Shape optimization for the Helmholtz equation with complex Robin boundary conditions
Frédéric Magoulès, Thi Phuong Kieu Nguyen, Pascal Omnès, Anna Rozanova-Pierrat · HAL (Le Centre pour la Communication Scientifique Directe) · 2017
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, containing self-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.