A Node-Based Smoothed Finite Element Method with Linear Gradient Fields for Elastic Obstacle Scattering Problems

Abstract

In this paper, a node-based smoothed finite element method (NS-FEM) with linear gradient fields (NS-FEM-L) is presented to solve elastic wave scattering by a rigid obstacle. By using Helmholtz decomposition, the problem is transformed into a boundary value problem with coupled boundary conditions. In numerical analysis, the perfectly matched layer (PML) and transparent boundary condition (TBC) are introduced to truncate the unbounded domain. Then, a linear gradient is constructed in a node-based smoothing domain (N-SD) by using a complete order of polynomial. The unknown coefficients of the smoothed linear gradient function can be solved by three linearly independent weight functions. Further, based on the weakened weak formulation, a system of linear equation with the smoothed gradient is established for NS-FEM-L with PML or TBC. Some numerical examples also demonstrate that the presented method possesses more stability and high accuracy. It turns out that the modified gradient makes the NS-FEM-L-PML and NS-FEM-L-TBC possess an ideal stiffness matrix, which effectively overcomes the instability of original NS-FEM. Moreover, the convergence rates of $L^2$ and $H^1$ semi-norm errors for the two NS-FEM-L models are also higher.