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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>3.2 Test 5: Elasticity Problem</TITLE><LINK REL=STYLESHEET TYPE="text/css" HREF="./Modulef.css" TITLE="Modulef CSS"><meta name="description" value="3.2 Test 5: Elasticity Problem"><meta name="keywords" value="Guide1"><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=tex2html468 HREF="node22.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/previous_motif.gif" ALT="previous"></A><A NAME=tex2html474 HREF="node21.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/up_motif.gif" ALT="up"></A><A NAME=tex2html476 HREF="node24.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/next_motif.gif" ALT="next"></A><A NAME=tex2html478 HREF="node2.html"><IMG BORDER=0 ALIGN=BOTTOM SRC="../icons/contents_motif.gif" ALT="contents"></A><A HREF="../Guide1-18/node23.html"><IMG BORDER=0 SRC="../icons/zoom18.gif" ALIGN=BOTTOM ALT="[BIG]"></A><A HREF="../Guide1-14/node23.html"><IMG BORDER=0 SRC="../icons/zoom14.gif" ALIGN=BOTTOM ALT="[Normal]"></A><A HREF="../Guide1-10/node23.html"><IMG BORDER=0 SRC="../icons/zoom10.gif" ALIGN=BOTTOM ALT="[small]"></A><BR><B> Next: </B> <A NAME=tex2html477 HREF="node24.html">3.3 Test 6: Fluid Mechanics Problem</A><B>Up: </B> <A NAME=tex2html475 HREF="node21.html">3 Batch tests</A><B> Prev: </B> <A NAME=tex2html469 HREF="node22.html">3.1 Test 4: Thermal Problem</A><B><A HREF="node2.html" >Contents</A></B><HR SIZE=3 WIDTH="75%"><H1><A NAME=SECTION04320000000000000000>3.2 Test 5: Elasticity Problem</A></H1><P><P><P><H2><A NAME=SECTION04321000000000000000>3.2.1 Description</A></H2><P><P><P>Calculate the displacements and stresses of a clamped beam subject to a vertical traction at its extreme.<P><P><A NAME=2856> </A><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img220.gif"><BR><STRONG>Figure 3.4:</STRONG> <i> Two-dimensional elasticity problem</i><A NAME=figtes5> </A><BR><P><P>The beam problem is a two-dimensional elasticity problem with plane deformation. We consider a homogeneous isotropic material.<P><b> Remark:</b> It is favorable that the nodes at corners <b>(20, -2)</b> and <b>(20, 2)</b> on <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img221.gif"> belong to two elements, due to discontinuities. For example, on the boundary<DIV ALIGN=center><IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img222.gif"></DIV>whereas on boundary<DIV ALIGN=center><IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img223.gif"></DIV><P><P><P><H2><A NAME=SECTION04322000000000000000>3.2.2 The boundary value problem</A></H2><P><P><P>Find the displacement <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img224.gif"> in <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img225.gif"> and <IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img226.gif"> which satisfies:<P><DIV ALIGN=center><IMG BORDER=0 ALIGN=MIDDLE ALT="" SRC="img227.gif"></DIV><P>where<DL COMPACT><DT><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img178.gif"><DD> are the Lamé coefficients, and <DT><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img179.gif"><DD> is Young's modulus and the Poisson ratio, respectively.<P> </DL><P>Assume that<DL COMPACT><DT><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img228.gif"><DD> = 50 000, and <DT><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img229.gif"><DD> = 0.25.<P> </DL><P>The exact solution is not known.<P><P><P>The type of finite element utilized in order to solve this problem: <b> TRIA 2P2D</b> (P2 Lagrange triangle)<P><P><A NAME=2865> </A><IMG BORDER=0 ALIGN=BOTTOM ALT="" SRC="img230.gif"><BR><STRONG>Figure 3.5:</STRONG> <i> Suggested mesh</i><A NAME=fig15> </A><BR><P><P><P><P><H2><A NAME=SECTION04323000000000000000>3.2.3 Execution of the test</A></H2><P><P><P><b> Step 1:</b> Mesh<P><TABLE COLS=3 RULES=GROUPS><COL ALIGN=LEFT><COL ALIGN=LEFT><COL ALIGN=LEFT><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>Execute program: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> D5NOXX</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Modules used: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> APNOPO</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> ... 2D mesh generator [<A HREF="node65.html#mod_104"><A NAME=tex2html37 HREF="../Guide3/welcome.html">MODULEF User Guide - 3</A></A>] </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> CONOPO</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> ... manual mesh generator </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> MODNOP</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> ... translation </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> RETRIN</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> ... subdivision of mesh </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> RECOLC</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> ... 'glue together' of meshes </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> ADPNOP</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> ... addition of nodes </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> GIBBS</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> ... renumbering of nodes </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Libraries: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP COLSPAN=2> <b> NOP2</b>, <b> NOPO</b>, <b> UTSD</b>, <b> UTII</b> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Input data file: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> D5NOPO.D</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Output data structure: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> T5NOPO</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR></TABLE><P><P><P><b> Step2:</b> Interpolation [<A HREF="node65.html#mod_13">13</A>]<P><TABLE COLS=2 RULES=GROUPS><COL ALIGN=LEFT><COL ALIGN=LEFT><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>Execute program: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> D5COXX</b> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Module used: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> COMACO</b> [<A HREF="node65.html#mod_13">13</A>] </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Libraries: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> ELAS</b>, <b> COSD</b>, <b> UTSD</b>, <b> UTII</b> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Element type: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> TRIA 2P2D</b> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Output data structures: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> T5MAIL</b> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> T5COOR</b> </TD></TR></TABLE><P><P><P><b> Step 3:</b> Calculation of the element arrays and description of boundary conditions<P><TABLE COLS=3 RULES=GROUPS><COL ALIGN=LEFT><COL ALIGN=LEFT><COL ALIGN=JUSTIFY WIDTH="0.33"><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>Execute program: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> D5CAXX</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Modules used: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> ELASCT</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... calculates element arrays for the constant case [<A HREF="node65.html#mod_14">14</A>] </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> CONDL1</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... construction of numbers of the degrees of freedom associated with each node [<A HREF="node65.html#mod_20">20</A>] </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> COBDC1</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... introduction of boundary conditions [<A HREF="node65.html#mod_18">18</A>] </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Libraries: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> COSD</b>, <b> ELAS</b>, <b> ELA2</b>, <b> UTII</b>, <b> UTSD</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Output data structures: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> T5TAE</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> T5BDCL</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"></TD></TR></TABLE><P><P><P><b> Step 4:</b> Solution<P><TABLE COLS=3 RULES=GROUPS><COL ALIGN=LEFT><COL ALIGN=LEFT><COL ALIGN=JUSTIFY WIDTH="0.33"><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>Execute program: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> D5REXX</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... solution by a Crout direct method after assembly of the matrix and RHS vector and incorporation of boundary conditions [<A HREF="node65.html#mod_5"><A NAME=tex2html39 HREF="../Guide5/welcome.html">MODULEF User Guide - 5</A></A>] </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Modules used: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> PREPAC</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... allocation of pointers </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> ASSMUA</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... assembly of matrix in main memory (m.m.) </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> ASEMBV</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... assembly of RHS vector in m.m.</TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> CLIMPC</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... introduction of boundary conditions </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> CROUPC</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... factorization by Crout </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> DRCRPC</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"> ... forward and backward substitution </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Libraries: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP COLSPAN=2> <b> RESB</b>, <b> RESO</b>, <b> RESD</b>, <b> UTSD</b>, <b> UTII</b></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Output data structure: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> T5B</b> </TD><TD VALIGN=BASELINE ALIGN=LEFT WIDTH="33%"></TD></TR></TABLE><P>The number of words necessary to store the matrix in main memory is 38 127.<P>The number of words necessary to store the corresponding right-hand-side (RHS) vector is 1377.<P><P><P><b> Step 5:</b> Calculation of stresses [<A HREF="node65.html#mod_90">90</A>]<P><TABLE COLS=2 RULES=GROUPS><COL ALIGN=LEFT><COL ALIGN=LEFT><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>Execute program: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> D5STXX</b> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Modules: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> STRESS</b> [<A HREF="node65.html#mod_90">90</A>] </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP></TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Output data structure: </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <b> T5TAES</b> </TD></TR></TABLE><P><P><P><H2><A NAME=SECTION04324000000000000000>3.2.4 Memory usage and Execution time</A></H2><P><P><P>The table below represents the approximate maximum values of memory usage and execution time.<P><P><P><TABLE COLS=3 BORDER FRAME=BOX RULES=GROUPS><COLGROUP><COL ALIGN=LEFT><COLGROUP><COL ALIGN=RIGHT><COLGROUP><COL ALIGN=RIGHT><TBODY><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>Step </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> Dimension of </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> Execution time </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> the super array <b> LM</b> </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> (DPS8 Multics) </TD></TR></TBODY><TBODY><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP>Mesh </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> 15 000 </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> 23.0 </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Interpolation </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> 8 000 </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> 3.5 </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Element arrays </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> 8 000 </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> 12.7 </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Solution </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> 45 000 </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> 25.0 </TD></TR><TR><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> Stresses </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> 6 000 </TD><TD VALIGN=BASELINE ALIGN=RIGHT NOWRAP> 7.0 </TD></TR></TBODY></TABLE><P><P><P>The super array <b> M</b> is dimensioned to 45 000 words in all the programs.<P><P><P><H2><A NAME=SECTION04325000000000000000>3.2.5 Results</A></H2><P><P><P>The output of the D.S. <b> B</b>, containing the displacements at all the nodes in the mesh,is found in the file <b> S5B.D</b> in library <b> TESD</b>. The stress values of the first 45 elements arefound in file <b> S5TAES.D</b> in library <b> TESD</b>.<P>The remainder of this section contains the partial or total printout of the various D.S. created by the different programs. These data structures can be viewed by calling preprocessor <b> IMAGXX</b> andspecifying the names of the files containing the desired data structures.<P><P><P><PRE> &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& PRINT OUT OF D.S. NOPO OF LEVEL 0 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&⌨️ 快捷键说明
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