1、39 1 ZVol.39 No.12009 M125 ADVANCES IN MECHANICS Jan.25, 2009E #*f1;y 1;2 11 bvt?t b,1000842 Department of Mechanical Engineering, Northwestern University, U.S.A.K1W% M ?ZE $# 5 =.EG ,E .dKaH ZE V ?CM e w,BtKaHZE 4 z) 58C+ .F ELB , X30E, E1 uY P F E f .W Z( MKl= a as a_f a a1 # )E F E( Ta Ta TaF Kl=
2、 THsT),i) sZEHHq) 5.N$ “d9 E Ia.lavMaa %a 3 , 5,Z U EM.d ZE .1oME,E, , Z,vM1KE “ - % S5Kr ZEB, p t+ y5 H9i% J. , E p a Ia Za %a #+vM15 H,K V ?3 e w,1,7 OY; I 5,A T HWs | %KKlj,y7 e w| P HWsVl,v9F 9 T ; Z5, ZZ_ ? Y5 ,yN9 V1s E ZV;5,91s a8M;K ,yNA 4) SLB vM, H9 4r) ; 6, K -) q ?Z,v K1 3 G . Bt ZE, KsE
3、aHE9 i 5.kKaH ZEt J,S=9 V20 W90 M 7S E 15.KZE,EBF p u, / f , Vsh“,1 Ss, V 9 ,7 O Vhl9 4. 7,E9iBt% J. , f B (, 9 v;v f +,yNEHHq F1KEN.psZ ZE Vs v .B ZE psZMHq , KsE; 6B ZE n5ysZ#HqMrs T,N$ y E, F E. E N,30,VF l : 2008-01-15, : 2008-12-23SE1 S(10672088), W 9(NCET-04-0091)9 L i “y E-mail: 2 Z2009 M39 E
4、 A, E1 uY I “F E kf (trial fun-tion). , E(element free Galerkinmethod, EFGM) E,7KE(flnitepoint method, FPM)5 (collocation)E,HE(boundary node method), O MKl= (moving least square, MLS) y kf .F E6, 7L,“d E # a Za Ia %Z Z.2 f KE,E f YVBF xI(I = 1;2; ;N) y ,G .f u(x) V u(x) uh(x) =nXI=1NI(x)uI = N(x)u (
5、1)uI = u(xI) f u(x)xI),NI(x)xIf , nf x)09 , N(x) = N1(x);N2(x); ;Nn(x),u = u1;u2; ;unT.5, f T(1) Vui(x) uhi (x) =nXI=1NI(x)uiI = N(x)u (2)uiI = ui(xI), N(x) = N1(x)I;N2(x)I; ;Nn(x)I, u = u11;u21;u31;u12; ;u3nT, I . f f.KE,v f +,yNuIB kf uh(x)xI),uh(xI) 6= uI, NI(xJ) 6= IJ.2.1MKl= pf u(x)9 x #x = V u
6、h(x;x) =mXi=1pi(x)ai(x) (3)x 9 x #x = bWUS,pi(x) f , m f , ai(x) “ .Y P TTf ,9 V P f , sf f .MKl= (moving least squares,MLS)8,“ ai(x) | P f uh(x;x)9 x #x(9 xl) = pf u(x) Kl=il/KD. xI)lB f (93f )wI(x) = w(xxI), xIBK uI( f wI(x)xIY)v ,7 # ,. 9 xlx n, f uh(x;x)tx = xI)F ZJ =nPI=1wI(x)uh(x;xI)u(xI)2 =nP
7、I=1wI(x) mPi=1pi(xI)ai(x)uI2(4)7J |Kl,“ ai(x), T(3),uh(x;x) = N(x; x)u (5)N(x;x) 8Vr T VnD42.2.1. f uh(x;x) pf u(x)9 xlx =F Kl=il/ KD . p x V #x =y pf u(x) KD ,t f uh(x;x)x = x)“ pf u(x) p = f uh(x)( m1 U),u(x) uh(x) = uh(x;x)flflx=x = N(x)u (6)f N(x) = N(x;x)jx=x.m1 1 f:E #3f K! T k = 0H, MLSf N(x)
8、Shepardf . TMLS | f wI(x)I = |1,I |0,5MLS |SKl= (LSQ).MLS V c f pi(x)(PnI=1 NI(x)pi(xI) = pi(x),yNV| p5+f c,V74 l q. , Z5,f V |9pT(x) =1;x;y;prcos 2;prsin 2;prsin 2 sin ;prcos 2 sin (7)r U , U L LC.MKl= 1 p f ) ZKl, f . Atluri 10s x H,1p f # ) ZKl,y 0 HH(x) = 1,5H(x) = 0. ffi(x) = minx2fikxxksign(n+
9、(xx)| f , x fi xK, n+ E_ O .T s:, xI, U )xI,5F I 35 7xlx = c f zM,n = m,y7Z , V p“ ai, T(3)i 7x = x, V T(1) f , f HqNI(xJ) = IJ,uh(xI) = uI,yN FHHq. I |NZEE(point interpolationmethod, PIM).2.71 #1 # Sibson36, 37dSibson38, 9 + Voronoim y f , O Kronecker Hq,yN kf VZL FHHq.=5 , p u VNxIM1 jHTI ,HTI xI1
10、 xJ(J 6= I)( m2(a) U).HTIxIVoronoiH,HTI(I = 1;2; ;N)“xI(I = 1;2; ;N)Voronoim(BVoronoim). 9 V/“Voronoim, ,“=Voronoim| u sM #xIxJM1 HTIJ.HTIJ ,xIxJsY KQK. Vn,xIxJVoronoiHTITJM # H,TIJ b./9 x f ,|xFm2(a)“,i 3“Voronoim( m2(b) U), HABCDxVoronoiHTx.|TxM #H cx1 #,5m2(b)x 41 #.A ,HABFE xxI=VoronoiHTxI, 9 “V
11、oronoiHTI“VoronoiHTx u.m2 Voronoim UiSibson361 #USTf . 7(x)TxLebesgue(Ba= sYa 8,m2(b) HABCD ), I(x)TxILebesgue(m2(b) HABFE ),5|NI(x) = I(x)(x) (28)xM 1 #xI1 #US.A PnI=1 NI(x) = 1, nx1 # .Sibsonf “ C0 , C1 36, 39.Voronoim1 , a M f .B5, SibsonLKN40. Sibson VL, p sZ H) V 1 f H,1B, H1Sibson/. Farin39, 4
12、14 C1Sibson, Gonzalez424 F f , X8 5;dSibson38 Belikov 4.7tIJV UTI = TI TITJ = TJ TJ“,5XJ6=IjtIJjxJd(xI;xJ) = xIXJ6=IjtIJjd(xI;xJ) (29)1 f:E #7jtIJj tIJLebesgue.dSibsonf lNI(x) = fiI(x)nPJ=1fiJ(x); fiI(x) = jtxIjd(x;xI)(30)jtxIj txILebesgue.=5,jtxIjm2(b)ABH.Sibson1 ,dSibson bW By f ,9 l;6, Sibson FHH
13、q H, TdjH,5 =f H ,/B,7dSibsonfH L, ajHH.“ ) f ,EKriging4345d ( B“H(non uniform rational B-spline, NURBS)46, 47.3F E 5 , eZ:Z: ij;j + fi = 0; =(31)HHq: ijnj ti = 0;t (32)MHHq: ui = ui;u (33)+ Z: “ij = 12(ui;j +uj;i); =(34) Z: ij = Dijkl“kl; =(35)Tfi8 , t H, ti , njHtELZ_?, uMH, uiM. Dijklf .57, T(31)
14、E p, ? p. !uhi (x) T(31)B , kf (trial function).A , uhi (x)B ? sZ(31).f ui(x)KD , Z T PsZ(31) ,.F E1 pZ F ( ,Zwi( ij;j + fi)d = 0 (36)Tf wi_f (test function),F E_f .3.1 E E_f kf |1Bf bW. |wi = ui, T(36)(u) =Zui( ij;j + fi)d = 0 (37) Tss,i Z(35)HHq(32), H I nuiju = 0,(u) =Z(“ij ij +ui fi)d+Ztuitid =
15、0 (38)| f (1) T,i I nuiI i,Ku = P (39)TK = R BTDBd,P = R NTfd +RtNTtd,BM ,D . E K .E E y pZ, E(EFG)48, 49aE(RKPM)12, 50ahp vE15, 51asE(PUM)16, 17a1 E37;5254aE(PIM)35, 33.E(material point method,MPM)55, 569F E, ,Bt+ y) .MPM,kf ,KLf B“.s 9 , s.68M,z ,7Y ?5, B HW9 i. Bardenhagen57 Z MPM kf _f 4S,Petrov
16、-Galerkin Z cMPMBBZE,4I 103, 104 u =H) E,7NeumannH( H, 1H) 4- E(n3.3),y rC. I5 #+ dLa dLHHqdL,d, ZE4 vG.d KM1,ZE|8 , ,E vM/M5. 6,K,K 4) I5C r, C.5K SK(erosion) E V| r , H9 5.7ZE V 41 .1 f:E #15E,?ZK*SPH n5$ E 511;146148,SPHV 0E T e,r q ,B 5 z -.%8 8 E?CSPHi %5, % T/, _ “v, H/ 149.Swegle150Balsara151
17、SPH s,s %y. 6SPHf !59$ t f /y.t5SPH4 Z. Dyka6449F ZE % %5. Randles655 z. Chen4CSPH152, %H5. Monaghan15349F E % %. 5SPH E,+Y c , %5ijA,1y / qv %?Z q65.(5 VB15.SPHBY V8 9 6B8 ,yN 8( VF+ yI n,SPHi ,yN( V ? 3,i. Campbell1544 f 0- 0( E. Vignjevic1554( E1 y ( ,$A8 .Johnson149, 156SPHZE, 5 B“ T. Johnson4 S
18、PH ZE184, 192 . ra r sa r s T1 f:E #175 E,9 ? r 1. ?0 (functionally gradient materi-als, FGM) B bWUS M . RaoRahman197|EFGMFGM 5 E,i4 BFGMTs9 r5,TA UM s5Z5W Z Yv,7Z %5 r T. IskeKaser120 Z2009 M39 aLagrange T AMMoCZE291 E F%, L dM 5. York292 MPM1 M(+,MPM E 5 n 8MT5.5.6 3 E 3 T 1. 3 # V,yN ZE1 p,+Y 3 5
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