Shape optimization in acoustic–structure interaction
ISSN: 0264-4401
Article publication date: 21 October 2021
Issue publication date: 1 February 2022
Abstract
Purpose
Motivated 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/approach
The 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.
Findings
The 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/value
The 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.
Keywords
Acknowledgements
The second and third authors acknowledge financial support from the DFG Research Center Matheon “Mathematics for key technologies” in Berlin, through Project C37 “Shape/Topology optimization methods for inverse problems” and Project D26 “Asymptotic analysis of the wave-propagation in realistic photonic crystal wave-guides”, respectively. The second author also acknowledges financial support from the Brazilian National Council for Scientific and Technological Development (Conselho Nacional de Desenvolvimento Científico e Tecnológico–CNPq) through the process: 408175/2018–4 “Otimização de forma não suave e controle de problemas de fronteira livre” and through the program “Bolsa de Produtividade em Pesquisa–PQ 2018”, process: 304258/2018–0.
Citation
Kliewe, P., Laurain, A. and Schmidt, K. (2022), "Shape optimization in acoustic–structure interaction", Engineering Computations, Vol. 39 No. 1, pp. 172-200. https://doi.org/10.1108/EC-07-2021-0379
Publisher
:Emerald Publishing Limited
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