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Top1. Introduction
Established numerical methods such as finite element methods FEM and finite volume methods FVM that are routinely used to resolve complex multiphysics interactions require significant effort in mesh generation. In fact, for most models of geometrically intricate components commonly encountered in engineering analysis, mesh generation is very time consuming, far from automated, and most often the most taxing part of the modeling process. The term meshless methods refer to a class of numerical techniques that rely on global interpolation on no ordered spatial point distributions that, as such, offer the hope of reducing the effort devoted to model preparation. These techniques have been under much development over the past few years (Belytscho, Lu, & Gu, 1994; Atluri & Shen, 2002; Atluri & Zhu, 1998; Liu, 2003; Melenk & Babuska, 1996; Kansa, 1990; Kansa, 1990; Kansa & Hon, 2000). Meshless methods use a nodal or point distribution that is not required to be uniform or regular in their spatial distribution due to the fact that most rely on global radial-basis functions RBF (Powell, 1992; Buhmann, 2003; Dyn, Levin, & Rippa, 1986). Moreover, care must be taken in the evaluation of derivatives in global RBF meshless methods. Although, very promising, these techniques can also be computationally intensive.
The basic idea of RBFCM is the construction of an unknown function, by the information that we get from the problem. An RBF depends on the number of nodes at which we will solve our problem and distances between these nodes. The use of RBF as a meshless procedure for numerical solution of PDEs is based on the collocation schemes. Due to the collocation technique, this method does not need to evaluate any integral (Cheng, Golberg, Kansa, & Zammito, 2003; Hoon, 2002; Kansa, 1990; Larsson & Fomberg, 2003; Hermite-Birkhoff, 1992). The RBFs methods performed well in many calculations including the numerical experiments that were reported by Franke (1982). For more information about the meshless method, we refer readers to Fasshuaer (2007) and the references therein.