Frequency range non-Lipschitz parametric optimization of a noise absorption

TL;DR

This paper develops a theoretical and numerical framework for optimizing noise absorption through shape optimization of absorbing materials with potentially fractal, non-Lipschitz boundaries, demonstrating improved performance over full coverage.

Contribution

It introduces a shape optimization approach for non-Lipschitz boundaries in wave energy absorption, including existence proofs and numerical methods for optimal distribution.

Findings

Optimal distribution reduces energy more effectively than full coverage.

Gradient descent and genetic algorithms achieve similar performance.

Non-Lipschitz, fractal boundaries are feasible for optimization.

Abstract

In the framework of the optimal wave energy absorption, we solve theoretically and numerically a parametric shape optimization problem to find the optimal distribution of absorbing material in the reflexive one defined by a characteristic function in the Robin-type boundary condition associated with the Helmholtz equation. Robin boundary condition can be given on a part or the all boundary of a bounded ($\epsilon$, $\infty$)-domain of R n . The geometry of the partially absorbing boundary is fixed, but allowed to be non-Lipschitz, for example, fractal. It is defined as the support of a d-upper regular measure with d $\in$]n -2, n[. Using the well-posedness properties of the model, for any fixed volume fraction of the absorbing material, we establish the existence of at least one optimal distribution minimizing the acoustical energy on a fixed frequency range of the relaxation problem. Thanks to the shape derivative of the energy functional, also existing for non-Lipschitz boundaries, we implement (in the two-dimensional case) the gradient descent method and find the optimal distribution with 50% of the absorbent material on a frequency range with better performances than the 100% absorbent boundary. The same type of performance is also obtained by the genetic method.