Wave Enhancement Through Optimization of Boundary Conditions

Ammari, Habib; Bruno, Oscar P.; Imeri, Kthim; Nigam, Nilima

doi:10.1137/19m1274651

Cited by 8 publications

(12 citation statements)

References 16 publications

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“…For example in [2], Steklov boundary conditions are studied. Moreover, mixed boundaries as they are studied in [1] move the singularity in k of the scattered wave away from k = 0. Thus with mixed boundaries we might achieve a better control of the resonance effect.…”

Section: Discussionmentioning

confidence: 99%

A mathematical and numerical framework for gradient meta-surfaces built upon periodically repeating arrays of Helmholtz resonators

Ammari¹,

Imeri²

2020

Preprint

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In this paper a mathematical model is given for the scattering of an incident wave from a surface covered with microscopic small Helmholtz resonators, which are cavities with small openings. More precisely, the surface is built upon a finite number of Helmholtz resonators in a unit cell and that unit cell is repeated periodically. To solve the scattering problem, the mathematical framework elaborated in [Ammari et al., Asympt. Anal., 114 (2019), 129-179] is used. The main result is an approximate formula for the scattered wave in terms of the lengths of the openings. Our framework provides analytic expressions for the scattering wave vector and angle and the phaseshift. It justifies the apparent absorption. Moreover, it shows that at specific lengths for the openings and a specific frequency there is an abrupt shift of the phase of the scattered wave due to the subwavelength resonances of the Helmholtz resonators. A numerically fast implementation is given to identify a region of those specific values of the openings and the frequencies.

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Section: Discussionmentioning

confidence: 99%

A mathematical and numerical framework for gradient meta-surfaces built upon periodically repeating arrays of Helmholtz resonators

Ammari¹,

Imeri²

2020

Preprint

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“…The main purpose of this section is to prove the following theorem, which establishes that the quasi-eigenvalues introduced in Definitions 2. 3 We will prove Theorem 4.1 by constructing an appropriate sequence of quasimodes which we first define in a very general setting.…”

Section: General Approachmentioning

confidence: 99%

Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons

Levitan

Parnovski

Polterovich

et al. 2022

Proceedings of London Math Soc

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We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions of curvilinear polygons in terms of their side lengths and angles. These formulae are quite precise: the errors tend to zero as the spectral parameter tends to infinity. The Steklov problem on planar domains with corners is closely linked to the classical sloshing and sloping beach problems in hydrodynamics; as we show it is also related to quantum graphs. Somewhat surprisingly, the arithmetic properties of the angles of a curvilinear polygon have a significant effect on the boundary behaviour of the Steklov eigenfunctions. Our proofs are based on an explicit construction of quasimodes. We use a variety of methods, including ideas from spectral geometry, layer potential analysis, and some new techniques tailored to our problem.M S C 2 0 2 0 35P20 (primary), 34B45 (secondary) Contents

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“…From [26,Theorem 4.10], and the fact that the first mixed Dirichlet -Neumann eigenvalue is not zero [6], as long as Γ S is not empty, we have that Z Γ S (x S , ·) ∈ H 1 (Ω) exists. From Lemma 3.1, we see that U λ Γ S (x S , ·) ∈ H 1 (Ω) exists.…”

Section: Asymptotics For the Steklov-neumann Functionmentioning

confidence: 99%

Optimization of Steklov-Neumann eigenvalues

Ammari

Imeri

Nigam

2020

Journal of Computational Physics

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This paper examines the Laplace equation with mixed boundary conditions, the Neumann and Steklov boundary conditions. This models a container with holes in it, like a pond filled with water but partly covered by immovable pieces on the surface. The main objective is to determine the right extent of the covering pieces, so that any shock inside the container yields a resonance. To this end, an algorithm is developed which uses asymptotic formulas concerning perturbations of the partitioning of the boundary pieces. Proofs for these formulas are established. Furthermore, this paper displays some results concerning bounds and examples with regards to the governing problem. Mathematics Subject Classification (MSC2000). 35R30, 35C20.Keywords. Steklov eigenvalue problem, boundary integral operators, mixed boundary conditions, perturbations formulas.

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Wave Enhancement Through Optimization of Boundary Conditions

Cited by 8 publications

References 16 publications

A mathematical and numerical framework for gradient meta-surfaces built upon periodically repeating arrays of Helmholtz resonators

A mathematical and numerical framework for gradient meta-surfaces built upon periodically repeating arrays of Helmholtz resonators

Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons

Optimization of Steklov-Neumann eigenvalues

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