第二小組 邊界單元法

1-0StudyoftheBEM

邊界元法學(xué)習(xí)成員：高成路、郭焱旭、梅潔、李銘、金純、劉克奇、匡偉、高松、李崴1-1巖土工程的數(shù)值方法工程問(wèn)題數(shù)學(xué)模型偏微分方程的邊值問(wèn)題或初值問(wèn)題邊界積分方程問(wèn)題解析方法數(shù)值方法解析方法數(shù)值方法FDMFEMEFM其它BEM其它KeywordsaboutBEMCharacteradvantage/disadvantageApplicationandtransformation

oftheBEMBasicconceptsDevelopmentoftheBEMBasicconceptsoftheBEM目錄1-2StudyoftheBEMKeywords1-3applicability適用性

stressanddeformationanalysis應(yīng)力和變形分析

integralstatement

功互等定理

kernels核函數(shù)quadraticelements二次單元

discretization離散化

approximation近似值shapefunctions形函數(shù)

intrinsiccoordinate本征坐標(biāo)Gaussianquadrature高斯正交

singularity奇異性，奇異點(diǎn)

CauchyPrincipalValue柯西主值.variationalformulation變分公式化，變分表述1-4numericalintegration數(shù)值積分

sparseandsymmetricmatrices稀疏對(duì)稱矩陣

fullypopulatedandasymmetricmatrices全充填非對(duì)稱矩陣Weightedresidualprinciple加權(quán)余量法

isoparametricelements等參單元undergroundexcavations地下開(kāi)挖

fracturingprocesses破裂過(guò)程

In-situstress原位應(yīng)力

permeabilitymeasurements滲透性觀測(cè)coupledthermo-mechanical熱力耦合materialheterogeneity材料各向異性Somigliana’sidentity索米利亞納恒等式hybridmodel混合模型Keywords1-5damageevolutionprocesses損傷演化過(guò)程

homogeneousandlinearlyelasticbodies.各向同性線彈性體sourcedensities原密度

fractureanalysis斷裂分析

fieldpoint場(chǎng)點(diǎn)globalstiffnessmatrices整體剛度矩陣

normalderivative法向?qū)?shù)

fracturepropagationproblems裂隙傳播問(wèn)題boreholestability鉆孔穩(wěn)定性

rockspalling巖石開(kāi)裂

stressintensityfactors(SIF)應(yīng)力強(qiáng)度因子

maximumtensilestrength最大抗拉強(qiáng)度microscopic微觀的Keywords1-6heatgradients熱力梯度

sharpcorners鈍化邊角

degreesoffreedom自由度

potentialfunction勢(shì)函數(shù)

meshlesstechnique無(wú)單元技術(shù)

movingleastsquares移動(dòng)最小二乘法simplificationoftheintegration積分簡(jiǎn)化

leastsquaremethod最小二乘法analyticalintegrationofdomainintegrals.積分域的解析解Fourierexpansionofintegrandfunctions.被積函數(shù)的傅里葉展開(kāi)higherorderfundamentalsolutions.高階基本解theDualReciprocityMethod(DRM).雙重互易法KeywordsKeywordsaboutBEMCharacteradvantage/disadvantageApplicationandtransformation

oftheBEMBasicconceptsDevelopmentoftheBEMBasicconceptsoftheBEM目錄1-7StudyoftheBEMBasicconcepts1-8UnliketheFEMandFDMmethods,theBEMapproachinitiallyseeksaweaksolutionatthegloballevelthroughanintegralstatement,basedonBetti’sreciprocaltheoremandSomigliana’sidentity.Foralinearelasticityproblemwithdomain?;boundaryΓofunitoutwardnormalvectorn?

，andconstantbodyforcef?,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目錄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目錄1-17StudyoftheBEM1-18advantageThemainadvantageoftheBEMisthereductionofthecomputationalmodeldimensionbyone,withmuchsimplermeshgenerationandthereforeinputdatapreparation,comparedwithfulldomaindiscretizationmethodssuchastheFEMandFDM.TheBEMisoftenmoreaccuratethantheFEMandFDM,duetoitsdirectintegralformulation.優(yōu)點(diǎn):降低求解問(wèn)題的維數(shù),3D問(wèn)題變?yōu)?D問(wèn)題,2D變?yōu)?D問(wèn)題.具有較高的精度,原因:僅僅對(duì)邊界進(jìn)行離散,域內(nèi)點(diǎn)的值采用邊界上的已知量計(jì)算得到.1-19disadvantagetheBEMisnotasefficientastheFEMindealingwithmaterialheterogeneity,becauseitcannothaveasmanysub-domainsaselementsintheFEM.TheBEMisalsonotasefficientastheFEMinsimulatingnon-linearmaterialbehaviour,suchasplasticityanddamageevolutionprocesses,becausedomainintegralsareoftenpresentedintheseproblems.KeywordsaboutBEMCharacteradvantage/disadvantageApplicationandalternativeformulation

oftheBEMBasicconceptsDevelopmentoftheBEMBasicconceptsoftheBEM目錄1-20StudyoftheBEM1-21ApplicationofBEM—FractureanalysiswithBEMToapplystandarddirectBEMforfractureanalysis,thefracturesmustbeassumedtohavetwooppositesurfaces,exceptattheapexofthefracturetipwherespecialsingulartipelementsmustbeused.DenoteΓcasthepathofthefracturesinthedomain?withitstwooppositesurfacesrepresentedbyΓc+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目錄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