On Smoothed Finite Element Methods

Abstract

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 combined scheme of edge-based and node-based smoothed finite element methods for Reissner–Mindlin flat shells

“…In ENS-FEM, the membrane, the bending and the shear strain fields are smoothed by the means of a gradient smoothing technique over the smoothing domains constructed based on a combination of ES-FEM and NS-FEM. ES-FEM produces usually lower bound solutions in strain energy of the exact solution as presented in numerical results [14,16,32,33], and NS-FEM possesses the upper bound property when a reasonably fine mesh is used [14,17,32,33]. Hence, ENS-FEM model could be softer than ES-FEM model and stiffer than NS-FEM model.…”

An edge-based smoothed finite element method (ES-FEM) for dynamic analysis of 2D Fluid-Solid interaction problems

“…On this aspect, the ES-FEM-T3 shows the excellent properties in both spacially and temporally stable (Liu and Nguyen-Thoi, 2010;Liu et al, 2009b). Hence, it is very suitable to apply the ES-FEM-T3 for free and forced vibration analyses of the fluidsolid interaction problems.…”

“…However, the FEM-T3 elements result in the overestimated stiffness matrix that leads to the poor accuracy of solutions and the locking phenomena in the problems related to the incompressible material or bending domination. In order to overcome this issue of the FEM-T3, Liu and Nguyen-Thoi (2010) have developed a series of "soften" models namely Smoothed FEM (S-FEMs) in which the strain smoothing technique of meshfree methods (Chen et al, 2001) is incorporated into the standard compatible FEM. The key point of TECHNICAL NOTE the methods is to replace the standard compatible strain by the smoothed strain calculated based on smoothing domains which can be created easily from the element mesh of the FEM.…”

Symmetric global partition polynomials for reproducing kernel elements

“…Further, our formulation retains the same simplicity of meshing as finite element triangular and tetrahedral meshes [6,14] but with higher order continuity. The latter being an important practical issue for the meshing of complex domains and adaptive meshing analysis [8]. The basic concept is to automatically generate a single interior node at the centroid of the element (see Fig.…”