Frequency range non-Lipschitz parametric optimization of a noise absorption

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.