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.
@article{M2AN_2013__47_6_1691_0,
author = {Laurens, S. and Tordeux, S. and Bendali, A. and Fares, M. and Kotiuga, P. R.},
title = {Lower and upper bounds for the Rayleigh conductivity of a perforated plate},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {47},
year = {2013},
pages = {1691-1712},
doi = {10.1051/m2an/2013082},
mrnumber = {3123372},
zbl = {1283.35088},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2013__47_6_1691_0}
}
Laurens, S.; Tordeux, S.; Bendali, A.; Fares, M.; Kotiuga, P. R. Lower and upper bounds for the Rayleigh conductivity of a perforated plate. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) pp. 1691-1712. doi : 10.1051/m2an/2013082. http://gdmltest.u-ga.fr/item/M2AN_2013__47_6_1691_0/
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