Dans cette note, on introduit une approche formelle visant à évaluer la sensibilité d’une fonction du domaine par rapport à la greffe d’un ligament très fin sur celui-ci. Dans le contexte modèle des structures élastiques, nous approchons cette question par un problème de petite inclusion tubulaire : on étudie la sensibilité de la solution d’une équation aux dérivées partielles posée dans un milieu ambiant, ainsi que celle d’une quantité d’intérêt associée, par rapport à l’inclusion d’un tube fin contenant un matériau distinct de celui du milieu ambiant. On obtient une formule explicite pour cette sensibilité, qui se prête à l’implémentation numérique. Cette idée est illustrée par deux applications en optimisation structurale.
In this note, we propose a formal framework accounting for the sensitivity of a function of the domain with respect to the addition of a thin ligament. To set ideas, we consider the model setting of elastic structures, and we approximate this question by a thin tubular inhomogeneity problem: we look for the sensitivity of the solution to a partial differential equation posed inside a background medium, and that of a related quantity of interest, with respect to the inclusion of a thin tube filled with a different material. A practical formula for this sensitivity is derived, which lends itself to numerical implementation. Two applications of this idea in structural optimization are presented.
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@article{CRMATH_2020__358_2_119_0,
author = {Dapogny, Charles},
title = {A connection between topological ligaments in shape optimization and thin tubular inhomogeneities},
journal = {Comptes Rendus. Math\'ematique},
pages = {119--127},
publisher = {Acad\'emie des sciences, Paris},
volume = {358},
number = {2},
year = {2020},
doi = {10.5802/crmath.3},
language = {en},
url = {http://www.numdam.org/articles/10.5802/crmath.3/}
}
TY - JOUR AU - Dapogny, Charles TI - A connection between topological ligaments in shape optimization and thin tubular inhomogeneities JO - Comptes Rendus. Mathématique PY - 2020 SP - 119 EP - 127 VL - 358 IS - 2 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.3/ DO - 10.5802/crmath.3 LA - en ID - CRMATH_2020__358_2_119_0 ER -
%0 Journal Article %A Dapogny, Charles %T A connection between topological ligaments in shape optimization and thin tubular inhomogeneities %J Comptes Rendus. Mathématique %D 2020 %P 119-127 %V 358 %N 2 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.3/ %R 10.5802/crmath.3 %G en %F CRMATH_2020__358_2_119_0
Dapogny, Charles. A connection between topological ligaments in shape optimization and thin tubular inhomogeneities. Comptes Rendus. Mathématique, Tome 358 (2020) no. 2, pp. 119-127. doi : 10.5802/crmath.3. http://www.numdam.org/articles/10.5802/crmath.3/
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