High order asymptotics for wave propagation across thin periodic interfaces

In this work, we present a new solution representation for the Helmholtz transmission problem in a bounded domain in R 2 with a thin and periodic layer of finite length. The layer may consists of a periodic pertubation of the material coefficients or it is a wall modelled by boundary conditions with an periodic array of small perforations. We consider the periodicity in the layer as the small variable δ and the thickness of the layer to be at the same order. Moreover we assume the thin layer to terminate at re-entrant corners leading to a singular behaviour in the asymptotic expansion of the solution representation. This singular behaviour becomes visible in the asymptotic expansion in powers of δ where the powers depend on the opening angle. We construct the asymptotic expansion order by order. It consists of a macroscopic representation away from the layer, a boundary layer corrector in the vicinity of the layer, and a near field corrector in the vicinity of the end-points. The boundary layer correctors and the near field correctors are obtained by the solution of canonical problems based, respectively, on the method of periodic surface homogenization and on the method of matched asymptotic expansions. This will lead to transmission conditions for the macroscopic part of the solution on an infinitely thin interface and corner conditions to fix the unbounded singular behaviour at its end-points. Finally, theoretical justifications of the second order expansion are given and illustrated by numerical experiments. The solution representation introduced in this article can be used to compute a highly accurate approximation of the solution with a computational effort independent of the small periodicity δ. Keywords Helmholtz equation, thin periodic interface, method of matched asymptotic expansions, method of periodic surface homogenization.

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“…[32] and Section 5 of Ref. [16]). More specifically, Problem (2.3) has a finite dimensional kernel of dimension 2, spanned by the functions…”

Section: Existence and Uniqueness Results For The Boundary Layer Problemmentioning

confidence: 96%

On the homogenization of the Helmholtz problem with thin perforated walls of finite length

2018

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“…We emphasize that formal expansions (29) and (30) will be justified a posteriori by the error analysis (Section 7). Note also that this kind of two-scale expansion is well known (cf.…”

Section: Formal Asymptotic Expansionmentioning

confidence: 99%

“…It is unbounded with respect to X 3 . The expansions (29) and (30) are assumed to be valid in two overlapping areas…”

Section: Formal Asymptotic Expansionmentioning

confidence: 99%

“…The behavior of far field terms, given in the following proposition, directly results from a Taylor expansion in the vicinity of . Although technical, the proof simply follows from an identification process: we plug Taylor's expansions (43) into the homogeneous Maxwell's equations, and we then collect the terms associated with the different powers of x 3 (see [29] and [25] for a detailed proof of this kind of result).…”

Section: 21mentioning

confidence: 99%

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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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“…6(c) for the two dimensions of the holes. The calculations are performed for an interval of the wave number k ∈ [5,35] and for ε 0 = 0.0125 which corresponds to N = 24.…”

Section: Validation Test -Acoustic Field In Fluidmentioning

confidence: 99%

Homogenization of the vibro–acoustic transmission on perforated plates

Rohan

Lukeš

2019

Applied Mathematics and Computation

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The paper deals with modelling of acoustic waves which propagate in inviscid fluids interacting with perforated elastic plates. The plate can be replaced by an interface on which transmission conditions are derived by homogenization of a problem describing vibroacoustic fluid-structure interactions in a transmission layer in which the plate is embedded. The Reissner-Mindlin theory of plates is adopted for periodic perforations designed by arbitrary cylindrical holes with axes orthogonal to the plate midplane. The homogenized model of the vibroacoustic transmission is obtained using the two-scale asymptotic analysis with respect to the layer thickness which is proportional to the plate thickness and to the perforation period. The nonlocal, implicit transmission conditions involve a jump in the acoustic potential and its normal one-side derivatives across the interface which represents the plate with a given thickness. The homogenized model was implemented using the finite element method and validated using direct numerical simulations of the non-homogenized problem. Numerical illustrations of the vibroacoustic transmission are presented.

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High order asymptotics for wave propagation across thin periodic interfaces

Cited by 15 publications

References 27 publications

On the homogenization of the Helmholtz problem with thin perforated walls of finite length

On the homogenization of the Helmholtz problem with thin perforated walls of finite length

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

Homogenization of the vibro–acoustic transmission on perforated plates

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