Abstract
An original approach to solve 2D time harmonic diffraction problems involving locally perturbed gratings is proposed. The propagation medium is composed of a periodically stratified half-space and a homogeneous half-space containing a bounded obstacle. Using Fourier and Floquet transforms and integral representations, the diffraction problem is formulated as a coupled problem of Fredholm type with two unknowns: the trace of the diffracted field on the interface separating the two half-spaces on one hand, and the restriction of the diffracted field to a bounded domain surrounding the obstacle, on the other hand.
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References
T. Abboud, Etude mathématique et numérique de quelques problèmes de diffraction d'ondes electromagnétiques, Ph.D. dissertation, Ecole Polytechnique Palaiseau (1991).
G. Bao, Variational approximation of Maxwell's equations by periodic structures, SIAM J. Numer. Anal. 32 (1995) 1155-1169.
A.-S. Bonnet-Bendhia and F. Starling, Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem, Math. Methods Appl. Sci. 17 (1994) 305-338.
A.-S. Bonnet-Bendhia and A. Tillequin, A generalized mode matching method for scattering problems with unbounded obstacles, J. Comp. Acoustics, to appear.
D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Krieger Publishing Company, 1992).
D.C. Dobson, A variational method for electromagnetic diffraction in biperiodic structures, Model. Math. Anal. Numer. 28 (1994) 419-439.
D.M. Eidus, The principle of limiting absorption, Amer. Math. Soc. Trans. 47 (1965) 157-191.
J. Elschner, R. Hinder, G. Schmidt and F. Penzel, Existence, uniqueness and regularity for solutions of the conical diffraction problem, Math. Mod. Methods Appl. Sci. 10(3) (2000) 317-341.
A. Friedman, Mathematics in Industrial Problems, Part 7, The IMA Volumes in Mathematics and its Applications, Vol. 67 (Springer, New York, 1995).
D. Huet, Décomposition Spectrale et Opérateurs (resses Universitaires de France, 1976).
P. Kuchment and P. Floquet, Theory for Partial Differential Equations (Birkhaüser, Basel, 1993).
M. Lenoir and A. Jami, A variational formulation for exterior problems in linear hydrodynamics, Comput. Methods Appl. Mech. Engrg. 16 (1978) 341-359.
J.-C. Nedelec and F. Starling, Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell's equations, SIAM J. Math. Anal. 22(6) (1991) 1679-1701.
R. Petit, Electromagnetic Theory of Gratings, Topics in Current Physics, Vol. 22 (Springer, Berlin, 1980).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. IV (Academic Press, New York, 1978).
J. Sanchez-Hubert and E. Sanchez-Palencia, Vibration and Coupling of Continuous Systems (Springer, Berlin, 1989).
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Bonnet-Bendhia, AS., Ramdani, K. Diffraction by an Acoustic Grating Perturbed by a Bounded Obstacle. Advances in Computational Mathematics 16, 113–138 (2002). https://doi.org/10.1023/A:1014437813575
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DOI: https://doi.org/10.1023/A:1014437813575