Model Order Reduction Methods for Meshfree Approximation of Problems with Singularities and ...

134 Pages·2013·4.52 MB·English

UCLA
UCLA Electronic Theses and Dissertations
Title
Model Order Reduction Methods for Meshfree Approximation of Problems with Singularities
and Discontinuities
Permalink
https://escholarship.org/uc/item/44n4c72n
Author
Marodon, Camille
Publication Date
2013

Peer reviewed|Thesis/dissertation
eScholarship.org Powered by the California Digital Library
University of California
UNIVERSITYOFCALIFORNIA
LosAngeles
ModelOrderReductionMethodsforMeshfreeApproximation
ofProblemswithSingularitiesandDiscontinuities
Adissertationsubmittedinpartialsatisfactionofthe
requirementsforthedegreeDoctorofPhilosophy
inCivilEngineering
by
CamilleMarodon
2014
(cid:13)c Copyrightby
CamilleMarodon
2014
ABSTRACTOFTHEDISSERTATION
ModelOrderReductionMethodsforMeshfreeApproximation
ofProblemswithSingularitiesandDiscontinuities
by
CamilleMarodon
DoctorofPhilosophyinCivilEngineering
UniversityofCalifornia,LosAngeles,2014
ProfessorJiun-ShyanChen,Chair
The modeling of problems with singularities and discontinuities, such as cracks in elastic me-
dia, is one of the most challenging problems in computational mechanics. The numerical solution
of such problems necessitates sufficient resolution in the discretization, which is very costly. In
thiswork,modelorderreduction(MOR)techniquesaredevelopedforproblemswithsingularities
basedonmeshfreemethods. Theoreticalinvestigationontheperformanceoftheproposedmethods
viaerrorestimation,stabilityanalysis,andcomplexityanalysisisalsocarriedoutinthisresearch.
In the first part of this work, the full discrete model is constructed using meshfree approxima-
tion along with enrichment techniques to properly approximate singularities and discontinuities.
This, on the other hand requires high order quadrature rules for sufficiently accurate integration
ofenrichmentfunctionsandtheirderivative. Instead,anintegratedsingularbasisfunctionmethod
(ISBFM)isintroducedtocircumventthisdifficulty. Iftheenrichmentbasisfunctionsareproperly
selectedsothattheysatisfythegoverningequationandtheboundaryconditionsontheboundaries
connected to the singularity point, the resulting ISBFM Galerkin formulation considers only the
integration of the enrichment functions on the boundaries away from the singularity point. This
ii
ISBFMGalerkinformulationalsoallowseffectiveMORproceduresproposedinthiswork.
Two MOR methods via projection of a discrete system obtained from the ISBFM Galerkin
formulationhavebeenproposed. ThesearetheISBFM-URmethodbasedonauniformprojection,
and the ISBFM-DR method based on the decomposed projection of the full scale system. The
ISBFM-UR method projects all smooth and non-smooth DOF to the reduced order space, while
the ISBFM-DR method projects smooth and non-smooth DOF separately in order to retain the
singularbehaviorofthefullscalesolution.
Error estimation and stability analysis show that the reduced order solution from the ISBFM-
DR method with the decomposed projection can achieve a much-enhanced accuracy with only
slightincreaseofconditionnumbercomparedtotheISBFM-URmethodwithuniformproject. Itis
shownintheerrorestimationandconfirmedbythenumericalresultsthattheerrorsofthetwoMOR
methods is mainly controlled by the right hand side vectors of the full and reduced order systems.
While the stability analysis shows that the two MOR methods provide better conditioning that the
full scale system, it also suggests that the conditioning of the ISBFM-DR method is somewhat
compromisedbyitsbetteraccuracycomparedtothatoftheISBFM-URmethod. Nevertheless,the
complexityanalysisshowsthattheISBFM-DRmethoddoesprovideaslightlybetterefficiencyin
additiontotheenhancedaccuracycomparedtotheISBFM-URmethod.
TheproposedMORmethodsareappliedPoissonproblemswithsingularitiesandLEFMprob-
lems. The numerical tests validate the effectiveness of the proposed MOR methods, and the nu-
merical results on accuracy and stability of the proposed methods are also shown to be in good
agreementwiththeanalyticalpredictionscarriedinthisresearch.
iii
ThedissertationofCamilleMarodonisapproved.
StanleyB.Dong
ErtugrulTaciroglu
JosephM.Teran
Jiun-ShyanChen,CommitteeChair
UniversityofCalifornia,LosAngeles
2014
iv
TomyparentsMartineandClaude,
v
Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 LiteratureReview 4
2.1 ReviewofMeshfreeMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 ReproducingKernelParticleMethod . . . . . . . . . . . . . . . . . . . . 6
2.1.3 NumericalTreatmentsofSingularitiesandCracks . . . . . . . . . . . . . 9
2.2 ReviewofModelOrderReduction . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 SVD-basedMORMethods . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Moment-MatchingMORMethods . . . . . . . . . . . . . . . . . . . . . 13
3 MeshfreeISBFMGalerkinFormulationforPoissonProblemswithSingularities 16
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1 ModelProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2 NumericalTreatmentsofSingularities . . . . . . . . . . . . . . . . . . . . 19
3.2 IntegratedSingularBoundaryFunctionMethod(ISBFM) . . . . . . . . . . . . . . 20
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3.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.2 ISBFMGalerkinFormulation . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.3 DiscretizationofISBFMGalerkinFormulation . . . . . . . . . . . . . . . 24
3.3 NumericalExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 ISFBMGalerkinbasedMORMethodforPoissonProblemswithSingularities 32
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1.2 SingularValueDecomposition . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1.3 ProperOrthogonalDecompositionMOR . . . . . . . . . . . . . . . . . . 35
4.1.4 ApplicationtoLinearStaticSystem . . . . . . . . . . . . . . . . . . . . . 38
4.2 ReducedModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 NumericalExamplesandDiscussions . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3.1 ReducedOrderModelingofaL-shapedDomainPoissonProblem . . . . . 41
4.3.2 ReducedOrderModelingofaCrackedBeamPoissonProblem . . . . . . . 46
5 MeshfreeISBFMGalerkinFormulationforLinearElasticFractureMechanics 50
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.1.1 BasicEquationsinLinearElasticFractureMechanics . . . . . . . . . . . . 50
5.1.2 NumericalTreatmentsofCracks . . . . . . . . . . . . . . . . . . . . . . . 54
5.2 ExtensionofISBFMtoLEFMProblems . . . . . . . . . . . . . . . . . . . . . . . 55
5.2.1 ISBFMGalerkinFormulation . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2.2 DiscretizationofISBFMGalerkinFormulation . . . . . . . . . . . . . . . 57
5.2.3 DiscreteSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.4 NumericalExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6 ISBFMGalerkinbasedMORMethodsforLinearElasticFractureMechanics 71
6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
vii
6.2 ReducedModelsforLEFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3 NumericalExampleandDiscussions . . . . . . . . . . . . . . . . . . . . . . . . . 73
7 Accuracy, Stability and Complexity Analysis of the Proposed ISBFM based MOR
Methods 77
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.2 ErrorBoundEstimationoftheReducedSolutions . . . . . . . . . . . . . . . . . . 77
7.3 StabilityAnalysisoftheReducedSystems . . . . . . . . . . . . . . . . . . . . . . 81
7.4 ComplexityAnalysisoftheMORProcedures . . . . . . . . . . . . . . . . . . . . 84
7.5 NumericalExamplesandDiscussions . . . . . . . . . . . . . . . . . . . . . . . . 87
7.5.1 L-shapedDomainPoissonProblem . . . . . . . . . . . . . . . . . . . . . 87
7.5.2 CrackedBeamPoissonProblem . . . . . . . . . . . . . . . . . . . . . . . 94
7.5.3 LinearElasticFractureProblem . . . . . . . . . . . . . . . . . . . . . . . 99
8 Conclusion 104
8.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.2 RecommandationsforFutureResearch . . . . . . . . . . . . . . . . . . . . . . . . 106
viii