Matching of asymptotic expansions for waves propagation in media with thin slots II: The error estimates

Joly, Patrick; Tordeux, Sébastien

doi:10.1051/m2an:2008004

Cited by 27 publications

(33 citation statements)

References 15 publications

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“…The idea of the proof follows the lines of [19] and consists first in constructing a global approximate solution using the first three terms (and not, curiously, the first two only) of the expansions (13) and (12), that coincides with the truncated far field expansion…”

Section: Main Steps Of the Proofmentioning

confidence: 99%

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Approximate models for wave propagation across thin periodic interfaces

Delourme

Haddar

Joly

2012

Journal de Mathématiques Pures et Appliquées

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101

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This work deals with the scattering of acoustic waves by a thin ring that contains regularly spaced inhomogeneities. We first explicit and study the asymptotic of the solution with respect to the period and thickness of the inhomogeneities using so-called matched asymptotic expansions. We then build simplified models replacing the thin ring with Approximate Transmission Conditions that are accurate up to third order with respect to the layer width. We pay particular attention to the study of these approximate models and the quantification of their accuracy. RésuméCet article traite de la diffraction d'une onde acoustique par un structure constituée d'un anneau mince de diélectrique contenant un grand nombre d'hétérogénéités disposées périodiquement. On commence par écrire et justifier un développement asymptotique de la solution en fonction de l'épaisseur de l'anneau et de la distance entre les hétérogénéités. Puis, on construit des modèles approchés stables et bien posés d'ordres 2 et 3 dans lesquels l'anneau périodique est remplacé par une condition de transmission.

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Section: Main Steps Of the Proofmentioning

confidence: 99%

“…The latter method has been developed in [17] to treat singular perturbation problems which arise in fluid mechanics. A standard work on the matched asymptotic expansions applied to the Helmholtz equation can be found in [18,19] and complex situations are studied in [20]. For recent applications, we refer the reader to [21,22].…”

Section: Introductionmentioning

confidence: 99%

Approximate models for wave propagation across thin periodic interfaces

Delourme

Haddar

Joly

2012

Journal de Mathématiques Pures et Appliquées

Self Cite

101

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“…We only sketch the proof (a detailed proof can be found in , similar results can be found in ). This estimation is obtained in three main steps: In a first step, for any

n MathClass-rel\in double-struckN

, we build a global approximation of the exact solution that coincides with the first n terms of the far‐field expansion far from the thin layer

{bold-italicE}_{e MathClass-punc, δ}^{n} MathClass-punc: MathClass-rel= {MathClass-op\sum}_{k MathClass-rel= 0}^{n} δ^{k} {bold-italicE}_{k} MathClass-punc,

and with the first n terms of the near‐field expansion in the vicinity of the periodic thin layer:

ϵ_{i MathClass-punc, δ}^{n} MathClass-punc: MathClass-rel= {MathClass-op\sum}_{k MathClass-rel= 0}^{n} δ^{k} ϵ_{k} MathClass-punc.

This approximation is built with the help of a truncation function χ that satisfies

χ (s) MathClass-rel= ({, \begin{matrix} falsenonefalsearray \\ arrayaxis 1 if | s | \leq 1, \\ arrayaxis 0 if | s | \geq 2, \end{matrix})

and a positive distance function η ( δ ) such that

\underset{normallim}{msub} η MathClass-rel= 0 and \underset{normallim}{msub} \frac{η}{δ} MathClass-rel= MathClass-bin+ MathClass-rel\infty MathClass-punc.

Then, considering

χ_{η} (x_{3}) MathClass-punc: MathClass-rel= χ ((), \frac{x_{3}}{η})

…”

Section: Error Estimates: Asymptotic Expansion Justification (Step 5)mentioning

confidence: 99%

High‐order asymptotics for the electromagnetic scattering by thin periodic layers

Delourme

2014

Math Methods in App Sciences

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This work deals with the scattering of electromagnetic waves by a thin periodic layer made of an array of regularly spaced obstacles. The size of the obstacles and the spacing between two consecutive ones are of the same order δ, which is much smaller than the wavelength of the incident wave. We provide a complete description of the asymptotic behavior of the solution with respect to the small parameter δ: we use a method that combines matched asymptotic expansions and homogenization techniques. We pay particular attention to the construction of the near‐field terms. Indeed, they satisfy electrostatic problems posed in an infinite 3D strip, which require a careful analysis. Error estimates are carried out to justify the accuracy of our expansion. Copyright © 2014 John Wiley & Sons, Ltd.

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“…Some more recent treatments concern the numerical analysis (see [17][18][19]). We are not aware of any discussion of the problem in our scaling, i.e.…”

Section: O(e)mentioning

confidence: 99%

The low-frequency spectrum of small Helmholtz resonators

Schweizer

2015

Proc. R. Soc. A.

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We analyse the spectrum of the Laplace operator in a complex geometry, representing a small Helmholtz resonator. The domain is obtained from a bounded set Ω ⊂ R n by removing a small obstacle Σ ε ⊂ Ω of size ε > 0. The set Σ ε essentially separates an interior domain Ω inn ε (the resonator volume) from an exterior domain Ω out ε , but the two domains are connected by a thin channel. For an appropriate choice of the geometry, we identify the spectrum of the Laplace operator: it coincides with the spectrum of the Laplace operator on Ω, but contains an additional eigenvalue μ −1 ε . We prove that this eigenvalue has the behaviour μ ε ≈ V ε L ε /A ε , where V ε is the volume of the resonator, L ε is the length of the channel and A ε is the area of the cross section of the channel. This justifies the well-known frequency formula ω HR = c 0 A/(LV) for Helmholtz resonators, where c 0 is the speed of sound.

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Matching of asymptotic expansions for waves propagation in media with thin slots II: The error estimates

Cited by 27 publications

References 15 publications

Approximate models for wave propagation across thin periodic interfaces

Approximate models for wave propagation across thin periodic interfaces

High‐order asymptotics for the electromagnetic scattering by thin periodic layers

The low-frequency spectrum of small Helmholtz resonators

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