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Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots

Published online by Cambridge University Press:  23 February 2006

Patrick Joly
Affiliation:
UMR POEMS, CNRS-INRIA, France. patrick.joly@inria.fr
Sébastien Tordeux
Affiliation:
Seminar für Angewandte Mathematik, ETHZ, Switzerland. tordeux@math.ethz.ch
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Abstract

In this article, we derive a complete mathematical analysis of acoupled 1D-2D model for 2D wave propagation in media including thin slots. Our error estimates are illustrated by numerical results.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2006

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References

T. Abboud and F. Starling, 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.
Artola, M. and Cessenat, M., 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) 231236.
Assous, F., Ciarlet, P., Jr., and J. Segré, Numerical solution to the time-dependent Maxwell equations in two-dimensional singular domains: the singular complement method. J. Comput. Phys. 161 (2000) 218249. CrossRef
Beale, J., Scattering frequencies of reasonators. Comm. Pure Appl. Math. 26 (1973) 549563. CrossRef
Bendali, A. and Lemrabet, K., The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation. SIAM J. Appl. Math. 56 (1996) 16641693. CrossRef
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland Publishing Co., Amsterdam. Stud. Math. Appl. 5 (1978).
Buffa, A. and Christiansen, S.H., The electric field integral equation on Lipschitz screens: definitions and numerical approximation. Numer. Math. 94 (2003) 229267. CrossRef
C. Butler and D. Wilton, General analysis of narrow strips and slots. IEEE Trans Ant. and Propag. 28 (1980).
P.G. Ciarlet, Plates and junctions in elastic multi-structures, 14 Recherches en Mathématiques Appliquées [Res. Appl. Math.], Masson, Paris (1990). An asymptotic analysis.
F. Collino and F. Millot, 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).
D. Colton and R. Kress, Integral equation methods in scattering theory, John Wiley & Sons Inc., New York, Pure Appl. Math. (1983). A Wiley-Interscience Publication.
Conca, C. and Zuazua, E., Asymptotic analysis of a multidimensional vibrating structure. SIAM J. Math. Anal. 25 (1994) 836858. CrossRef
D. Crighton, A. Dowling, J.F. Williams, M. Heckl and F. Leppington, Modern Methods in Analytical acoustics. Lect. Notes, Springer-Verlag, London (1992). An asymptotic analysis.
R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 3. Masson, Paris (1985).
B. Engquist and J.C. Nédélec, Effective boundary conditions for acoustic and electromagnetic scattering in thin layers. Tech. Report 278, École Polytechnique CMAP (France) (1993).
J. Gilbert and R. Holland, Implementation of the thin-slot formalism in the finite-difference code threedii. IEEE Trans. Nuc. Sci. 28 (1981).
Givoli, D., Patlashenko, I. and Keller, J., Discrete Dirichlet-to-Neumann maps for unbounded domains. Comput. Methods Appl. Mech. Engrg. 164 (1998) 173185. Exterior problems of wave propagation (Boulder, CO, 1997; San Francisco, CA, 1997). CrossRef
P. Grisvard, Elliptic problems in nonsmooth domains 24, Pitman (Advanced Publishing Program), Boston, MA, Monographs Stud. Math. (1985).
P. Grisvard, Singularities in boundary value problems 22, Recherches en Mathématiques Appliquées [Res. Appl. Math.], Masson, Paris (1992).
Guiney, D., Noye, B. and Tuck, E., Transmission of water waves through small apertures. J. Fluid Mech. 55 (1972) 149161. CrossRef
Harari, I., Patlashenko, I. and Givoli, D., Dirichlet-to-Neumann maps for unbounded wave guides. J. Comput. Phys. 143 (1998) 200223. CrossRef
Harrington, P. and Auckland, D., Electromagnetic transmission through narrow slots in thick conducting screens. IEEE Trans. Antenna Propagation 28 (1980) 616622. CrossRef
R. Holland and L. Simpson, Finite-difference analysis EMP coupling to thin struts and wires. IEEE Trans. Electromagn. Compat. 23 (1981).
P. Joly and S. Tordeux, Modèles asymptotiques pour la propagation des ondes dans les milieux comportant des fentes, Tech. Report RR-5568, INRIA, http://www.inria.fr (May 2005).
Keller, J. and Givoli, D., Exact nonreflecting boundary conditions. J. Comput. Phys. 82 (1989) 172192. CrossRef
Kriegsmann, G., The flanged waveguide antenna: discrete reciprocity and conservation. Wave Motion 29 (1999) 8195. CrossRef
H. Le Dret, 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].
N. Lebedev, Special functions and their applications, Revised English edition. Translated and edited by Richard A. Silverman, Prentice-Hall Inc., Englewood Cliffs, NJ (1965).
J.-L. Lions and E. Magenes, 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.
D. Martin, Documentation de la librairie éléments finis melina. http://perso.univ-rennes1.fr/daniel.martin/melina, 1996–2003.
McIver, P. and Rawlins, A.D., Two-dimensional wave-scattering problems involving parallel-walled ducts. Quart. J. Mech. Appl. Math. 46 (1993) 89116. CrossRef
J.C. Nédélec, Acoustic and electromagnetic equations. Springer-Verlag, New York. Appl. Math. Sci. 144 (2001). Integral representations for harmonic problems.
F. Rogier, 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).
E. Sánchez-Palencia, Nonhomogeneous media and vibration theory. Lect. Notes Phys. 127 (1980).
T. Senior and J. Volakis, Approximate Boundary Conditions in Electromagnetics. IEE Pres, New York and London (1995).
A. Taflove, Computational electrodynamics. Artech House Inc., Boston, MA (1995). The finite-difference time-domain method.
Taflove, A., Umashankar, K. and Becker, B., Calculation and experimental validation of induced currents on coupled wires in an arbitrary shaped cavity. IEEE Trans Antenna Propag. 35 (1987) 12481257.
Taflove, A., Umashankar, K., Becker, B., Harfoush, F. and Yee, K., Detailed fdtd analysis of electromagnetic fields penetrating narrow slots ans lapped joints in thick conducting screens. IEEE Trans Antenna Propag. 36 (1988) 247257. CrossRef
F. Tatout, 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).
S. Tordeux, Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes. Ph.D. thesis, Université de Versailles (2005).
E. Tuck, Matching problems involving flow through small holes. Academic Press, New York Adv. Appl. Mechanics 15 (1975) 89–158.
G. Watson, Bessel functions and Kapteyn series. Proc. London Math. Soc. (April 1916) 150–174.