13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- A Stabilized Sequential Coupling Algorithm for Hydro-mechanical Systems using Reproducing Kernel Particle Method Yongning XIE1, Gang WANG1,* 1 Department of Civil and Environmental Engineering The Hong Kong University of Science and Technology, Hong Kong SAR, China * Corresponding author: gwang@ust.hk Abstract Sequential coupling scheme is a flexible numerical scheme to solve a coupled hydro-mechanical system. However, it suffers from severe numerical instability. Stability analysis of a sequential coupling scheme is performed in this paper. It is derived that the numerical scheme can be unconditionally stable with a stabilization term introduced to the fluid equation, while it is only conditionally stable without the stabilization. Reproducing Kernel Particle Method (RKPM) is used for spatial discretization. One dimensional consolidation in an elastic medium is conducted to verify the sequential scheme, and the convergence behavior during the iterations is presented. Keywords hydro-mechanical coupling, sequential, meshfree, stability 1. Introduction Sequential coupling scheme has been extensively studied in the past decades to solve the coupled hydro-mechanical system (e.g. [1-4]). It has great advantage over the traditional fully-coupled numerical scheme due to its modularity such that the fluid and the mechanical solver for corresponding governing equations can be executed separately without many extra manipulations. The coupling effect is fulfilled through the information exchange between two solvers. Modularity is particularly advantageous in practical applications. However, the convenience does not come without any price. The numerical stability and convergence is one problem frequently encountered using the sequential scheme. As has been demonstrated in literatures, different sequential algorithms require different stability conditions [4]. One part of this work is to seek a stable sequential couplings scheme. It is worth pointing out that the current study only focuses on the stability and convergence problem in the temporal space. To achieve this goal, a stabilization term is introduced to the fluid equation. Using the conventional stability analysis, it is found that the scheme can be unconditionally stable with a suitably chosen relaxation parameter. Another part of the work is to employ one type of meshfree methods, Reproducing Kernel Particle Method (RKPM), for spatial discretization. Due to their high order interpolations and mesh-free nature, meshfree methods are generally considered to have advantage in handling problems with large strains or strain localization. These features are not demonstrated in this paper as the emphasis is more on the formulation and stability of the sequential scheme. However, the numerical results do suggest the applicability of RKPM to the coupling scheme developed. More interesting features of RKPM may show up if nonlinear constitutive models are used and more complicated boundary value problems considered. This paper is arranged as follows: Governing equations for the hydro-mechanical system are presented first, followed by the spatial discretization by RKPM and algorithm of sequential scheme. Stability analysis of the numerical scheme is presented subsequently. Numerical simulations for verification of the stability conditions are then demonstrated. Throughout the paper, letter in bold face denotes tensor or vector. , () i denotes partial derivative with respect to coordinate. A superscript dot over a variable . () denotes the time derivative of that variable. ij is the Kronecker delta tensor.
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