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/disadvantageApplicationandtransformationoftheBEMBasicconceptsDevelopmentoftheBEMBasicconceptsoftheBEM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/disadvantageApplicationandtransformationoftheBEMBasicconceptsDevelopmentoftheBEMBasicconceptsoftheBEM目录1-7StudyoftheBEMBasicconcepts1-8UnliketheFEMandFDMmethods,theBEMapproachinitiallyseeksaweaksolutionatthegloballevelthroughanintegralstatement,basedonBetti’sreciprocaltheoremandSomigliana’sidentity.ForalinearelasticityproblemwithdomainΏ;boundaryΓofunitoutwardnormalvectornί，andconstantbodyforcefί,forexample,theintegralstatementiswrittenasi(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/disadvantageApplicationandtransformationoftheBEMBasicconceptsDevelopmentoftheBEMBasicconceptsoftheBEM目录1-14StudyoftheBEM1-15ThedevelopmentofBEMIn1963,JaswonandSymmgavetheboundaryintegralequationmethodforsolvingpotentialproblems.In1967,RizzoandCrusegotthebreakthroughforstressanalysisinsolids.In1978,Crusestudiedforfracturemechanicsapplications,basedonBetti’sreciprocaltheorem(Betti,1872)andSomigliana’sidentityinelasticitytheory(Somigliana,1885).In1977,BrebbiaandDominguezwrittenthebasicequationsusingtheweightedresidualprinciple.Watson(1976)gavetheintroductionofisoparametricelementsusingdifferentordersofshapefunctionsinthesamefashionasthatinFEM,greatlyenhancedtheBEM’sapplicabilityforstressanalysisproblems.1-16CrouchandFairhurst(1973),BradyandBray(1978)takenmostnotableoriginaldevelopmentsofBEMapplicationinthefieldofrockmechanics.Intheearly80s,PanandMaier(1997),Elzein(2000)andGhassemistartedtoconcernBEMformulationsforcoupledthermo-mechanicalandhydro-mechanicalprocesses.KuriyamaandMizuta(1993),Kuriyama(1995)andCayolandCornet(1997)reported3-DapplicationsduetotheBEM’sadvantageinreducingmodeldimensions,,especiallyusingDDMforstressanddeformationanalysis.ThedevelopmentofBEMKeywordsaboutBEMCharacteradvantage/disadvantageApplicationandtransformationoftheBEMBasicconceptsDevelopmentoftheBEMBasicconceptsoftheBEM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.1-19disa