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Comptes Rendus. Mathématique

Optimisation de forme, Analyse numérique
A connection between topological ligaments in shape optimization and thin tubular inhomogeneities
Comptes Rendus. Mathématique, Tome 358 (2020) no. 2, pp. 119-127.

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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DOI : https://doi.org/10.5802/crmath.3
@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},
}
Charles Dapogny. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.3/

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