Voir la notice de l'article provenant de la source Numdam
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.
Dapogny, Charles 1
CC-BY 4.0
@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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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 | Zbl | MR
[3] Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., Volume 194 (2004) no. 1, pp. 363-393 | Zbl | MR | DOI
[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 | Zbl | MR
[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 | Zbl | MR | DOI
[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 | Zbl | mathdoc-id | MR | DOI
[8] The finite element method for elliptic problems, 40, Society for Industrial and Applied Mathematics, 2002 | Zbl | MR
[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 | Zbl | MR | DOI
[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 | Zbl | DOI
[15] Self-adjoint extensions of differential operators and exterior topological derivatives in shape optimization, Control Cybern., Volume 34 (2005), pp. 903-925 | Zbl | MR
[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 | mathdoc-id | MR | Zbl
[17] Optimal shape design for elliptic systems, Springer, 1982 | Zbl
[18] Introduction to shape optimization, Springer, 1992 | Zbl
Cité par Sources :