第二小组 边界单元法

1-0StudyoftheBEM

边界元法学习成员：高成路、郭焱旭、梅洁、李铭、金纯、刘克奇、匡伟、高松、李崴1-1岩土工程的数值方法工程问题数学模型偏微分方程的边值问题或初值问题边界积分方程问题解析方法数值方法解析方法数值方法FDMFEMEFM其它BEM其它KeywordsaboutBEMCharacteradvantage/disadvantageApplicationandtransformation

oftheBEMBasicconceptsDevelopmentoftheBEMBasicconceptsoftheBEM目录1-2StudyoftheBEMKeywords1-3applicability适用性

stressanddeformationanalysis应力和变形分析

integralstatement

功互等定理

kernels核函数quadraticelements二次单元

discretization离散化

approximation近似值shapefunctions形函数

intrinsiccoordinate本征坐标Gaussianquadrature高斯正交

singularity奇异性，奇异点

CauchyPrincipalValue柯西主值.variationalformulation变分公式化，变分表述1-4numericalintegration数值积分

sparseandsymmetricmatrices稀疏对称矩阵

fullypopulatedandasymmetricmatrices全充填非对称矩阵Weightedresidualprinciple加权余量法

isoparametricelements等参单元undergroundexcavations地下开挖

fracturingprocesses破裂过程

In-situstress原位应力

permeabilitymeasurements渗透性观测coupledthermo-mechanical热力耦合materialheterogeneity材料各向异性Somigliana’sidentity索米利亚纳恒等式hybridmodel混合模型Keywords1-5damageevolutionprocesses损伤演化过程

homogeneousandlinearlyelasticbodies.各向同性线弹性体sourcedensities原密度

fractureanalysis断裂分析

fieldpoint场点globalstiffnessmatrices整体刚度矩阵

normalderivative法向导数

fracturepropagationproblems裂隙传播问题boreholestability钻孔稳定性

rockspalling岩石开裂

stressintensityfactors(SIF)应力强度因子

maximumtensilestrength最大抗拉强度microscopic微观的Keywords1-6heatgradients热力梯度

sharpcorners钝化边角

degreesoffreedom自由度

potentialfunction势函数

meshlesstechnique无单元技术

movingleastsquares移动最小二乘法simplificationoftheintegration积分简化

leastsquaremethod最小二乘法analyticalintegrationofdomainintegrals.积分域的解析解Fourierexpansionofintegrandfunctions.被积函数的傅里叶展开higherorderfundamentalsolutions.高阶基本解theDualReciprocityMethod(DRM).双重互易法KeywordsKeywordsaboutBEMCharacteradvantage/disadvantageApplicationandtransformation

oftheBEMBasicconceptsDevelopmentoftheBEMBasicconceptsoftheBEM目录1-7StudyoftheBEMBasicconcepts1-8UnliketheFEMandFDMmethods,theBEMapproachinitiallyseeksaweaksolutionatthegloballevelthroughanintegralstatement,basedonBetti’sreciprocaltheoremandSomigliana’sidentity.ForalinearelasticityproblemwithdomainΏ;boundaryΓofunitoutwardnormalvectornί

，andconstantbodyforcefί,forexample,theintegralstatementiswrittenas

(8)ThesolutionoftheintegralEq.(8)requiresthefollowingsteps:1-9(1)DiscretizationoftheboundaryΓwithafinitenumberofboundaryelements.Basicconcepts

(9)1-10(2)Approximationofthesolutionoffunctionslocallyatboundaryelementsby(trial)shapefunctions,inasimilarwaytothatusedforFEM.Thedisplacementandtractionfunctionswithineachelementarethenexpressedasthesumoftheirnodalvaluesoftheelementnodes:Basicconcepts

(10)1-11SubstitutionofEqs.(10)into(9)andforEq.(8)canbewritteninmatrixformasBasicconcepts

(11)

(12)1-12(3)EvaluationoftheintegralsTij,UijandBiwithpointcollocationmethodbysettingthesourcepointPatallboundarynodessuccessively.(4)Incorporationofboundaryconditionsandsolution.IncorporationoftheboundaryconditionsintothematrixEq.(12)willleadtofinalmatrixequationBasicconcepts

(14)1-13(5)Evaluationofdisplacementsandstressesinsidethedomain.Forpracticalproblems,itisoftenthestressesanddisplacementsatsomepointsinsidethedomainofinterestthathavespecialsignificance.UnliketheFEMinwhichthedesireddataareautomaticallyproducedatallinteriorandboundarynodes,whethersomeofthemareneededornot,inBEMthedisplacementandstressvaluesatanyinteriorpoint,P,mustbeevaluatedseparatelybyBasicconcepts

(16)(15)KeywordsaboutBEMCharacteradvantage/disadvantageApplicationandtransformation

oftheBEMBasicconceptsDevelopmentoftheBEMBasicconceptsoftheBEM目录1-14StudyoftheBEM1-15ThedevelopmentofBEMIn1963,JaswonandSymmgavetheboundaryintegralequationmethodforsolvingpotentialproblems.In1967,RizzoandCrusegotthebreakthroughforstressanalysisinsolids.In1978,Crusestudiedforfracturemechanicsapplications,basedonBetti’sreciprocaltheorem(Betti,1872)andSomigliana’sidentityinelasticitytheory(Somigliana,1885).In1977,BrebbiaandDominguezwrittenthebasicequationsusingtheweightedresidualprinciple.Watson(1976)gavetheintroductionofisoparametricelementsusingdifferentordersofshapefunctionsinthesamefashionasthatinFEM,greatlyenhancedtheBEM’sapplicabilityforstressanalysisproblems.1-16CrouchandFairhurst(1973),BradyandBray(1978)takenmostnotableoriginaldevelopmentsofBEMapplicationinthefieldofrockmechanics.Intheearly80s,PanandMaier(1997),Elzein(2000)andGhassemistartedtoconcernBEMformulationsforcoupledthermo-mechanicalandhydro-mechanicalprocesses.KuriyamaandMizuta(1993),Kuriyama(1995)andCayolandCornet(1997)reported3-DapplicationsduetotheBEM’sadvantageinreducingmodeldimensions,,especiallyusingDDMforstressanddeformationanalysis.ThedevelopmentofBEMKeywordsaboutBEMCharacteradvantage/disadvantageApplicationandtransformation

oftheBEMBasicconceptsDevelopmentoftheBEMBasicconceptsoftheBEM目录1-17StudyoftheBEM1-18advantageThemainadvantageoftheBEMisthereductionofthecomputationalmodeldimensionbyone,withmuchsimplermeshgenerationandthereforeinputdatapreparation,comparedwithfulldomaindiscretizationmethodssuchastheFEMandFDM.TheBEMisoftenmoreaccuratethantheFEMandFDM,duetoitsdirectintegralformulation.优点:降低求解问题的维数,3D问题变为2D问题,2D变为1D问题.具有较高的精度,原因:仅仅对边界进行离散,域内点的值采用边界上的已知量计算得到.1-19disadvantagetheBEMisnotasefficientastheFEMindealingwithmaterialheterogeneity,becauseitcannothaveasmanysub-domainsaselementsintheFEM.TheBEMisalsonotasefficientastheFEMinsimulatingnon-linearmaterialbehaviour,suchasplasticityanddamageevolutionprocesses,becausedomainintegralsareoftenpresentedintheseproblems.KeywordsaboutBEMCharacteradvantage/disadvantageApplicationandalternativeformulation

oftheBEMBasicconceptsDevelopmentoftheBEMBasicconceptsoftheBEM目录1-20StudyoftheBEM1-21ApplicationofBEM—FractureanalysiswithBEMToapplystandarddirectBEMforfractureanalysis,thefracturesmustbeassumedtohavetwooppositesurfaces,exceptattheapexofthefracturetipwherespecialsingulartipelementsmustbeused.DenoteΓcasthepathofthefracturesinthedomainΏwithitstwooppositesurfacesrepresentedbyΓc+andΓc-,respectively,Somigliana’sidentity(whenthefieldpointisontheboundary)canbewrittenas

(17)1-22TwonewtechniqueswereproposedforfractureanalysiswithBEM.ThefirstoneisDualBoundaryElementMethod(DBEM),whichwasfirstpresentedbyPortela(1992),andwasextendedto3-DcrackgrowthproblemsbyMiandAliabadi(1992,1994).Theessenceofthistechniqueistoapplydisplacementboundaryequationsatonesurfaceofafractureelementandtractionboundaryequationsatitsoppositesurface,althoughthetwoopposingsurfacesoccupypracticallythesamespaceinthemodel.Thegeneralmixedmodefractureanalysiscanbeperformednaturallyinasingledomain.DBEM—FractureanalysiswithBEM1-23ThesecondoneisDDM.TheDDMhasbeenwidelyappliedtosimulatefracturingprocessesinfracturemechanicsingeneralandinrockfracturepropagationproblemsinparticularduetotheadvantagethatthefracturescanberepresentedbysinglefractureelementswithoutneedforseparaterepresentationoftheirtwooppositesurfaces,asshouldbedoneinthedirectBEMsolutions.DDM—FractureanalysiswithBEM1-24ApplicationofBEM—FractureanalysiswithBEMButtherearestillgreatboundednessinanalyzingfracturingprocessesusingBEM,especiallyforrockmechanicsproblems.Ontheonehand,whathappensexactlyatthefracturetipsinrocksstillremainstobeadequatelyunderstood,Ontheotherhand,complexnumericalmanipulationsarestillneededforre-meshingfollowingthefracturegrowthprocesssothatthetipelementsareaddedtowherenewfracturetipsarepredicted.Duetotheabovedifficulties,fracturegrowthanalysesinrockmechanicshavenotbeenwidelyapplied.KeywordsaboutBEMCharacteradvantage/disadvantageAlternativeformulation

oftheBEMBasicconceptsDevelopmentoftheBEMBasicconceptsoftheBEM目录1-25StudyoftheBEM1-26AlternativeformulationsassociatedwithBEMThestandardBEM,DBEMandDDMaspresentedabovehaveacommonfeature:thefinalcoefficientmatricesoftheequationsarefullypopulatedandasymmetric,duetothetraditionalnodalcollocationtechnique.Thismakesthestorageoftheglobalcoefficientmatrixandsolutionofthefinalequationsystemlessefficient,comparedwithFEM.Andthismethodneedsspecialtreatmentfortheproblemwithsharpcornersontheboundarysurfaces(curves)oratthefractureintersections,andartificialcornersmoothing,additionalnodesorspecialcornerelementsareusuallythetechniquesappliedtosolvethisparticulardifficulty.1-27GalerkinBoundaryElementMethodTheGBEMproducesasymmetriccoefficientmatrixbymultiplyingthetraditionalboundaryintegralbyaweightedtrailfunctionandintegratesitwithrespecttothesourcepointontheboundaryforasecondtime,inaGalerkinsenseofweightedresidualformulation.

(19)1-28TheGBEMisanattractiveapproachduetothesymmetryofitsfinalsystemequation,whichpavesthewayforthevariationalformulationofBEMforsolvingnon-linearproblems.GalerkinBoundaryElementMethod1-29BoundaryContourMethodTheBoundaryContourMethod(BCM)involvesrearrangingthestandardBEMintegralEq.(8)sothatthedifferenceofthetwointegralsappearingontheright-handsideofEq.(8)canberepresentedbyavectorfunctionFi=Uij*tj–tij*ujwhichisdivergencefree

(8)(22)1-30TheBCMapproachisattractivemainlybecauseofitsfurtherreductionofcomputationalmodeldimensionsandsimplificationoftheintegration.Thesavingsinpreprocessingofthesimulationsareclear.Treatmentoffracturesandmaterialnon-homogeneityhasnotbeenstudiedinBCM;thesemaylimititsapplicationstorockmechanicsproblemsconsideringthepresentstate-ofthe-art.BoundaryContourMethod1-31BoundaryNodeMethodThemethodisacombinationoftraditionalBEMwithameshlesstechniqueusingthemovingleastsquaresforestablishingtrialfunctionswithoutanexplicitmeshofboundaryelements.Itfurthersimplifiesthemeshgenerationtasks.Itsapplicationsconcentrateonshapesensitivityanalysisatpresentandsolutionofpotentialproblems,butcanbeextendedtogeneralgeom