Programming The Finite Element Method With Matlab.pdf

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1、ProgramingtheFiniteElementMethodwithMatlabJackChessa3rdOctober20021IntroductionThegoalofthisdocumentistogiveaverybriefoverviewanddirectioninthewritingofniteelementcodeusingMatlab.Itisassumedthatthereaderhasabasicfamiliaritywiththetheoryoftheniteelementmethod,andouratte

2、ntionwillbemostlyontheimplementation.AnexampleniteelementcodeforanalyzingstaticlinearelasticproblemswritteninMatlabispresentedtoillustratehowtoprogramtheniteelementmethod.Theexampleprogramandsupportinglesareavailableathttp://www.tam.northwestern.edu/jfc795/Matlab/1.1No

3、tationForclarityweadoptthefollowingnotationinthispaper;thebolditalicsfontvdenotesavectorquantityofdimensionequaltothespacialdimensionoftheproblemi.e.thedisplacementorvelocityatapoint,theboldnon-italicizedfontddenotesavectorormatrixwhichisofdimensionofthenumberofunknownsin

4、thediscretesystemi.e.asystemmatrixlikethestinessmatrix,anuppercasesubscriptdenotesanodenumberwhereasalowercasesubscriptingeneraldenotesavectorcomponentalongaCartesianunitvector.So,ifdisthesystemvectorofnodalunknowns,uIisadisplacementvectorofnodeIanduIiisthecomponentofthe

5、displacementatnodeIintheidirection,oruI
ei.OftenMatlabsyntaxwillbeintermixedwithmathematicalnotationGraduateResearchAssistant,NorthwesternUniversity(j-chessa@northwestern.edu)1whichhopefullyaddsclaritytotheexplanation.Thetypewriterfont,font,isusedtoindicatethatMatlabsynt

6、axisbeingemployed.2AFewWordsonWritingMatlabProgramsTheMatlabprogramminglanguageisusefulinillustratinghowtoprogramtheniteelementmethodduetothefactitallowsonetoveryquicklycodenumericalmethodsandhasavastpredenedmathematicallibrary.Thisisalsoduetothefactthatmatrix(sparseand

7、dense),vectorandmanylinearalgebratoolsarealreadydenedandthedevelopercanfocusentirelyontheimplementationofthealgorithmnotdeningthesedatastructures.Theextensivemathematicsandgraphicsfunctionsfurtherfreethedeveloperfromthedrudgeryofdevelopingthesefunctionsthemselvesorndin

8、gequivalentpre-existinglibraries.AsimpletwodimensionalniteelementprograminMatlabneedonlybeafewh