Research Paper A cell-based smoothed point interpolation method for flow-deformation analysis of saturated porous media A. Tootoonchi a , A. Khoshghalb a,⇑ , G.R. Liu b , N. Khalili a a School of Civil and Environmental Engineering, UNSW Australia, Sydney, NSW 2052, Australia b Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH 45221, USA article info Article history: Received 29 October 2015 Received in revised form 6 January 2016 Accepted 29 January 2016 Keywords: Flow-deformation analysis Meshfree methods Smoothing gradient operation Point interpolation methods abstract A group of cell-based smoothed point interpolation methods based on the generalised gradient smooth- ing technique are proposed for the numerical modelling of saturated porous media. In the methods pro- posed, 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 ðW 2 Þ formulation referred to as the Generalised Smoothed Galerkin method. Both dis- placement 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 tech- nique, 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 ele- ment methods. Ó 2016 Elsevier Ltd. All rights reserved. 1. Introduction The Finite Element Method (FEM) [1–3] is ubiquitously used in numerical modelling of boundary value geotechnical engineering problems [4–6]. Despite its convenience and flexibility in use, the FEM suffers from many inherent problems, such as overly stiff behaviour, volumetric locking, poor accuracy in derivative solu- tions and less flexibility in solving problems involving large defor- mation and discontinuities [7]. The root cause of these problems can invariably be traced to the strong reliance of the FEM on qual- ity mesh to generate interpolation functions and discretised sys- tem of equations using weak formulations. To overcome at least part of these mesh-related difficulties, meshfree methods (MMs) have been developed. MMs date back to as early as 1970s when the smoothed particle hydrodynamics (SPH) method was introduced by Lucy [8] and Gingold and Mon- aghan [9] to solve astrodynamics problems. Since then, many different MMs have been proposed, including the diffuse element method (DEM) [10], element-free Galerkin method (EFGM) [11], reproducing kernel particle method (RKPM) [12] and the meshless local Petrov–Galerkin method (MLPG) [13], amongst the others. Overviews on the development of MMs can be found in [7,14–16]. To date, MMs have been applied to solve many geotechnical engineering problems including two-dimensional contaminant transport through saturated porous media [17]; prediction of sub- sidence over compacting reservoirs [18]; stress analysis for crack evolution [19]; consolidation analysis in saturated porous media [20]; soil collapse and erosion process in excavations [21]; and analysis of slope stability and discontinuous mechanics [22], to name a few. Excerpt for particle MMs (e.g. SPH) and some of MMs that are based on strong form of governing equations (e.g. collocation method [23]) many MMs are not completely free of mesh [11]. They often require a mesh, commonly referred to as the back- ground mesh, for the Gaussian numerical integration of system matrices. Nevertheless, in MMs, as opposed to the FEM, the numer- ical operations extend beyond the background mesh and the http://dx.doi.org/10.1016/j.compgeo.2016.01.027 0266-352X/Ó 2016 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Computers and Geotechnics 75 (2016) 159–173 Contents lists available at ScienceDirect Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo