计算流体力学课件 chap05a.ppt

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1、Chapter5WeightedResidualMethods(WRMs)WeightedResidualMethodsFiniteDifferencesDiscretenodalvaluesDifferentialformulationforeachnodeTaylorseriesexpansiononastructuredgridTruncationerrorsAccuracy:reducetruncationerrorWeightedresidualsContinuousshapefunctionIntegralf

2、ormulationforeachelementMinimizeweightedresidualforanarbitrarycontrolvolumeInterpolationerrorsAccuracy:higher-orderinterpolation,optimizecoefficientsforminimumresidualsxzyC.V.WeightedResidualMethodsAssumethesolutioncanberepresentedanalytically(withinthefinitev

3、olume)ExpressthesolutionsintermsoftrialfunctionsFinite-volumeFiniteelementSpectralBoundaryelementSpectralelementGlobalmethods–fullmatricesLocalmethods–bandedmatricesWeightedResidualMethodsStartwiththeintegralformofgoverningequationsAssumefunctionalformfortrial(in

4、terpolation,shape)functionsMinimizeerrors(residuals)withselectedweightingfunctionsPowerseriesFourierseriesLagrangeHermiteChebychevWeightedResidualMethodsAssumecertainprofile(trialorshapefunction)betweennodesResidualWeightedResidualWeightedResidualMethodsIngeneral

5、,wedealwiththenumericalintegrationoftrialorinterpolationfunctionsTrialfunctions–constant,linear,quadratic,sinusoidal,Chebychevpolynomial,….Weightingfunctions–subdomain,collocation,leastsquare,Galerkin,….5.1GeneralFormulationWeightedResidualMethods(WRMs)Constructa

6、napproximatesolutionSteadyproblems–systemofalgebraicequationsfortrialfunctionj(x,y,z)Transientproblems–systemofODEsintimeChosentosatisfyI.C./B.C.sifpossibleWeightedResidualConsiderone-dimensionaldiffusionequationIngeneral,R0withincreasingJ(higher-order)Weakform

7、–integralform,discontinuityallowed(discontinuousfunctionand/orslope)ExactsolutionApproximationWeightedResidualMethodsWeakform–integralformulationR0,but“weightedR”=0Choicesofshapeorinterpolationfunctions?Choicesofweightingfunctions?SubdomainMethodEquivalenttofini

8、tevolumemethodDm:numericalelement(arbitrarycontrolvolume)DmmaybeoverlappedCollocationMethodZeroresidualsatselectedlocations(xm,ym,zm)Nocontrolontheresidualsbet