011J A smoothed Hermite radial point interpolation

ArchApplMech(2011)81:1–18DOI10.1007/s00419-009-0392-0ORIGINALXiangyangCui·GuirongLiu·GuangyaoLiAsmoothedHermiteradialpointinterpolationmethodforthinplateanalysisReceived:21July2009/Accepted:14October2009/Publishedonline:4November2009©Springer-Verlag2009AbstractAsmoothedHermiteradialpointinterpolationmethodusinggradientsmoothingoperationisformu-latedforthinplateanalysis.TheradialbasisfunctionsaugmentedwithpolynomialbasisareusedtoconstructtheshapefunctionsthathavetheimportantDeltafunctionproperty.ThesmoothedGalerkinweakformisadoptedtodiscretizethegoverningpartialdifferentialequations,andacurvaturesmoothedoperationisdevel-opedtorelaxthecontinuityrequirementandachieveaccuratebendingsolutions.Theapproximationbasedonbothdeflectionandrotationvariablesmaketheproposedmethodveryeffectiveinenforcingtheessentialboundaryconditions.Theeffectsofdifferentnumbersofsub-smoothing-domainscreatedbasedonthetrian-gularbackgroundcellareinvestigatedindetail.Anumberofnumericalexampleshavebeenstudiedandtheresultsshowthatthepresentmethodisverystableandaccurateevenforextremelyirregularbackgroundcells.KeywordsNumericalmethods·Meshfreemethods·Radialpointinterpolationmethod·Thinplate·Gradientsmoothingoperation1IntroductionInthepastdecades,variousmeshfreemethods,suchassmoothparticlehydrodynamics[1],diffuseelementmethod(DEM)[2],elementfreeGalerkin(EFG)method[3],reproducingkernelparticlemethod(RKPM)[4],finitepointmethod(FPM)[5],H-Pclouds[6],partitionofunitymethod(PUM)[7],naturalelementmethod(NEM)[8],meshlesslocalPetrov–Galerkin(MLPG)method[9],pointinterpolationmethod(PIM)[10–12],andradialpointinterpolationmethod(RPIM)[13–15],havebeenproposedandappliedtosolveengineeringandscientificproblems[16–21].Numericalmethodsforplateshavebeencontinuouslyreceivingmuchattention,duetothewideappli-cationofplatestructuresincivil,mechanical,andaerospaceengineering.AstheapproximationofthesomeX.Cui·G.Li(B)StateKeyLaboratoryofAdvancedDesignandManufacturingforVehicleBody,HunanUniversity,410082Changsha,People’sRepublicofChinaE-mail:[e-mail].:[phone]Fax:[phone]X.CuiE-mail:[e-mail]·G.LiuDepartmentofMechanicalEngineering,CentreforAdvancedComputationsinEngineeringScience(ACES),NationalUniversityofSingapore,9EngineeringDrive1,117576Singapore,SingaporeG.LiuSingapore-MITAlliance(SMA),E4-04-10,4EngineeringDrive3,117576Singapore,Singapore2X.Cuietal.meshfreemethodscaneasilyachieveanarbitraryorderofcontinuityusingreasonablesupportnodes,thethinplateformulationcanbeeasilyperformedusingthesemeshfreemethods.Anumberofworksonmeshfreemethodsforplateshavebeenreported.KryslandBelytschko[22]consideredEFGtoanalyzetheKirchhoffplateproblemswhereC1continuitycanbeeasilyachievedusingmovingleast-squares(MLS)approximation.LiuandChen[23]extendedtheEFGforstaticandfreevibrationanalysesofthinplates.MLPGisanotherattractivemeshlessmethodusedfortheanalysisofplateproblems.GuandLiu[24]haveextendedtheMLPGformulationforstaticandfreevibrationanalysesofthinplates.LongandAtluri[25]extendedMLPGmethodforsolvingthebendingproblemofathinplate.Intheweakformmesh-freemethods,backgroundcellsarecommonlyusedtoimplementtheGaussianintegration.DuetothecomplexityinvolvedintheGaussintegration,thetechniqueshavebeendevelopedbyperformingintegralsbasedonthenodes.BeisselandBelytschko[26]demonstratedthatthenodalintegrationofEFGresultedinaspatialinstabilityduetotheunderintegrationoftheweakform,andproposedastabilizedproceduretoovercomethespatialinstability.BonetandKulasegaram[27]presentedaleast-squarestabilizationtechniquetoeliminatespuriousmodeinnodalintegration.Chenetal.[28]proposedastabilizedconformingnodalintegration(SCNI)fornodalintegratedmeshfreemethodusingMLSandRKPMshapefunctions.TheSCNIwasfurtherextendedforanalyzingMindlin–Reissenerplates[29]andsheardeformableshellstructures[30].Intheseworks,theshearandmembranelockingweretreatedusingquadraticbasisfunctionsthatsatisfytheKirchhoffmodereproducingconditionandacurvaturesmoothingstabilizationtechnique.Zhangetal.[31]proposedamovingleast-squaresapproximationwithdiscontinuousderivativebasisfunctionsforshellstructureswithslopediscontinuities.Recently,aHermitereproducingkernelapproximationhasalsobeenproposedbyWangandChen[32]forthin-plateanalysiswithsub-domainstabilizedconformingintegration.BothMLSandRKPMshapefunctionsusedintheabovemethodsdonotpossesstheKroneckerdeltafunc-tionproperties,andhencespecialtechniquesareneededforhandlingtheessentialboundaryconditions.ThePIMandRPIMshapefunctionsusinglocalnodeshavetheKroneckerdeltafunctionproperties;theessentialboundaryconditionscanbeeasilyimposedforPIMandRPIMmodels.AconformingRPIMforthickshellstructureswasproposedin[33]andthePIMwasextendedforspatialgeneralshellsstructures[34].Liuetal.[35]proposedtheconformingradialpointinterpolationmethodforstaticandfreevibrationanalysisofplatesusingstabilizedconformingnodalintegrationtechnique.AmeshfreeHermite-typeradialpointinterpolationmethodwasalsopresentedforKirchhoffplateproblemsin[36].Inthispaper,weproposedasmoothedHermiteradialpointinterpolationmethod(Hermite-RPIM)f