Fast Boundary Element Methods in Engineering and Industrial Applications

278 Pages·2012·2.836 MB·English

by  Langer U., Wendland W.L., et al. (eds.)

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Fast Boundary Element
Methods in Engineering and
Industrial Applications

Ulrich Langer, Martin Schanz,
Olaf Steinbach, and Wolfgang L. Wendland (Eds.)

123
Editors
Prof. Dr. Ulrich Langer Prof. Dr. Olaf Steinbach
Institut für Numerische Mathematik Institut für Numerische Mathematik
Johannes Kepler Universität Linz Technische Universität Graz
4040 Linz, Austria 8010 Graz, Austria

Prof. Dr. -Ing. Martin Schanz Prof. Dr. -Ing. Dr. h. c. Wolfgang L. Wendland
Institut für Baumechanik Institut für Angewandte Analysis und
Technische Universität Graz Numerische Simulation
8010 Graz, Austria Universität Stuttgart
70569 Stuttgart, Germany

ISSN: 1613-7736 e-ISSN: 1860-0816
ISBN: 978-3-642-25669-1 e-ISBN: 978-3-642-25670-7
DOI 10.1007/ 978-3-642-25670-7
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Library of Congress Control Number: 2011944752
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Preface
This volume on mathematical aspects and applications of fast boundary element
methodsinengineeringandindustrycontainseightcontributionsonthestateofthe
art in this field. This book is strongly related to the annual So¨llerhaus workshops
onFastBoundaryElementMethodsinIndustrialApplications1whererecenttrends
and new methodologiesare discussed to solve todays challengingproblemsin al-
mostallareasofapplications.ThestimulatingatmosphereoftheSo¨llerhauswork-
shopscontributedalottonewdevelopmentsandnewinterdisciplinarycooperations
whicharealsodocumentedwithinthisbook.Thisspiritofstrongcooperationsbe-
tween mathematicians and engineers, with direct applicationsin industry, follows
an already long–ongoinghistory. To underline this, we just mention the volumes
BoundaryElementTopics(Springer1997),BoundaryElementAnalysis.Mathemat-
icalAspectsandApplications(Springer2007),andthespecialissueofComputing
andVisualizationinScience(Volume8,2005),whichindicatethedevelopmentof
themathematicalfoundationsofboundaryintegralequationmethodsandtheappli-
cationsoffastboundaryelementmethods.
Nowadays,fastboundaryelementmethodsareapowerfultoolforthesimulation
of physical phenomena in different fields of applications. In particular, boundary
integral equation techniques are well suited for the solution of partial differential
equations in unbounded exterior domains, or for problems which are considered
in complicated geometries, but with simple physical model assumptions. The lat-
teralsoinvolvesapplicationswithnonlinearinterfaceortransmissionconditions,as
theyappearinmultiphysicssimulations.Severaloftheseaspectsarecoveredwithin
thisbook.Anefficientandaccuratenumericalsimulationoftime–dependentprob-
lemsbothintimeandfrequencydomainbelongsstilltothemostchallengingprob-
lems. This book includes contributionson the mathematical analysis of boundary
integralformulations,thenumericalanalysisofboundaryelementmethodsandthe
construction of robust and efficient preconditioningstrategies, and the design and
implementationoffastboundaryelementmethodstosolvechallengingproblemsof
interest.
1seehttp://www.numerik.math.tu-graz.ac.at/tagungen
VI Preface
Theaimofthisbookistopresentsomeofthecurrentdevelopmentsoffastbound-
aryelementmethodsandtheirapplications.Weareawarethatsuchabookcannot
coverallaspectsintheanalysisandapplicationsoffastboundaryelementmethods.
Therearenocontributions,e.g.,foradaptivefastboundaryelementmethods.Other
missing topics include the use of fast boundary element methods for the simula-
tion of complexmultiphysicsproblemsincludingthe couplingwith finite element
methods, as well as related inverse and shape optimization problems. In fact, this
bookmay serve to presentsome of the basic tools to handlethe abovementioned
problems. The ongoing work on the solution of these problems will be reported
onfutureworkshopsandconferences,andtheresultswillbedocumentedinfuture
publicationsaswell.
We wouldliketothankallauthorsfortheircontributionstothisvolume.More-
over,wealsothankallanonymmousrefereesfortheirwork,theircriticism,andtheir
suggestions.Thesehintswereveryhelpfultoimprovethecontributions.Finally,we
wouldliketothankDr.T.DitzingerofSpringerHeidelbergforthecontinuingsup-
portandpatiencewhilepreparingthisvolume.
Graz,Linz,Stuttgart UlrichLanger
September2011 MartinSchanz
OlafSteinbach
WolfgangL.Wendland
Contents
Differential Forms and Boundary Integral Equations for
Maxwell-TypeProblems .............................................. 1
StefanKurz,BernhardAuchmann
1 Introduction............................................. 1
2 DifferentialForms–Preliminaries .......................... 3
2.1 BasicDefinitions................................. 3
2.2 IntegralTransformations .......................... 11
2.3 FundamentalSolutionoftheHelmholtzEquation...... 14
2.4 Single-LayerandDouble-LayerPotentials............ 16
3 SobolevSpacesofDifferentialForms ....................... 18
3.1 SobolevSpacesontheDomain ..................... 18
3.2 SobolevSpacesontheBoundary.................... 21
4 RepresentationFormula................................... 26
4.1 Maxwell-TypeProblems,SolutionSpaces,andTrace
Operators ....................................... 27
4.2 AsymptoticConditions............................ 28
4.3 RepresentationFormulaforMaxwellSolutions........ 31
4.4 JumpRelationsoftheLayerPotentials............... 42
5 BoundaryIntegralOperators............................... 45
5.1 SymmetryProperties.............................. 47
5.2 EllipticityProperties.............................. 50
6 BoundaryIntegralEquations............................... 53
6.1 Caldero´nProjectorforInteriorandExteriorProblems .. 53
6.2 Equivalent Maxwell-Type Problems, Dual
Transformations.................................. 55
7 Conclusions............................................. 60
References.................................................... 61
VIII Contents
DiscreteElectromagnetismwithShapeFormsofHigherPolynomial
Degree .............................................................. 63
MarvinFleck,SergejRjasanow
1 Introduction............................................. 63
2 TheMagnetoquasistaticApproximationofMaxwell’s
Equations............................................... 66
2.1 ClassicalDerivation .............................. 66
2.2 DifferentialForms................................ 70
3 TheLowOrderVersion ................................... 77
3.1 DualComplex ................................... 77
3.2 DerivativeandTraceOperators ..................... 78
3.3 HodgeOperators ................................. 79
3.4 PairingMatrices ................................. 80
3.5 DiscreteEquation ................................ 80
4 LocalizationofDegreesofFreedom......................... 81
4.1 SmallSimplices.................................. 81
4.2 Higher-DimensionalSimplices ..................... 84
5 HigherOrderDEM....................................... 84
6 NonlinearMaterials ...................................... 88
7 NumericalResults........................................ 89
8 Conclusions............................................. 91
References.................................................... 91
Additive Schwarz Methods for the hp Version of the Boundary
ElementMethodinR3 ................................................ 93
FlorianLeydecker,ErnstP.Stephan
1 Introduction............................................. 93
2 AdditiveSchwarzMethod–GeneralSetting.................. 96
3 AdditiveSchwarzMethodforthehp-VersionBEMforthe
HypersingularIntegralEquationonRectangularMeshes ....... 96
4 AdditiveSchwarzMethodforthe p-VersionBEMforthe
HypersingularIntegralEquationonTriangles................. 100
5 AdditiveSchwarzMethodforthehp-VersionBEMforthe
WeaklyIntegralEquation.................................. 106
References.................................................... 108
Fast Boundary Element Methods for Industrial Applicationsin
Magnetostatics ...................................................... 111
ZoranAndjelic,Gu¨ntherOf,OlafSteinbach,PeterUrthaler
1 Introduction............................................. 111
2 BoundaryIntegralFormulationsforTransmissionProblems..... 112
2.1 ModelProblem .................................. 113
2.2 Steklov–Poincare´OperatorInterfaceEquation ........ 114
2.3 SingleLayerPotentialFormulation.................. 117
2.4 DoubleLayerPotentialFormulation................. 118
Contents IX
2.5 EquivalenceandUniqueSolvabilityoftheBoundary
IntegralEquations................................ 121
2.6 EvaluationoftheMagneticField.................... 121
3 BoundaryElementMethods ............................... 124
3.1 Steklov–Poincare´OperatorInterfaceEquation ........ 125
3.2 SingleLayerPotentialFormulation.................. 127
3.3 DirectDoubleLayerPotentialFormulation........... 128
3.4 IndirectDoubleLayerPotentialFormulation.......... 130
4 NumericalExamples ..................................... 131
4.1 Sphere.......................................... 131
4.2 Cube ........................................... 132
4.3 Ring ........................................... 133
4.4 RingwithGap ................................... 138
4.5 ControllableReactor.............................. 139
5 Conclusions............................................. 141
References.................................................... 141
WavePropagationProblemsTreatedwithConvolutionQuadrature
andBEM ............................................................ 145
LehelBanjai,MartinSchanz
1 Introduction–StateoftheArt.............................. 145
2 TimeDependentBoundaryIntegralEquations ................ 147
2.1 GoverningEquations.............................. 148
2.2 IntegralEquations................................ 152
3 ConvolutionQuadrature................................... 155
3.1 LinearMultistepBasedConvolutionQuadrature....... 156
3.2 Runge-KuttaBasedConvolutionQuadrature.......... 157
3.3 Implementation .................................. 159
4 ConvolutionQuadratureAppliedtoHyperbolicInitialValue
Problems ............................................... 163
4.1 BoundsintheLaplaceDomain ..................... 164
4.2 PropertiesofConvolutionWeights .................. 166
4.3 DissipationandDispersion......................... 168
5 SpaceDiscretization...................................... 169
5.1 GalerkinandCollocationinSpace .................. 170
5.2 FastData-SparseMethodsinFrequencyDomain ...... 171
6 NumericalExample ...................................... 172
Appendix..................................................... 177
References.................................................... 179
FastNystro¨mMethodsforParabolicBoundaryIntegralEquations .... 185
JohannesTausch
1 Introduction............................................. 185
2 HeatPotentialsasAbelIntegralOperators ................... 187
3 TimeDependentIntegralOperators ......................... 189
3.1 ProjectionMethods............................... 190