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JournalofMaterialsProcessingTechnology140(2003)616621EfficiencyenhancementinsheetmetalforminganalysiswithameshregularizationmethodJ.H.Yoon,H.HuhDepartmentofMechanicalEngineering,KoreaAdvancedInstituteofScienceandTechnologyScienceTown,Daejeon305-701,SouthKoreaAbstractThispapernewlyproposesameshregularizationmethodfortheenhancementoftheefficiencyinsheetmetalforminganalysis.Theregularizationmethodsearchesfordistortedelementswithappropriatesearchingcriteriaandconstructspatchesincludingtheelementstobemodified.Eachpatchisthenextendedtoathree-dimensionalsurfaceinordertoobtaintheinformationofthecontinuouscoordinates.Inconstructingthesurfaceenclosingeachpatch,NURBS(non-uniformrationalB-spline)surfaceisemployedtodescribeathree-dimensionalfreesurface.Onthebasisoftheconstructedsurface,eachnodeisproperlyarrangedtoformunitelementsascloseastoasquare.Thestatevariablescalculatedfromitsoriginalmeshgeometryaremappedintothenewmeshgeometryforthenextstageorincrementalstepofaforminganalysis.Theanalysisresultswiththeproposedmethodarecomparedtotheresultsfromthedirectforminganalysiswithoutmeshregularizationinordertoconfirmthevalidityofthemethod.2003ElsevierB.V.Allrightsreserved.Keywords:Meshregularization；Distortedelement；NURBS；Patch；Finiteelementanalysis1.IntroductionNumericalsimulationofsheetmetalformingprocessesenjoysitsprosperitywithaburstofdevelopmentofthecom-putersandtherelatednumericaltechniques.Thenumericalanalysishasextendeditscapabilitiesforsheetmetalformingofcomplicatedgeometrymodelsandmulti-stageforming.Inthecaseofacomplicatedgeometrymodel,however,severelocaldeformationoccurstoinducetheincreaseofthecom-putingtimeanddeterioratetheconvergenceoftheanalysis.Distortionandseveredeformationofthemeshgeometryhasaneffectonthequalityofforminganalysisresultsespeciallyinthecaseofmulti-stageforminganalysiswhenthemeshgeometryformedbytheforminganalysisatthefirststageisusedfortheforminganalysisatthenextstage.Thisillbehaviorofthedistortedmeshcanbeavoidedbytherecon-structionofthemeshsystemsuchasthetotalortheadaptiveremeshingtechniques.Theadaptiveremeshingtechniqueisknowntobeanefficientmethodtoreducedistortionofelementduringthesimulation,butitstillneedstremen-douscomputingandputsrestrictionsamongsubdividedelements.Correspondingauthor.Tel.:+82-42-869-3222；fax:+82-42-869-3210.E-mailaddress:hhuhkaist.ac.kr(H.Huh).Effectivemethodstoconstructameshsystemhavebeenproposedbymanyresearchers.Typicalmethodscouldber-method1inwhichnodalpointsareproperlyrearrangedwithoutthechangeofthetotaldegreesoffreedomofthemeshsystem,h-method2inwhichthenumberofmeshesisincreasedwithelementsofthesamedegreesoffreedom,andp-method3inwhichthetotaldegreesoffreedomofthemeshsystemisincreasedtoenhancetheaccuracyofso-lutions.SluiterandHansen4andTalbertandParkinson5constructedtheanalysisdomainasacontinuousloopandcreatedelementsinsub-loopsdividedfromthemainloop.Lo6constructedtriangularelementsinthewholedomainandthenconstructedrectangularelementsbycom-biningadjacenttriangularelements.Inthispaper,ameshregularizationmethodisnewlyproposedinordertoenhancetheefficiencyoffiniteele-mentanalysesofsheetmetalforming.Themeshregular-izationmethodautomaticallyfindsoutdistortedelementswithsearchingcriteriaproposedandcomposespatchestobemodified.Eachpatchisthenextendedtothree-dimensionalsurfacesinordertoobtaintheinformationofthecontinuouscoordinatesonthethree-dimensionalsurface.Thesurfaceenclosingeachpatchisdescribedasathree-dimensionalfreesurfacewiththeuseofNURBS(non-uniformrationalB-spline).Onthebasisoftheconstructedsurface,eachnodeisproperlyarrangedtocomposeregularelementsclosetoasquare.Thestatevariablescalculatedfromitsoriginalmesh0924-0136/$seefrontmatter2003ElsevierB.V.Allrightsreserved.doi:10.1016/S0924-0136(03)00801-XJ.H.Yoon,H.Huh/JournalofMaterialsProcessingTechnology140(2003)616621617geometryaremappedintothenewmeshgeometryfortheforminganalysisatthenextstage.Numericalresultscon-firmtheefficiencyoftheproposedmethodandtheaccuracyoftheresult.Itisalsonotedthatthepresentmethodiseffec-tiveinthecrashanalysesofsheetmetalmembersobtainedfromtheformingsimulation.2.RegularizationofthedistortedelementTheregularizationproceduretomodifydistortedele-mentsisintroducedinordertoenhancetheefficiencyofanalysisforthenextfiniteelementcalculation.Thedistortedelementsareselectedwithappropriatesearchingcriteriaandallocatedtoseveralpatchesforregularization.Thepatchesareextendedtothree-dimensionalsurfaceswiththeuseofNURBSforfullinformationofthecontinuouscoor-dinatesonthethree-dimensionalsurface.Onobtainingthenewcoordinatesofeachnode,thedistortedelementsareregularizedtoaregularelementthatisclosetoasquare.2.1.ThecriterionofmeshdistortionDistortedmeshesareselectedwiththetwogeometricalcriteria:oneistheinnerangle；andtheotheristheaspectratiooftheelementside.2.1.1.InnerangleTheinnerangleofaquadrilateralelementshouldbeclosetotherightangleforgoodresultsfromfiniteelementcalcu-lation.Zhuetal.7definedthereasonableelementwhenthefourinneranglesareformedwiththeangleof9045whileLoandLee8proposedtheinnerangleof9052.5asthesamecriterion.ThecriterionofmeshdistortionfortheinnerangleisdeterminedbyconstitutingEq.(1).AmeshisregardedasdistortedwhenEq.(1)islessthan/3or(i)maxinEq.(3)9isgreaterthan/6.Thecriterionisratherstrictinordertoavoidthegeometricallimitationincaseofapplyingtheregularizationmethodinconfinedregions:vectorfQ=1e1+2e2+3e3+4e4(1)|vectorfQ|=radicaltpradicalvertexradicalvertexradicalbt4summationdisplayi=1(i)2(2)i=vextendsinglevextendsinglevextendsingle12ivextendsinglevextendsinglevextendsingle(3)2.1.2.AspectratiooftheelementTheidealaspectratiooftheelementsideshouldbeunitywhenthefoursidesofanelementhavethesamelength.TheaspectratioisdefinedasEq.(4)andthenthedistortionisdefinedwhenitislessthan5thatcouldbemuchlessforastrictcriterion:maxr12,r23,r34,r41minr12,r23,r34,r41(4)whererijisthelengthofeachelementside.Fig.1.Processforconstructionofapatch.2.2.Domainconstruction2.2.1.ConstructionofthepatchDistortedelementsselectedbythecriteriaofmeshdistor-tionaredistributedinvariousregionsaccordingtothecom-plexityoftheshapeofformedgeometry.Theseelementsareallocatedtopatchesconstructedfortheefficiencyofthealgorithm.Theshapeofpatchesismadeupforrectangu-larshapesincludingalldistortedelementsforexpandingtheregionofregularizationandapplyingtoNURBSsurfaceex-plainedinnextsection.ThisprocedureisshowninFig.1.Whenholesandedgesarelocatedbetweendistortedele-ments,theregionsarefilleduptomakepatchesarectangularshape.Thepatchisthenmappedtoathree-dimensionalfreesur-facebyusingNURBSsurface.Theprocedureisimportanttoobtainentireinformationofthecontinuouscoordinatesonthethree-dimensionalsurface.NURBSsurfacecande-scribethecomplexshapequicklybyusinglessdatapointsanddoesnotchangetheentiredomaindataduetothelocalchange.2.2.2.NURBSsurfaceNURBSsurfaceisgenerallyexpressedbyEq.(5)asthep-orderintheu-directionandtheq-orderinthev-direction10:S(u,v)=summationtextni=0summationtextmj=0Ni,p(u)Nj,q(v)wi,jPi,jsummationtextni=0summationtextmj=0Ni,p(u)Nj,q(v)wi,j(5)wherePi,jisthecontrolpointsastheu-,v-direction,wi,jtheweightfactorandNu,p(u),Nj,q(u)thebasisfunctionthatareexpressedbyEq.(6):Ni,0=braceleftBigg1ifuiuui+1,0otherwise,Ni,p(u)=uuiui+puiNi,p(u)+ui+p+1uui+p+1ui+1Ni+1,p1(u)(6)Inordertomapthenodesfromthepatchesontothecon-structedsurface,anumberofpointsarecreatedfortheircoordinatesontheNURBSsurface.Thelocationofeach618J.H.Yoon,H.Huh/JournalofMaterialsProcessingTechnology140(2003)616621Fig.2.Selectingdirectionofdistortedelements.movingnodebyapplyingaregularizationmethodisdeter-minedsuchthatthelocationofapointhastheminimumdistancebetweennodesonNURBSsurface.Theinforma-tiononthecoordinatesofthenodalpointstobemovedisstoredtoconstructanewmeshsystem.2.3.RegularizationprocedureTheregularizationmethodiscarriedoutwiththeunitofapatchthatformsarectangularshape.FiniteelementstoberegularizedisselectedbytheorderofFig.2.Eachselectedelementisdividedbytwotriangularelementsandthenthedividedelementismadeofarighttriangularelementbyrelocatingthevertexonthecirclehavingthediameterfromvectorx1tovectorx2asshowninEq.(7)andFig.3.Whentheprocedureterminates,thesameprocedureisrepeatedintheoppositedirection:vectorx1+vectorx22=vectorxcen,|vectorx1vectorx2|2=r,vectorxcurvectorxcen=vectorxdir,vectorxnew=vectorxdir|vectorxdir|rfactor+vectorxcen(7)Thefinallocationofanoderelocatedbyusingthereg-ularizationmethodissubstitutedforthelocationofapointonNURBSsurface.Aftertheregularizationproce-dureisfinished,asimplesoothingprocedureiscarriedoutbyEq.(8)fortheroughregiongeneratedduringtheFig.3.Rcedure:PN=summationtextNi=1AiCisummationtextNi=1Ai(8)wherePNisthecoordinateofanewnode,AitheareasofadjacentelementsandCithecentroidoftheadjacentelements.2.4.LevelofdistortionAsadistortionfactor,levelofdistortion(LD)isnewlyproposed.LDcanbeusedtoevaluatethedegreeofimprove-mentintheelementquality:LD=AB(9)whereA=summationtext4i=1|sini|4,B=tanhparenleftbigkBprimeparenrightbig(10)Bprime=minr12,r23,r34,r41maxr12,r23,r34,r41,k=tanh1()(11)LDhasthevaluebetween0and1；whenLD=1,theele-mentisanidealelementofasquareandwhenLD=0,thequadrilateralelementbecomesatriangularelement.iarethefourinneranglesofanelement,soAisthefactorfortheinnerangle.BisthefactorfortheaspectratioofelementsidesandisdefinedbythehyperbolictangentfunctioninordertomakeLDlesssensitivetothechangeofB.Forex-ample,whenthereasonableaspectratiooftheelementsideis1:4,thevalueofBcanbeadjustedbyapplying=0.25and=0.6suchthattheslopeofthefunctionBischangedabruptlyaroundthevalueofBprime=0.25.Consequently,thevalueofLDdecreasesrapidlywhentheaspectratioBprimeislessthan0.25whilethevalueofLDincreasesslowlywhentheBprimeisgreaterthan0.25.Thisschemecanregulatethein-nerangleandtheaspectratiotohavetheequaleffectontheLD.2.5.MappingofthestatevariablesWhentheregularizedmeshsystemisusedforthenextcalculationoftheforminganalysisorthestructuralanaly-sis,mappingofthestatevariablesisneededformoreaccu-rateanalysisconsideringthepreviousforminghistory.Themappingprocedureistomapthecalculatedstatevariablesintheoriginalmeshsystemontotheregularizedmeshsys-tem.AsshowninFig.4,asphereisconstructedsurround-inganewnodesuchthatthestatevariablesofnodesinthespherehaveaneffectonthestatevariablesofthenewnode.ThestatevariablesofthenewnodearedeterminedfromthestatevariablesoftheneighboringnodesinthespherebyimposingtheweightingfactorinverselyproportionaltothedistancebetweenthetwonodesasshowninEq.(12):Vc=summationtextmj=1Vj/rjsummationtextmi=11/ri(12)J.H.Yoon,H.Huh/JournalofMaterialsProcessingTechnology140(2003)616621619Fig.4.Controlsphereformappingofthestatevariables.whereVjisthestatevariablecalculatedontheoriginalmeshsystem,andrjthedistancebetweenthenewnodeandtheneighboringnodes.3.Numericalexamples3.1.ForminganalysisofanoilpanWhileoilpansareusuallyfabricatedwithatwo-stageprocessinthepressshop,thepresentanalysisiscarriedoutwithasingle-stageprocessasshowninFig.5thatdescribesthepunchanddieset.Theregularizationmethodcanbeappliedtothefiniteele-mentmeshsystemwheneverneededforenhancementofthecomputationefficiency.Inthisexamplefordemonstration,themethodisappliedtotheanalysisofoilpanformingattwoformingintervalsforregularizationofdistortedmeshesasdirectedinFig.6.Fig.7explainstheprocedureoftheregularizationmethod.Fig.7(a)showsthedeformedshapeatthepunchstrokeofFig.5.Punchanddiesetforoilpanforming.60%forming6080%forming100%formingRegularizationRegularizationFig.6.Applyingtheregularizationmethodtotheforminganalysis.Fig.7.Procedureofregularization:(a)searchingdistortedelements；(b)constructingpatchesfordistortedelements；(c)regularizationofdistortedelements.60%andthreepartsofmeshdistortionbytheformingpro-cedure.Itindicatesthatthenumberofpatchestobecon-structedis3.Distortedmeshesareselectedaccordingtothetwogeometricalcriteriaformeshdistortio