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.

本文提出了一种基于稳定节点的带有PML的平滑有限元方法（SNS-FEM-PML）来解决二维刚性障碍物对时谐弹性平面波的散射问题。在算法中，稳定性项是通过梯度的泰勒展开来构造的，以解决原始NS-FEM的不稳定性。梯度相对于x和y的线性变化包含在稳定性项中，它是使用基于节点的平滑域的等效圆中的四个积分点计算的。同时，完美匹配层（PML）技术用于截断无界域。进一步推导了SNS-FEM-PML模型的平滑Galerkin弱公式，并为具有耦合边界的Navier方程和Helmholtz方程构建了线性平滑梯度的线性代数系统。此外，我们还从理论上证明了 SNS-FEM 模型的软化效果和收敛性。几个数值例子验证了SNS-FEM模型的有效性和准确性。结果表明，SNS-FEM模型的L2和H1半范数误差的收敛阶次与FEM理论一致，