Inverse acoustic scattering by solid obstacles: topological sensitivity and its preliminary application

Abstract

A topological sensitivity approach is developed for the imaging of penetrable solid obstacles embedded in a fluid background medium by way of inverse acoustic scattering. To this end, an asymptotic form for the scattered field caused by a nucleating spherical solid obstacle in the reference fluid medium is derived within the boundary integral equation framework, where the required limiting behaviour of the acceleration field inside the perturbation is established based on solutions of two simplified fluid-solid interaction problems. With this result, the equivalence of acoustic scattering by fluid and solid scatterers in terms of asymptotic behaviour is observed and numerically validated. The direct and adjoint-field topological sensitivity expressions for transient and time-harmonic excitations are obtained accordingly. The utility of the proposed method as a tool for preliminary obstacle reconstruction and bulk modulus characterization is illustrated through numerical examples and an exploratory experimental study. On the basis of adjusted topological sensitivity formulas for fluid reference domains containing pre-existing solid inhomogeneities, an iterative 3D obstacle reconstruction approach is established and its performance with synthetic data is demonstrated.

Topological Imaging Methods for the Iterative Detection of Multiple Impedance Obstacles

“…where J |n| is the derivative of the Bessel function J |n| . Using now (44) and (46), we find the asymptotic expansions for J |n| (kε) and J |n| (kε), from where we deduce for y ∈ ∂ app :…”

“…As a consequence, various TD-based iterative methods have been investigated in the literature. Iterative methods completely based on the computation of TDs were first introduced for optimal design of structures [22,25], and then adapted for inverse scattering problems for the Helmholtz equation in 2D in [13][14][15][16] and 3D in [12,46]. The TD can also be used to generate initial guesses in shape derivatives-based methods such as the level set method [20,21].…”

Theoretical Identification of Coupling Effect and Performance Analysis of Single-Source Direct Sampling Method

“…Hence, it is natural to consider the generation of a good initial guess before the iteration procedure. Due to this reason, various noniterative techniques have been investigated and applied to inverse scattering problems and microwave imaging, for example, MUltiple SIgnal Classifi-cation (MUSIC) algorithm [19,20], Kirchhoff and subspace migrations [21,22], Factorization method [23,24], topological sensitivity [25,26], and linear sampling method [27,28].…”

Lamb Waves Topological Imaging of Multiple Blind Defects in an Isotropic Plate

“…Topological gradient method has been used to achieve a high resolution imaging. 9,10 However, the method requires many iterations. Subsequently, the researchers began to explore the use of other fast imaging methods, for instance the time domain topological energy method proposed by Dominguez and Gibiat 11 and frequency domain topological imaging by Rodriguez et al 12 Owing to the refocusing properties of the time-reversal [13][14][15] for the scatterers, an accurate image could be readily obtained by solving two wave propagation problems in an homogeneous medium.…”