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<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 3.2 Final//FR"><!-- Converted with LaTeX2HTML 95.1 (Fri Jan 20 1995) --><!-- by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds --><!-- Modified Simulog 03/97 --><HTML><HEAD><TITLE>1.2 The symmetric model problem</TITLE><LINK REL=STYLESHEET TYPE="text/css" HREF="./Modulef.css" TITLE="Modulef CSS"><meta name="description" value="1.2 The symmetric model problem"><meta name="keywords" value="Guide5"><meta name="resource-type" value="document"><meta name="distribution" value="global"></HEAD><BODY BGCOLOR="#FFFFFF"><P> <IMG SRC="../icons/smallmod.gif" WIDTH=211 HEIGHT=50 ALIGN=BOTTOM ALT="Modulef"><A NAME=tex2html791 HREF="node41.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/previous_motif.gif" ALT="previous"></A><A NAME=tex2html797 HREF="node40.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/up_motif.gif" ALT="up"></A><A NAME=tex2html799 HREF="node43.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/next_motif.gif" ALT="next"></A><A NAME=tex2html801 HREF="node2.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/contents_motif.gif" ALT="contents"></A><A HREF="../Guide5-18/node42.html"><IMG BORDER=0 SRC="../icons/zoom18.gif" ALIGN=BOTTOM ALT="[BIG]"></A><A HREF="../Guide5-14/node42.html"><IMG BORDER=0 SRC="../icons/zoom14.gif" ALIGN=BOTTOM ALT="[Normal]"></A><A HREF="../Guide5-10/node42.html"><IMG BORDER=0 SRC="../icons/zoom10.gif" ALIGN=BOTTOM ALT="[small]"></A><BR><B> Next: </B> <A NAME=tex2html800 HREF="node43.html">1.3 The non-symmetric model problem</A><B>Up: </B> <A NAME=tex2html798 HREF="node40.html">1 Examples</A><B> Prev: </B> <A NAME=tex2html792 HREF="node41.html">1.1 Introduction</A><B><A HREF="node2.html" >Contents</A></B><HR SIZE=3 WIDTH="75%"><H1><A NAME=SECTION05120000000000000000>1.2 The symmetric model problem</A></H1><P> The example chosen is the solution of a realistic linear elasticity problem by the finite element method: a hook, greased on part of its boundary, is submitted to a vertical traction force on the utility part. Given the envisaged boundary conditions, and the material assumed homogeneous isotropic, the problem exhibits a symmetry plane, so that the domain is reduced to a half-hook (see Figure 1).<P> For more details about the variational formulation of this problem, as well as its finite element approximation and the different aspects of the computer implementation, consult also [<A HREF="node44.html#magnifiquedocument">1</A>]. Other numerical experiences concerning the solution of this problem are also found in this reference.<P> During numerical implementation, several distinctive steps appear before the actual solution: <UL><LI> construction of the triangulation <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img80.gif"> (data structure NOPO) by module APNOPO, <LI> construction of data structures MAIL and COOR by module COMACO, <LI> construction of data structures FORC and MILI by module COFOMI, <LI> generation of the element arrays (structure TAE) by module THELAS, using data structures MAIL, COOR, FORC and MILI, <LI> description of the problem's boundary conditions (data structure BDCL) by module COBDC1. </UL><P> Consult the corresponding guides for the utilization of these modules.<P> <P><P> <P><A NAME=2229> </A><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img81.gif"><BR><STRONG>Figure 1.1:</STRONG> <i> Mesh of the domain</i><A NAME=figmaillage> </A><BR><P><P> The geometry of the model problem is described by a NOPO data structure, and a view of the object is presented in Figure <A HREF="node42.html#figmaillage">1.1</A>. The mesh contains 1006 elements and 937 nodes, the finite elements used being of degree 1.<P> At each of the nodes, we need to compute the three displacement components, so that the resulting linear system has 2811 unknowns. The structure of the matrix of the linear system is represented in figure <A HREF="node42.html#figmatr">1.2</A>.<P> <P><A NAME=2238> </A><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img82.gif"><BR><STRONG>Figure 1.2:</STRONG> <i> Matrix structure</i><A NAME=figmatr> </A><BR><P><P> To solve the linear system corresponding to the model problem, we need to: <UL><LI> initialize the data structure containing the matrix, to calculate the memory space required for storing it, eg compute the pointers, <LI> assemble the element matrices and right-hand-sides, <LI> take the boundary conditions into account, <LI> solve the linear system (by a direct or iterative method). </UL><P> We have already mentioned that each of these steps are performed by a distinct module; to simplify the utilization of the MODULEF code, sequences of these modules are programmed for each option available (the preprocessors of library PPAL). The user can also create his/her own sequences from the MODULEF programs.<P> We will now look at the different possibilities:<P> <P><P> <H2><A NAME=SECTION05121000000000000000>1.2.1 Solution by Cholesky in main memory</A></H2><P> <P><P> Main program <b> CHOLXX</b> successively calls modules PREPAC, ASSMUA, ASEMBV, CHOLPC and, lastly, DRCHPC. The listing of the data of <b> CHOLXX</b> is shown below:<P><PRE> mail $ FILE NAME 1 $ AND LEVEL OF THE D.S. MAIL 1 5 3 $ NDSM NTYP ND tae $ FILE NAME 1 $ AND LEVEL OF THE D.S. TAE 1 $ 1 IF BDCL IS USED , 0 IF NOT bdcl $ FILE NAME 1 $ AND LEVEL OF THE D.S. BDCL 0 $ 1 IF BOUNDARY CONDITIONS I.T.O. $ LINEAR RELATIONS EXIST B $ FILE NAME 1 $ AND LEVEL OF THE D.S. B 3 $ IMPREB</PRE><P> The results corresponding to an execution on a Hewlett-Packard Apollo series 400 workstation is given below:<P><PRE> M M OOO DDDD U U L EEEEE FFFFF MM MM O O D D U U L E F M M M O O D D U U L EEEE FFFF M M O O D D U U L E F M M OOO DDDD UUU LLLLL EEEEE F VERSION 93 DATE : 14/06/93 AUTEUR : dutoit ************************************** EXECUTING THE CHOLESKY SOLUTION METHOD ************************************** -- CREATE THE DATA =-= EXECUTE MODULE (DATA EXISTING) =-= END =-= ? - - ---E -- NAME OF DATA FILE ? dchol.data_gb -- PRINT PARAMETER FOR THE EXECUTION ? 3 ++ OPEN(11,FILE='mail',SPEC='OLD,UNFORMATTED',RECL=0) ++ OPEN(12,FILE='tae',SPEC='OLD,UNFORMATTED',RECL=0) ++ OPEN(13,FILE='bdcl',SPEC='OLD,UNFORMATTED',RECL=0) ++ OPEN(14,FILE='crochet.bchol',SPEC='UNFORMATTED',RECL=0) &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& MODULE PREPAC : NUMBER OF D.O.F. PER NODE (ND) : 3 NUMBER OF LOAD CASES (NDSM) : 1 PRESENCE OF LINEAR RELATIONS (NCLRL) : 0 MATRIX STORAGE CODE (NCODSA) : 1 TYPE OF TABLES (NTY) : 5 INPUT D.S. MAIL (NFMAIL ET NIMAIL) : 11 1 INPUT D.S. NDL1 (NFNDL1 ET NINDL1) : 0 0 INPUT D.S. BDCL (NFBDCL ET NIBDCL) : 13 1 OUTPUT D.S. MUA (NFMUA ET NIMUA ) : 0 0 NUMBER OF ASSOCIATED TABLES (NTMUA) : 0 NUMBER OF WORDS OF THE MATRIX (LMUA5) : 737202 MAX DIFFERENCE BETWEEN 2 NODES OF AN ELEMENT (LBDP) : 70 MAX DIFFERENCE BETWEEN 2 D.O.F. OF AN ELEMENT (LBDPDL) : 210 MAX NUMBER OF NODES PER ELEMENT (NNOMAX) : 8 NUMBER OF RHS VECTORS (NDSM) : 1 NUMBER OF WORDS IN C.M. FOR D.S. NDL1 (MCNDL1) : 0 NUMBER OF WORDS IN C.M. FOR D.S. MUA (MCMUA) : 740060 NUMBER OF WORDS IN C.M. FOR D.S. B (MCB) : 5667 END OF MODULE PREPAC &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& MODULE ASSMUA : ARRAY OF TAE TO ASSEMBLE (NOT) : 1 INPUT D.S. TAE (NFTAE AND NITAE) : 12 1 INPUT D.S. NDL1 (NFNDL1 AND NINDL1) : 0 0 INPUT D.S. MUA (NFMUAE AND NIMUAE) : 0 0 OUTPUT D.S. MUA (NFMUAS AND NIMUAS) : 0 0 ASSEMBLY OF ELEMENT ARRAY 1 HAS 1 INDEX(CES), WITH STORAGE CODE 1 STORAGE CODE OF THE GLOBAL MATRIX 1 END OF MODULE ASSMUA &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& MODULE ASEMBV : ARRAY TO ASSEMBLE (NOT) : 2 NUMBER OF LOAD CASES (NDSM) : 1 INPUT D.S. TAE (NFTAE AND NITAE) : 12 1 INPUT D.S. NDL1 (NFNDL1 AND NINDL1) : 0 0 OUTPUT D.S. B (NFB AND NIB) : 0 1 NUMBER OF ASSOCIATED TABLES (NTB) : 0 NUMBER OF WORDS IN CENTRAL MEMORY FOR THE D.S. B : 5667 END OF MODULE ASEMBV &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& PROCESSING OPTION (NIVO) : 3⌨️ 快捷键说明
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