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JOURNALOFMATERIALSPROCESSINGTECHNOLOGY1402003616621EFFICIENCYENHANCEMENTINSHEETMETALFORMINGANALYSISWITHAMESHREGULARIZATIONMETHODJHYOON,HHUHDEPARTMENTOFMECHANICALENGINEERING,KOREAADVANCEDINSTITUTEOFSCIENCEANDTECHNOLOGYSCIENCETOWN,DAEJEON305701,SOUTHKOREAABSTRACTTHISPAPERNEWLYPROPOSESAMESHREGULARIZATIONMETHODFORTHEENHANCEMENTOFTHEEFFICIENCYINSHEETMETALFORMINGANALYSISTHEREGULARIZATIONMETHODSEARCHESFORDISTORTEDELEMENTSWITHAPPROPRIATESEARCHINGCRITERIAANDCONSTRUCTSPATCHESINCLUDINGTHEELEMENTSTOBEMODIFIEDEACHPATCHISTHENEXTENDEDTOATHREEDIMENSIONALSURFACEINORDERTOOBTAINTHEINFORMATIONOFTHECONTINUOUSCOORDINATESINCONSTRUCTINGTHESURFACEENCLOSINGEACHPATCH,NURBSNONUNIFORMRATIONALBSPLINESURFACEISEMPLOYEDTODESCRIBEATHREEDIMENSIONALFREESURFACEONTHEBASISOFTHECONSTRUCTEDSURFACE,EACHNODEISPROPERLYARRANGEDTOFORMUNITELEMENTSASCLOSEASTOASQUARETHESTATEVARIABLESCALCULATEDFROMITSORIGINALMESHGEOMETRYAREMAPPEDINTOTHENEWMESHGEOMETRYFORTHENEXTSTAGEORINCREMENTALSTEPOFAFORMINGANALYSISTHEANALYSISRESULTSWITHTHEPROPOSEDMETHODARECOMPAREDTOTHERESULTSFROMTHEDIRECTFORMINGANALYSISWITHOUTMESHREGULARIZATIONINORDERTOCONFIRMTHEVALIDITYOFTHEMETHOD2003ELSEVIERBVALLRIGHTSRESERVEDKEYWORDSMESHREGULARIZATIONDISTORTEDELEMENTNURBSPATCHFINITEELEMENTANALYSIS1INTRODUCTIONNUMERICALSIMULATIONOFSHEETMETALFORMINGPROCESSESENJOYSITSPROSPERITYWITHABURSTOFDEVELOPMENTOFTHECOMPUTERSANDTHERELATEDNUMERICALTECHNIQUESTHENUMERICALANALYSISHASEXTENDEDITSCAPABILITIESFORSHEETMETALFORMINGOFCOMPLICATEDGEOMETRYMODELSANDMULTISTAGEFORMINGINTHECASEOFACOMPLICATEDGEOMETRYMODEL,HOWEVER,SEVERELOCALDEFORMATIONOCCURSTOINDUCETHEINCREASEOFTHECOMPUTINGTIMEANDDETERIORATETHECONVERGENCEOFTHEANALYSISDISTORTIONANDSEVEREDEFORMATIONOFTHEMESHGEOMETRYHASANEFFECTONTHEQUALITYOFFORMINGANALYSISRESULTSESPECIALLYINTHECASEOFMULTISTAGEFORMINGANALYSISWHENTHEMESHGEOMETRYFORMEDBYTHEFORMINGANALYSISATTHEFIRSTSTAGEISUSEDFORTHEFORMINGANALYSISATTHENEXTSTAGETHISILLBEHAVIOROFTHEDISTORTEDMESHCANBEAVOIDEDBYTHERECONSTRUCTIONOFTHEMESHSYSTEMSUCHASTHETOTALORTHEADAPTIVEREMESHINGTECHNIQUESTHEADAPTIVEREMESHINGTECHNIQUEISKNOWNTOBEANEFFICIENTMETHODTOREDUCEDISTORTIONOFELEMENTDURINGTHESIMULATION,BUTITSTILLNEEDSTREMENDOUSCOMPUTINGANDPUTSRESTRICTIONSAMONGSUBDIVIDEDELEMENTSCORRESPONDINGAUTHORTEL82428693222FAX82428693210EMAILADDRESSHHUHKAISTACKRHHUHEFFECTIVEMETHODSTOCONSTRUCTAMESHSYSTEMHAVEBEENPROPOSEDBYMANYRESEARCHERSTYPICALMETHODSCOULDBERMETHOD1INWHICHNODALPOINTSAREPROPERLYREARRANGEDWITHOUTTHECHANGEOFTHETOTALDEGREESOFFREEDOMOFTHEMESHSYSTEM,HMETHOD2INWHICHTHENUMBEROFMESHESISINCREASEDWITHELEMENTSOFTHESAMEDEGREESOFFREEDOM,ANDPMETHOD3INWHICHTHETOTALDEGREESOFFREEDOMOFTHEMESHSYSTEMISINCREASEDTOENHANCETHEACCURACYOFSOLUTIONSSLUITERANDHANSEN4ANDTALBERTANDPARKINSON5CONSTRUCTEDTHEANALYSISDOMAINASACONTINUOUSLOOPANDCREATEDELEMENTSINSUBLOOPSDIVIDEDFROMTHEMAINLOOPLO6CONSTRUCTEDTRIANGULARELEMENTSINTHEWHOLEDOMAINANDTHENCONSTRUCTEDRECTANGULARELEMENTSBYCOMBININGADJACENTTRIANGULARELEMENTSINTHISPAPER,AMESHREGULARIZATIONMETHODISNEWLYPROPOSEDINORDERTOENHANCETHEEFFICIENCYOFFINITEELEMENTANALYSESOFSHEETMETALFORMINGTHEMESHREGULARIZATIONMETHODAUTOMATICALLYFINDSOUTDISTORTEDELEMENTSWITHSEARCHINGCRITERIAPROPOSEDANDCOMPOSESPATCHESTOBEMODIFIEDEACHPATCHISTHENEXTENDEDTOTHREEDIMENSIONALSURFACESINORDERTOOBTAINTHEINFORMATIONOFTHECONTINUOUSCOORDINATESONTHETHREEDIMENSIONALSURFACETHESURFACEENCLOSINGEACHPATCHISDESCRIBEDASATHREEDIMENSIONALFREESURFACEWITHTHEUSEOFNURBSNONUNIFORMRATIONALBSPLINEONTHEBASISOFTHECONSTRUCTEDSURFACE,EACHNODEISPROPERLYARRANGEDTOCOMPOSEREGULARELEMENTSCLOSETOASQUARETHESTATEVARIABLESCALCULATEDFROMITSORIGINALMESH09240136/SEEFRONTMATTER2003ELSEVIERBVALLRIGHTSRESERVEDDOI101016/S092401360300801XJHYOON,HHUH/JOURNALOFMATERIALSPROCESSINGTECHNOLOGY1402003616621617GEOMETRYAREMAPPEDINTOTHENEWMESHGEOMETRYFORTHEFORMINGANALYSISATTHENEXTSTAGENUMERICALRESULTSCONFIRMTHEEFFICIENCYOFTHEPROPOSEDMETHODANDTHEACCURACYOFTHERESULTITISALSONOTEDTHATTHEPRESENTMETHODISEFFECTIVEINTHECRASHANALYSESOFSHEETMETALMEMBERSOBTAINEDFROMTHEFORMINGSIMULATION2REGULARIZATIONOFTHEDISTORTEDELEMENTTHEREGULARIZATIONPROCEDURETOMODIFYDISTORTEDELEMENTSISINTRODUCEDINORDERTOENHANCETHEEFFICIENCYOFANALYSISFORTHENEXTFINITEELEMENTCALCULATIONTHEDISTORTEDELEMENTSARESELECTEDWITHAPPROPRIATESEARCHINGCRITERIAANDALLOCATEDTOSEVERALPATCHESFORREGULARIZATIONTHEPATCHESAREEXTENDEDTOTHREEDIMENSIONALSURFACESWITHTHEUSEOFNURBSFORFULLINFORMATIONOFTHECONTINUOUSCOORDINATESONTHETHREEDIMENSIONALSURFACEONOBTAININGTHENEWCOORDINATESOFEACHNODE,THEDISTORTEDELEMENTSAREREGULARIZEDTOAREGULARELEMENTTHATISCLOSETOASQUARE21THECRITERIONOFMESHDISTORTIONDISTORTEDMESHESARESELECTEDWITHTHETWOGEOMETRICALCRITERIAONEISTHEINNERANGLEANDTHEOTHERISTHEASPECTRATIOOFTHEELEMENTSIDE211INNERANGLETHEINNERANGLEOFAQUADRILATERALELEMENTSHOULDBECLOSETOTHERIGHTANGLEFORGOODRESULTSFROMFINITEELEMENTCALCULATIONZHUETAL7DEFINEDTHEREASONABLEELEMENTWHENTHEFOURINNERANGLESAREFORMEDWITHTHEANGLEOF9045WHILELOANDLEE8PROPOSEDTHEINNERANGLEOF90525ASTHESAMECRITERIONTHECRITERIONOFMESHDISTORTIONFORTHEINNERANGLEISDETERMINEDBYCONSTITUTINGEQ1AMESHISREGARDEDASDISTORTEDWHENEQ1ISLESSTHAN/3ORIMAXINEQ39ISGREATERTHAN/6THECRITERIONISRATHERSTRICTINORDERTOAVOIDTHEGEOMETRICALLIMITATIONINCASEOFAPPLYINGTHEREGULARIZATIONMETHODINCONFINEDREGIONSVECTORFQ1E12E23E34E41|VECTORFQ|RADICALTPRADICALVERTEXRADICALVERTEXRADICALBT4SUMMATIONDISPLAYI1I22IVEXTENDSINGLEVEXTENDSINGLEVEXTENDSINGLE12IVEXTENDSINGLEVEXTENDSINGLEVEXTENDSINGLE3212ASPECTRATIOOFTHEELEMENTTHEIDEALASPECTRATIOOFTHEELEMENTSIDESHOULDBEUNITYWHENTHEFOURSIDESOFANELEMENTHAVETHESAMELENGTHTHEASPECTRATIOISDEFINEDASEQ4ANDTHENTHEDISTORTIONISDEFINEDWHENITISLESSTHAN5THATCOULDBEMUCHLESSFORASTRICTCRITERIONMAXR12,R23,R34,R41MINR12,R23,R34,R414WHERERIJISTHELENGTHOFEACHELEMENTSIDEFIG1PROCESSFORCONSTRUCTIONOFAPATCH22DOMAINCONSTRUCTION221CONSTRUCTIONOFTHEPATCHDISTORTEDELEMENTSSELECTEDBYTHECRITERIAOFMESHDISTORTIONAREDISTRIBUTEDINVARIOUSREGIONSACCORDINGTOTHECOMPLEXITYOFTHESHAPEOFFORMEDGEOMETRYTHESEELEMENTSAREALLOCATEDTOPATCHESCONSTRUCTEDFORTHEEFFICIENCYOFTHEALGORITHMTHESHAPEOFPATCHESISMADEUPFORRECTANGULARSHAPESINCLUDINGALLDISTORTEDELEMENTSFOREXPANDINGTHEREGIONOFREGULARIZATIONANDAPPLYINGTONURBSSURFACEEXPLAINEDINNEXTSECTIONTHISPROCEDUREISSHOWNINFIG1WHENHOLESANDEDGESARELOCATEDBETWEENDISTORTEDELEMENTS,THEREGIONSAREFILLEDUPTOMAKEPATCHESARECTANGULARSHAPETHEPATCHISTHENMAPPEDTOATHREEDIMENSIONALFREESURFACEBYUSINGNURBSSURFACETHEPROCEDUREISIMPORTANTTOOBTAINENTIREINFORMATIONOFTHECONTINUOUSCOORDINATESONTHETHREEDIMENSIONALSURFACENURBSSURFACECANDESCRIBETHECOMPLEXSHAPEQUICKLYBYUSINGLESSDATAPOINTSANDDOESNOTCHANGETHEENTIREDOMAINDATADUETOTHELOCALCHANGE222NURBSSURFACENURBSSURFACEISGENERALLYEXPRESSEDBYEQ5ASTHEPORDERINTHEUDIRECTIONANDTHEQORDERINTHEVDIRECTION10SU,VSUMMATIONTEXTNI0SUMMATIONTEXTMJ0NI,PUNJ,QVWI,JPI,JSUMMATIONTEXTNI0SUMMATIONTEXTMJ0NI,PUNJ,QVWI,J5WHEREPI,JISTHECONTROLPOINTSASTHEU,VDIRECTION,WI,JTHEWEIGHTFACTORANDNU,PU,NJ,QUTHEBASISFUNCTIONTHATAREEXPRESSEDBYEQ6NI,0BRACELEFTBIGG1IFUIUUI1,0OTHERWISE,NI,PUUUIUIPUINI,PUUIP1UUIP1UI1NI1,P1U6INORDERTOMAPTHENODESFROMTHEPATCHESONTOTHECONSTRUCTEDSURFACE,ANUMBEROFPOINTSARECREATEDFORTHEIRCOORDINATESONTHENURBSSURFACETHELOCATIONOFEACH618JHYOON,HHUH/JOURNALOFMATERIALSPROCESSINGTECHNOLOGY1402003616621FIG2SELECTINGDIRECTIONOFDISTORTEDELEMENTSMOVINGNODEBYAPPLYINGAREGULARIZATIONMETHODISDETERMINEDSUCHTHATTHELOCATIONOFAPOINTHASTHEMINIMUMDISTANCEBETWEENNODESONNURBSSURFACETHEINFORMATIONONTHECOORDINATESOFTHENODALPOINTSTOBEMOVEDISSTOREDTOCONSTRUCTANEWMESHSYSTEM23REGULARIZATIONPROCEDURETHEREGULARIZATIONMETHODISCARRIEDOUTWITHTHEUNITOFAPATCHTHATFORMSARECTANGULARSHAPEFINITEELEMENTSTOBEREGULARIZEDISSELECTEDBYTHEORDEROFFIG2EACHSELECTEDELEMENTISDIVIDEDBYTWOTRIANGULARELEMENTSANDTHENTHEDIVIDEDELEMENTISMADEOFARIGHTTRIANGULARELEMENTBYRELOCATINGTHEVERTEXONTHECIRCLEHAVINGTHEDIAMETERFROMVECTORX1TOVECTORX2ASSHOWNINEQ7ANDFIG3WHENTHEPROCEDURETERMINATES,THESAMEPROCEDUREISREPEATEDINTHEOPPOSITEDIRECTIONVECTORX1VECTORX22VECTORXCEN,|VECTORX1VECTORX2|2R,VECTORXCURVECTORXCENVECTORXDIR,VECTORXNEWVECTORXDIR|VECTORXDIR|RFACTORVECTORXCEN7THEFINALLOCATIONOFANODERELOCATEDBYUSINGTHEREGULARIZATIONMETHODISSUBSTITUTEDFORTHELOCATIONOFAPOINTONNURBSSURFACEAFTERTHEREGULARIZATIONPROCEDUREISFINISHED,ASIMPLESOOTHINGPROCEDUREISCARRIEDOUTBYEQ8FORTHEROUGHREGIONGENERATEDDURINGTHEFIG3REGULARIZATIONSCHEMEBYMOVINGNODESPROCEDUREPNSUMMATIONTEXTNI1AICISUMMATIONTEXTNI1AI8WHEREPNISTHECOORDINATEOFANEWNODE,AITHEAREASOFADJACENTELEMENTSANDCITHECENTROIDOFTHEADJACENTELEMENTS24LEVELOFDISTORTIONASADISTORTIONFACTOR,LEVELOFDISTORTIONLDISNEWLYPROPOSEDLDCANBEUSEDTOEVALUATETHEDEGREEOFIMPROVEMENTINTHEELEMENTQUALITYLDAB9WHEREASUMMATIONTEXT4I1|SINI|4,BTANHPARENLEFTBIGKBPRIMEPARENRIGHTBIG10BPRIMEMINR12,R23,R34,R41MAXR12,R23,R34,R41,KTANH111LDHASTHEVALUEBETWEEN0AND1WHENLD1,THEELEMENTISANIDEALELEMENTOFASQUAREANDWHENLD0,THEQUADRILATERALELEMENTBECOMESATRIANGULARELEMENTIARETHEFOURINNERANGLESOFANELEMENT,SOAISTHEFACTORFORTHEINNERANGLEBISTHEFACTORFORTHEASPECTRATIOOFELEMENTSIDESANDISDEFINEDBYTHEHYPERBOLICTANGENTFUNCTIONINORDERTOMAKELDLESSSENSITIVETOTHECHANGEOFBFOREXAMPLE,WHENTHEREASONABLEASPECTRATIOOFTHEELEMENTSIDEIS14,THEVALUEOFBCANBEADJUSTEDBYAPPLYING025AND06SUCHTHATTHESLOPEOFTHEFUNCTIONBISCHANGEDABRUPTLYAROUNDTHEVALUEOFBPRIME025CONSEQUENTLY,THEVALUEOFLDDECREASESRAPIDLYWHENTHEASPECTRATIOBPRIMEISLESSTHAN025WHILETHEVALUEOFLDINCREASESSLOWLYWHENTHEBPRIMEISGREATERTHAN025THISSCHEMECANREGULATETHEINNERANGLEANDTHEASPECTRATIOTOHAVETHEEQUALEFFECTONTHELD25MAPPINGOFTHESTATEVARIABLESWHENTHEREGULARIZEDMESHSYSTEMISUSEDFORTHENEXTCALCULATIONOFTHEFORMINGANALYSISORTHESTRUCTURALANALYSIS,MAPPINGOFTHESTATEVARIABLESISNEEDEDFORMOREACCURATEANALYSISCONSIDERINGTHEPREVIOUSFORMINGHISTORYTHEMAPPINGPROCEDUREISTOMAPTHECALCULATEDSTATEVARIABLESINTHEORIGINALMESHSYSTEMONTOTHEREGULARIZEDMESHSYSTEMASSHOWNINFIG4,ASPHEREISCONSTRUCTEDSURROUNDINGANEWNODESUCHTHATTHESTATEVARIABLESOFNODESINTHESPHEREHAVEANEFFECTONTHESTATEVARIABLESOFTHENEWNODETHESTATEVARIABLESOFTHENEWNODEAREDETERMINEDFROMTHESTATEVARIABLESOFTHENEIGHBORINGNODESINTHESPHEREBYIMPOSINGTHEWEIGHTINGFACTORINVERSELYPROPORTIONALTOTHEDISTANCEBETWEENTHETWONODESASSHOWNINEQ12VCSUMMATIONTEXTMJ1VJ/RJSUMMATIONTEXTMI11/RI12JHYOON,HHUH/JOURNALOFMATERIALSPROCESSINGTECHNOLOGY1402003616621619FIG4CONTROLSPHEREFORMAPPINGOFTHESTATEVARIABLESWHEREVJISTHESTATEVARIABLECALCULATEDONTHEORIGINALMESHSYSTEM,ANDRJTHEDISTANCEBETWEENTHENEWNODEANDTHENEIGHBORINGNODES3NUMERICALEXAMPLES31FORMINGANALYSISOFANOILPANWHILEOILPANSAREUSUALLYFABRICATEDWITHATWOSTAGEPROCESSINTHEPRESSSHOP,THEPRESENTANALYSISISCARRIEDOUTWITHASINGLESTAGEPROCESSASSHOWNINFIG5THATDESCRIBESTHEPUNCHANDDIESETTHEREGULARIZATIONMETHODCANBEAPPLIEDTOTHEFINITEELEMENTMESHSYSTEMWHENEVERNEEDEDFORENHANCEMENTOFTHECOMPUTATIONEFFICIENCYINTHISEXAMPLEFORDEMONSTRATION,THEMETHODISAPPLIEDTOTHEANALYSISOFOILPANFORMINGATTWOFORMINGINTERVALSFORREGULARIZATIONOFDISTORTEDMESHESASDIRECTEDINFIG6FIG7EXPLAINSTHEPROCEDUREOFTHEREGULARIZATIONMETHODFIG7ASHOWSTHEDEFORMEDSHAPEATTHEPUNCHSTROKEOFFIG5PUNCHANDDIESETFOROILPANFORMING60FORMING6080FORMING100FORMINGREGULARIZATIONREGULARIZATIONFIG6APPLYINGTHEREGULARIZATIONMETHODTOTHEFORMINGANALYSISFIG7PROCEDUREOFREGULARIZATIONASEARCHINGDISTORTEDELEMENTSBCONSTRUCTINGPATCHESFORDISTORTEDELEMENTSCREGULARIZATIONOFDISTORTEDELEMENTS60ANDTHREEPARTSOFMESHDISTORTIONBYTHEFORMINGPROCEDUREITINDICATESTHATTHENUMBEROFPATCHESTOBECONSTRUCTEDIS3DISTORTEDMESHESARESELECTEDACCORDINGTOTHETWOGEOMETRICALCRITERIAFORMESHDISTORTIONANDTHENTHEPATCHESOFARECTANGULARSHAPEAREFORMEDTOINCLUDEALLDISTORTEDELEMENTSASSHOWNINFIG7BFINALLY,THEELEMENTSINTHEPATCHESAREREGULARIZEDASSHOWNINFIG7CINORDERTOEVALUATETHEDEGREEOFIMPROVEMENTINTHEELEMENTQUALITYAFTERAPPLYINGTHEREGULARIZATIONMETHOD,THEVALUEOFLDFORTHEREGULARIZEDMESHSYSTEMISCOMPAREDTHEONEFORTHEORIGINALMESHSYSTEMTHELDVALUESFORTHEREGULARIZEDMESHSYSTEMHAVEUNIFORMDISTRIBUTIONTHROUGHOUTTHEELEMENTSWHILETHOSEFORTHEORIGINALMESHSYSTEMHAVEWIDEVARIATIONASSHOWNINFIG8ITMEANSTHATTHEQUALITYOFTHEREGULARIZEDMESHSYSTEMISENHANCEDWITHTHESAMELEVELDISTORTIONCONSEQUENTLY,EXPLICITFINITEELEMENTCOMPUTATIONWITHTHEREGULARIZEDMESHSYSTEMCANBEPRECEDEDWITHALARGERINCREMENTALTIMESTEPASSHOWNINFIG9INTHISANALYSISOFOILPANFORMING,THECOMPUTINGTIMEWITHTHEREGULARIZEDMESHSYSTEMISREDUCEDABOUT12EVENAFTERTWOTIMESOFREGULARIZATIONTHEAMOUNTOFREDUCTIONINTHECOMPUTINGTIMECANBEINCREASEDWITHMOREFREQUENTREGULARIZATION32CRASHANALYSISOFAFRONTSIDEMEMBERTHECRASHANALYSISISUSUALLYCARRIEDOUTWITHOUTCONSIDERINGTHEFORMINGEFFECTANDADOPTSTHEMESHSYSTEMAPARTFORMTHEFORMINGANALYSISINCASETHATTHEFORMINGEFFECTISCONSIDEREDTOIMPROVETHEACCURACYANDRELIABILITYOFTHEANALYSISRESULTS,THEMESHSYSTEMFORTHEFORMINGANALYSISCOULDBEDIRECTLYUSEDINTHECRASHANALYSISFORTHEEFFICIENCYOFTHEANALYSISTHEMESHSYSTEMAFTERTHEFORMINGANALYSIS,620JHYOON,HHUH/JOURNALOFMATERIALSPROCESSINGTECHNOLOGY1402003616621204060801001201401601800002040608LEVELOFDISTORTIONELEMENTNUMBER204060801001201401601800002040608LEVELOFDISTORTIONELEMENTNUMBERABFIG8COMPARISONOFLDAORIGINALMESHBREGULARIZEDMESHHOWEVER,HASTOOMANYDISTORTEDMESHESDUETOSEVEREDEFORMATIONANDDISTORTIONBEUSEDDIRECTLYINTHECRASHANALYSISWITHOUTREMESHINGONEREMEDYISTOCONSTRUCTANEWMESHSYSTEMANDTHEOTHERISTOMODIFYTHEMESHSYSTEMAFTERTHEFORMINGANALYSISTHELATTERCANBEVERYEFFICIENTIFTHEREMESHINGPROCESSISSUCCESSFULLYCARRIEDOUTFORANEFFICIENTREMESHINGPROCESS,THEREGULARIZATIONMETHODCANBEAPPLIEDTOTRANSFORMDISTORTEDMESHESINTOANEARSQUAREINTHISEXAMPLE,APARTOFTHEFRONTSIDEMEMBER,NAMEDFRONTREINFORCEMENT,INFIG10ISSELECTEDFORTHECRASHANALYSISTHEIRREGULARFINITEELEMENTSINTHELOCALDISTORTEDREGIONOFTHEMEMBERAFTERFORMINGANALYSISAREMODIFIEDTOAREGULARELEMENTUSINGTHEREGULARIZATIONMETHODASSHOWNINFIG11ANDTHENTHEREGULARIZEDMESHSYSTEMISUSEDINTHECRASHANALYSISTHEANALYSISCONDITIONISDEPICTEDINFIG12THECRASHANALYSISWITHTHEREGULARIZEDMESHSYSTEMCOULDBECARRIEDOUTWITHALAGERTIMESTEPASSHOWNINFIG13WITHOUTSACRIFICINGTHEACCURACYOFTHEANALYSISRESULTTHECOMPUTINGTIMEFORTHECRASHANALYSISWASREDUCEDBY40OFTHETIMEWITHTHEORIGINALMESHSYSTEMTHEANALYSISRESULTWASENCOURAGINGINBOTHTHECOMPUTINGTIMEANDTHEACCURACYOFTHECOMPUTATIONRESULTANDPROVEDTHATTHEREGULARIZEDMESHSYSTEMCOULDBEUSEDEFFECTIVELYTOENHANCETHEEFFICIENCYOFTHENUMERICALANALYSISFIG9COMPARISONOFTHETIMESTEPSIZEWITHRESPECTTOTHEPUNCHSTROKEFIG10ASSEMBLYOFAFRONTSIDEMEMBERFIG11REGULARIZATIONOFAFRONTREINFORCEMENTFIG12CONDITIONOFTHEIMPACTANALYSISJHYOON,HHUH/JOURNALOFMATERIALSPROCESSINGTECHNOLOGY1402003616621621FIG13COMPARISONOFTHETIMESTEPSIZEBETWEENTHEORIGINALANDREGULARIZEDMESHSYSTEM4CONCLUSIONAMESHREGULARIZATIONMETHODISNEWLYPROPOSEDINORDERTOENHANCETHEEFFICIENCYOFFINITEELEMENTANALYSESOFSHEETMETALFORMINGMESHESINTHESHEETMETALFORMINGANALYSISAREDISTOR