Time-dependent acoustic scattering from generalized impedance boundary conditions via boundary elements and convolution quadrature

TL;DR

This paper develops a boundary element and convolution quadrature method for simulating time-dependent acoustic scattering from materials with complex boundary conditions, ensuring stability and convergence.

Contribution

It introduces a novel numerical approach combining boundary elements and convolution quadrature for generalized impedance boundary conditions in acoustic scattering.

Findings

The method is proven to be stable and convergent with optimal order.

Numerical experiments confirm the effectiveness of the approach.

The approach handles complex material interactions with multiple scales.

Abstract

Generalized impedance boundary conditions are effective, approximate boundary conditions that describe scattering of waves in situations where the wave interaction with the material involves multiple scales. In particular, this includes materials with a thin coating (with the thickness of the coating as the small scale) and strongly absorbing materials. For the acoustic scattering from generalized impedance boundary conditions, the approach taken here first determines the Dirichlet and Neumann boundary data from a system of time-dependent boundary integral equations with the usual boundary integral operators, and then the scattered wave is obtained from the Kirchhoff representation. The system of time-dependent boundary integral equations is discretized by boundary elements in space and convolution quadrature in time. The well-posedness of the problem and the stability of the numerical discretization rely on the coercivity of the Calder\'on operator for the Helmholtz equation with frequencies in a complex half-plane. Convergence of optimal order in the natural norms is proved for the full discretization. Numerical experiments illustrate the behaviour of the proposed numerical method.