Numerical solution of the cavity scattering problem for flexural waves on thin plates: linear finite element methods

TL;DR

This paper introduces two novel finite element methods, IP-FEM and BP-FEM, with transparent boundary conditions, to accurately and stably solve the challenging flexural wave scattering problem on infinite thin plates with cavities.

Contribution

The paper develops and analyzes two new numerical methods that effectively handle the fourth-order plate-wave equation on unbounded domains using boundary truncation techniques.

Findings

Both methods successfully suppress oscillations at the cavity boundary.

IP-FEM and BP-FEM demonstrate superior stability over regular FEM.

Methods accurately simulate flexural wave scattering on infinite plates.

Abstract

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.