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.
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.
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Charles Dapogny 1
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@article{CRMATH_2020__358_2_119_0,
author = {Charles Dapogny},
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},
}
TY - JOUR AU - Charles Dapogny 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 DO - 10.5802/crmath.3 LA - en ID - CRMATH_2020__358_2_119_0 ER -
Charles Dapogny. A connection between topological ligaments in shape optimization and thin tubular inhomogeneities. Comptes Rendus. Mathématique, Volume 358 (2020) no. 2, pp. 119-127. doi : 10.5802/crmath.3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.3/
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