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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.

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.

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.

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.