C H 0 U W Z LOGICAL B DIMENSION DCORVG 3 KNPR KVEL NVEL DO 10 IVT 1 NVT INPR KNPR IVT IF INPR EQ 0 GOTO 10 IEL KVEL 4 IVT PX DCORVG 1 IVT PY DCORVG 2 IVT PZ DCORVG 3 IVT IF ABS PZ 0 0D0 GT 1D 8 AND ABS PZ 0 41D0 GT 1D 8 AND ABS PX 0 0DO GT 1D 8 AND ABS PX 2 50D0 GT 1D 8 AND ABS PY 0 0DO GT 1D 8 AND ABS PY 0 41D0 GT 1D 8 THEN 0 5000 0 2000 0 0500 DL SQRT PX PXM 2 2 DCORVG 1 IVT PXM RAD DL PX PXM DCORVG 2 IVT PYM RAD DL PY PYM GOTO 10 ENDIF IF ABS PZ 0 0D0 LE 1D 8 OR ABS PZ 0 41D0 LE 1D 8 THEN 0 5000 0 2000 0 0500 RADH 0 05D0 1D 8 IF ABS PX PXM LE RADH AND ABS PY PYM LE RADH AND IEL
2. SUBROUTINE PTSDAT TIMENS DNU Data for Point output for fpost ok 2 K oe oe ok A 3K ke A 2 3K k oe ke 2 kK oe oe ok 2 K 2 oe oe 2 ok ge ke ok ooo eoe 2 2K OK IMPLICIT DOUBLE PRECISION A C H 0 U W Z LOGICAL COMMON NSPTS KPU 2 KPP 4 SAVE ee Points for velocity and pressure ee KPU 1 7263 KPU 2 3444 1 3491 2 3557 3 3575 4 3444 99999 END In NEUDAT the boundary parts containing natural boundary conditions are defined This is done by prescribing the cartesian coordinates for this manifold Analogously in BDPDAT the coordinates for the pressure integral boundary parts are set and in BDFDAT the coordinates for calculating lift and drag If not desired set the corresponding IFLAG parameters to 0 Again in PTSDAT certain mesh points are defined in which some velocity components KPU 1 KPU 2 and some pressure values KPP 1 KPP 2 KPP 3 KPP 4 are printed for the runtime protocol 2 2 input data of FEATFLOW parq2d f This the F file is either a standard FEAT parametrization file which is created by your own see EAT2D manual or the special OMEGA2D parametrization file In this case the file parpre f has to be copied onto parq2d f parg3d f 99999 99999 99999 C
3. IA IK IMPLICIT DOUBLE PRECISION A C H 0 U W Z LOGICAL B Case 0 Set number of Neumann boundary parts ee IF INPART EQ 0 THEN INPART 1 ENDIF IF INPART GT O THEN C IF INPART EQ 1 THEN DPARN1 1DO DPARN2 2DO INPRN 1 ENDIF C ENDIF C 99999 END C SUBROUTINE PTSDAT TIMENS DNU Data for Point output for fpost and bdpres and bdforc SEFC IK ak IMPLICIT DOUBLE PRECISION A C H 0 U W Z LOGICAL B COMMON NSPTS KPU 2 KPP 4 KPX 4 KP1 2 DPI 2 2 DPF 2 SAVE EXTERNAL UE PU 1 22 PU 2 2 1 42 2 46 PP 3 2 4 22 1 41 PX 2 10 Parameters for 2 integral pressures bdpres f 1 2 DPI 1 1 0DO DPI 2 1 1DO 2 2 DPI 1 2 0 00D0 DPI 2 2 0 25D0 22 Structure of FEATFLOW Parameters for lift DFW and drag DAW in bdforc f INPR 2 dfw 2 int dpf 1 dut dn ny p n x ds dpf 2 C daw 2 int dpf 1 dut dn n x p n y ds dpf 2 C Q222222222222z2222
4. 0 16m 0 0 mc 0 1m y 0 41m p 0 15m X 0 41m cm e Ro wt Inflow plane 0 0 0 0 0 H Figure 3 8 3D domain channel with cylinder corresponding coarse mesh which is simply a 3D extension of our c2d2 tri grid can be obtained by using TR2TO3 First of all do the following cd FEATFLOW application example input files make f tr2to3 m mv tr2to3 make f trigen3d m mv trigen3d cd FEATFLOW application example Next the corresponding 3D coarse mesh adc c3d2 tri see data tr2to3 dat is gen erated by executing TR2TO3 Here in z direction 8 non equidistant layers are defined The following Avs picture can be generated by performing TRIGEN3D Now we are ready to start our 3D Navier Stokes calculation Again let us begin with the stationary example for Reynolds number 20 In this case we use the data file input files indat3d f stat and the following already well known procedure has to be performed cd FEATFLOW application example input files cp indat3d f stat indat3d f make f cc3d m mv cc3d cd FEATFLOW application example 3 3 3D example 61 221 E Figure 3 9 Coarse mesh c3d2 tri This data file input files indat3d f stat contains the following definitions The inflow velocity profile at 2 0 0 is parabolic with a maximum value Umar 0 45 but leading to the same mean velocity as in the 2D case There is one boundary
5. FEATFLOW Navier Stokes tree Discrete Proj Scheme Nonlinear solvers Figure 1 2 The solver structure of FEATFLOW 10 Mathematical Background 1 2 1 Discretization schemes Spatial discretization nonconforming nonparametric rotated bi trilinear finite elements with edge oriented d o f s for velocity piecewise constant finite elements for pressure a priori adapted coarse meshes with exact boundary adaptation during successive refinements adaptive refinement leading to conforming triangulations possible adaptive FE upwinding for convective terms Samarskij upwind ing adaptive streamline diffusion for convective terms natural do nothing b c s flux and pressure drop b c s Temporal discretization One step 0 scheme Implicit Euler Crank Nicolson scheme Fractional step 0 scheme as strongly A stable implicit time step ping scheme of 2nd order adaptive time stepping by estimating the local discretization error 3 substeps with and 1 substep with 3At extrapolation of partial solutions for higher accuracy 1 2 Discretization and solution techniques FEATFLOW 11 1 2 2 Nonlinear solvers Nonlinear iteration for stationary and nonstationary problems of Navier Stokes or Burgers type fixed point iteration quasi Newton with adaptive step length con trol by nonlinear defect minimization addit
6. U V 0 0 H 0 16m U V 0 outlet 0 45m to im inlet y 0 15m U V 0 0 0 E gt 2 5m Figure 3 1 2D domain channel with cylinder This can be easily done with OMEGA2D requiring a computer with PC emulation at least an AT 286 or higher and WINDOWS 3 1 or higher We start in OMEGA2D with describing the two boundary components first we prescribe the outer rectangle by setting the first point at the origin and then proceeding in counter clockwise sense until this boundary curve is closed Second we draw a circle with start ing point at 0 45 0 20 and center point 0 50 0 20 end point is again at 0 45 0 20 and it is very important to perform a negative circle that means the parametrization is proceeding in a clockwise sense For more details look at the OMEGA2D manual Next we define mesh points at the boundary components and in the interior It is re markable that boundary points cannot be set arbitrarily but next to another existing boundary point and then they can be moved along the boundary curve Furthermore you are always able to undo your last actions or to remove points and even makros and boundary components when needed In parallel to defining mesh points makros that means quadrilateral elements can be defined by clicking at four mesh points to form an element Finally this session has to be saved let us assume as c2d geb c2d prm and c2d tri and the files 2 ge
7. REL U1 REL U2 DEF U1 DEF U2 RHONL OMEGNL RHOMG1 RHOMG2 EHE HERE HE HE HERE EHE HERE H HERE HH HH EIER EHE EH HERE 1 68 TIME 0 295D 01 REL2 P 0 151D 01 RELM P 0 581D 01 REL U1 REL U2 DEF U1 DEF U2 RHONL OMEGNL RHOMG1 RHOMG2 0 0 69D 04 0 94D 04 1 0 93D 01 0 12D 00 0 21D 05 0 10D 05 0 23D 01 0 10D 01 0 52D 02 0 41D 02 2 0 72D 02 0 62D 02 0 66D 07 0 80D 07 0 29D 01 0 10D 01 0 13D 01 0 26D 01 IT DIV L2 RHOMGP 2 68 0 2960 01 REL2 P 0 153D 01 RELM P 0 537 01 EHE HERE HE HE EHE EE EIER HE IT REL U1 REL U2 DEF U1 DEF U2 RHONL OMEGNL RHOMG1 RHOMG2 EHE HERE 3 68 TIME 0 297D 01 REL2 P 0 279D 01 RELM P 0 961D 01 000000000000000000000000000000000000000000000000000000000000000000000000000020000 OLD DT 0 96D 02 U L2 0 13D 02 U MX 0 51D 02 P L2 0 45D 02 P MX 0 95D 02 CHOICE 0 68 NEW DT 0 954D 02 OLD DT 0 960D 02 000000000000000000000000000000000000000000000000000000000000000000000000000020002 P VELO 0 12490D 01 0 45097D 00 0 10542D 01 0 67446D 00 P PRES 0 17141D 01 0 41863D 00 0 64658D 00 0 17289D 00 58 Examples for the use of FEATFLOW I PRES P FLUX total mavec konv bdry LC ILU U mg P mg STATIS 0 30836D 00 0 41448D 02 I FORCE 0 29159D 01 0 54744D 00 0 20401D 00 SHSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS 68
8. FEATFLOW Finite element software for the incompressible Navier Stokes equations User Manual helease 1 1 5 Turek Chr Becker University of Heidelberg Institute for Applied Mathematics Im Neuenheimer Feld 294 D 69120 Heidelberg Germany Heidelberg 02 01 1998 Contents Short description 1 Mathematical Background I Introduction D WON a EU oues 12 Discretization and solution techniques in FEATFLOW 1 2 1 1 2 2 1 2 3 1 2 4 Discretization schemes Nonlinear solvers ee Linear solvers i cok Gee ow bite Bod deor hus ome pen Ned Discrete projection method 1 3 The Navier Stokes solver packages in FEATFLOW 2 Structure of FEATFLOW 21 programming structure of FEATFLOW llle 2 2 The input data of FEATFLOW 2 2 2 2 2 2 2 3 2 2 4 2 2 5 2 2 6 2 2 7 General comments for user input 2 22 indat2d f indat3d f parq2d f parq3d f data files for user trigen2d dat parameter flle for TRIGEN2D tr2to3 dat parameter file for 2 03 trigen3d dat parameter file for TRIGEN3D pp2d dat pp3d dat parameter files for PP2D and PP3D cc2d dat cc3d dat parameter files for CC2D and CC3D 2 3 The file structure of FEATFLOW 4 4 s e ess 2 9 1 2 3 2 2 3
9. 272 TIME 0 297D 01 RELU L2 0 25D 01 RELP L2 0 20D 01 REL 0 25D 01 SHSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS time time time time time time time time TICS NWORK IWORKG IWMAXG IWORKI IWMAXI IWORK IWMAX P mg subst mg P nonli img U time eps near 3364 2498697899 309 36813378334 406 31425285339 2 0668334960938 87 866734504700 323 13556003571 1063 4651870728 1141 2314567566 1300000 116168 119323 967903 967903 1039351 1203452 5327 6336588524 5320 4342214540 3 6333331540227 38 716011047363 1797 2014658451 487 26168847084 647 08451652527 3 0346069335938 139 76541614532 520 05523777008 1709 3506851196 1771 5327262878 428 2326 U MULTIGRID COMPONENTS in percent smoothing solver defect calc prolongation restr P MULTIGRID COMPONENTS in percent iction smoothing solver defect calc prolongation restr iction 42 218687909096 2 2564278443300 32 417565448349 20 300764952555 2 6163307795102 50 378172496605 4 9000111811739 16 941610100891 19 692759428133 7 8014891041156 3 2 2D example 59 Some explanations ILEV NVT NMT NEL NVBD ILEV NU NA NB as before ILEV NP NC means number of pressure unknowns and nonzero matrix entries for pressure matrix We perform macro time step 68 at absolute time 2 94s and the actually chosen time step is t 0 0096 As predicto
10. IBLOC EQ 2 THEN FDATIN ODO ENDIF IF IBLOC EQ 3 THEN FDATIN ODO ENDIF ENDIF ee IF ITYP EQ 5 THEN 0 0 ENDIF Case 6 Right hand side for momentum equation IF ITYP EQ 6 THEN IF IBLOC EQ 1 THEN FDATIN ODO ENDIF IF IBLOC EQ 2 THEN FDATIN ODO ENDIF IF IBLOC EQ 3 THEN FDATIN ODO ENDIF ENDIF Case 7 Right hand side for continuity equation ee IF ITYP EQ 7 THEN 0 0 ENDIF C Case 8 Mean pressure values ee IF ITYP EQ 8 THEN FDATIN ODO 2 2 input data of FEATFLOW 25 C ENDIF C 99999 END C akk kk k ak oko k k k k k oko 2k ak 2k k ok ok ok oko k k ok K 2K 3k 2k 2k k 2k k ook ok oko oe K k oe 2k ak ook ok oko oko K k k 3k 2k 2 2 2 k k k k K K K 2K ok 2 ok SUBROUTINE NEUDAT IEL INPR PX PY PZ TIMENS IFLAG Neumann boundary part ak k ak ak ak ak 2k ak 3K ak 3k 2k IMPLICIT DOUBLE PRECISION A C H 0 U W Z LOGICAL B ee Set Neumann boundary parts IF PX EQ 2 5D0 THEN IFLAG 1 ENDIF 99999 END SUBROUTINE BDPDAT IEL INPR PX PY PZ TIMENS IF
11. Ludmilla Rivkind University of Heidelberg rivkind gaia iwr uni heidelberg de Friedhelm Schieweck University of Magdeburg Friedhelm Schieweck Mathematik Uni Magdeburg de Rainer Schmachtel University of Heidelberg rainer gaia iwr uni heidelberg de Peter Schreiber University of Heidelberg schreib gaia iwr uni heidelberg de Stefan Turek University of Heidelberg tureQgaia iwr uni heidelberg de John Wallis University of Heidelberg wallisQgaia iwr uni heidelberg de Owen Walsh UBC Vancouver owen math ubc ca 71 C Appendix Future projects in FEATFLOW In the following we list the current projects which are under development and which shall be added to one of the next releases of FEATFLOW Concerning the improvement of the discretization and solution schemes we are just or will soon begin with developing software containing The mixed coupled solvers CP2D and CP3D Periodic and moving boundaries A Galerkin method with adaptive error control in space space time Galerkin method with adaptive error control A parallel version of all codes for shared and distributed memory machines An improved FEAT version FEAST Software for a improved preprocessing OMEGA2D as CAD tool under JAVA see the DEVISOR OMEGA3D as CAD tool under JAVA see the DEVISOR TRIGEN2D with mixing of refinement types TRIGEN3D with adaptive refinement strategies INTPOL2D and INTPOL3D for interpolation
12. but with extrapolation in time TIMEMX maximum absolute time for stopping DTMIN parameter for smallest time step during adaptive control DTMAX parameter for largest time step during adaptive control DTFAC factor for largest possible time step changes TIMEIN absolute time for start procedure EPSADI parameter for time error limit in start phase EPSADL parameter for time error limit after start phase EPSADU upper limit for acceptance for ABS IADTIM 3 IEPSAD parameter for type of error control 1 control of u L2 2 control of u MAX 3 control of p L2 4 control of p MAX 5 control of MAX u L2 p L2 6 control of MAX u MAX p MAX 7 control of MAX u L2 p L2 u MAX p MAX 8 control of MIN u L2 p L2 u MAX p MAX IADIN parameter for error control in start phase 0 EPSADL EPSADI for T TIMEIN 1 EPSADL linear combination EPSADI EPSADL for T lt TIMEIN 2 EPSADL logarithmic combination EPSADI EPSADL for T TIMEIN IREPIT maximim number of repetitions for ABS IADTIM 3 PRDIF1 parameter for reactive preconditioner usual PRDIF1 1 PRDIF2 parameter for diffusive preconditioner usual PRDIF2 0 or 1 2 2 input data of FEATFLOW 39 2 2 7 cc2d dat cc3d dat parameter files for CC2D and The input parameter for the 2D and the 3D version are almost identical The parameter files cc2d dat cc3d dat and pp2d dat pp3d dat differ with respect to a few parameters only Specials in c
13. nsdef f m011 f xmrout f dtkt f bbuild f projma f i dfkt f Li gupwd f mgrout f supwg f bndry f m011 f optcnl f It mgrout f Figure 2 2 The PP2D PP3D structure Beside the explained solver tools there are some other programs belonging to FEATFLOW which are essential for the preprocessing description of domain coarsest triangula tion pre adapted meshes output for graphics and for the preparation of an application generation and interpolation of a start solution These are in detail OMEGA2D OMEGA2D is a graphical preprocessor which provides tools for describing and defining the boundary components of a 2D domain and for generating a first and very coarse tri angulation of this domain This process is fully interactive and mouse oriented however actually running under DOS and WINDOWS 3 1 only It has been created by Matthies Schieweck in Magdeburg and further development is under progress In the next chapter we demonstrate how to use it and the complete manual of Version 1 6 can be found in the manual directory of FEATFLOW TRIGEN2D TRIGEN2D is preprocessing tool for designing better 2D coarse triangulations and to write the corresponding data in some special formats onto hard disc It reads in a parametrization in standard FEAT or in OMEGA2D format and a corresponding mesh in FEAT format and refines it locally in a successive manner The user has different possibilities for mesh
14. see the file MACHINES and have to be modified for computers or compilers not belonging to this list Each of these machine dependent directories contains some shell scripts beginning with make_ which have to be executed for installation see the section on installation In the next versions there will be some more computers added to our list containing optimized machine dependent compiler options Finally the directory libraries contains the compiled system libraries which have to be linked to each user application The actual FEATFLOW libraries created by the installa tion depending on the makefile used is in the subdirectory libgen while in libspec machine and compiler dependent system libraries are offered in correspondance to the special makefiles see above 2 3 4 Subdirectory application applications under FEATFLOW application is a directory thought to be the place for user applications The idea is to use a separate subdirectory for each application New subdirectories should be generated by linking or copying at least the input files and the data directory FEATFLOW is installed with 5 subdirectories user start comp dat example example and workspace user start is typical working directory for the user containing all needed input pa rameter files in data and a special subdirectory input files In this directory the user can find all makefiles copied during installation and adapted to the machine used and
15. tions on the reference element P VELO I IFORCE are analogously defined as in the 2D case As before we obtain files for graphical output namely in avs the file u1 inp and in byu the BYU files u1 vec velocity and p1 scl pressure A typical example for a AVS velocity plot in the midplane is shown in the following picture Figure 3 10 Velocity plot in the midplane z 0 205 for the stationary 3D calculation 3 3 3D example 65 Next we perform a nonstationary calculation again for Reynolds number 100 We use the data file input files indat3d f non and the same well known procedure has to be performed cd FEATFLOW application example input files cp indat3d f non indat3d f make f pp3d m mv cd FEATFLOW application example This data file input files indat3d f non is almost identical to the stationary one inflow velocity profile at 2 0 0 is again parabolic but with a maximum value Umax 2 25 UMEAN is changed for defining drag and lift Now we can execute PP3D with given parameter file data pp3d dat A solution is cal culated on level NLMAX 3 with stationary start solution file data DX3_stat Now the chosen discretization scheme for the convective parts is done by upwinding and we per form our time stepping with the fractional step scheme and fully adaptive time step control until TIMEMX 9DO Files for graphical output namely for avs and byu are w
16. 0 F cycle 1 V cycle 2 W cycle parameter for smallest number of mg steps for pressure parameter for largest number of mg steps for pressure parameter for mg interpolation for pressure 1 constant interpolation 2 rotated trilinear interpolation 3 trilinear interpolation 3 modified optimized rotated trilinear interpolation parameter for mg smoother for pressure 1 Jacobi 2 SOR 3 SSOR 4 ILU parameter for mg solver for pressure 1 SOR 2 CG 3 ILU 4 CG ILU parameter for number of mg smoothing steps for pressure parameter for number of mg solving steps for pressure 2 2 input data of FEATFLOW 37 NSMPFA RE UPSAM OMGMIN OMGMAX OMGINI EPSUR EPSUD DMPUD DMPUMG DMPUSL RLXSMU RLXSLU AMINU AMAXU EPSP DMPPMG DMPPSL RLXSMP RLXSLP AMINP AMAXP IPROJ NITNS EPSNS TIMENS THETA TSTEP IFRSTP INSAV INSAVN DTFILM DTAVS DTBYU IFUSAV parameter for change of number of mg smoothing steps on coarser levels for pressure n NSMP nNiMAXK ILEV smoothing steps if on level parameter for viscosity 1 NU parameter for convectice discretization gt 0 upwind parameter for Samarskij upwinding usual UPSAM 1 gt 0 SD leading coefficient with spatial adaptivity usual UPSAM 1 0 upwind simple first order upwinding 0 SD leading coefficient without spatial adaptivity usual UPSAM 1 parameter for lower limit for optimal r
17. 1 2 These two methods belong to the group of one step 0 schemes The CN scheme occasionally suffers from unexpected instabilities because of its only weak damping property not strongly A stable while the BE scheme is of first order accuracy only Another method which seems to have the potential to excel in this competition is the Fractional step 0 scheme FS It uses three different values for 0 and for the time step k at each time level For a realistic comparison we define a macro time step with K tn 1 tn as a sequence of 3 time steps of possibly variable size k Then in the case of the Backward Euler or the Crank Nicolson scheme we perform 3 substeps with the same 0 as above and time step k K 3 1 2 Discretization and solution techniques FEATFLOW 7 For the Fractional step 0 scheme we proceed as follows Choosing 0 1 n 0 1 26 and a 12 8 1 the macro time step tn gt tn K is split into three consecutive substeps with 0 a0 K 66 K I N u u OK Vp t I 80K N u u 0Kf Wai 0 I 0 0 1 8 1 9 0 KV prt K N u t jur 9 K 1 9 V eu Hl 9 0 I OKWVp zs o BOKN utt Junt 0Kf n 1 9 0 1 0 and in the following we use more compact form for diffusive advective part N u u vAu u Vu 1 4 Being a strongly A stable scheme the FS method possesses th
18. EQ 0 THEN DL SQRT PX PXM 2 PY PYM 2 DCORVG 1 IVT PXM RAD DL PX PXM DCORVG 2 IVT PYM RAD DL PY PYM GOTO 10 ENDIF ENDIF 28 Structure of FEATFLOW C 10 CONTINUE C END The functions parx pary parz simply prescribe the cartesian coordinates while the subroutine trparv provides transformations to more complex domains In this case a channel with an innner cylinder around 0 5 0 2 and radius r 0 05 is prescribed This file corresponds to the example in chapter 3 2 2 The input data of FEATFLOW 29 2 2 3 trigen2d dat parameter file for TRIGEN2D MT ICHECK IMESH CPARM IBDCHK IBYU IAVS NLEV IFMT CFILEI CFILEO ITYPEL FEAT parameter for output 0 no output gt 0 output see FEAT2D manual FEAT parameter for terminal output 0 no output gt 0 output see FEAT2D manual FEAT parameter for subroutine tracing 0 no tracing gt 0 tracing see FEAT2D manual parameter for type of parametrization 0 FEAT parametrization OMEGA2D parametrization name of OMEGA2D parametrization file parameter for boundary checking for further refinements 0 no checking gt 0 number of finer boundary points for check of boundary consistency for further refinement level for BYU output level for AVS output number of refined levels 1 parameter for level of output i formatted output of mesh data until level i i unformatted output of me
19. J Numer Anal 31 1312 1335 1994 M ller S Prohl A Rannacher R Turek S Implicit time discretization of the nonstationary incompressible Navier Stokes equations Proc 10th GAMM Seminar Kiel January 14 16 1994 G Wittum W Hackbusch eds Vieweg 67 68 BIBLIOGRAPHY 22 23 25 26 27 Rannacher R Numerical analysis of the Navier Stokes equations Appl Math 38 361 380 1993 Rannacher R On Chorin s projection method for the incompressible Navier Stokes equations in Navier Stokes Equations Theory and Numerical Meth ods R Rautmann et al eds Proc Oberwolfach Conf August 19 23 1991 Springer 1992 Rannacher R Turek S A simple nonconforming quadrilateral Stokes element Numer Meth Part Diff Equ 8 97 111 1992 Schafer M Turek S with support by F Durst E Krause R Rannacher Benchmark computations of laminar flow around cylinder in E H Hirschel edi tor Flow Simulation with High Performance Computers II Volume 52 of Notes on Numerical Fluid Mechanics 547 566 Vieweg 1996 Schieweck F A parallel multigrid algorithm for solving the Navier Stokes equa tion Impact Comp Sci Engnrg 5 345 378 1993 Schreiber P A new finite element solver for the nonstationary incompressible Navier Stokes equations in three dimensions Thesis University of Heidelberg 1995 Schreiber P Turek S An efficient finite element solver
20. causes your trouble Another reason might be if you have more than 1 boundary component that you must have positioned at least 3 mesh points on every boundary component I have problems with reading an unformatted solution file This happens on some computers Use formatted output The code starts correctly but before finishing the first iteration step it stops with an error Be sure that the corresponding provided memory size NNWORK is large enough The code starts correctly but the multigrid rates become almost identical 1 Perhaps your mesh is so bad But in most cases you prescribed an inflow profile only and your definition of the boundary part containing natural b c s is wrong so your flow cannot get incompressible Check this in the data file indat2d f analogously in 3D My solution schemes linear muligrid nonlinear schemes time stepping schemes are crashing Perhaps your problem is so bad If you perform a stationary calculation try the same with the nonstationary code with adaptively chosen time step size If you still have problems then your mesh might be too hard too large aspect ratios Check your triangulation Take the parameter files belonging to the most robust version However in most cases it is sufficient to check again the data and the parameter file 4 Known problems and errors during postprocessing Q A I have none of the proposed graphic tools That s a really hard proble
21. double precision built up every time parameter for gradient and divergence matrices 0 single precision with usual quadrature matrix in RAM double precision with usual quadrature matrix in RAM not yet 2 double precision exact matrix entries elementwise application not yet 3 single precision with exact evaluation matrix in RAM 4 double precision with exact evaluation matrix in RAM not yet quadrature formula for real mass matrix quadrature formula for Laplacian matrix quadrature formula for convective matrix quadrature formula for gradient matrix quadrature formula for right hand side parameter for smallest number of nonlinear steps matrix 1 linear extrapolation in time if INLMAX 1 constant extrapolation in time if INLMAX 1 parameter for largest number of nonlinear steps matrix 1 linear extrapolation in time if INLMIN 1 constant extrapolation in time if INLMIN 1 parameter for mg cycle 0 F cycle 1 V cycle 2 W cycle parameter for smallest number of mg steps parameter for largest number of mg steps parameter for mg interpolation 2 2 input data of FEATFLOW Al ISM ISL NSM NSL NSMFAC RE UPSAM OMGMIN OMGMAX OMGINI EPSD EPSDIV EPSUR EPSPR DMPD DMPMG EPSMG DMPSL EPSSL RLXSM RLXSL AMINMG AMAXMG ISTAT NITNS EPSNS TIMENS THETA TSTEP IFRSTP INSAV INSAVN 1 rotated trilinear for
22. for level of output i formatted output of mesh data until level i i unformatted output of mesh data until level i name of input coarse mesh name of created output coarse mesh 34 Structure of FEATFLOW 2 2 6 pp2d dat pp3d dat parameter files for PP2D and PP3D The input parameter for the 2D and the 3D version are almost identical The parameter files pp2d dat pp3d dat and cc2d dat cc3d dat differ with respect to a few parameters only Specials pp2d dat pp3d dat IMASSL ICUBML ISORTU ICYCU ILMINU ILMAXU IINTU ISMU ISLU NSMU NSLU NSMUFA ISORTP ICYCP ILMINP ILMAXP IINTP ISMP ISLP NSMP NSLP NSMPFA EPSUR EPSUD DMPUD DMPUMG DMPUSL RLXSMU RLXSLU AMINU AMAXU EPSP DMPPMG DMPPSL RLXSMP AMINP AMAXP IPROJ PRDIF1 PRDIF2 IMESH parameter for type of parametrization 0 FEAT parametrization 1 OMEGA2D parametrization active only in PP2D IRMESH input of mesh data 0 create mesh gt 0 read mesh created by TRIGEN2D gt 1 formatted mesh data 1 unformatted CPARM name of OMEGA2D parametrization file CMESH name of coarse mesh not longer than 15 characters CFILE name of protocol file ISTART input of start vector 0 start with homogeneous vector only b c s 1 read unformatted start vector from same level 1 formatted 2 read unformatted start vector from level 1 2 formatted CSTART name of start vector file not longer than 15 characters ISO
23. in FEATFLOW source feat2d src If I start the makefiles for CC2D or CC3D in my application I get the error message that the VANCA routines are not linked This happens on some computers Copy the file FEATFLOW source cc2d src vanca f to FEATFLOW source cc2d mg Then start there again the shell script cc2d mg m Do the analogous procedure for the 3D case 69 70 Appendix Troubleshooting with FEATFLOW 2 Known problems and errors during preprocessing PO PO PO have problems with OMEGA2D Check your PC emulation and that you have at least WINDOWS 3 1 If so ask the author for more advice I have problems with the german manual of OMEGA2D Sorry but in summer 1998 our first version of the DEVISOR will be finished I have problems with TRIGEN2D Check in your manual of the actual release that you marked an admissible set of elements for your desired adaptive refinement strategy 3 Known problems and errors during solution process oro I have problems while seemingly the triangulations on every level are created The most usual error is that your parametrization or coarse mesh is wrong For instance both parametrization curves follow the same direction error or the starting point of your parametrization is not captured by a mesh point Additionally be sure that all mesh points which were created while using OMEGA2D do really belong to an element In most cases one of these errors
24. of Van Kan 27 resp Chorin 4 In contrast 2 CC3D the upcoming CP2D CP3D analogously perform one nonlinear coupled solution step in a different way The file nsdef f handles the solution process being a solver of the nonlinear coupled equation with performing Oseen equations lin earized Navier Stokes equations as preconditioner in each nonlinear step m011 f with a special block Gauss Seidel smoother and following defect minimization optcnl f versions CP2D CP3D will differ in solving the linearized coupled equations by a spe cial multilevel scheme involving the linear discrete projection method multilevel discrete projection method This approach guarantees the fully coupled solution and hence a larger stability and im proved accuracy compared to PP2D PP3D However the numerical effort increases and in this actual release CC2D CC3D is preferrable in the case of low Reynolds numbers and on moderate meshes only Following these remarks the programming structure of CC2D CC3D and PP2D PP3D can be represented by the following diagrams init1 f mgstp f fpost f error f gdat f nsdef f xmrout f bbuild f dfkt f gupwd f supwg f bndry f 7 7 s A m011 f optcnl f gd vanca f mgrout f Figure 2 1 The cc2D CC3D structure 2 1 programming structure of FEATFLOW 17 ppad t pp3d f init1 f prostp f fpost f VANA error f s a 4 ace
25. part containing natural boundary conditions outflow This is the boundary segment with x 2 5 where the corresponding parameter IFLAG is activated in subroutine NEUDAT integral mean pressure is calculated over the full and half of the circular cylinder We define similar parameters for the calculation of drag and lift as in 2D now over the full cylinder surface RHO is a density parameter DIST a typical length scale and UMEAN is mean velocity We show the velocity values at the mesh points KPU 1 3421 corresponds to the coordinates 0 65 0 20 0 205 and KPU 2 2677 0 85 0 20 0 205 and the pres sure values at KPP 1 687 0 45 0 20 0 205 KPP 2 690 0 55 0 20 0 205 KPP 3 23498 0 50 0 25 0 205 and KPP 4 677 0 85 0 20 0 205 The dif ference of the pressure values for 1 687 and 2 690 can be used again for determing a typical pressure difference on the cylinder Now we can execute CC3D with given parameter file data cc3d dat solution is calculated on level NLMAX 3 and the corresponding solution vector is saved as unfor matted file tdata 4DX3 stat The chosen discretization scheme for the convective parts is the streamline diffusion ansatz and the stopping criterions are 1 107 for the maxi mum of relative changes It is remarkable that the number of smoothing steps NSM 64 is surprisingly large due to the chosen anisotropic mesh in combi
26. refinement and in this release the elements to be refined have to be given in a list We hope to provide a graphical interface for this marking process in the next version and additionally to use this tool in a fully adaptive mesh refinement 18 Structure of FEATFLOW process Furthermore a boundary check on consistency for complex domains is included The formats for data output are formatted or unformatted FEAT style BYU style and AVS ucd format TR2TO3 TR2TO3 is a tool providing 3D mesh generation from 2D meshes all in FEAT format This tool is designed for applications when the 3D mesh is simply constructed by popping 2D mesh layers in sandwich like technique For instance problems in ducts around cylinders or other prism like bodies are belonging to this class see the example in the next chapter TRIGENSD TRIGENSD is doing the same job as TRIGEN2D but in 3 dimensions However up to now only FEAT format is supported that means the boundary of 2 must be defined by the standard FEAT parametrization file We hope to add some appropriate CAD tools in future for being able to model much more complex domains Additionally the adaptive grid refinement procedure is not finished yet and will be part of the next version However at least the data output is performing analogously as in 2D INTPOL2D and INTPOL3D These programs will be tools for interpolating a given solution vector on a given mesh to an arbi
27. the directory featflow object makefiles example There wait two tasks for you 1 Edit the shell scripts make_copy and make lib and change the shell variable FEATFLOW containing the right location of FEATFLOW and the variable MAKEFILES where you are 2 Edit the makefiles and modify if necessary the compiler options These are the lines beginning with COMOPT This can be done by hand or by using the shell script make change Analogously the shell variable FEATFLOW has to be defined correctly as above using make change Step 2c Installation Assuming that you are in featflow object makefiles examples or any other directory containing your edited makefiles and shell scripts of step 2 Then you have to do the following 0 1 Be sure that you use a C shell bin csh 0 2 Be sure that featflow object libraries libgen exists 0 3 Be sure that the correct ztime f file in featflow source feat2d src is taken 1 0 Execute the shell script make copy 2 0 Execute the shell script make lib 2 4 Installation of FEATFLOW 47 This last step takes between 10 and 60 minutes libraries will be generated and all makefiles needed for applications are copied to featflow application user_start featflow application example and featflow application comp Step 3 Test and application As described in subsection 2 3 4 go to featflow application user_start to featflow application example or to featflow application comp and start
28. the input files parq2d f parq3d f indat2d f and indat3d f see above These files have to be edited and modified according to the actual application and the corresponding makefile has to be executed Additionally before compiling the corresponding storage has to be defined by defining the corresponding inc file see above and the later subsection on the workspace subdirectory After compiling the tool by executing the makefile the compiled program should be copied or moved to the parent directory user start and the application may be started there after editing the corresponding parameter file in data All other applica tions should follow these instructions to make life easier In a similar way comp is a pre installed application directory performing test calculcations similar to the DFG benchmark configuration of 1995 see 16 These applications are thought to provide reference results and CPU times for verifying the installed FEATFLOW version and to obtain benchmark results for different computer types for comparisons These results can be found in results 2 4 Installation of FEATFLOW 45 example is a pre installed application directory containing the example programs of Chap ter 3 Finally there is the directory workspace containing some shell scripts Their only use is to select the corresponding include files containing the size of storage amount For instance if the directory user start is in progress a l
29. velocity constant for pressure 2 rotated trilinear for velocity pressure parameter for mg smoother 1 parameter for mg solver 1 Vanca parameter for number of mg smoothing steps parameter for number of mg solving steps parameter for change of number of mg smoothing steps on coarser levels n NSM nNbMAXK ILEV smoothing steps if on level LEV parameter for viscosity 1 NU parameter for convectice discretization gt 0 upwind parameter for Samarskij upwinding usual UPSAM 1 gt 0 SD leading coefficient with spatial adaptivity usual UPSAM 1 lt 0 upwind simple first order upwinding lt 0 SD leading coefficient without spatial adaptivity usual UPSAM 1 parameter for lower limit for optimal relaxation in nonlinear iteration gt 0 relative changes are calculated lt 0 no relative changes are calculated if OMGMIN OMGMAX parameter for upper limit for optimal relaxation in nonlinear iteration gt 0 relative changes are calculated lt 0 no relative changes are calculated if OMGMIN OMGMAX parameter for start value for calculation of optimal relaxation in nonlinear iteration stopping criterion for defect in velocity in nonlinear iteration stopping criterion for defect in divergence in nonlinear iteration stopping criterion for relative changes in velocity in nonlinear iteration stopping criterion for relative changes in pressure in nonlinear iteration stopping criterion for d
30. we denote the lumped mass matrix which is diagonal Two possible approaches for solving these discrete nonlinear problems are 1 We first treat the nonlinearity by an outer nonlinear iteration of fixed point or quasi Newton type or by a linearization technique through extrapolation in time and we obtain linear indefinite subproblems of Oseen type which can be solved by a coupled versions CC2D CC3D or a splitting approach versions CP2D CP3D 2 We first split the coupled problem and obtain definite problems in u Burgers equations as well as in p linear pressure Poisson resp quasi Poisson problems Then we treat the nonlinear problems in u by an appropriate nonlinear iteration or a linearization tech nique versions PP2D PP3D In our applications the nonlinear problems are solved by the adaptive fired point defect correction method see 23 while for the solution of the coupled problems resp for their decoupling the discrete projection method formalism is used see 22 To understand completely the algorithm we will shortly present other important tools of the solver namely the multilevel solvers for the corresponding linear subproblems Oseen problems convection diffusion problems for u quasi Poisson problems for p the adaptive time step control and defect optimization procedures In compact form the components of the solvers are explained in the following subsections Discretization schemes Linear solvers
31. 0000000000 Level for velocity IFUSAV 0 Level for pressure IFPSAV 0 Level for streamlines IFXSAV 0 Level for AVS IAVS 3 Level for BYU IBYU 2 Start file IFINIT 1 Type of adaptivity IADTIM 1 Max Time TIMEMX 5 0000000000000 Min Timestep DTMIN 1 0000000000000D 06 Max Timestep DTMAX 9 0000000010000 Timestep change DTFACT 9 0000000010000 Time for start procedure TIMEIN 1 0000000000000 EPS for start procedure EPSADI 0 12500000000000 EPS for acceptance EPSADL 1 25000000000000 03 EPS for not acceptance EPSADU 0 50000000000000 Acceptance criterion IEPSAD 3 Start procedure IADIN 2 Max numbers of repetitions IREPIT t ILEV NVT NAT NEL NELO NEL1 NEL2 1 1485 3836 1184 1120 0 64 ILEV NVT NAT NEL NELO NEL1 NEL2 2 10642 29552 9472 8864 96 512 ILEV NVT NAT NEL NELO NEL1 NEL2 3 80388 231872 75776 70512 1168 4096 time for grid initialization 36 883334092796 ILEV NU NA NB 1 3836 39356 7104 LEV NU NA NB 2 29552 313712 56832 ILEV NU NA NB 3 231872 2505152 454656 H time for initialization of linear operators 62 316661834717 RELU RELP DEF U DEF DIV DEF TOT RHONL OMEGNL RHOMG 0 0 26D 03 0 68D 03 0 72D 03 1 0 10D 01 0 10D 01 0 93D 04 0 48D 05 0 93D 04 0 13D 00 0 10D 01 0 12D 00 2 0 51D 00 0 42D 00 0 14D 03 0 84D 06 0 14D 03 0 43D 00 0 10D 01 0 18D 00 3 0 26D 00 0 26D 00 0 28D 04 0 11D 06 0 28D 04 0 34D 00 0 10D 01 0 77D 01 4 0 10D 00 0 39D 01 0 34D 05 0
32. 1 maximum of linear mg steps ILMAX 5 type of interpolation IINT 2 type of smoother ISM 1 type of solver ISL 1 number of smoothing steps NSM 4 number of solver steps NSL 100 factor sm steps on coarser lev NSMFAC 2 KPRSM KPOSM ON LEVEL 1 32 32 KPRSM KPOSM ON LEVEL 2 16 16 KPRSM KPOSM ON LEVEL 3 8 8 KPRSM KPOSM ON LEVEL 4 4 4 KPRSM KPOSM ON LEVEL 5 4 4 KPRSM KPOSM ON LEVEL 6 4 4 KPRSM KPOSM ON LEVEL 7 4 4 KPRSM KPOSM ON LEVEL 8 4 4 KPRSM KPOSM ON LEVEL 9 4 4 Real parameters of RPARM etc Viscosity parameter 1 NU 1000 00000000000 parameter for Samarskij upwind UPSAM 0 50000000000000 lower limit for optimal OMEGA OMGMIN 0 upper limit for optimal OMEGA OMGMAX 2 0000000000000 start value for optimal OMEGA OMGINI 1 0000000000000 limit for U defects B EPSD 1 0000000000000D 05 limit for DIV defects EPSDIV 1 0000000000000D 08 limit for U changes EPSUR 1 0000000000000D 03 limit for P changes EPSPR 1 0000000000000D 03 defect improvement DMPD 1 00000000000000 01 damping of MG residuals DMPMG 1 0000000000000D 01 limit for MG residuals EPSMG 1 0000000000000D 01 damping of residuals for solving DMPSL 1 0000000000000D 01 limit of changes for solving EPSSL 1 0000000000000D 01 relaxation for the U smoother RLXSM 1 0000000000000 1 0000000000000 10 0000000000000 10 0000000000000 relaxation for the U solver RLXSL lower limit optimal MG ALPHA AMINM
33. 2 2 and KPI 2 are defined appropriately And finally some constants for the calculation of lift and drag on boundary component 2 are prescribed If not desired set DPF 1 DPF 2 to 0 or KPI 1 KPI 2 to 0 2 2 input data of FEATFLOW 23 indat3d f DOUBLE PRECISION FUNCTION FDATIN ITYP IBLOC X Y Z TIMENS RE Prescribed data for files coeff f and bndry f IMPLICIT DOUBLE PRECISION A C H 0 U W Z LOGICAL PARAMETER PI 3 1415926535897931D0 FDATIN ODO IF ITYP EQ 1 THEN IF IBLOC EQ 1 THEN IF X EQ ODO FDATIN 16DO0 0 45D0 0 41D0 4 Y 0 41D0 Y Z 0 41D0 Z ENDIF IF IBLOC EQ 2 THEN IF 0 0 000 FDATIN ODO ENDIF IF IBLOC EQ 3 THEN IF 0 0 000 FDATIN ODO ENDIF 0 2 C IF IBLOC EQ 1 THEN IF 0 0 000 FDATIN ODO ENDIF C IF IBLOC EQ 2 THEN IF 0 0 000 FDATIN ODO ENDIF C IF IBLOC EQ 3 THEN IF 0 0 000 FDATIN ODO ENDIF IF ITYP EQ 3 THEN IF IBLOC EQ 1 THEN FDATIN ODO ENDIF IF IBLOC EQ 2 THEN FDATIN ODO ENDIF IF IBLOC EQ 3 THEN FDATIN ODO ENDIF Structure of FEATFLOW 24 ENDIF ee Case 4 Velocity z derivative of exact solution ee IF ITYP EQ 4 THEN IF IBLOC EQ 1 THEN FDATIN ODO ENDIF IF
34. 2 2 22222222222 2222 22222222222 2 2 22222 22 222 2 2 C RHO 1 000 DIST 0 100 UMEAN O 2DO C DPF 1 RHO DNU DPF 2 RHO DIST UMEAN 2 C 99999 END Most command lines have not to be explained more in detail even being written in FORTRANT7 they explain themselves only a few comments are necessary In the case IF ITYP EQ 7 THEN in FDATIN the mean pressure values on boundary parts corresponding to natural b c s can be described This is done by prescribing the starting parameter value DPAR1 and the ending parameter value DPAR2 corresponding to the parametrization which is used for describing the geometrical boundary Additionally the number of the boundary component has to be defined INPR In NEUDAT the boundary parts containing natural boundary conditions are defined This is done by prescribing the number of such boundary parts by setting INPART and then again by setting the starting parameter value DPARN1 the ending parameter value DPARN2 and the corresponding boundary component INPRN In PTSDAT certain mesh points are defined in which some velocity components KPU 1 KPU 2 some pressure values KPP 1 KPP 2 KPP 3 KPP 4 and a flux value defined as difference of streamfunction values KPX 1 and KPX 2 are printed for the runtime protocol Furthermore 2 integral pressure values on some fixed boundary parts parameter values DPI 1 1 DPI 2 1 and boundary component 1 resp DPI 1 2 DPI
35. 20D 06 0 34D 05 0 26D 00 0 99D 00 0 22D 00 5 0 40D 01 0 31D 01 0 29D 05 0 79D 07 0 29D 05 0 33D 00 0 99D 00 0 69D 00 EHE HERE HE HE HERE EIER HE H H H HEHH H HH HE HHH H H HHHH H H HE HERR EP EP EEE EEE 1 1 TIME 0 000D 00 NORM U 0 2160847D 00 NORM P 0 7659621D 01 P VELO 0 32444D 01 0 12893D 02 0 21419D 04 0 20393D 00 0 16592D 02 0 28822D 04 P PRES 0 19791D 00 0 36067D 01 0 19174D 02 0 45314D 01 I PRES 0 61808D 01 0 95865D 01 I FORCE 0 63585D 01 0 38097D 02 STATISTICS NWORK 10000000 IWORKG 1339516 IWMAXG 3450964 IWORKI 9732825 64 Examples for the use of FEATFLOW IWMAXI 9732825 IWORK 9732825 IWMAX 9893601 total time 21627 316276040 appr time 21626 999564104 grid time 36 883334092796 post time 51 515625000000 lin time 1290 7828254700 gt mavec time 109 70226669312 gt konv time 1106 0506057739 gt bdry time 0 75054168701172 LC time 74 279411315918 mg time 20247 817779541 substeps 1 nonlinear 5 mg 6 MULTIGRID COMPONENTS in percent smoothing 98 178536576809 solver 0 46786725951263 defect calc 0 70117728389642 prolongation 0 59852068627434 restriction 5 3898193507251 02 Only a few differences with respect to the 2D example have to be explained ILEV NELO NEL1 NEL2 appears in combination with streamline diffusion only It presents the number of element requiring fully trilineaylinear amp xiparallel transforma
36. 298564 ILEV NU NA NB 1 313 2089 592 ILEV NU NA NB 2 1218 8322 2368 ILEV NU NA NB 3 4804 33220 9472 ILEV NU NA NB 4 19080 132744 37888 time for initialization of linear operators 2 3499999046326 RELU RELP DEF U DEF DIV DEF TOT RHONL OMEGNL RHOMG 0 0 58D 02 0 26D 01 0 26D 01 1 0 10D 01 0 10D 01 0 34D 02 0 34D 04 0 34D 02 0 13D 00 0 99D 00 0 20D 00 2 0 41D 00 0 49D 00 0 25D 03 0 85 05 0 25D 03 0 98D 01 0 10D 01 0 87D 01 3 0 21D 00 0 14D 00 0 87D 04 0 82D 06 0 87D 04 0 15D 00 0 11D 01 0 24D 00 4 0 64D 01 0 52D 01 0 20D 04 0 27D 06 0 20D 04 0 17D 00 0 11D 01 0 31D 00 0 13D 01 0 45D 02 0 58D 05 0 44D 07 0 58D 05 0 19D 00 0 11D 01 0 33D 00 6 0 33D 02 0 19D 02 0 14D 05 0 16D 07 0 14D 05 0 19D 00 0 11D 01 0 29D 00 7 0 62D 03 0 39D 03 0 33D 06 0 38D 08 0 33D 06 0 20D 00 0 11D 01 0 31D 00 EHE HERE HERE HH HHHH HEHEHHE H H HH HEHHEHE H H HER HE HERR HR HR EP EEE EE EEE 1 1 TIME 0 000D 00 NORM U 0 2092401D 00 NORM P 0 4859169D 01 P VELO 0 20913 01 0 14274D 02 0 16732D 00 0 14309D 02 P PRES 0 12008D 00 0 14282D 01 0 81432D 02 0 28207D 01 I PRES 0 47304D 01 0 29039D 01 I FORCE 0 53363D 01 0 66688D 02 P FLUX 0 39419D 01 STATISTICS NWORK 1000000 IWORKG 116168 IWMAXG 119323 IWORKI 685234 IWMAXI 685234 IWORK 685234 IWMAX 704450 total time 212 94999999553 appr time 212 83334153146 grid time 3 6833333298564 post time 7 5666656494141 lin time 49 299993515015 g
37. 3 equidistant 6 element is partitioned into 3 finer elements for only elements belonging to 2 boundary component allowed 30 Structure of FEATFLOW for corner element DELMA distance of element boundary to new inner vertex in percentage DELMB no use usual DELMA 0 5 equidistant NELMOD name of elements to be adaptively refined list of elements to be refined follows after DELMB one element per line separated by return DELMA parameter for distance in refinement process 0 no adaptive refinement strategy 1 unformatted output of mesh data until level i DELMB parameter for distance in refinement process 0 no adaptive refinement strategy i unformatted output of mesh data until level i Figure 2 3 ITYPEL 1 Refinement into 9 finer elements DELMA INT INT BOUNDARY DELMB INTERIOR Figure 2 4 ITYPEL 2 Refinement into 3 finer elements BOUNDARY INT INTERIOR Not allowed 111 Figure 2 5 ITYPEL 3 Refinement into 2 finer elements 2 2 input data of FEATFLOW 31 BOUNDARY B INT INT B INTERIOR DELMA DELMA Not allowed 111 Figure 2 6 ITYPEL 4 Refinement into 3 finer elements DELMA Figure 2 7 ITYPEL 5 Refinement into 5 finer elements INTERIOR INT B BOUNDARY BOUNDARY DELMA Not allowed 1 11 Figure 2 8 ITYPEL 6 Refinement into 3 finer elements 32 Structure of FEATFLOW 2 2 4 tr2to3 dat par
38. 3 2 3 4 2 3 5 2 3 6 Subdirectory source source code for FEATFLOW Subdirectory manual manuals for FEATFLOW Subdirectory object system software for FEATFLOW Subdirectory application applications under FEATFLOW Subdirectory graphic graphic tools for FEATFLOW Subdirectory utility utilities for FEATFLOW 2 4 Installation of FEATFLOW 3 Examples for the use of FEATFLOW 3 1 The installation 2 3 2 Lhe 2D example sa eene oe EL 3 3 The sD example 2 ue run T Bibliography A Appendix B Appendix C Appendix Troubleshooting with FEATFLOW The FEATFLOW group Future projects in FEATFLOW 10 11 11 12 14 15 15 18 18 19 29 32 33 34 39 43 43 43 43 44 45 45 45 48 48 50 60 67 69 71 72 Short description The progam package FEATFLOW is both a user oriented as well as a general purpose subroutine system for the numerical solution of the incompressible Navier Stokes equations in two and three space dimensions FEATFLOW is part of the FEAST project which has the aim to develop software which realizes our new mathematical and algorithmic ideas in combination with high performance computational techniques For more information about this project ask the authors or look at the Internet URL http
39. 8 min max ns V va _ gt qy gt 0 1 6 pn Ln 0 Vvallo Many stable pairs of finite element spaces have been proposed the literature Our favorite candidate is a quadrilateral element which in 2D uses piecewise rotated bilin ear shape functions for the velocities spanned by x y x y 1 resp rotated trilinear shape functions in 3D spanned by 22 y x z x y z 1 and piecewise constant pressure approximations see Figure 1 1 The nodal values are the mean values of the 8 Mathematical Background u v u v Figure 1 1 Nodal points of the nonconforming finite element pair in 2D velocity vector over the element edges faces element type 30 or in the corresponding midpoints element type 31 and the mean values of the pressure over the elements ren dering this approach nonconforming This element is the natural quadrilateral analogue of the well known triangular Stokes element of Crouzeix Raviart see 5 A convergence analysis of both the parametric elements E030 E031 and the nonparametric versions elements EM30 EM31 which are preferrable on non tensor product grids or meshes with large aspect ratios is given in 15 and very promising computational results are reported in 19 23 21 This element pair has several important features It admits beside the typical Galerkin streamline diffusion technique simple upwind strategies which lead to matrices with cer
40. ABS IADTIM 3 2 3 file structure of FEATFLOW 43 2 3 The file structure of FEATFLOW In reversed order we arrive at the point to explain the internal structure of FEATFLOW FEATFLOW consists of 6 directories application graphic manual object source and utility We want to explain their tasks in detail not following the literal order 2 3 1 Subdirectory source source code for FEATFLOW The directory source contains almost all source code for the preprocessing and solver tools These are OMEGA2D TRIGEN2D TRIGEN3D TR2TO3 INTPOL2D INTPOL3D CC2D CC3D PP2D and PP3D Each of them is split into two resp three directories namely src and dev resp mg idea is that src contains all information to build up the corresponding system libraries see object and the instructions for installation These are needed for running an user application mg is similar and is needed for building the corresponding libraries containing the multigrid components for solving if necessary They are only needed by the solvers CC2D CC3D PP2D and PP3D directory dev is corresponding directory containing all source and makefiles and it is thought to be developer directory which is needed by the experienced user to modify the FEATFLOW software Furthermore the source directory contains the code for BLAS FEAT2D FEAT3D and OMEGA2D In an analogous way src contains all files to build up the corresponding system l
41. D CC3D We use the adaptive fixed point defect correction method as outer iteration while for the linear coupled subproblems we choose C S and 1 Hence the linear equations in the nonlinear process are solved in one iteration step meaning that L 1 This solver is a multilevel based approach see 19 with a block Gauf Seidel scheme as smoothing operation Vanca smoother There may be problems on very anisotropic meshes and there is no gain in efficiency for fully nonstationary problems if At 0 This solver is actually preferrable for the case of low Reynolds numbers stationary or quasi stationary problems Mixed solver CP2D CP3D We use again the adaptive fired point defect correction method as outer iteration and select the preconditioner C We will obtain the same solutions as by CC2D CC3D however in a more robust way First tests show that this solver works in a very efficient way independent of the mesh and Re number being very efficient for fully nonstationary problems The essential tool is a so called multilevel discrete projection algorithm for the linear coupled subproblems This solver will be added to FEATFLOW in the next release Projection solver PP2D PP3D First we apply a decoupling step for u and p as outer iteration nonlinear discrete projection scheme We choose again C Mj but perform only L 1 iteration in each time step In the fully nonlinear case we use the
42. G upper limit optimal MG ALPHA AMAXMG Parameters of NS Time dependency ISTAT 0 Number of time steps NITNS 9 limit for time derivative EPSNS 1 0000000000000D 05 Total time TIMENS 0 Theta THETA 1 0000000000000 Time step TSTEP 1 00000000000000 02 Fractional step IFRSTP 1 Stepsize for nonsteady savings INSAV 0 Number of files INSAVN 0 Time step for Film DIFILM 0 Time step for AVS DTAVS 1 0000000000000 Time step for BYU DTBYU 1 0000000000000 Level for velocity IFUSAV 0 Level for pressure IFPSAV 0 Level for streamlines IFXSAV 0 Level for AVS IAVS 4 Level for BYU IBYU 4 Start file IFINIT 1 Type of adaptivity IADTIM 3 54 Examples for the use of FEATFLOW Max Time TIMEMX 100 000000000000 Min Timestep DTMIN 1 0000000000000D 06 Max Timestep DTMAX 1 0000000010000 Max Timestep change DTFACT 9 0000000010000 Time for start procedure TIMEIN 0 50000000000000 EPS for start procedure EPSADI 0 12500000000000 EPS for acceptance EPSADL 1 2500000000000D 03 EPS for not acceptance EPSADU 0 50000000000000 Acceptance criterion IEPSAD t Start procedure IADIN 2 Max numbers of repetitions IREPIT 3 ILEV NVT NMT NEL NVBD 1 165 313 148 34 ILEV NVT NMT NEL NVBD 2 626 1218 592 68 ILEV NVT NMT NEL NVBD 3 2436 4804 2368 136 ILEV NVT NMT NEL NVBD 4 9608 19080 9472 272 time for grid generation 3 6833333
43. L output of solution vector 0 no output unformatted output on finest level gt 1 formatted output on finest level CSOL name of solution vector file not longer than 15 characters M FEAT parameter for output 0 no output gt 0 output see FEAT2D manual MT FEAT parameter for terminal output 0 no output gt 0 output see FEAT2D manual ICHECK FEAT parameter for subroutine tracing 0 no tracing gt 0 tracing see FEAT2D manual MSHOW parameter for protocol 0 reduced protocol in file CFILE 1 expanded protocol in file CFILE 2 expanded protocol in file CFILE and terminal 3 full protocol in file CFILE 4 full protocol in file CFILE and terminal NLMIN parameter for smallest level number NLMAX parameter for highest level number IELT parameter for element type 2 2 input data of FEATFLOW 35 ISTOK IRHS IBDR IERANA IMASS IMASSL IUPW IPRECA IPRECB ICUBML ICUBM ICUBA ICUBN ICUBB ICUBF INLMIN INLMAX 0 E031 parametric E030 parametric 2 E031 nonparametric 3 E030 nonparametric parameter for Stokes calculation 1 Stokes calculation lt gt 1 Navier Stokes calculation parameter for right hand side 0 homogeneous r h s 1 steady inhomogeneous r h s 2 nonsteady inhomogeneous r h s parameter for boundary 0 Dirichlet Neumann boundary values 1 Dirichlet Neumann pressure drop boundary values 2 1 t
44. LAG1 IFLAG2 Pressure integral boundary part I I A I I a 2K K 1 IMPLICIT DOUBLE PRECISION A C H 0 U W Z LOGICAL B IF PZ GT 0 00D0 AND PZ LT 0 41D0 AND PX GE 0 45D0 AND PX LE 0 55D0 AND PY GE 0 15DO AND PY LE 0 25DO THEN IFLAG1 1 ENDIF C IF PZ GT 0 00DO AND PZ LT 0 41D0 AND PX EQ 0 45D0 AND PY GE 0 15D0 AND PY LE 0 25D0 THEN IFLAG2 1 ENDIF C 99999 END C SUBROUTINE BDFDAT IEL INPR PX PY PZ TIMENS DNU IFLAG DPF1 DPF2 lift and drag data IMPLICIT DOUBLE PRECISION A C H 0 U W Z LOGICAL Parameters for lift DFW and drag DAW bdforc f INPR 2 C C C dfw 2 int_s dpf 1 dut dn ny p n x ds dpf 2 C daw 2 int_s dpf 1 dut dn n x p n y ds dpf 2 C C RHO 1 0D0 DIST 0 041D0 UMEAN 0 2D0 C DPF1 RHO DNU DPF2 RHO DIST UMEAN 2 C 26 Structure of FEATFLOW C IF PZ GT 0 00DO AND PZ LT 0 41D0O AND PX GE 0 45D0 AND PX LE 0 55DO AND PY GE 0 15D0 AND PY LE 0 25D0 THEN IFLAG 1 ENDIF 99999 END C ok
45. LOC EQ 1 THEN IF X EQ 0 0D0 FDATIN ODO ENDIF IF IBLOC EQ 2 THEN IF X EQ 0 0D0 FDATIN ODO ENDIF ENDIF Case 3 Velocity y derivative of exact solution 20 Structure of FEATFLOW IF ITYP EQ 3 THEN IF IBLOC EQ 1 THEN FDATIN ODO ENDIF IF IBLOC EQ 2 THEN FDATIN ODO ENDIF ENDIF C Case 4 Exact pressure solution IF ITYP EQ 4 THEN 0 0 ENDIF C Case 5 Right hand side for momentum equation ee IF ITYP EQ 5 THEN IF IBLOC EQ 1 THEN FDATIN ODO ENDIF IF IBLOC EQ 2 THEN FDATIN ODO ENDIF 0 6 FDATIN 0DO ENDIF Case 7 Mean pressure values IF 0 7 THEN DPAR X INPR IBLOC IF DPAR GT 1DO AND DPAR LT 2DO AND INPR EQ 1 THEN FDATIN ODO ENDIF IF DPAR GT 3DO AND DPAR LT 4DO AND INPR EQ 1 THEN 0 0 ENDIF ENDIF 99999 END 2 2 input data of FEATFLOW 21 SUBROUTINE NEUDAT INPART INPRN DPARN1 DPARN2 TIMENS Neumann boundary part FEO
46. RL http www iwr uni heidelberg de featflow which contains most of our research activities including the Virtual Album of Fluid Motion Heidelberg August 1998 1 Mathematical Background 1 1 Introduction The following mathematical description for the implemented software is part of our paper archive see the given URL or of our book Efficient solvers for incompressible flow problems An algorithmic approach in view of computational aspects which will appear by Springer Verlag There the reader will find much more details We consider numerical solution techniques for the nonstationary or stationary without the term u incompressible Navier Stokes equations u vAu u Vu Vp f V u 0 inQ x 0 7 1 1 for a given force f and viscosity v with any prescribed boundary values on the boundary OQ of Dirichlet or Neumann type see 10 and an initial condition at t 0 Solving this problem numerically still seems to be a considerable task in the case of long time calculations and higher Reynolds numbers particularly in 3D The corresponding discretized nonlinear systems of equations may be treated by a coupled approach in u and p see 23 which promises the best stability behaviour but also entails the largest numerical effort Another variant known as projection method see 7 decouples pressure and velocity which reduces the problem to the solution of a sequence of simple scalar p
47. alistic applications too We hope to demonstrate that this approach is not only faster and more efficient than most other programs but also applicable for the same problems Therefore one of our main goals is to demonstrate the practical flexibil ity of our software A first proof for these statements could be found by performing the DFG benchmarks of the project Flow Simulation with High Performance Computers CONTENTS 3 Mainly active members of the FEAST project are the numeric group in Heidelberg around Rannacher and Turek A list containing almost all participants helping practically and theoretically can be found in the Appendix FEATFLOW 1 1 is based on the FORTRAN 77 finite element packages FEAT2D and FEAT3D which are not user oriented systems They only provide subroutines for several main steps in a finite element program The user should be familiar with the mathematical formulation of the discrete problems The data structure of FEAT is transparent so that modifications or augmentations of the program package are very easy for instance the implementation of new elements or the application of error control For details concern ing tools and data structures of these finite element codes the reader is referred to the manuals 3 and 9 We consider to use other more sophisticated finite element tools in FORTRAN90 under NETSCAPE in the future This manual has the following structure Chapter 1 shows the mathematica
48. ameter file for TR2TO3 M FEAT parameter for output 0 no output gt 0 output see FEAT2D manual MT FEAT parameter for terminal output 0 no output gt 0 output see FEAT2D manual ICHECK FEAT parameter for subroutine tracing 0 no tracing gt 0 tracing see FEAT2D manual IMESH parameter for type of parametrization 0 FEAT parametrization OMEGA2D parametrization CPARM name of OMEGA2D parametrization file IBDCHK parameter for boundary checking for further refinements 0 no checking gt 0 number of finer boundary points for check of boundary consistency for further refinement IBYU level for BYU output IAVS level for AVS output CFILEI name of input coarse mesh CFILEO name of created output coarse mesh NPZ parameter for number of z slides list of interior z coordinates follows after PZMAX one element per line separated by return PZMIN parameter for minimum z coordinate PZMAX parameter for maximum z coordinate 2 2 input data of FEATFLOW 33 2 2 5 trigen3d dat parameter file for TRIGEN3D MT ICHECK IBYU IAVS NLEV IFMT CFILEI CFILEO FEAT parameter for output 0 no output gt 0 output see FEAT2D manual FEAT parameter for terminal output 0 no output gt 0 output see FEAT2D manual FEAT parameter for subroutine tracing 0 no tracing gt 0 tracing see FEAT2D manual level for BYU output level for AVS output number of refined levels 1 parameter
49. an application or create a new directory for your application as described above 3 Examples for the use of FEATFLOW This chapter demonstrates explicitely the use of FEATFLOW performing first the installa tion and then applying FEATFLOW for solving a 2D and 3D problem which represents the typical application of a flow in a channel around a cylinder The values to be computed are lift and drag and pressure differences on the cylinder surface in a stationary as well as non stationary configuration These examples are almost identical with the DFG benchmark 1995 see 16 only the 2D case is a little bit different longer channel We show in short form how FEATFLOW is manually installed followed by designing a first coarse mesh with OMEGA2D This triangulation is refined a pre adaptive way by TRIGEN2D generating the full sequence of nested meshes which are neccessary for the multigrid solvers We perform a stationary by CC2D and a nonstationary calculation by PP2D and demonstrate how to interpret the data output and how to use Avs for visualization These 2D calculations are followed by a 3D application We first use TR2TO3 to construct an adequate 3D coarse mesh for this channel configuration We demonstrate the use of TRIGEN3D to generate the 3D multigrid structure and finally we perform the same calculations with CC3D and PP3D as in the 2D case 3 1 The installation example We assume that you got the binary file
50. arskij upwind UPSAM 1 0000000000000 lower limit for optimal OMEGA OMGMIN 0 upper limit for optimal OMEGA OMGMAX start value for optimal OMEGA OMGINI limit for U defects EPSD limit for DIV defects EPSDIV limit for U changes EPSUR limit for P changes EPSPR 0000000000000D 02 defect improvement DMPD 0000000000000D 01 damping of MG residuals DMPMG 0 50000000000000 limit for MG residuals EPSMG 0 50000000000000 damping of residuals for solving DMPSL 1 0000000000000D 01 limit of changes for solving EPSSL 1 0000000000000D 01 relaxation for the U smoother RLXSM 1 0000000000000 relaxation for the U solver RLXSL 0 80000000000000 lower limit optimal MG ALPHA AMINMG 10 0000000000000 upper limit optimal MG ALPHA AMAXMG 10 0000000000000 Parameters of NS oa NNNWN 20 M or o NoN PUN 0000000000000 0000000000000 00000000000000 05 00000000000000 05 0000000000000D 02 M eononrerrere 3 3 3D example 63 Time dependency ISTAT 0 Number of time steps NITNS 50 limit for time derivative EPSNS 1 0000000000000D 05 Total time TIMENS 0 Theta THETA 1 0000000000000 Time step TSTEP 1 0000000000000D 02 Fractional step IFRSTP 1 Stepsize for nonsteady savings INSAV 0 Number of files INSAVN 0 Time step for Film DIFILM 0 Time step for AVS DTAVS 1 0000000000000 Time step for BYU DTBYU 1 000
51. b c2d prm and c2d tri have to be transferred by ftp or whatever to the subdirectory FEATFLOW application example pre and additionally the coarse mesh c2d tri as c2d0 tri to FEATFLOW application example adc Now we are ready to proceed with FEATFLOW on our UNIX workstation Next we want to improve our coarse mesh c2d0 tri to obtain a better triangulation Therefore we go toFEATFLOW application example input files execute the make files ending with m and move the compiled objects to the parent directory namely to FEATFLOW application example More in detail for performing the 2D example 3 2 2D example 51 Figure 3 2 Coarsest mesh c2d0 tri 30 elements FEATFLOW application example input files make f trigen2d m mv trigen2d cd FEATFLOW application example Next we want to refine our mesh c2d0 tri around and before and behind the circle This is done by TRIGEN2D If we perform cp data trigen2d dat_O data trigen2d dat time trigen2d we obtain in adc a new coarse mesh c2d1 tri which is refined in an appropriate way with ITYPEL 1 that means the marked elements are refined into 9 elements The next steps generate the final coarse mesh c2d2 tri which is additionally refined around the circle only with ITYPEL 3 cp data trigen2d dat_1 data trigen2d dat time trigen2d Figure 3 3 Refined coarse mesh
52. between arbitrary meshes Software for a improved postprocessing Free modules of AVS EXPRESS Cooperation with other graphic packages 72 Appendix Future projects FEATFLOW 73 Software for more complex models Adding and testing of Boussinesq and more complex turbulence models Weakly compressible flows Non newtonian and multiphase flows Shape optimization with respect to lift and drag for instance And finally for the users Other FORTRAN compilers GNU for instance FORTRANO0O version see FEAST C version Industrial and commercial applications If you have more suggestions please let them be known the authors
53. c2d dat cc3d dat IMASSL ICYCLE ILMIN ILMAX IINT ISM ISL NSM NSL NSMFAC EPSD EPSDIV EPSUR EPSPR DMPD DMPMG EPSMG DMPSL EPSSL RLXSM RLXSL AMINMG AMAXMG ISTAT IMESH parameter for type of parametrization 0 FEAT parametrization 1 OMEGA2D parametrization active only in CC2D IRMESH input of mesh data 0 create mesh gt 0 read mesh created by TRIGEN2D gt 1 formatted mesh data 1 unformatted CPARM name of OMEGA2D parametrization file CMESH name of coarse mesh not longer than 15 characters CFILE name of protocol file ISTART input of start vector 0 start with homogeneous vector only b c s 1 read unformatted start vector from same level 1 formatted 2 read unformatted start vector from level 1 2 formatted CSTART name of start vector file not longer than 15 characters ISOL output of solution vector 0 no output 1 unformatted output on finest level gt 1 formatted output on finest level CSOL name of solution vector file not longer than 15 characters M FEAT parameter for output 0 no output gt 0 output see FEAT2D manual MT FEAT parameter for terminal output 0 no output gt 0 output see FEAT2D manual ICHECK FEAT parameter for subroutine tracing 0 no tracing gt 0 tracing see FEAT2D manual MSHOW parameter for protocol 0 reduced protocol in file CFILE 1 expanded protocol in file CFILE 2 expanded protocol in file CFILE an
54. c2d1 tri 130 elements Figure 3 4 Further refined coarse mesh c2d2 tri 148 elements 52 Examples for the use of FEATFLOW Now we are ready to start a Navier Stokes calculation Let us begin with a station ary example for Reynolds number 20 Doing that we use the data file input files indat2d f stat and the following procedure has to be performed cd FEATFLOW application example input files cp indat2d f stat indat2d f make f cc2d m mv cc2d cd FEATFLOW application example This data file input files indat2d f stat contains the following definitions The inflow velocity profile at x 0 0 is parabolic with a maximum value Umag 0 3 There is one boundary part containing natural boundary conditions outflow This is the boundary segment with x 2 5 and the corresponding parameter values range from DPARN1 1D0 until DPARN1 2DO We show the velocity values at the mesh points KPU 1 264 corresponds to the coordinates 0 65 0 20 and KPU 2 17 0 85 0 20 the pressure values at KPP 1 27 0 45 0 20 KPP 2 30 0 55 0 20 KPP 3 316 0 50 0 25 and KPP 4 17 0 85 0 20 and the flux between the mesh points KPX 1 31 and 2 6 under the circle The difference of the pressure values for 1 27 and KPP 2 30 determines a typical pressure difference Additionally our runtime pro tocol will give us the mean pressure over the whole
55. circle by setting DPI 1 1 0DO and DPI 2 1 0D0 and over half of the circle by DPI 2 2 0 5D0 Finally we de fine parameters for the calculation of drag and lift where RHO is a density parameter DIST a typical length scale and UMEAN is mean velocity Now we can execute CC2D with given parameter file data cc2d dat A solution is cal culated on level NLMAX 4 and the corresponding solution vector is saved as unformatted file data DX4_stat The chosen discretization scheme for the convective parts is the streamline diffusion ansatz and the stopping criterions are 107 for the maximum of rel ative changes As result we obtain the protocol file data cc2d stat which has to be explained in more detail Protocol file data cc2d stat Parametrization file pre c2d prm Coarse grid file adc c2d2 tri Integer parameters of IPARM etc minimum mg level NLMIN maximum mg level NLMAX element type 3 Stokes calculation ISTOK 0 RHS Boundary generation Error evaluation mass evaluation generation 0 0 0 0 3 2 2D example lumped mass eval convective part Accuracy for ST Accuracy for B ICUB mass matrix ICUB diff matrix ICUB conv matrix ICUB matrices B1 B2 ICUB right hand side 1 minimum of nonlinear iterations INLMIN 1 maximum of nonlinear iterations INLMAX 10 type of mg cycle ICYCLE 0 minimum of linear mg steps ILMIN
56. d terminal 3 full protocol in file CFILE 4 full protocol in file CFILE and terminal NLMIN parameter for smallest level number NLMAX parameter for highest level number IELT parameter for element type 0 E031 parametric E030 parametric 40 Structure of FEATFLOW ISTOK IRHS IBDR IERANA IMASS IMASSL IUPW IPRECA IPRECB ICUBM ICUBA ICUBN ICUBB ICUBF INLMIN INLMAX ICYCLE ILMIN ILMAX IINT 2 E031 nonparametric 3 E030 nonparametric parameter for Stokes calculation 1 Stokes calculation lt gt 1 Navier Stokes calculation parameter for right hand side 0 homogeneous r h s 1 steady inhomogeneous r h s 2 nonsteady inhomogeneous r h s parameter for boundary 0 Dirichlet Neumann boundary values 1 Dirichlet Neumann pressure drop boundary values 2 1 time dependent Dirichlet Neumann conditions parameter for error analysis 0 no error analysis gt 0 error analysis with quadrature formula IERANA parameter for mass matrix type 0 lumped mass matrix 1 real mass matrix parameter for mass matrix lumping 0 usual lumping 10 diagonal lumping parameter for convective terms 0 streamline diffusion 1 upwinding parameter for diffusive and reactive matrices 0 single precision in RAM 1 double precision in RAM 2 single precision from hard disc 3 double precision from hard disc not yet 4
57. de execution and explanation of output data Graphical presentation of the data At the end of this manual the Appendix contains a discussion of the most usual errors and problems with FEATFLOW and how to survive them followed by a list with names and contact addresses of involved persons and a remark how to get FEATFLOW Additionally there is a short section which shows projects being under development for future versions of FEATFLOW We hope that you enjoy this software and are very grateful for every comment concerning bugs and critical aspects or subjects to be added In fact this version FEATFLOW 1 1 contains only slight modifications compared to the original version FEATFLOW 1 0 However we may announce still for 1998 the new ver sion FEATFLOW 2 0 which will contain the essentially improved solvers CP2D and some tools for Boussinesq and or nonnewtonian flows including also the improved pre and processing tools together with freely downloadable AVS EXPRESS modules for high end visualization For a mathematical description of these software releases we recommend the papers our paper archive see the following URL or our book Efficient solvers for incompressible flow problems An algorithmic approach in view of computational aspects which will appear by Springer Verlag Further we hope to present a first version of our new FEAST package For more information about these project take a look at the Internet U
58. e dimensional is demonstrated and their input parameters are explained The systems CP2D CP3D are not finished yet and will be added to the next release in fact you may find in the recent version a test version of CP2D together with some other software tools postprocessing part finally contains files and shell scripts for supported graphi cal packages These are in this version GNUPLOT for 1D pictures and MOVIE BYU or CQUEL BYU resp AVS for 2D and 3D graphics Moreover it has shown to be easy to incoorporate output files for GRAPE or some other appropriate graphics and to add our own particle tracing tool PTRAC At the end of this chapter we show the content of the other subdirectories of FEATFLOW and explain the structure of FEATFLOW with its installation backup makefile and library parts 4 CONTENTS In addition most important for the user sample programs for some standard applications installation preprocessing and the solution of a 2D and 3D problem are included in Chapter 3 These can be used as starting programs which may be modified for the actual application There are several main levels to be explained in detail Installation of FEATFLOW editing of makefiles and other preparation for the use Generation of coarse and refined triangulations for the solution process Input data data files and input parameter as for instance boundary conditions and right hand side Co
59. e full smoothing property which is important in the case of rough initial or boundary values Further it contains only very little numerical dissipation which is crucial in the computation of non enforced temporal oscillations in the flow A rigorous theoretical analysis of the FS scheme see 11 12 applied to the Navier Stokes problem establishes second order accuracy for this special choice of 0 Corresponding numerical tests are performed in 21 Therefore this scheme can combine the advantages of both the classical CN scheme 2nd order accuracy and the BE scheme strongly A stable but with the same numerical effort So in each time step we have to solve nonlinear problems of the following type Given u parameters k k tn41 0 O tn41 and 0 Oiltn 1 i 1 3 then solve for u and p I 6kN u u kVp I 4 kN u u 05kf O3kf V u 0 1 5 For spatial discretization we choose a finite element approach In setting up a finite element model of the Navier Stokes equations one starts with a variational formulation On the finite mesh triangles quadrilaterals or their analogues in covering the domain Q with local element width h one defines polynomial trial functions for velocity and pressure These spaces and Lp should lead to numerically stable approximations as h 0 i e they should satisfy the Babuska Brezzi condition with mesh independent constant y see
60. efect improvement in nonlinear iteration stopping criterion for defect improvement in linear mg iteration stopping criterion for defect limit in linear mg iteration stopping criterion for defect improvement for solver in linear mg iteration stopping criterion for defect limit for solver in linear mg iteration relaxation parameter for mg smoother relaxation parameter for mg solver lower limit for optimal correction for mg solver upper limit for optimal correction for mg solver parameter for type of problem 0 steady Navier Stokes problem 1 nonsteady Navier Stokes problem maximum number of macro time steps stopping criterion for time derivative parameter for absolute start time parameter for time stepping value 1 Implicit Euler first order if IFRSTP 0 0 5 Crank Nicolson second order if IFRSTP 0 starting time step parameter for time stepping scheme 0 one step schemes 1 fractional step scheme second order parameter for unformatted saving of solution vector 0 no saving gt 0 macro step size for saving procedure ns modulo number of files for unformatted saving of solution vector maximum 10 42 Structure of FEATFLOW DTFILM time difference for unformatted film output DTAVS time difference for AVS output DTBYU time difference for BYU output IFUSAV level for unformatted velocity film output IFPSAV level for unformatted pressure film output IFXSAV level for unformatted streamline film ou
61. elaxation in nonlinear iteration gt 0 relative changes are calculated lt 0 no relative changes are calculated if OMGMIN OMGMAX parameter for upper limit for optimal relaxation in nonlinear iteration gt 0 relative changes are calculated 0 no relative changes are calculated if OMGMIN OMGMAX parameter for start value for calculation of optimal relaxation in nonlinear iteration stopping criterion for relative changes in velocity in nonlinear iteration stopping criterion for defect in velocity in nonlinear iteration stopping criterion for defect improvement in velocity in nonlinear iteration stopping criterion for defect improvement in velocity in linear mg iteration stopping criterion for defect improvement for solver in velocity in linear mg iteration relaxation parameter for mg smoother in velocity relaxation parameter for mg solver in velocity lower limit for optimal correction for mg solver in velocity upper limit for optimal correction for mg solver in velocity stopping criterion for divergence of velocity in pressure equation stopping criterion for defect improvement in pressure in linear mg iteration stopping criterion for defect improvement for solver in pressure in linear mg iteration relaxation parameter for mg smoother in pressure relaxation parameter for mg solver in pressure lower limit for optimal correction for mg solver in pressure upper limit for optimal correction for mg solver in pressure parame
62. elocity p1 scl pressure and x1 vec streamfunction A typical example for a AVS isoline plot of the pressure can be found in the following picture Figure 3 5 Pressure isolines for the stationary 2D calculation 56 Examples for the use of FEATFLOW Next we perform a nonstationary calculation for Reynolds number about 100 We use the data file input files indat2d f non and the following procedure has to be performed cd FEATFLOW application example input files cp indat2d f non indat2d f make f pp2d m mv pp2d cd FEATFLOW application example This data file input files indat2d f non is almost identical to the preceeding one The inflow velocity profile at z 0 0 is again parabolic but with a maximum value Umar 125 UMEAN is changed for defining drag and lift Now we can execute PP2D with given parameter data pp2d dat solution is calculated on level NLMAX 4 with stationary start solution file data DX4_stat Now the chosen discretization scheme for the convective parts is the upwinding ansatz and we perform our time stepping with the fractional step scheme and fully adaptive time step control until TIMEMX 4D0 Files for graphical output namely for avs and byu are written out all 1 second As result we obtain the protocol file data pp2d non which is very similar to data cc2d stat but containing information for all time steps performed We explain this protocol file for only one
63. elop very efficient solution schemes of both coupled and projection type with a special non linear or linearized treatment of the advection The resulting solutions are coincident as soon as the time steps are small enough and no spurious pressure oscillations occur This approach is the basis of our following theoretical and numerical investigations 1 2 Discretization and solution techniques in FEATFLOW We first discretize the time derivative the Navier Stokes equations 1 1 one of the usual time stepping schemes with prescribed boundary values for every time step Given u and the time step k tn41 tn then solve for u u t and p u u n4l i 6 vAu u Vu Vp g Vu 0 ing 1 2 with right hand side gt 1 8 1 0 vAu u Vu 1 3 In the past explicit time stepping schemes have been commonly used in nonstationary flow calculations but because of the severe stability problems inherent in this approach the required small time steps prohibited the long time solution of really time dependent flows Due to the high stiffness one seems to be limited to implicit schemes in the choice of time stepping methods for solving this problem Since implicit methods have become feasible thanks to more efficient linear solvers the schemes most frequently used are either the simple first order Backward Euler scheme BE with 0 1 or the second order Crank Nicolson scheme CN with 0
64. featflow tar gz and that you created the direc tory home people example featflow in which you should move this binary file Being there you have to perform the following commands gunzip featflow tar gz tar xvf featflow tar rm featflow tar Now the complete source code of FEATFLOW is installed In most cases all makefiles are already prepared if you work on a normal platform SUN SGI DEC HP PC LINUX etc such that the following installation step is very easy Read the README file and execute the installation script install feat 48 3 1 The installation example 49 After selecting the supported platform the installation procedure starts You con tinue with the 2D and 3D examples If you work on a nonstandard platform whatsoever you must manually perform the following two steps For the following we assume you have a SUN Sparc 10 or compatible with the SUN FORTRAN77 compiler version 2 0 Next you go to home people example featflow object makefiles and do the following copy cp rp sun_sparc10_2 0 my compiler cd my_compiler Edit for instance the file pp2d m to figure out how the shell variables FEATFLOW and COMOPT are defined FEATFLOW should be set to FEATFLOW home people example featflow while COMOPT has to be modified for other machines or compilers only These changes could be done by hand or by using the shell script make_change which performs after appropria
65. for the nonstationary incompressible Navier Stokes equations in two and three dimensions Proc Work shop Numerical Methods for the Navier Stokes Equations Heidelberg Oct 25 28 1993 Vieweg Thomasset F Implementation of Finite Element Methods for Navier Stokes Equations Springer New York 1981 Turek S A comparative study of time stepping techniques for the incompressible Navier Stokes equations From fully implicit nonlinear schemes to semi implicit projection methods Int J Numer Meth Fluids 22 987 1011 1996 Turek S On discrete projection methods for the incompressible Navier Stokes equations An algorithmical approach Comput Methods Appl Mech Engrg 143 271 288 1997 Turek S Tools for simulating nonstationary incompressible flow via discretely divergence free finite element models Int J Numer Meth Fluids 18 71 105 1994 Turek S Multilevel Pressure Schur Complement techniques for the numerical solution of the incompressible Navier Stokes equations Habilitation Thesis Uni versity of Heidelberg 1997 Turek S Multigrid techniques for a divergence freefinite element discretization East West J Numer Math Vol 2 No 3 229 255 1994 Turek S Efficient solvers for incompressible flow problems An algorithmic ap proach in view of computational aspects Springer 1998 Van Kan J A second order accurate pressure correction scheme for viscous in compressible flow SIAM J Sc
66. i Stat Comp 7 870 891 1986 A Appendix Troubleshooting with FEATFLOW In the following we list some problems which may apparently occur as far as the author knows during the installation and execution of FEATFLOW The following list cannot con tain everything and the author will be very grateful for showing him even more problems but hopefully with solution strategies 1 Known problems and errors during installation Q A gt o gt O EO ue um O I have no gunzip to decompress my FEATFLOW binary file The author may tell you how to get gzip gunzip or can send you the FEATFLOW data in another format I have no FORTRAN 77 compiler The author may tell you how to get an older but free FORTRAN77 or the GNU FORTRANT77 compiler I get tar error after decompression Your gzip gunzip command may not work well or you did forget to activate the binary mode during ftp transfer I have problems with the execution of the make shell scripts while installing the system libraries Be sure that you start the installation in a bin csh C shell The installation stops directly after calling make 115 Be sure that you the directories FEATFLOW object libraries libgen exist I get errors during the compilation process with make lib Check your compiler options used perhaps together with your system administrator My application stops with an error immediately after starting Check your ztime f subroutine
67. ibraries during installation Additionally there are two testdir directories containing a small test program to get familiar with FEAT Additionally we think about adding the GNU FORTRAN77 compiler to provide a FORTRAN compiler for everybody 2 3 2 Subdirectory manual manuals for FEATFLOW The directory manual contains almost all LATEX source files in src and postscript files in ps for needed manuals These are OMEGA2D FEAT2D FEAT3D and this FEAT FLOW manual 2 3 3 Subdirectory object system software for FEATFLOW The directory object contains the system software of FEATFLOW which is needed for installation for user applications and for generating a compressed binary data file con taining the FEATFLOW code There are three subdirectories extract makefiles and libraries extract is a directory having shell scripts for generating a compressed tarfile containing the FEATFLOW package There are several levels of FEATFLOW data files 44 Structure of FEATFLOW Level 00 complete FEATFLOW even with all libraries Level 01 complete FEATFLOW sources but without compiled libraries Level 02 partial FEATFLOW sources without dev directories Level 10 partial FEATFLOW sources libraries without multigrid sources Level 20 FEATFLOW libraries without any sources The subdirectory makefiles contains all makefiles for building up the system libraries and for user applications They are available for a class of machines
68. ime dependent Dirichlet Neumann conditions parameter for error analysis 0 no error analysis gt 0 error analysis with quadrature formula IERANA parameter for mass matrix type 0 lumped mass matrix 1 real mass matrix parameter for element type of lumped mass matrix 0 E031 parametric 1 E030 parametric 2 E031 nonparametric 3 E030 nonparametric parameter for convective terms 0 streamline diffusion 1 upwinding parameter for diffusive and reactive matrices 0 single precision in RAM 1 double precision in RAM 2 single precision from hard disc 3 double precision from hard disc 4 double precision built up every time parameter for gradient and divergence matrices 0 single precision with usual quadrature matrix in RAM 1 double precision with usual quadrature matrix in RAM 2 double precision exact matrix entries elementwise application 3 single precision with exact evaluation matrix in RAM 4 double precision with exact evaluation matrix in RAM quadrature formula for lumped mass matrix gt 0 usual lumping lt 0 diagonal lumping quadrature formula for real mass matrix quadrature formula for Laplacian matrix quadrature formula for convective matrix quadrature formula for gradient matrix quadrature formula for right hand side parameter for smallest number of nonlinear steps matrix 1 linear extrapolation in time if INLMAX 1 constant extrapolation
69. in time if INLMAX 1 parameter for largest number of nonlinear steps matrix 1 linear extrapolation in time if INLMIN 1 constant extrapolation in time if INLMIN 1 36 Structure of FEATFLOW ISORTU ICYCU ILMINU ILMAXU IINTU ISMU ISLU NSMU NSLU NSMUFA ISORTP ICYCP ILMINP ILMAXP IINTP ISMP ISLP NSMP NSLP parameter for renumbering of edges 1 with respect to x coordinate 2 with respect to y coordinate 3 with Cuthill McKee algorithm parameter for mg cycle for velocity 0 F cycle 1 V cycle 2 W cycle parameter for smallest number of mg steps for velocity parameter for largest number of mg steps for velocity parameter for mg interpolation for velocity 1 E031 parametric or nonparametric 2 E030 parametric or nonparametric parameter for mg smoother for velocity 1 Jacobi 2 SOR 3 SSOR 4 ILU parameter for mg solver for velocity 1 SOR 2 BICGSTAB 3 ILU 4 BiCGSTAB ILU parameter for number of mg smoothing steps for velocity parameter for number of mg solving steps for velocity parameter for change of number of mg smoothing steps on coarser levels for velocity n NSMU nNiMAXK ILEV smoothing steps if on level LEV parameter for renumbering of elements 1 with respect to x coordinate 2 with respect to y coordinate 3 with Cuthill McKee algorithm parameter for mg cycle for pressure
70. ink has to be set to the corresponding inc files in the directory user start This is done by executing the appropriate links user_start file or other files analogously This process has to be done before compiling 2 3 5 Subdirectory graphic graphic tools for FEATFLOW This directory provides graphic features which are helpful for FEATFLOW In the present version there are the subdirectories avs byu and gnuplot These features will be massively expanded in future versions avs contains a avsrc file as link to avsrc and a directory appl containing special applications This avsrc file has to be edited and adapted to the existing configuration Up to now AVS has to be executed over network if available and the manual has to be taken from there too The subdirectories byu and gnuplot contain some helpful files for handling the packages MOVIE BYU or CQUEL BYU over network if available and for working with GNUPLOT which may be used for 1D graphics 2 3 6 Subdirectory utility utilities for FEATFLOW This directory contains software which can be helpful for the use of FEATFLOW In the present version there are only programs which provide an a priori estimate for the needed storage amount They simply have to be compiled f77 file f o file and to be moved to the directory used for the application for instance to application user start Then it reads the corresponding parameter file and gives an estimate for the NNWORK va
71. ional defect correction possible Linearization for nonstationary problems only semi implicit treatment of nonlinear convective terms by linear extrapolation in time of 2nd order 1 2 3 Linear solvers Multigrid for velocity and pressure simultanously Oseen nonconforming mesh adapted makro elementwise interpolation for grid transfer adaptive step length control for correction step with F cycle Vanca like block Gauf Seidel scheme as smoother and solver Multigrid for velocity Burgers and pressure quasi Poisson nonconforming mesh adapted makro elementwise interpolation for grid transfer adaptive step length control for correction step with ILU SOR scheme as smoother with special renumbering strategies 12 Mathematical Background 1 2 4 Discrete projection method For linear or nonlinear coupled problems Su kBp g Blu 0 Given p solve for l 1 L 1 p a BTC 1B 1 BTS 1 Bp Cem Heg Set p and determine u through BTu 0 Su g 4 5 pr Classical Richardson scheme for Schur complement formulation with preconditioner P BT C B 1 C S L 1 for CC2D CC3D L gt 1 for PP2D PP3D If C 2 P corresponds to discrete Laplacian with minimal matrix stencil 5 in 2D 7 in 3D independent of the mesh Further improvements Additional preconditi
72. l background and explains the temporal and spatial discretizations Additionally subjects like nonlinear solution schemes multigrid solvers for linear subproblems discrete projection schemes for coupled systems adaptive time step control and other optimization tools are treated ex plicitly Inbetween there is much more literature cited which explains all pointed subjects more in detail We describe the discretization and solution process in form of flow charts since they can be directly translated into the programming structure of FEATFLOW This structure of programs and subroutines is explained in Chapter 2 We divide all software into three classes Preprocessing solver and postprocessing The preprocessing part contains the programs OMEGA2D description of 2D domains graphical generation of coarse tringulations developed by Matthies Schieweck in Magde burg and which will be soon replaced by our own JAVA based preprocessing tool TRI GEN2D adaptively refined meshes in 2D output of triangulations TR2TO3 generation of 3D meshes out of 2D meshes and TRIGEN3D analogous as TRIGEN2D but in 3D These programs mainly provide coarse triangulations for the following solution and graphical output routines The solution part contains the solver packages PP2D PP3D as purely time dependent projection schemes and CC2D CC3D solving stationary and nonstationary problems a fully coupled way The use of both solvers two as well as thre
73. lue needed 2 4 Installation of FEATFLOW The usual installation process is the following Step 1 Unzip and tar Create a directory called featflow or similar which will contain all FEATFLOW data Usually the user has obtained a binary file featflow tar gz or levi file tar gz i stands for level see above This has to be moved to the created directory in which the FEATFLOW tree structure will be installed It has to be decompressed gunzip featflow tar gz 46 Structure of FEATFLOW and then a tar has to be started tar xvf featflow tar This generates the complete FEATFLOW tree structure and the file featflow tar may be removed In most cases all makefiles are already prepared if you work on a normal platform SUN IBM SGI DEC PC LINUX etc such that the following installation step is very easy Step 2a Automatic installation Read the README file and execute the installation script install_feat After selecting the supported platform the installation procedure starts You may con tinue with Step 3 If you work on a nonstandard platform whatsoever you must manually perform the following two steps Step 2b Editing of makefiles and installation scripts In the next step go to featflow object makefiles There are different subdirectories corresponding to various computer types and compiler options If not copy one to create your owen or modify the makefiles in example Let us assume you are in
74. m H Harig J M ller S Turek S FEAT2D Finite element analysis tools User Manual Release 1 3 Technical report University Heidelberg 1992 Chorin A J Numerical solution of the Navier Stokes equations Math Comp 22 745 762 1968 Crouzeix M Raviart P A Conforming and non conforming finite element meth ods for solving the stationary Stokes equations R A IL R O R 3 77 104 1973 Cuvelier C Segal A Steenhoven A Finite element methods and Navier Stokes equations D Reidel Publishing Company Dordrecht 1986 Gresho P M On the theory of semi implicit projection methods for viscous in compressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix Part 1 Theory Int J Numer Meth Fluids 11 587 620 1990 Part 2 Implementation Int J Numer Meth Fluids 11 621 659 1990 Girault V Raviart P A Finite Element Methods for Navier Stokes Equations Springer Berlin Heidelberg 1986 Harig J Schreiber P Turek S FEAT3D Finite element analysis tools in 3 dimensions User Manual Release 1 1 Technical report University Heidelberg 1993 Heywood J Rannacher R Turek S Artificial boundaries and flux and pressure conditions for the incompressible Navier Stokes equations to appear in Int J Numer Meth Fluids Kloucek P Rys F S On the stability of the fractional step 0 scheme for the Navier Stokes equations SIAM
75. m Try to find another one which has similar features or ask your local computer center or ask the author for the hopefully soon freely available AVS EXPRESS modules Appendix FEATFLOW group We are indebted to the following persons who were involved with theoretical and practical help or suggestions to develop FEATFLOW We hope they won t be too angry about arising questions because of some misleading comments in this manual However this is still the first version and source code and man pages of FEATFLOW will grow hopefully Whoever is interested in getting FEATFLOW or whoever has questions or trouble please send an email to S Turek Chr Becker or to featflow gaia iwr uni heidelberg de List of involved persons Christian Becker University of Heidelberg cbecker gaia iwr uni heidelberg de Roland Becker University of Heidelberg roland gaia iwr uni heidelberg de Heribert Blum University of Dortmund blum math uni dortmund de Phil Gresho LLNL Livermore Joachim Harig University of Heidelberg joachim gaia iwr uni heidelberg de Jaroslav Hron Charles University of Prague hron karlin mff cuni cz John Heywood UBC Vancouver heywood math ubc ca Susanne Kilian University of Heidelberg susanne gaia iwr uni heidelberg de Steffen M ller Urbaniak FORD Koln Hubertus Oswald University of Heidelberg oswald gaia iwr uni heidelberg de Rolf Rannacher University of Heidelberg rannacher Qgaia iwr uni heidelberg de
76. macro time step Protocol file data pp2d non ILEV NVT NMT NEL NVBD 1 165 313 148 34 ILEV NVT NMT NEL NVBD 2 626 1218 592 68 ILEV NVT NMT NEL NVBD 3 2436 4804 2368 136 ILEV NVT NMT NEL NVBD 4 9608 19080 9472 272 time for grid generation 3 6333331540227 ILEV NU NA NB 1 313 2089 592 ILEV NU NA NB 2 1218 8322 2368 ILEV NU NA NB 3 4804 33220 9472 ILEV NU NA NB 4 19080 132744 37888 ILEV NP NC 1 148 706 ILEV NP NC 2 592 2892 ILEV NP NC 3 2368 11704 ILEV NP NC 4 9472 47088 SHSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS MACRO STEP 68 AT TIME 0 294D 01 WITH 1 STEP DT1 0 288D 01 SHSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSHSSSSSSSSSSSSSSSSSSSSSSS REL U1 REL U2 DEF U1 DEF U2 RHONL OMEGNL RHOMG1 RHOMG2 3 2 2D example 57 0 0 17D 03 0 23D 03 1 0 20D 00 0 26D 00 0 14D 04 0 77D 05 0 62D 01 0 10D 01 0 18D 01 0 13D 01 2 0 45D 01 0 28D 01 0 11D 05 0 14D 05 0 78D 01 0 10D 01 0 37D 01 0 57D 01 EHE HERE HHH HHHH HH HHHH EE HEHEH H H HH HHHH H H HHH H EHE HERR EH HOHER HER RR HOO HERR 1 68 TIME 0 297D 01 REL2 P 0 705D 00 RELM P 0 221D 01 EHE HERR HE HER HERE EHE HERE HERE HERE EE HERE HIERHER PEPE EH HER EEE PEEP SHSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS MACRO STEP 68 AT TIME 0 294D 01 WITH 3 STEPS DT3 0 960D 02 SHSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS
77. nation with the streamline diffusion method As result of CC3D we obtain the protocol file data cc3d stat which is almost identical to the 2D version data cc2d stat 62 Examples for the use of FEATFLOW Protocol file data cc3d stat Parametrization file pre c3d prm Coarse grid file adc c3d2 tri Integer parameters of IPARM etc minimum mg level NLMIN maximum mg level NLMAX element type 3 Stokes calculation ISTOK 0 RHS generation 0 Boundary generation Error evaluation M wor mass evaluation lumped mass eval convective part Accuracy for ST Accuracy for B ICUB mass matrix ICUB diff matrix ICUB conv matrix ICUB matrices B1 B2 ICUB right hand side 1 minimum of nonlinear iterations INLMIN maximum of nonlinear iterations INLMAX type of mg cycle ICYCLE 0 minimum of linear mg steps ILMIN maximum of linear mg steps ILMAX type of interpolation IINT 2 type of smoother ISM 1 type of solver ISL 1 number of smoothing steps NSM 64 number of solver steps NSL 500 factor sm steps on coarser lev NSMFAC 8 KPRSM KPOSM ON LEVEL 1 4096 4096 KPRSM KPOSM ON LEVEL 512 512 KPRSM KPOSM ON LEVEL 64 64 KPRSM KPOSM ON LEVEL 64 64 KPRSM KPOSM ON LEVEL 64 64 KPRSM KPOSM ON LEVEL 64 64 KPRSM KPOSM ON LEVEL 64 64 KPRSM KPOSM ON LEVEL 64 64 KPRSM KPOSM ON LEVEL 9 64 64 Real parameters of RPARM etc Viscosity parameter 1 NU 1000 00000000000 parameter for Sam
78. oner possible of diffusive type Elimination of pressure boundary layers Multilevel Discrete Projection scheme CP2D CP3D fastest solver discretization for fully nonstationary problems robust solver discretization for quasi stationary problems All proposed discretization and solution steps can be represented by the following flow chart the Navier Stokes tree leading to the methods mentioned above It is remarkable that most of all existing Navier Stokes solvers can be interpreted by this tree structure This is a very powerfull tool for the understanding and then optimization of existing algorithms and software 1 2 Discretization and solution techniques FEATFLOW 13 NSE uy N u u Vp f V u 0 in 20 30 B C I OkN u u kVp 9 0 FEM FV FD M OKN Up Un KBpp gn BTu 0 a S Un Un m a Uh Outer N nonlinear steps Outer L DPM steps C M Inner L DPM steps Oseen Inner N nonlinear steps Burgers C S 1 1 L gt 1 L 1 L gt 1 4 CG2D CC3D CP2D CPSD PP2D PP3D E Van Kan 10 N 1 N 1 Galerkin schemes Projection schemes Chorin 0 Figure 1 3 The Navier Stokes tree 14 Mathematical Background 1 3 The Navier Stokes solver packages in FEATFLOW The different solution schemes for the Navier Stokes equations we can apply are the fol lowing ones Coupled solver CC2
79. p2d inc containing the needed storage size for the applied program for instance NNWORK 1 000 000 means 1 million double precision ele ments needed or resp a storage amount of 8 Mbyte These files are usually located in the directory input files The best way is to collect all data files together in a subdirectory corresponding to the actual application This is usually done by FEATFLOW for instance in the directories application user start application example and application comp Furthermore the corresponding makefiles are also located in input files such that all data files are concentrated at one place 2 2 2 indat2d f indat3d f parq2d f parq3d f data files for user indat2d f DOUBLE PRECISION FUNCTION FDATIN ITYP IBLOC X Y TIMENS RE Prescribed data for files coeff f bndry f IMPLICIT DOUBLE PRECISION A C H 0 U W Z LOGICAL B PARAMETER PI 3 1415926535897931D0 FDATIN ODO IF ITYP EQ 1 THEN IF IBLOC EQ 1 THEN IF 000 FDATIN 4D0 0 3D0 0 1681D0 Y 0 41D0 Y SIN O 5DO PI MIN TIMENS 1D0 1D0 ENDIF IF IBLOC EQ 2 THEN IF X EQ 0 0D0 FDATIN ODO ENDIF ENDIF es Case 2 Velocity x derivative of exact solution IF ITYP EQ 2 THEN IF IB
80. r step we perform a calculation with a three times larger time step DT1 0 288D 01 and then three substeps with the actual time step size DT3 0 960D 02 Anyway we perform first the nonlinear iteration for the Burgers step which multigrid convergence rates RHOMG1 and RHOMG2 for each convection diffusion equation Then the corresponding pressure equation is solved with multigrid convergence rate RHOMGP estimated time error in different norms L and 1 9 for velocity and pressure are written out followed by the new adaptively chosen time step value RELU L2 0 25D 01 and RELP L2 0 20D 01 are measures for the time derivative of the velocity resp the pressure Additionally the point values for P VELO P FLUX are printed separately into cor responding files in the subdirectory points and can be visualized by GNUPLOT for instance Furthermore we obtain other files for graphical output namely in avs and in byu written with a 1 second delay In the following picture the corresponding pressure plot for 4 file avs u91 inp is shown ca value time Figure 3 6 Lift values for the nonstationary 2D calculation Figure 3 7 Pressure isolines for the nonstationary 2D calculation for T 4 60 Examples for the use of FEATFLOW 3 3 The 3D example In this example we want to perform similar calculations in the following 3D domain Outflow plane U V W 0
81. r task is to initialize all data to build triangulations linear matrices pointer structures and right hand sides on all grid levels all in the file initi f and to generate all vectors needed also in initi f After selecting all parameters for the chosen time stepping scheme and corresponding adaptive time step size the discretization process is finished and the time stepping loop may begin That means everything is prepared for solving the pointed nonlinear coupled generalized stationary Navier Stokes problems corresponding to each time level After doing this in the subroutine prostp f resp mgstp f the main program cc2d f or cc3d f pp2d f pp3d f continues with the time step control selecting a new time step size for the next macro step or repeating the last one if necessary then evaluates the actual solution vector with some extrapolation in time if needed and enters the output subroutines fpost f or error f These main programs provide most of the output for monitoring all actions and give the needed workspace and computer times for the performed application The subroutine on the next level is prostp f resp mgstp f solving or approximating only in the case of the pure projection schemes PP2D PP3D a number of generalized stationary Navier Stokes equations corresponding to the actual macro time level number of these problems and their parameters depend on the scheme and the time step control chosen Furthermore the co
82. ritten out all 1 second We obtain the protocol file data pp3d non which is almost identical to the preceeding protocol files Therefore we renounce a more precise explanation Again the point values for P VELO are printed separately into correspond ing files in the subdirectory points and be visualized by GNUPLOT for instance see Figure 3 6 Furthermore we obtain other files for graphical output namely in avs and in byu written with a 1 second delay In the following Figure 3 12 the corresponding velocity plot for 9 file avs u110 inp is shown 0 08 0 06 0 04 0 02 0 0 02 0 04 0 06 0 08 0 ca value 1 2 3 4 5 6 7 8 9 time Figure 3 11 Lift values for the nonstationary 3D calculation We hope that the presented examples are helpful in understanding the features of FEAT FLOW and give a first feeling how to use it For more questions the author is hopefully prepared to your problems 66 Examples for the use of FEATFLOW Figure 3 12 Velocity plot in the midplane z 0 205 for T 9 Bibliography Axelsson O Barker V A Finite Element Solution of Boundary Value Problems Academic Press 1984 Becker R Rannacher R Finite element discretization of the Stokes and Navier Stokes equations on anisotropic grids Proc 10th GAMM Seminar Kiel January 14 16 1994 G Wittum W Hackbusch eds Vieweg Blu
83. roblems However at the same time it leads to smaller time steps due to the inherently more explicit character and often suffers from spurious pressure boundary layers These different approaches lead to a large variety of schemes all of which are occuring in practice since years see 22 and 21 Theoretical considerations could provide some ideas concerning stability of these schemes convergence rates for subproblems necessary time step sizes or qualitative behaviour for large Reynolds numbers but a complete analysis or quantitative prediction is not possible even today Therefore the only way to make a judgement was to perform numerical tests at least for some classes of problems which seem to be representative This was done in 21 6 Mathematical Background What we reached is a finite element discretization and a solution procedure such that 1 The finite element spaces u and p are stable i e satisfy the LBB condition 8 2 A robust and efficient coupled solver is available 3 A robust and efficient solver of projection type is available 4 An efficient nonlinear solution strategy is available 5 An efficient time step control is available The method which seems to satisfy all these requirements consists of discrete projection schemes with nonconforming linear or rotated multilinear finite elements for u and piece wise constant approximations for p see 15 22 21 With this approach we can dev
84. rresponding boundary conditions are set in this file by bndry f As shown in the previous tree diagram at this point there are the differences between the codes 2 PP3D and 2 cc3p While the fully coupled versions 2 CC3D perform an outer linearization by applying first the adaptive fixed point control as non linear solver PP2D PP3D first decouple velocity and pressure and then solve a nonlinear Burgers equation and a linear quasi Poisson problem In detail PP2D PP3D do the following for solving one nonlinear coupled equation With given pressure as right hand side a nonlinear Burgers equation is solved in nsdef f with adaptive defect optimization optcnl f and multigrid solvers for the linear convection diffusion problems m011 f With the divergence of this intermediate velocity as right hand side a corresponding update problem of quasi Poisson type for the pressure is solved 15 16 Structure of FEATFLOW by a multigrid scheme again m011 f and the auxiliary solution is added to the old pressure to obtain the new one Finally the diffusive preconditioner may be additionally obtained to improve the new pressure This approach corresponds to the special version L 1 of the nonlinear discrete projection scheme We perform this decoupling process only once and hence we obtain an approximate solution in each time step only However that is very similar to the idea of the classical projection schemes
85. same fixed point iteration as explained above for the transport diffusion step with nonlinear operator S Burgers equations In an analogous way we treat by extrapolation the linearized schemes All versions of CC2D CC3D and CP2D CP3D and even 2 PP3D for L large enough lead to the same solutions if the numbers of nonlinear steps N and discrete projection steps L are large enough The complexity analysis in 22 shows that in 2D one single iteration of the coupled scheme C S can be expected to cost at least 10 2D until 20 3D times more than one iteration of the operator splitting method C M At the same time we have to use more nonlinear sweeps or smaller time steps for C due to the more explicit character of the scheme However extensive numerical tests in 21 show that the projection solvers are superior in most cases compared to CC2D CC3D especially for highly nonstationary flows In the next chapter one will see that FEATFLOW follows exactly this flow chart the Navier Stokes tree and that the programming structure is a direct translation of this tree structure 2 Structure of FEATFLOW 2 1 The programming structure of FEATFLOW As pointed in the last section the programming structure of FEATFLOW follows directly the tree structure in diagram 1 3 The corresponding programs CC2D CC3D and PP2D PP3D CP2D CP3D are not yet finished are the main programs which form the body of the code Thei
86. sh data until level i name of input coarse mesh name of created output coarse mesh parameter for type of element refinement 0 no adaptive refinement strategy element is partitioned into 9 finer elements for all elements allowed DELMA distance of element boundary to first refinement line in percentage DELMB no use usual DELMA 0 3333333 equidistant 2 element is partitioned tangential to domain boundary into finer stripes for only elements belonging to 1 boundary component allowed DELMA distance of domain boundary to first refinement line in percentage DELMB distance of domain boundary to second refinement line in percentage usual DELMA 0 3333333 DELMA 0 6666667 equidistant 3 element is partitioned tangential to domain boundary into 2 finer stripes for only elements belonging to 1 boundary component allowed DELMA distance of domain boundary to first refinement line in percentage DELMB no use usual DELMA 0 5 equidistant 4 element is partitioned normal to domain boundary into finer stripes for only elements belonging to 1 boundary component allowed DELMA distance of element boundary to first refinement line in percentage DELMB no use usual DELMA 0 3333333 equidistant 5 element is partitioned into 5 finer elements for all elements allowed DELMA distance of element boundary to first refinement line in percentage DELMB no use usual DELMA 0 333333
87. t mavec time 6 1333212852478 gt konv time 39 716608047485 gt bdry time 4 9992084503174D 02 gt LC time 3 4000720977783 mg time 152 28334903717 substeps 1 3 2 2D example 55 nonlinear 7 MULTIGRID COMPONENTS in percent smoothing 74 324147510282 solver 6 0085270812218 defect calc 8 7993897069504 prolongation 9 7515820090053 restriction 1 0944625011425 Most statement have not to be explained only a few ones ILEV NVT NMT NEL NVBD means number of vertices midpoints elements boundary points on level ILEV ILEV NU NA NB means number of midpoints and nonzero matrix entries for the Laplacian gradient matrix on level ILEV RHONL convergence rate of nonlinear iteration OMEGNL optimally chosen relaxation parameter for nonlinear iteration RHOMG multigrid convergence rate for Oseen equation P VELO contains the u and v velocity for both grid points defined in input files indat2d f PC PRES contains the 4 pressure values for the grid points defined before ICPRES contains the 2 integral mean pressure values defined before I FORCE contains the drag and lift values defined before P FLUX contains the flux value defined before IWMAX contains the value for NNWORK needed NWORK shows the actually defined parameter Additionally we obtain files for graphical output namely in avs the file u1 inp and in byu the BYU files u1 vec v
88. tain M matrix properties Further efficient multigrid solvers are available which work satisfactorily over the whole range of relevant Reynolds numbers 1 lt Re lt 10 and also on nonuniform meshes In 22 we have even shown by a complexity analysis that this pair of elements is most efficient compared to other finite element pairs especially in the case of highly nonstationary flows In combination with the discrete projection methods see 22 it works very robust and efficient in a multigrid code also on highly stretched and anisotropic grids Using the same symbols u and p also for the coefficient vectors in the nodal representation for the functions u and p the discrete version of problem 1 5 may be written as a nonlinear algebraic system of the form Given u a right hand side g and a time step k then solve for u and p Su kBp g Blu 0 1 7 with matrix S and right hand side g such that 6kN u u 6 kN u u 02kf t 1 8 is the mass matrix and N the advection matrix containing the diffusive and convective parts corresponding to the nonlinear form in 1 4 For dominant transport the advection part may include some stabilization for instance some upwind mechanism see 23 or the streamline diffusion method 26 B is the gradient matrix and 1 2 Discretization and solution techniques FEATFLOW 9 the transposed divergence matrix With
89. te replacements an sed command on all files It is necessary that you use a C shell bin csh Additionally the variable MAKEFILES in make lib should be set appropriate From now on we use FEATFLOW as abbreviation for home people example featflow Before we continue be sure that FEATFLOW object libraries libgen exists if not cre ate them by the mkdir command and that the right subroutine for time measurements are taken That means for SUN SPARC 10 and many others cd FEATFLOW source feat2d src ztime sun sparc ztime f cd FEATFLOW object makefiles my compiler Now do the following Execute the shell script make copy by typing make copy Execute the shell script make lib by typing time make lib This last step takes between 10 and 60 minutes the time command is not necessary FEATFLOW libraries will be generated and all makefiles needed for applications are copied to FEATFLOW application example and to FEATFLOW application user start and FEATFLOW application comp Next go to FEATFLOW application workspace and run the shell script links example which activates the applications in FEATFLOW application example more precise the inc files in FEATFLOW application example input files will be used for defining the storage amount Now we are ready to start the 2D and 3D examples 50 Examples for the use of FEATFLOW 3 2 The 2D example We start with describing our 2D domain which shall look like
90. ter for type of projection scheme 0 first order Chorin gt 0 second order Van Kan lt 0 steps of first order then second order maximum number of macro time steps stopping criterion for time derivative parameter for absolute start time parameter for time stepping value 1 Implicit Euler first order if IFRSTP 0 0 5 Crank Nicolson second order if IFRSTP 0 starting time step parameter for time stepping scheme 0 one step schemes 1 fractional step scheme second order parameter for unformatted saving of solution vector 0 no saving gt 0 macro step size for saving procedure in ns modulo number of files for unformatted saving of solution vector maximum 10 time difference for unformatted film output time difference for AVS output time difference for BYU output level for unformatted velocity film output 38 Structure of FEATFLOW IFPSAV level for unformatted pressure film output IFXSAV level for unformatted streamline film output active only in PP2D IAVS level for AVS output IBYU level for BYU output IFINIT start file number for film output IADTIM parameter for adaptive time step control 0 no control fixed time step TSTEP is used 1 prediction without repetition 2 prediction with repetition if nonlinear stopping criteria too large 3 prediction with repetition if time error or nonlinear stopping criteria too large lt 0 the same as ABS IADTIM
91. tput active only in PP2D IAVS level for AVS output IBYU level for BYU output IFINIT start file number for film output IADTIM parameter for adaptive time step control 0 no control fixed time step TSTEP is used 1 prediction without repetition 2 prediction with repetition if nonlinear stopping criteria too large 3 prediction with repetition if time error or nonlinear stopping criteria too large 0 the same as ABS IADTIM but with extrapolation in time TIMEMX maximum absolute time for stopping DTMIN parameter for smallest time step during adaptive control DTMAX parameter for largest time step during adaptive control DTFAC factor for largest possible time step changes TIMEIN absolute time for start procedure EPSADI parameter for time error limit in start phase EPSADL parameter for time error limit after start phase EPSADU upper limit for acceptance for ABS IADTIM 3 IEPSAD parameter for type of error control 1 control of u L2 2 control of u MAX 3 control of p L2 4 control of p MAX 5 control of MAX u L2 p L2 6 control of MAX u MAX p MAX 7 control of MAX u L2 p L2 u MAX p MAX 8 control of MIN u L2 p L2 u MAX p MAX IADIN parameter for error control in start phase 0 EPSADL EPSADI for lt TIMEIN 1 EPSADL linear combination EPSADI EPSADL for T lt TIMEIN 2 EPSADL logarithmic combination EPSADI EPSADL for T lt TIMEIN IREPIT maximim number of repetitions for
92. trary different triangulation in 2D as well as in 3D Both data and grid have to be given in FEAT format These tools will be available in the next release 2 2 The input data of FEATFLOW In this section we explain all input data which are needed to start an application with FEATFLOW Let s start with the preprocessing tools and with some general comments In the next chapter we demonstrate explicitely the meaning of all input data 2 2 1 General comments for user input First of all files containing pure input parameter for user applications have to be in the directory data The file name is the name of the corresponding program ended with dat for instance data pp2d dat These files will be explained in detail in the following subsections Examples for different styles of parameter parameter files concerning ro bustness efficiency and small stoarge amount can be found in the directory application data example Furthermore the only source files which have to be modified by editing or copying are the parametrization files parq2d f resp parq3d f and the data files indat2d f resp 2 2 input data of FEATFLOW 19 indat3d f They contain all information about boundary values right hand sides ex act solutions and their derivatives and output constants like lift and drag or integral or pointwise values in some special coordinates The last file which has to be edited is a corresponding inc file for instance p
93. www iwr uni heidelberg de featflow which contains most of our research activities including the Virtual Album of Fluid Motion FEATFLOW is designed for the following three classes of applications Education Students coming from Mathematics Physics Engineering sciences or Computer sciences learn to handle modern numerical software This helps for future work in industry or for doing Diploma or Ph D works since the basics of today s numerical software engineer ing concerning software and hardware are provided Actually we realize this education program as Software Praktikum at the University of Heidelberg Research This software and hence the underlying Mathematics is running at several universities in Europe and North America As a basic tool it helps to verify own codes by comparisons with reference configurations and it can be used for adding new components to solve more complex problems For instance in our group it is the basic solver which is used after slight modifications for exploring new algorithms for compressible media turbulence modelling multi phase and non newtonian flows or shape optimization Industry Since this software is purely based on mathematically optimized approach it should be able if everything is implemented in a correct way to be much more efficient than most actual software tools used by industry Therefore we try not only to solve mathematical test problems but re

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