Shape optimization in acoustic–structure interaction

Abstract

PurposeMotivated by the acoustics of motor vehicles, a coupled fluid–solid system is considered. The air pressure is modeled by the Helmholtz equation, and the structure displacement is described by elastodynamic equations. The acoustic–structure interaction is modeled by coupling conditions on the common interface. First, the existence and uniqueness of solutions are investigated, and then, after recalling fundamental notions of shape optimization, the tensor form of the distributed shape derivative is obtained for the coupled problem. It is then applied to the minimization of the sound pressure by variation of the structure shape through the positioning of beads.Design/methodology/approachThe existence and uniqueness of solutions up to eigenfrequencies are shown by the Fredholm–Riesz–Schauder theory using a novel decomposition into an isomorphism and a compact operator. For the design optimization, the distributed shape derivative is obtained using the averaged adjoint method. It is then used in a closed 3D optimization process of the position of a bead for noise reduction. In this process, the C++ library concepts are used to solve the differential equations on hexahedral meshes with the finite element method of higher order.FindingsThe existence and uniqueness of solutions have been shown for the case without absorption, where the given proof allows for extension to the case with absorption in the domain or via boundary conditions. The theoretical results show that the averaged adjoint can be applied to compute distributed shape derivatives in the context of acoustic–structure interaction. The numerical results show that the distributed shape derivative can be used to reduce the sound pressure at a chosen frequency via rigid motions of a nonsmooth shape.Originality/valueThe proof of shape differentiability and the calculation of the distributed shape derivative in tensor form allows to consider nonsmooth shapes for the optimization, which is particularly relevant for the optimal placement of beads or stampings in a structural-acoustic system.

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].…”