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JOURNALOFMATERIALSPROCESSINGTECHNOLOGY1402003616–621EFFICIENCYENHANCEMENTINSHEETMETALFORMINGANALYSISWITHAMESHREGULARIZATIONMETHODJHYOON,HHUH∗DEPARTMENTOFMECHANICALENGINEERING,KOREAADVANCEDINSTITUTEOFSCIENCEANDTECHNOLOGYSCIENCETOWN,DAEJEON305701,SOUTHKOREAABSTRACTTHISPAPERNEWLYPROPOSESAMESHREGULARIZATIONMETHODFORTHEENHANCEMENTOFTHEEFFICIENCYINSHEETMETALFORMINGANALYSISTHEREGULARIZATIONMETHODSEARCHESFORDISTORTEDELEMENTSWITHAPPROPRIATESEARCHINGCRITERIAANDCONSTRUCTSPATCHESINCLUDINGTHEELEMENTSTOBEMODIFIEDEACHPATCHISTHENEXTENDEDTOATHREEDIMENSIONALSURFACEINORDERTOOBTAINTHEINFORMATIONOFTHECONTINUOUSCOORDINATESINCONSTRUCTINGTHESURFACEENCLOSINGEACHPATCH,NURBSNONUNIFORMRATIONALBSPLINESURFACEISEMPLOYEDTODESCRIBEATHREEDIMENSIONALFREESURFACEONTHEBASISOFTHECONSTRUCTEDSURFACE,EACHNODEISPROPERLYARRANGEDTOFORMUNITELEMENTSASCLOSEASTOASQUARETHESTATEVARIABLESCALCULATEDFROMITSORIGINALMESHGEOMETRYAREMAPPEDINTOTHENEWMESHGEOMETRYFORTHENEXTSTAGEORINCREMENTALSTEPOFAFORMINGANALYSISTHEANALYSISRESULTSWITHTHEPROPOSEDMETHODARECOMPAREDTOTHERESULTSFROMTHEDIRECTFORMINGANALYSISWITHOUTMESHREGULARIZATIONINORDERTOCONFIRMTHEVALIDITYOFTHEMETHOD2003ELSEVIERBVALLRIGHTSRESERVEDKEYWORDSMESHREGULARIZATION;DISTORTEDELEMENT;NURBS;PATCH;FINITEELEMENTANALYSIS1INTRODUCTIONNUMERICALSIMULATIONOFSHEETMETALFORMINGPROCESSESENJOYSITSPROSPERITYWITHABURSTOFDEVELOPMENTOFTHECOMPUTERSANDTHERELATEDNUMERICALTECHNIQUESTHENUMERICALANALYSISHASEXTENDEDITSCAPABILITIESFORSHEETMETALFORMINGOFCOMPLICATEDGEOMETRYMODELSANDMULTISTAGEFORMINGINTHECASEOFACOMPLICATEDGEOMETRYMODEL,HOWEVER,SEVERELOCALDEFORMATIONOCCURSTOINDUCETHEINCREASEOFTHECOMPUTINGTIMEANDDETERIORATETHECONVERGENCEOFTHEANALYSISDISTORTIONANDSEVEREDEFORMATIONOFTHEMESHGEOMETRYHASANEFFECTONTHEQUALITYOFFORMINGANALYSISRESULTSESPECIALLYINTHECASEOFMULTISTAGEFORMINGANALYSISWHENTHEMESHGEOMETRYFORMEDBYTHEFORMINGANALYSISATTHEFIRSTSTAGEISUSEDFORTHEFORMINGANALYSISATTHENEXTSTAGETHISILLBEHAVIOROFTHEDISTORTEDMESHCANBEAVOIDEDBYTHERECONSTRUCTIONOFTHEMESHSYSTEMSUCHASTHETOTALORTHEADAPTIVEREMESHINGTECHNIQUESTHEADAPTIVEREMESHINGTECHNIQUEISKNOWNTOBEANEFFICIENTMETHODTOREDUCEDISTORTIONOFELEMENTDURINGTHESIMULATION,BUTITSTILLNEEDSTREMENDOUSCOMPUTINGANDPUTSRESTRICTIONSAMONGSUBDIVIDEDELEMENTS∗CORRESPONDINGAUTHORTEL82428693222;FAX82428693210EMAILADDRESSHHUHKAISTACKRHHUHEFFECTIVEMETHODSTOCONSTRUCTAMESHSYSTEMHAVEBEENPROPOSEDBYMANYRESEARCHERSTYPICALMETHODSCOULDBERMETHOD1INWHICHNODALPOINTSAREPROPERLYREARRANGEDWITHOUTTHECHANGEOFTHETOTALDEGREESOFFREEDOMOFTHEMESHSYSTEM,HMETHOD2INWHICHTHENUMBEROFMESHESISINCREASEDWITHELEMENTSOFTHESAMEDEGREESOFFREEDOM,ANDPMETHOD3INWHICHTHETOTALDEGREESOFFREEDOMOFTHEMESHSYSTEMISINCREASEDTOENHANCETHEACCURACYOFSOLUTIONSSLUITERANDHANSEN4ANDTALBERTANDPARKINSON5CONSTRUCTEDTHEANALYSISDOMAINASACONTINUOUSLOOPANDCREATEDELEMENTSINSUBLOOPSDIVIDEDFROMTHEMAINLOOPLO6CONSTRUCTEDTRIANGULARELEMENTSINTHEWHOLEDOMAINANDTHENCONSTRUCTEDRECTANGULARELEMENTSBYCOMBININGADJACENTTRIANGULARELEMENTSINTHISPAPER,AMESHREGULARIZATIONMETHODISNEWLYPROPOSEDINORDERTOENHANCETHEEFFICIENCYOFFINITEELEMENTANALYSESOFSHEETMETALFORMINGTHEMESHREGULARIZATIONMETHODAUTOMATICALLYFINDSOUTDISTORTEDELEMENTSWITHSEARCHINGCRITERIAPROPOSEDANDCOMPOSESPATCHESTOBEMODIFIEDEACHPATCHISTHENEXTENDEDTOTHREEDIMENSIONALSURFACESINORDERTOOBTAINTHEINFORMATIONOFTHECONTINUOUSCOORDINATESONTHETHREEDIMENSIONALSURFACETHESURFACEENCLOSINGEACHPATCHISDESCRIBEDASATHREEDIMENSIONALFREESURFACEWITHTHEUSEOFNURBSNONUNIFORMRATIONALBSPLINEONTHEBASISOFTHECONSTRUCTEDSURFACE,EACHNODEISPROPERLYARRANGEDTOCOMPOSEREGULARELEMENTSCLOSETOASQUARETHESTATEVARIABLESCALCULATEDFROMITSORIGINALMESH09240136/–SEEFRONTMATTER2003ELSEVIERBVALLRIGHTSRESERVEDDOI101016/S092401360300801XJHYOON,HHUH/JOURNALOFMATERIALSPROCESSINGTECHNOLOGY1402003616–621617GEOMETRYAREMAPPEDINTOTHENEWMESHGEOMETRYFORTHEFORMINGANALYSISATTHENEXTSTAGENUMERICALRESULTSCONFIRMTHEEFFICIENCYOFTHEPROPOSEDMETHODANDTHEACCURACYOFTHERESULTITISALSONOTEDTHATTHEPRESENTMETHODISEFFECTIVEINTHECRASHANALYSESOFSHEETMETALMEMBERSOBTAINEDFROMTHEFORMINGSIMULATION2REGULARIZATIONOFTHEDISTORTEDELEMENTTHEREGULARIZATIONPROCEDURETOMODIFYDISTORTEDELEMENTSISINTRODUCEDINORDERTOENHANCETHEEFFICIENCYOFANALYSISFORTHENEXTFINITEELEMENTCALCULATIONTHEDISTORTEDELEMENTSARESELECTEDWITHAPPROPRIATESEARCHINGCRITERIAANDALLOCATEDTOSEVERALPATCHESFORREGULARIZATIONTHEPATCHESAREEXTENDEDTOTHREEDIMENSIONALSURFACESWITHTHEUSEOFNURBSFORFULLINFORMATIONOFTHECONTINUOUSCOORDINATESONTHETHREEDIMENSIONALSURFACEONOBTAININGTHENEWCOORDINATESOFEACHNODE,THEDISTORTEDELEMENTSAREREGULARIZEDTOAREGULARELEMENTTHATISCLOSETOASQUARE21THECRITERIONOFMESHDISTORTIONDISTORTEDMESHESARESELECTEDWITHTHETWOGEOMETRICALCRITERIAONEISTHEINNERANGLE;ANDTHEOTHERISTHEASPECTRATIOOFTHEELEMENTSIDE211INNERANGLETHEINNERANGLEOFAQUADRILATERALELEMENTSHOULDBECLOSETOTHERIGHTANGLEFORGOODRESULTSFROMFINITEELEMENTCALCULATIONZHUETAL7DEFINEDTHEREASONABLEELEMENTWHENTHEFOURINNERANGLESAREFORMEDWITHTHEANGLEOF90◦45◦WHILELOANDLEE8PROPOSEDTHEINNERANGLEOF90◦525◦ASTHESAMECRITERIONTHECRITERIONOFMESHDISTORTIONFORTHEINNERANGLEISDETERMINEDBYCONSTITUTINGEQ1AMESHISREGARDEDASDISTORTEDWHENEQ1ISLESSTHANΠ/3ORΔΘIMAXINEQ39ISGREATERTHANΠ/6THECRITERIONISRATHERSTRICTINORDERTOAVOIDTHEGEOMETRICALLIMITATIONINCASEOFAPPLYINGTHEREGULARIZATIONMETHODINCONFINEDREGIONSVECTORFQΔΘ1ˆE1ΔΘ2ˆE2ΔΘ3ˆE3ΔΘ4ˆE41||VECTORFQ||RADICALTPRADICALVERTEXRADICALVERTEXRADICALBT4SUMMATIONDISPLAYI1ΔΘI22ΔΘIVEXTENDSINGLEVEXTENDSINGLEVEXTENDSINGLE12Π−ΘIVEXTENDSINGLEVEXTENDSINGLEVEXTENDSINGLE3212ASPECTRATIOOFTHEELEMENTTHEIDEALASPECTRATIOOFTHEELEMENTSIDESHOULDBEUNITYWHENTHEFOURSIDESOFANELEMENTHAVETHESAMELENGTHTHEASPECTRATIOISDEFINEDASEQ4ANDTHENTHEDISTORTIONISDEFINEDWHENITISLESSTHAN5THATCOULDBEMUCHLESSFORASTRICTCRITERIONMAX{R12,R23,R34,R41}MIN{R12,R23,R34,R41}4WHERERIJISTHELENGTHOFEACHELEMENTSIDEFIG1PROCESSFORCONSTRUCTIONOFAPATCH22DOMAINCONSTRUCTION221CONSTRUCTIONOFTHEPATCHDISTORTEDELEMENTSSELECTEDBYTHECRITERIAOFMESHDISTORTIONAREDISTRIBUTEDINVARIOUSREGIONSACCORDINGTOTHECOMPLEXITYOFTHESHAPEOFFORMEDGEOMETRYTHESEELEMENTSAREALLOCATEDTOPATCHESCONSTRUCTEDFORTHEEFFICIENCYOFTHEALGORITHMTHESHAPEOFPATCHESISMADEUPFORRECTANGULARSHAPESINCLUDINGALLDISTORTEDELEMENTSFOREXPANDINGTHEREGIONOFREGULARIZATIONANDAPPLYINGTONURBSSURFACEEXPLAINEDINNEXTSECTIONTHISPROCEDUREISSHOWNINFIG1WHENHOLESANDEDGESARELOCATEDBETWEENDISTORTEDELEMENTS,THEREGIONSAREFILLEDUPTOMAKEPATCHESARECTANGULARSHAPETHEPATCHISTHENMAPPEDTOATHREEDIMENSIONALFREESURFACEBYUSINGNURBSSURFACETHEPROCEDUREISIMPORTANTTOOBTAINENTIREINFORMATIONOFTHECONTINUOUSCOORDINATESONTHETHREEDIMENSIONALSURFACENURBSSURFACECANDESCRIBETHECOMPLEXSHAPEQUICKLYBYUSINGLESSDATAPOINTSANDDOESNOTCHANGETHEENTIREDOMAINDATADUETOTHELOCALCHANGE222NURBSSURFACENURBSSURFACEISGENERALLYEXPRESSEDBYEQ5ASTHEPORDERINTHEUDIRECTIONANDTHEQORDERINTHEVDIRECTION10SU,VSUMMATIONTEXTNI0SUMMATIONTEXTMJ0NI,PUNJ,QVWI,JPI,JSUMMATIONTEXTNI0SUMMATIONTEXTMJ0NI,PUNJ,QVWI,J5WHEREPI,JISTHECONTROLPOINTSASTHEU,VDIRECTION,WI,JTHEWEIGHTFACTORANDNU,PU,NJ,QUTHEBASISFUNCTIONTHATAREEXPRESSEDBYEQ6NI,0BRACELEFTBIGG1IFUI≤U≤UI1,0OTHERWISE,NI,PUU−UIUIP−UINI,PUUIP1−UUIP1−UI1NI1,P−1U6INORDERTOMAPTHENODESFROMTHEPATCHESONTOTHECONSTRUCTEDSURFACE,ANUMBEROFPOINTSARECREATEDFORTHEIRCOORDINATESONTHENURBSSURFACETHELOCATIONOFEACH618JHYOON,HHUH/JOURNALOFMATERIALSPROCESSINGTECHNOLOGY1402003616–621FIG2SELECTINGDIRECTIONOFDISTORTEDELEMENTSMOVINGNODEBYAPPLYINGAREGULARIZATIONMETHODISDETERMINEDSUCHTHATTHELOCATIONOFAPOINTHASTHEMINIMUMDISTANCEBETWEENNODESONNURBSSURFACETHEINFORMATIONONTHECOORDINATESOFTHENODALPOINTSTOBEMOVEDISSTOREDTOCONSTRUCTANEWMESHSYSTEM23REGULARIZATIONPROCEDURETHEREGULARIZATIONMETHODISCARRIEDOUTWITHTHEUNITOFAPATCHTHATFORMSARECTANGULARSHAPEFINITEELEMENTSTOBEREGULARIZEDISSELECTEDBYTHEORDEROFFIG2EACHSELECTEDELEMENTISDIVIDEDBYTWOTRIANGULARELEMENTSANDTHENTHEDIVIDEDELEMENTISMADEOFARIGHTTRIANGULARELEMENTBYRELOCATINGTHEVERTEXONTHECIRCLEHAVINGTHEDIAMETERFROMVECTORX1TOVECTORX2ASSHOWNINEQ7ANDFIG3WHENTHEPROCEDURETERMINATES,THESAMEPROCEDUREISREPEATEDINTHEOPPOSITEDIRECTIONVECTORX1VECTORX22VECTORXCEN,|VECTORX1−VECTORX2|2R,VECTORXCUR−VECTORXCENVECTORXDIR,VECTORXNEWVECTORXDIR|VECTORXDIR|RFACTORVECTORXCEN7THEFINALLOCATIONOFANODERELOCATEDBYUSINGTHEREGULARIZATIONMETHODISSUBSTITUTEDFORTHELOCATIONOFAPOINTONNURBSSURFACEAFTERTHEREGULARIZATIONPROCEDUREISFINISHED,ASIMPLESOOTHINGPROCEDUREISCARRIEDOUTBYEQ8FORTHEROUGHREGIONGENERATEDDURINGTHEFIG3REGULARIZATIONSCHEMEBYMOVINGNODESPROCEDUREPNSUMMATIONTEXTNI1AICISUMMATIONTEXTNI1AI8WHEREPNISTHECOORDINATEOFANEWNODE,AITHEAREASOFADJACENTELEMENTSANDCITHECENTROIDOFTHEADJACENTELEMENTS24LEVELOFDISTORTIONASADISTORTIONFACTOR,LEVELOFDISTORTIONLDISNEWLYPROPOSEDLDCANBEUSEDTOEVALUATETHEDEGREEOFIMPROVEMENTINTHEELEMENTQUALITYLDAB9WHEREASUMMATIONTEXT4I1|SINΘI|4,BTANHPARENLEFTBIGKBPRIMEPARENRIGHTBIG10BPRIMEMIN{R12,R23,R34,R41}MAX{R12,R23,R34,R41},KTANH−1ΒΑ11LDHASTHEVALUEBETWEEN0AND1;WHENLD1,THEELEMENTISANIDEALELEMENTOFASQUAREANDWHENLD0,THEQUADRILATERALELEMENTBECOMESATRIANGULARELEMENTΘIARETHEFOURINNERANGLESOFANELEMENT,SOAISTHEFACTORFORTHEINNERANGLEBISTHEFACTORFORTHEASPECTRATIOOFELEMENTSIDESANDISDEFINEDBYTHEHYPERBOLICTANGENTFUNCTIONINORDERTOMAKELDLESSSENSITIVETOTHECHANGEOFBFOREXAMPLE,WHENTHEREASONABLEASPECTRATIOOFTHEELEMENTSIDEIS14,THEVALUEOFBCANBEADJUSTEDBYAPPLYINGΑ025ANDΒ06SUCHTHATTHESLOPEOFTHEFUNCTIONBISCHANGEDABRUPTLYAROUNDTHEVALUEOFBPRIME025CONSEQUENTLY,THEVALUEOFLDDECREASESRAPIDLYWHENTHEASPECTRATIOBPRIMEISLESSTHAN025WHILETHEVALUEOFLDINCREASESSLOWLYWHENTHEBPRIMEISGREATERTHAN025THISSCHEMECANREGULATETHEINNERANGLEANDTHEASPECTRATIOTOHAVETHEEQUALEFFECTONTHELD25MAPPINGOFTHESTATEVARIABLESWHENTHEREGULARIZEDMESHSYSTEMISUSEDFORTHENEXTCALCULATIONOFTHEFORMINGANALYSISORTHESTRUCTURALANALYSIS,MAPPINGOFTHESTATEVARIABLESISNEEDEDFORMOREACCURATEANALYSISCONSIDERINGTHEPREVIOUSFORMINGHISTORYTHEMAPPINGPROCEDUREISTOMAPTHECALCULATEDSTATEVARIABLESINTHEORIGINALMESHSYSTEMONTOTHEREGULARIZEDMESHSYSTEMASSHOWNINFIG4,ASPHEREISCONSTRUCTEDSURROUNDINGANEWNODESUCHTHATTHESTATEVARIABLESOFNODESINTHESPHEREHAVEANEFFECTONTHESTATEVARIABLESOFTHENEWNODETHESTATEVARIABLESOFTHENEWNODEAREDETERMINEDFROMTHESTATEVARIABLESOFTHENEIGHBORINGNODESINTHESPHEREBYIMPOSINGTHEWEIGHTINGFACTORINVERSELYPROPORTIONALTOTHEDISTANCEBETWEENTHETWONODESASSHOW
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