The fast solution of boundary integral equations

284 Pages·2007·9.889 MB·English

by  Sergej Rjasanow, Olaf Steinbach

The Fast Solution of Boundary
Integral Equations
MATHEMATICAL AND ANALYTICAL
TECHNIQUES WITH APPLICATIONS
TO ENGINEERING
SeriesEditor
AlanJeffrey
Theimportanceofmathematicsinthestudyofproblemsarisingfromtherealworld,
andtheincreasingsuccesswithwhichithasbeenusedtomodelsituationsranging
from the purely deterministic to the stochastic, in all areas of today’s Physical
SciencesandEngineering,iswellestablished.Theprogressinapplicablemathematics
has been brought about by the extension and development of many important
analytical approaches and techniques, in areas both old and new, frequently aided
bytheuseofcomputerswithoutwhichthesolutionofrealisticproblemsinmodern
Physical Sciences and Engineering would otherwise have been impossible. The
purposeoftheseriesistomakeavailableauthoritative,uptodate,andself-contained
accounts of some of the most important and useful of these analytical approaches
and techniques. Each volume in the series will provide a detailed introduction to a
specificsubjectareaofcurrentimportance,andthenwillgobeyondthisbyreviewing
recentcontributions,therebyservingasavaluablereferencesource.
SeriesTitles:
THEFASTSOLUTIONOFBOUNDARYINTEGRALEQUATIONS
SergejRjasanow&OlafSteinbach,ISBN978-0-387-34041-8
THEORYOFSTOCHASTICDIFFERENTIALEQUATIONSWITHJUMPSAND
APPLICATIONS
RongSitu,ISBN978-0-387-25083-0
METHODSFORCONSTRUCTINGEXACTSOLUTIONSOFPARTIAL
DIFFERENTIALEQUATIONS
S.V.Meleshko,ISBN978-0-387-25060-1
INVERSEPROBLEMS
AlexanderG.Ramm,ISBN978-0-387-23195-2
SINGULARPERTURBATIONTHEORY
RobinS.Johnson,ISBN978-0-387-23200-3
INVERSEPROBLEMSINELECTRICCIRCUITSANDELECTROMAGNETICS
N.V.Korovkin,ISBN978-0-387-33524-7
The Fast Solution of Boundary
Integral Equations
Sergej Rjasanow
UniversitätdesSaarlandes
Olaf Steinbach
TechnischeUniversitätGraz
SergejRjasanow
Fachrichtung6.1–Mathematik
UniversitätdesSaarlandes
Postfach151150
D-66041Saarbrücken
GERMANY
OlafSteinbach
InstitutfürNumerischeMathematik
TechnischeUniversitätGraz
Steyrergasse30
A-8010Graz
AUSTRIA
LibraryofCongressControlNumber:2006927233
ISBN978-0-387-34041-8 e-ISBN978-0-387-34042-5
ISBN0-387-34041-6 e-ISBN0-387-34042-4
Printedonacid-freepaper.
(cid:2)c 2007SpringerScience+BusinessMedia,LLC
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Preface
BoundaryElementMethods(BEM)playanimportantroleinmodernnumer-
icalcomputationsintheappliedandengineeringsciences.Suchalgorithmsare
often more convenient than the traditional Finite Element Method (FEM),
sincethecorrespondingequationsareformulatedontheboundary,and,there-
fore,asignificantreductionofdimensionalitytakesplace.Especiallywhenthe
physicaldescriptionoftheproblemleadstoanunboundeddomain,traditional
methods like FEM become unalluring.
A numerical procedure, called Boundary Element Methods (BEM), has
been developed in the physics and engineering community since the 1950s.
This method turns out to be a powerful tool for numerical studies of various
physical phenomena. The most prominent examples of such phenomena are
the potential equation (Laplace equation) in electromagnetism, gravitation
theory,andinperfectfluids.AfurtherexampleleadingtotheLaplaceequation
is the steady state heat flow. One of the most popular applications of the
BEM is, however, the system of linear elastostatics which can be considered
inbothboundedandunboundeddomains.Asimplemodelforafluidflow,the
Stokessystem,canalsobesolvedbytheuseoftheBEM.Themostimportant
examplesfortheHelmholtzequationaretheacousticscatteringandthesound
radiation.
It has been known for a long time that boundary value problems for el-
liptic partial differential equations can be reformulated in terms of boundary
integral equations. The trace of the solution on the boundary and its co-
normal derivative (Cauchy data) can be found by solving these equations
numerically. The solution of the problem as well as its gradients or even high
order derivatives are then given by the application of Green’s third formula
(representation formula); this method based on Green’s formula is called the
directBEMapproach.Anotherpossibilityistousethepropertythatsingleor
double layer potentials solve the partial differential equation exactly for any
given density function. Thus, this function can be used in order to fulfill the
boundary conditions. The density function obtained this way has, in general,
VI
no physical meaning. Therefore, these boundary element methods are called
indirect.
When boundary integral equations are approximated and solved numer-
ically, the study of stability and convergence is the most important issue.
The most popular numerical methods are the Galerkin methods which per-
fectly fit to the variational formulation of the boundary integral equations.
The theoretical study of the Galerkin methods is now completed and pro-
videsapowerfultheoreticalbackgroundforBEM.Traditionally,however,the
collocation methods were widely used, especially in the engineering commu-
nity. These methods provide an easier practical implementation compared
with the Galerkin methods. However, the stability and convergence theory
for collocation methods is available only for two-dimensional problems. Fur-
thermore, the error analysis of the collocation methods for three-dimensional
problems, when assuming their stability, shows that the rate of convergence
oftheGalerkinmethodsisbetter,whenassumingthatthesolutionissmooth
enough.
Inanycase,anumericalprocedureappliedtotheboundaryintegralequa-
tionleadstoalinearsystemofalgebraicequations.Thematrixofthissystem
isingeneraldense,i.e.almostallitsentriesaredifferentfromzero,and,there-
fore, have to be stored in computer memory. It is clear that this is the main
disadvantage of the BEM compared with FEM which leads to sparse matri-
ces.Thisquadraticamountofcomputermemorysetsverystrong,unattractive
bounds for the discretisation parameters and, often, force the user to switch
to the out–of–core programming. However, so called fast BEM have been de-
veloped in the last two decades. The original methods are the Fast Multipole
Method and the Panel Clustering; another example is the use of wavelets.
Furthermore, the Adaptive Cross Approximation (ACA) was introduced and
successfully applied to many practical problems in the last years.
The purpose of this book is twofold. The first goal is to give an exact
mathematical description of various mathematical formulations and numer-
ical methods for boundary integral equations in the three-dimensional case
in an uniform and possibly compact form. The second goal is a systematic
numerical treatment of a variety of boundary value problems for the Laplace
equation, for the linear elastostatics system, and for the Helmholtz equation.
This study will illustrate both the convergence of the Galerkin methods cor-
responding to the theory and the fast realisation of BEM based on the ACA
method.Werestrictournumericalteststosomemoreorlessartificialsurface
examples. The simplest one is the surface of the unit sphere. Furthermore,
two TEAM examples (Testing Electromagnetic Analysis Methods) will be
considered besides some other non-trivial surfaces.
Thisbookissubdividedintofourparts.Chapter1providesanoverviewof
the direct and indirect reformulations of second order boundary value prob-
lemsbyusingboundaryintegralequations,anditdiscussesthemappingprop-
erties of all boundary integral operators involved. From this, the unique solv-
abilityoftheresultingboundaryintegralequationsandthecontinuousdepen-
VII
dence of the solution on the given boundary data can be deduced. Chapter
2 is concerned with boundary element methods, especially with the Galerkin
method.ThediscreteversionoftheboundaryintegralequationsfromChapter
1 and their variational formulations lead to systems of linear equations with
different matrices. The entries of these matrices are explicitly derived for all
integral operators involved. Chapter 3 describes the Adaptive Cross Approx-
imation of dense matrices and provides, in addition to the theory, some first
numerical examples. The largest part of the book, Chapter 4, contains some
results of numerical experiments. First, the Laplace equation is considered,
where we study Dirichlet, Neumann, and mixed boundary value problems as
wellasaninhomogeneousinterfaceproblem.Then,twomixedboundaryvalue
problemsoflinearelastostaticswillbepresented,and,finally,manyexamples
for the Helmholtz equation are described. We consider again Dirichlet and
Neumann, interior and exterior boundary value problems as well as multifre-
quency analysis. Many auxiliary results are collected in three appendices.
The chapters are relatively independent of one another. Necessary nota-
tions and formulas are not only cross-referred to other chapters but usually
repeated at the appropriate places.
In 2003, Prof. Allan Jeffrey approached us with the idea to write a book
about fast solutions of boundary integral equations. It has been delightful to
writethisbookandwearealsoverythankfulforhisprovidingtheopportunity
to get this book published.
WewouldliketothankourcolleaguesfromtheBEMcommunityformany
useful discussions and suggestions. We are grateful to our home institutions,
the University of Saarland in Saarbru¨cken and the Technical University in
Graz, for providing an excellent scientific environment and financial funding
to our research.
We appreciate the help of Ju¨rgen Rachor, who read the manuscript and
made valuable comments and corrections. Furthermore, the authors would
very much like to express their appreciation to Richard Grzibovski for his
help in performing numerical tests.
Saarbru¨cken and Graz Sergej Rjasanow
March 2007 Olaf Steinbach
Contents
Preface ........................................................ V
1 Boundary Integral Equations .............................. 1
1.1 Laplace Equation........................................ 2
1.1.1 Interior Dirichlet Boundary Value Problem ........... 10
1.1.2 Interior Neumann Boundary Value Problem .......... 13
1.1.3 Mixed Boundary Value Problem..................... 17
1.1.4 Robin Boundary Value Problem..................... 19
1.1.5 Exterior Dirichlet Boundary Value Problem........... 21
1.1.6 Exterior Neumann Boundary Value Problem.......... 22
1.1.7 Poisson Problem .................................. 24
1.1.8 Interface Problem ................................. 26
1.2 Lam´e Equations......................................... 27
1.2.1 Dirichlet Boundary Value Problem .................. 35
1.2.2 Neumann Boundary Value Problem.................. 36
1.2.3 Mixed Boundary Value Problem..................... 37
1.3 Stokes System .......................................... 40
1.4 Helmholtz Equation ..................................... 44
1.4.1 Interior Dirichlet Boundary Value Problem ........... 49
1.4.2 Interior Neumann Boundary Value Problem .......... 50
1.4.3 Exterior Dirichlet Boundary Value Problem........... 52
1.4.4 Exterior Neumann Boundary Value Problem.......... 54
1.5 Bibliographic Remarks ................................... 56
2 Boundary Element Methods ............................... 59
2.1 Boundary Elements...................................... 59
2.2 Basis Functions ......................................... 61
2.3 Laplace Equation........................................ 65
2.3.1 Interior Dirichlet Boundary Value Problem ........... 65
2.3.2 Interior Neumann Boundary Value Problem .......... 72
2.3.3 Mixed Boundary Value Problem..................... 77
X Contents
2.3.4 Interface Problem ................................. 84
2.4 Lame Equations......................................... 87
2.5 Helmholtz Equation ..................................... 91
2.5.1 Interior Dirichlet Problem .......................... 91
2.5.2 Interior Neumann Problem ......................... 93
2.5.3 Exterior Dirichlet Problem ......................... 97
2.5.4 Exterior Neumann Problem......................... 98
2.6 Bibliographic Remarks ................................... 99
3 Approximation of Boundary Element Matrices ............101
3.1 Hierarchical Matrices ....................................101
3.1.1 Motivation .......................................101
3.1.2 Hierarchical clustering .............................108
3.2 Block Approximation Methods ............................112
3.2.1 Analytic Form of Adaptive Cross Approximation ......112
3.2.2 Algebraic Form of Adaptive Cross Approximation .....119
3.3 Bibliographic Remarks ...................................129
4 Implementation and Numerical Examples..................131
4.1 Geometry Description....................................131
4.1.1 Unit Sphere ......................................131
4.1.2 TEAM Problem 10 ................................131
4.1.3 TEAM Problem 24 ................................134
4.1.4 Relay ............................................134
4.1.5 Exhaust manifold .................................135
4.2 Laplace Equation........................................135
4.2.1 Analytical solutions................................135
4.2.2 Discretisation, Approximation and Iterative Solution...137
4.2.3 Generation of Matrices.............................139
4.2.4 Interior Dirichlet Problem ..........................143
4.2.5 Interior Neumann Problem .........................149
4.2.6 Interior Mixed Problem ............................155
4.2.7 Inhomogeneous Interface Problem ...................160
4.3 Linear Elastostatics......................................162
4.3.1 Generation of Matrices.............................162
4.3.2 Relay ............................................163
4.3.3 Foam ............................................166
4.4 Helmholtz Equation .....................................168
4.4.1 Analytical Solutions ...............................169
4.4.2 Discretisation, Approximation and Iterative Solution...169
4.4.3 Generation of Matrices.............................170
4.4.4 Interior Dirichlet Problem ..........................171
4.4.5 Interior Neumann Problem .........................185
4.4.6 Exterior Dirichlet Problem .........................191
4.4.7 Exterior Neumann Problem.........................196
Contents XI
A Mathematical Foundations.................................199
A.1 Function Spaces.........................................199
A.2 Fundamental Solutions ...................................208
A.2.1 Laplace Equation..................................208
A.2.2 Lame System .....................................209
A.2.3 Stokes System ....................................210
A.2.4 Helmholtz Equation ...............................212
A.3 Mapping Properties......................................213
B Numerical Analysis........................................225
B.1 Variational Methods .....................................225
B.2 Approximation Properties ................................231
C Numerical Algorithms .....................................239
C.1 Numerical Integration....................................239
C.2 Analytic Integration .....................................244
C.2.1 Single Layer Potential for the Laplace operator........246
C.2.2 Double Layer Potential for the Laplace operator.......249
C.2.3 Linear Elasticity Single Layer Potential ..............252
C.3 Iterative Solution Methods ...............................256
C.3.1 Conjugate Gradient Method (CG)...................256
C.3.2 Generalised Minimal Residual Method (GMRES)......263
References.....................................................269
Index..........................................................277