Technische Universit at Chemnitz-Zwickau SPC
Contents
1. Lus 19 out file 6 19 37 39 std file 6 15 19 22 38 39 amw22 std 28 30 cubel92 std 27 cubel92a std 28 cube384 std 27 cube48 std 27 cube768 std 27 cube96 std 27 cubusl std 25 cubus2 std 27 cubusu std 26 cubusug std 26 druck std 32 etest std Ln 33 etestl std 33 etestlu std 33 etestd std 33 etestdu std 33 etestu std 33 fem std 3l fichera std 31 lame22d std 34 zug std luu 32 PARDEST 4 PPCDEST 4 archi 4 5 A architecture 4 5 B boundary conditions 19 23 bsp f 2 2 2 2 5 bspamw 28 bsp etest 2 2 2 0 33 bsp lame 34 bsp xy 27 28 31 bsp z 2 2 2 2 25 27 bsp z2 s 26 bsPp 23 sens 26 C change 14 of the maximal number of iterations 14 of the preconditioner 14 of the stop tolerance 14 of the variable ion 14 control quad 6 1 9 control tet 6 1 9 E element types 3 4 epsilon 1 9 F f3sgi ssn 12 38 f3sun 12 38 femakkvar 7 G GRAPE 12 35 38 39 graphics switch off 5 I installation 4 ION 2 onen 7 iter oe ee eee ee ee 7 L Lam system 32. Bei Mehrfachdefinition gilt die Letzte Reihenfolge im File Bei Nichtdefinition kommen die Werte aus control f zur Anwendung femakkvar 1 lin quad 1 vertvar 1 loesvar 4 nint2ass 34 nint2error 34 nint3ass 111 211 nint3error 511 511 ion 1 iter 500 epsilon 1 e 10 ndiag 150 200 verf 0 5 21 12 1995 15 32 Release 1 of User s Manual 10 CHAPTER 2 BASIC DESCRIPTION 2 4 Output information a typical run of the program Output information can be classified into two groups e information that is printed in dependence of the variable ion see Table 2 1 e information that can be called by choosing a menu item We explain this information by following a typical run with ion 1 After calling the program we get an introduction screen with the number of the version the names of main authors the length of the working vector and the number of processors used Then we get a copy of the control parameters and the input request for a problem file tap kain fem xr8 tet ppc run Requesting network by calling nrm run Creating 4 2 descriptor by calling mkdesc run Starting D Server at kain link 3 RHEIN TER H SSSS PPPPP CCCC PPPPP M M PPPPP 333 SS SS PP PP CC CC PP PP MM MM PP PP 33 33 SS PP PP CC PP PP MMM MMM PP PP 33 SSSS PPPPP CC PPPPP MM MMM MM PPPPP 000 333 SS PP CC PP MM M MM PP 00 00 33 SS SS PP cc CC PP MM MM PP 00 00 33 33 SSSS PP CCCC PP MM MM
3. 0 00000 0 00000D 00 4 10 000 0 000 0 000 0 00000 0 00000 0 00000D 00 5 10 000 0 0001 7 500 0 75000 0 75000 0 39456D 05 6 10 0001 0 0001 5 000 0 50000 0 50000 0 43561D 06 7 10 0001 0 0001 2 500 0 24998 0 25000 0 23232D 04 8l 7 500 2 500 10 000 1 00000 1 00000 0 00000D 00 l 9l 5 000 5 000 10 000 1 00000 1 00000 0 00000D 00 l 101 2 500 7 500 10 000 1 00000 1 00000 0 00000D 00 1411 7 500 0 000 7 500 0 74999 0 75000 0 62478D 05 12 5 000 0 000 5 000 0 50000 0 50000 0 15223D 05 191 0 000 2 500 2 500 0 25002 0 25000 0 22938D 04 20 2 5001 0 000 0 000 0 00000 0 00000 0 00000D 00 Abbruch J N j If the output is on screen it can be terminated by entering any character at a enter request Menu item 5 gives the results of local and global error calculations estimations kokok akak ak akak ak akak ak akak ak akak ak akak ok ok ok akak akak ak akak akak akak ok ok ak ok ok ok ok ok OK kakak akak KK oe RE 2K K k KK 2 AUSGABEMENUE kk Kok ok ok ok ok akak ok ok ok ok akak ok ok ak akak ak ok ok ok akak ok ok ok akak ok ok kok ok ak ok kok 2k k k 2 2 2 akak k k k k k k kok k k k k O WEITER 1 GRAPE 2 AUSGABE DER NETZDATEN 3 AUSGABE DER RANDKETTENDATEN 4 AUSGABE DER LOESUNG 5 AUSGABE VON FEHLERNORMEN ak akak kakak akak ak akak ak akak ke ok ak ak akak akak ak akak ak ok ak akak akak akak akak ak akak ak akak ak
4. 4 BPX without coarse grid solver 5 BPX with coarse grid solver 14 31 bling Neumann boundary data nint2ass UU Ut 1 digit quadrilaterals see Table 2 2 2 4 digit triangles see Table 2 3 in the assembling 5lll 1 digit tetrahedra see Table 2 4 nint3ass 311 151 Au 214 digit hexahedra bricks see Table 2 6 3 digit pentahedra triangular prisms see Table 2 5 nint3error 311 131 as nint3ass but used for the integration of 3D integrals in the error calculation gt 0 message after each ion th CG iteration lt 0 no information about the iteration no startup screen and no problem info integer no information on numbers of coupling faces edges nodes lt no menus lt no input request messages maximal number of iterations in the CG algorithm stop criterion for the CG relative decrease of the epsilon 1 LE 4 real gt 0 norm of the residual upper estimate for the number of nonzero entries In any row of the stiffness matrix If it is chosen too integer gt 0 large the program may suffer from lack of memory and if it is chosen too small the number is itera i increased waste of time refinement parameter for a certain class o realc 0 examples see Subsection 4 1 7 0 no change of the mesh Table 2 1 Variables in control tet control quad 21 12 1995 15 32 Release 1 of User s Manual 8 CHAPTER 2 BASIC DESCRIPTION Formula Number of exact for
5. BASIC DESCRIPTION Figure 2 1 Finite elements implemented in SPC PM Po 3D The finite element stiffness matrix and the right hand side are generated locally in the subdomains by approximating the integrals using a quadrature rule see Sections 4 1 and 4 2 in 4 The resulting system of equations is solved using a parallel version of the conjugate gradient method with Jacobi Yserentant hierarchical basis or BPX preconditioning which are described in 4 Chapter 5 It is planned to include also a multigrid method The postprocessing includes a simple variant of error assessing If in special test examples the exact solution of the problem is known then the error in L and H norms are calculated by numerical integration additionally the error is measured in the discrete maximum norm see 4 Subsection 4 4 1 In general the exact solution of the problem is not available thus we must rely on an error estimator We plan to implement an improved variant of the residual type error estimator see 14 2 2 Installation Provided AFS the Andrew File System is installed any user can install the package by using the shellscript afs tu chemnitz de home urz p pester bin install3d name of destdir where name of destdir should be a name which does not yet exist For a quick start do the following 1 Edit the Makefile in name_of_destdir and adjust the variables PARDEST and PPCDEST ensure that these directories exists at the co
6. Chapter 2 Basic description 2 1 Mathematical background Consider the Poisson problem in the notation Au f in QCR u ug on ON Ou o 7 on OOo Ou 0 on o0 0f 005 On or the Lam problem for u uw u 9 u T pAu A u grad div u in OCR on IN i 1 2 3 on N i 1 2 3 1 0 on on ap an t l 2 3 o e SS where 1 10 100 1907 S u n is the normal stress the stress tensor S u ij ja is defined with g a z 9 2 by Ou Out si H 5 PE T5AV u n is the outward normal and is the Kronecker delta The domain 2 C IR must be bounded In the present version curved boundaries can not be treated by the refinement procedure thus Q is restricted to be a polyhedron The boundary value problem is solved by a standard finite element method using either tetrahedral or brick elements with linear or quadratic shape functions of the serendipity class see Figure 2 1 The initial mesh must be generated outside SPC PM Po 3D After the file input it is distributed to the processors using a spectral bisection algorithm 20 That means the domain 2 is decomposed in non overlapping subdomains the basis for our parallel algorithms Then the elements are hierarchically refined to generate the final finite element mesh for a description of the algorithm see Chapter 3 in 4 21 12 1995 15 32 Release 1 of User s Manual page 3 4 CHAPTER 2
7. 5 lt enter gt At this stage the coarse mesh data are read in and distributed to the processors the mesh is 21 12 1995 15 32 Release 1 of User s Manual 12 CHAPTER 2 BASIC DESCRIPTION hierarchically refined and the stiffness matrix as well as the coarse grid matrix are assembled After an the system of equation is solved giving information on the convergence and on times for communication and arithmetics Finally the program stops in the next menu IT r w As s ALFA BETA 1 6 964812E 01 2 071824E 02 3 361682E 01 0 000000E 00 2 6 968691E 01 9 348962E 02 7 453972E 02 1 000557E 00 3 1 382542E 01 1 788189E 02 7 731519E 02 1 983933E 01 4 2 990811E 00 3 549269E 01 8 426555E 02 2 163270E 01 5 4 965188E 01 4 576239E 00 1 084993E 01 1 660148E 01 6 1 323842E 01 1 472371E 00 8 991227E 02 2 666248E 01 7 3 502404E 02 3 375455E 01 1 037610E 01 2 645636E 01 8 8 619178E 03 5 656719E 02 1 523706E 01 2 460932E 01 9 2 135504E 03 1 573281E 02 1 357357E 01 2 477620E 01 10 5 290934E 04 3 136041E 03 1 687138E 01 2 477604E 01 11 2 299528E 04 1 639463E 03 1 402611E 01 4 346167E 01 12 4 114933E 05 3 600621E 04 1 142840E 01 1 789468E 01 13 7 486498E 06 5 780739E 05 1 295076E 01 1 819349E 01 14 7 883535E 07 6 721151E 06 1 172944E 01 1 053034E 01 15 1 570714E 07 6 721151E 06 1 172944E 01 1 992398E 01 IT 15 Zeiten fuer Warten Kommunikation s Prozessor log phys input in 4 output in 4 gesamt 0 0 0 0 0 08 30 79 0 07 25 36 0 28 1 O
8. 52 21 21 98 58 57 22 69 70 23 23 191 80 80 25 97 25 25 364 101 98 26 cube384 cube768 LoesVar LoesVar Level 3 5 2 azoo 8 33 27 27 1 46 41 32 24 67 61 38 35 95 54 42 38 Table 4 6 Iteration numbers for different preconditioners in different examples Input Output time FEMAKKVar 1 Input Output time FEMAKKVar 2 Number of processors Number of processors Level 16 16 64 Level 16 1 64 8 nodes 16 nodes 32 nodes 8 nodes 16 nodes 32 nodes 0 25 0 50 0 99 2 58 Total time FEMAKKVar 1 Total time FEMAKKVar 2 Number of processors Number of processors Level 16 16 64 Level 16 1 64 8 nodes 16 nodes 32 nodes 8 nodes 16 nodes 32 nodes Table 4 7 Comparison of two data accumulation algorithms 2 for different numbers of processors and different problem sizes running on Parsytec GCPP time in seconds 21 12 1995 15 32 Release 1 of User s Manual 30 CHAPTER 4 EXAMPLES zm Elements Elements 12 12 24 18 48 30 96 45 96 45 192 15 384 135 168 225 168 225 1536 405 3072 765 6144 1377 6144 1377 12288 2601 24576 5049 49152 9537 49152 9537 98304 18513 196608 36465 393216 70785 393216 70785 186432 139425 1572864 276705 3145728 545025 3145728 545025 Table 4 8 Number of nodes for different refinement levels a b c Figure 4 2 a coarse mesh with z 0 b one refinement step with u 0 c one refinement step with u 1 For this and only this domain
9. O 1d 0 13 45 22 0 15 53 75 0 28 2 0 0 3 0 11 38 12 0 16 58 00 0 28 3 0 0 2 0 24 85 56 0 03 10 59 0 28 4 010 0 08 28 84 0 19 66 89 0 28 5 O 1 1 0 23 82 91 0 03 13 73 0 28 6 O 1 3 0 21 75 40 0 05 19 94 0 28 7 01 2 0 24 83 99 0 03 12 30 0 28 reine Arithmetikzeit max 0 12 kk akak ak akak ak akak ak akak ak ak akak ok ok ak akak kakak kak kak ak akak ok ok ok ok k ok ok ok akak akak kok akak oe akak akak kakak kakak AUSGABEMENUE kk Kok kokok ok akak ok ok ok ok akak ok ok kok akak ok kok akak ok akak akak 2 22 2 ok akak akak 2k ok k k ok ok kok ok k k k k k k k kok k k k O WEITER 1 GRAPE 2 AUSGABE DER NETZDATEN 3 AUSGABE DER RANDKETTENDATEN 4 AUSGABE DER LOESUNG 5 AUSGABE VON FEHLERNORMEN A kakak akak ak oke kakak ak ak akak akak ak akak ak akak akak akak ak akak kakak akak akak ak akak akak akak akak kakak ak akak akak 2K K 2K EINGABE 4 With item 0 we exit the menu with item 1 we are asked for the host name for displaying then we start the data transfer to the interactive graphics package GRAPE see 18 provided the program f3_sun or f3_sgi runs at the own workstation host name In this case a control and a graphics window will appear in order to display the grid and or solution One solution starting with the first degree of freedom can appear at one time Using the control window we can make visible the other degrees of freedom by pressing the buttons with the names of the corresponding
10. OF MESHES VIA OLDNETZ 21 Input parameter w internal angle of the sector u mesh refinement parameter here u 0 4 number of circular arcs number of nodes at the edge N K R radius of the circular edge the middle A B circle of the torus z coordinate of this middle circle radius of of the sector in the cross FAT section IS number of sectors for mesh genera tion here 4 Figure 3 4 3 d family sector of a torus with arbitrary internal angle w Input parameter w internal angle of the sector u mesh refinement parameter here u 0 6 N number of circular arcs K number of nodes at the edge R radius of the cylinder A height of the cylinder IS number of sectors for mesh genera tion here 4 WV y gt b i amp Figure 3 5 4 family sector of a cylinder with arbitrary internal angle w Input parameter N reciprocal value of the mesh size 0 1 2 3 Figure 3 6 5 e family Fichera corner 21 12 1995 15 32 Release 1 of User s Manual 22 CHAPTER 3 MESHES AND BOUNDARY CONDITIONS to enter the type and the data of the boundary condition once for the whole group This procedure is repeated for each degree of freedom To define the group one enters conditions of the form rci y y z z Te LS rE Ty TV lt r lt ri 8 N A AA 7 3 RB eke B Lue xe 8 N In this way all nodes are marked which satis
11. Q a value verf Z 0 in control tet is useful in order to control an anisotropic mesh refinement The following coordinate transformation is carried out 1 refinement level verf grading parameter e e 2 1 r zy Du p ifr lt t p ie Inew hk VT old Ynew h Yold For u verf 1 we get a change in the coordinates only for points with r gt t that means they are moved on the curved boundary see Figure 4 2 In Tables 4 9 and 4 10 we show some results for the error behavior for different values of u verf The tests were carried out with amw22d std and the following parameters os u 111 for linear elements Cu URP 211 for quadratic elements Nint3error 511 Nint2ass 11 11 for linear elements Nint2error 12 for quadratic elements 21 12 1995 15 32 Release 1 of User s Manual 42 LAME SYSTEM 31 Level linear elements quadratic elements 4 0797e 1 1 6819e 1 2 3825e4 0 2 0854e 1 3 5542e 2 9 2351e 1 3 3811e 1 7 6697e 2 1 5922e 0 1 4325e 1 1 3205e 2 5 6153e 1 2 3133e 1 3 2269e 2 1 0164e 0 9 3024e 2 4 8989e 3 3 4802e 1 1 5039e 1 1 3063e 2 6 4116e 1 5 9467e 2 1 8467e 3 2 1748e 1 9 5848e 2 5 1834e 3 4 0279e 1 Table 4 9 Discretization error for verf p 1 lincar elements 2 8163e 1 1 3671e 1 1 2327e 0 1 324le 1 4 9917e 2 4 3328e 1 6 0739e 2 2 2336e 2 1 5496e 1 2 5
12. ads 12 EN o 1 0 END OF DATA Be N IN o HG L LG HG B H G Gp G oHLBHB HH HH HB oH SB HiupB Cr 0irngusb K ne5rnx go iooo0d 0t 2rm 0ltnuiu NNGONNGDWOU ONO ON ONB d QN EN Table 3 3 The file Cubusl std HEADER vertices edges faces solids regions dirfaces neumfaces materials Note that the backslash marks a continuation of the line dirfaces and neumfaces means the number of faces with Dirichlet and Neumann data respectively The actual data blocks follow now in any permutation A block consists of a key word line and a number of data lines Note that the key word line may contain an integer The key words and the structure of the data lines is summarized in Table 3 2 for a full explanation see 15 The file Cubusl std see Table 3 3 may serve as an introductory example which describes the partition of a cube Q 0 1 into 6 congruent tetrahedra compare Table 3 4 and Figure 3 1 for the understanding of the topology Here no REGION is defined a region name is useful to point to an internal table of materials If undefined all elements belong to one region with the name 1 21 12 1995 15 32 Release 1 of User s Manual 18 CHAPTER 3 MESHES AND BOUNDARY CONDITIONS Names of edge 1 13 15 1 2 15 5 13 15 6 16 1 15 19 14 16 11 14 10 14 19 9 2 13 17 3 4 13 16 17 8 18 4 16 19 17 18 12 14 11 17 19 12 Table 3 4 Names of faces edges and nodes of the 6 tetrahedra in Cubusl s
13. bsp f are called There is a third pair of files in this family etestl std and etestlu std which differs from the first two pairs by the Neumann condition T y on z 00 z 0 or z 10 z 0 I elsewhere In this case the exact solution is not known 21 12 1995 15 32 Release 1 of User s Manual 34 CHAPTER 4 EXAMPLES 4 2 5 lame22d std with bsp lame This is a test with a known solution which has the typical behavior near an edge The domain and the meshes are the same as in 4 1 7 the exact solution is rl 3 cos sp cos Sy 5sin sp sin Sp w 33 sin o sin By cos y cos By r2 3 sin 20 and with v we get f 0 in Q and u 0 on the faces forming the edge In lame22d std there are Dirichlet boundary conditions defined on the whole boundary the values are taken from bsp f bsp lame Unfortunately the system is very badly conditioned because v is close to i l v which results in very high iteration numbers Examples of the error behavior for different values for verf are given in the file mesh3 lame22d txt The numbers are not really promising may be there is still an er ror anywhere Hints are welcome 21 12 1995 15 32 Release 1 of User s Manual Appendix A Mesh generation and related programs Our research group has been developing several programs for the automatic generation of meshes 2D 3D sequential parallel and their visualization Due to historical re
14. bsp xy used as bsp f and different preconditioners The files cubena std define 0 ON and were used in 2 to compare two communi cation routines The results are not reproducible because since then a scaling error in the hierarchical list was discovered and removed which influenced the number of iterations In Table 4 7 we give some results with the correct version of preconditioner The tests were carried out with bsp xy as bsp f which means Au 0 in N u zy on OO linear shape functions cubel92a std Epsilon 1074 LoesVar 2 Yserentant without coarse grid solver 4 1 7 amw std with bsp amw The amw family of meshes describes the domain 3 3 Q z r cos y rsiny z E R 0 lt r lt 1 O lt lt 5 0 lt z lt 1 which was used extensively in the papers 3 5 6 but on serial computers The two digits in the filename gives the number of intervals in r and z direction that means their reciprocal value corresponds to the mesh size The d as the last letter of the base name stands for global Dirichlet boundary conditions 004 ON Contrary in amw22 std we have 00 u E 00 2 0 The meshes are useful in connection with bsp amw where the exact solution is given by Au 2 u 10 z r sin y A FE 21 12 1995 15 32 Release 1 of User s Manual 4 1 POISSON EQUATION 29 cube48 cube192 LoesVar LoesVar 5 Level 3 5 19 14 13 18 18 33 1T 18 3 21 36 27 25 22 17 52 40 38 21 AT
15. includes Makefiles and meshes 2 3 The files control tet and control quad The mesh and the boundary conditions are described in files with the extension std see Subsection 3 2 Additionally there is a couple of variables controlling the execution of the program They are described together with their standard values in Table 2 1 5ome of the variables contain numbers of quadrature formulas They are given for the different types of elements in Tables 2 2 2 6 Note that the standard values may change during the evolution of the program These standard values can be overwritten by defining other values in a file control tet or control quad respectively The lines in this file have the form variable value or variable value lin value quad The is relevant variable must be written in lower case There is no check of the usefulness of the value Different values for the linear and the quadratic case can be given 21 12 1995 15 32 Release 1 of User s Manual 2 3 THE FILES CONTROL TET AND CONTROL QUAD 7 values value ol shape functions 1 linear shape functions 2 quadratic shape functions of coarse grid partitioning vertvar 2 trivial partitioning 2 partitioning via recursive spectral bisection f kk 3 ere are two variants of accumulation of dis emakkvar BER data see 2 choice of the preconditioner 1 Jacobi 2 Yserentant without coarse grid solver loesvar 3 Yserentant with coarse grid solver
16. of full multigrid techniques January 1994 U Groh Lokale Realisierung von Vektoroperationen auf Parallelrechnern March 1994 U Groh Chr Israel St Meinel and A Meyer On the numerical simulation of coupled transient problems on MIMD parallel systems April 1994 G Globisch On an automatically parallel generation technique for tetrahedral meshes April 1994 M Pester and T Steidten Parallel implementation of the Fourier Finite Element Method June 1994 A Meyer and M Pester Verarbeitung von Sparse Matrizen in Kompaktspeicherform KLZ KZU June 1994 Th Apel and F Milde Realization and comparison of various mesh refinement strategies near edges August 1994 M Pester On line visualization in parallel computations November 1994 M Pester Grafik Ausgabe vom Parallelrechner f r 2D Gebiete November 1994 M Meisel A Meyer Implementierung eines parallelen vorkonditionierten Schur Komplement CG Verfahrens in das Programmpaket FEAP Januar 1995 M Meyer Grafik Ausgabe vom Parallelrechner f r 3D Gebiete Januar 1995 T Apel G Haase Meyer M Pester Parallel solution of finite element equation systems efficient inter processor communication Februar 1995 U Groh Ein technologisches Konzept zur Erzeugung adaptiver hierarchischer Netze f r FEM Schemata Mai 1995 P Kunkel V Mehrmann W Rath J Weickert GELDA A Software Package for the Solution of General Linear Differential Algebraic Equa
17. or changed The file input is stopped either by reaching the end of the file or the statement amp END OF DATA After the VERSION statement there may be optional information statements see Table 3 1 for a selection Moreover it is possible to redefine some internal array dimensions via such statements see 15 The information part and the data part of the file are separated by a HEADER statement It determines the maximal number of data lines of the different types 21 12 1995 15 32 Release 1 of User s Manual page 15 16 CHAPTER 3 MESHES AND BOUNDARY CONDITIONS Statement DESCRIPTION string description of the file for cataloging DATE date USER username HOST hostname PROGRAM name DIMENSION 3D date of creation of the file Login name of the creator of the file name of the host where the file was created name of the creating program geometrical dimension of the problem here only 3D useful EQN_TYPE string problem type defines e g the meaning of the material data DEG_OF_FREE integer number of degrees of freedom standard 5 Table 3 1 Selection of information statements in the input file Key word Ine VERTEX EDGE amp SOLID REGION DIRICHLET name zxcoord ycoord zcoord I R R R name type start end middle pointer data I I I I I I arbitrary type 1 straight edge type 2 arc of a circle name type m edgel edgen pointer data I I I I I
18. variable archi see also below The file variante archi is included in the main source file and defines the length of a long vector for storing all vector data its length must be adapted to the size of the memory of the machine to be used The file makefile archi is included in the main makefile and contains specific options and directories which are machine dependent The variable GRAF can be set to Graf or NoGraf thus the graphic 21 12 1995 15 32 Release 1 of User s Manual 6 CHAPTER 2 BASIC DESCRIPTION libraries are linked or not which results in a considerable difference in the length of the executable file A couple of meshes for tests are contained in the directories mesh3 tetrahedral meshes and mesh4 cuboidal meshes std The file structure is described in Section 3 2 These directories are linked to afs tucz home urz t tap fem mesh 3 4 in order to prevent that the data files exist several times In some cases there is a file name txt which gives some information about the corresponding problem name std These AFS directories are readable and executable for any user Th Apel is administrating these directories and can include further AFS users to a list of people who are allowed to add files in these directories The directory mesh3 contains also a couple of files with the extension out These files were created with the mesh generator PARMESH3D see 11 and can be processed with the program mesh3 renfindsun on a SUNA work
19. 111 in the linear case Nint3 eee ass 211 in the quadratic case Nint3error 211 u un is quadratic 26 CHAPTER 4 EXAMPLES Level linear elements ee pam 1 84e 13 1 63e 12 6 35e 13 3 85e 9 5 66e 9 1 37e 8 1 05e 8 Tevet LLL near elements pi RZ KA PANA KA LANA Table 4 2 Discretization error for u z 4 1 3 cubusu std with bsp z bsp z2 and bsp z3 With these examples we test the discretization error orders Again we have 2 0 10 with Dirichlet boundary conditions at 00 x Q z 0 or z 10 but this time the boundary values are taken from the corresponding function in the file bsp f If we copy bsp z u to bsp f we get no discretization error see Subsection 4 1 2 With bsp z2 the exact solution is u z f 2 and we get an error with linear elements but no error with quadratic elements The third example bsp z3 corresponds with u z f 6z and we observe in both cases the optimal order of the error 77 k degree of the shape functions m order of the Sobolev space H N m 0 1 to measure the error Tables 4 1 and 4 2 contains the values The tests were carried out with 211 for linear elements f is quadratic for u z Nint3 i l kih 311 for quadratic elements f y is cubic for u z 511 u uj is of degree 6 but 5 is the best formula Nint3error programmed Epsilon 10710 LoesVar
20. 277e 2 1 2162e 2 5 5923e 2 1 0240e 2 Table 4 10 Discretization error for verf ji 0 5 with linear elements and verf u 0 3 with quadratic elements Epsilon 10710 LoesVar 4 BPX without coarse grid solver 4 1 8 fichera std The domain mesh is 1 1 V 0 1 which is known as a Fichera corner It was used with the sequential code for the tests in 7 but not yet on the parallel computer The digit means in analogy to 4 1 7 the reciprocal of the meshsize 4 1 9 fem std The domain consists of the letters FEM which have a different size in the third direction The coarse mesh consists of 93 elements with 122 nodes We have Dirichlet boundary conditions at the bottom face Q z 0 z 0 In Figure 4 4 we demonstrate isolines at the surface of the domain calculated with bsp xy Level 2 4 2 Lam system 4 2 1 Introduction We consider the Lam equation system pAu A u grad divu f for u uu u 9 T with the boundary conditions ue ul on 000 1 3 21 12 1995 15 32 Release 1 of User s Manual 32 CHAPTER 4 EXAMPLES Figure 4 3 Fichera corner Figure 4 4 Isolines on FEM t g on An pesos 0 on an anl an i 1 3 where 109 10 19 7 is the normal stress 4 2 2 druck std This is again the cube 2 0 10 divided into six tetrahedra The boundary conditions are u 0 on z 00 z 0 0 u 0 o
21. 31 LIBLISTE 5 lin quad T loesvar 2 1 0 eee eee 7 M make 6 CLEAN 6 clean 6 quad 6 tar celeres 6 tet eee ee eee 6 Makefile 4 6 makefile archi 5 menu 11 12 14 meshes l l 6 N ndiag 22 22 202020 7 nint2ass s 7 nint2error 7 nint3ass 7 nint3error css 7 O oldnetz 6 19 35 P parmesh3d 19 27 35 37 Poisson equation 3 25 postprocessing 4 12 Q quadrature formulas 8 quick start 4 R renedgsun 19 38 renfindsun 6 19 28 35 38 renumerate nodes 19 S scale coarse grid matrix 14 set boundary condition 19 23 solver 222222 4 V variante archi 5 verf 2 22 22 7 30 31 34 vertvar 222 een 7 X xbe 6 22 35 39 21 12 1995 15 32 Release 1 of User s Manual Other titles in the SPC series concerning implementation 93_2 93 3 94 1 94 2 94 5 94 6 94 10 94 12 94 15 94 23 9424 95_2 95_4 95_5 95 13 95 15 95 19 95 20 95 26 95 27 95 33 95 34 M Pester and S Rjasanow A parallel version of the preconditioned conjugate gradient method for boundary element equations June 1993 G Globisch PARMESH a parallel mesh generator June 1993 J Weickert and T Steidten Efficient time step parallelization
22. 4 BPX without coarse grid solver 4 1 4 cubusug std The example differs from cubusu std only by the boundary conditions Again we have Dirichlet boundary conditions on 0 x Q z 0 or z 10 but on the remaining part of the boundary we have Neumann conditions 05 N V 0991 The values of uo and 21 12 1995 15 32 Release 1 of User s Manual 4 1 POISSON EQUATION 27 27 12 10 16 Table 4 3 Numbers of iterations for cubus2 std bsp xy and 8 or 16 processors Yserentant and BPX without coarse grid solver here 27 45 65 123 205 125 225 369 725 1305 729 1377 2465 4905 9265 4913 9537 17985 39931 69729 35937 70785 137345 274593 540865 274625 545025 1073409 2146625 2146689 Table 4 4 Number of nodes for different refinement levels g are taken from bsp f The use of bsp z yields no discretization error which can be used as a test 4 1 5 cubus2 std The domain and the mesh cubus2 std are identical to cubusl The boundary conditions are 00 z 0 2 0 005 00 N where the values of ug and g are taken from bsp f For example one can link with bsp xy as bsp f which corresponds to ug ry f g 0 Table 4 3 shows the number of iterations for different preconditioners in this case We used Epsilon 1074 and linear elements 4 1 6 cube std with bsp xy The family of meshes cube48 std cube96 std cubel92 std cube384 std and cube768 std was generated in order to have test example
23. Description number points x y with midpoint center of gravity 2x2 Gaussian points Table 2 2 Quadrature formulas for quadrilaterals Formula Number of exact for Description number points x y with center of gravity midpoints of the edges Gaussian points Gaussian points 3x3 Gaussian points Table 2 3 Quadrature formulas for triangles Formula Number of exact for Description number points x z with center of gravity Gaussian points Gaussian points Gaussian points Gaussian points Table 2 4 Quadrature formulas for tetrahedra Formula Number of the formula is a cross product of the formulas exact for number points for triangle for interval z direction x y z with Table 2 5 Quadrature formulas for pentahedra center of gravity midpoints of edges 4 Gaussian points midpoints of edges 4 Gaussian points 4 Gaussian points 7 Gaussian points 7 Gaussian points midpoint midpoint midpoint 2 Gaussian points 2 Gaussian points 3 Gaussian points 2 Gaussian points 3 Gaussian points Formula Number of exact for Description number points x y z with midpoint center of gravity 2x2x2 Gaussian points Table 2 6 Quadrature formulas for hexahedra 3x3x3 Gaussian points midpoints of the faces Irons formula 21 12 1995 15 32 Release 1 of User s Manual 2 3 THE FILES CONTROL TET AND CONTROL QUAD 9 for all integer variables This is especially usef
24. I arbitrary type 1 plain face name type n facel facem pointer data I I I I I I arbitrary type 1 parameter is not used yet name type n solid solidn I I I I I type parameter is not used yet name I type data pointer data I R I arbitrary type 0 no Dirichlet condition for this d o f type 1 constant value given in data type 2 boundary values are given by a linear function in one line per d o f global coordinates uo z y z data l x data 2 y data 3 z data 4 type gt 100 function pointer boundary values are taken from function subroutine in bsp f NEUMANN in analogy to DIRICHLET MATERIAL Tab name nmn datal datan I I R R le 3 2 Structure of the data blocks in the input file 21 12 1995 15 32 Release 1 of User s Manual 3 2 STRUCTURE OF THE INPUT FILE STD 17 VERSION 1 0 18 DESCRIPTION 6 kongruente Tetraeder DATE 13 7 1995 USER Thomas Apel DIMENSION 3D EQN_TYPE Poisson DEG_OF_FREE 1 HEADER 8 8 19 18 VERTEX 8 r o ND O0 d WN OO NO O1 DEN EN o EN EN EN o r N r N m ds c1 r a r a EN o Or rO0Or ro m o KR OO HM KO O KAR HOO O SO e w r I FB RnB HR pH pg mR gn xxn H HH H H m o EN 18 SOLID r I 3 3 3 3 3 3 3 3 3 3 3 3 3 7 3 3 3 3 3 6 4 4 4 4 4 4 1 DIRICHLET 1 0 ND an d IN Bee w Ne 11 e e e
25. PC 93 1 G Kunert Ein Residuenfehlerschatzer fur anisotrope Tetraedernetze und Dreiecksnetze in der Finite Elemente Methode Preprint SPC 95 10 TU Chemnitz Zwickau 1995 D Lohse Datenschnittstelle I Standardfile Preprint SPC95_Z TU Chemnitz Zwickau 1995 In preparation M Meisel and A Meyer Implementierung eines parallelen vorkonditionierten Schur Komplement CG Verfahrens in das Programmpaket FEAP Preprint SPC 95 2 TU Chemnitz Zwickau 1995 A Meyer and M Pester Verarbeitung von Sparse Matrizen in Kompaktspeicherform KLZ KZU Preprint SPC 94 12 TU Chemnitz Zwickau 1994 M Meyer Grafik Ausgabe vom Parallelrechner f r 3D Gebiete Preprint SPC 95 4 TU Chemnitz Zwickau 1995 W Queck FEMGP Finite Element Multi Grid Package Programmdokumenta tion und Nutzerinformation Report TU Chemnitz Zwickau Fachbereich Mathematik 1993 C Walshaw and M Berzins Dynamic load balancing for PDE solvers on adaptive unstructured meshes Research Report 92 32 University of Leeds School of Computer Studies 1992 A Wierse and M Rumpf GRAPE Eine objektorientierte Visualisierungs und Nu merikplattform Informatik Forsch Entw 7 145 151 1992 21 12 1995 15 32 Release 1 of User s Manual Index Index 4 The italic numbers denote the pages where the corresponding entry is described numbers underlined point to the definition all others indicate the places where it is used Symbols edg file
26. PP 000 333 RHEIN TER H Programm Modul 3D Potentialprobleme Version 1 95 DFG Forschergruppe SPC TU Chemnitz Zwickau Fakultaet fuer Mathematik Th Apel A Meyer M Meyer F Milde M Pester M Thess 16 MB Variante 3600000 Worte bis zu 1024 Prozessoren in Benutzung 8 Prozessor en Gelinkt mit bsp z RHEIN TER H kak ok akak ak akak ak akak ak akak kak akak kok akak ok akak ok ok ak akak akak ak OK kak akak akak ak akak OK KK eoe kakak Belegung der Steuerparameter kann mittels File control tet angepasst werden akak akak akak ke ke ak akak akak ke akak akak akak ak akak ke ke ak akak akak akak kakak kakak akak k kk vertvar 2 lin_quad 1 nen2d 3 nen3d 4 femakkvar 2 loesvar 5 nint2ass 14 nint3ass 311 nint2error 11 nint3error 311 iter 200 epsilon 0 10E 03 ion 1 ndiag TO Verzeichnis fuer Netze mesh3 kak ok akak ak akak ak akak ak akak kak akak kok akak ok akak ok ok ak akak akak ak OK kak akak akak ak akak OK KK eoe kakak Filename cubusi 21 12 1995 15 32 Release 1 of User s Manual 2 4 OUTPUT INFORMATION A TYPICAL RUN OF THE PROGRAM 11 The file name is typed in here cubus the input of a question mark generates a ls command for the appropriate directory Then we are asked for the number of refinement steps There is also the possibility to escape by typing 1 for a new mesh or 2 to q
27. Technische Universitat Chemnitz Zwickau DFG Forschergruppe SPC Fakult t f r Mathematik Thomas Apel SPC PM Po 3D User s Manual Acknowledgement The package SPC PM Po 3D has been developed in the re search group SPC at the Fakult t f r Mathematik of the Technische Universitat Chemnitz Zwickau under the supervision of A Meyer and Th Apel Other main contributors are G Globisch D Lohse M Meyer F Milde M Pester and M Thef Section 3 4 and Appendix A of this documentation were written together with F Milde and G Globisch respectively Section 3 5 was written by D Lohse The tests in Section 4 were partially carried out by A Meyer and U Reichel The manuscript was influenced by remarks of A Meyer and it was typed by U Reichel The research group SPC is supported by Deutsche Forschungsgemeinschaft Ger man Research Foundation No La 767 3 All this collaboration and support is gratefully acknowledged Preprint Reihe der Chemnitzer DFG Forschergruppe Scientific Parallel Computing SPC 95_33 December 1995 Contents 1 Introduction 2 Basic description 2 1 Mathematical background ll 2 2 Installation 4 Coon nn 2 3 The files control tet and control quad nn nn 2 4 Output information a typical run of the program 3 Meshes and boundary conditions 3 1 General remarks 2222s 3 2 Structure of the input file std aa 3 3 The tools renfindsun and
28. ackage for solving the Poisson equation over 3D domains on sequential com puters see 3 based on internal mesh generation and on another file structure ada Th Apel F Milde SPC PM CFD AFS workcfd pmhi ppc px Parallel simulation of fluid dynamics in 2D St Meinel A Meyer SPC PM EL2D AFS workel pmhi ppc px Parallel simulation of elasticity in 2D A Meyer SPC PM Po2D AFS worksy pmhi ppc px Parallel simulation of potential problems in 2D A Meyer 21 12 1995 15 32 Release 1 of User s Manual 38 APPENDIX A MESH GENERATION AND RELATED PROGRAMS 21 12 1995 15 32 Release 1 of User s Manual Bibliography 1 2 10 11 12 Th Apel G Haase A Meyer and M Pester Numerical comparison of two commu nication algorithms Documentation TU Chemnitz Zwickau 1995 Th Apel G Haase A Meyer and M Pester Parallel solution of finite element equation systems efficient inter processor communication Preprint SPC95 5 TU Chemnitz Zwickau 1995 Th Apel and F Milde Realization and comparison of various mesh refinement strate gies near edges Preprint 5PC94 15 TU Chemnitz Zwickau 1994 Th Apel F Milde and M Thess SPC PM Po 3D Programmer s Manual Preprint SPC95_34 TU Chemnitz Zwickau 1995 Th Apel R Mucke and J R Whiteman An adaptive finite element technique with a priori mesh grading Technical Report 9 BICOM Institute of Computational Math em
29. ak akak ak akak kakak akak a 2K OK EINGABE 5 21 12 1995 15 32 Release 1 of User s Manual 14 CHAPTER 2 BASIC DESCRIPTION AUSGABE VON FEHLERNORMEN LOKAL PROZ MAX NORM L2 NORM H1 NORM 0 0 00000E 00 0 00000E 00 0 00000E 00 1 0 00000E 00 0 00000E 00 0 00000E 00 2 0 23232E 04 0 78504E 04 0 93344E 04 3 0 22938E 04 0 89531E 04 0 72902E 04 4 0 10585E 04 0 42743E 04 0 46008E 04 5 0 22938E 04 0 40395E 04 0 64121E 04 6 0 15438E 04 0 78381E 04 0 54767E 04 7 0 23232E 04 0 56625E 04 0 55416E 04 AUSGABE VON FEHLERNORMEN GLOBAL MAX NORM L2 NORM H1 NORM 0 23232E 04 0 16428E 03 0 16225E 03 kk akak ak akak ak akak ak akak ak ak akak ok ok ak akak kakak kak kak ak akak ok ok ok ok k ok ok ok akak akak kok akak oe akak akak kakak kakak AUSGABEMENUE kk Kok kokok ok akak ok ok ok ok akak ok ok kok akak ok kok akak ok akak akak 2 22 2 ok akak akak 2k ok k k ok ok kok ok k k k k k k k kok k k k O WEITER 1 GRAPE 2 AUSGABE DER NETZDATEN 3 AUSGABE DER RANDKETTENDATEN 4 AUSGABE DER LOESUNG 5 AUSGABE VON FEHLERNORMEN A kakak akak ak oke kakak ak ak akak akak ak akak ak akak akak akak ak akak kakak akak akak ak akak akak akak akak kakak ak akak akak 2K K 2K gt EINGABE 0 GEWUENSCHTE ZAHL VON VERFEINERUNGSSCHRITTEN 1 NEUES NETZ 2 PROGRAMM BEENDEN EINGABE 2 kk akak ak akak ak akak ak akak ak ak akak ok ok ak akak kakak kak
30. asons the pre main and postprocessing tools use input and output files with different data structure Therefore a few little programs for converting the files from one structure into the other have been made available This is useful for reusing meshes in other programs for example for benchmark tests A survey of the programs and tools is given in Figure A 1 stressing their connection with respect to the data structure A detailed description of the programs is beyond the scope of this manual we restrict ourselves to the following list Note that AFS stands for afs tu chemnitz de home urz p pester bin such programs can be accessed from all computers with AFS installed Graphical editors GRAFED f femtools grafedv2 exe Graphical editor for describing geometrical data in 2D at PC storing them as file inp M Fritz see also 8 NETS k util nets net exe as GRAFED but storing data as file net M Seibt M Pe ster Automatic mesh generation PARMESH3D AFS parix parmesh3d px AFS ppc parmesh3d px Automatic parallel 2D 3D mesh generation 10 11 output files have structure file out or 2D file wqf G Globisch PREMESH AFS SUNA premeshg fAfemtools premesh exe Sequential 2D grid genera tion in a UNIX and DOS version 19 M Goppold Converting data structures GRAFEDSUN AFS SUNA4 grafedsun Converting file inp see GRAFED into file bsp see FEM BEM and vice versa G Haase GUNDOLFSUN AFS SUNA gundolf
31. atics 1993 Th Apel and S Nicaise Elliptic problems in domains with edges anisotropic regularity and anisotropic finite element meshes Preprint SPC94_16 TU Chemnitz Zwickau 1994 Th Apel A M Sandig and J R Whiteman Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non smooth domains Technical Report 12 BICOM Institute of Computational Mathematics 1993 To appear in Math Meth Appl Sci M Fritz Grafischer Editor f r die Netzmanipulation Diplomarbeit TU Chemnitz Sektion Mathematik 1991 G Globisch Der Algorithmus HYBRID zur Knotenumordnung Praktikumsarbeit TH Karl Marx Stadt Sektion Mathematik 1985 G Globisch On an automatically parallel generation technique for tetrahedral meshes Preprint SPC 94 6 TU Chemnitz Zwickau 1994 G Globisch PARMESH a parallel mesh generator Parallel Computing 21 3 509 524 1995 G Haase B Heise M Jung and M Kuhn FEMooBEM A parallel solver for linear and nonlinear coupled FE BE equations Report nr 96 16 DFG Schwerpunkt Randelementmethoden 1994 21 12 1995 15 32 Release 1 of User s Manual page 39 40 13 14 15 16 17 18 19 APPENDIX A MESH GENERATION AND RELATED PROGRAMS G Haase Th Hommel A Meyer and M Pester Bibliotheken zur Entwicklung paral leler Algorithmen Preprint SPC 95 20 TU Chemnitz Zwickau 1995 Updated version of SPC 94 4 and S
32. city with in general mixed boundary conditions of Dirichlet and Neumann type see Section 2 1 The domain Q C IR can be an arbitrary bounded polyhedron The input is a coarse mesh a description of the data and some control parameters The program distributes the elements of the coarse mesh to the processors refines the elements generates the system of equations using linear or quadratic shape functions solves this system and offers graphical tools to display the solution Further the behavior of the algorithms can be monitored arithmetic and communication time is measured the discretization error is measured different preconditioners can be compared We plan to extend the program in the next future by including a multigrid solver an error estimator and the treatment of coupled thermo elastic problems The program has been developed for MIMD computers it has been tested on Parsytec machines GCPowerPlus 128 with Motorola Power PC601 processors and GCel 192 on transputer basis and on workstation clusters using PVM The special case of only one processor is included that means the package can be compiled for single processor machines without any change in the source files We point out that the implementation is based on a special data structure which allows that all components of the program run with almost optimal performance O N or O N In N In this documentation we use slanted style for really existing paths and filenames italic style fo
33. functions Pressing the continue button in the control window the program on the parallel computer is forced to continue for example to compute a new solution During this time the graphical program may go on displaying the old data until 21 12 1995 15 32 Release 1 of User s Manual 2 4 OUTPUT INFORMATION A TYPICAL RUN OF THE PROGRAM 13 the FE3D neu button is pressed to receive new data from the parallel computer again via menu item 1 With Exit we can finish the graphics program The choice of item 2 leads to the output of the local mesh data to files netzred number_of_processor dat one file per processor The same is done by item 3 with the coordinates of the nodes stored in Kettes for the term Kette see 2 they are stored in files kettinf Pnumber_of_processor dat With menu item 4 we get a table of values into a file loesung dat or on screen The table includes the local node numbers their coordinates the calculated solution the solution using the function u from bsp f probably the previously known exact solution and their difference for each processor see the printout AUSGABE DER WERTETABELLE DER LOESUNG AUSGABE IN FILE LOESUNG DAT J N PROZESSOR 0 NUMNP 0 PROZESSOR 1 NUMNP 0 PROZESSOR 2 NUMNP 35 NR X Y Z BER LOESUNG EXAKTE LSG DIFFERENZ 1l 10 000 0 000 10 000 1 00000 1 00000 0 00000D 00 2l 0 000 10 000 10 000 1 00000 1 00000 0 00000D 00 3 0 000 0 000 0 000 0 00000
34. fy all the conditions given The group consists of all faces which have only marked nodes Note the special case when no condition is entered then all boundary faces are in the group After defining the group of faces the user is asked for e the kind of boundary condition 1 Dirichlet 2 Neumann e the type and the data for the boundary conditions see Table 3 2 for the explanation Then the next group of boundary faces can be defined or one may exit this menu In the second case one is asked for a filename to store the data and the program terminates Note that faces can be included in groups several times then the boundary condition is always redefined for these faces This feature can be use for correcting errors or to enter complicated boundary data For example if all faces but one have Dirichlet conditions one can first enter the Dirichlet condition for all faces and then redefine the exceptional face 3 5 The program xbc 3 5 1 Description xbc was planned as a tool to check the integrity of files std and as test environment for routines managing standard files It is grown up to a visualization tool for objects stored in the standard file format general polyhedra in boundary representation as well as 3D meshes with the capability to create and to manipulate boundary values on that objects The program needs an X View environment there is no plain X Windows nor Motif based version 3 5 2 Command Line Parameters All
35. itches to the view window To save a file it s necessary to choose the Save File button in the File menu and to enter the file name manually There is no command line parameter for a standard save file 3 5 4 The View Window The view window is used to visualize the object and to choose faces to set boundary condi tions There are two buttons and two menus in the view window e The Back button switches to the main window e The Repaint button is reserved for a general hidden line algorithm that will be imple mented soon e The Settings menu is used to control the behavior of xbc e The BC menu contains tools to manipulate the boundary conditions The object in the view window can be rotated by moving the mouse holding the middle mouse button down A single click with this button forces a refresh of the viewport Faces without boundary conditions are shown in gray faces with Dirichlet conditions in red and faces with Neumann conditions blue The presence of both types of boundary conditions is represented by violet color 3 5 5 The BC Menu In the current release V0 9 only the Set BC button of the BC menu is active It is used to manipulate values of boundary conditions Pressing this button opens an Object Selection window and enables the selection mode for the mouse buttons Pressing the left mouse button in the viewport selects the visible face at the mouse pointer The right mouse button unselects the face It is also possible t
36. kak ak akak ok ok ok ok k ok ok ok akak akak kok akak oe akak akak kakak kakak PROGRAMMENDE A kakak akak ak akak kakak ak ak akak akak ak akak ak akak akak akak ak akak kakak akak akak ak akak akak akak akak kakak ak akak akak 2K K 2K run Returning network by calling nrm run Terminating with result 0 tapOkain femA The choice of item 0 led to the main menu see above Some of the information is also written in the files fort 08 and fort 09 but this is only for test reasons and permanently changing Furthermore we note that at the stage Start der Simulation Vorkonditionierung Nr 4 lt enter gt some special letters can be entered to control the PCCG iteration process for a change of the preconditioner loesvar for a change of the maximal number of iterations iter for a change of the stop tolerance epsilon for a scaling of the coarse grid matrix N amp 0 H 4 for a change of the variable ion These corrections are valid only during the following CG iteration and do not overwrite the standard values of these variables see Subsection 2 3 An exception is ion 21 12 1995 15 32 Release 1 of User s Manual Chapter 3 Meshes and boundary conditions 3 1 General remarks The program SPC PM Po 3D has not been designed to generate coarse meshes or boundary data It is assumed that these data are prepared before and stored in a file with extension Std The structure of such files is described in 15 we summa
37. ked AFS SGI5 f3_sgi dynamically linked AFS SGI5 f3grape sgi statically linked Visualization of 3D data received via socket connection 18 based on GRAPE M Meyer GRAFEM f femtools grafem exe Visualization of 2D FEM data including the solution data type file wgl 19 G Haase 1see also Graphical editors above 21 12 1995 15 32 Release 1 of User s Manual 37 GRAPE AFS SUN4 grape_sun Visualization of 3D data files file out node related structure based on GRAPE 21 Th Hommel SHOWNET AFS SUN4 shownet Visualization of 2D FEM data including the solution isolines data type file out possible output as ps file F Brauer VINP f femtools vinp exe AFS SUN4 vinp Visualization of 2D data files file inp 19 M Goppold KBO AFS SUN4 xbc Visualization of 3D data files file std edge related and modifi cation of boundary conditions D Lohse see also Section 3 5 Other preprocessing DECOMP AFS SUN4 decomp Spectral graph partitioning of finite element meshes for parallel computations M Goppold Main processing FEMGP Package for solving 2D boundary value problems on sequential computers see 19 based on files file wgf and partially file out M Jung T Steidten W Queck and others FEMQDBEM Package for solving 2D boundary value problems using a coupled FEM BEM strategy on parallel computers based on files file bsp see 12 G Haase M Jung and others FEMPS3D P
38. n z 00 z 10 2 t 0 elsewhere The exact solution is not known With v 0 3 E 2 10 we get a deformation as shown in Figure 4 5 4 2 3 zug std and zugl std The two files zug std and zugi std describe the same example but they were created by different programs We have again the cube 2 0 10 The boundary conditions are u 0 on z E0N z 0 0 0 on z 00 z 10 1 t 0 elsewhere The exact solution is not known We calculated again with v 0 3 E 2 10 the result is shown in Figure 4 6 21 12 1995 15 32 Release 1 of User s Manual 42 LAME SYSTEM 33 Figure 4 5 Cube under pressure Figure 4 6 Cube under pull 4 2 4 The etest family For tests of the validity of the computer results we use the following example which is described by bsp etest consequently f 0 t n There is no discretization error thus the error is in the range of the error of the solver We prepared two test examples In etestd std and etestdu std the whole boundary is of Dirichlet type while in etest std and etestu std also Neumann boundary conditions appear on z 00 z 0 or z 10 3 ww elsewhere The files with and without the u at the end of the basename differ by the way the boundary conditions are described In the version without the u the data of the conditions are defined in the file whereas in the version with u the functions from
39. n in a file file std with the data structure as given in Section 3 2 The user is requested to enter the number of the family and the corresponding parameters for a short description see Figures 3 2 3 6 For refining meshes using the parameter u see 3 5 6 7 3 4 2 Setting boundary conditions If a mesh contains not only a few elements then it is boring to enter the boundary conditions face by face Thus a dialog with the user was programmed to define groups of faces and 21 12 1995 15 32 Release 1 of User s Manual 20 CHAPTER 3 MESHES AND BOUNDARY CONDITIONS 0 r b r b Input parameter 1 u mesh refinement parameter here u 2 0 4 N number of circular arcs 3 K number of nodes at the edge R outer radius radius of the middle cir 4 cle of the torus A z coordinate of this middle circle B radius of of the sector in the cross section DEN Figure 3 2 Description of the 1 family a 90 sector of a torus perspective view top view and cross section b Input parameter u mesh refinement parameter here u 1 1 N number of slices in the cross section see figure K number of nodes at the edge 2 R outer radius radius of the middle cir cle of the torus 3 A z coordinate of this middle circle B length of the cathete in the cross 4 N section Figure 3 3 2 family as before but with another cross section 21 12 1995 15 32 Release 1 of User s Manual 3 4 GENERATION
40. ntually some real values see Table 3 2 The mesh and the boundary conditions can be visualized by means of the program xbc which is also capable to impose change boundary conditions see Section 3 5 Moreover the user can determine whether he she wants to renumerate the nodal points of the mesh in order to reduce the bandwidth profile of the corresponding matrix adja cency matrix to the edge graph The corresponding algorithm is implemented to be an efficient combination of minimal degree ordering and nested dissection see 9 The numer ical expense is O N for two dimensional meshes where N denotes the number of nodes in the three dimensional case we were not able to prove an estimate Note that files which have already the structure file std can be renumerated by the program renedgsun even a repeated application of renedgsun can further reduce the bandwidth profile The mesh generator parmesh3d can also construct meshes consisting of tetrahedra having curved boundaries The corresponding internal data structure is given in 10 But to date there is no agreement about the file structure for curved elements The corresponding extension of the related programs will be done in the future 3 4 Generation of meshes via oldnetz 3 4 1 Mesh generation The program oldnetz is compiled for a SUN4 workstation and can be used interactively to generate 5 different families of meshes to describe the boundary conditions and to store this informatio
41. o select or unselect faces by editing the Face Name line in the Object Selection window The Reset button in this window unselects all Selected faces change their color to yellow The Cancel button terminates the whole value setting process the OK button finishes the selection process and opens the Set BC Values window This window allows the simple choice of the kind of boundary condition Dirichlet Neumann as well as the input of the 21 12 1995 15 32 Release 1 of User s Manual 24 CHAPTER 3 MESHES AND BOUNDARY CONDITIONS actual number of the degree of freedom the equation type Poisson or Lam and the set of values of that boundary condition If more than one degree of freedom is used for some faces it is necessary to set all degrees of freedom in one step by using the Apply button of the Set BC Values window A Set BC procedure finished by the OK button overwrites all settings of the chosen faces Cancel stops the whole setting process The meaning of the equation types are e Free No BC is given at the current degree e Const The BC is constant on the surface the value is given in the field Value 1 e Lin Glob The BC is given by u V1 a V2 y V3 z V4 x y z are the coordinates in the global system e User The values of the BC will be given by special routines of the user program 3 5 6 The Settings Menu The Settings menu controls the general behavior of xbc It includes two buttons the View Control button and
42. on routines are different e archi SUNA is set after calling setpvm on a SUN4 workstation The executable files are tet SUN4 and quad SUN4 they can run under pvm or without the daemon of pvm as single processor variant at a SUN workstation e archi HPPA is set by calling setpvm on a HP workstation e archi parix is set by setparix The executable files run at Parsytec transputer machines as the GCel 192 under the operating system PARIX e archi ppc is the setting after calling setppc which causes the compilation of an executable file for Parsytec machines based on the Motorola Power PC601 chip as the Xplorer or the GCPowerPlus 128 under the operating system PARIX After the installation there is a file structure as given in Figure 2 2 The directories Assem Grafik Netz and Solve contain source files links to some include files and a Makefile which works together with the file LIBLISTE A call of make in these subdirectories is done by a make in the main directory If a user wants to include additional source files he she should add it in the file LIBLISTE Sometimes it is necessary to describe problem data by function subroutines right hand sides exact solution if available These routines are contained in the file Assem bsp f Our approach is to save example data in files bsp example name and to copy the appropriate file to bsp f The directory Makedir contains some architecture specific files which are distinguished by the
43. r program parameters sans serif style to characterize buttons and menu items of programs with a graphical user interface and typewriter style for the names of variables 21 12 1995 15 32 Release 1 of User s Manual page 1 Dr Thomas Apel Dr Gerhard Globisch Dag Lohse Prof Arnd Meyer Dr Magdalene Meyer Frank Milde Dr Matthias Pester Uwe Reichel Michael St bner Michael Thef CHAPTER 1 INTRODUCTION List of contributors email apel mathematik tu chemnitz de contribution supervision assembly error assessment com munication tests email globischQmathematik tu chemnitz de contribution PARMESH3D renfindsun see 3 3 email lohse mathematik tu chemnitz de contribution input files xbc see 3 2 3 5 email a meyerQmathematik tu chemnitz de contribution supervision solver communication email m meyer mathematik tu chemnitz de contribution interface to GRAPE email milde physik tu chemnitz de contribution general frame of program mesh refinement tests and oldnetz see 3 4 email pester amp mathematik tu chemnitz de contribution maintaining the libraries time measurement communication email reichel amp mathematik tu chemnitz de contribution domain decomposition via recursive spectral bisection tests email m stuebner mathematik tu chemnitz de contribution error norms email thess mathematik tu chemnitz de contribution solver 21 12 1995 15 32 Release 1 of User s Manual
44. renedgsun lens 3 4 Generation of meshes via oldnetz nn 3 5 The program abc hne 4 Examples 4 1 Poisson equation llle ns 4 2 Lam system 2 2 2 2 2 2 2 2 22 2522222 9 aos A Mesh generation and related programs Bibliography Index 21 12 1995 15 32 Release 1 of User s Manual page 0 25 25 31 35 39 41 Chapter 1 Introduction At present time much effort is being spent in both developing and implementing parallel algorithms The experimental package SPC PM Po 3D is part of the ongoing research of the Chemnitz research group Scientific Parallel Computing SPC into finite element meth ods for problems over three dimensional domains Special emphasis is paid to choose finite element meshes which exhibit an optimal order of the discretization error to develop precon ditioners for the arising finite element system based on domain decomposition and multilevel techniques and to treat problems in complicated domains as they arise in practice The package SPC PM Po 8D is based on a set of libraries which are still under devel opment They are documented in the Programmer s Manual 4 and in other separate papers 13 16 17 18 The aim of this User s Manual is to provide an overview over the program its capabilities its installation and handling Moreover test examples are explained In Version 2 0 the program can solve the Poisson equation and the Lam system of linear elasti
45. rize it briefly in Section 3 2 There are several ways to create such an input file For the easiest domains one can just create it with an editor Moreover several mesh generators have been programmed in the past Because they use different file structures there have been developed adapter programs see Appendix A In Section 3 3 we describe the adapter program mesh3 renfindsun author G Globisch which writes files of the structure appropriate for SPC PM Po 3D This program has two additional features renumeration of the nodes to minimize the profile of the coarse grid matrix and an interactive definition of boundary conditions For five classes of meshes which were used already with the sequential program FEM PS3D there is the tool mesh3 oldnetz author F Milde which is described in Section 3 4 In Section 3 5 we introduce the tool mesh3 xbc author D Lohse which is an XView application to visualize meshes and boundary conditions which are stored in std files Furthermore it is possible to re define boundary conditions with this tool 3 2 Structure of the input file std The input file is a 7 bit ASCII file which contains data lines control lines and key word lines both starting with a and comment lines starting with see for example Cubusl std in Table 3 3 The file starts with a control line defining the version VERSION 1 0 in order to circumvent incompatibilities when the data structure 1s extended
46. rresponding machines More over it is useful to copy the directories mesh3 and mesh4 to the remote machine or link the directories afs tucz home urz t tap fem mesh3 and mesh4 to di rectories mesh3 and mesh4 in the working directory of the remote maschine Choose the architecture you want to work with by calling one of the shellscripts usr global bin setpvm usr global bin setparix or usr global bin setppc 5ome variables including archi are now defined 3 Call make 21 12 1995 15 32 Release 1 of User s Manual 2 2 INSTALLATION 5 Assem Makedir Allgemein Netz Quader name of destdir Solve Tetraeder mesh3 linked to afs tucz home urz t tap fem mesh3 mesh4 linked to afs tucz home urz t tap fem mesh4 Figure 2 2 File structure after installation of SPC PM Po 3D Then after successful compilation the executable files tet archi for tetrahedral meshes and quad archi for cuboidal meshes should be contained in your directory and for archi parix and archi ppc in the directories on the remote machines Before we are going to describe in some detail the use of the various files which were created during the installation we explain the diverse values of the variable archi It is used to distinguish the different architectures for which an executable file shall be compiled and linked because the compiler libraries and especially the communicati
47. s with equidistributed coarse meshes on any number 2 k 0 7 of processors and with numbers of nodes as large as possible see Table 4 4 The number of elements of cuben in level is n 2 The domain is the cube 0 2 The meshes were generated using the mesh generator PARMESH3D 11 from 2D refer ence meshes see Figure 4 1 which are reproduced several times into the third dimension Thus prisms with triangular basis can be formed and divided in three tetrahedra each The corresponding reference meshes and the number of their reproduction is given in Ta ble 4 5 Note that cube384 and cube768 represent different meshes than cube48 and cube96 respectively with one refinement step though they have the same number of elements 21 12 1995 15 32 Release 1 of User s Manual 28 CHAPTER 4 EXAMPLES a b c Figure 4 1 2D reference meshes for the cube family 2D reference a number of reproductions 2 Table 4 5 The cube family and their corresponding reference meshes The mesh data are stored in the files cube out which can be processed by the program renfindsun in order to describe boundary conditions and to create the data structure of standard files The boundary conditions in the standard files cuben std are OQ x Q z 0 or z 2 where the boundary conditions are taken from the function u in bsp f In Table 4 6 we document tests with these meshes with
48. station This program produces a file with the right data structure and with boundary conditions which are set by a dialog with the user Moreover renfindsun can optionally renumerate the nodes to minimize the bandwidth of the resulting stiffness matrix see Section 3 3 The program mesh3 oldnetz produces a restricted class of tetrahedral meshes see Sub section 3 4 The program mesh3 xbc in an XView application to view meshes and to set or to change boundary conditions interactively see Section 3 5 In the main directory name_of_destdir there is the main Makefile some more FOR TRAN source files include files and the files control tet control quad which are described in Section 2 3 The Makefile is used to compile source files to create libraries to link the executable file and to copy it to the appropriate machine george informatik or kain hrz The destination for the remote copy is defined by two variables PARDEST and PPCDEST in the Makefile which should be adjusted by the user see above Note that it is possible to link only tet archi or quad archi by calling make tet or make quad respectively The Makefile can also be used to remove the libraries tar files and executable files make clean removes the target files for the current architecture and make CLEAN removes them for all architectures Only the files of the installation as well as user created files remain The additional option make tar creates a archive with all sources
49. sun Converting file net see NETS into file bsp see FEM BEM and vice versa G Haase 21 12 1995 15 32 Release 1 of User s Manual page 35 36 APPENDIX A MESH GENERATION AND RELATED PROGRAMS FEM BEM 2D file bsp 2D file bsp E ME GUNDOLFSUN ENDEN IGRAFEDSUN 2D file net 2D file inp 3D file inp SE reg TRANSFERSUN L L 2D file inp GRAFED 2D file inp 2D 3D file inp 2D file out E DECOMP VINP SHOWNET PREMESHG PREMESH EXE Oo main processing mesh generation visualization 7 converting data structures Figure A 1 Connection of the tools corresponding to the data structure POS2NET AFS SUN4 pos2net Converting file wgf into file net involving a renum bering of the nodes and the setting of boundary conditions G Globisch RENEDGSUN AFS SUN4 renedgsun Renumbering the nodes to minimize the matrix profile input and output are files of std structure G Globisch RENFINDSUN AFS SUN4 renfindsun Conversion of file out 3D node related see 10 19 into file std 3D edge related see 3 2 involving a renumbering of the nodes to minimize the matrix profile and the interactive setting of boundary con ditions G Globisch see also 3 3 3 4 2 TRANSFERSUN AFS SUNA transfersun Converting file net see NETS into file inp see GRAFED G Globisch Visualization F3 AFS SUN4 f3_sun dynamically linked AFS SUNA4 f3grape sun statically lin
50. td T 11 10 9 6 nodes 6 7 j edges 4 15 faces 1 Figure 3 1 View of the cube which is described in Cubusl std 21 12 1995 15 32 Release 1 of User s Manual 3 3 THE TOOLS RENFINDSUN AND RENEDGSUN 19 3 3 The tools renfindsun and renedgsun Because of the importance of the files file std for the package SPC PM Po 3D the program renfindsun shall be described in more detail here The program renfindsun converts the ASCII output file out see 10 19 for a description of the structure of the file of the parallel mesh generator parmesh3d tetrahedral meshes into the file std see 3 2 for this data structure This means a change of the node related data structure into the edge face structure Note that renfindsun may also store the output data as file edg This is another file type for the edge related data structure see 10 It organized similarly to file out This transfer includes the setting of boundary conditions type and data to the boundary faces by a dialog with the user There are two possibilities namely face by face or by defining face groups The second variant is described in 3 4 2 The first possibility consists in the facewise screen output of the coordinates of the three nodes and in prompting for the description of the related boundary condition for each degree of freedom For both methods this information consists of the kind Dirichlet Neumann 3 kind Robin the type and eve
51. the Zoom menu The View Control button opens a window which allows to choose the drawing method Solid Hidden Line Wire Frame This window is also used to control the visibility of the names e g integers of objects like vertices edges or faces The Zoom menu offers some standard zoom factors and the capability to enter user defined factors using the Other button 21 12 1995 15 32 Release 1 of User s Manual Chapter 4 Examples 4 1 Poisson equation 4 1 1 Introduction We consider the Poisson equation with in general mixed Dirichlet and Neumann boundary conditions Au f inQ u ug on OQ Ou an 7 9 en OOo Ou 0 on on 004 05 On In the next subsections we describe some test examples which demonstrate that our code gives the right result and works very effectively 4 1 2 cubusl std with bsp z The file cubusl std describes a cube Q 0 10 with Dirichlet boundary conditions uo 0 at the bottom face x N z 0 and uo 1 at the top face x Q z 10 That means the boundary conditions are not taken from bsp f but directly from the file and for the successful test the program should be linked with bsp z This means a setting f 0 for the right hand side and for the exact solution which is used to calculate error norms In this example there is no discretization error thus the error is proportional to error tolerance in the solver If not check first the integration rules for example
52. the standard XView command line parameters are available e g display displayname or fg colorname xbc help shows a list of these parameters Although all of these param eters work there is no test of bad usage implemented Two additional parameters allow a quick file access e InPath pathname is the main path for the input files if no nPath is present the actual working directory is used as path for the input files e InFile filename is the name of the input file relative to the input path The menu offers also boundary condition of 3 kind but SPC PM Po 3D can not treat them yet 21 12 1995 15 32 Release 1 of User s Manual 3 5 THE PROGRAM XBC 23 A list of all implemented parameters is shown by xbc Help All other parameters will be interpreted as file names If no input file is specified the user has to enter file name and path manually in the File menu 3 5 3 Loading and Saving Files If a file name is specified in the command line xbc loads this file automatically The user can enter the file name manually by opening the File menu and choosing the Load File button Loading a file xbc first reads the information part shows this information and asks for confirmation During the loading process xbc checks the integrity of the data Any problems will be shown in error messages and the user will be asked for continuing the reading procedure After a successful load procedure the Show button becomes available It sw
53. tion February 1995 A Meyer D Michael Some remarks on the simulation of elasto plastic problems on parallel com puters March 1995 B Heinrich B Lang B Weber Parallel computation of Fourier finite element approximations and some experiments May 1995 M Meisel A Meyer Kommunikationstechnologien beim parallelen vorkonditionierten Schur Komplement CG Verfahren Juni 1995 G Haase T Hommel A Meyer and M Pester Bibliotheken zur Entwicklung paralleler Algorithmen Juni 1995 C Israel NETGEN69 Ein hierarchischer paralleler Netzgenerator August 1995 M Jung Parallelization of multi grid methods based on domain decomposition ideas November 1995 Th Apel SPC PM Po3D User s manual December 1995 Th Apel F Milde M The SPC PM Po3D Programmer s manual December 1995 Some papers can be accessed via anonymous ftp from server ftp tu chemnitz de directory pub Local mathematik SPC Note the capital L in Local The complete list of current and former preprints is available via http www tu chemnitz de pester sfb spc95pr html
54. uit the program GEWUENSCHTE ZAHL VON VERFEINERUNGSSCHRITTEN 1 NEUES NETZ 2 PROGRAMM BEENDEN EINGABE 2 After this we get information on the current state of the program and to problem data EINLESEN DER NETZDATEN AUS mesh3 cubusi std Wuerfel Kantenlaenge 10 oben unten Dirichlet Gerhard Globisch 07 11 1994 Poisson Gleichung PARMESH RENFINDSUN 3D copy of the information of the input file extension std EINLESEN BEENDET IER 0 VERTEILUNG DER TETRAEDER DURCH REKURSIVE SPEKTRALBISEKTION Anzahl der Elemente in den Prozessoren information on the progress of 2 2 2 2 4 4 4 4 0 0 2 2 2 2 2 2 0 0 1 1 1 1 1 1 the recursive spectral bisec tion NETZ VERFEINERT VFS 1 NETZ VERFEINERT VFS 2 START GENERIEREN ASSEMBLIEREN ASSEMBLIEREN BEENDET Coars Grid Matrix Generation ler 0 Groesse der Matrix VBZ 30 matrix information on the coarse grid Probleminformationen lokal Prozessor P globale Anzahl Crosspoints 8 Anzahl der Knoten lokal 35 davon lok Crosspoints 4 Summe der Randketten 30 information on data on pro Koppelknoten 34 cessor 0 innere Knoten 1 Anzahl der Koppelkanten Anzahl der Koppelflaechen 4 Probleminformationen global Anzahl der Prozessoren 8 Anzahl der Knoten 125 davon Koppelknoten 119 global information interne Knoten 6 gt Gesamtanzahl der Freiheitsgrade 125 Start der Simulation Vorkonditionierung Nr
55. ul for the quadrature rules and for ndiag If a variable appears more than once in the file then the last value is taken Note that these files can be omitted if only standard values shall be used As an example consider the case that the user likes to change the stop criterion in the CG method to e lt 107190 He she has two possibilities Either one can change this during the execution see the last paragraph in Section 2 4 Or he she introduces the file control tet or control quad with one line epsilon 1 E 10 As an example we display here the file control tet as it is contained in the distribution of SPC PM Po 3D File zur Anpassung von Standardwerten fuer PFEM Kommentarzeilen sollten mit beginnen Datenzeilen haben die Form schluesselwort wert Der Doppelpunkt ist wichtig Grosz Kleinschreibung ist signifikant Die Richtigkeit der Werte wird nicht ueberprueft Folgende Schluesselworte sind zulaessig ihr Name entspricht der zu besetzenden Variable deren Bedeutung und zulaessige Werte gehen aus dem Quelltext standard f hervor lin quad 1 vertvar 2 femakkvar 2 loesvar 3 nint2ass 14 nint3ass 311 nint2error 11 nint3error 311 ion 1 iter 200 epsilon 1E 4 ndiag 70 verf 0 Fuer alle Integer Werte koennen zwei Werte fuer linear quadratisch angegeben werden Trennzeichen erforderlich Diese Liste musz bei Veraenderung von standard f gegebenenfalls aktualisiert werden
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