Inverse homogenization using the topological derivative

Abstract

PurposeThe purpose of this study is to solve the inverse homogenization problem, or so-called material design problem, using the topological derivative concept.Design/methodology/approachThe optimal topology is obtained through a relaxed formulation of the problem by replacing the characteristic function with a continuous design variable, so-called density variable. The constitutive tensor is then parametrized with the density variable through an analytical interpolation scheme that is based on the topological derivative concept. The intermediate values that may appear in the optimal topologies are removed by penalizing the perimeter functional.FindingsThe optimization process benefits from the intermediate values that provide the proposed method reaching to solutions that the topological derivative had not been able to find before. In addition, the presented theory opens the path to propose a new framework of research where the topological derivative uses classical optimization algorithms.Originality/valueThe proposed methodology allows us to use the topological derivative concept for solving the inverse homogenization problem and to fulfil the optimality conditions of the problem with the use of classical optimization algorithms. The authors solved several material design examples through a projected gradient algorithm to show the advantages of the proposed method.

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).…”

A New Non-Iterative Reconstruction Method for Solving a Class of Inverse Problems

“…The topological derivative method has applications in shape and topology optimization [Novotny et al, 2007, Amstutz and, inverse problems [Canelas et al, 2015, Ferreira and, image processing [Auroux et al, 2007, Amstutz et al, 2014, multi-scale material design and mechanical modelling, including damage [Allaire et al, 2011] and fracture [Xavier et al, 2017] evolution phenomena. See, for instance, the book by Novotny et al [2019a] and the special issue on the topological derivative method and its applications in computational engineering recently published in the Engineering Computations Journal , 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, Louër and Rapún, 2022a synthesis and optimal design of metamaterials [Ferrer andGiusti, 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].…”