Volume 47, Number 6, November-December 2013
|Page(s)||1691 - 1712|
|Published online||11 October 2013|
Lower and upper bounds for the Rayleigh conductivity of a perforated plate
CERFACS – EMA, 42 avenue Gaspar
2 INRIA Bordeaux Sud-Ouest – Magique 3D
3 Université de Pau, LMA (UMR-CNRS 5142), avenue de l’Université, 64013 Pau, France
4 INSA-Mathematical Institute of Toulouse (UMR-CNRS 5219), 135 avenue de Rangueil, 31077 Toulouse, France
5 CERFACS - EMA, 42 avenue Gaspar Coriolis, 31100 Toulouse, France
6 Boston University, Department of Electrical and Computer Engineering, 8 Saint Mary’s Street, Boston MA, 02215, USA
Received: 10 April 2012
Revised: 22 February 2013
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.
Mathematics Subject Classification: 35Q35 / 35J05 / 35J25
Key words: Rayleigh conductivity / perforated plate / Kelvin principle / Dirichlet principle
© EDP Sciences, SMAI, 2013
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