A cell-based smoothed point interpolation method for flow-deformation analysis of saturated porous media

A group of cell-based smoothed point interpolation methods based on the generalised gradient smoothing technique are proposed for the numerical modelling of saturated porous media. In the methods proposed, the problem domain is first discretised with the use of a simple triangular background mesh. The purpose of the background mesh is twofold: (i) it is used to select the supporting nodes for each point of interest for the construction of nodal shape functions, and (ii) it provides cells to serve as the smoothing domains. Spatial discretisation of the coupled partial differential equations is derived by applying the weakened weak (W2) formulation referred to as the Generalised Smoothed Galerkin method. Both displacement and pressure fields are interpolated using the point interpolation shape functions (polynomial and radial). Shape function differentiations are effected through the use of the smoothed gradient technique, leading to smoothed strains and pressure gradients. Temporal discretisation is performed with a three-point time discretisation scheme with variable time steps. A host of node selection schemes, known as T-schemes, are adopted to guarantee the non-singularity of the moment matrices in creating shape functions. The proposed methods are thoroughly examined by simulation of a number of benchmark examples with analytical or semi-analytical solutions. The accuracy and convergence rate of the methods are investigated through comparison of the numerical results of the proposed methods with those obtained using analytical/semi-analytical solutions, point interpolation methods, and standard finite element methods.