An Overview on Meshless Methods and Their Applications

Abstract

Meshless methods can be traced back to 1977 when [Lucy (1977)] and [Gingold and Monaghan (1977)] proposed a smooth particle hydrodynamics (SPH) method that was used for modeling astrophysical phenomena without boundaries, such as exploding stars and dust clouds. Extensive developments have been made in several varieties since then and with many different names: SPH [Monaghan, 1982, Monaghan, 1988, Monaghan, 1992], generalized finite difference method [(Liszka and Orkisz, 1980)], diffuse element method [(Nayroles et al., 1992)], particle in cell method [(Sulsky et al., 1992)], wavelet galerkin method [(Qian and Weiss, 1993)], reproducing kernel particle method (RKPM) [(Liu et al., 1995a, [(Liu et al., 1995b], element-free Galerkin (EFG) [(Belytschko et al., 1994)], partition of unity (PU) [(Babuska and Melenk, 1995, (Babuska and Melenk, 1996], Hp clouds [(Duarte and Oden, 1995), (Duarte and Oden, 1996)], finite point method [(Onate et al., 1996a), (Onate et al., 1996b)], free-mesh method [(Yagawa and Furukawa, 2000)], meshless local boundary integration equation method, meshless local Petrov-Galerkin method (MLPG) [(Atluri and Zhu, 2000]; Zhu, 1999)], and multiscale methods [(Liu et al., 1996a, Liu et al., 1996b, Liu et al.,1997, Liu et al.,2000)]. This chapter is to give an overview of the development of meshless methods, with emphasis on the approximation functions, the numerical implementation, and the applications.