Optimization of Steklov-Neumann eigenvalues

Ammari, Habib; Imeri, Kthim; Nigam, Nilima

doi:10.1016/j.jcp.2019.109211

Cited by 9 publications

(7 citation statements)

References 28 publications

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“…Thus different boundary conditions might model the physical problem more accurately. 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.…”

Section: Discussionmentioning

confidence: 99%

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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%

“…2 ), this follows from describing Γ k + as an infinite sum of Hankel functions of the first kind of order 0, see[4, Section 3.1], and the singularity term of that Hankel function, see [3, Section 2.3]. Moreover, we have that for k → 0, Γ k + → Γ 0 + , where Γ 0 + is the fundamental solution to the Laplace equation, compare [4, Section 3.1].…”

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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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“…As we conduct the numerical experiments purely for illustrative purposes in order to demonstrate the practical effectiveness of the asymptotics, the use of uniform meshes already gives very good results as shown above. For an alternative method of calculating Steklov or mixed Steklov-Dirichlet-Neumann eigenvalues, see, for example, [3,4].…”

Section: On-diagonal Rectanglesmentioning

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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“…Here we demonstrate another method, which relies on using asymptotic expansions of solutions to partial differential equation. Such expansions have already been studied in different papers [1,2,4,5]. With that tool we can position and scale objects, on which the waves scatter, such that the resulting structure satisfies the desired requirements.…”

Section: Introductionmentioning

confidence: 99%

Optimal design of optical analog solvers of linear systems

Imeri

2021

Partial Differ. Equ. Appl.

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In this paper, given a linear system of equations $$\mathbf {A}\, \mathbf {x}= \mathbf {b}$$ A x = b , we are finding locations in the plane to place objects such that sending waves from the source points and gathering them at the receiving points solves that linear system of equations. The ultimate goal is to have a fast physical method for solving linear systems. The issue discussed in this paper is to apply a fast and accurate algorithm to find the optimal locations of the scattering objects. We tackle this issue by using asymptotic expansions for the solution of the underlying partial differential equation. This also yields a potentially faster algorithm than the classical BEM for finding solutions to the Helmholtz equation.

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Optimization of Steklov-Neumann eigenvalues

Cited by 9 publications

References 28 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

Optimal design of optical analog solvers of linear systems

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