The smoothed finite element methods (S-FEM) are a family of methods formulated through carefully designed combinations of the standard FEM and some of the techniques from the meshfree methods. Studies have proven that S-FEM models behave softer than the FEM counterparts using the same mesh structure, often produce more accurate solutions, higher convergence rates, and much less sensitivity to mesh distortion. They work well with triangular or tetrahedral mesh that can be automatically generated, and hence are ideal for automated computations and adaptive analyses. Some S-FEM models can also produce upper bound solution for force driving problems, which is an excellent unique complementary feature to FEM. Because of these attractive properties, S-FEM has been applied to numerous problems in the disciplines of material mechanics, biomechanics, fracture mechanics, plates and shells, dynamics, acoustics, heat transfer and fluid–structure interactions. This paper reviews the developments and applications of the S-FEM in the past ten years. We hope this review can shed light on further theoretical development of S-FEM and more complex practical applications in future.

Smoothed Finite Element Methods (S-FEM): An Overview and Recent Developments

This review paper presents a methodological study on possible and existing meshfree methods for solving the partial differential equations (PDEs) governing solid mechanics problems, based mainly on the research work in the past two decades at the authors group. We start with a discussion on the general steps in a meshfree method based on nodes, with the displacements as the primary variables. We then examine the major techniques used in each of these steps: (1) techniques for displacement function approximations using nodes, (2) approximation of the gradient of the displacements or strains based on nodes and a background T-cells that can be automatically generated and refined, and (3) formulation techniques for producing algebraic equations. The function approximation techniques include node-based interpolation methods, cell-based interpolation methods, function smoothing techniques, and moving least squares approximation techniques. The gradient approximation includes direct differentiation, gradient smoothing, and special strain construction. Formulation techniques include strong-form, weakform, local weakform, weak-strong-form, and weakened weakform (W2). In theory, a meshfree method can be developed using a combination of function approximation, gradient approximation, and formulation techniques, which can lead to matrix of a large number of possible methods. This review attempts to provide an overall methodological review, rather than a usual review of comparing different methods. We hope to show readers the differences between the forests, and just between the trees.

An Overview on Meshfree Methods: For Computational Solid Mechanics

Hybrid smoothed finite element method for two-dimensional underwater acoustic scattering problems

In this work, the smoothed finite element method using four-node quadrilateral elements (SFEM-Q4) is employed to resolve underwater acoustic radiation problems. The SFEM-Q4 can be regarded as a com...

Application of Smoothed Finite Element Method to Two-Dimensional Exterior Problems of Acoustic Radiation

A truly meshfree method for solving acoustic problems using local weak form and radial basis functions

The smoothed finite element method (S-FEM) has been recently developed as an effective solver for solid mechanics problems. This paper represents an effective approach to compute the lower bounds of vibration modes or eigenvalues of elasto-dynamic problems, by making use of the important softening effects of node-based S-FEM (NS-FEM). We first use NS-FEM, FEM and the analytic approach to compute the eigenvalues of transverse free vibration in strings and membranes. It is found that eigenvalues by NS-FEM are always smaller than those by FEM and the analytic method. However, NS-FEM produces spurious unphysical modes because of overly soft behavior. A technique is then proposed to remove them by analyzing their vibration shapes (eigenvectors). It is observed that spurious modes with excessively large wave numbers, which are unrelated to the physical deflection shapes but related to the discretization density, therefore can be easily removed. The final results of NS-FEM become the lower bound of eigenvalues and the accuracy can be improved via mesh refinement. And NS-FEM solutions (softer) are more reliable, because the large wave number component can be used as an indicator, which is available in FEM (stiffer), on the quality of the numerical solutions. The proposed NS-FEM procedure offers a viable and practical computational means to effectively compute the lower bounds of eigenvalues for solid mechanics problems.

Lower Bound of Vibration Modes Using the Node-Based Smoothed Finite Element Method (NS-FEM)

This paper represents some basic mathematic theories for Gs spaces of functions that can be used for weakened weak (W2) formulations, upon which the smoothed finite element methods (S-FEMs) and the smoothed point interpolation methods (S-PIMs) are based for solving mechanics problems. We first introduce and prove properties of Gs spaces, such as the lower boundedness and convergence of the norms, which are in contrast with H1 spaces. We then prove the equivalence of the Gs norms and its corresponding semi-norms. These mathematic theories are important and essential for the establishment of theoretical frame and the development of relevant numerical approaches. Finally, numerical examples are presented by using typical S-FEM models known as the NS-FEM and αS-FEM to examine the properties of a smoothed method based on Gs spaces, in comparison with the standard FEM with weak formulation.

Mathematical Basis of G Spaces

This paper briefs some of recent works on applications of the smoothed finite element method (SFEM) to fracture problems. In the SFEM formulation for simulating the singular filed near the crack-tip, it is known that one can directly enrich the filed by adding in proper terms needed to simulate the singularity, which can be done in a number of ways (see the SFEM book). In this work a generalized technique called the "enriched linear PIM" for constructing shape functions is used to formulate a special "five-node (T5) singular crack-tip element" that can produce a proper order of stress singularity near the crack-tip. One layer of T5 crack-tip singular elements are used in a basic mesh of linear elements leading to a very simple model that can be crated automatically for complicated geometry. Because the SFEM uses weakened weak formulations, such a simple model does not require transition elements and the compatibility is ensured. In addition, the singular terms of functions as well as mapping procedures are no longer necessary to compute the stiffness matrix. Thus, the singular SFEM method is straightforward and can be easily implemented in the existing codes. The stress intensity factors of mix-modes can also be easily evaluated by an appropriate treatment during the domain form of the interaction integral, due to the use of simple triangular mesh. The effectiveness of the present singular T5 element is demonstrated via benchmark examples.

S-fem for fracture problems, theory, formulation and application

This paper presents a novel and effective cell-based smoothed alpha radial point interpolation method (CS-αRPIM) using αPIM shape functions for approximating displacement and cell-based smoothed strains for displacement gradient construction. Using a scaling factor α ∈ [0, 1], the αPIM shape functions are combinations of the condensed RPIM (RPIM-Cd) shape functions and the linear PIM shape functions, where the former often leads to a "softer" CS-RPIM model, and the latter a "stiffer" linear CS-RPIM model (which is as same as linear FEM), compared to the exact one. Through adjusting the value of α in our new CS-αRPIM, the stiffness of the model can be "designed" for desired purposes, such as for seeking nearly exact solutions in strain energy norm (or possibly other norms). A simple and practical procedure to search for such an α has also been presented. Some 2D and 3D numerical examples are studied to examine various properties of the present method in terms of accuracy, convergence and computational efficiency.

Meshfree cell-based smoothed alpha radial point interpolation method (CS-α RPIM) for solid mechanics problems

Recently, the smoothed point interpolation method (S-PIM) regarded as a weakened weak ( W 2 ) formulation method has been developed for solving engineering mechanics problems. It works well with distorted meshes. The G space theory offers the theoretical base for all the W 2 methods that use smoothing operations. In this paper, we first prove mathematically that if a function is of Lipschitz continuity and its interpolated function is established using PIM shape functions, then the interpolated function belongs to a G h , 0 s space. Our proofs work for smoothing operations that are the node-based, cell-based and a mixture of both smoothing domains. In addition, when mesh is refined under a given regularity condition, a sufficiently smooth target function can be approximated by its interpolated function with arbitrary accuracy, meaning that the interpolation error norm approaches to zero. Therefore, the stability and convergence of a W 2 method using PIM shape functions and G space theory can be ensured.

Ming Li

Papers

Lower Bound of Vibration Modes Using the Node-Based Smoothed Finite Element Method (NS-FEM)

Mathematical Basis of G Spaces

S-fem for fracture problems, theory, formulation and application

Meshfree cell-based smoothed alpha radial point interpolation method (CS-α RPIM) for solid mechanics problems

Proofs of the stability and convergence of a weakened weak method using PIM shape functions