Asymptotic modelling of conductive thin sheets

Kersten Schmidt 1 Sébastien Tordeux 2
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
Abstract : We derive and analyse models which reduce conducting sheets of a small thickness ε in two dimensions to an interface and approximate their shielding behaviour by conditions on this interface. For this we consider a model problem with a conductivity scaled reciprocal to the thickness ε, which leads to a nontrivial limit solution for ε → 0. The functions of the expansion are defined hierarchically, i.e. order by order. Our analysis shows that for smooth sheets the models are well defined for any order and have optimal convergence meaning that the H1-modelling error for an expansion with N terms is bounded by O(ε^{N+1}) in the exterior of the sheet and by O(ε^{N+1/2}) in its interior. We explicitly specify the models of order zero, one and two. Numerical experiments for sheets with varying curvature validate the theoretical results.
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Submitted on : Tuesday, October 19, 2010 - 4:47:27 PM
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Kersten Schmidt, Sébastien Tordeux. Asymptotic modelling of conductive thin sheets. Zeitschrift für Angewandte Mathematik und Physik, Springer Verlag, 2010, 61 (4), pp.603-626. ⟨10.1007/s00033-009-0043-x⟩. ⟨inria-00527608⟩



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