[Optimisation de la structure d'une poutre fine en torsion et ensembles de Cheeger]
On considère le problème d'optimisation suivant : une quantité fixée d'un matériau élastique isotrope donné doit être placée dans un cylindre droit de manière à maximiser sa résistance à un chargement donné tendant à provoquer un mouvement de torsion. Lorsque le rayon et le taux de remplissage du cylindre tendent tous deux vers zéro, on montre que la distribution optimale de matière se concentre dans chaque section sur le bord de l'ensemble de Cheeger.
We consider the variational problem which consists in minimizing the compliance of a prescribed amount of isotropic elastic material placed into a given design region when it is subjected to a given load. We perform the asymptotics of this problem when the design region is a straight cylinder with infinitesimal cross section. The results presented in this Note concern the pure torsion regime and state the existence of optimal shapes for the limit problem. When the filling ratio tends in turn to zero, these optimal shapes concentrate on the boundary of the Cheeger set of the section of the design region.
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@article{CRMATH_2010__348_7-8_467_0,
author = {Bouchitt\'e, Guy and Fragal\`a, Ilaria and Seppecher, Pierre},
title = {The optimal compliance problem for thin torsion rods: {A} {3D-1D} analysis leading to {Cheeger-type} solutions},
journal = {Comptes Rendus. Math\'ematique},
pages = {467--471},
publisher = {Elsevier},
volume = {348},
number = {7-8},
year = {2010},
doi = {10.1016/j.crma.2010.01.006},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2010.01.006/}
}
TY - JOUR AU - Bouchitté, Guy AU - Fragalà, Ilaria AU - Seppecher, Pierre TI - The optimal compliance problem for thin torsion rods: A 3D-1D analysis leading to Cheeger-type solutions JO - Comptes Rendus. Mathématique PY - 2010 SP - 467 EP - 471 VL - 348 IS - 7-8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2010.01.006/ DO - 10.1016/j.crma.2010.01.006 LA - en ID - CRMATH_2010__348_7-8_467_0 ER -
%0 Journal Article %A Bouchitté, Guy %A Fragalà, Ilaria %A Seppecher, Pierre %T The optimal compliance problem for thin torsion rods: A 3D-1D analysis leading to Cheeger-type solutions %J Comptes Rendus. Mathématique %D 2010 %P 467-471 %V 348 %N 7-8 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2010.01.006/ %R 10.1016/j.crma.2010.01.006 %G en %F CRMATH_2010__348_7-8_467_0
Bouchitté, Guy; Fragalà, Ilaria; Seppecher, Pierre. The optimal compliance problem for thin torsion rods: A 3D-1D analysis leading to Cheeger-type solutions. Comptes Rendus. Mathématique, Tome 348 (2010) no. 7-8, pp. 467-471. doi : 10.1016/j.crma.2010.01.006. http://www.numdam.org/articles/10.1016/j.crma.2010.01.006/
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