Location and shape reconstructions of sound-soft obstacles buried in penetrable cylinders

Abstract

In this paper, we present a method for the solution of a location and shape reconstruction problem related to sound-soft obstacles buried in arbitrarily shaped penetrable cylinders in two dimensions. The direct problem considered here is to obtain the scattered near/far-field in the case of a single time-harmonic acoustic plane wave insonification. The aim of the inverse problem is to find the shape as well as the location of the buried scatterer from the limited/full aperture measurements of the scattered field via Newton iterations. Both for the direct and inverse scattering problems, different potential (layer) approaches are used to derive a system of boundary integral equations. The integral equations are evaluated by using Nyström and collocation methods. Moreover, in order to obtain a stable solution of the first kind of Fredholm-type integral equations Tikhonov regularization is employed. We test the applicability and the effectiveness of the inversion algorithm also with noisy limited far-field data and obtain satisfactory numerical results as illustrated in the last section.

Inhomogeneity reconstructions in tendon ducts via boundary integral equations

“…We note that regularization parameters α in (3.7), β for the equation system (3.14)-(3.17), γ for Eq. (3.15) and m in (4.2) are chosen by trial and error as a typical procedure which is applied in [23][24][25][26][27][28]. However, it is important to emphasize that the selection of the mentioned parameters do not affect the quality of the reconstructions if Table 1.…”

“…Afterwards, scattered field data is calculated on a measurement circle through density functions. As an advantage of the method, an exponential convergence is obtained as it was exhibited in the several publications, see [32,27,30]. We refer to the paper [27] for a more detailed presentation of forward solver and for the ideas on the test cases for the accuracy.…”

“…Locations of the sound-soft buried objects are found by a simple testing algorithm which depends on the finding minimum value of the residue ξ ξ ¼ [27] as an alternative approach for the location reconstructions if a priori information exists on the approximate location and size of the unknown buried object.…”

“…Then the update step is performed via [24,27]. (ii) Shape reconstruction for IP 2: In the second step of IP2, we seek for updates not only for the initially guessed shape but also for the density functions.…”

“…Density functions and related integral representations allow us to calculate the scattered and the total fields in each region. Then we employ iterative reconstruction algorithms introduced in [24,23] which were tested for shape reconstructions of objects in free-space [24][25][26] and for the ones buried in bounded domains [27,28].…”