no. 7
Équations aux dérivées partielles, Analyse numérique
Electromagnetic shielding by thin periodic structures and the Faraday cage effect
[Blindage électromagnétique par des structures fines et périodiques]
Delourme, Bérangère1  ; Hewett, David P.2
1 Université Sorbonne Paris Nord, Laboratoire Analyse Géométrie et Applications (UMR 7539), 93430 Villetaneuse, France
2 Department of Mathematics, University College London, London, United Kingdom
Comptes Rendus. Mathématique, Tome 358 (2020) no. 7, pp. 777-784.

Dans cette note, nous nous intéressons à la diffraction des ondes électromagnétiques (équations de Maxwell en régime harmonique) par une nappe perforée plane constituée de petit obstacles parfaitement conducteurs placée à l’interface entre deux milieux homogènes. La taille des obstacles et la distance séparant deux obstacles consécutifs sont du même ordre de grandeur δ, δ supposé petit. En étudiant trois configurations modèles ((i) obstacles « discrets », (ii) fils parallèles, (iii) maillage constitué de deux nappes de fils parallèles), nous montrons que la limite de la solution quand δ tend vers 0 dépend de la forme des obstacles constituant la nappe périodique, le phénomène de «  cage de Faraday » n’apparaissant que dans le cas du maillage de fils.

In this note we consider the scattering of electromagnetic waves (governed by the time-harmonic Maxwell equations) by a thin periodic layer of perfectly conducting obstacles. The size of the obstacles and the distance between neighbouring obstacles are of the same small order of magnitude δ. By deriving homogenized interface conditions for three model configurations, namely (i) discrete obstacles, (ii) parallel wires, (iii) a wire mesh, we show that the limiting behaviour as δ→0 depends strongly on the topology of the periodic layer, with full shielding (the so-called “Faraday cage effect”) occurring only in the case of a wire mesh.

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DOI : 10.5802/crmath.59
Delourme, Bérangère 1 ; Hewett, David P. 2

1 Université Sorbonne Paris Nord, Laboratoire Analyse Géométrie et Applications (UMR 7539), 93430 Villetaneuse, France
2 Department of Mathematics, University College London, London, United Kingdom 
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Delourme, Bérangère; Hewett, David P. Electromagnetic shielding by thin periodic structures and the Faraday cage effect. Comptes Rendus. Mathématique, Tome 358 (2020) no. 7, pp. 777-784. doi : 10.5802/crmath.59. http://www.numdam.org/articles/10.5802/crmath.59/

[1] Amrouche, Chérif; Bernardi, Christine; Dauge, Monique; Girault, Vivette Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., Volume 21 (1998) no. 9, pp. 823-864 | DOI | MR | Zbl

[2] Chapman, S. Jonathan; Hewett, David P.; Trefethen, Lloyd N. Mathematics of the Faraday cage, SIAM Rev., Volume 57 (2015) no. 3, pp. 398-417 | DOI | MR | Zbl

[3] Delourme, Bérangère High-order asymptotics for the electromagnetic scattering by thin periodic layers, Math. Methods Appl. Sci., Volume 38 (2015) no. 5, pp. 811-833 | DOI | MR | Zbl

[4] Delourme, Bérangère; Haddar, Houssem; Joly, Patrick On the well-posedness, stability and accuracy of an asymptotic model for thin periodic interfaces in electromagnetic scattering problems, Math. Methods Appl. Sci., Volume 23 (2013) no. 13, pp. 2433-2464 | DOI | MR | Zbl

[5] Faraday, Michael Experimental researches in electricity. Vol 1., Richard and John Edward Taylor, 1849 (reprinted from Philosophical Transactions of 1831–1838, http://www.gutenberg.org/ebooks/14986)

[6] Hewett, David P.; Hewitt, Ian J. Homogenized boundary conditions and resonance effects in Faraday cages, Proc. R. Soc. Lond., Ser. A, Volume 472 (2016) no. 2189, 20160062, 28 pages correction in ibid. 473 (2017), no. 2202, article ID 20170331 (2 pages) | Zbl

[7] Holloway, Christopher L.; Kuester, Edward F. Generalized sheet transition conditions for a metascreen. A fishnet metasurface, IEEE Trans. Antennas Propag., Volume 66 (2018) no. 5, pp. 2414-2427 | DOI

[8] Holloway, Christopher L.; Kuester, Edward F.; Dienstfrey, Andrew A homogenization technique for obtaining generalized sheet transition conditions for an arbitrarily shaped coated wire grating, Radio Sci., Volume 49 (2014) no. 10, pp. 813-850 | DOI

[9] Marigo, Jean-Jacques; Maurel, Agnès Two-scale homogenization to determine effective parameters of thin metallic-structured films, Proc. R. Soc. Lond., Ser. A, Volume 472 (2016) no. 2192, 20160068, 21 pages | MR | Zbl

[10] Maz’ya, Vladimir; Nazarov, Sergeĭ; Plamenevskiĭ, Boris Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, Birkhäuser, 2012

[11] Monk, Peter Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, 2003 | MR | Zbl

[12] Nédélec, Jean-Claude Acoustic and electromagnetic equations: integral representations for harmonic problems, Applied Mathematical Sciences, 144, Springer, 2001 | Zbl

[13] Schweizer, Ben; Urban, Maik Effective Maxwell’s equations in general periodic microstructures, Appl. Anal., Volume 97 (2018) no. 13, pp. 2210-2230 | DOI | MR | Zbl

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