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.
Mots-clés : Rayleigh conductivity, perforated plate, Kelvin principle, Dirichlet principle
@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 },
pages = {1691--1712},
publisher = {EDP-Sciences},
volume = {47},
number = {6},
year = {2013},
doi = {10.1051/m2an/2013082},
mrnumber = {3123372},
zbl = {1283.35088},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2013082/}
}
TY - JOUR AU - Laurens, S. AU - Tordeux, S. AU - Bendali, A. AU - Fares, M. AU - Kotiuga, P. R. TI - Lower and upper bounds for the Rayleigh conductivity of a perforated plate JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1691 EP - 1712 VL - 47 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013082/ DO - 10.1051/m2an/2013082 LA - en ID - M2AN_2013__47_6_1691_0 ER -
%0 Journal Article %A Laurens, S. %A Tordeux, S. %A Bendali, A. %A Fares, M. %A Kotiuga, P. R. %T Lower and upper bounds for the Rayleigh conductivity of a perforated plate %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1691-1712 %V 47 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013082/ %R 10.1051/m2an/2013082 %G en %F 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 , Tome 47 (2013) no. 6, pp. 1691-1712. doi : 10.1051/m2an/2013082. http://www.numdam.org/articles/10.1051/m2an/2013082/
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