# Shape optimization for the Helmholtz equation with complex Robin boundary conditions

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1 MAS - Mathématiques Appliquées aux Systèmes - EA 4037 2 CEA-DEN - CEA-Direction de l-Energie Nucléaire

Abstract : In the aim to find the simplest and most efficient shape of a noise absorbing wall to dissipate the energy of a sound wave, we consider a frequency model the Helmholtz equation with a damping on the boundary. The damping on the boundary is firstly related with the damping in the volume, knowing the macroscopic parameters of a fixed porous medium.Once the well-posedness results are proved for the time-dependent and the frequency models in the class of bounded $n$-sets for instance, locally uniform domains with a $d$-set boundary, containingself-similar fractals or Lipschitz domains as examples, the shape optimization problem of minimizing the acoustical energy for a fixed frequency is considered. To obtain an efficient wall shape for a large range of frequencies, we define the notion of $\epsilon$-optimal shapes and prove their existence in a class of multiscale Lipschitz boundaries when we consider energy dissipation on a finite range of frequencies, and in a class of fractals for an infinite frequency range. The theory is illustrated by numerical results.

Keywords : wave propagation Absorbing wall shape optimization fractals.

Autor: Frédéric Magoulès - Thi Phuong Kieu Nguyen - Pascal Omnes - Rozanova-Pierrat Anna -

Fuente: https://hal.archives-ouvertes.fr/

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