Lower and upper bounds for the Rayleigh conductivity of a perforated plate

¹ CERFACS – EMA, 42 avenue Gaspar Coriolis, 31100 Toulouse, France
sophie.laurens@insa-toulouse.fr
² INRIA Bordeaux Sud-Ouest – Magique 3D  
³ Université de Pau, LMA (UMR-CNRS 5142), avenue de l’Université, 64013 Pau, France
sebastien.tordeux@inria.fr
⁴ INSA-Mathematical Institute of Toulouse (UMR-CNRS 5219), 135 avenue de Rangueil, 31077 Toulouse, France
abderrahmane.bendali@insa-toulouse.fr
⁵ CERFACS - EMA, 42 avenue Gaspar Coriolis, 31100 Toulouse, France
fares@cerfacs.fr
⁶ Boston University, Department of Electrical and Computer Engineering, 8 Saint Mary’s Street, Boston MA, 02215, USA
prk@bu.edu

Received: 10 April 2012
Revised: 22 February 2013

Abstract

Lower and upper bounds for the Rayleigh conductivity of a perforation in a thick plate are usually derived from intuitive approximations and by physical reasoning. This paper addresses a mathematical justification of these approaches. As a byproduct of the rigorous handling of these issues, some improvements to previous bounds for axisymmetric holes are given as well as new estimates for tilted perforations. The main techniques are a proper use of the Dirichlet and Kelvin variational principlesin the context of Beppo-Levi spaces. The derivations are validated by numerical experiments in 2D for the axisymmetric case as well as for the full three-dimensional problem.