On Solving Nonlinear Moving Boundary Problems with Heterogeneity Using the Collocation Meshless Method

Abstract

In this article, a solution to nonlinear moving boundary problems in heterogeneous geological media using the meshless method is proposed. The free surface flow is a moving boundary problem governed by Laplace equation but has nonlinear boundary conditions. We adopt the collocation Trefftz method (CTM) to approximate the solution using Trefftz base functions, satisfying the Laplace equation. An iterative scheme in conjunction with the CTM for finding the phreatic line with over–specified nonlinear moving boundary conditions is developed. To deal with flow in the layered heterogeneous soil, the domain decomposition method is used so that the hydraulic conductivity in each subdomain can be different. The method proposed in this study is verified by several numerical examples. The results indicate the advantages of the collocation meshless method such as high accuracy and that only the surface of the problem domain needs to be discretized. Moreover, it is advantageous for solving nonlinear moving boundary problems with heterogeneity with extreme contrasts in the permeability coefficient.

A Spacetime Meshless Method for Modeling Subsurface Flow with a Transient Moving Boundary

“…For example, considering the no-flux and the zero pressure head boundary conditions, the unknowns are the coordinates of collocation points. From Equations (23) and 24, it is found that we may solve a nonlinear system of equations to obtain the coordinates of collocation points for the given time. The moving boundary problem may, therefore, exhibit the nonlinear characteristic.…”

“…The Picard iteration first begins from the initial guess of the location for the moving boundary. The iteration may be achieved by applying Equations (23) and (24).…”

“…It is found that the location of the nonlinear moving boundary using the proposed method agrees very well with the results from the previous studies at the final steady-state time. Since several numerical approaches have been applied to solve this problem, we further compare the final time solutions of our method with those of Aitchison (1972) [31], Chen et al (2007) [9], and [23], as depicted in Figure 10. It is found that the location of the nonlinear moving boundary using the proposed method agrees very well with the results from the previous studies at the final steady-state time.…”

“…The discretization of the domain for the collocation approaches is relatively simple because the arbitrary points are assigned only on the boundary if we find the basis functions, which must satisfy the governing equation [23,24]. This idea was developed by Erich Trefftz and known as the Trefftz method [25].…”

“…Previous studies have demonstrated that applications of the Trefftz method may be limited to linear as well as stationary problems. Recently, a study on solving subsurface flow problems with free and moving boundaries governed by the Laplace governing equation adopting the Trefftz method has been developed [23]. However, the engineering applications of the Trefftz method with complete Trefftz functions for dealing with transient problems are still hardly found, where solving transient moving boundary problems adopting the Trefftz method rarely exist.…”