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Anefficientadaptiveanalysisprocedureusingtheedge-basedsmoothedpointinterpolationmethod(ES-PIM)for2Dand3DproblemsQ.Tanga,b,G.Y.Zhangc,n,G.R.Liud,Z.H.Zhonga,Z.C.Hea,baStateKeyLaboratoryofAdvancedDesignandManufacturingforVehicleBody,HunanUniversity,Changsha410082,ChinabDepartmentofMechanicalEngineering,NationalUniversityofSingapore,9EngineeringDrive1,117576Singapore,SingaporecSchoolofMechanicalandChemicalEngineering,TheUniversityofWesternAustralia,35StirlingHighway,WA6009,AustraliadSchoolofAerospaceSystems,UniversityofCincinnati,Cincinnati,OH45221-0070,USAarticleinfoArticlehistory:Received18December2011Accepted6March2012Availableonline13April2012Keywords:MeshfreemethodsErrorindicatorAdaptiveanalysisPointinterpolationmethodDelaunaytriangulationabstractInthispaper,anefficientadaptiveanalysisprocedureisproposedusingthenewlydevelopededge-basedsmoothedpointinterpolationmethod(ES-PIM)forbothtwodimensional(2D)andthreedimensional(3D)elasticityproblems.TheES-PIMworkswellwiththree-nodetriangularandfour-nodetetrahedralmeshes,iseasytobeimplementedforcomplicatedgeometry,andcanobtainnumericalresultsofmuchbetteraccuracyandhigherconvergenceratethanthestandardfiniteelementmethod(FEM)withthesamesetofmeshes.Alltheseimportantfeaturesmakeitanidealcandidateforadaptiveanalysis.Inthepresentadaptiveprocedure,anovelerrorindicatorisdevisedforES-PIMsettings,whichevaluatesthemaximumdifferenceofstrainenergyvaluesamongthevertexesofeachbackgroundcell.Asimpleh-typelocalrefinementschemeisadoptedtogetherwithameshgeneratorbasedonDelaunaytechnology.Intensivenumericalstudiesof2Dand3Dexamplesindicatethattheproposedadaptiveprocedurecaneffectivelycapturethestressconcentrationandsolutionsingularities,carryoutlocalrefinementautomatically,andhenceachievemuchhigherconvergenceforthesolutionsinstrainenergynormcomparedtothegeneraluniformrefinement.&2012ElsevierLtd.Allrightsreserved.1.IntroductionAdaptiveanalysisisimportantinthecomputationalmethods,toachievedesiredhighaccuracywithminimumcomputationalcost.InthetraditionalFEM,adaptivemeshrefinementtechniquesalongwithpropererroranalysishavebeenwellstudied[1–5],howeverformeshfreemethods[6–9],itisstillanopentopic.Themeshfreeedge-basedpointinterpolationmethod(ES-PIM)[10]hasbeenrecentlydevelopedusingthegeneralizedsmoothedGalerkin(GS-Galerkin)weakform[11]withthepointinterpola-tionmethod(PIM)forfieldvariableapproximation[12]andtheedge-basedgradientsmoothingoperationforstrainconstruction.IntheES-PIM,PIMshapefunctionsareconstructedwithasetofsmallnumberofnodeslocatedinalocalsupportdomainandpossessKroneckerDeltafunctionpropertywhichallowsstraight-forwardimpositionofpointessentialboundaryconditions.Thegeneralizedgradientsmoothingtechnique[11]extendedformthestrainsmoothingoperation[13]allowstheuseofdiscontinuousfunctions.Itcanprovidetheso-called‘‘softening’’effecttothenumericalmodelandhencesolvetheoverly-stiffproblemexistinginadisplacement-basedfullycompatibleFEMmodel[11,14].AsthetheoreticalbasisofES-PIM,LiuandcoworkershavedevelopedtheGspacetheoryandtheweakenedweak(W2)formulation[11,15,16]foraunifiedformulationofawideclassofcompatibleandincompatiblemethods.ThesemethodsincludethepresentES-PIM,thenode-basedsmoothedpointinterpolationmethod(NS-PIMorLC-PIMoriginally)whichcanprovideupperboundsolutionsinenergynormfortheforcedrivenelasticityproblems[14,17,18],thecell-basedsmoothedpointinterpolationmethod(CS-PIM)whichobtainshighlyaccurateandconvergentsolutions[16,46]andstrainconstructedpointinterpolationmethod(SC-PIM)[47,48].FortheES-PIM,ortheedge-basedsmoothedfiniteelementmethod(ES-FEM)whichisaspecialcaseofES-PIMusinglinearshapefunctions[19],theedge-basedstrainsmoothingoperationcanproperlysoftenthemodelandmakethenumericalmodelhaveaquiteclose-to-exactstiffnessbyevenusinglineartriangularelements[10,19].ThusnumericalsolutionsbytheES-PIMaregenerallyofmuchbetteraccuracy,higherconvergencerateandefficiencythanthestandardFEMusingthesamemesh.Further-more,theformulationoftheES-PIMisstraightforward,theimplementationisverysimpleandthemethodworkswellparticularlyforlow-orderlinearelements.AllthesefeaturesmaketheES-PIManexcellentcandidateforadaptiveanalyses.ContentslistsavailableatSciVerseScienceDirectjournalhomepage:://dx.doi.org/10.1016/j.enganabound.2012.03.007nCorrespondingauthor.E-mailaddress:[e-mail](G.Y.Zhang).EngineeringAnalysiswithBoundaryElements36(2012)1424–1443Inanadaptiveanalysisanappropriateerrorindicatorandassociatedmeshrefinementstrategyaretwocrucialissues.Ingeneral,twodistincttypesofproceduresarecurrentlyavailableforderivingerrorindicators:therecoverybasederrorindicatorandtheresidualbasederrorindicator.ZienkiewiczandZhu[5]firstlyintroducedthewell-knownrecoverybasederrorindicatorin1987,whichusesrecoveredstressesasreferencesolutionstocalculatethedomainerrorinenergynorm.Therearesomeexcellentstudiesonthisrecoverymethodology[20–22].Residualbasederrorindicatorsmakeuseoftheresidualofthenumericalapproxim