Advances in Meshfree Techniques

321 Pages·2007·3.43 MB·English

by  C. Armando Duarte, Dae-Jin Kim, Ivo Babuška (auth.), V. M. A. Leitão, C. J. S. Alves, C. Armando Duarte (eds.)

ADVANCESINMESHFREETECHNIQUES
Computational Methods in Applied Sciences
Volume5
SeriesEditor
E.Oñate
International CenterforNumericalMethodsinEngineering (CIMNE)
TechnicalUniversityofCatalunya(UPC)
EdificioC-1,CampusNorteUPC
GranCapitán,s/n
0834Barcelona, Spain
[email protected]
www.cimne.com
Advances in
Meshfree Techniques
Editedby
V.M.A.Leitão
InstitutoSuperiorTécnico,
Lisboa,Portugal
C.J.S. Alves
InstitutoSuperiorTécnico,
Lisboa,Portugal
and
C. Armando Duarte
UniversityofIllinoisatUrbana-Champaign,
Urbana,IL,USA
AC.I.P.CataloguerecordforthisbookisavailablefromtheLibraryofCongress.
ISBN978-1-4020-6094-6(HB)
ISBN978-1-4020-6095-3(e-book)
PublishedbySpringer,
P.O.Box17,3300AADordrecht,TheNetherlands.
www.springer.com
Printedonacid-freepaper
AllRightsReserved
©2007Springer
Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmitted
inanyformorbyanymeans,electronic,mechanical,photocopying,microfilming,recording
orotherwise,withoutwrittenpermissionfromthePublisher,withtheexception
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andexecutedonacomputersystem,forexclusiveusebythepurchaserofthework.
Table of Contents
Preface vii
AGlobal-LocalApproachfortheConstructionofEnrichmentFunctions
fortheGeneralizedFEMandItsApplicationtoThree-DimensionalCracks 1
C.ArmandouDarte,Dae-JinKimandIvoBabukša
StudyofSomeOptimalXFEM TypeMethods 27
ElieChahine,PatrickLaborde,JulienPommier,YvesRenardand
MichelSalaün
GeneralizedFiniteElementMethodinMixedVariationalFormulation:
AStudyofConvergenceandSolvability 39
W.GóisandS.P.B.Proença
StabilityinLagrangianandSemi-LagrangianReproducingKernel
DiscretizationsUsingNodalIntegrationinNonlinearSolidMechanics 55
Jiun-ShyanChenandYoucaiWu
SimulationofFormingProcessesbytheα-Shapes-BasedNaturalElement
Method 77
I.Alfaro,E.Cueto,F.ChinestaandM.Doblaré
NewAdvancesinMeshlessMethods:CouplingNaturalElementand
MovingLeastSquaresTechniques 97
F.Chinesta,J.Yvonnet,P.iVllon,P.Breitkopf,P.Joyot,I.Alfaroand
E.Cueto
v
vi TableofContents
EliminatingShear-LockinginMeshlessMethods:ACriticalOverviewanda
NewFrameworkforStructuralTheories 123
CarlosTiagoandiVtorM.A.Leitão
FEM/SPHCouplingTechniquesforHighVelocityImpactSimulations 147
LeventAktayandAlastairF.Johnson
OntheConstructionofMassConservativeandMeshlessAdaptiveParticle
AdvectionMethods 169
ArminIske
Spectral-LikeAccuracyinSpaceofaMeshlessVortexMethod 187
L.A.Barba
AHybridMeshless/Spectral-ElementMethodfortheShallowWater
EquationsontheSphere 199
ChristopherD.Blakely
IteratedApproximateMovingLeastSquaresApproximation 221
GregoryE.FasshauerandJackG.hZang
AKansaTypeMethodUsingFundamentalSolutionsAppliedtoElliptic
PDEs 241
CarlosJ.S.AlvesandSvilenS.aVltchev
FromGlobaltoLocalRadialBasisFunctionCollocationMethodfor
TransportPhenomena 257
BoiždarŠarler
ComputationofStaticDeformationsandNaturalFrequenciesofShear
DeformablePlatesbyanRBF-PseudospectralMethodwithanOptimal
ShapeParameter 283
A.J.M.FerreiraandG.E.Fasshauer
AuthorIndex 311
SubjectIndex 313
Preface
Over the past decade, mesh reduction techniques (meshless or meshfree methods)
haveemergedaseffectivenumericaltechniquesforsolvingscienceandengineering
problems. Meshing difficulties of existing numerical techniques like the finite ele-
mentandboundaryelementmethods,weretheinitialthrustforthedevelopmentof
meshfree methods.Severaltechniquesdevoted to the simulationof crack propaga-
tion,movingmaterialinterfaces,largedeformationandshearbandlocalizationprob-
lems,amongothers,wereproposedduringthelastdecade.Problemswithhighlyos-
cillatory solutions such as Helmholtz equationsand multiscale problemshave also
spurredgreatinterestonthisclassofmethods.Thevarietyofproblemsthatarenow
beingaddressedbythesetechniquescontinuestoexpandandthequalityoftheresults
obtaineddemonstratestheeffectivenessofmanyofthemethodscurrentlyavailable.
Variousscientificmeetingshavebeenorganizedinrecentyearsthatweretotally
oratleastpartiallydevotedtothisareaofknowledgeandalargenumberofpapers
have been published in prestigious journals. The number of methods that have re-
centlybeenproposedisalsoanevidenceofthegrowinginterestoftheengineering
andmathematicscommunityworldwideonthesetypesofnumericaltechniques.
Theobjectiveofthisbookistocollectstate-of-the-artresearch,methodsandnew
ideasonthesubjectofmeshreductiontechniques,andtocontributetothedevelop-
mentofthisareaofknowledge.Thebookcontains15invitedcontributionswritten
byparticipantstooneoftheThematicConferencessponsoredbytheEuropeanCom-
munityonComputationalMethodsinAppliedSciences, theECCOMAS Thematic
ConferenceonMeshless Methodsheldin Lisbon,Portugal,fromJuly 11thto 14th
2005.Thevitalityandmulti-disciplinarityoftheresearchcommunityinthisfieldis
reflected on the broad range of methods discussed in this book. Each contribution
in this book is an extended and revised version of the paper presented at the con-
ference.Theypresenta sampleofthestate ofthe artinthefield withmethodsthat
vii
viii Preface
havereachedacertainlevelofmaturitywhilealsoaddressingmanyopenissues.The
listofcontributorsrevealsafortunatemixofhighlydistinguishedauthorsaswellas
quiteyoungbutveryactiveandpromisingresearchers.
Itis the editors’wish thatthis bookmayconstitute a valuablereferenceforre-
searchersinterestedinthefieldofmeshreductiontechniques,onethatmayhelppro-
motingtheuseofthese techniquesandtoextenditsapplicationtoanevenbroader
varietyofengineeringandscienceproblems.
Theeditorswouldliketotakethisopportunitytothankallauthorsforsubmitting
theircontributions.
V.M.A.Leitão
C.J.S.Alves
C.A.Duarte
A Global-Local Approach for the Construction of
Enrichment Functions for the Generalized FEM and
Its Application to Three-Dimensional Cracks
C.ArmandoDuarte1,Dae-JinKim1andIvoBabuška2
1DepartmentofCivilandEnvironmentalEngineering,UniversityofIllinoisat
Urbana-Champaign,205NorthMathewsAvenue,Urbana,IL61801,U.S.A.;
E-mail:[email protected],[email protected]
2InstituteforComputationalEngineeringandSciences,TheUniversityofTexasatAustin,
1UniversityStation,C0200,Austin,TX78712,U.S.A.;E-mail:[email protected]
Abstract. Existing generalized or extended finite element methods for modeling cracks in
three-dimensions require theuse of asufficientlyrefinedmesh around thecrackfront. This
offsetssomeof theadvantages of thesemethods speciallyinthecaseof propagating three-
dimensional cracks.Inthispaper,astrategytoovercomethislimitationisinvestigated.The
approach involves thedevelopment of enrichment functions that are computed using anew
global-local approach. Thisstrategyallowstheuseof afixedglobal mesharoundthecrack
front and is specially appealing for non-linear or time dependent problems since it avoids
mappingofsolutionsbetweenmeshes.Theresultingtechniqueenjoysthesameflexibilityof
theso-calledmeshfreemethodsforthisclassofproblemwhilebeingmorecomputationally
efficient.
TheproposedgeneralizedFEMwithglobal-localfunctions,bynumericallyconstructing
theenrichmentfunctions,bringsthebenefitsofexistinggeneralizedFEMtoabroaderclassof
problems.Theprocedureisappliedtothesolutionofthree-dimensionallinearelasticfracture
mechanicsproblems.Numericalexperimentsdemonstratingthecomputationalefficiencyand
accuracyofthemethodarepresented.
Key words: Generalized finite element method, extended finite element method, meshfree
method,global-localmethod,fracturemechanics.
1 Introduction
The generalized finite element method (GFEM) [4, 15, 31, 38] and the eXtended
finite element method (XFEM) [7, 26] are examples of the so-called partition of
unity method which originated in the works of Babuška et al. [3] and Duarte and
1
V.M.A.Leitãoetal.(eds),AdvancesinMeshfreeTechniques,1–26.
©2007Springer.PrintedintheNetherlands.
2 C.A.Duarteetal.
Fig.1.ConstructionofGFEMshapefunctionsusing(a)polynomialand(b)non-polynomial
enrichmentfunctions.Here,ϕα arethefunctionsatthetop,theenrichmentfunctions,L,are
the functions in the middle, and the generalized FE shape functions, φα, are the resulting
bottomfunctions.
Oden[17,19].Theshapefunctionsinthisclassofmethodsarebuiltfromtheproduct
ofapartitionofunity,ϕ ,andenrichmentfunctions,L,
α
φ (x) ϕ (x)L(x) (1)
α α
:=
where ϕ , α 1,...,N, N being the number of functions, constitute a parti-
α
=
tion of unity, i.e., a set of functions defined in a domain (cid:5) with the property that
N ϕ (x) 1forallxin(cid:5).Figure1illustratestheconstructionofGFEMshape
funαc=ti1onαsusin=gapolynomialandanon-polynomialenrichmentfunction.
(cid:2)
Thepartitionofunitypropertyoffunctionsϕ , α 1,...,N,impliesthat
α
=
N N
φ (x) ϕ (x)L(x) L(x) (2)
α α
= =
α 1 α 1
(cid:3)= (cid:3)=
AnyenrichmentfunctionLcanberepresentedexactlythroughlinearcombinations
ofgeneralizedFEshapefunctions.Therefore,iftheenrichmentfunctionLcanap-
proximate the solution u of a boundary value problem, the corresponding GFEM
shapefunctionalsowill.Thereisconsiderablefreedominthechoiceoftheenrich-
mentfunctionsandthecorrespondingapproximationspacesusedintheGFEM.Fig-
ure1b,forexample,illustrates theconstructionofa GFEM shape functionusinga
highlyoscillatorynon-polynomialenrichmentfunction.
Customizedenrichmentfunctionscanbeusedtomodellocalfeaturesinadomain
likecracks[11,16,26,27,30,32,42],edgesingularities[15],boundarylayers[13],
inclusions [41], voids [37, 41], microstructures[25, 34], etc., instead of a strongly