Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 1, p. 63-97
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.
Classification:  35J05,  74J05,  78A45,  78M30,  78M35
Keywords: slit, slot, wave equation, Helmholtz equation, approximate model
     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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {1},
     year = {2006},
     pages = {63-97},
     doi = {10.1051/m2an:2006008},
     zbl = {pre05038393},
     mrnumber = {2223505},
     language = {en},
     url = {}
Joly, Patrick; Tordeux, Sébastien. Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 1, pp. 63-97. doi : 10.1051/m2an:2006008.

[1] 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. | Zbl 0819.35132

[2] M. Artola and M. Cessenat, 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 0743.35081

[3] F. Assous, P. Ciarlet, 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) 218-249. | Zbl 1007.78014

[4] J. Beale, Scattering frequencies of reasonators. Comm. Pure Appl. Math. 26 (1973) 549-563. | Zbl 0254.35094

[5] A. Bendali and K. Lemrabet, 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 0869.35068

[6] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland Publishing Co., Amsterdam. Stud. Math. Appl. 5 (1978). | MR 503330 | Zbl 0404.35001

[7] A. Buffa and S.H. Christiansen, The electric field integral equation on Lipschitz screens: definitions and numerical approximation. Numer. Math. 94 (2003) 229-267. | Zbl 1027.65188

[8] C. Butler and D. Wilton, General analysis of narrow strips and slots. IEEE Trans Ant. and Propag. 28 (1980).

[9] P.G. Ciarlet, Plates and junctions in elastic multi-structures, 14 Recherches en Mathématiques Appliquées [Res. Appl. Math.], Masson, Paris (1990). An asymptotic analysis. | MR 1071376 | Zbl 0706.73046

[10] 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, (Août 1998).

[11] 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. | MR 700400 | Zbl 0522.35001

[12] C. Conca and E. Zuazua, Asymptotic analysis of a multidimensional vibrating structure. SIAM J. Math. Anal. 25 (1994) 836-858. | Zbl 0806.35128

[13] 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.

[14] R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 3. Masson, Paris (1985). | MR 902802 | Zbl 0749.35005

[15] 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).

[16] J. Gilbert and R. Holland, Implementation of the thin-slot formalism in the finite-difference code threedii. IEEE Trans. Nuc. Sci. 28 (1981).

[17] D. Givoli, I. Patlashenko and J. Keller, 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 0944.65122

[18] P. Grisvard, Elliptic problems in nonsmooth domains 24, Pitman (Advanced Publishing Program), Boston, MA, Monographs Stud. Math. (1985). | MR 775683 | Zbl 0695.35060

[19] P. Grisvard, Singularities in boundary value problems 22, Recherches en Mathématiques Appliquées [Res. Appl. Math.], Masson, Paris (1992). | MR 1173209 | Zbl 0766.35001

[20] D. Guiney, B. Noye and E. Tuck, Transmission of water waves through small apertures. J. Fluid Mech. 55 (1972) 149-161. | Zbl 0246.76017

[21] I. Harari, I. Patlashenko and D. Givoli, Dirichlet-to-Neumann maps for unbounded wave guides. J. Comput. Phys. 143 (1998) 200-223. | Zbl 0928.65140

[22] P. Harrington and D. Auckland, Electromagnetic transmission through narrow slots in thick conducting screens. IEEE Trans. Antenna Propagation 28 (1980) 616-622.

[23] R. Holland and L. Simpson, Finite-difference analysis EMP coupling to thin struts and wires. IEEE Trans. Electromagn. Compat. 23 (1981).

[24] P. Joly and S. Tordeux, Modèles asymptotiques pour la propagation des ondes dans les milieux comportant des fentes, Tech. Report RR-5568, INRIA, (May 2005).

[25] J. Keller and D. Givoli, Exact nonreflecting boundary conditions. J. Comput. Phys. 82 (1989) 172-192. | Zbl 0671.65094

[26] G. Kriegsmann, The flanged waveguide antenna: discrete reciprocity and conservation. Wave Motion 29 (1999) 81-95. | Zbl 1074.78527

[27] 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]. | MR 1130395 | Zbl 0744.73027

[28] N. Lebedev, Special functions and their applications, Revised English edition. Translated and edited by Richard A. Silverman, Prentice-Hall Inc., Englewood Cliffs, NJ (1965). | MR 174795 | Zbl 0131.07002

[29] 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. | MR 350177 | Zbl 0223.35039

[30] D. Martin, Documentation de la librairie éléments finis melina., 1996-2003.

[31] P. Mciver and A.D. Rawlins, Two-dimensional wave-scattering problems involving parallel-walled ducts. Quart. J. Mech. Appl. Math. 46 (1993) 89-116. | Zbl 0777.76016

[32] J.C. Nédélec, Acoustic and electromagnetic equations. Springer-Verlag, New York. Appl. Math. Sci. 144 (2001). Integral representations for harmonic problems. | MR 1822275 | Zbl 0981.35002

[33] 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).

[34] E. Sánchez-Palencia, Nonhomogeneous media and vibration theory. Lect. Notes Phys. 127 (1980). | MR 578345 | Zbl 0432.70002

[35] T. Senior and J. Volakis, Approximate Boundary Conditions in Electromagnetics. IEE Pres, New York and London (1995). | Zbl 0828.73001

[36] A. Taflove, Computational electrodynamics. Artech House Inc., Boston, MA (1995). The finite-difference time-domain method. | MR 1338377 | Zbl 0840.65126

[37] A. Taflove, K. Umashankar and B. Becker, Calculation and experimental validation of induced currents on coupled wires in an arbitrary shaped cavity. IEEE Trans Antenna Propag. 35 (1987) 1248-1257.

[38] A. Taflove, K. Umashankar, B. Becker, F. Harfoush and K. Yee, 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] 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).

[40] S. Tordeux, Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes. Ph.D. thesis, Université de Versailles (2005).

[41] E. Tuck, Matching problems involving flow through small holes. Academic Press, New York Adv. Appl. Mechanics 15 (1975) 89-158.

[42] G. Watson, Bessel functions and Kapteyn series. Proc. London Math. Soc. (April 1916) 150-174. | JFM 46.0576.03