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 = {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/
[1] Shape optimization with a level set based mesh evolution method, Comput. Methods Appl. Mech. Eng., Volume 282 (2014), pp. 22-53 | DOI | MR
[2] Structural optimization using topological and shape sensitivity via a level set method, Control Cybern., Volume 34 (2005) no. 1, p. 59 | MR | Zbl
[3] Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., Volume 194 (2004) no. 1, pp. 363-393 | DOI | MR | Zbl
[4] Thin cylindrical conductivity inclusions in a three-dimensional domain: a polarization tensor and unique determination from boundary data, Inverse Probl., Volume 25 (2009) no. 6, p. 065004 | MR | Zbl
[5] An asymptotic formula for the displacement field in the presence of thin elastic inhomogeneities, SIAM J. Math. Anal., Volume 38 (2006) no. 4, pp. 1249-1261 | DOI | MR | Zbl
[6] Asymptotic formulas for steady state voltage potentials in the presence of thin inhomogeneities. A rigorous error analysis, J. Math. Pures Appl., Volume 82 (2003) no. 10, pp. 1277-1301 | DOI | MR | Zbl
[7] A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, ESAIM, Math. Model. Numer. Anal., Volume 37 (2003) no. 1, pp. 159-173 | DOI | Numdam | MR | Zbl
[8] The finite element method for elliptic problems, 40, Society for Industrial and Applied Mathematics, 2002 | MR | Zbl
[9] The topological ligament in shape optimization: an approach based on thin tubular inhomogeneities asymptotics (2020) (in preparation)
[10] Uniform asymptotic expansion of the voltage potential in the presence of thin inhomogeneities with arbitrary conductivity, Chin. Ann. Math., Ser. B, Volume 38 (2017) no. 1, pp. 293-344 | DOI | MR | Zbl
[11] Null space gradient flows for constrained optimization with applications to shape optimization (2019) (submitted, https://hal.archives-ouvertes.fr/hal-01972915/)
[12] Shape variation and optimization. A geometrical analysis, EMS Tracts in Mathematics, 28, European Mathematical Society, 2018 | Zbl
[13] Topological derivative of the energy functional due to formation of a thin ligament on a spatial body, Folia Math., Volume 12 (2005), pp. 39-72 | MR | Zbl
[14] The topological derivative of the Dirichlet integral due to formation of a thin ligament, Sib. Math. J., Volume 45 (2004) no. 2, pp. 341-355 | DOI | Zbl
[15] Self-adjoint extensions of differential operators and exterior topological derivatives in shape optimization, Control Cybern., Volume 34 (2005), pp. 903-925 | MR | Zbl
[16] A representation formula for the voltage perturbations caused by diametrically small conductivity inhomogeneities. Proof of uniform validity, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 26 (2009) no. 6, pp. 2283-2315 | DOI | Numdam | MR | Zbl
[17] Optimal shape design for elliptic systems, Springer, 1982 | Zbl
[18] Introduction to shape optimization, Springer, 1992 | Zbl
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