Numerical analyses of steady-state seepage problems using the interpolation finite difference method

Abstract

Seepage analyses have mainly been executed using the finite element method; numerical analyses using the finite difference method (FDM) have been limited to cases where the calculation domains are comparatively simple. This limitation is observed because FDM is considered to be inappropriate for application in calculations over complex domains. However, by applying the so-called “interpolation FDM (IFDM)”, we can now freely solve two- and three-dimensional elliptic partial differential equations (PDEs) over complex domains with high speed and high accuracy. By adopting this procedure, named the boundary polynomial interpolation, all of the numerical analyses of elliptic PDEs reduce to Dirichlet problems over regular domains. This method is also effective in the calculation of a flow net where mixed Dirichlet and Neumann conditions exist. By giving the coordinate values of changing points regarding the polygonal line of a domain and boundary conditions, grid generation is automatically carried out and numerical solutions are promptly obtained. In this paper, the method of saturated seepage analyses with a fixed domain is first formulated and then expanded to unconfined domain problems, namely, free surface problems. While analytical solutions of the PDE are highly limited, there is an analytical solution for the location of the free surface in a rectangular dam. The numerical solutions obtained using the IFDM are compared with the analytical ones, and it is shown that the proposed method has adequate accuracy in practice and wide applicability as a general method of numerically solving seepage problems.