计算流体力学课件 chap05a

Chapter5WeightedResidualMethods(WRMs)WeightedResidualMethodsFiniteDifferencesDiscretenodalvaluesDifferentialformulationforeachnodeTaylorseriesexpansiononastructuredgridTruncationerrorsAccuracy:reducetruncationerrorWeightedresidualsContinuousshapefunctionIntegralformulationforeachelementMinimizeweightedresidualforanarbitrarycontrolvolumeInterpolationerrorsAccuracy:higher-orderinterpolation,optimizecoefficientsforminimumresidualsxzyC.V.WeightedResidualMethodsAssumethesolutioncanberepresentedanalytically(withinthefinitevolume)ExpressthesolutionsintermsoftrialfunctionsFinite-volumeFiniteelementSpectralBoundaryelementSpectralelementGlobalmethods–fullmatricesLocalmethods–bandedmatricesWeightedResidualMethodsStartwiththeintegralformofgoverningequationsAssumefunctionalformfortrial(interpolation,shape)functionsMinimizeerrors(residuals)withselectedweightingfunctions)()(),(xtatxTJ1jjjPowerseriesFourierseriesLagrangeHermiteChebychevsin,cos()()()()jjjjjxjxjxxLxHxTxWeightedResidualMethodsAssumecertainprofile(trialorshapefunction)betweennodes)()(),(xtatxTJ1jjj0dxTWLdxtxWR0txRTL)(),(but,),()(ResidualWeightedResidualWeightedResidualMethodsIngeneral,wedealwiththenumericalintegrationoftrialorinterpolationfunctionsTrialfunctions–constant,linear,quadratic,sinusoidal,Chebychevpolynomial,….Weightingfunctions–subdomain,collocation,leastsquare,Galerkin,….0d)T(WLdxdydz)t,z,y,x(R)z,y,x(W5.1GeneralFormulationWeightedResidualMethods(WRMs)ConstructanapproximatesolutionSteadyproblems–systemofalgebraicequationsfortrialfunctionj(x,y,z)Transientproblems–systemofODEsintimeJ1jjjozyxtatzyxTtzyxT),,()(),,,(),,,(ChosentosatisfyI.C./B.C.sifpossibleWeightedResidualConsiderone-dimensionaldiffusionequationIngeneral,R0withincreasingJ(higher-order)Weakform–integralform,discontinuityallowed(discontinuousfunctionand/orslope)0txRTL0TTTLxxt),()()(ExactsolutionApproximationM21m0dxdydztzyxRzyxWm,,,,),,,(),,(WeightedResidualMethodsWeakform–integralformulationR0,but“weightedR”=0Choicesofshapeorinterpolationfunctions?Choicesofweightingfunctions?0dxdydzTL(x,y,z)W0dxdydzTLW(x,y,z)0TLm)(:tionDiscretiza)(:FormIntegralExact)(:FormalDifferentiSubdomainMethodEquivalenttofinitevolumemethodDm:numericalelement(arbitrarycontrolvolume)Dmmaybeoverlapped0RTL0dxdydzTL(x,y,z)Wm)(;)(0d)z,y,x(Rdxdydz)T(LWDoutside,0Din,1WmDmmmmCollocationMethodZeroresidualsatselectedlocations(xm,ym,zm)Nocontrolontheresidualsbetweennodes0RTL0dxdydzTL(x,y,z)Wm)(;)(mmmmmmmmz,y,xRxRd)x(RxxdxRxWxxxWLeastSquareMethodMinimizethesquareerror0RTL0dxdydzTL(x,y,z)Wm)(;)(0xdRa21xdxRxWRW2aRR2aRaRxW2mmmmm2mm0R2SquareerrorR20GalerkinMethodWeightingfunction=trial(interpolation)functionFororthogonalpolynomials,theresidualRisorthogonaltoeverymemberofacompleteset!0RTL0dxdydzTL(x,y,z)Wm)(;)(xdxRxxdxRxWxxWmmmmNumericalAccuracyHowdowedeterminethemostaccuratemethod?Howshouldtheerrorbe“weighted”?Zeroaverageerror?Leastsquareerror?Leastrmserror?Minimumerrorwithinselecteddomain?Minimum(zero)erroratselectedpoints?Minimax–minimizethemaximumerror?Somefunctionshavefairlyuniformerrordistributionscomparingtotheothers5.1.1ApplicationtoanODEConsiderasimpleODE(Initialvalueproblem)UseglobalmethodwithonlyoneelementSelectatrialfunctionoftheformofAutomaticallysatisfytheauxiliaryconditionaj=constant,notafunctionoftimexey10y1x00ydxyd)(,N1jjjxa1yApplicationtoanODEConsideracubicinterpolationfunctionwithN=3QUESTION:Whichcubicpolynomialgivesthebestfittotheexact(exponentialfunction)solution?Definitionofbestfit?Zeroaverageerror,leastsquare,leastrms,…?33221N1jjjxaxaxa1xa1yResidualSubstitutethetrialfunctionintogoverningequationForcubicinterpolationfunctionN=3TheresidualisacubicpolynomialR0Determinetheoptimalvaluesofajtominimizetheerror(underpre-selectedweightingfunctions))()(xjxa1xa1xjaydxdyyLRN1j1jjN1jjjN1j1jj0xaxaa3xaa21axx3axx2ax1a1xR33223121323221)()()()()()()(SubdomainMethodZeroaverageerrorineachsubdomainNote:R(0)=0.01560,R(1)=0.01550281304219001561a31a324163a8126a1810Rdx3m31a32469a8120a1830Rdx2m31a32411a818a1850Rdx1mx4ax3aax2aax1a0dxxRRdxWi3211323213231321310xx433232121xx10mm1mm1m...:::)()()()(////D1D2D3Uniformspacingx0x1x2x3»A=[5/188/8111/324;3/1820/8169/324;1/1826/81163/324]A=0.27780.09880.03400.16670.24690.21300.05560.32100.5031»b=[1/3;1/3;1/3]b=0.33330.33330.3333»a=A\ba=1.01560.42190.2812SubdomainMethod3232x28120x42190x1719001560Rx28120x42190x015611y.......Cubicapproximationfunctiony(x)TheresidualR(x)isalsoacubicpolynomial»x1=0:1/30:1/3;R1=(a(1)-1)+(2*a(2)-a(1))*x1+(3*a(3)-a(2))*x1.^2-a(3)*x1.^3;»x2=1/3:1/30:2/3;R2=(a(1)-1)+(2*a(2)-a(1))*x2+(3*a(3)-a(2))*x2.^2-a(3)*x2.^3;»x3=2/3:1/30:1;R3=(a(1)-1)+(2*a(2)-a(1))*x3+(