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User`s Manual - Technische Universität Chemnitz

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1. 26 3 6 The progra xpe Imes acs dre a dede pe 28 3 7 Mesh construction and manipulation with geo conv 31 4 Examples 33 4 1 Poisson equation 33 ZO IIO Syst m sre ere a en bt wee Eee bie D ele ee es 40 A Mesh generation and related programs 43 Bibliography 47 Index 49 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 now part of SFB393 into finite element methods 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 preconditioners 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 3D is based on a set of libraries which are still under de velopment They are documented in the Programmer s Manual 4 and in other separate papers 13 17 18 19 The aim of this User s Manual is to provide an overview over the pro gram its capabilities its installation and handling Moreover test examples are explained This new release describes the changes from version 2 to version 3 3 the new standard file vers
2. NAECHSTER PROZESSOR ABBRUCH SONST WEITER NUMNP 35 a If the output is on screen it stops after displaying 20 nodes and the user is prompted how to proceed You can switch to the next processor by pressing N cancel output with A or proceed displaying with any other key Menu item 7 gives the results of local and global error calculations estimations PK K 2K K 2K ok ok ok ok ok ok ok ok ok ok K 2g oe ook oe ok ok 2k 2k FK ok K ok K FK oe oe oe ok ok ok ke K 2k ok ok ok ok ook ok ok ok ok ok ok 2K 2K 2K 2 eoe ok ok AUSGABEMENUE aK KK kK FK FK K kK FK K kK FK FK K kK FK K K K K FK FK K K K kK FK K K FK K K K kK K K K K K FK K K K K K FK K K K kK FK K K FK K K K K K K O WEITER 1 3D GRAFIK MIT GRAPE 2 2D GRAFIK SCHNITT OBERFLAECHE 4 AUSGABE DER NETZDATEN 5 AUSGABE DER RANDKETTENDATEN 6 AUSGABE DER LOESUNG 7 AUSGABE VON FEHLERNORMEN kk okck ok aka a ok okek ok ae ak ae 2A k ok 3K 3K ak K EK ak ke ae ak ak ak fe ae ak ke ea ke Ek ak ak ke ae aka Ek ak OE ak ak ok gt EINGABE 7 AUSGABE VON FEHLERNORMEN LOKAL PROZ MAX NORM L2 NORM H1 NORM Ol 0 00000E 00 0 00000E 00 0 00000E 00 1 0 10760E 04 0 32158E 04 0 40438E 04 2 5 OUTPUT INFORMATION A TYPICAL RUN OF THE PROGRAM 17 2 0 24787E 04 0 72468E 04 0 69347E 04 31 0 24787E 04 0 82577E 04 0 60800E 04 4 0 00000E 00 0 00000E 00 0 00000E 00 51 0 73617E 05 0 39079E 04 0 35490E 04
3. 0 1 y 0 0 5 1 and y z 0 1 x 0 0 5 1 l The menu offers also boundary condition of 3 4 kind but SPC PM Po 3D can not treat them yet 3 5 MESH GENERATION WITH QNET 27 a ordinary cube b Fichera corner c piano d tripod Quee e cube with a slit f SPC mesh Figure 3 7 Mesh families offered by qnet 28 CHAPTER 3 MESHES AND BOUNDARY CONDITIONS Figure 3 8 Example of a general cutting area with a hole in y direction and moved planes Each of this planes could now be moved along an axis After choosing the plane orientation qnet offers all possible planes in a menu and you can choose one and change its height This could be continued until you cancel Then the mesh will be saved As an example see Figure 3 8 If you want to create a so called SPC mesh you will be asked for the kind 1 to 3 which makes a difference in the used font and in element count Then you must enter the section height and the number of sections for each letter The SPC meshes have no real practical use they are only for demonstration The generated meshes are all version 2 standard files with no boundary conditions set To set boundary conditions the xbc program must be used 3 6 The program xbc 3 6 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 form
4. 6 0 78932E 05 0 26933E 04 0 28801E 04 7 0 24787E 04 0 52470E 04 0 63799E 04 AUSGABE VON FEHLERNORMEN GLOBAL MAX NORM L2 NORM H1 NORM 0 24787E 04 0 13457E 03 0 12767E 03 aK gt kK K ook oe oe oe oe oe oe oe oe oe oe oe oe ok ok 2k 2k 2k ok ok ok oko oe oe oe ok ok ok ok ok 2k ok ok ok ok oko ook ok ok ok ok ok ok 2K 2 oe oe oe ok ok ek AUSGABEMENUE aK kK kK k K kK kK K k kK k K k kK K K kK FK K K K kK FK K kK FK K K K K K K K K k FK FK K K K K K kK K K kK K K K FK K K K K K K K K K O WEITER 1 3D GRAFIK MIT GRAPE 2 2D GRAFIK SCHNITT OBERFLAECHE 4 AUSGABE DER NETZDATEN 5 AUSGABE DER RANDKETTENDATEN 6 AUSGABE DER LOESUNG 7 AUSGABE VON FEHLERNORMEN a ae ae ae ak ak a ae fe ae ae ak ak a ae ae ee ae ae ae ae ae aK aK ae a ae ae ae ae ae ae ee ae 2K ae aa aa a ak ak ak EINGABE O GEWUENSCHTE ZAHL VON VERFEINERUNGSSCHRITTEN 1 NEUES NETZ PROGRAMM BEENDEN 3 NEUE PARAMETER EINGABE 2 1 PROGRAMMENDE ok kk ae ae I I kk ok ok ae aka kk kk ok ae ok ae ae ok ae ook ok ae ae ae ae kk ook 2k ke K A A AK ak ak 2K run Returning network by calling nrm run Terminating with result 0 ureiOkain fem The choice of item 0 led to the main menu see above Some of the information is also w
5. 8 nodes 16 nodes 32 nodes 8 nodes 16 nodes 32 nodes 0 48 0 25 1 12 0 50 1 94 0 99 4 10 2 58 Total time FEMAKKVar 1 Total time FEMAKKVar 2 Number of processors Number of processors 16 16 64 Level 16 16 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 38 CHAPTER 4 EXAMPLES 12 12 24 18 48 30 96 45 96 45 192 79 384 135 768 225 768 225 1536 405 3072 765 6144 1377 6144 1377 12288 2601 24576 5049 49152 9537 49152 9537 98304 18513 196608 36465 393216 70785 393216 70785 786432 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 0 c one refinement step with u 1 i zia Ya TE 1 de een rte df r t pct ifr gt t Tnew h Told Ynew h Youd For 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 behaviour for different values of verf The tests were carried out with amw22d std and the following parameters 121 for linear elements Nint3ass 231 quadratic el
6. libFehler a 6 CHAPTER 2 BASIC DESCRIPTION Bsp linked to afs tucz project sfb393 FEM Bsp include linked to afs tucz project sfb393 F EM include libs linked to afs tucz project sfb393 FEM libs mylibs Makedir cid graph linked to afs tucz project sfb393 FEM graph ass4 linked to afs tucz project sfb393 FEM ass4 ass3 linked to afs tucz project sfb393 F EM ass3 mesh3 linked to afs tucz project sfb393 F EM mesh3 L mesh4 linked to afs tucz project sfb393 FEM mesh4 Figure 2 2 File structure after installation of SPC PM Po 3DV3 2 The way how to modify the library sources is described in section 2 3 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 bsp f Our approach is to save example data in files Bsp 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 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
7. see 3 2 3 6 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 pesterQmathematik tu chemnitz de contribution maintaining the libraries time measurement communication handling of curved boundaries email reichel mathematik tu chemnitz de contribution domain decomposition via recursive spectral bisection interface to chaco tests and qnet see 3 5 email m stuebnerQmathematik tu chemnitz de contribution error norms email thessQ mathematik tu chemnitz de contribution solver Chapter 2 Basic description 2 1 Mathematical background Consider the Poisson problem in the notation u f in lt 0 EAR u ug on 09 oi on O9 g 25 Ou On 0 on OQ OQ X 9 the Lam problem for u u u T pAu grad divu f in 2000 99 t g on 000 i 1 2 3 Um lt Q cos OD Noo de 12 where t t2 S u n is the normal stress the stress tensor S u 1 is defined with x 200 202 2 by Ou ud Sij Bs oa U n is the outward normal and is the Kronecker delta The domain C IR must be bounded In the present version curved boundaries can not be
8. 0 38 3 2 0 0 29 78 01 0 06 17 08 0 38 4 1 0 19 50 94 0 17 46 18 0 38 5 1 1 0 29 76 30 0 07 18 96 0 38 6 3 1 0 24 65 05 0 11 29 36 0 38 7 2 1 0 29 77 64 0 06 17 08 0 38 reine Arithmetikzeit max 0 03 HB2BPX max 0 00 Filename for PS output 2 k K kK ok K K oe K K oe K oe oe oe oe oe oe ok ok 2k K 2k ok ok ok oko oe oe K oe ok ok ok ok 2k K ok ok ok oko oe ke ok ok ok 2k 2K ok 2K 2 oe 2 oe oe ok K K AUSGABEMENUE X aK gt ok ook ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok ok K K 2k oko O WEITER 1 3D GRAFIK MIT GRAPE 2 2D GRAFIK SCHNITT OBERFLAECHE 4 AUSGABE DER NETZDATEN 5 AUSGABE DER RANDKETTENDATEN 6 AUSGABE DER LOESUNG 7 AUSGABE VON FEHLERNORMEN a k ae ak ak ae ak ae ae ak ak ea a 3K ea ak ak fe ae ak ae ea ak ak ae ae 2 ae ae ak 2 ak 3K ae ak 2A ak ak fe ae 2A 2 2k aka ea ak ak ak gt EINGABE 6 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 19 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 o
9. 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 75 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 23 f 6z and we observe in both cases the optimal order of the error A 1 k degree of the shape functions m order of the Sobolev space H Q m 0 1 to measure the error Tables 4 1 and 4 2 contains the values The tests were carried out with 221 linear elements f p is quadratic for u 2 DNS 331 for quadratic elements f is cubic for u 2 Giese Bey u uj is of degree 6 but 5 is the best formula B programmed Epsilon 10710 LoesVar 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 00 x z 0 or z 10 but on the remaining part of the boundary we have Neumann conditions 0 0 004 The values of ug and g are taken from bsp f The use of bsp z yields no discretization error which can be used as a test 4 1 POISSON EQUATION 35 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
10. HB nS H r r gt N SOLID 8 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 6 4 4 4 4 4 4 1 SDIRICHLET ONOOBRWNHFAQANOOKBWHNDY 1 0 END_OF_DATA op op u HH RP BRB BB enore gt gt gt NN ONNOWWONAOANDAN Hr CO Table 3 3 The file Cubusl std Names of edges 1 5 13 15 1 2 15 13 7 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 Cubus1 std 3 3 THE TOOLS RENFINDSUN AND RENEDGSUN 23 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 22 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 condition
11. N tion 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 1 1 N number of slices in the cross section see figure 9 number of nodes at the edge 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 sec 4 N tion Figure 3 3 2 family as before but with another cross section 3 4 GENERATION OF MESHES VIA OLDNETZ 25 Input parameter w internal angle of the sector u mesh refinement parameter here jj 0 4 N number of circular arcs number of nodes at the edge R radius of the circular edge the middle circle of the torus z coordinate of this middle circle B radius of of the sector in the cross sec tion IS number of sectors for mesh genera tion here 4 Figure 3 4 3 family sector of a torus with arbitrary internal angle w SX Input parameter w internal angle of the sector u mesh refinement parameter here jj 0 6 N number of circular arcs number of nodes at the edge R radius of the cylinder A IS II N lt CNW SR y y SS N li j HY SS height of the cylinder number of sectors for mesh genera tion here 4 j SS I A Input parameter N reciprocal
12. 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 S ndig and J R Whiteman Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non smooth domains Math Methods Appl Sci 19 63 85 1996 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 SPC94 6 TU Chemnitz Zwickau 1994 G Globisch PARMESH a parallel mesh generator Parallel Computing 21 509 524 1995 Haase B Heise M Jung and M Kuhn FEM BEM A parallel solver for linear and nonlinear coupled FE BE equations Report Nr 96 16 DFG Schwerpunkt Randelementmethoden 1994 G Haase Th Hommel A Meyer and M Pester Bibliotheken zur Entwicklung paral leler Algorithmen Preprint SPC95 20 TU Chemnitz Zwickau 1995 Updated version of SPC94 4 and SPC93 1 AT 48 14 15 16 17 18 19 20 21 22 23 24 25 BIBLIOGRAPHY G Kunert Error estimation for anisotropic tetrahedral and triangular finite element meshes Preprint SFB393 97 17 TU Chemnitz 1997
13. Nu merikplattform Informatik Forsch Entw 7 145 151 1992 Index Index 49 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 edge fle nu 23 out file 7 23 43 45 std file 6 19 23 28 44 45 amw22 std 36 38 cubel92 std 35 cubel92a std 36 cube384 std 35 cube48 std 35 cube768 std 35 cube96 std 35 cubusl std 33 cubus2 std 35 cubusu std 34 cubusug std 34 druck std 40 etest std irn tex cus 41 etestl std 41 etestlu std 41 etestd std 41 etestdu std 41 etestu std 41 fem std 39 fichera std 38 lame22d std 41 zig ia d tc utes 40 boundary conditions set 23 6 setup ppC 6 std2graph sun 6 control quad 9 control tet 9 makefile archi 6 variante archi 6 PARDEST 4 AFSDEST 4 PPCDEST 4 ioa del dex 4 6 architecture 4 6 boundary conditions set ber US EE teda 23 29 os san LES 6 bsp amw 36 bsp etest 41 bsplame 41 42 bsp xy 35 36 39 orate yd 33 34 bSp z2 Reds
14. a cube 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 join some objects to one area So defined regions could get special properties in the program If undefined all elements belong to one region with the name 1 Note that the meaning of the REGION block has substantially changed from version 1 to version 2 standard files see 15 16 Another essential change in version 2 of the standard file is the block FACE GEO referred by the face type This feature provides a wide range of possibilities for easily creating meshes with curved boundaries For a detailed explanation we refer to 21 3 2 STRUCTURE OF THE INPUT FILE STD 21 Statement DESCRIPTION string description of the file for cataloguing DATE date date of creation of the file USER username Login name of the creator of the file PROGRAM name name of the creating program DIMENSION 3D 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 line VERTEX name zcoord ycoord zcoord I R R R EDGE name type start end middle pointer data I I I I I I arbitrary type 1 ignored name type m edgel edgen
15. and the user will be asked for continuing the reading procedure After a successful load procedure the Show button becomes available It switches 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 6 4 The View Window The view window figure 3 9 is used to visualize the object and to choose faces to set boundary conditions 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 behaviour 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 6 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 Pressi
16. pointer data I I I I I I arbitrary type 1 0 plain face type gt 1 pointer to a face geometrie in FACE GEO SOLID name type m facel facen pointer data I I I I I I arbitrary type 1 0 standard material type gt 1 pointer to material name in MATERIAL REGION name type m solidl solidn I I I I I type 1 ignored DIRICHLET 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 global coordinates uo x y 2 data 1 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 name n datal datan I I R R FACE_GEO name type ofgeo n datal datan I I I R R for possible values of type_of_geo see 21 one line per d o f Table 3 2 Structure of the data blocks in the input file 22 CHAPTER 3 MESHES AND BOUNDARY CONDITIONS VERSION 1 0 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 6 0 4 0 0 VERTEX 8 m gt WE m HG gt Per Gn e e ON SD HH HH
17. the data structure is extended or changed The latest version of the standard file format is 2 1 described in citelohse 98a The file input is stopped either by reaching the end of the file or the statement 19 20 CHAPTER 3 MESHES AND BOUNDARY CONDITIONS nodes j edges k faces Figure 3 1 View of the cube which is described in Cubus1 std 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 16 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 HEADER count vertices edges faces solids regions dirfaces neumfaces materials facegeoms where count 4 9 is the number parameters in the header 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 16 The file Cubus1 std see Table 3 3 may serve as an introductory example which describes the partition of
18. treated by the refinement procedure thus 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 24 or external information 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 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 3 4 CHAPTER 2 BASIC DESCRIPTION Figure 2 1 Finite elements implemented in SPC PM Po 3D 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 There exists also a special version using a multigrid method M Jung The post processing includes a simple variant of error assessing If in special test ex amples the exact solution of the problem is known then the error in L and H norms are ca
19. value of the mesh size VS i PN 5 y R w z Figure 3 6 5 family Fichera corner 26 CHAPTER 3 MESHES AND BOUNDARY CONDITIONS To define the group one enters conditions of the form qu we up ceu COE Oe pu ox qq n ce T T L re ap In this way all nodes are marked which satisfy 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 Mesh generation with qnet The program qnet is compiled for SUN4 and HPPA workstations and provides the interactive generation of 6 kind
20. x z 0 and 1 at the top face x z 101 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 1 m uU 0 u 0 10 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 es 121 in the linear case 231 in the quadratic case Nint3error 211 u up is quadratic 33 34 CHAPTER 4 EXAMPLES Lov LL nens elements Ve a LZ De 5 77e 2 1 82e 2 1 84e 13 1 63e 12 6 35e 13 1 42e 2 9 05 1 85e 21 2 25e 8 1 07e 0 3 60e 1 4 54e 1 66e 28e 3 59e 8 3 89e 1 9 07e 0 2 27e 1 31 5 70e 8 1 26e 1 2 27e 0 1 14e 1 05e 28e 1 0le 7 3 89e 2 5 70e 1 5 70e 0 oe Table 4 1 Discretization error for z Towel LLL near elements te T Oe 8 72e 3 2 36e4 3 6 31e4 2 1 62e 2 4 11 1 1 03 1 Table 4 2 Discretization error for u 2 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 0 10 with Dirichlet boundary conditions at 20 x Q z
21. zulaessige Werte gehen aus dem Quelltext Netz Tetraeder control f hervor lin quad 1 vertvar 3 femakkvar 3 loesvar 5 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 Bei Mehrfachdefinition gilt die Letzte Reihenfolge im File Bei Nichtdefinition kommen die Werte aus control f zur Anwendung loesvar 5 lin quad 1 nint2ass 34 nint2error 34 nint3ass 121 232 nint3error 531 531 ion 10 iter 500 epsilon 1 e 6 ndiag 150 200 verf 0 5 vertvar 2 12 CHAPTER 2 BASIC DESCRIPTION 2 5 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 ureiOkain fem A xr8 tet ppc run a pp8 tet ppc run Requesting network by c
22. 34 bsp z3 ede ee ea 34 C Ls iu Vu Ex 6 7 45 Change 2e RR 17 of the maximal number of iterations 17 of the preconditioner 17 of the stop tolerance 17 of the variable ion 17 control quad 8 controltet amp OMS e Boo ee 7 E element types 3 4 epsilon 8 9 F 9 591 e oops 15 44 f sun iik 15 44 femakkvar 9 G ZeO_cOnV 31 GRAPE 2 15 44 45 graphics switch off 6 I installation 4 TOU Lore lk or sees 9 HOP xfi s ELA 9 L Lame system 40 Lam system 3 52 222 7 linquad 9 loesvar 9 M make nes T CLEAN 7 elean sss iiam 7 ggquad 7 uis rano 7 help TRES 7 ed Rx 7 CAT ek y RAMS 7 let veces URS 7 Makefile 4 6 7 menu 13 15 17 meshes 6 N Bdiag o4 RN 9 nint2asS 9 nint2error 9 nintdass 9 nintderror 9 oldnetz 2 7 23 P parmesh3d 2 23 35 43 Poisson equation 3 33 post processing 4 15 Q nel he AA 26 quadrature formulas 10 quick start 4 R re numerate nodes 23 renedgsun 23 44 renfindsun 2 7 23 35 44 S scaling of the coarse grid matrix 17 set b
23. 65 123 205 125 225 369 725 1305 729 1377 2465 4905 9265 4913 9537 17985 35931 69729 35937 70785 137345 274593 540865 274625 545025 1073409 2146625 2146689 Table 4 4 Number of nodes for different refinement levels 4 1 5 cubus2 std The domain and the mesh cubus2 std are identical to cubus1 The boundary conditions are 00 x 0 2 20H 00 9 N O3 where the values of uy and 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 10 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 examples 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 27 The domain is the cube 0 2 3 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 me
24. BE DER RANDKETTENDATEN a k ae ak ak ae ak A aK 3K ak ak ea ke 2K ak ak fe ae ak ae 3K 3K ke EK ak ke ae ak a EE ae ak ee ae ak ak fe 2 2 2k ak ak ea ak ak ak gt EINGABE 0 14 CHAPTER 2 BASIC DESCRIPTION Now the possibility to visualize the generated mesh either in 3D or 2D is given We go on choosing 0 or just pressing enter START GENERIEREN ASSEMBLIEREN Zeiten fuer Warten Kommunikation s Prozessor log phys input in 4 output 4 gesamt 0 0 0 0 00 0 00 0 00 0 00 0 00 1 1 0 0 00 0 00 0 00 0 00 0 01 time information for as 2 3 0 0 00 0 00 0 00 0 00 0 01 sembling process 3 2 0 0 00 0 00 0 00 0 00 0 01 4 0 1 0 00 0 00 0 00 0 00 0 00 5 1 1 0 00 0 00 0 00 0 00 0 01 6 3 1 0 00 0 00 0 00 0 00 0 01 7 2 1 0 00 0 00 0 00 0 00 0 01 reine Arithmetikzeit max 0 01 ASSEMBLIEREN BEENDET Coars Grid Matrix Generation 0 a tion gA Phe coarse Groesse der Matrix VBZ 30 grid matrix 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 Koppelknoten 34 processor 0 innere Knoten 1 Anzahl der Koppelkanten 6 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 5 l
25. D Lohse Datenschnittstelle I Standardfile Preprint SPC95 7 TU Chemnitz Zwickau 1995 In preparation D Lohse Ein Standard File f r 3D Gebietsbeschreibungen Definition des Fileformats V 2 1 Preprint SFB393 98 11 TU Chemnitz April 1998 M Meisel and A Meyer Implementierung eines parallelen vorkonditionierten Schur Komplement CG Verfahrens in das Programmpaket FEAP Preprint SPC95 2 TU Chemnitz Zwickau 1995 A Meyer and M Pester Verarbeitung von Sparse Matrizen in Kompaktspeicherform KLZ KZU Preprint SPC94 12 TU Chemnitz Zwickau 1994 M Meyer Grafik Ausgabe vom Parallelrechner f r 3D Gebiete Preprint SPC95 4 TU Chemnitz Zwickau 1995 M Pester Grafik Ausgabe vom Parallelrechner f r 2D Gebiete Preprint SPC94_24 TU Chemnitz Zwickau 1994 M Pester Behandlung gekr mmter Oberflachen in einem 3D FEM Programm f r Parallelrechner Preprint SFB393 97 10 TU Chemnitz April 1997 W Queck FEMGP Finite Element Multi Grid Package Programmdokumenta tion und Nutzerinformation Report TU Chemnitz Zwickau Fachbereich Mathematik 1993 U Reichel Partitionierung von Finite Elemente Netzen Preprint SFB393 96 18 TU Chemnitz Zwickau 1996 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
26. Graf or NoGraf thus the graphic libraries are linked or not which results in a considerable difference in the size of the executable file Both the handling of the bsp example_name files and the setting of the GRAF variable are provided by the shell script setup ppc This script did all the needed changes automatically after the users choice 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 project sfb393 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 Corresponding to the mesh 3 4 directories exists the directories ass 3 4 containing as signment informations for an optimal distribution of the mesh among the processors These informations are considered if vertvar 3 is chosen in the control file see 2 4 The as signment files are created by chaco sun and must follow the naming convention described in 23 The expected input for chaco is stored in the graph directory as name graph For new meshes a graph file could be generated from the std using the tool std2graph sun All these AFS directories are readable and executable for any user M Pester is ad ministrating these directories and can include further AFS users to a list of people w
27. Technische Universitat Chemnitz Sonderforschungsbereich 393 Numerische Simulation auf massiv parallelen Rechnern Thomas Apel Uwe Reichel SPC PM Po 3D V3 3 User s Manual Preprint SFB393 99 06 Acknowledgement The package SPC PM Po 3D has been developed in the Son derforschungsbereich 393 of the Technische Universit t Chemnitz under the supervi sion of A Meyer and Th Apel Other main contributors are G Globisch D Lohse F Milde M Pester U Reichel and M Thef Section 3 4 and Appendix A of this documentation were written together with F Milde and G Globisch respectively Section 3 6 was written by D Lohse The tests in Section 4 were partially carried out by A Meyer and U Reichel Preprint Reihe des Chemnitzer SFB 393 SFB393 99 06 Februar 1999 Contents 1 Introduction 1 2 Basic description 3 2 1 Mathematical background 3 223 Ln 4 2 3 Modifying source files under the CVS T 2 4 The files control tet and control quad 8 2 5 Output information a typical run of the program 12 3 Meshes and boundary conditions 19 3 1 1 19 3 2 Structure of the input file std 19 3 3 The tools renfindsun and 23 3 4 Generation of meshes via oldnetz 23 3 5 Mesh generation with
28. To link now the program with the local libraries instead of the global ones some modi fications in the global Makefile are needed 1 Refer to all local directories you want to process during compilation behind the variable MYLIBS at line 41 8 CHAPTER 2 BASIC DESCRIPTION 2 Substitute ALIBDIR with MYLIBDIR in the lines 136 to 143 to link with the local version of a library After doing so a simple make in the main directory should create the default 4 executa bles for the chosen architecture archi All makefiles work together with the LIBLISTE in their directory a text file providing the source files to be included in the library If a user wants to include additional source files he she should add it in the corresponding file LIBLISTE The CVS must also be used to redistribute modified sources In the following a short description of possible transactions cvs co MODULNAME creates a subdirectory MODULNAME including the latest sources the Makefile and LIBLISTE for the corresponding library in the subdirectory MODULNAME cvs update updates the local source files with the global ones cvs n update shows the changes since the last cvs co or cvs update cvs diff D today files shows differences between the local files and the last version checked in also D yesterday is possible use cvs diff D today D yesterday to compare two archived versions cvs commit files checks in the named files cvs release propagates
29. Visualization of 3D data files file out node related structure based on GRAPE 25 Th Hommel SHOWNET AFS SUN4 shownet Visualization of 2D FEM data including the solution isolines data type file out possible output as ps file Brauer VINP f femtools vinp exe AFS SUN4 vinp Visualization of 2D data files file inp 22 M Goppold XBC AFS SUNA xbc AFS SUNA xbc Visualization of 3D data files file std edge related and modification of boundary conditions D Lohse see also Section 3 6 Other preprocessing DECOMP AFS SUNA decomp Spectral graph partitioning of finite element meshes for parallel computations M Goppold CHACO AFS SUN4 chaco AFS HPPA chaco AFS LINUX chaco Partitioning of fi nite element meshes for loadbalacing of parallel computations see 23 and the citations therein STD2GRAPH AFS SUNA std2graph AFS HPPA std2graph Generating of input files for chaco from std files U Reichel Q2T AFS SUN4 q2t AFS HPPA q2t Converting hexahedral std files into tedrahe dral std files U Reichel Main processing FEMGP Package for solving 2D boundary value problems on sequential computers see 22 based on files file wqf 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 other
30. alling nrm run Creating 4 2 descriptor by calling mkdesc run Starting D Server at kain link 5 H 8555 CCCC PPPPP M M 333 SS SS PP PP CC CC PP PP MM MM PP 33 33 SS PP PP CC PP PP MMM MMM PP 33 4 SSSS PPPPP CC PPPPP MM MM 000 333 SS PP MM M MM PP 00 00 33 4 SS SS PP CC PP MM MM 00 00 33 33 SSSS PP CCCC PP MM MM PP 000 333 H Programm Modul 3D Potentialprobleme Version 3 30 TU Chemnitz Sonderforschungsbereich 393 Th Apel A Meyer M Meyer F Milde M Pester M Thess Fakultaet fuer Mathematik 16 MB Variante 3500000 Worte bis zu 1024 Prozessoren in Benutzung 8 Proz Gelinkt mit bsp z H dE aaa ae ak I GA ae I 3K ak ae ae I aK 3K K aK 2K 3K K aK 9K 2K A I I 3K 3K a A aa a ak ak ak ak ak ak Belegung der Steuerparameter kann mittels File control tet angepasst werden kk kkk akak kk a ae ak kak k kk akk k ak ak kk ak akk kak vertvar 2 li
31. ass but used for the integration of 3D integrals in the error calculation controls the amount of output ol the program gt 0 message after each ion th CG iteration X 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 no input request messages iter 200 integer gt 0 TUNE number of iterations in the CG algorithm stop criterion for the CG relative decrease of the epsilon E 4 real gt 0 norm of the residual upper estimate for the number of nonzero entries 1n any row of the stiffness matrix If it is chosen too ndiag 70 integer gt 0 large the program may suffer from lack of memory and if it is chosen too small the number is itera tively increased waste of time mesh refinement parameter for a certain class o verf realc 0 1 examples see Subsection 4 1 7 0 no change of the mesh Table 2 1 Variables in control tet control quad 10 CHAPTER 2 BASIC DESCRIPTION Formula Number of Description number points exact for zy with midpoint center of gravity 2x2 Gaussian points 3x3 Gaussian points Table 2 2 Quadrature formulas for quadrilaterals Formula Number of Description mmber points center of gravity midpoints of the edges 3 4 Gaussian points 4 T Gaussian points Table 2 3 Quadrature formulas for triangles a ee Description points xy z with cent
32. at 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 XView environment there is no plain X Windows nor Motif based version 3 6 2 Command Line Parameters All 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 3 6 THE PROGRAM XBC 29 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 6 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
33. by special routines of the user program 3 6 6 The Settings Menu The Settings menu controls the general behaviour of xbc It includes two buttons the View Control button and the Zoom menu The View Control button opens a window which allows to choose the drawing method Solid Hidden Line Wire Frame see right side of figure 3 10 This window is also used to control the visibility of the names e g integers of objects like vertices edges or faces 3 7 MESH CONSTRUCTION AND MANIPULATION WITH GEO_CONV 31 Figure 3 10 Left The Set BC Values window Right The View Control window The Zoom menu offers some standard zoom factors and the capability to enter user defined factors using the Other button 3 7 Mesh construction and manipulation with geo conv The program geo conv was designed to join two or more objects provided as standard files to one new object and to save this in a new standard file To do so the input objects could be displaced rotated and mirrored Notes edges and faces which are joined could be identified and fusioned The program is text oriented menu driven and offers an online help It is also able to operate in batch mode A program call should be as follows geo conv v t c lt cfgfile gt p lt protfile gt e lt exefile gt lt filename gt lt filename gt is no filename given the user is prompted for a filename after the program start The configuration f
34. cture appropriate for SPC PM Po 3D This program has two additional features re numeration 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 To generate hexahedral meshes exists the tool mesh4 qnet author U Reichel In Section 3 5 we describe its functionality and the mesh classes it provides In Section 3 6 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 Another tool written by D Lohse is the program geo_conv which is explained in Sec tion 3 7 It gives the possibility to move a given mesh in space translation rotation and to combine two or more meshes to a new one So it is relative easy to construct complex forms from simple geometric solids 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 Cubus1 std in Table 3 3 The file starts with a control line defining the version VERSION 2 1 in order to circumvent incompatibilities when
35. d direction The coarse mesh consists of 93 elements with 122 nodes We have Dirichlet boundary conditions at the bottom face 00 x 0 z 0 In Figure 4 4 we demonstrate isolines at the surface of the domain calculated with bsp xy Level 2 Figure 4 3 Fichera corner Figure 4 4 Isolines on FEM 40 CHAPTER 4 EXAMPLES 4 2 Lam system 4 2 1 Introduction We consider the Lam equation system pAu grad divu f for u u 9 7 with the boundary conditions An u g ond i 1 3 0 ondQ an aay i 1 3 where t 19 10 1 T is the normal stress 4 2 2 druck std This is again the cube 0 10 divided into six tetrahedra The boundary conditions are u 0 on z z 0 u 0 on x 00 z 10 2 t 0 elsewhere The exact solution is not known With 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 zugl std describe the same example but they were created by different programs We have again the cube 0 10 The boundary conditions are u 0 on z 00 2 0 4 SS S N N N S Figure 4 5 Cube under pressure Figure 4 6 Cube under pull 4 2 LAME SYSTEM Al Ie 0 0 on z 00 z 10 1 0 Lm elsewhere The exact solution is n
36. djusted by the user see above Note that it is possible to link only a special program by calling make tet make ggtet make quad or make ggquad 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 includes Makefiles and meshes Some more information about the Makefile could be obtain by calling make help 2 3 Modifying source files under the CVS The whole source tree of SPC PM Po 3D is under CVS control to keep the source manage ment easy and transparent As mentioned in section 2 2 the standard installation includes no source files except the main program pfem f the function subroutines bsp f and getdofs f which influences the behaviour of the 2D graphic To modify the sources of the libraries the source tree of each one had to be checked out of the CVS using the shell command cvs co MODULNAME where MODULNAME is one of NetzA NetzT NetzQ Solve Assem Elem3D Fehler Before performing this checkout the correct environment must be set by executing one of the scripts setpvm setppc or setparix The cvs command creates a directory MODULNAME containing the latest sources the LIBLISTE and a CVS directory needed by the version control
37. e drive letter f for DOS files stands for riemann_home2 public numwork 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 f femtools premesh exe Sequential 2D grid genera tion in a UNIX and DOS version 22 M Goppold Converting data structures GRAFEDSUN AFS SUNA grafedsun Converting file inp see GRAFED into file bsp see FEM BEM and vice versa G Haase GUNDOLFSUN AFS SUN4 gundolfsun Converting file net see NETS into file bsp see FEM BEM and vice versa G Haase 43 44 APPENDIX A MESH GENERATION AND RELATED PROGRAMS FEM BEM 2D file bsp 2D file bsp ge ee princ ER eee GUNDOLFSUN RENEDGSUN GRAFEDSUN GRA poe 2D file net 2D file inp NETS 3D file inp ne ADAPMESH mo ERE 2D file i 2D file waf TRANSFERSUN hand SEPA BALMESH Tan e pom J 2D file inp GRAFED H 2D file inp PREMESHG PREMESH EXE 2D 3D file inp 2D file out DECOMP VINP SHOWNET 3 main processing mesh
38. ements Nint3error 511 Nint2ass 11 11 for linear elements Nint2error 12 for quadratic elements Epsilon 10710 LoesVar 4 BPX without coarse grid solver 4 1 8 fichera std The domain mesh is Q 1 1 V 0 1 which is known as 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 POISSON EQUATION 39 linear elements 4 0797e 1 1 6819e 1 2 3825e 0 2 0854e 1 3 5542e 2 9 2351e 1 3 3811e 1 7 6697e 2 1 5922 0 1 4325e 1 1 3205e 2 5 6153e 1 2 3133e 1 3 2269e 2 1 0164e4 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 a Table 4 9 Discretization error for verf u 1 incar elements 2 8163e 1 1 9642e 1 1 3677e 1 7 8165e 2 1 2327e 0 1 3241e 1 7 2266e 2 4 9917e 2 2 0029e 2 4 3328e 1 6 0739e 2 1 8634e 2 2 2336e 2 4 4908e 3 1 5496e 1 2 527 1e 2 5 0494e 3 1 2162e 2 8 2507e 4 5 5923e 2 1 0240e 2 1 3128e 3 E Table 4 10 Discretization error for verf 0 5 with linear elements and verf 0 3 with quadratic elements 4 1 9 fem std The domain consists of the letters FEM which have a different size in the thir
39. er of gravity lt 1 Gaussian points Eg s 2 Gaussian points E3q k x3 Gaussian points lt 4 Gaussian points i j k lt 5 exact for zy with Table 2 4 Quadrature formulas for tetrahedra Formula Number of the formula is a cross product of the formulas for triangle for interval z direction center of gravity midpoint midpoints of edges midpoint exact for xy z with Table 2 5 Quadrature formulas for pentahedra 4 Gaussian points midpoints of edges 4 Gaussian points 4 Gaussian points 7 Gaussian points 7 Gaussian points midpoint 2 Gaussian points 2 Gaussian points 3 Gaussian points 2 Gaussian points 3 Gaussian points Formula Number of for Description number points vy zt 2 with midpoint center of gravity Table 2 6 Quadrature formulas for hexahedra 2x2x2 Gaussian points 3x3x3 Gaussian points midpoints of the faces Irons formula 2 4 THE FILES CONTROL TET AND CONTROL QUAD 11 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
40. es describes the domain 3 x rcosy rsiny z 0 lt lt 1 0 lt p lt 5m 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 0 Contrary in amw22 std we have 00 u 00 2 0 The meshes are useful in connection with bsp amw where the exact solution is given by Aus 2 u 10 2z r snAy A 3 For this and only this domain 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 L refinement level verf grading parameter 1 1 2 ce 2 4 1 POISSON EQUATION 37 cube48 cubel92 LoesVar LoesVar Level 3 Level 3 5 14 18 18 18 5 0 1 1 19 21 13 1 27 25 22 2 33 36 17 17 2 52 40 38 3 47 52 21 21 3 98 58 9T 22 1 69 70 23 23 4 191 80 80 25 5 93 97 25 25 5 364 101 98 26 cube384 cube768 LoesVar LoesVar 5 3 31 3 5 0 1 37 31 33 27 27 18 53 45 45 46 41 32 24 76 60 60 67 61 38 35 110 1 95 84 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 16 16 6 16 1 64
41. generation visualization t 1 converting data structures Figure A 1 Connection of the tools corresponding to the data structure POS2NET AFS SUN4 pos2net Converting file wqfinto 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 22 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 Globisch see also 3 3 3 4 2 TRANSFERSUN AFS SUNA transfersun Converting file net see NETS into file inp see GRAFED G Globisch NET4STD AFS SUN4 net4std AFS HPPA net4std AFS LINUX net4std Takes net files as surface and extends them into the third dimension M Pester Visualization AFS SUN4 f3_sun dynamically linked AFS SUN4 f3grape sun statically linked AFS SGI5 f3_sgi dynamically linked AFS SGI5 f3grape sgi statically linked Visualization of 3D data received via socket connection 19 based on GRAPE M Meyer see also Graphical editors above 45 GRAFEM f femtools grafem exe Visualization of 2D FEM data including the solution data type file wql 22 Haase GRAPE AFS SUN4 grape sun
42. ho are allowed to add files in these directories 2 8 MODIFYING SOURCE FILES UNDER THE CVS 7 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 SUN4 workstation 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 re numerate 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 6 In the main directory name_of_destdir there is the main Makefile the essential FOR TRAN source files some text files with helpful informations the executables f3_sgi and f3_sun providing the graphical interface GRAPE see A the executable chaco sun for pre loadbalacing see 23 and the files control tet control quad which are described in Section 2 4 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 a
43. hoose the architecture you want to work with by calling one of the shell scripts usr global bin setpvm usr global bin setparix or usr global bin setppc Some variables including archi are now defined 2 2 INSTALLATION 5 4 Call make Then after successful compilation the executable files tet archi for tetrahedral meshes quad archi for cuboidal meshes and ggtet archi ggquad archi both us ing Globisch Nepomnyaschikh method for embedding unstructured meshes should be con tained in your directory If defined a copy should be in AFSDEST 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 communication routines are different e archi SUN4 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 SUNMP is set by calling setpvm on a Sun multiprocessor workstation e archi HPPA is set by calling setpvm on a HP workstation e archi HPPAMP is set by calling setpvm on a HP multiprocessor workstation e arch
44. i SGI5 is set by calling setpvm on a SGI workstation running under Irix 5 x e archi SGI64 is set by calling setpvm on SGI workstation running under Irix 6 4 e archi LINUX is set by calling setpvm on a PC or other machines running under Linux 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 e archi ppcmpi could be set by the user after calling setppc which causes the compi lation of an executable file for Parsytec machines based on the Motorola Power PC601 chip as the Xplorer or the GCPowerPlus 128 using the message passing interface MPI instead of the PARIX routines In case of our latest version V3 2 after the installation a file structure as shown in Figure 2 2 is given Except the files pfem f bsp f and getdofs f the directory name of destdir contains no source files All needed code is included in precompiled static link libraries which could be found in the libs directory separated by architectures The default search path for this libraries is given in the Makefile by the variable LIBDIR The SPC PM Po 3D specific link libraries are libNetzA a libNetzT a libNetzQ a libSolve a libAssem a libElem3D a
45. ile lt cfgfile gt contain set commands and additional types for face geometries All executed commands its outputs and error messages could be saved in the protocol file lt protfile gt An existing protocol file could be used as control file lt execfile gt in batch mode With switch t the program operates in trace mode which means the every line in the control file must be confirmed before execution Is no control file given the switch remains meaningless The switch v must given for displaying the current program version Note that the order of the given parameters is essential for there right recognition For a short command list type help at the command prompt after start For a detailed description of the commands the protocol control and config file we refer to the manual which could be found under afs tucz project stb393 FEM doc geo conv in various forms 32 CHAPTER 3 MESHES AND BOUNDARY CONDITIONS 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 u ug on 9 Qu nS g on 05 Qu 0 oQ OQ d O5 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 cubus1 std describes a cube 0 10 with Dirichlet boundary conditions ug 0 at the bottom face
46. ion 2 1 and new related tools In Version 3 x the program can solve the Poisson equation and the Lam system of linear elasticity with in general mixed boundary conditions of Dirichlet and Neumann type see Section 2 1 The domain Q C IR can be a various curved bounded polyhedron see 21 The input is 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 behaviour of the algorithms can be monitored arithmetic and communication time is measured the discretization error is measured different preconditioners can be compared There exists special versions of SPC PM Po 8D using a multigrid solver M Jung having an error estimator Kunert or using Globisch Nepomnyaschikh mesh transformation technique in the solver G Globisch All these versions are adapted from the SPC PM Po 3D package but include major changes Thats why only some of this features are part of the official distribution The version 3 x documented here is the last version with uniform mesh refinement It is not developed further only bugs will be fixed The currently developed version 4 will include adaptive mesh refinement and dynamic load balancing The program has been developed for MIMD computers i
47. ith v I 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 42 CHAPTER 4 EXAMPLES from bsp f bsp lame Unfortunately the system is very badly conditioned because is close to I 2 v which results in very high iteration numbers Examples of the error behaviour 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 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 reasons 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 project sfb393 FEM bin such programs can be accessed from all computers with AFS installed th
48. lculated 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 It exists a special version of SPC PM Po 3D with an improved variant of the residual type error estimator see 14 At the moment this estimator is only available for linear tetrahedral elements and not part of the official release 2 2 Installation Provided AFS the Andrew File System is installed any user can install the package by using the shell script afs tu chemnitz de home urz p pester bin install3d name_of_destdir version v m where name of destdir should be a name which does not yet exist and the optional pa rameter version provides the possibility to install different versions of the package An explanation of various versions can be obtained by calling the script without parameters For a quick start do the following 1 Install the package in name of destdir by calling install3d 2 If the installation of SPC PM Po 3D is already on AFS no changes are necessary Oth erwise edit the Makefile in name_of_destdir and adjust the variables PARDEST and PPCDEST or more common the variable AFSDEST ensure that these directories exists at the corresponding machines Moreover it is useful to copy or link the directories mesh3 and mesh4 to the working directory of the remote machines or in AFS 3 C
49. n_quad 1 nen2d 3 nen3d 4 femakkvar 3 loesvar 5 nint2ass 34 nint3ass 121 nint2error 34 nint3error 531 iter 500 epsilon 0 10E 03 ion 10 ndiag 70 Verzeichnis fuer Netze mesh3 aK k ok kK k oe oe oe oe oe oe oe oe oe oe oe oe oe ok ok 2k 2 ok ok 2K oko oe oe oe ook ok 2 2 oko ok ok ok oe K ok ok ok KK K 2K FK Filename cubusi 2 5 OUTPUT INFORMATION A TYPICAL RUN OF THE PROGRAM 13 The file name is typed in here cubus the input of a question mark generates ls command for the appropriate directory Then we are asked for the number of refinement steps It is also possible to escape by typing 1 for a new mesh 2 to quit or 3 for new parameters GEWUENSCHTE ZAHL VON VERFEINERUNGSSCHRITTEN NEUES NETZ PROGRAMM BEENDEN 3 NEUE PARAMETER EINGABE 2 1 Ne After this we get information on the current state of the program and to the input mesh EINLESEN DER NETZDATEN AUS mesh3 cubusi std Wuerfel Kantenlaenge 10 oben unten Dirichlet Typ 2 u z 10 Gerhard Globisch 07 11 1994 Poisson Gleichung 5 PARMESH RENFINDSUN the input file extension 3D std data_read done No error copy of the information of EINLESEN BEENDET IER 0 VERTEILUNG DER TETRAEDER DURCH REKURSIVE SPEKTRALBISEKTION Anzahl der Elemente in den Prozessoren 3 3 3 3 3 3 3 8 information on the progress 1 1 2 2 1 1 2 2 f the recursi ral bi b Sor OU ONE of the recursi
50. ng 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 to 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 see left side of figure 3 10 This window allows the simple choice of the kind of boundary condition Dirichlet Neumann as well as the input of the actual number of the degree of freedom the equation type Poisson or Lam and the set of values of that boundary condition 30 CHAPTER 3 MESHES AND BOUNDARY CONDITIONS Figure 3 9 The View Window of xbc 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 Vl 7 V2 y V3x 2z VA x y z are the coordinates in the global system e User The values of the BC will be given
51. on and their difference for each processor see the printout AUSGABE DER WERTETABELLE DER LOESUNG AUSGABE IN FILE LOESUNG DAT J N n PROZESSOR 0 NUMNP 0 PROZESSOR 1 NUMNP 35 NR X Y Z BER LOESUNG EXAKTE LSG DIFFERENZ 1 10 000 0 000 10 000 1 00000 1 00000 0 00000D 00 2 10 000 10 000 10 000 1 00000 1 00000 0 00000D 00 3 0 000 10 000 10 000 1 00000 1 00000 0 00000D 00 4 10 000 0 000 0 000 0 00000 0 00000 0 00000 00 5 10 000 0 000 7 500 0 74999 0 75000 0 77712D 05 6 10 000 0 000 5 000 0 50000 0 50000 0 20307 05 7T 10 000 0 000 2 500 0 24999 0 25000 0 10760D 04 8 10 000 2 5001 10 0001 1 00000 1 00000 0 00000D 00 9 10 000 5 000 10 0001 1 00000 1 00000 0 00000D 00 10 10 000 7 500 10 000 1 00000 1 00000 0 00000D 00 11 7 500 2 500 10 0001 1 00000 1 00000 0 00000D 00 12 5 000 5 0001 10 0001 1 00000 1 00000 0 00000D 00 13 2 500 7 5001 10 0001 1 00000 1 00000 0 00000D 00 14 10 000 7 500 7 500 0 75000 0 75000 0 14338D 05 15 10 000 5 000 5 000 0 50000 0 50000 0 26510D 05 16 10 000 2 500 2 500 0 24999 0 25000 0 59909D 05 17 7 500 10 000 10 000 1 00000 1 00000 0 00000D 00 18 5 000 10 000 10 000 1 00000 1 00000 0 00000D 00 19 2 5001 10 000 10 000 1 00000 1 00000 0 00000D 00 20 2 500 7 500 7 500 0 75001 0 75000 0 59876 05
52. ot known We calculated again with v 0 3 E 2 10 the result is shown in Figure 4 6 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 A p 02 z consequently f 0 1 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 u y on re 00 z 20 or z 10 2 n t 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 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 u y on z 00 z 0 or z 10 z 0 elsewhere In this case the exact solution is not known 4 2 5 lame22d std with bsp lame This is a test with a known solution which has the typical behaviour near an edge The domain and the meshes are the same as in 4 1 7 the exact solution is r9 9L 3 cos 2 ge 0 5 sin gy Fsin r P V3 3sin sin By cos cos r 3 sin 2o and w
53. oundary conditions 23 set boundary conditions Po Pins ea Se 23 29 SOLVERS anodes ann 4 V verf 9 36 38 39 42 vertvar sess 9 X XD 2 2 24 2 al 2 7 28 45 Other titles in the SFB393 series 99 01 P Kunkel V Mehrmann W Rath Analysis and numerical solution of control problems in descriptor form January 1999 99 02 A Meyer Hierarchical preconditioners for higher order elements and applications in computational mechanics January 1999 99 03 T Apel Anisotropic finite elements local estimates and applications Habilitationsschrift January 1999 99 04 C Villagonzalo R A Romer M Schreiber Thermoelectric transport properties in disordered systems near the Anderson transition February 1999 99 05 D Michael Notizen zu einer geometrisch motivierten Plastizitatstheorie Februar 1999 The complete list of current and former preprints is available via http www tu chemnitz de sfb393 preprints html
54. ritten 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 5 enter 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 for a change of the variable ion 4 These corrections are valid only during the following CG iteration and do not overwrite the standard values of these variables see Subsection 2 4 An exception is ion 18 CHAPTER 2 BASIC DESCRIPTION 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 16 we summarize 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 stru
55. rved 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 information 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 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 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 24 CHAPTER 3 MESHES AND BOUNDARY CONDITIONS 0 r b i b Input parameter 1 u mesh refinement parameter here u 2 0 4 N number of circular arcs 3 number of nodes at the edge R outer radius radius of the middle cir A cle of the torus A z coordinate of this middle circle B radius of of the sector in the cross sec 5
56. s FEMPS3D Package 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 tucz home urz p pester workcfd pmhi ppc px Parallel simulation of fluid dynamics in 2D St Meinel A Meyer SPC PM EL 2D afs tucz home urz p pester workel pmhi ppc px Parallel simulation of elasticity in 2D A Meyer SPC PM Po2D afs tucz home urz p pester worksy pmhi ppc px Parallel simulation of potential problems in 2D A Meyer 46 APPENDIX A MESH GENERATION AND RELATED PROGRAMS 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 Comparison of several mesh refinement strategies near edges Comm Numer Methods Engrg 12 373 381 1996 Th Apel F Milde and M Thef SPC PM Po 3D Programmer s Manual Preprint SPC95 34 TU Chemnitz Zwickau 1995 Th Apel R M cke and J R Whiteman An adaptive finite element technique with a priori mesh grading Technical Report 9 BICOM Institute of Computational Math ematics 1993 Th Apel and S
57. s 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 face wise 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 eventually 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 6 Moreover the user can determine whether he she wants to re numerate 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 re numerated 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 cu
58. s of hexahedral meshes The generated meshes can be stored in std file with the structure as given in Section 3 2 After calling the program it provides a menu and the user is prompted for the mesh type Appropriate choices are shown in Figures 3 7 a 3 7 f In addition to this 6 kinds of meshes qnet provides the possibility to create a general cutting area which means it can create any mesh which could be obtained by cutting elements from a cube mesh After a mesh type was chosen qnet asks for the z y z Dimension and for the number of sections in each direction In case of the cube with a split it asks further for the angle of the split If the general cutting area was chosen first a cube is generated and the user is asked if elements should be caught Saying yes the user must determine a range of elements to be caught This is done by giving the from and to section of each direction As an example you have created a cube with three sections in each direction and now you want to cut a hole in y direction The correct choice is to cut from 2 to 2 in z from 1 to 3 in y and from 2 to 2 in z direction After each cut the user will be asked if he want to do another cut If not the program goes on offering a plane move For the meaning of this feature another example Say you have generated a cube 0 113 with 2 sections in each direction So you have a cube consisting of 9 planes The cube planes are x y 0 1 z 0 0 5 1
59. shes than cube48 and cube96 respectively with one refinement step though they have the same number of elements 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 99 x Q z 0 z 2 where the boundary conditions are taken from the function u in bsp f 36 CHAPTER 4 EXAMPLES a b c Figure 4 1 2D reference meshes for the cube family mesh 2D reference a a b number of reproductions 2 4 Table 4 5 The cube family and their corresponding reference meshes In Table 4 6 we document tests with these meshes with bsp xy used as bsp f and different preconditioners The files cubena std define 0 N and were used 1 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 Q u ry on 0 linear shape functions cube192a std Epsilon 10 LoesVar 2 Yserentant without coarse grid solver 4 1 7 amw std with bsp amw The amw family of mesh
60. t enter gt Now 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 After providing the possibility to save the graphic of time usage the program finally stops in the next menu IT r w As s ALFA BETA Eta 1 3 850645 01 5 718772E 01 6 733342E 01 0 000000 00 1 00 2 2 147706E 01 1 432139E 02 1 499650E 01 5 577524E 01 0 56 3 3 863126E 00 2 177399E 01 1 774193E 01 1 798722E 01 0 32 4 5 871520E 01 2 966601E 00 1 979208E 01 1 519888E 01 0 25 5 1 974870E 01 1 042542E 00 1 894283E 01 3 363473E 01 0 27 6 4 499389E 02 2 041502E 01 2 203960E 01 2 278322 01 0 26 7 7 271924E 03 2 561211E 02 2 839252E 01 1 616203E 01 0 24 8 1 445351E 03 5 632380E 03 2 566146E 01 1 987577E 01 0 23 9 2 842315E 04 9 748493E 04 2 915645E 01 1 966522E 01 0 23 10 9 355177E 05 2 828425E 04 3 307558bE 01 3 291394E 01 0 24 11 2 334666E 05 1 145204E 04 2 038646E 01 2 495588E 01 0 24 12 2 409204E 06 1 059402E 05 2 274118E 01 1 031926E 01 0 22 2 5 OUTPUT INFORMATION A TYPICAL RUN OF THE PROGRAM 15 13 4 041727E 07 1 625228 06 2 486868E 01 1 677619E 01 0 22 14 6 576520E 08 1 625228 06 2 486868E 01 1 627156E 01 0 21 IT 14 Zeiten fuer Warten Kommunikation s Prozessor log phys input in output in 4 gesamt 0 0 18 47 11 0 17 43 91 0 38 1 10 0 26 69 20 0 09 25 98 0 38 2 3 0 0 23 59 90 0 13 34 40
61. t 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 CHAPTER 1 INTRODUCTION In this documentation we use slanted style for really existing paths and filenames italic style for program parameters sans serif style to characterize buttons and menu items of programs with a graphical user interface and typewriter style forthe names of variables Dr Thomas Apel Dr Gerhard Globisch Dr Michael Jung Dr Gerd Kunert Dag Lohse Prof Arnd Meyer Dr Magdalene Meyer Frank Milde Dr Matthias Pester Uwe Reichel Michael St bner Michael Thef List of contributors email apelQmathematik tu chemnitz de contribution supervision assembly error assessment com munication tests email globisch mathematik tu chemnitz de contribution PARMESH3D renfindsun see 3 3 email jung mathematik tu chemnitz de contribution multigrid solver email gkunert mathematik tu chemnitz de contribution error estimator email lohseQ mathematik tu chemnitz de contribution input files xbc
62. te that these files can be omitted ifonly 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 10 19 He she has two possibilities Either can change this during the execution see the last paragraph in Section 2 5 Or he she introduces the file control tet or control quad with one line epsilon 1 E 10 2 4 THE FILES CONTROL TET AND CONTROL QUAD 9 values value of shape functions linear shape functions 2 quadratic shape functions of coarse grid partitioning 1 trivial partitioning as 2 3 2 partitioning via recursive spectral bisection 3 read partitioning from file there are three variants of accumulation dis P 1 2 3 tributed data see 2 C of the preconditioner 5 Yserentant without coarse grid solver loesvar 5 T Yserentant with coarse grid solver 4 BPX without coarse grid solver 5 BPX with coarse grid solver int2 14 31 bling Neumann boundary data SES idet 1 digit quadrilaterals see Table 2 2 2 d digit triangles see Table 2 3 in the assembling ea 1 digit tetrahedra see Table 2 4 mantsase EI 27d digit hexahedra bricks see Table 2 6 and digit pentahedra triangular prisms see Table 2 5 14 31 E as nint2ass but used in the error estimator for the integration a the jump of the normal derivatives nint errer 521 581 as nint3
63. that no changes will be made in the next time For further details on the CVS see http www loria fr molli cvs doc cvs toc html http www tu chemnitz de pester cvs cvs exp html or the manpages 2 4 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 Some 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 exactness of the quadrature formulas belongs to the master element it may change for the actual element through transformation by multiplication with a nonconstant jacobian The standard values have changed 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 for all integer variables This is especially useful for the quadrature rules and for ndiag If a variable appears more than once in the file then the last value is taken No
64. ther degrees of freedom by pressing the buttons with the names of the corresponding 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 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 On item 2 the window of our 2D interactive graphics interface is opened and asked for the kind of visualization Possible choices are the solution plot on the surface the solution plot of a chosen plane of intersection or quit In the first two cases the user is prompted for some information on the perspective of the plot and related things For more details how to operate with the 2D interface see 20 The choice of item 4 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 4 with the coordinates of the nodes stored in Kettes for the term Kette see 2 they are stored in files 16 CHAPTER 2 BASIC DESCRIPTION kettinf Pnumber_of_processor dat With menu item 6 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 soluti
65. ve spectral b section gt in 2 Tetr 2 mal Flaechen getauscht NETZ VERFEINERT VFS 1 NETZ VERFEINERT VFS 2 aK gt K oko oe oe oe oe oe oe oe oe oe oe ok oe ok ok 2k 2k 2k ok ok ok oko oe oe oe ok ok ok ok ok 2k ok ok ok ok 2K K ok ok ok ok ok ok ok ok 2K k 3K oe oe ok ok ek AUSGABEMENUE aK kK k K K kK kK K k kK kK K K kK K K Fk FK K K K kK FK K K FK K K K K K K K FK K K FK K K K K FK K K K kK K K K K K K K K K K K K K O WEITER 4 AUSGABE DER NETZDATEN 5 AUSGABE DER RANDKETTENDATEN 8 AUSGABE DER NETZDATEN IN STANDARDFILE kk kk ak ea ke ae ak ak ea a ae fe ak ak ak fe ae ea ak ak ae ae 2 ke ae ak ak EE ae ak ee ae ak ak fe ae aka Ek 2k ak a ea ak ak ak gt EINGABE 0 At this stage the coarse mesh data are read in and distributed to the processors the mesh is hierarchically refined Now the possibility is given to print out some informations about the refined mesh and the chain fields and to store the refined mesh as std file Choosing we go on aK k oko oe oe oe oe oe oe oe oe oe oe ok oe ok ok 2k 2k 2k ok ok ok oko oe oe oe oe ok ok ok ok 2k ok ok 2g 2g oko ok ok ok ook 2 2K ok K 2 oe oe oe ok 2 27223 AUSGABEMENUE aK kK k k K kK k K k kK k K k kK kK K kK FK K K K kK FK K K FK K K K k K K kK K K K FK K K K K FK K K K k FK K K K K K K K K K K K K K WEITER 1 3D GRAFIK MIT GRAPE 2 2D GRAFIK SCHNITT OBERFLAECHE 4 AUSGABE DER NETZDATEN 5 AUSGA

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