A Node-Based Smoothed Finite Element Method with Linear Gradient Fields for Elastic Obstacle Scattering Problems (Advances in Applied Mathematics and Mechanics, 2023)
In this paper, a node-based smoothed finite element method (NS-FEM) withlinear gradient fields (NS-FEM-L) is presented to solve elastic wave scattering by arigid obstacle. By using Helmholtz decomposition, the problem is transformed into aboundary 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 constructedin 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 bythree linearly independent weight functions. Further, based on the weakened weakformulation, a system of linear equation with the smoothed gradient is established forNS-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 stiffnessmatrix, which effectively overcomes the instability of original NS-FEM. Moreover, theconvergence rates of $L^2$ and $H^1$ semi-norm errors for the two NS-FEM-L models arealso higher.