Xiao, Jing-En (National Taiwan Ocean University) | Ku, Cheng-Yu (National Taiwan Ocean University) | Chien, Lien-Kwei (National Taiwan Ocean University) | Huang, Wei-Po (National Taiwan Ocean University)
In this article, the numerical solution for solving subsurface flow problems in heterogeneous soil using a novel meshless method is presented. The numerical solutions are approximated by a set of non-singular basis function of the Laplace equation from the collocation Trefftz method. To deal with the subsurface flow problems in heterogeneous soil, the domain decomposition method is applied. The novel meshless method is validated for several test problems. Application examples are also performed. The results reveal that the proposed method has resolved one of the major issues which are finding the satisfactory location for the source points in the method of fundamental solutions. In addition, the proposed method has great numerical accuracy for solving subsurface flow problems in layered heterogeneous soil even with extreme contrasts in the hydraulic conductivity.
In the past, numerical approaches to the simulation of various subsurface flow phenomena using the conventional mesh-based methods such as the finite difference method (FDM) (Clement et al., 1994; Fukuchi, 2016; Todsen, 1971; Pollock, 1988, Ku, 2013), the boundary element method (BEM) (Fan, 1992; France, 1974) and the finite element method (FEM) (Chen and Tompkins, 1994; Chen et al., 2000) were well documented. Since the meshless method has the advantages that it does not need the mesh generation, the meshless method has attracted considerable attention in recent years in solving practical problems involving complex geometry in subsurface flow problems. Several meshless methods have been reported in literature, such as the method of fundamental solutions (MFS) (Kupradze and Aleksidze, 1964; Katsurada, 1996), the collocation Trefftz method (CTM) (Trefftz, 1926; Yeih et al., 2010), element free Galerkin methods (EFGM) (Belytschko et al., 1994), radial basis function collocation method (RBFCM) (Amaziane et al., 2004; Chan and Fan, 2013), generalized finite difference method (GFDM) (Benitoet et al., 2001; Fan et al., 2014). Among these meshless methods, the method of fundamental solutions was proposed by Kupradze and Aleksidze in 1964 and the Trefftz method was proposed by Trefftz in 1926 are two important representative boundary-type meshless methods. Subsurface flow problems are governed by second-order partial differential equation. Problems involving regions of irregular geometry are generally intractable analytically. For such problems, the use of the boundary-type meshless method, to obtain approximate solutions is advantageous. In this study, we adopt the domain decomposition method (DDM) to deal with the subsurface flow problems in layered heterogeneous soil because the DDM is natural from the physics of the problem where different hydraulic conductivity in different subdomains.