基于移动最小二乘法的曲线曲面拟合-曾清红

2002102110102020973G19990328051978-(xi,yi),i=1,2,,nxiy=f(x)f()()()f(x)()[1],[2]()()(MLS)()LancasterSalkauskasMLS[3]Belytschko[4]1a(x)p(x)a(x)x2CompactSupportxyxxxw(x)()JOURNALOFENGINEERINGGRAPHICSNo.1230027(MovingLeast-SquaresMLS)(LS)MLSTP391A1003-0158(2004)01-0084-0611.1f(x)∑===miiixxpxpxxf1T)()()()()(aa1T21)](,),(),([)(xaxaxaxmΛ=axT21)](,),(),([)(xpxpxpxpmΛ=km3,],,1[)(T==myxxp2a6,],,,,,1[)(T22==myxyxyxxp2bL2(],,[21nxxxΛ=xL212221niix/⎛⎞⎜⎟⎜⎟=⎝⎠=∑x)∑∑==--=--=nIIIInIIIyxxpxxwyxfxxwJ12T12])()()[(])()[(a3n()fxIyI=xx()IIyy=x()Iw-xxIx()xa33a0)()()(=-=∂∂yxBxxAJaa4yxBxAx)()()(1-=a5∑=-=nIIIIxpxpxxwxA1T)()()()(6)]()(,),()(),()([)(2211nnxpxxwxpxxwxpxxwxB---=Λ7],,,[21TnyyyyΛ=8(5)(1)MLS∑===nIkIkIyxÖyxxf1)()()(9)(xÖkk)()()(],,,[)(1T21xBxAxpxÖknkkk-==Λ100k=(){1}px=Shepard∑==-=nJJIIxxwxxwxÖ1Shepard)()()(11p(x)9f(x)rCp∈swC∈min()rsfC,∈1.2w(xxI)x(x)(1)smaxxw(xxI)2Ixx-w(xxI)C11Cs=xxImaxsss=122⎪⎪⎪⎩⎪⎪⎪⎨⎧-+-+-=)1(0)121(344434)21(4432)(3232sssssssssw126A(x)13maxsx1229()1112x(a)x(b)x(c))(xÖk(d)x34()33.1(MLS)x=[0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0]y=[0,4,5,14,15,14.5,14,12,10,5,4]p(x)=[1,x]T33(a)~(c)MLS13a23b33c3MLSX0.00.20.40.60.81.00.00.51.01.52.0x0.00.20.40.60.81.00246810121416x0.00.20.40.60.81.00.00.20.40.60.81.0x0.00.20.40.60.81.00246810121416x0.00.20.40.60.81.00.00.10.20.30.40.50.60.7x0.00.20.40.60.81.00246810121416x3.2W:{3.0x3.0,3.0y3.0}(,)fxy])1(exp[31]exp[)5(10])1(exp[)1(2),(222253222yxyxyxxyxxyxf-+------+---=13W4545151956p(x,y)=[1,x,y]T6c3ba[]∫-=ÙyxfyxfÙd),(),(error2MLSEXACT14EXACT(,)fxyMLS(,)fxy44LSerror23.1484=MLSerror15.8278=MLSLSLS5412()345()[1],,,.[J].,1999,20(3):41~46.[2],.[J].,1990,(1):40~41.[3]LancasterP,SalkauskasK.Surfacesgeneratedbymovingleastsquaresmethods[J].MathematicsofComputation,1981,37(155):141~158.[4]BelytschkoT,LuYY,GuL.Element-freegalerkinmethods[J].Int.J.Numer.Meth.Engng.,1994,(37):229~256.46CurveandSurfaceFittingBasedonMovingLeast-SquaresMethodsZENGQing-hong,LUDe-tang(DepartmentofMechanicsandMechanicalEngineering,UniversityofScienceandTechnologyofChina,Hefei230027,China)Abstract:Anewcurveandsurfacefittingmethodbasedonmovingleast-squares(MLS)methodsisintroduced.Themethodimprovestraditionalleast-squaresmethod,whichmakestheapproximatecurveorsurfaceoccupysomeadvantages,suchashighdegreeofaccuracyandhighsmoothness.Theprinciple,applicationandcharacteristicaredescribedandtheflowchartforcurveandsurfacefittingbasedonMLSisgiven.Twofittingcasesaregiveninthepaper,oneiscurvefitting,theotherisfittingsurfacetoscattereddata.TheMLSfittingresultsarecomparedwithLSfittingresultsandthesmoothnessandfittingqualityareanalyzed.Thefittingresultsshowthatusingmovingleast-squaresmethodstofitcurveandsurfaceissuperiorandeffective.Keywords:applicationofcomputer;movingleast-squaresmethods;curveandsurfacefitting;weightfunction[2003]0089920022VRML