Boundary element methods with applications to nonlinear problems

733 Pages·2010·15.572 MB·English

by  Goong Chen, Jianxin Zhou

ATLANTIS STUDIES IN MATHEMATICS FORENGINEERING AND SCIENCE
VOLUME7
SERIES EDITOR:C.K.CHUI
Atlantis Studies in
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Boundary Element Methods
with Applications to Nonlinear
Problems
Goong Chen
ProfessorofMathematicsandAerospace Engineering,TexasA&M
University
CollegeStation, Texas77843,USA
and
DistinguishedProfessor ofMathematics
NationalTaiwan University
Taipei,Taiwan, RepublicofChina
Jianxin Zhou
ProfessorofMathematics
TexasA&MUniversity
CollegeStation,Texas77843,USA
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Preface to the 1st edition
Boundaryelementmethods(BEM)haveundergonerapidadvancementinrecentyears.Be-
cause of their numerous advantages, such as ease of coding, small memory requirement
andcomputationalefficiency,theyhavebecomeamajornumericaltool.Theanalysisand
rigor of BEM have been strengthened through the work of several mathematicians. It is
nowpossiblefortheauthorstofittogetherthemanyscatteredcontributionsintoacompre-
hensiveaccount.Inthisbook,wepresentmathematicalformulationsofboundaryintegral
equations(BIE)forseveralofthemostimportantlinearellipticboundaryvalueproblems
(BVP), discuss their computationalalgorithms and the accuracy of their solutions, illus-
trate the numerical solutions and show some applications. We wrote this monograph as
a referencesourcefor researcherswhoare concernedwith numericalsolutionsof partial
differentialequations(PDE),andasagraduatetextforacourseinthissubject.
IthasbeenknownforalongtimethatitispossibletoformulateBIEwithsingularkernelsto
solveBVPofthelinearelliptictype.Theunknownboundarydatainthesolutiondependon
theprescribedboundarydatathroughthoseBIE.Oncealltheboundarydata(ortheCauchy
data)becomeavailable,thesolutionofthegivenPDEisobtainedbyanapplicationofthe
Green’sformula.ComplexvariabletechniquesforsolvingBIEonplanarcomplexcontours
wereperfectedbytheRussianschool(seeMuskhelishvili[137])intheearly1950s,andare
stillwidelyusedtodayfortwo-dimensionalpotentialandelasticityproblems.Amonumen-
talachievement,developingageneraltheoryofBIEusingthefundamentalsolutionrather
thanthecomplexCauchykernel,ispresentedinV.D.Kupradze’sbook[117],publishedin
1965.Inthatdecade,withelectroniccomputersgaininglargerpowerandfasterspeed,and
being in wider use in the U.S., T.A. Cruse, F.J. Rizzo [58, 160, 59] and othersstarted to
computesolutionstomanyBIEarisingincontinuummechanicsandobtainedexcellentnu-
mericalresults.TheBIEtheyusedatthattimewereprimarilyformulatedfromtheGreen’s
formula,whichiscommonlyreferredtoasthedirectapproachintheliterature.
vii
viii BoundaryElementMethodswithApplicationstoNonlinearProblems
Actually,forellipticBVPanalternativewayofformulatingsimplersystemsofBIEispos-
sible. In a paper [72] published in 1961 G. Fichera used ideas from potential theory to
represent the solution as a simple-layer potential. This ansatz leads to greatly simplified
systems of BIE whose solutions, as a consequence, are much easier to compute. Never-
theless, Fichera’spaperremainedlargelyobscureuntilmorethana decadelater, whenin
1973G.C.HsiaoandR.C.MacCamy[95]adaptedhisideasandlucidlyexemplifiedthem
forthePoissonequationonanexteriordomain,andforthesecondbiharmonicproblemin
two-dimensionallinear elasticity. The argumentsin [95] are now standardin provingthe
existenceanduniquenessofsolutionsofBIE,asareshownatseveralplacesinourbook.
WhenBIEareapproximatednumerically,thequestionsofconvergenceanderroranalysis
need to be addressed. The mathematical foundation of such an analysis was laid down
in 1977 in a paper [96] by G.C. Hsiao and W.L. Wendland. There, they used the strong
ellipticity of the simple-layer boundary operator to derive sharp error estimates for the
solutions of the Galerkin approximations of BIE. Their proof works in arbitrary space
dimensionsforanystronglyellipticpseudodifferentialequationsarisingfromellipticBVP.
In practice, however, BIE are more easily approximated by point collocation. Error
analysis questions concerning collocation approximations were essentially resolved by
D.N.Arnold,J.SaranenandW.L.Wendlandin[9,165,10]inthecaseoftwo-dimensional
problems,whereinthe culminatingpaper[10] the maintechniquewas a delicateFourier
series analysis. Nevertheless, as of this writing, no analogoussuccess has been achieved
forcollocationanalysisofthree-dimensionalproblems.
Theauthors’interestsinBEMstemfromnumericalsolutionstovariousappliedproblems
inscienceandengineering.InourpursuitofnumericalsolutionsofPDE,wemarvelledat
the efficacy of the BEM and we were gratified by the accuracy of the numericalresults.
During those years we have also received growing demands from graduate students in
manyengineeringdepartmentsaskingustoteachcoursescontainingBEMmaterial.While
teachingthose courses, we beganto realize the needsof developinga graduateleveltext
and monographthat is sufficiently self-contained, emphasizes more mathematical rigor,
andincludesmanyusefulexamplesofapplicationsandillustrationstostimulatetheaudi-
ence’s interests. Indeed,these are the three major goalswe have been striving to achieve
throughoutthevolume—wehopewithsomesuccess.
The organizationand selectionof the materialcarrythe strongbiasof the authors.There
are10chaptersinthebook.Thefirstfiveconsistofpreparatorymaterial:
Prefacetothe1stedition ix
(1) Chapter 1, an introduction, gives readers some quick ideas as how BEM work and
howtheycomparewithothernumericalPDEmethods.
(2) SobolevspacesarefundamentalinthemoderntheoryofPDE.InChapter2,weex-
plainsomeoftheirbasicpropertiesthatareessentialinunderstandingthesolvability
ofBIE.
(3) Hypersingularintegralsoccurnaturallyinboundaryelementcomputations.Themain
objectiveofChapter3istoenablereaderstounderstandhowtoregularizesuchinte-
gralsbaseduponthetheoryofdistributions.
(4) Boundary integral operators can be studied in an elegant way using the theory of
pseudodifferentialandFredholmoperators.InChapter4,wegiveaconciseaccount
ofthisapproachanditsapplicationstoBIE.
(5) Themathematicaltheoryoffinite elementsisa prerequisiteforBEM.InChapter5,
wedescribethecommonlyusedfiniteelementspacesandtheirbasicproperties.
ThesubsequentchaptersdealwiththecorpusofBEMforPDE:
(6) Chapter6studiesBIEforthepotentialequation.
(7) Chapter7studiesBIEfortheHelmholtzequation.
(8) Chapter8studiesBIEforthethinplateequation.
(9) Chapter9studiesBIEforthelinearelastostaticsystems.
(10) Finally,Chapter10containserrorestimatesforGalerkinandcollocationsolutionsof
generalBIE.
InChapters6–9,foreachspecifictypeofPDEtreated,thephysicalbackgroundisfirstde-
scribed,andfromtherefollowthedifferentlayerformulationsforvarioustypesofbound-
aryconditions.Numericalexamplesandcomputergraphicsarethenpresentedtoillustrate
thetheory.
Fromourcomputationalexperience,ourfavoriteapproachisthe(augmented)simple-layer
representationcollocatedbypiecewiseconstantboundaryelements.Amongdifferentad-
missible layer representations and discretization schemes, this approach is certainly the
easiest to programon a computer.It yieldsrather smoothprofiles of solutionsdue to the
smoothingpropertyofthesimple-layerboundaryoperator.Inthediscussionofthecompu-
tationalaspectsoftheexamplesinChapters6–9,wehaveavoidedminutetechnicaldetails
of discretizationand quadrature,as readerscan easily improviseon their own, also since