chpt03-fundamentals for finite element method

1FiniteElementMethodTHEFINITEELEMENTMETHODAPracticalCourseG.R.LiuandS.S.QuekCHAPTER3:2CONTENTSSTRONGANDWEAKFORMSOFGOVERNINGEQUATIONSHAMILTON’SPRINCIPLEFEMPROCEDURE–Domaindiscretization–Displacementinterpolation–FormationofFEequationinlocalcoordinatesystem–Coordinatetransformation–AssemblyofFEequations–Impositionofdisplacementconstraints–SolvingtheFEequationsSTATICANALYSISEIGENVALUEANALYSISTRANSIENTANALYSISREMARKS3STRONGANDWEAKFORMSOFGOVERNINGEQUATIONSSystemequations:strongform,difficulttosolveWeakform:requiresweakercontinuityonthedependentvariables(u,v,winthiscase).Weakformisoftenpreferredforobtaininganapproximatedsolution.Formulationbasedonaweakformleadstoasetofalgebraicsystemequation-FEMFEMcanbeappliedforpracticalproblemswithcomplexgeometryandboundaryconditions.4HAMILTON’SPRINCIPLEOfalltheadmissibletimehistoriesofdisplacementthemostaccuratesolutionmakestheLagrangianfunctionalaminimum.Anadmissibledisplacementmustsatisfy:–Thecompatibilityequations–Theessentialorthekinematicboundaryconditions–Theconditionsatinitial(t1)andfinaltime(t2)5HAMILTON’SPRINCIPLEMathematically021dtLttwhereL=T-P+WfVUUTTVd21VcVΠTVTVddεε21σε21fsTSbTVfSfUVfUWfdd(Kineticenergy)(Potentialenergy)(Workdonebyexternalforces)6FEMPROCEDUREStep1:DomaindiscretizationStep2:DisplacementinterpolationStep3:FormationofFEequationinlocalcoordinatesystemStep4:CoordinatetransformationStep5:AssemblyofFEequationsStep6:ImpositionofdisplacementconstraintsStep7:SolvingtheFEequations7Step1:DomaindiscretizationThesolidbodyisdividedintoNeelementswithproperconnectivity–compatibility.Alltheelementsformtheentiredomainoftheproblemwithoutanyoverlapping-compatibility.Therecanbeofdifferenttypesofelementwithdifferentnumberofnodes.Thedensityofthemeshdependsontheaccuracyrequirementoftheanalysis.Themeshisusuallynotuniform,andfinermeshisoftenusedintheareawherethedisplacementgradientislarger.8Step2:DisplacementinterpolationBasesonlocalcoordinatesystem,thedisplacementwithinelementisinterpolatedusingnodaldisplacements.eiinizyxzyxzyxddNdNU),,(),,(),,(112displacementcomponent1displacementcomponent2displacementcomponentfinfdddnd12displacementsatnode1displacementsatnode2displacementsatnodedendndddd9Step2:DisplacementinterpolationNisamatrixofshapefunctions12(,,)(,,)(,,)(,,)fornode1fornode2fornodedndxyzxyzxyzxyznNNNNfiniiiNNN00000000000021NwhereShapefunctionforeachdisplacementcomponentatanode10DisplacementinterpolationConstructingshapefunctions–Considerconstructingshapefunctionforasingledisplacementcomponent–Approximateintheform1()()()dnhiiiTupxxpxα123={,,,......,}dTnαpT(x)={1,x,x2,x3,x4,...,xp}(1D)11Pascaltriangleofmonomials:2Dxyx2x3x4x5y2y3y4y5x2yx3yx4yx3y2xy2xy3xy4x2y3x2y2Constantterms:1xy1Quadraticterms:3Cubicterms:4Quarticterms:5Quinticterms:6Linearterms:23terms6terms10terms15terms21terms22()(,)1,,,,,,...,,TTppxyxyxyxyxypxp12Pascalpyramidofmonomials:3Dxx2x3x4yy2y3y4xyzxzyzx2yxy2x2zzy2z2xz2yz2xyzz3x3yx3zx2y2x2z2x2yzxy3zy3z2y2xy2zxyz2xz3z4z3y1Constantterm:1Linearterms:3Quadraticterms:6Cubicterms:10Quarticterms:154terms10terms20terms35terms222()(,,)1,,,,,,,,,,...,,,TTpppxyzxyzxyyzzxxyzxyzpxp13Displacementinterpolation–Enforceapproximationtobeequaltothenodaldisplacementsatthenodesdi=pT(xi)i=1,2,3,…,ndorde=Pwhere12=denddddT1T2T()()()dnpxpxPpx,14Displacementinterpolation–Thecoefficientsincanbefoundbye-1αPd–Therefore,uh(x)=N(x)de12111112()()()12()()()()()()()()nTTTTnNNNnNNN----xxxNxpxPpxPpxPpxPxxx15DisplacementinterpolationSufficientrequirementsforFEMshapefunctions1,1,2,,0,,1,2,,dijijdijjnNijijnx1.(Deltafunctionproperty)1()1niiNx2.(Partitionofunityproperty–rigidbodymovement)1()dniiiNxxx3.(Linearfieldreproductionproperty)16Step3:FormationofFEequationsinlocalcoordinatesSinceU=NdeTherefore,e=LUe=LNde=BdeStrainmatrixeTeΠkdd21orwhere(Stiffnessmatrix)eTVeTeeTTeVeTVeVcVcVcΠddBBddBdBdd)(2121εε21VcTVeedBBk17Step3:FormationofFEequationsinlocalcoordinatesSinceU=NdeeUNdoreeTeTdmd21where(Massmatrix)111dd(d)222eeeTTTTTeeeeVVVTVVVUUdNNddNNddeTeVVmNNdeTeVVmNN18Step3:FormationofFEequationsinlocalcoordinateseTesTebTefWFdFdFdsbeFFf(Forcevector)dd(d)(d)eeeeTTTTTTTTfebesebesVSVSWVSVSdNfdNfdNfdNfdeTbbVVFNfdeTssSSFNf19Step3:FormationofFEequationsinlocalcoordinates0d)(21-teTeeeTeeeTettFddkddmd)(dd)dd(TeTeTettdddttteettTeeettTetteeTeeettTeddd21212121dmddmddmddmd--0d)(21--teeeeTettFkddmd0d)2121(21-teTeeeTeeeTettFddkddmdeeeeefdmdkFEEquation(Hamilton’sprinciple)20Step4:Coordinatetransformationeeeefdmkdxyx'y'y'x'LocalcoordinatesystemsGlobalcoordinatesystemseeTDdeeeeeFDMDKTkTKeTeTmTMeTeeTefTF,,where(Local)(Global)21Step5:AssemblyofFEequationsDirectassemblymethod–AddingupcontributionsmadebyelementssharingthenodeFDMKDFKD(Static)22Step6:ImposedisplacementconstraintsNoconstraintsrigidbodymovement(meaninglessforstaticana