An Efficient and High-Order Accurate Boundary Integral Solver

192 Pages·2004·3.02 MB·English

An Efficient and High-Order Accurate Boundary Integral Solver
for the Stokes Equations in Three Dimensional Complex
Geometries
by
Lexing Ying
Adissertationsubmittedinpartialfulfillment
oftherequirementsforthedegreeof
DoctorofPhilosophy
DepartmentofComputerScience
NewYorkUniversity
May2004
DenisZorin
(cid:1)c LexingYing
AllRightsReserved,2004
Acknowledgments
This dissertation would not have been possible without the help and support from
manypeopletowhomIamgreatlyindebted.
First of all, I would like to thank my advisor Denis Zorin for his guidance, en-
couragement and collaboration. It is Denis who taught me how to do research and
his influence will be timeless. He has demonstrated great devotion to the sciences
andalsogreatcareforhisstudents.
I am grateful to my collaborator George Biros for being a great teacher and a
constantsupporter.
I would like to thank Leslie Greengard, Michael Shelley and Olof Widlund for
serving on my thesis committee and for providing valuable discussion and sugges-
tion.
I wish to thank the people I have worked with at MRL for making a wonderful
place to work. Special thanks to Henning Biermann, Eitan Grinspun, Aaron Hertz-
mann,HarperLangston,JianboPengandElifTosun.
Iwouldliketothankthefacultyandstaffofthecomputersciencedepartmentfor
providingagreatenvironmentforacademicresearch.
IamgreatlyindebtedtoYiZhouforhercare,supportandencouragement. With-
outher,mydaysasagraduatestudentwouldhavebeencolorlessandboring.
iii
Finally, I want to thank my parents, Lizhen Ying and Lihua Wang, for their love,
careandencouragement. Theyprovidemewiththebesteducationpossible. Without
them,Iwouldbenothing.
iv
Abstract
Thisdissertationpresentsanefficientandhigh-orderboundaryintegralsolverforthe
Stokes equations in complex 3D geometries. The targeted applications of this solver
are the flow problems in domains involving moving boundaries. In such problems,
traditional finite element methods involving 3D unstructured mesh generation expe-
rience difficulties. Our solver uses the indirect boundary integral formulation and
discretizestheequationusingtheNystro¨mmethod.
Although our solver is designed for the Stokes equations, we show that it can be
generalizedtootherconstantcoefficientellipticpartialdifferentialequations(PDEs)
withnon-oscillatorykernels.
First, we present a new geometric representation of the domain boundary. This
scheme takes quadrilateral control meshes with arbitrary geometry and topology as
input,andproducessmoothsurfacesapproximatingthecontrolmeshes. Oursurfaces
are parameterized over several overlapping charts through explicit nonsingular C∞
parameterizations,dependlinearlyonthecontrolpoints,havefixed-sizelocalsupport
forbasisfunctions,andhavegoodvisualquality.
Second, we describe a kernel independent fast multipole method (FMM) and its
parallel implementation. The main feature of our algorithm is that it is based only
onkernelevaluationanddoesnotrequirethemultipoleexpansionsoftheunderlying
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kernel. We have tested our method on kernels from a wide range of elliptic PDEs.
Our numerical results indicate that our method is efficient and accurate. Other ad-
vantagesincludethesimplicityoftheimplementationanditsimmediateextensionto
other elliptic PDE kernels. We also present an MPI based parallel implementation
whichscaleswelluptothousandsofprocessors.
Third, we present several algorithms to evaluate the singular integrals in our
solver. A singular integral is decomposed into a smooth far field part and a local
part that contains the singularity. The smooth part of the integral is integrated us-
ingthetrapezoidalruleoveroverlappingcharts,andthesingularpartisintegratedin
the polar coordinates which removes or decreases the order of singularity. We also
describe a new algorithm to integrate the nearly singular integrals coming from the
evaluationatpointsclosetotheboundary.
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Contents
Acknowledgments iii
Abstract v
ListofFigures xi
ListofTables xiii
ListofAppendices xv
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 ProblemStatement . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 NeumannBoundaryConditionandOtherPDEs . . . . . . . . . . . 8
1.5 ContributionsandThesisOrganization . . . . . . . . . . . . . . . . 10
1.6 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6.1 DiscretizationMethods . . . . . . . . . . . . . . . . . . . . 12
1.6.2 TechniquesforFastSolvers . . . . . . . . . . . . . . . . . 14
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2 High-orderSurfaceRepresentationScheme 16
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 ManifoldStructure . . . . . . . . . . . . . . . . . . . . . . 19
2.3.2 OverviewoftheConstruction . . . . . . . . . . . . . . . . 21
2.3.3 ChartsandTransitionMaps . . . . . . . . . . . . . . . . . 22
2.3.4 PartitionofUnity . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.5 Defininggeometry. . . . . . . . . . . . . . . . . . . . . . . 26
2.3.6 Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 KernelIndependentFastMultipoleMethod 34
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 ReviewoftheFastMultipoleMethod . . . . . . . . . . . . . . . . 40
3.3 TheNewAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.1 DensityTranslations . . . . . . . . . . . . . . . . . . . . . 45
3.3.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.3 TheCompleteAlgorithm . . . . . . . . . . . . . . . . . . . 55
3.3.4 ImplementationIssues . . . . . . . . . . . . . . . . . . . . 59
3.3.5 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 ErrorAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.1 FMMFactorization . . . . . . . . . . . . . . . . . . . . . . 65
3.4.2 DiscretizationError . . . . . . . . . . . . . . . . . . . . . . 68
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3.4.3 SingleStepError . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.4 TotalDiscretizationError . . . . . . . . . . . . . . . . . . 75
3.5 NumericalResults . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.5.1 AccuracyontheEquivalentDensityApproximation . . . . 77
3.5.2 OverallApproximationError . . . . . . . . . . . . . . . . . 81
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4 ParallelImplementationofKernelIndependentFMM 93
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 TheParallelAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2.1 DataPartitioningandTreeGeneration . . . . . . . . . . . . 96
4.2.2 InteractionCalculation . . . . . . . . . . . . . . . . . . . . 98
4.3 ScalabilityResults . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5 Nystro¨mIntegration 113
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2 DiscretizationandSingularIntegralEvaluationforVelocity . . . . . 114
5.3 ErrorAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.4 EfficientImplementation . . . . . . . . . . . . . . . . . . . . . . . 125
5.5 SingularIntegrationforPressureandStress . . . . . . . . . . . . . 128
5.6 EvaluationofNearlySingularIntegrals . . . . . . . . . . . . . . . 134
5.7 SingularandNearlySingularIntegralEvaluationforOtherEquations 139
5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6 ResultsandApplications 142
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6.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7 Conclusion 152
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.2 FutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Appendices 158
Bibliography 165
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