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)值方法工程問題數(shù)學(xué)模型偏微分方程的邊值問題或初值問題邊界積分方程問題解析方法數(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稀疏對稱矩陣

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

isoparametricelements等參單元undergroundexcavations地下開挖

fracturingprocesses破裂過程

In-situstress原位應(yīng)力

permeabilitymeasurements滲透性觀測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損傷演化過程

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

fractureanalysis斷裂分析

fieldpoint場點(diǎn)globalstiffnessmatrices整體剛度矩陣

normalderivative法向?qū)?shù)

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

rockspalling巖石開裂

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勢函數(shù)

meshlesstechnique無單元技術(shù)

movingleastsquares移動最小二乘法simplificationoftheintegration積分簡化

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):降低求解問題的維數(shù),3D問題變?yōu)?D問題,2D變?yōu)?D問題.具有較高的精度,原因:僅僅對邊界進(jìn)行離散,域內(nèi)點(diǎn)的值采用邊界上的已知量計算得到.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