A-99-line-topology-optimization-code-written-in-Ma

EducationalarticleStructMultidiscOptim21,120–127Springer-Verlag2001A99linetopologyoptimizationcodewritteninMatlabO.SigmundAbstractThepaperpresentsacompactMatlabim-plementationofatopologyoptimizationcodeforcom-plianceminimizationofstaticallyloadedstructures.ThetotalnumberofMatlabinputlinesis99includingopti-mizerandFiniteElementsubroutine.The99linesaredividedinto36linesforthemainprogram,12linesfortheOptimalityCriteriabasedoptimizer,16linesforamesh-independencyfilterand35linesforthefiniteelementcode.Infact,excludingcommentlinesandlinesassoci-atedwithoutputandfiniteelementanalysis,itisshownthatonly49Matlabinputlinesarerequiredforsolvingawell-posedtopologyoptimizationproblem.Byaddingthreeadditionallines,theprogramcansolveproblemswithmultipleloadcases.Thecodeisintendedforedu-cationalpurposes.ThecompleteMatlabcodeisgivenintheAppendixandcanbedown-loadedfromtheweb-sitefieldoftopologyoptimizationcandown-loadthecodefromtheweb-page’intuitionforoptimaldesign.Advancedstudentsmaybeaskedtoguesstheop-timaltopologyforgivenboundaryconditionandvolumeReceivedOctober22,1999O.SigmundDepartmentofSolidMechanics,Building404,TechnicalUni-versityofDenmark,DK-2800Lyngby,Denmarke-mail:[e-mail]ramshowsthecorrectoptimaltopologyforcomparison.Intheliterature,onecanfindamultitudeofapproachesforthesolvingoftopologyoptimizationproblems.IntheoriginalpaperBendsøeandKikuchi(1988)aso-calledmicrostructureorhomogenizationbasedapproachwasused,basedonstudiesofexistenceofsolutions.Thehomogenizationbasedapproachhasbeenadoptedinmanypapersbuthasthedisadvantagethatthedeter-minationandevaluationofoptimalmicrostructuresandtheirorientationsiscumbersomeifnotunresolved(fornoncomplianceproblems)andfurthermore,theresultingstructurescannotbebuiltsincenodefinitelength-scaleisassociatedwiththemicrostructures.However,theho-mogenizationapproachtotopologyoptimizationisstillimportantinthesensethatitcanprovideboundsonthetheoreticalperformanceofstructures.Analternativeapproachtotopologyoptimizationistheso-called“power-lawapproach”orSIMPapproach(SolidIsotropicMaterialwithPenalization)(Bendsøe1989;ZhouandRozvany1991;Mlejnek1992).Here,ma-terialpropertiesareassumedconstantwithineachelem-entusedtodiscretizethedesigndomainandthevariablesaretheelementrelativedensities.Thematerialproper-tiesaremodelledastherelativematerialdensityraisedtosomepowertimesthematerialpropertiesofsolidma-terial.Thisapproachhasbeencriticizedsinceitwasar-guedthatnophysicalmaterialexistswithpropertiesde-scribedbythepower-lawinterpolation.However,arecentpaperbyBendsøeandSigmund(1999)provedthatthepower-lawapproachisphysicallypermissibleaslongassimpleconditionsonthepoweraresatisfied(e.g.p≥3forPoisson’sratioequalto13).Toensureexistenceofso-lutions,thepower-lawapproachmustbecombinedwithaperimeterconstraint,agradientconstraintorwithfil-teringtechniques(seeSigmundandPetersson1998,foranoverview).Thepower-lawapproachtotopologyop-timizationhasbeenappliedtoproblemswithmultipleconstraints,multiplephysicsandmultiplematerials.Whereasthesolutionoftheabovementionedap-proachesisbasedonmathematicalprogrammingtech-niquesandcontinuousdesignvariables,anumberofpa-pershaveappearedonsolvingthetopologyoptimizationproblemasanintegerproblem.Beckers(1999)success-121fullysolvedlarge-scalecomplianceminimizationprob-lemsusingadual-approachbutotherapproachesbasedongeneticalgorithmsorothersemi-randomapproachesrequirethousandsoffunctionevaluationsevenforsmallnumberofelementsandmustbeconsideredimpractical.Apartfromabovementionedapproaches,whichallsolvewelldefinedproblems(e.g.minimizationofcom-pliance)anumberofheuristicorintuitionbasedap-proacheshavebeenshowntodecreasecomplianceorotherobjectivefunctions.Amongthesemethodsareso-calledevolutionarydesignmethods(seee.g.XieandSteven1997;Baumgartneretal.1992).Apartfrombe-ingveryeasytounderstandandimplement(atleastforthecomplianceminimizationcase),themainmoti-vationfortheevolutionaryapproachesseemstobethatmathematicallybasedorcontinuousvariableapproaches“involvesomecomplexcalculusoperationsandmathe-maticalprogramming”(citationfromLietal.1999)andtheycontain“mathematicalmethodsofsomecomplex-ity”(citationfromZhaoetal.1998)whereastheevo-lutionaryapproach“takesadvantageofpowerfulcom-putingtechnologyandintuitiveconceptsofevolutionprocessesinnature”(citationfromLietal.1999).Twothingscanbearguedagainstthis.First,theevolutionaryapproachesbecomecomplicatedthemselves,oncemorecomplexobjectivesthancomplianceminimizationareconsideredandsecond,asshowninthispaper,the“math-ematicallybased”approachesforcomplianceminimiza-tionaresimpletoimpleme