Boundary-Finite Element discretization of time dependent acoustic scattering by elastic obstacles with piezoelectric behavior

TL;DR

This paper develops a coupled boundary element and finite element method for simulating time-dependent acoustic scattering by elastic, piezoelectric obstacles, providing stability analysis and second-order convergence results.

Contribution

It introduces a novel coupled BEM/FEM formulation for piezoelectric scattering, with stability proofs and second-order convergence for the time discretization methods.

Findings

Stable semi-discretization in the Laplace domain

Second-order convergence of BDF2-based method

Numerical validation of the proposed approach

Abstract

A coupled BEM/FEM formulation for the transient interaction between an acoustic field and a piezoelectric scatterer is proposed. The scattered part of the acoustic wave is represented in terms of retarded layer potentials while the elastic displacement and electric potential are treated variationally. This results in an integro-differential system. Well posedness of a general Galerkin semi-discretization in space of the problem is shown in the Laplace domain and translated into explicit stability bounds in the time domain. Trapezoidal-Rule and BDF2 Convolution Quadrature are used in combination with matching time stepping for time discretization. Second order convergence is proven for the BDF2-based method. Numerical experiments are provided for BDF2 and Trapezoidal Rule based time evolution.