A stable node-based smoothed finite element method with PML technique for the elastic wave obstacle scattering

Abstract

In this paper, a stable node-based smoothed finite element method with PML (SNS-FEM-PML) is proposed to solve the scattering problem of a time-harmonic elastic plane wave by a rigid obstacle in two dimensions. In the algorithm, the stability term is constructed by the Taylor expansion of the gradient to cure the instability of the original NS-FEM. The linear variations of the gradient with respect to x and y are included in the stability term, which are calculated using four integral points in an equivalent circle of node-based smoothing domain. Meanwhile, the perfectly matched layer (PML) technique is used to truncate the unbounded domain. Furtherly, the smoothed Galerkin weak formulations of SNS-FEM-PML model are derived and the linear algebra system with the linear smoothed gradient is constructed for the Navier equation and Helmholtz equations with coupled boundaries. Besides, we also prove theoretically the softening effect and convergence of the SNS-FEM model. Several numerical examples verify the effectiveness and accuracy of SNS-FEM model. The results suggest that the convergence order of L2 and H1 semi-norm errors of the SNS-FEM model is consistent with the theory of FEM, and convergence rate of the relative error is higher than that of the FEM.