The paper presents an overview of the smoothed finite element methods (S-FEM) which are formulated by combining the existing standard FEM with the strain smoothing techniques used in the meshfree methods. The S-FEM family includes five models: CS-FEM, NS-FEM, ES-FEM, FS-FEM and α-FEM (a combination of NS-FEM and FEM). It was originally formulated for problems of linear elastic solid mechanics and found to have five major properties: (1) S-FEM models are always “softer” than the standard FEM, offering possibilities to overcome the so-called overly-stiff phenomenon encountered in the standard the FEM models; (2) S-FEM models give more freedom and convenience in constructing shape functions for special purposes or enrichments (e.g, various degree of singular field near the crack-tip, highly oscillating fields, etc.); (3) S-FEM models allow the use of distorted elements and general n-sided polygonal elements; (4) NS-FEM offers a simpler tool to estimate the bounds of solutions for many types of problems; (5) the αFEM can offer solutions of very high accuracy. With these properties, the S-FEM has rapidly attracted interests of many. Studies have been published on theoretical aspects of S-FEMs or modified S-FEMs or the related numerical methods. In addition, the applications of the S-FEM have been also extended to many different areas such as analyses of plate and shell structures, analyses of structures using new materials (piezo, composite, FGM), limit and shakedown analyses, geometrical nonlinear and material nonlinear analyses, acoustic analyses, analyses of singular problems (crack, fracture), and analyses of fluid-structure interaction problems.

A linearly conforming radial point interpolation method (LC-RPIM) is presented for stress analysis of two-dimensional solids. In the LC-RPIM method, each field node is enclosed by a Voronoi polygon, and the displacement field function is approximated using RPIM shape functions of Kronecker delta function property created by simple interpolation using local nodes and radial basis functions augmented with linear polynomials to guarantee linear consistency. The system equations are then derived using the Galerkin weak form and nodal integration techniques, and the essential boundary conditions are imposed directly as in the finite element method. The LC-RPIM method is verified via various numerical examples and an extensive comparison study is conducted with the conventional RPIM, analytical approach and FEM. It is found that the presented LC-RPIM is more stable, more accurate in stress and more efficient than the conventional RPIM.

To simulate the contact nonlinearity in 2D solid problems, a contact analysis approach is formulated using incremental form of the subdomain parametric variational principle (SPVP). The formulation is based on a linearly conforming radial point interpolation method (LC-RPIM) using nodal integration technique. Contact interface equations are also presented using a modified Coulomb frictional contact model and discretized by contact point-pairs. In the present approach, the global discretized system equations are transformed into a standard linear complementarity problem (LCP) that can be solved readily using the Lemke method. The present approach can simulate various contact behaviors including bonding/debonding, contacting/departing, and sticking/slipping. An intensive numerical study is performed to validate the proposed method via comparison with the ABAQUS ® and to investigate the effects of the various parameters used in computations. These parameters include normal and tangential adhesions, frictional coefficient, nodal density, the dimension of local nodal support domain, nodal irregularity, shape parameters used in the radial basis function and the external load. The numerical results have demonstrated that the present approach is accurate and stable for contact analysis of 2D solids.

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A novel Lagrangian gradient smoothing method (L-GSM) is developed to solve "solid-flow" (flow media with material strength) problems governed by Lagrangian form of Navier-Stokes equations. It is a particle-like method, similar to the smoothed particle hydrodynamics (SPH) method but without the socalled tensile instability that exists in the SPH since its birth. The L-GSM uses gradient smoothing technique to approximate the gradient of the field variables, based on the standard GSM that was found working well with Euler grids for general fluids. The Delaunay triangulation algorithm is adopted to update the connectivity of the particles, so that supporting neighboring particles can be determined for accurate gradient approximations. Special techniques are also devised for treatments of 3 types of boundaries: no-slip solid boundary, free-surface boundary, and periodical boundary. An advanced GSM operation for better consistency condition is then developed. Tensile stability condition of L-GSM is investigated through the von Neumann stability analysis as well as numerical tests. The proposed L-GSM is validated by using benchmarking examples of incompressible flows, including the Couette flow, Poiseuille flow, and 2D shear-driven cavity. It is then applied to solve a practical problem of solid flows: the natural failure process of soil and the resultant soil flows. The numerical results are compared with theoretical solutions, experimental data, and other numerical results by SPH and FDM to evaluate further L-GSM performance. It shows that the L-GSM scheme can give a very accurate result for all these examples. Both the theoretical analysis and the numerical testingresults demonstrate that the proposed L-GSM approach restores first-order accuracy unconditionally and does not suffer from the tensile instability. It is also shown that the L-GSM is much more computational efficient compared with SPH, especially when a large number of particles are employed in simulation.