Adaptive analysis using the node-based smoothed fi

INTERNATIONALJOURNALFORNUMERICALMETHODSINBIOMEDICALENGINEERINGInt.J.Numer.Meth.Biomed.Engng.2011;27:198–218Publishedonline16July2009inWileyOnlineLibrary(wileyonlinelibrary.com).DOI:10.1002/cnm.1291COMMUNICATIONSINNUMERICALMETHODSINENGINEERINGAdaptiveanalysisusingthenode-basedsmoothedfiniteelementmethod(NS-FEM)T.Nguyen-Thoi1,3,∗,†,G.R.Liu1,2,H.Nguyen-Xuan2,3andC.Nguyen-Tran31CenterforAdvancedComputationsinEngineeringScience(ACES),DepartmentofMechanicalEngineering,NationalUniversityofSingapore,9EngineeringDrive1,Singapore117576,Singapore2Singapore-MITAlliance(SMA),E4-04-10,4EngineeringDrive3,Singapore,117576,Singapore3DepartmentofMathematicsandComputerScience,UniversityofNaturalSciences,VietnamNationalUniversity—HCM,VietnamSUMMARYThepaperpresentsanadaptiveanalysiswithintheframeworkofthenode-basedsmoothedfiniteelementmethod(NS-FEM)usingtriangularelements.Anerrorindicatorbasedontherecoverystrainisusedandshowntobeasymptoticallyexactbyaneffectivityindexandnumericalresults.Asimplerefinementstrategyusingthenewestnodebisectionisbrieflypresented.Thenumericalresultsofsomebenchmarkproblemsshowthatthepresentadaptiveprocedurecanaccuratelycatchtheappearanceofthesteepgradientofstressesandtheoccurrenceofrefinementisconcentratedproperly.TheenergyerrornormsofadaptivemodelsforbothNS-FEMandFEMobtainhigherconvergenceratecomparedwiththeuniformlyrefinedmodels,buttheresultsofNS-FEMarebetterandachievehigherconvergenceratethanthoseofFEM.TheeffectivityindexofNS-FEMisalsocloserandapproachestounityfasterthanthatofFEM.TheupperboundpropertyinthestrainenergyofNS-FEMisalwaysverifiedduringtheadaptiveprocedure.Copyrightq2009JohnWiley&Sons,Ltd.Received21October2008;Revised11April2009;Accepted22May2009KEYWORDS:finiteelementmethod(FEM);meshfreemethods;node-basedsmoothedfiniteelementmethod(NS-FEM);upperbound;errorindicator;adaptiveanalysis1.INTRODUCTIONAdaptiveanalysishasbeenusedinthetraditionalfiniteelementmethod(FEM)andvariousproce-duresforerrorestimateandrefinementhavebeendeveloped.Amongerrorestimators,residual-basedandrecovery-basedonesarethemostpopular.Theresidual-basederrorestimatorshavebeendevelopedbyconsideringlocalresidualsofthenumericalsolutions,inapatchofelementsorinasingleelement.ThistypeoferrorestimatorswasoriginallyintroducedbyBabuskaandRheinboldt[1,2],andthendevelopedbymanyothersresearcherssuchasBankandWeiser[3],AinsworthandOden[4,5].Recovery-basederrorestimatorshavebeenstudiedbyusingoftherecoverysolutionsderivedfromaposterioritreatmentofthenumericalresultstoobtainmoreaccuraterepresentationoftheunknowns.Thistypeoferrorestimatorswasintroducedanddevel-opedbyZienkiewiczandZhu[6–8]andhasbeenwidelyusedintheFEM.Inaddition,errorestimatorsbasedontheconstructionofastaticallyadmissiblestressfieldwerealsointroducedbyLadev`eze[9–11].Oncetheerrorestimatorprocesshasbeensetup,itisnaturaltoseeka∗Correspondenceto:T.Nguyen-Thoi,CenterforAdvancedComputationsinEngineeringScience(ACES),DepartmentofMechanicalEngineering,NationalUniversityofSingapore,9EngineeringDrive1,Singapore117576,Singapore.†E-mail:[e-mail],[e-mail]2009JohnWiley&Sons,Ltd.ADAPTIVEANALYSISUSINGTHENS-FEM199refinementschemebywhichthedesigncanbeimproved.Therearevariousproceduresoftherefinementandtheymaybebroadlyclassifiedintothreecategories:h-typerefinement,p-typerefinementandr-typerefinement[12,13].Inanh-typerefinement,thesameclassofelementswillcontinuetobeusedbutmoreelementsareneededatthenecessarypositionstoprovidemaximumeconomyinreachingthedesiredsolution.Inap-typerefinement,thesameelementsareusedbuttheorderofthepolynomialfunctionsisincreased.Inar-typerefinement,thenodesofelementsarerelocatedbutthemeshconnectivityiskeptunchanged[13].Recently,ane-typerefinement(enrichmentadaptivity)thatusesanextendedglobalderivativerecoveryforenrichedFEMssuchasextendedfiniteelementmethod(XFEM)isalsoproposed[14–17].Thee-typerefinementisshowntobesimpleandsuitabletoindustrialapplications.Intheotherfrontofdevelopmentofnumericalmethods,aconformingnodalintegrationtechniquehasbeenproposedbyChenetal.[18]tostabilizethesolutionsinthecontextofthemeshfreemethodandthenappliedinthenatural-elementmethod[19].Liuetal.haveappliedthistechniquetoformulatethelinearconformingpointinterpolationmethod(LC-PIM)[20]andthelinearlyconformingradialpointinterpolationmethod[21].ApplyingthesameideatotheFEM,anelement-basedsmoothedfiniteelementmethod(CS-FEMorSFEM)[22–25],andnode-basedsmoothedfiniteelementmethod(NS-FEM)[24]havealsobeenformulated.IntheCS-FEM,thestrainsmoothingoperationandtheintegrationoftheweakformareperformedoversmoothingcells(SCs)locatedinsidethequadrilateralelements,asshowninFigure1.TheCS-FEMhasbeendevelopedforgeneraln-sidedpolygonalelements[26],dynamicanalyses[27],incompressiblematerialsusingselectiveintegration[28,29],andfurtherextendedforplateandshellanalyses[30–34],respectively.Inaddition,CS-FEMhasalsobeencoupledtotheXFEM[35]tosolvefracturemechanicsproblemsin2Dcontinuumandplates[36].IntheNS-FEM,thestrainsmoothingoperationandtheintegrationoftheweakformareperformedoverthesmoothingcellsassociatedwithnodes,andm