Trapped modes in thin and infinite ladder like domains. Part 2 : asymptotic analysis and numerical application

Bérangère Delourme 1 Sonia Fliss 2 Patrick Joly 2 Elizaveta Vasilevskaya 1
1 LAGA
LAGA - Laboratoire Analyse, Géométrie et Applications
2 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : We are interested in a 2D propagation medium obtained from a localized perturbation of a reference homogeneous periodic medium. This reference medium is a " thick graph " , namely a thin structure (the thinness being characterized by a small parameter ε > 0) whose limit (when ε tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. In the first part of this work, we proved that such a geometrical perturbation is able to produce localized eigenmodes (the propagation model under consideration is the scalar Helmholtz equation with Neumann boundary conditions). This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We used a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough. The objective of the present work is to complement the previous one by constructing and justifying a high order asymptotic expansion of these eigenvalues (with respect to the small parameter ε) using the method of matched asymptotic expansions. In particular, the obtained expansion can be used to compute a numerical approximation of the eigenvalues and of their associated eigenvectors. An algorithm to compute each term of the asymptotic expansion is proposed. Numerical experiments validate the theoretical results.
Type de document :
Pré-publication, Document de travail
2018
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Bérangère Delourme, Sonia Fliss, Patrick Joly, Elizaveta Vasilevskaya. Trapped modes in thin and infinite ladder like domains. Part 2 : asymptotic analysis and numerical application. 2018. 〈hal-01822437〉

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