加权残数法理论基础的初步探讨-豆柴文库

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加权残数法理论基础的初步探讨
Introduction
TheWeightedResidualsMethod(WRM)isanumericalanalysistechniquethatinvolvesdividingastructureintoaseriesoffiniteelementsandsolvingindividualequationsforthefiniteelementsthroughaweightedresidualmethod.TheWRMisacommonlyusedmethodinstructuralmechanicstosolvethegoverningequationsofanengineeringsystem.
ThepurposeofthispaperistoexplorethetheoreticalfoundationsoftheWRMinfiniteelementanalysis.Wewillexaminethebasicprinciplesbehindthemethod,itspracticalapplications,anditsadvantagesoverothernumericalmethods.
BasicPrinciplesoftheWeightedResidualsMethod
TheWRMisbasedonthefundamentalprincipleoftheweightedresidualtechnique,whichinvolvesmultiplyingthegoverningequationofasystembyaweightingfunctionandintegratingitoverthedomainofthesystem.Thisresultsinasetofalgebraicequationsthatcanbesolvediterativelytoobtainthesolutiontothesystem.
TheWRMinvolvessubdividingthedomainofthesystemintosmallfiniteelements,andsolvingthegoverningequationofeachelementusingtheweightedresidualtechnique.Theequationsforalltheelementsarethencombinedtoformasystemofalgebraicequationsthatcanbesolvedtoobtainthesolutiontotheentiresystem.
Theweightedresidualtechniqueinvolvesmultiplyingthegoverningequationofeachelementbyaweightingfunctionandintegratingitoverthedomainoftheelement.Theweightingfunctionischosensuchthatitsatisfiestheboundaryconditionsoftheproblem.Thesolutiontothesystemisobtainedbyminimizingtheresidualerrorbetweenthecalculatedandmeasuredvaluesofthesystem.
ApplicationsoftheWeightedResidualsMethod
TheWRMisusedinawiderangeofengineeringapplications,includingsolidmechanics,fluidmechanics,andelectromagnetics.Itisparticularlyusefulinproblemswhereexactsolutionsaredifficulttoobtain,orwheretheproblemislargeandcomplex.
Insolidmechanics,theWRMisusedtoanalyzethebehaviorofstructuresunderloads.Itisusedtocalculatestresses,deformations,anddisplacementsofstructures,andtooptimizethedesignofthestructureforaparticularloadingcondition.
Influidmechanics,theWRMisusedtoanalyzethebehavioroffluidsincomplexgeometr