Optimal Absorption of Acoustic Waves by a Boundary

Abstract

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. The well-posedness of the model is shown in a class of domains with d-set boundaries (N − 1 ≤ d < N ). We introduce a class of admissible Lipschitz boundaries, in which an optimal shape of the wall exists in the following sense: We prove the existence of a Radon measure on this shape, greater than or equal to the usual Lebesgue measure, for which the corresponding solution of the Helmholtz problem realizes the infimum of the acoustic energy defined with the Lebesgue measure on the boundary. If this Radon measure coincides with the Lebesgue measure, the corresponding solution realizes the minimum of the energy. For a fixed porous material, considered as an acoustic absorbent, we derive the damping parameters of its boundary from the corresponding time-dependent problem described by the damped wave equation (damping in volume).

Generalization of Rellich-Kondrachov theorem and trace compacteness in the framework of irregular and fractal boundaries

“…Let us generalize the Green formula formulated for d-sets in [4] (initially proposed by Lancia [26,Thm. 4.15] for a Von Koch curve) and the integration by parts from Appendix A Theorem A.3 [29] (see also the proof of formula (4.11) of Theorem 4.5 in [9]).…”

Boundary value problems on non-Lipschitz uniform domains: Stability, compactness and the existence of optimal shapes

“…fields like acoustical engineering, biomimetic architecture and surface design. In [MNORP21] mixed boundary value problems for the Helmholtz equation had been considered. It had been observed that since integrals over the boundary are involved, classes of Lipschitz admissible shapes may not contain the shape realizing the infimum energy over the class, see also [HMRP+21,Section 5].…”

“…Note that also [BG16,Theorem 3.2] used the idea to enlarge the classes of admissible shapes beyond Lipschitz, although in a methodologically different context. The equations discussed here and in [HRPT21,HMRP+21,MNORP21] are linear. In [DRPT21] the authors proved well-posedness results for damped linear wave equations and for the nonlinear Westervelt equation, they also established approximation theorems in specific situations.…”

On the existence of optimal shapes in architecture

“…When varying the shape, the natural notion of convergence for these measures is weak convergence, and this is delicate in the sense that the weak limit of a sequence of codimension one Hausdorff measures (which are the 'natural' surface measures on the boundary of a Lipschitz domain) is not necessarily a Hausdorff measure itself. Using the method in [22] one can show the existence of an optimal shape which realizes the infimum of the energy over a class of bounded Lipschitz domains, Theorem 4. Employing results from [19] one can verify the existence of an optimal shape in a larger class of bounded uniform domains which then realizes the minimum of the energy over this class, Theorem 5.…”