Time-Dependent Wave-Structure Interaction Revisited: Thermo-Piezoelectric Scatterers

George C Hsiao,Tonatiuh Sánchez-Vizuet

doi:10.3390/fluids6030101

George C Hsiao, Tonatiuh Sánchez-Vizuet

Open Access

https://doi.org/10.3390/fluids6030101

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Abstract

In this paper, we are concerned with a time-dependent transmission problem for a thermo-piezoelectric elastic body that is immersed in a compressible fluid. It is shown that the problem can be treated by the boundary-field equation method, provided that an appropriate scaling factor is employed. As usual, based on estimates for solutions in the Laplace-transformed domain, we may obtain properties of corresponding solutions in the time-domain without having to perform the inversion of the Laplace-domain solutions.

Highlights

Wave-Structure Interaction Revisited: The mathematical description of the interaction between an acoustic wave and an elastic body is of central importance in applied mathematics and engineering, as attested, for instance, by its usage for the detection and identification of submerged objects
In order to sidestep the challenge of undboundedness, we will resort to a formulation of the transmission problem that is defined by Equations (12)–(15) that will couple boundary integral equations with partial differential equations
Let us first define a class of admissible symbols in order to state the result that will allow us to transfer our previous analysis in the Laplace domain back in to the time domain, following [18]

Summary

Introduction

Wave-Structure Interaction Revisited: The mathematical description of the interaction between an acoustic wave and an elastic body is of central importance in applied mathematics and engineering, as attested, for instance, by its usage for the detection and identification of submerged objects. One of the main reasons behind the use of the boundary-field equation method for treating time-harmonic wave-structure problems is to reduce the transmission problem, posed originally in an unbounded domain, to one set in the bounded domain Ω that was determined by the elastic scatterer (see Figure 1). In the present paper, being inspired by the work of Estorff and Antes [17], we will apply the boundary-field equation method not to a time-harmonic problem, but rather one in the transient regime This will require the treatment of the wave equation, as opposed to the Helmholtz equation that is used in the frequency domain.

Formulations of the Problem

A Nonlocal Boundary Problem

Variational Solutions

Results in the Time Domain

Concluding Remarks