Meshfree and particle methods and their applicatio

MeshfreeandparticlemethodsandtheirapplicationsShaofanLiDepartmentofCivil&EnvironmentalEngineering,UniversityofCalifornia,BerkeleyCA94720;[e-mail]fMechanicalEngineering,NorthwesternUniversity,2145SheridanRd,EvanstonIL60208;[e-mail]eshfreeandparticlemethodsandtheirapplicationsinappliedmechan-icsaresurveyed.Threemajormethodologieshavebeenreviewed.First,smoothedparticlehydrodynamics~SPH!isdiscussedasarepresentativeofanon-localkernel,strongformcollo-cationapproach.Second,mesh-freeGalerkinmethods,whichhavebeenanactiveresearchareainrecentyears,arereviewed.Third,someapplicationsofmoleculardynamics~MD!inappliedmechanicsarediscussed.Theemphasesofthissurveyareplacedonsimulationsoffinitedeformations,fracture,strainlocalizationofsolids;incompressibleaswellascompress-ibleflows;andapplicationsofmultiscalemethodsandnano-scalemechanics.Thisreviewar-ticleincludes397references.@DOI:10.1115/1.1431547#1INTRODUCTIONSincetheinventionofthefiniteelementmethod~FEM!inthe1950s,FEMhasbecomethemostpopularandwidelyusedmethodinengineeringcomputations.AsalientfeatureoftheFEMisthatitdividesacontinuumintodiscreteele-ments.Thissubdivisioniscalleddiscretization.InFEM,theindividualelementsareconnectedtogetherbyatopologicalmap,whichisusuallycalledamesh.Thefiniteelementin-terpolationfunctionsarethenbuiltuponthemesh,whichensuresthecompatibilityoftheinterpolation.However,thisprocedureisnotalwaysadvantageous,becausethenumericalcompatibilityconditionisnotthesameasthephysicalcom-patibilityconditionofacontinuum.Forinstance,inaLa-grangiantypeofcomputations,onemayexperiencemeshdistortion,whichcaneitherendthecomputationaltogetherorresultindrasticdeteriorationofaccuracy.Inaddition,FEMoftenrequiresaveryfinemeshinproblemswithhighgradientsoradistinctlocalcharacter,whichcanbecompu-tationallyexpensive.Forthisreason,adaptiveFEMhasbe-comeanecessity.Today,adaptiveremeshingproceduresforsimulationsofimpact/penetrationproblems,explosion/fragmentationprob-lems,flowpassobstacles,andfluid-structureinteractionproblemsetchavebecomeformidabletaskstoundertake.Thedifficultiesinvolvedarenotonlyremeshing,butalsomappingthestatevariablesfromtheoldmeshtothenewmesh.Thisprocessoftenintroducesnumericalerrors,andfrequentremeshingisthusnotdesirable.Therefore,thesocalledArbitraryLagrangianEulerian~ALE!formulationshavebeendeveloped~see,eg@1–4#!.Foracompletedescrip-tiononthissubject,readersmayconsultChapter7ofthebookbyBelytschko,Liu,andMoran@5#.TheobjectiveoftheALEformulationistomakethemeshindependentofthematerialsothatthemeshdistortioncanbeminimized.Un-fortunately,incomputersimulationsofverylargedeforma-tionand/orhigh-speedmechanicalandstructuralsystems,evenwiththeALEformulation,adistortedmeshintroducessevereerrorsinnumericalcomputations.Furthermore,theconvectivetransporteffectsinALEoftenleadtospuriousoscillationthatneedstobestabilizedbyartificialdiffusionoraPetrov-Galerkinstabilization.Inothercases,ameshmaycarryinherentbiasinnumericalsimulations,anditspresencebecomesanuisanceincomputations.Awellknownexampleisthesimulationofthestrainlocalizationproblem,whichisnotoriousforitsmeshalignmentsensitivity@6,7#.Therefore,itwouldbecomputationallyefficacioustodiscretizeacon-tinuumbyonlyasetofnodalpoints,orparticles,withoutmeshconstraints.Thisistheleitmotifofcontemporarymesh-freeGalerkinmethods.Theadvantagesofthemeshfreeparticlemethodsmaybesummarizedasfollows:1!Theycaneasilyhandleverylargedeformations,sincetheconnectivityamongnodesisgeneratedaspartofthecomputationandcanchangewithtime;2!ThemethodologycanbelinkedmoreeasilywithaCADdatabasethanfiniteelements,sinceitisnotnecessarytogenerateanelementmesh;3!Themethodcaneasilyhandledamageofthecomponents,suchasfracture,whichshouldproveveryusefulinmod-elingsofmaterialfailure;TransmittedbyAssociateEditorJNReddyASMEReprintNoAMR319$26.00ApplMechRevvol55,no1,January2002©2002AmericanSocietyofMechanicalEngineers14!Accuracycanbecontrolledmoreeasily,sinceinareaswheremorerefinementisneeded,nodescanbeaddedquiteeasily~h-adaptivity!;5!Thecontinuummeshfreemethodscanbeusedtomodellargedeformationsofthinshellstructures,suchasnano-tubes;6!Themethodcanincorporateanenrichmentoffinescalesolutionsoffeatures,suchasdiscontinuitiesasafunctionofcurrentstressstates,intothecoarsescale;and7!Meshfreediscretizationcanprovideaccuraterepresenta-tionofgeometricobject.Ingeneral,particlemethodscanbeclassifiedbasedontwodifferentcriteria:physicalprinciples,orcomputationalformulations.Accordingtothephysicalmodeling,theymaybecategorizedintotwoclasses:thosebasedondeterministicmodels,andthosebasedonprobabilisticmodels.Ontheotherhand,accordingtocomputationalmodelings,theymaybecategorizedintotwodifferenttypesaswell:thoseservingasapproximationsofthestrongformsofpartialdifferentialequations~PDEs!,andthoseservingasapproximationsoftheweakformsofPDEs.Inthissurvey,theclassificationbasedoncomputationalstrategiesisadopted.ToapproximatethestrongformofaPDEusingaparticlemethod,thepartialdifferentialequationisusuallydiscretizedbyaspecificcollocationtechnique.Exa