Application of Edge Based Smoothed Finite Element Method in Solution of Seepage and Foundation Settlement Problems

Smoothed finite element method (SFEM) was introduced by application of the strain smoothing technic in the conventional finite element method (FEM). The strain smoothing technic was previously used in mesh-free methods to overcome the numerical instabilities due to nodal integration. SFEM has three main types: 1-Cell-based SFEM (CSFEM), 2-Node-based SFEM (NSFEM) and 3-Edge-based SFEM (ESFEM). In these methods, problem domain is first discretized into a mesh of elements, similar to the FEM, and then based on these elements, domains are created to perform the strain smoothing operation on them. These domains are called “Smoothing Domains”. The difference between SFEM types is in the method of creating these smoothing domains. Different smoothing domains, can give results with different qualities. Among them, the edge-based method can give results that are ultra-accurate and super-convergent. Due to their interesting features, SFEMs have been used to solve different problems. Problems such as, mechanics of solids and piezoelectrics, fracture mechanics and crack propagation, heat transfer, structural acoustics, nonlinear and contact problems, adaptive analysis, phase change problem and many more. In this paper, first idea and formulation of SFEM is reviewed, with special consideration on the edge-based method. Detailed instructions are given for creation of edge-based smoothing domains, and strain and stiffness matrices for this method are derived. After that, the algorithm for creation of a SFEM code is introduced. Based on these formulations and algorithm, an edge-based smoothed finite element code is created, that is used for analysis of some numerical examples. Two problems, based on two different practical geotechnical engineering applications, are solved using the ESFEM and also FEM with 3-node and 6-node triangular elements. Using same mesh for all three methods, makes comparison possible, and performance of the ESFEM will be investigated. First problem, is a steady state seepage problem, where seepage below a sheet pile barrier is modeled, with the assumption of plane strain condition. Since there is no analytical solution for this problem, FEM results using 6-node triangular elements are considered as the more accurate results for comparison. Investigating the results reveals that implementation of the strain smoothing technic in FEM using 3-node triangular elements, can make the results closer to those of the FEM using 6-node triangular elements, while the degrees of freedom remain constant. Edge-based smoothed finite element method can give results for steady state seepage problem, that have errors less than half of the conventional FEM results errors, with the same mesh and number of degrees of freedom. The other problem, is calculating the elastic settlement of a circular foundation, to investigate the performance of the ESFEM in axisymmetric problems, compared with the FEM. Again, the problem is solved using three ESFEM, FEM with 3-node triangular elements, and FEM with 6-node triangular element, with the latter as the most accurate. Surface deformation of the problem domain, after imposition of the foundation load is studied. It is seen that ESFEM results match the FEM results. A closer look reveals that the ESFEM results for settlement of the foundation, is closer to the FEM results using 6-node triangular elements, than the FEM using 3-node elements and are more accurate.