Optimal and efficient shapes in acoustic boundary absorption

Frédéric Magoulès (1) , Thi Phuong Kieu Nguyen , Pascal Omnes (2, 3) , Anna Rozanova-Pierrat (4)

1 CentraleSupélec
2 LAGA - Laboratoire Analyse, Géométrie et Applications
3 CEA-DES (ex-DEN) - CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire)
4 MICS - Mathématiques et Informatique pour la Complexité et les Systèmes

In the aim to find the simplest and most efficient shape of a noise absorbing wall to dissipate the acoustical energy of a sound wave, we consider a frequency model described by the Helmholtz equation with a damping on the boundary. For the case of a regular boundary we provide the shape derivative of an objective function, chosen to describe the acoustical energy. Using the gradient method for the shape derivative, combined with the finite volume and level set methods, we find numerically the optimal shapes for a fixed frequency. We show the stability of the numerical algorithm and the non-uniqueness of the optimal shape, which can be explained by the non-uniqueness of the geometry providing the same spectral properties. We find numerically the most efficient shape in a range of frequencies, which contains different geometrical scales. Thus we show that if we simplify the obtained optimal shape, by deleting the smaller scales of the geometry, the new shape is efficient in the frequencies corresponding to its characteristic geometry scale length, but no more efficient in the higher frequencies.

Mots clés: Shape optimization; Helmholtz equation; wave absorption; level set method; finite differences

Domaines: Mathématiques [math]; Physique mathématique [math-ph]; Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]

Frédéric Magoulès, Thi Phuong Kieu Nguyen, Pascal Omnes, Anna Rozanova-Pierrat. Optimal and efficient shapes in acoustic boundary absorption. 2019. ⟨hal-02543993⟩