Flexural wave scattering plays a crucial role in optimizing and designing structures for various engineering applications. Mathematically, the flexural wave scattering problem on an infinite thin plate is described by a fourth-order plate-wave equation on an unbounded domain, making it challenging to solve directly using the regular linear finite element method (FEM). In this paper, we propose two numerical methods, the interior penalty FEM (IP-FEM) and the boundary penalty FEM (BP-FEM) with a transparent boundary condition (TBC), to study flexural wave scattering by an arbitrary-shaped cavity on an infinite thin plate. Both methods decompose the fourth-order plate-wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions at the cavity boundary. A TBC is then constructed based on the analytical solutions of the Helmholtz and modified Helmholtz equations in the exterior domain, effectively truncating the unbounded domain into a bounded one. Using linear triangular elements, the IP-FEM and BP-FEM successfully suppress the oscillation of the bending moment of the solution at the cavity boundary, demonstrating superior stability and accuracy compared to the regular linear FEM when applied to this problem.

翻译：弯曲波散射在优化和设计各类工程结构时起着关键作用。从数学上看，无限薄板上的弯曲波散射问题由定义在无界域上的四阶板波方程描述，这使得直接使用常规线性有限元方法（FEM）求解十分困难。本文提出了两种数值方法——带有透明边界条件（TBC）的内部惩罚有限元法（IP-FEM）和边界惩罚有限元法（BP-FEM），用于研究无限薄板上任意形状空腔对弯曲波的散射。这两种方法将四阶板波方程分解为亥姆霍兹方程和修正亥姆霍兹方程，并在空腔边界处施加耦合条件。随后，基于外部区域中亥姆霍兹方程和修正亥姆霍兹方程的解析解构建TBC，从而将无界域有效截断为有界域。利用线性三角形单元，IP-FEM和BP-FEM成功抑制了空腔边界处解的弯矩振荡，与常规线性FEM相比，在处理该问题时展现出更优越的稳定性和准确性。