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Asymptotic modelling of conductive thin sheets

by K. Schmidt and S. Tordeux

(Report number 2008-28)

Abstract
We derive and analyse models which reduce conducting sheets of a small thickness $\epsilon$ 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 $\epsilon$, which leads a nontrivial limit solution for $\epsilon\to0$. The functions of the expansion are defined hierarchically, i.e. order by order. Our analysis shows that for smooth sheets the model 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 $O(\epsilon^{N+1})$ in the exterior of the sheet and by $O(\epsilon^{N+1/2})$ in the interior. We explicitly specify the models of order zero, one and two. Numerical experiments for sheets with varying curvature validate the theoretical results.

Keywords:

@Techreport{ST08_391,
  author = {K. Schmidt and S. Tordeux},
  title = {Asymptotic modelling of conductive thin sheets},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2008-28},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2008/2008-28.pdf },
  year = {2008}
}

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