In this article, we derive a complete mathematical analysis of a coupled 1D-2D model for 2D wave propagation in media including thin slots. Our error estimates are illustrated by numerical results.
Mots-clés : slit, slot, wave equation, Helmholtz equation, approximate model
@article{M2AN_2006__40_1_63_0,
author = {Joly, Patrick and Tordeux, S\'ebastien},
title = {Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {63--97},
publisher = {EDP-Sciences},
volume = {40},
number = {1},
year = {2006},
doi = {10.1051/m2an:2006008},
mrnumber = {2223505},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an:2006008/}
}
TY - JOUR AU - Joly, Patrick AU - Tordeux, Sébastien TI - Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2006 SP - 63 EP - 97 VL - 40 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2006008/ DO - 10.1051/m2an:2006008 LA - en ID - M2AN_2006__40_1_63_0 ER -
%0 Journal Article %A Joly, Patrick %A Tordeux, Sébastien %T Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 63-97 %V 40 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2006008/ %R 10.1051/m2an:2006008 %G en %F M2AN_2006__40_1_63_0
Joly, Patrick; Tordeux, Sébastien. Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 1, pp. 63-97. doi : 10.1051/m2an:2006008. http://archive.numdam.org/articles/10.1051/m2an:2006008/
[1] and, Scattering of an electromagnetic wave by a screen, in Boundary value problems and integral equations in nonsmooth domains (Luminy, 1993), Dekker, New York. Lect. Notes Pure Appl. Math. (1995) 167 1-17. | Zbl
[2] and, Diffraction d'une onde électromagnétique par une couche composite mince accolée à un corps conducteur épais. I. Cas des inclusions fortement conductrices. C. R. Acad. Sci. Paris Sér. I Math. 313 (1991) 231-236. | Zbl
[3] ,, and, Numerical solution to the time-dependent Maxwell equations in two-dimensional singular domains: the singular complement method. J. Comput. Phys. 161 (2000) 218-249. | Zbl
[4] , Scattering frequencies of reasonators. Comm. Pure Appl. Math. 26 (1973) 549-563. | Zbl
[5] and, The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation. SIAM J. Appl. Math. 56 (1996) 1664-1693. | Zbl
[6] , and, Asymptotic analysis for periodic structures. North-Holland Publishing Co., Amsterdam. Stud. Math. Appl. 5 (1978). | MR | Zbl
[7] and, The electric field integral equation on Lipschitz screens: definitions and numerical approximation. Numer. Math. 94 (2003) 229-267. | Zbl
[8] and, General analysis of narrow strips and slots. IEEE Trans Ant. and Propag. 28 (1980).
[9] , Plates and junctions in elastic multi-structures, 14 Recherches en Mathématiques Appliquées [Res. Appl. Math.], Masson, Paris (1990). An asymptotic analysis. | MR | Zbl
[10] and, Fils et méthodes d'éléments finis pour les équations de Maxwell. Le modèle de Holland revisité. Tech. Report 3472, INRIA, http://www.inria.fr (Août 1998).
[11] and, Integral equation methods in scattering theory, John Wiley & Sons Inc., New York, Pure Appl. Math. (1983). A Wiley-Interscience Publication. | MR | Zbl
[12] and, Asymptotic analysis of a multidimensional vibrating structure. SIAM J. Math. Anal. 25 (1994) 836-858. | Zbl
[13] ,,, and, Modern Methods in Analytical acoustics. Lect. Notes, Springer-Verlag, London (1992). An asymptotic analysis.
[14] and, Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 3. Masson, Paris (1985). | MR | Zbl
[15] and, Effective boundary conditions for acoustic and electromagnetic scattering in thin layers. Tech. Report 278, École Polytechnique CMAP (France) (1993).
[16] and, Implementation of the thin-slot formalism in the finite-difference code threedii. IEEE Trans. Nuc. Sci. 28 (1981).
[17] , and, Discrete Dirichlet-to-Neumann maps for unbounded domains. Comput. Methods Appl. Mech. Engrg. 164 (1998) 173-185. Exterior problems of wave propagation (Boulder, CO, 1997; San Francisco, CA, 1997). | Zbl
[18] , Elliptic problems in nonsmooth domains 24, Pitman (Advanced Publishing Program), Boston, MA, Monographs Stud. Math. (1985). | MR | Zbl
[19] , Singularities in boundary value problems 22, Recherches en Mathématiques Appliquées [Res. Appl. Math.], Masson, Paris (1992). | MR | Zbl
[20] , and, Transmission of water waves through small apertures. J. Fluid Mech. 55 (1972) 149-161. | Zbl
[21] , and, Dirichlet-to-Neumann maps for unbounded wave guides. J. Comput. Phys. 143 (1998) 200-223. | Zbl
[22] and, Electromagnetic transmission through narrow slots in thick conducting screens. IEEE Trans. Antenna Propagation 28 (1980) 616-622.
[23] and, Finite-difference analysis EMP coupling to thin struts and wires. IEEE Trans. Electromagn. Compat. 23 (1981).
[24] and, Modèles asymptotiques pour la propagation des ondes dans les milieux comportant des fentes, Tech. Report RR-5568, INRIA, http://www.inria.fr (May 2005).
[25] and, Exact nonreflecting boundary conditions. J. Comput. Phys. 82 (1989) 172-192. | Zbl
[26] , The flanged waveguide antenna: discrete reciprocity and conservation. Wave Motion 29 (1999) 81-95. | Zbl
[27] , Problèmes variationnels dans les multi-domaines 19, Recherches en Mathématiques Appliquées [Res. Appl. Math.], Masson, Paris (1991). Modélisation des jonctions et applications. [Modeling of junctions and applications]. | MR | Zbl
[28] , Special functions and their applications, Revised English edition. Translated and edited by Richard A. Silverman, Prentice-Hall Inc., Englewood Cliffs, NJ (1965). | MR | Zbl
[29] and, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York (1972). Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. | MR | Zbl
[30] , Documentation de la librairie éléments finis melina. http://perso.univ-rennes1.fr/daniel.martin/melina, 1996-2003.
[31] and, Two-dimensional wave-scattering problems involving parallel-walled ducts. Quart. J. Mech. Appl. Math. 46 (1993) 89-116. | Zbl
[32] , Acoustic and electromagnetic equations. Springer-Verlag, New York. Appl. Math. Sci. 144 (2001). Integral representations for harmonic problems. | MR | Zbl
[33] , Problèmes mathématiques et numériques liés à l'approximation de la géométrie d'un corps diffractant dans les équations de l'électromagnétisme, Ph.D. thesis, Université de Paris 6 (1989).
[34] , Nonhomogeneous media and vibration theory. Lect. Notes Phys. 127 (1980). | MR | Zbl
[35] and, Approximate Boundary Conditions in Electromagnetics. IEE Pres, New York and London (1995). | Zbl
[36] , Computational electrodynamics. Artech House Inc., Boston, MA (1995). The finite-difference time-domain method. | MR | Zbl
[37] , and, Calculation and experimental validation of induced currents on coupled wires in an arbitrary shaped cavity. IEEE Trans Antenna Propag. 35 (1987) 1248-1257.
[38] ,,, and, Detailed fdtd analysis of electromagnetic fields penetrating narrow slots ans lapped joints in thick conducting screens. IEEE Trans Antenna Propag. 36 (1988) 247-257.
[39] , Propagation d'une onde électromagnétique dans une fente mince. Propagation et réflexion d'ondes en élasticité. Application au contrôle. Ph.D. thesis, École normale supérieure de Cachan (Dec. 1996).
[40] , Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes. Ph.D. thesis, Université de Versailles (2005).
[41] , Matching problems involving flow through small holes. Academic Press, New York Adv. Appl. Mechanics 15 (1975) 89-158.
[42] , Bessel functions and Kapteyn series. Proc. London Math. Soc. (April 1916) 150-174. | JFM
Cité par Sources :
