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Asymptotic modelling of conductive thin sheets

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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 H 1-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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Authors and Affiliations

  1. Project POEMS, INRIA Paris-Rocquencourt, BP 105, 78153, Le Chesnay, France

    Kersten Schmidt

  2. Mathematical Institute of Toulouse (UMR CNRS 5219), Toulouse University, INSA-Toulouse, 135 avenue de Rangueil, 31077, Toulouse Cedex 4, France

    Sébastien Tordeux

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  1. Kersten Schmidt
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  2. Sébastien Tordeux
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Correspondence to Kersten Schmidt.

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Schmidt, K., Tordeux, S. Asymptotic modelling of conductive thin sheets. Z. Angew. Math. Phys. 61, 603–626 (2010). https://doi.org/10.1007/s00033-009-0043-x

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  • DOI: https://doi.org/10.1007/s00033-009-0043-x

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