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

Semin, Adrien and Delourme, Bérangère and Schmidt, Kersten :
On the homogenization of the Helmholtz problem with thin perforated walls of finite length.
In: ESAIM Math. Model. Numer. Anal., 52 (1) pp. 29-67.
[Article] , (2018)

Abstract

In this work, we present a new solution representation for the Helmholtz transmission problem in a bounded domain in \mathbbR² 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.

Item Type: Article
Erschienen: 2018
Creators: Semin, Adrien and Delourme, Bérangère and Schmidt, Kersten
Title: On the homogenization of the Helmholtz problem with thin perforated walls of finite length
Language: German
Abstract:

In this work, we present a new solution representation for the Helmholtz transmission problem in a bounded domain in \mathbbR² 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.

Journal or Publication Title: ESAIM Math. Model. Numer. Anal.
Volume: 52
Number: 1
Uncontrolled Keywords: Helmholtz equation, thin periodic interface, method of matched asymptotic expansions, method of periodic surface homogenization
Divisions: 04 Department of Mathematics
04 Department of Mathematics > Numerical Analysis and Scientific Computing
Date Deposited: 19 Nov 2018 20:56
DOI: 10.1051/m2an/2017030
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