001J SFEM for linear and geometrically nonlinear

Copyrightc2008TechSciencePressCMES,vol.28,no.2,pp.109-125,2008ASmoothedFiniteElementMethod(SFEM)forLinearandGeometricallyNonlinearAnalysisofPlatesandShellsX.Y.Cui1,2,G.R.Liu2,3,G.Y.Li1,X.Zhao2,T.T.Nguyen2andG.Y.Sun1Abstract:Asmoothedfiniteelementmethod(SFEM)ispresentedtoanalyzelinearandgeo-metricallynonlinearproblemsofplatesandshellsusingbilinearquadrilateralelements.Theformu-lationisbasedonthefirstordersheardeformationtheory.InthepresentSFEM,theelementsarefur-therdividedintosmoothingcellstoperformstrainsmoothingoperation,andthestrainenergyineachsmoothingcellisexpressedasanexplicitformofthesmoothedstrain.Theeffectofthenumberofdivisionsofsmoothingcellsinelementsisinvesti-gatedindetail.Itisfoundthatusingthreesmooth-ingcellsforbendingstrainenergyintegrationandonesmoothingcellforshearstrainenergyinte-grationachievemostaccurateresultsandhencethesenumbersrecommendedforplatesandshellsinthisstudy.Inthegeometricallynonlinearanal-ysis,thetotalLagrangianapproachisadopted.Thearc-lengthtechniqueinconjunctionwiththemodifiedNewton-Raphsonmethodisutilizedtosolvethenonlinearequations.Thenumericalex-amplesdemonstratethatthepresentSFEMpro-videsverystableandmostaccurateresultswiththesimilarcomputationaleffortcomparedtotheexistingFEMtechniquestestedinthiswork.Keyword:smoothedfiniteelementmethod,plateandshell,mid-rectanglerule,nonlinear,smoothedweakform.1StateKeyLaboratoryofAdvancedDesignandManu-facturingforVehicleBody,HunanUniversity,Changsha,410082,PRChina.2CentreforAdvancedComputationsinEngineeringSci-ence(ACES),DepartmentofMechanicalEngineering,NationalUniversityofSingapore,9EngineeringDrive1,117576Singapore.3Singapore-MITAlliance(SMA),E4-04-10,4EngineeringDrive3,117576,Singapore1IntroductionInthepastdecades,thefiniteelementmethod(FEM)hasbeenplayingaveryimportantroleinsolvingvariousproblemsinengineeringandsci-ence,includingmechanicsproblemsofplatesandshells[ZienkiewiczandTaylor(2000);LiuandQuek(2003)].Anumberofplateandshellele-mentshavebeendevelopedforlinearandnonlin-earanalysis.BatozandTahar(1982)proposedadiscreteKirchhoffquadrilateralelement(DKQ),whichcangiveefficientresultsforbendingprob-lemsforthinplates,butnotforthickplates.BatheandDvorkin(1985)presenteda4-nodeplateel-ementbasedonMindlin-Reissnertheoryusingmixedinterpolatedtensorialcomponents(MITC).BelytschkoandLeviathan(1994)developedaonepointquadraturequadrilateralshellelementwithphysicalhourglasscontrol.Inrecentyears,mesh-freemethodshavebeendevelopedandachievedremarkableprogressforsolvingplateandshellproblems,andmanyworksaresummarizedinthebookbyLiu(2002),Atluri(2005).ChenandLiu(2001)hadusedtheelementfreeGalerkin(EFG)methodforsolvingstaticanddynamicproblemofthinplateofcomplicatedshape.Compositelam-inatedplateshadalsobeenstudiedforvibrationproblems,bucklingproblems[Chenetal.(2002,2003)].ShellshadalsobeenanalyzedusingEFG[Liuetal.(2002)].WangandChen(2004)pre-sentedaMindlin-Reissnerplateformulationusingastabilizedconformingnodalintegrationtomit-igatetheshearlocking.Aradialpointinterpola-tionmethod[Liuetal.(2008)]wasformulatedforplateproblemsusingthesmoothednodalintegra-tion.MeshlessLocalPetrov-Galerkin(MLPG)Method[AtluriandShen(2002);Atluri(2004)]wasalsoemployedforplatesandshellsanalysisbymanyresearchers[GuandLiu(2001);Long110Copyrightc2008TechSciencePressCMES,vol.28,no.2,pp.109-125,2008andAtluri(2002);Soricetal.(2004);Lietal.(2005);Sladeketal.(2006);Jaraketal.(2007)],andAtlurietal.(2004)proposedameshlessfi-nitevolumemethodthroughtheMLPG“mixed”approach.Fornonlinearanalysisofplatesandshells,HorrigmoeandBergan(1978)presentedageneralformulationforgeometricallynonlin-earanalysisofshellsusingflatfiniteelements.HughesandLiu(1981)presentedageneralnon-linearfiniteelementformulationusinguniformreducedintegrationforshellanalysis.Recently,Leeetal.(2002)introducedageometricallynon-linearassumedstrainformulationofanine-nodesolidshellelement.WenandHon(2007)formu-latedaReissner-Mindlinplateelementforgeo-metricallynonlinearanalysisbyusingameshlesscollocationmethod.TheotherworksofplatesandshellsanalysisincludethosegivenbyBasarandKintzell(2003),Qianetal.(2003),Suetake(2006).AlthoughasignificantmountofworkshavebeendoneusingFEM,someinherentproblemsrelatedtoelementdistortionstillremainunsolved.Liuetal.(2007a)proposedasmoothedfiniteelementmethod(SFEM)bycombiningthestandardFEMwiththestrainsmoothingtechniqueusedinmesh-freemethods[Chenetal.(2005)].TheSFEMfurtherdividestheelementsintosmoothingcellsandcomputestheintegralsalongtheedgesofthesmoothingcellsbasedontheGreen’sdiver-gencetheorem.Then-sidedpolygonalelementcanbeeasilycarriedoutusingtheSFEM[Daietal.(2007)].Liuetal.(2007b)gavedetailedtheo-reticalaspectsincludingstability,boundpropertyandconvergenceabouttheSFEMandrevealedanumberofattractivefeaturesresultedfromthe“softening”effectsofthestrainsmoothingtech-nique.OtherproblemssolvedusingSFEMin-cludingfreeandforcedvibrationanalysis[DaiandLiu(2007)],piezoelectricelementanalysisoftwo-dimensionalsmartstructures[Nguyen-Vanetal.(2008)].Aplateelement[Nguyen-Xuanetal.(2008)]hasbeenformulatedusingtheSFEM.I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