A Gradient Stable Node-Based Smoothed Finite Element Method for Solid Mechanics Problems

Guangsong Chen,Jia Ma,Linfang Qian

doi:10.1155/2019/8610790

Abstract

This paper presents a gradient stable node-based smoothed finite element method (GS-FEM) which resolves the temporal instability of the node-based smoothed finite element method (NS-FEM) while significantly improving its accuracy. In the GS-FEM, the strain is expanded at the first order by Taylor expansion in a node-supported domain, and the strain gradient is then smoothed within each smoothing domain. Therefore, the stiffness matrix includes stable terms derived by the gradient of the strain. The GS-FEM model is softer than the FEM but stiffer than the NS-FEM and yields far more accurate results than the FEM-T3 or NS-FEM. It even has comparative accuracy compared with those of the FEM-Q4. The GS-FEM owns no spurious nonzero-energy modes and is thus temporally stable and well-suited for dynamic analyses. Additionally, the GS-FEM is demonstrated on static, free, and forced vibration example analyses of solids.

Highlights

Beissel and Belytschko [38] first proposed a stabilized nodal integration procedure to eliminate spurious nonzeroenergy modes by adding the square of the residual of the equilibrium equation to the potential energy function. is solution has been further extended to 2D and 3D problems to form a stabilization procedure for node-based smoothed finite element method (NS-FEM) [35, 39] with a recommended range for the stabilization parameter
Taylor series expansions of the displacement fields [40, 41] can be used to reduce the instability in direct node integration (DNI), but high-order derivatives appear in underling formulations resulting in an increase in its computational cost
Liu et al [29] proposed an α-FEM combining NS-FEM and standard FEM which can be used to stabilize the NS-FEM by introducing stiffening effects from the standard FEM stiffness matrix with a small α

Summary

Gradient Stabilization of NS-FEM

Equation (24) can be rewritten in the following compact form: ε(x) ≈ εL + εLx x − xL􏼁 + εLy y − yL􏼁. Note that the assumed displacement field used in this work does not have second-order derivatives over the whole problem domain as the FEM shape function is applied, i.e., ΦI,ij(x) ≡ 0; terms (b) and (c) of equation (24) do not contribute to the stabilization if ΦI,ij(x) is calculated directly. We replace equation (27) with the gradient smoothing technique presented, that is, ε(x) ≈ εL + εLx x − xL􏼁 + εLy y − yL􏼁. E smoothed divergence of strain tensors in equation (26) can be truncated in the following vector form: in which εL 􏽘 BIuI, εLx 􏽘 BIxuI, εLy 􏽘 BIyuI,. Equation (35) contains stable terms introduced by the smoothed gradient expansion of the strain at the node. AL, JLx, and JLy can be calculated as follows: AL 􏽚 F􏽢An1dΓ, F􏽢 Jx n1

Free and Forced Vibration Analyses

Numerical Implementation

Conclusions