Analysis of transient acoustic scattering by an elastic obstacle

Abstract

Consider the scattering of an acoustic plane wave by a bounded elastic obstacle which is immersed in an open space filled with a homogeneous medium. This paper concerns the mathematical analysis of the coupled two-and three-dimensional acoustic-elastic wave propagation problem in the time-domain. A compressed coordinate transformation is proposed to reduce equivalently the scattering problem into an initial-boundary value problem in a bounded domain over a finite time interval. The reduced problem is shown to have a unique weak solution by using the Galerkin method. The stability estimate and an a priori estimate with explicit time dependence are obtained for the weak solution. The reduced model problem is suitable for numerical simulations. The proposed method is applicable to many other time-domain scattering problems imposed in open domains.2010 Mathematics Subject Classification. 78A46, 65C30. Key words and phrases. Time domain, acoustic wave equation, elastic wave equation, fluid-structure interaction, well-posedness and stability, a priori estimate.

The solid-fluid transmission problem

“…Well posedness for solid-fluid systems is also shown in e.g. [LZ19], [dHHP17]. We actually take the extra step of identifying the domain of the self adjoint operator P explicitly.…”

“…The velocity-pressure system in a fluid was studied, e.g. in [BKPR98], and, coupled with a solid via transmission conditions, in [BGL18], [LZ19] (with constant Lamé parameters and densities). Displacement-pressure systems for a solid-fluid in the stationary formulation have been studied e.g.…”

Inverse obstacle scattering for acoustic waves in the time domain

“…For some recent developments on path reconstructions of moving point sources, we refer to [8,35,39,40]. For other direct scattering problems in time domain with acoustic, elastic or electromagnetic wave, we refer to [3,2,6,10,11,18,19,21,20,34,43]. As for the frequency domain inverse scattering problem, we refer to [13,14,31,32,33,36,41,42].…”

Time-domain analysis of forward obstacle scattering for elastic wave

“…Compared with the time-harmonic scattering problems, the time-domain scattering problems are more difficult due to time dependence, so the research results are fewer. In the scattering problems of time-domain acoustic, elastic and electromagnetic waves, including various structures, the results 4112 LU ZHAO, HEPING DONG AND FUMING MA are given in [1,3,4,7,8,9,13,14,15,16,20,22] for the forward scattering problem. In [3,13,14,15,16], the main idea is that by using TBC the model problem is reduced into an initial boundary value problem in a bounded domain, and it is transformed into s-domain by Laplace transform.…”

“…Finally, energy method is used to obtain the stability estimate for the solution in the time-domain. In [1,20], by using the fact that the scattered wave has a finite speed of propagation, a compressed coordinate transformation is proposed to reduce the problem into an initial boundary value problem in a bounded domain over a finite time interval. Based on the Galerkin method and energy estimate, the well-posedness and a priori estimate is obtained.…”