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JournalofMaterialsProcessingTechnology1402003616–621EfficiencyenhancementinsheetmetalforminganalysiswithameshregularizationmethodJ.H.Yoon,H.Huh∗DepartmentofMechanicalEngineering,KoreaAdvancedInstituteofScienceandTechnologyScienceTown,Daejeon305701,SouthKoreaAbstractThispapernewlyproposesameshregularizationmethodfortheenhancementoftheefficiencyinsheetmetalforminganalysis.Theregularizationmethodsearchesfordistortedelementswithappropriatesearchingcriteriaandconstructspatchesincludingtheelementstobemodified.Eachpatchisthenextendedtoathreedimensionalsurfaceinordertoobtaintheinformationofthecontinuouscoordinates.Inconstructingthesurfaceenclosingeachpatch,NURBSnonuniformrationalBsplinesurfaceisemployedtodescribeathreedimensionalfreesurface.Onthebasisoftheconstructedsurface,eachnodeisproperlyarrangedtoformunitelementsascloseastoasquare.Thestatevariablescalculatedfromitsoriginalmeshgeometryaremappedintothenewmeshgeometryforthenextstageorincrementalstepofaforminganalysis.Theanalysisresultswiththeproposedmethodarecomparedtotheresultsfromthedirectforminganalysiswithoutmeshregularizationinordertoconfirmthevalidityofthemethod.©2003ElsevierB.V.Allrightsreserved.KeywordsMeshregularizationDistortedelementNURBSPatchFiniteelementanalysis1.IntroductionNumericalsimulationofsheetmetalformingprocessesenjoysitsprosperitywithaburstofdevelopmentofthecomputersandtherelatednumericaltechniques.Thenumericalanalysishasextendeditscapabilitiesforsheetmetalformingofcomplicatedgeometrymodelsandmultistageforming.Inthecaseofacomplicatedgeometrymodel,however,severelocaldeformationoccurstoinducetheincreaseofthecomputingtimeanddeterioratetheconvergenceoftheanalysis.Distortionandseveredeformationofthemeshgeometryhasaneffectonthequalityofforminganalysisresultsespeciallyinthecaseofmultistageforminganalysiswhenthemeshgeometryformedbytheforminganalysisatthefirststageisusedfortheforminganalysisatthenextstage.Thisillbehaviorofthedistortedmeshcanbeavoidedbythereconstructionofthemeshsystemsuchasthetotalortheadaptiveremeshingtechniques.Theadaptiveremeshingtechniqueisknowntobeanefficientmethodtoreducedistortionofelementduringthesimulation,butitstillneedstremendouscomputingandputsrestrictionsamongsubdividedelements.∗Correspondingauthor.Tel.82428693222fax82428693210.Emailaddresshhuhkaist.ac.krH.Huh.Effectivemethodstoconstructameshsystemhavebeenproposedbymanyresearchers.Typicalmethodscouldbermethod1inwhichnodalpointsareproperlyrearrangedwithoutthechangeofthetotaldegreesoffreedomofthemeshsystem,hmethod2inwhichthenumberofmeshesisincreasedwithelementsofthesamedegreesoffreedom,andpmethod3inwhichthetotaldegreesoffreedomofthemeshsystemisincreasedtoenhancetheaccuracyofsolutions.SluiterandHansen4andTalbertandParkinson5constructedtheanalysisdomainasacontinuousloopandcreatedelementsinsubloopsdividedfromthemainloop.Lo6constructedtriangularelementsinthewholedomainandthenconstructedrectangularelementsbycombiningadjacenttriangularelements.Inthispaper,ameshregularizationmethodisnewlyproposedinordertoenhancetheefficiencyoffiniteelementanalysesofsheetmetalforming.Themeshregularizationmethodautomaticallyfindsoutdistortedelementswithsearchingcriteriaproposedandcomposespatchestobemodified.Eachpatchisthenextendedtothreedimensionalsurfacesinordertoobtaintheinformationofthecontinuouscoordinatesonthethreedimensionalsurface.ThesurfaceenclosingeachpatchisdescribedasathreedimensionalfreesurfacewiththeuseofNURBSnonuniformrationalBspline.Onthebasisoftheconstructedsurface,eachnodeisproperlyarrangedtocomposeregularelementsclosetoasquare.Thestatevariablescalculatedfromitsoriginalmesh09240136/–seefrontmatter©2003ElsevierB.V.Allrightsreserved.doi10.1016/S092401360300801XJ.H.Yoon,H.Huh/JournalofMaterialsProcessingTechnology1402003616–621617geometryaremappedintothenewmeshgeometryfortheforminganalysisatthenextstage.Numericalresultsconfirmtheefficiencyoftheproposedmethodandtheaccuracyoftheresult.Itisalsonotedthatthepresentmethodiseffectiveinthecrashanalysesofsheetmetalmembersobtainedfromtheformingsimulation.2.RegularizationofthedistortedelementTheregularizationproceduretomodifydistortedelementsisintroducedinordertoenhancetheefficiencyofanalysisforthenextfiniteelementcalculation.Thedistortedelementsareselectedwithappropriatesearchingcriteriaandallocatedtoseveralpatchesforregularization.ThepatchesareextendedtothreedimensionalsurfaceswiththeuseofNURBSforfullinformationofthecontinuouscoordinatesonthethreedimensionalsurface.Onobtainingthenewcoordinatesofeachnode,thedistortedelementsareregularizedtoaregularelementthatisclosetoasquare.2.1.ThecriterionofmeshdistortionDistortedmeshesareselectedwiththetwogeometricalcriteriaoneistheinnerangleandtheotheristheaspectratiooftheelementside.2.1.1.InnerangleTheinnerangleofaquadrilateralelementshouldbeclosetotherightangleforgoodresultsfromfiniteelementcalculation.Zhuetal.7definedthereasonableelementwhenthefourinneranglesareformedwiththeangleof90◦±45◦whileLoandLee8proposedtheinnerangleof90◦±52.5◦asthesamecriterion.ThecriterionofmeshdistortionfortheinnerangleisdeterminedbyconstitutingEq.1.AmeshisregardedasdistortedwhenEq.1islessthanπ/3orδθimaxinEq.39isgreaterthanπ/6.ThecriterionisratherstrictinordertoavoidthegeometricallimitationincaseofapplyingtheregularizationmethodinconfinedregionsvectorfQδθ1ˆe1δθ2ˆe2δθ3ˆe3δθ4ˆe41||vectorfQ||radicaltpradicalvertexradicalvertexradicalbt4summationdisplayi1δθi22δθivextendsinglevextendsinglevextendsingle12π−θivextendsinglevextendsinglevextendsingle32.1.2.AspectratiooftheelementTheidealaspectratiooftheelementsideshouldbeunitywhenthefoursidesofanelementhavethesamelength.TheaspectratioisdefinedasEq.4andthenthedistortionisdefinedwhenitislessthan5thatcouldbemuchlessforastrictcriterionmax{r12,r23,r34,r41}min{r12,r23,r34,r41}4whererijisthelengthofeachelementside.Fig.1.Processforconstructionofapatch.2.2.Domainconstruction2.2.1.ConstructionofthepatchDistortedelementsselectedbythecriteriaofmeshdistortionaredistributedinvariousregionsaccordingtothecomplexityoftheshapeofformedgeometry.Theseelementsareallocatedtopatchesconstructedfortheefficiencyofthealgorithm.TheshapeofpatchesismadeupforrectangularshapesincludingalldistortedelementsforexpandingtheregionofregularizationandapplyingtoNURBSsurfaceexplainedinnextsection.ThisprocedureisshowninFig.1.Whenholesandedgesarelocatedbetweendistortedelements,theregionsarefilleduptomakepatchesarectangularshape.ThepatchisthenmappedtoathreedimensionalfreesurfacebyusingNURBSsurface.Theprocedureisimportanttoobtainentireinformationofthecontinuouscoordinatesonthethreedimensionalsurface.NURBSsurfacecandescribethecomplexshapequicklybyusinglessdatapointsanddoesnotchangetheentiredomaindataduetothelocalchange.2.2.2.NURBSsurfaceNURBSsurfaceisgenerallyexpressedbyEq.5astheporderintheudirectionandtheqorderinthevdirection10Su,vsummationtextni0summationtextmj0Ni,puNj,qvwi,jPi,jsummationtextni0summationtextmj0Ni,puNj,qvwi,j5wherePi,jisthecontrolpointsastheu,vdirection,wi,jtheweightfactorandNu,pu,Nj,quthebasisfunctionthatareexpressedbyEq.6Ni,0braceleftBigg1ifui≤u≤ui1,0otherwise,Ni,puu−uiuip−uiNi,puuip1−uuip1−ui1Ni1,p−1u6Inordertomapthenodesfromthepatchesontotheconstructedsurface,anumberofpointsarecreatedfortheircoordinatesontheNURBSsurface.Thelocationofeach618J.H.Yoon,H.Huh/JournalofMaterialsProcessingTechnology1402003616–621Fig.2.Selectingdirectionofdistortedelements.movingnodebyapplyingaregularizationmethodisdeterminedsuchthatthelocationofapointhastheminimumdistancebetweennodesonNURBSsurface.Theinformationonthecoordinatesofthenodalpointstobemovedisstoredtoconstructanewmeshsystem.2.3.RegularizationprocedureTheregularizationmethodiscarriedoutwiththeunitofapatchthatformsarectangularshape.FiniteelementstoberegularizedisselectedbytheorderofFig.2.Eachselectedelementisdividedbytwotriangularelementsandthenthedividedelementismadeofarighttriangularelementbyrelocatingthevertexonthecirclehavingthediameterfromvectorx1tovectorx2asshowninEq.7andFig.3.Whentheprocedureterminates,thesameprocedureisrepeatedintheoppositedirectionvectorx1vectorx22vectorxcen,|vectorx1−vectorx2|2r,vectorxcur−vectorxcenvectorxdir,vectorxnewvectorxdir|vectorxdir|rfactorvectorxcen7ThefinallocationofanoderelocatedbyusingtheregularizationmethodissubstitutedforthelocationofapointonNURBSsurface.Aftertheregularizationprocedureisfinished,asimplesoothingprocedureiscarriedoutbyEq.8fortheroughregiongeneratedduringtheFig.3.Regularizationschemebymovingnodes.procedurePNsummationtextNi1AiCisummationtextNi1Ai8wherePNisthecoordinateofanewnode,AitheareasofadjacentelementsandCithecentroidoftheadjacentelements.2.4.LevelofdistortionAsadistortionfactor,levelofdistortionLDisnewlyproposed.LDcanbeusedtoevaluatethedegreeofimprovementintheelementqualityLDAB9whereAsummationtext4i1|sinθi|4,BtanhparenleftbigkBprimeparenrightbig10Bprimemin{r12,r23,r34,r41}max{r12,r23,r34,r41},ktanh−1βα11LDhasthevaluebetween0and1whenLD1,theelementisanidealelementofasquareandwhenLD0,thequadrilateralelementbecomesatriangularelement.θiarethefourinneranglesofanelement,soAisthefactorfortheinnerangle.BisthefactorfortheaspectratioofelementsidesandisdefinedbythehyperbolictangentfunctioninordertomakeLDlesssensitivetothechangeofB.Forexample,whenthereasonableaspectratiooftheelementsideis14,thevalueofBcanbeadjustedbyapplyingα0.25andβ0.6suchthattheslopeofthefunctionBischangedabruptlyaroundthevalueofBprime0.25.Consequently,thevalueofLDdecreasesrapidlywhentheaspectratioBprimeislessthan0.25whilethevalueofLDincreasesslowlywhentheBprimeisgreaterthan0.25.ThisschemecanregulatetheinnerangleandtheaspectratiotohavetheequaleffectontheLD.2.5.MappingofthestatevariablesWhentheregularizedmeshsystemisusedforthenextcalculationoftheforminganalysisorthestructuralanalysis,mappingofthestatevariablesisneededformoreaccurateanalysisconsideringthepreviousforminghistory.Themappingprocedureistomapthecalculatedstatevariablesintheoriginalmeshsystemontotheregularizedmeshsystem.AsshowninFig.4,asphereisconstructedsurroundinganewnodesuchthatthestatevariablesofnodesinthespherehaveaneffectonthestatevariablesofthenewnode.ThestatevariablesofthenewnodearedeterminedfromthestatevariablesoftheneighboringnodesinthespherebyimposingtheweightingfactorinverselyproportionaltothedistancebetweenthetwonodesasshowninEq.12Vcsummationtextmj1Vj/rjsummationtextmi11/ri12J.H.Yoon,H.Huh/JournalofMaterialsProcessingTechnology1402003616–621619Fig.4.Controlsphereformappingofthestatevariables.whereVjisthestatevariablecalculatedontheoriginalmeshsystem,andrjthedistancebetweenthenewnodeandtheneighboringnodes.3.Numericalexamples3.1.ForminganalysisofanoilpanWhileoilpansareusuallyfabricatedwithatwostageprocessinthepressshop,thepresentanalysisiscarriedoutwithasinglestageprocessasshowninFig.5thatdescribesthepunchanddieset.Theregularizationmethodcanbeappliedtothefiniteelementmeshsystemwheneverneededforenhancementofthecomputationefficiency.Inthisexamplefordemonstration,themethodisappliedtotheanalysisofoilpanformingattwoformingintervalsforregularizationofdistortedmeshesasdirectedinFig.6.Fig.7explainstheprocedureoftheregularizationmethod.Fig.7ashowsthedeformedshapeatthepunchstrokeofFig.5.Punchanddiesetforoilpanforming.60forming6080forming100formingRegularizationRegularizationFig.6.Applyingtheregularizationmethodtotheforminganalysis.Fig.7.Procedureofregularizationasearchingdistortedelementsbconstructingpatchesfordistortedelementscregularizationofdistortedelements.60andthreepartsofmeshdistortionbytheformingprocedure.Itindicatesthatthenumberofpatchestobeconstructedis3.Distortedmeshesareselectedaccordingtothetwogeometricalcriteriaformeshdistortion.AndthenthepatchesofarectangularshapeareformedtoincludealldistortedelementsasshowninFig.7b.Finally,theelementsinthepatchesareregularizedasshowninFig.7c.Inordertoevaluatethedegreeofimprovementintheelementqualityafterapplyingtheregularizationmethod,thevalueofLDfortheregularizedmeshsystemiscomparedtheonefortheoriginalmeshsystem.TheLDvaluesfortheregularizedmeshsystemhaveuniformdistributionthroughouttheelementswhilethosefortheoriginalmeshsystemhavewidevariationasshowninFig.8.Itmeansthatthequalityoftheregularizedmeshsystemisenhancedwiththesameleveldistortion.Consequently,explicitfiniteelementcomputationwiththeregularizedmeshsystemcanbeprecededwithalargerincrementaltimestepasshowninFig.9.Inthisanalysisofoilpanforming,thecomputingtimewiththeregularizedmeshsystemisreducedabout12evenaftertwotimesofregularization.Theamountofreductioninthecomputingtimecanbeincreasedwithmorefrequentregularization.3.2.CrashanalysisofafrontsidememberThecrashanalysisisusuallycarriedoutwithoutconsideringtheformingeffectandadoptsthemeshsystemapartformtheforminganalysis.Incasethattheformingeffectisconsideredtoimprovetheaccuracyandreliabilityoftheanalysisresults,themeshsystemfortheforminganalysiscouldbedirectlyusedinthecrashanalysisfortheefficiencyoftheanalysis.Themeshsystemaftertheforminganalysis,
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