layout optimization using the genetic algorithm

InternationalJournalofMachineTools&Manufacture40(2000)579–598MachiningfixturelayoutoptimizationusingthegeneticalgorithmKulankaraKrishnakumar,ShreyesN.Melkote*TheGeorgeW.WoodruffSchoolofMechanicalEngineering,GeorgiaInstituteofTechnology,Atlanta,GA30332-0405,USAReceived5November1998;receivedinrevisedform16June1999;accepted30July1999AbstractDimensionalandformaccuracyofaworkpieceareinfluencedbythefixturelayoutselectedforthemachiningoperation.Hence,optimizationoffixturelayoutisacriticalaspectofmachiningfixturedesign.Thispaperpresentsafixturelayoutoptimizationtechniquethatusesthegeneticalgorithm(GA)tofindthefixturelayoutthatminimizesthedeformationofthemachinedsurfaceduetoclampingandmachiningforcesovertheentiretoolpath.TheadvantagesoftheGA-basedmethodoverpreviouslyreportednon-linearprogrammingmethodsforfixturelayoutoptimizationarediscussed.TwoGA-basedfixturelayoutoptimizationapproachesareimplementedandcomparedbyapplyingthemtoseveraltwo-dimensionalexampleproblems.1999ElsevierScienceLtd.Allrightsreserved.Keywords:Fixturelayoutoptimization;Geneticalgorithm1.IntroductionThemainfunctionsofamachiningfixturearetolocate,constrain,andadequatelysupporttheworkpieceduringmachining.Thesefunctionsareachievedbystrategicallyplacinglocatorpins,clamps,andsupportsaroundtheworkpieceandapplyingtheappropriateclampingforces.Apoorchoiceofthepositionsofthefixtureelementsandclampingforce(s)canleadtoundesirableworkpiecedeformationandlowdimensional/formaccuracyoftheworkpiece.Consequently,animportantconsiderationinfixturedesignistooptimizethefixturelayouti.e.positionsoflocatorsandclamps,sothatworkpiecedeformationduetoclampingandmachiningforcesisminimized.Anumberoffixturedesignoptimizationapproacheshavebeenreportedbyotherresearchers.*Correspondingauthor.Tel.:[phone];fax:[phone].E-mailaddress:[e-mail](S.N.Melkote)0890-6955/00/$-seefrontmatter1999ElsevierScienceLtd.Allrightsreserved.PII:S0890-6955(99)00072-3580K.Krishnakumar,S.N.Melkote/InternationalJournalofMachineTools&Manufacture40(2000)579–598Theseapproachesalluseeithertherigidbodyorfiniteelementmethodtomodelthefixture-workpiecesystem.KingandHutter[1]presentedamethodforoptimalfixturelayoutdesignusingarigidbodymodelofthefixture-workpiecesystembutaccountingforthecontactstiffness.Non-linearoptimizationtechniqueswereusedtodetermineastaticallystablefixturelayout.DeMeter[2]usedarigidbodyfixture-workpiecemodelandthemin-maxloadcriterionforsynthesisoftheoptimalfixturelayoutandminimumclampactuationintensity.Hereagainnonlinearoptimiz-ationmethodswereusedandelasticdeformationoftheworkpiecewasneglected.LeeandHaynes[3]wereamongstthefirsttousethefiniteelementmethodforfixturedesignandanalysis.TheymodeledtheeffectofCoulombfrictionexactlyandanalyzeditseffectonfixturingparameterslikethemaximumworkdonebyclampingandmachiningforces,thework-piecedeformationindex,andthemaximumstressdevelopedintheworkpiece.MenassaandDeVries[4]alsousedthefiniteelementmethodandanonlinearoptimizationalgorithmtosynthe-sizesupportpositionsforaplate-typeworkpiece.Theirobjectivewastominimizethesummationofnodaldisplacementsatspecificpointsonthesurfacetobemachined.Caietal.[5]extendedtheworkofMenassaandDeVriestoincludethesynthesisoffixturelayoutsforsheetmetalassembly.Theyminimizedthesum-of-squaresofnodaldeflectionsnormaltothesheetmetalsurface.Pongetal.[6]alsoappliedasimilartechniqueandconsideredthemin–maxobjectivefunctiontominimizethemaximumdeflectionatselectedfiniteelementnodes.Recently,DeMeter[7]presentedafiniteelement-basedsupportlayoutoptimizationprocedurewithcomputationallyattractivequalities.Anon-linearoptimizationalgorithmwasusedbutthegenerallocatorandclamppositionsynthesisproblemwasnotconsideredinthiswork.Theabovestudieshavetwomajordrawbacks:(1)theyusenonlinearprogrammingmethodswhich,forthefixturelayoutproblem,donotgive“global”or“near-global”optimumsolutions,and(2)theyemployobjectivefunctionsthatarenotexplicitfunctionsofthedesignvariablesi.e.thelocatorandclamppositions.Therefore,thesemethodsatbestyieldanimprovedsolutionbutnotthe“global”optimum.Inaddition,solutionsobtainedfromthesemodelsareverysensitivetotheinitialfeasiblefixturelayoutusedasinputtotheoptimizationprocess.Theaforementioneddrawbackscanbeovercomebyusingthegeneticalgorithm(GA)[8]methodforfixturelayoutoptimization.Fewresearchershavereportedtheuseofthegeneticalgorithmmethodforthisproblem.WuandChan[9]usedthegeneticalgorithmtodeterminethemoststaticallystablefixturelayout.However,theyusedarigidbodyfixture-workpiecemodelandignoredelasticdeformationoftheworkpieceduetoclampingandmachiningforces.Asnotedearlier,workpieceaccuracyisgreatlyinfluencedbysuchdeformationsandhencecannotbeignored.IshikawaandAoyama[10]appliedthegeneticalgorithmtodeterminetheoptimalclamp-ingconditionforanelasticworkpiece.Theiranalysishoweverdoesnotconsiderthemoregeneralcaseoflocatorandclamplayoutsynthesis.Thispaperpresentsageneralmethodforfixturelayoutoptimizationthatusesthegeneticalgor-ithm.Theoptimizationobjectiveistosearchforafixturelayoutthatminimizesthemaximumelasticdeformationofthemachinedsurfacefortheentir