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ANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.LagrangeANSYScontact-Penaltyvs.Lagrange-HowtomakeitconvergeErkeWangCAD-FEMGmbH.GermanyANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.LagrangeVarietyofalgorithmsANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.LagrangePenaltymeansthatanyviolationofthecontactconditionwillbepunishedbyincreasingthetotalvirtualwork:PurepenaltymethoddAggTTTTNNNNggAugmentedLagrangemethod:dAggdVTTTNNNVT)(ggTheequationcanalsobewritteninFEform:FuGGKT)(ThisistheequationusedinFEAforthepurepenaltymethodwhereisthecontactstiffnessNFTNgTgANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.LagrangePurepenaltymethodThecontactspringwilldeflectanamount,suchthatequilibriumissatisfied:FuGGKT)(Somefiniteamountofpenetration,0,isrequiredmathematicallytomaintainequilibrium.However,physicalcontactingbodiesdonotinterpenetrate(=0).FThereisnooverconstrainingproblemIterativesolversareapplicable–largemodelsaredoable!Theconditionofthestiffnessmatrixcruciallydependsonthecontactstiffnessitself.GGKKTThereisnoadditionalDOF.FuGGKT)(NNFTNgTgANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.LagrangePurepenaltymethodSomefiniteamountofpenetration,0,isrequiredmathematicallytomaintainequilibrium.However,physicalcontactingbodiesdonotinterpenetrate(=0).Differenceind:0.281e-3/0.284e-7=1e4Differenceinstress:(3525-3501)/3525=0.7%FKN=1PENEStressFKN=1e4PENEStressistheResultfromFKNandtheequilibriumanalysis.Pressure=*=Stress100-timesDifferenceinFKNleadsto100-timesDifferenceinbutleadstoonlyabout1%DifferenceinContactpressureandtherelatedstress.ANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.LagrangePurepenaltymethodSomefiniteamountofpenetration,0,isrequiredmathematicallytomaintainequilibrium.However,physicalcontactingbodiesdonotinterpenetrate(=0).Tip:Aslongasthepenetrationdoesnotleadstothechangeofthecontactregion,ThepenetrationwillnotinfluencethecontactpressureandStressunderneaththecontactelementCaution:Forpre-tensionproblem,uselargeFKN1,Becausethesmallpenetrationwillstronglyinfluencethepre-tensionforce.ANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.LagrangePurepenaltymethodTheconditionofthestiffnessmatrixcruciallydependsonthecontactstiffnessitself.IterationnFIterationn+1FFContactFIterationn+2Ifthecontactstiffnessistoolarge,itwillcauseconvergencedifficulties.Themodelcanoscillate,withcontactingsurfacesbouncingoffofeachother.FKN=1FKN=0.01ANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.LagrangePurepenaltymethodTheconditionofthestiffnessmatrixcruciallydependsonthecontactstiffnessitself.Thisproblemisalmostsolvedsince8.1,withautomaticcontactstiffnessadjustment.KEYOPT(10)=2KEYOPT(10)=0KEYOPT(10)=2205iterations84iterationsANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.LagrangePurepenaltymethodTheconditionofthestiffnessmatrixcruciallydependsonthecontactstiffnessitself.Forbendingdominantproblem,youshouldstillusethe0.01forthestartingFKNandcombinewithKEYOPT(10)=2FKN=0.01,KEY(10)=0FKN=1:KEY(10)=0DivergenceFKN=0.01,KEY(10)=2203iterations43iterationsANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.LagrangePurepenaltymethodTheconditionofthestiffnessmatrixcruciallydependsonthecontactstiffnessitself.Tip:AlwaysuseKEYOPT(10)=2ForbendingproblemuseFKN=0.01andKEYOPT(10)=2ForbulkyproblemuseFKN=1andKEYOPT(10)=2Caution:Forpre-tensionproblem,uselargeFKN1.Becausethesmallpenetrationwillstronglyinfluencethepre-tensionforce.ANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.LagrangePurepenaltymethodThereisnoadditionalDOF.ThereisnooverconstrainingproblemIterativesolversareapplicable–largemodelsaredoable!Tip:AlwaysusePenaltyif:•Symmetriccontactorself-contactisused.•Multiplepartssharethesamecontactzone•3Dlargemodel(300.000DOFs),usePCGsolver.ANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.Lagrange•AnyviolationofthecontactconditionwillbefurnishedwithaLagrangemultiplier.PureLagrangemultipliersmethoddAgdVTNNVT)(gλTContactconstraintcondition:000NNNNggEnsurenopenetrationEnsurecompressivecontactforce/pressureNocontact,gapisnonzeroContact,contactforceisnonzero0N0Ng0=0gFλuGGKTTheequationislinear,incaseoflinearelasticandNode-to-Nodecontact.Otherwise,theequationisnonlinearandaniterativemethodisusedtosolvetheequation.UsuallytheNewton-Methodisused.Forlinearelasticproblems:ANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.LagrangePureLagrangemultipliersmethod0=0gFλuGGKTLagrangemultipliersareadditionalDOFstheFEmodelisgettinglarge.N+GZeromaindiagonalsinsystemmatrixNoiterativesolverisapplicable.ForsymmetriccontactoradditionalCP/CE,andboundaryconditions,theequationsystemmightbeover-constrainedSensitivetochatteringofthevariationofcontactstatusNoneedtodefinecontactstiffnessAccuracy-constraintissatisfiedexactly,therearenomatrixconditioningproblemsANSYS,Inc.Proprietary©2004ANSYS,Inc.Penaltyvs.LagrangePureLagrangemultipliersmethodLagrangemultipliersareadditionalDOFstheFEmodelisgettinglarge.Tip:AlwaysuseLagrangemultipliermethodif:•Themodelis2D.•3Dnonlinearmaterialproblemwith100.000DofsANSYS,Inc.Proprieta