Direct and inverse time-harmonic elastic scattering from point-like and extended obstacles

Guanghui Hu; Andrea Mantile; Mourad Sini; Tao Yin

doi:10.3934/ipi.2020054

Article Contents

2020, Volume 14, Issue 6: 1025-1056. Doi: 10.3934/ipi.2020054

This issue Previous Article Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation Next Article Interactions of semilinear progressing waves in two or more space dimensions

Direct and inverse time-harmonic elastic scattering from point-like and extended obstacles

1.
School of Mathematical Sciences, Nankai University, Tianjin 300071, China
2.
Laboratoire de Mathématiques de Reims, UMR 9008 CNRS, France
3.
RICAM, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040 Linz, Austria
4.
Department of Computing & Mathematical Sciences, California Institute of Technology, 1200 East California Blvd., CA 91125, United States

^* Corresponding author
^* Corresponding author

Received: December 2019

Revised: July 2020

Early access: August 2020

Published: December 2020

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Abstract

This paper is concerned with the time-harmonic direct and inverse elastic scattering by an extended rigid elastic body surrounded by a finite number of point-like obstacles. We first justify the point-interaction model for the Lamé operator within the singular perturbation approach. For a general family of pointwise-supported singular perturbations, including anisotropic and non-local interactions, we derive an explicit representation of the scattered field.

In the case of isotropic and local point-interactions, our result is consistent with the ones previously obtained by Foldy's formal method as well as by the renormalization technique. In the case of multiple scattering with pointwise and extended obstacles, we show that the scattered field consists of two parts: one is due to the diffusion by the extended scatterer and the other one is a linear combination of the interactions between the point-like obstacles and the interaction between the point-like obstacles with the extended one.

As to the inverse problem, the factorization method by Kirsch is adapted to recover simultaneously the shape of an extended elastic body and the location of point-like scatterers in the case of isotropic and local interactions. The inverse problems using only one type of elastic waves (i.e. pressure or shear waves) are also investigated and numerical examples are presented to confirm the inversion schemes.

Keywords:

Mathematics Subject Classification: 74B05, 78A45, 81Q10.

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Figure 1. The kite-shaped extended obstacle

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Figure 2. Reconstruction of the kite-shaped obstacle and 6 point-like scatterers for Example 1 with different polarization vectors $ \mathbf{a} = (\cos\beta,\sin\beta) $. We set $ \beta = 0 $ in (a, c, e) and $ \beta = \pi/2 $ in (b, d, f)

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Figure 3. Reconstruction of the kite-shaped obstacle and 11 point-like scatterers for Example 1 with different polarization vectors $ \mathbf{a} = (\cos\beta,\sin\beta) $. $ \alpha = 0 $ in (a, c, e), $ \beta = \pi/2 $ in (b, d, f)

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Figure 4. Reconstruction of the kite-shaped obstacle and 6 point-like scatterers for Example 2 with different "impedance'' coefficients $ \alpha _{j}, j = 1,\cdots,M $.

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Figure 5. Reconstruction of the kite-shaped obstacle and 20 point-like scatterers for Example 3 with different polarization vectors $ \mathbf{a} = (\cos\beta,\sin\beta) $

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