Frequency range non-Lipschitz parametric optimization of a noise absorption
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arXiv
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| Format: | Preprint |
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2024
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| author | Magoules, Frederic Menoux, Mathieu Rozanova-Pierrat, Anna |
| author_facet | Magoules, Frederic Menoux, Mathieu Rozanova-Pierrat, Anna |
| contents | 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 ($ε$, $\infty$)-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06292 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Frequency range non-Lipschitz parametric optimization of a noise absorption Magoules, Frederic Menoux, Mathieu Rozanova-Pierrat, Anna Analysis of PDEs Optimization and Control 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 ($ε$, $\infty$)-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. |
| title | Frequency range non-Lipschitz parametric optimization of a noise absorption |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2409.06292 |