Imaging of small penetrable obstacles based on the topological derivative method

Abstract

PurposeThe purpose of this paper is to present topological derivatives-based reconstruction algorithms to solve an inverse scattering problem for penetrable obstacles.Design/methodology/approachThe method consists in rewriting the inverse reconstruction problem as a topology optimization problem and then to use the concept of topological derivatives to seek a higher-order asymptotic expansion for the topologically perturbed cost functional. Such expansion is truncated and then minimized with respect to the parameters under consideration, which leads to noniterative second-order reconstruction algorithms.FindingsIn this paper, the authors develop two different classes of noniterative second-order reconstruction algorithms that are able to accurately recover the unknown penetrable obstacles from partial measurements of a field generated by incident waves.Originality/valueThe current paper is a pioneer work in developing a reconstruction method entirely based on topological derivatives for solving an inverse scattering problem with penetrable obstacles. Both algorithms proposed here are able to return the number, location and size of multiple hidden and unknown obstacles in just one step. In summary, the main features of these algorithms lie in the fact that they are noniterative and thus, very robust with respect to noisy data as well as independent of initial guesses.

Topology optimization of three-dimensional structures subject to self-weight loading

“…Thus, the topological derivative method has applications in many different fields. For a complete review of the topological derivative method and the most recent developments in this area, see the special issue on the topological derivative method and its applications in computational engineering recently published in the Engineering Computations Journal (Novotny et al ., 2022), covering various topics ranging from new theoretical developments (Amstutz, 2022; Baumann and Sturm, 2022; Delfour, 2022) to applications in structural and fluid dynamics topology optimization (Kliewe et al ., 2022; Romero, 2022; Santos and Lopes, 2022), geometrical inverse problems (Bonnet, 2022; Canelas and Roche, 2022; Fernandez and Prakash, 2022; Le Louër and Rapún, 2022a, b), synthesis and optimal design of metamaterials (Ferrer and Giusti, 2022; Yera et al ., 2022), fracture mechanics modelling (Xavier and Van Goethem, 2022), up to industrial applications (Rakotondrainibe et al ., 2022) and experimental validation of the topological derivative method (Barros et al ., 2022).…”

On the robustness of the topological derivative for Helmholtz problems and applications

“…The number of articles has increased tremendously, so that seeking a complete list of references is inordinate. See the special issue on the topological derivative method and its applications in computational engineering, recently published in the Engineering Computations Journal (Novotny, Giusti and Amstutz, 2022), covering various topics ranging from new theoretical developments (Amstutz, 2022;Baumann and Sturm, 2022;and Delfour, 2022) to applications in structural and fluid dynamics topology optimization (Kliewe, Laurain and Schmidt, 2022;Romero, 2022;and Santos and Lopes, 2022), geometrical inverse problems (Bonnet, 2022;Canelas and Roche, 2022;Fernandez and Prakash, 2022;Le Louër and Rapún, 2022a,b), synthesis and optimal design of metamaterials (Ferrer and Giusti, 2022;Yera et al, 2022), fracture mechanics modelling (Xavier and Van Goethem, 2022), up to industrial applications (Rakotondrainibe, Allaire and Orval, 2022) and experimental validation of the topological derivative method (Barros et al, 2022).…”

Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part I: the free space case

“…The reader might also find interesting the numerical experiments presented in Le Louër and Rapún (2019) for the 2D and 3D impedance case (not considered in the present paper), or in Rapún (2020), where scatterers of different nature simultaneously immersed in double-struckR2 are found without knowing their nature by topological energy methods. It also worths mentioning the very recent paper by Fernandez and Prakash (2021) where a noniterative method based on the computation of second-order TDs is applied for the identification of penetrable objects immersed in a two-dimensional bounded media (see the original paper by de Faria et al (2007) or Chapters 10 and 11 in Novotny et al (2019) for details on higher-order TDs).…”