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A Stabilization Procedure for Soil-water Coupled Problems Using the Mesh-free Method

T. Shibata
Dept. of Civil and Environmental Engineering, Matsue National College of Technology, Shimane, Japan
A. Murakami
Graduate School of Environmental Science, Okayama University, Okayama, Japan.

Keywords: Mesh-free method, soil-water coupled problem, stabilization procedure

ABSTRACT: The development of stability problems related to classical mixed methods has recently been observed. In this study, a soil-water coupled boundary-value problem, one type of stability problem, is presented using the Element-free Galerkin Method (EFG Method). In this soil-water coupled problem, anomalous behavior appears in the pressure field unless stabilization techniques are used. The remedy to such numerical instability has generally been to adopt a higher interpolation order for the displacements than for the pore pressure. As an alternative, however, an added stabilization term is incorporated into the equilibrium equation. The advantages of this stabilization procedure are as follows: (1) The interpolation order for the pore pressure is the same as that for the displacements. Therefore, the interpolation functions in the pore pressure field do not reduce the accuracy of the numerical results. (2) The stabilization term consists of first derivatives. The first derivatives of the interpolation functions for the EFG Method are smooth, and therefore, the solutions for pore pressure are accurate.In order to validate the above stabilization technique, some numerical results are given. It can be seen from the results that a good convergence is obtained with this stabilization term.
The development of numerical computation technologies has enabled a variety of engineering problems to be solved and has brought about remarkable progress in recent decades. Among the related findings, meshless and/or mesh-free methods in particular have been applied to some problems for which the usual finite element
method is ineffective in dealing with significant mesh distortion brought about by large deformations, crack growth, and moving discontinuities. 
Various meshless and/or mesh-free methods have been used for geotechnical problems, instead of the finite element method, to overcome the above-mentioned difficulties. Consolidation phenomena have been analyzed
by means of EFGM (Modaressi et al., 1998)(Nogami et al., 2004)(Murakami et al., 2005)(Wang et al., 2006), the point/radial point interpolation method (PIM/RPIM) (Wang et al., 2001)(Wang et al., 2002), the local RPIM (Wang et al., 2005), RKPM (Chen et al., 2001)(Zhang et al., 2005), and the natural neighbor method (Cai et al., 2005), the transient response of saturated soil has been dealt with under cyclic loading by means of EFGM (Karim et al., 2002)(Sato et al., 2006), wave-induced seabed response and instability have been examined by EFGM (Wang et al., 2007) and RPIM (Wang et al., 2004), slip lines have been modeled by geological materials using EFGM (Rabczuk et al., 2006), and a Bayesian inverse analysis has been carried out in conjunction with the meshless local Petrov-Galerkin method (Sheu, 2006).

However, unless certain requirements are met in dealing with soil-water coupled problems for the finite element computation, based on the coupled formulation becoming ill-conditioned, numerical instabilities will occur (Chapelle et al., 1993). In order to overcome these weaknesses, several strategies have been proposed (Pastor et al., 1999). For example, as a necessary condition for stability, the interpolation degree of the displacement field must be higher than that of the pore pressure field. An alternative means of stabilization was also proposed based on the Simo-Rifai enhanced strain method which even allows an equal order of interpolation degree for both variables. However, these strategies are not directly applicable to meshless/mesh-free methods, because all the nodal points simultaneously have the same degree of freedom for both the displacement field and the pore pressure field, and no information between the element and the nodes can be utilized. The purpose of this paper is to present a stabilization methodology for the mesh-free analysis of soil-water coupled problems by incorporating the stabilizing term into the weak form. The following sections deal with descriptions of the formulation, including the stabilization term. In Section 3, two applications of the strategy to soil-water coupled problems are analyzed, one being the saturated soil column test appearing in Mira et al.(2003),to demonstrate the effectiveness of the strategy, and the other being the foundation behavior under displacement-controlled condition, for which the feasibility of theanalysis will be thoroughly discussed while

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