Boundary Integral Formulations for Flexural Wave Scattering in Thin Plates

TL;DR

This paper introduces second kind integral formulations for flexural wave scattering in thin plates, enabling high-accuracy solutions for various boundary conditions including clamped, supported, and free plates.

Contribution

It develops novel integral equation formulations for free and supported plates, improving solution stability and accuracy for complex boundary conditions.

Findings

Fredholm integral equations of the second kind derived for different boundary conditions

Enhanced methods for analyzing far field scattering patterns

Capability to solve large-scale multiple scatterer problems

Abstract

In this paper, we develop second kind integral formulations for flexural wave scattering problems involving the clamped, supported, and free plate boundary conditions. While the clamped plate problem can be solved with layer potentials developed for the biharmonic equation, the free plate problem is more difficult due to the order and complexity of the boundary conditions. In this work, we describe a representation for the free plate problem that uses the Hilbert transform to cancel singularities of certain layer potentials, ultimately leading to a Fredholm integral equation of the second kind. Additionally, for the supported plate problem, we improve on an existing representation to obtain a second kind integral equation formulation. With these representations it is possible to solve flexural wave scattering problems with high-order-accurate methods, examine the far field patterns of scattering objects, and solve large problems involving multiple scatterers.