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Improvements of Generalized Finite Difference Method And View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Applied Mathematical Modelling 27 (2003) 831–847 www.elsevier.com/locate/apm Improvements of generalized finite difference method and comparison with other meshless method L. Gavete a,*, M.L. Gavete b, J.J. Benito c a Escuela Tecnica Superior de Ingenieros de Minas, Universidad Politecnica, c/Rios Rosas 21, 28003 Madrid, Spain b Facultad de Farmacia, Universidad Complutense, Avda Complutense s/n, 28040 Madrid, Spain c Escuela Tecnica Superior de Ingenieros Industriales, U.N.E.D., Apdo. Correos 60149, 28080 Madrid, Spain Received 3 December 2001; received in revised form 29 January 2003; accepted 19 February 2003 Abstract One of the most universal and effective methods, in wide use today, for approximately solving equations of mathematical physics is the finite difference (FD) method. An evolution of the FD method has been the development of the generalized finite difference (GFD) method, which can be applied over general or ir- regular clouds of points. The main drawback of the GFD method is the possibility of obtaining ill- conditioned stars of nodes. In this paper a procedure is given that can easily assure the quality of numerical results by obtaining the residual at each point. The possibility of employing the GFD method over adaptive clouds of points increasing progressively the number of nodes is explored, giving in this paper a condition to be accomplished to employ the GFD method with more efficiency. Also, in this paper, the GFD method is compared with another meshless method the, so-called, element free Galerkin method (EFG). The EFG method with linear approximation and penalty functions to treat the essential boundary condition is used in this paper. Both methods are compared for solving Laplace equation. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Meshless; Generalized finite difference method; Element free Galerkin method; Singularities 1. Introduction The objective of meshless methods is to eliminate, at least, a part of the structure of elements as in the finite element method (FEM) by constructing the approximation entirely in terms of nodes. * Corresponding author. Tel.: +34-913-366-466; fax: +34-913-363-230. E-mail addresses: [email protected] (L. Gavete), [email protected] (J.J. Benito). 0307-904X/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0307-904X(03)00091-X 832 L. Gavete et al. / Appl. Math. Modelling 27 (2003) 831–847 Although meshless methods were originated about twenty years ago, the research effort devoted to them until recently has been very small. One of the starting points is the smooth particle hy- drodynamics method [1] used for modelling astrophysical phenomena without boundaries such as exploding stars and dust clouds. Other path in the evolution of meshless methods has been the development of the generalized finite difference (GFD) method, also called meshless finite dif- ference (FD) method. The GFD method is included in the so named meshless methods (MM). One of the early contributors to the former were Perrone and Kao [2]. The bases of the GFD were published in the early seventies. Jensen [3] was the first to introduce fully arbitrary mesh. He considered Taylor series expansions interpolated on six-node stars in order to derive the FD formulae approximating derivatives of up to the second order. While he used that approach to the solution of boundary value problems given in the local formulation, Nay and Utku [4] extended it to the analysis of problems posed in the variational (energy) form. However, these very early GFD formulations were later essentially improved and extended by many other authors, but the most robust of these methods was developed by Liszka and Orkisz [5,6], using moving least squares (MLS) interpolation [7], and the most advanced version was given by Orkisz [8]. The explicit FD formulae used in the GFD method, as well as the influence of the main parameters involved, was studied by Benito et al. [9]. Other different MM have been proposed. The diffuse element method, developed by Nayroles et al. [10], was a new way for solving partial differential equations. Belytschko et al. [11] developed an alternative implementation using MLS approximation. They called their approach the element free Galerkin (EFG) method. The use of a constrained variational principle with a penalty function to alleviate the treatment of Dirichlet boundary conditions in (EFG) method has been proposed [12,13]. Liu et al. [14] have used a different kind of ‘‘griddles’’ multiple scale method based on reproducing kernel and wavelet analysis. Onnate~ et al. [15] focused on the application to fluid flow problems with a standard point collocation technique. Duarte and Oden [16], on the one hand and Babuska and Melenk [17] on the other, have shown how the denominated methods without mesh can be based on the partition of the unity. All these methods can be considered as MM. This paper is organized as follows. Firstly, in Section 2 the GFD method is briefly described. Secondly, in Section 3 several examples in the presence of singularities are given and the per- formance of the GFD method is analyzed using fixed or variable radius of influence for the weighting functions. Also in Section 3 the possibility of employing the GFD method over adaptive clouds of points is explored. Thirdly, the GFD method is compared to the EFG method in Section 4. And finally, in Section 5, some conclusions are obtained. 2. Generalized finite difference method For any sufficiently differentiable function f ðx; yÞ, in a given domain, the Taylor series ex- pansion around a point Pðx0; y0Þ may be expressed in the form of of h2 o2f k2 o2f o2f f ¼ f þ h 0 þ k 0 þ 0 þ 0 þ hk 0 þ oðq3Þð1Þ 0 ox oy 2 ox2 2 oy2 oxoy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 where f ¼ f ðx; yÞ, f0 ¼ f ðx0; y0Þ, h ¼ x À x0, k ¼ y À y0 and q ¼ h þ k . L. Gavete et al. / Appl. Math. Modelling 27 (2003) 831–847 833 Eq. (1) and all following formulae will be limited to second order approximations and two- dimensional problems. In any case, the extension to other problems is obvious. We consider norm B 2 XN of of o2f o2f o2f B ¼ f À f þ h 0 þ k 0 þ h2 0 þ k2 0 þ h k 0 w ð2Þ 0 i i ox i oy i ox2 i oy2 i i oxoy i i¼1 where fi ¼ f ðxi; yiÞ, f0 ¼ f ðx0; y0Þ, hi ¼ xi À x0, ki ¼ yi À y0, wi ¼ weighting function with compact support. The solution may be obtained by minimizing norm B, writing oB ¼ 0 ð3Þ ofDf g of of o2f o2f o2f fDf gT ¼ 0 ; 0 ; 0 ; 0 ; 0 ð4Þ ox oy ox2 oy2 oxoy we come to a set of five equations with five unknowns for each node. For example, the first equation is as follows XN XN of XN of XN o2f XN h3 f w2h À f w2h þ 0 w2h2 þ 0 w2h k þ 0 w2 i 0 i i i i i ox i i oy i i i ox2 i 2 i¼1 i¼1 i¼1 i¼1 i¼1 o2f XN k2h o2f XN þ 0 w2 i i þ 0 w2h2k ¼ 0 ð5Þ oy2 i 2 oxoy i i i i¼1 i¼1 this Eq. (5) and all following equations give us the following system of equations 0 18 9 h3 2 o 2 2 2 2 i 2 ki hi 2 2 > f0 > Rwi hi Rwi hiki Rwi Rwi Rwi hi ki > ox > B 2 2 C> > B 2 3 C> of0 > 2 2 2 2 hi ki 2 ki 2 2 > > B Rw h k Rw k Rw Rw Rw h k C> oy > B i i i i i i 2 i 2 i i i C<> => B 3 2 4 2 2 3 C o2f 2 h 2 kih 2 h 2 h k 2 h ki 0 B i i i i i i C o 2 B Rwi 2 Rwi 2 Rwi 4 Rwi 4 Rwi 2 C> x > > o2 > B 2 3 2 2 4 3 C> f0 > B 2 hiki 2 ki 2 hi ki 2 ki 2 hiki C> > @ Rw Rw Rw Rw Rw A> oy2 > i 2 i 2 i 4 i 4 i 2 > > 3 3 : o2f ; 2 2 2 2 2 h ki 2 hik 2 2 2 0 Rw h k Rw h k Rw i Rw i Rw h k oxoy 0i i i i i i i 2 1 i 2 i i i 2 2 Àf0Rw hi þ Rfiw hi B i i C B f Rw2k Rf w2k C B À 0 i i þ i i i C B h2 h2 C B f Rw2 i Rf w2 i C 6 ¼ B À 0 i 2 þ i i 2 C ð Þ B k2 k2 C @ 2 i 2 i A Àf0Rwi 2 þ Rfiwi 2 2 2 Àf0Rwi hiki þ Rfiwi hiki This system of linear equations (6) in resumed notation is given by APDfP ¼ bP ð7Þ where the AP are matrices of 5 · 5, and the vector DfP is 5 · 1. 834 L. Gavete et al. / Appl. Math. Modelling 27 (2003) 831–847 2 2 2 2 If we are interested in solving PoissonÕs equation, we can calculate o f0=ox , o f0=oy at each node according to (6) and then o2f o2f 0 þ 0 À gðx ; y Þ¼0 ð8Þ ox2 oy2 0 0 giving us a linear system of equations for the considered domain.
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