Weighted regularization for composite materials in electromagnetism

Patrick Ciarlet 1 François Lefèvre 2 Stéphanie Lohrengel 2 Serge Nicaise 3
1 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
2 LMR - Laboratoire de Mathématiques de Reims
3 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015
Abstract : In this paper, a weighted regularization method for the time-harmonic Maxwell equations with perfect conducting or impedance boundary condition in composite materials is presented. The computational domain Ω is the union of polygonal or polyhedral subdomains made of different materials. As a result, the electromagnetic field presents singularities near geometric singularities, which are the interior and exterior edges and corners. The variational formulation of the weighted regularized problem is given on the subspace of H(curl;Ω) whose fields u satisfy wα div(εu) ∈ L 2(Ω) and have vanishing tangential trace or tangential trace in L2(δΩ). The weight function w(x) is equivalent to the distance of x to the geometric singularities and the minimal weight parameter α is given in terms of the singular exponents of a scalar transmission problem. A density result is proven that guarantees the approximability of the solution field by piecewise regular fields. Numerical results for the discretization of the source problem by means of Lagrange Finite Elements of type P1 and P2 are given on uniform and appropriately refined two-dimensional meshes. The performance of the method in the case of eigenvalue problems is addressed. © EDP Sciences, SMAI 2009.
Mots-clés : Modèle mathématique Analyse numérique Problème valeur propre Calcul 2 dimensions Méthode élément fini Méthode discrétisation Solution régulière Problème transmission Fonction poids Champ électromagnétique Condition aux limites Équation Maxwell Électromagnétisme Matériau composite Méthode régularisation Régularisation Problème mal posé Algèbre linéaire numérique
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Submitted on : Wednesday, October 16, 2013 - 2:04:25 PM
Last modification on : Wednesday, July 3, 2019 - 10:48:03 AM

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Patrick Ciarlet, François Lefèvre, Stéphanie Lohrengel, Serge Nicaise. Weighted regularization for composite materials in electromagnetism. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2010, 44 (1), pp.75-108. ⟨10.1051/m2an/2009041⟩. ⟨hal-00873065⟩

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