Optimal absorption of acoustical waves by a boundary

Frédéric Magoulès (1) , Thi Phuong Kieu Nguyen (1) , Pascal Omnes (2) , Anna Rozanova-Pierrat (1)

1 MICS - Mathématiques et Informatique pour la Complexité et les Systèmes
2 CEA-DES (ex-DEN) - CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire)

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.

Mots clés: Absorbing wall; fractals.; shape optimization; wave propagation

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]; Analyse numérique [math.NA]; Physique mathématique [math-ph]; Optimisation et contrôle [math.OC]

Frédéric Magoulès, Thi Phuong Kieu Nguyen, Pascal Omnes, Anna Rozanova-Pierrat. Optimal absorption of acoustical waves by a boundary . 2017. ⟨hal-01558043v3⟩