力学声响振动研究室 (MSVLAB).ppt

Degeneratescaleanalysisformembraneandplateproblemsusingthemeshlessmethodandboundaryelementmethod 研究生 吳清森指導教授 陳正宗教授陳義麟博士 國立台灣海洋大學河海工程學系結構組碩士班論文口試日期 2004 06 1609 00 10 20 Frameofthethesis Chapter2Green sfunctionandPoissonintegralformula Laplaceproblem Chapte3BIEMandBEMfordegeneratescaleproblem Laplaceandbiharmonicproblem Chapter4Meshlessmethodfordegeneratescaleproblem Laplaceandbiharmonicproblem Literaturereview Engineeringbackground Literaturereview Mathematicalbackground Motivation 1 BIEM BEM 2 MFS TrefftzMethod Methods Techniques DegeneratekernelCirculants Membrane Laplaceequation Plate biharmonicequation Statics Degeneratescaleproblem 1 Dcase Eulerbeam S r x fieldpoint variables sourcepoint fixed Degeneratekernel AlternativederivationsforthePoissonintegralformula a DerivationofthePoissonintegralformula Traditionalmethod Imagesource Null fieldintegralequationinconjunctionwithdegeneratekernels B Boundarydensities Degeneratekernel Unknowncoefficients unknown specified Fundamentalsolution Green sidentity DegeneratescaleforplateanalysisusingtheBIEMandBEM Engineeringproblemgovernedbybiharmonicequation 1 Planeelasticity 2 Slowviscousflow Stokes Flow 3 Solidmechanics Plateproblem Problemstatement flexurerigidity uniformdistributedload domainofinterest Governingequation Boundarycondition Splittingmethod Governingequation Boundarycondition deflectionofthecircularplate Boundaryintegralequationsforplate 2 Slope 3 Normalmoment 4 Effectiveshearforce 1 Displacement Operators Slope Normalmoment Effectiveshearforce Kernelfunctions Fundamentalsolution Kernelfunctions Degeneratekernelsforbiharmonicoperator Mathematicalanalysis Discretemodel Fortheclampedcircularplate uand arespecified formulation Circulant a 1 2 3 4 5 2N 1 2N 2 2N 3 2N Eigenvaluesofthefourmatrices kernel kernel kernel kernel Determinant Degeneratescale a Degeneratescalesfortheclampedcase Degeneratescalesforthesimply supportedcase 6options Degeneratescale formulation formulation Degeneratescalesforthefreecase RelationshipbetweentheLaplaceproblemandbiharmonicproblem a translation b rotation constant NontrivialmodesinFEMandBEM Numberofdegeneratescales Laplaceproblem Laplaceproblem UTformulation LMformulation Nodegeneratescale Numberofdegeneratescales biharmonicproblem formulation formulation formulation Numberofdegeneratescales biharmonicproblem formulation formulation formulation Nodegeneratescaleoccurs Illustrativeexample JFM Mill1977 Weadoptthenull fieldintegralequationinconjunctionwithdegeneratekerneltoderivetheanalyticsolution Exactsolution M 20 M 50 OntheequivalenceoftheTrefftzmethodandMFSforLaplaceandbiharmonicequations TrefftzmethodandMFS isthenumberofcompletefunctions isthenumberofsourcepointsintheMFS StatementforLaplaceproblem Two dimensionalLaplaceproblemwithacirculardomain G E B C Interior Exterior Analyticalsolution Bymatchingtheboundaryconditionat Interiorproblem Exteriorproblem Derivationofunknowncoefficients Trefftzmethod Fieldsolution Interior Exterior T completesetfunctions Interior Exterior Degeneratekernel Interiorproblem Exteriorproblem Fieldsolution Interior Exterior Derivationofunknowncoefficients MFS Trefftz MFS Relationshipbetweenthetwomethods Interior Exterior Bysetting Trefftzmethod MFS Matrix Matrix ill posedproblem Degeneratescaleproblem ill posedproblem Degeneratescaleproblem Circulants where Interior Degeneratescaleproblem R 1 NumericalExamples D B a NumericalExample1 NumericalExample2 NumericalExample3 NumericalExample4 TrefftzmethodandMFSforbiharmonicequation Analyticalsolution RelationshipbetweentheTrefftzmethodandMFS CoefficientsoftheTrefftzmethod CoefficientsoftheMFS Mappingmatrix K DecompositionoftheKmatrix DiagonalmatrixTR Nonuniqueness innumericalaspect Degeneratescaleproblem O K Specialsize positionofthesourcepoints TheoccurrenceofthedegeneratescalesusingtheMFS OnthecompletesetoftheTrefftzmethodandtheMFSusingthedegeneratekernel T completefunctionsoftheTrefftzmethod DegeneratekerneloftheMFS Freetermsforthebiharmonicequationusingthedualboundaryintegralequation HistoryoffreetermsinthedualBEM 2 Dand3 DLaplaceproblem 2 Dand3 Delasticityproblem W C Chenthesis 2 Dbiharmonicproblem Bump contourmethodTaylorseriesexpansion Freeterms Dualboundaryintegralequation Improperintegrals Bump contourmethod Forasmoothboundary y B B B B B D Singularpoint Explicitformsforthesixteenkernelfunctions Taylorexpansionforboundarydensityfunctions Boundary Domain vectorcomponent FreetermsofdualBIEforLaplaceproblem 2 Dproblem 3 Dproblem Half Half Dualboundaryintegralequations F P denotesthefinitepart Singularbehaviorofthesixteenkernels Freetermsduetothebumpintegralforthebiharmonicequation Dualboundaryintegralequationsforthebiharmonicproblem Afterderivingthesixteenimproperintegrals wehave forasmoothboundary Conclusions NewmethodstoderivethePoissonintegralformulabyusingthedegeneratekernelsandthenull fieldintegralequations Theoccurringmechanismofdegeneratescalesdependsontheformulationinsteadoftheboundaryconditions ItisinterestingtofindthattheT completesetintheTrefftzmethodisimbeddedinthedegeneratekernelsofMFS Weadoptthebump contourmethodtoderivethefreetermssurroundingthesingularity Forasmoothboundary thesumoffreetermsarehalf Conclusions Thanksforyourkindattention Imagemethod known unknown Imagemethod Closed formGreen sfunction Closed formGreen sfunction Interiorproblem a Imagepoint Series formGreen sfunction degeneratekernels Closed formandseries formGreen sfunctionsforinteriorandexteriorproblems Closed form Series form Interiorproblem Exteriorproblem Poissonintegralformula Poissonintegralformu