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In this work, we present a new solution representation for the Helmholtz transmission problem in a bounded domain in ℝ2 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.
Semin, Adrien 1 ; Delourme, Bérangère 1 ; Schmidt, Kersten 1
@article{M2AN_2018__52_1_29_0,
author = {Semin, Adrien and Delourme, B\'erang\`ere and Schmidt, Kersten},
title = {On the homogenization of the {Helmholtz} problem with thin perforated walls of finite length},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {29--67},
publisher = {EDP-Sciences},
volume = {52},
number = {1},
year = {2018},
doi = {10.1051/m2an/2017030},
zbl = {1397.32004},
mrnumber = {3808152},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017030/}
}
TY - JOUR AU - Semin, Adrien AU - Delourme, Bérangère AU - Schmidt, Kersten TI - On the homogenization of the Helmholtz problem with thin perforated walls of finite length JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 29 EP - 67 VL - 52 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017030/ DO - 10.1051/m2an/2017030 LA - en ID - M2AN_2018__52_1_29_0 ER -
%0 Journal Article %A Semin, Adrien %A Delourme, Bérangère %A Schmidt, Kersten %T On the homogenization of the Helmholtz problem with thin perforated walls of finite length %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 29-67 %V 52 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017030/ %R 10.1051/m2an/2017030 %G en %F M2AN_2018__52_1_29_0
Semin, Adrien; Delourme, Bérangère; Schmidt, Kersten. On the homogenization of the Helmholtz problem with thin perforated walls of finite length. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 29-67. doi: 10.1051/m2an/2017030
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