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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.3
Dapogny, Charles 1

1 Univ. Grenoble Alpes, CNRS, Grenoble INP, LJK, 38000 Grenoble, France
@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/

[1] Allaire, Grégoire; Dapogny, Charles; Frey, Pascal Shape optimization with a level set based mesh evolution method, Comput. Methods Appl. Mech. Eng., Volume 282 (2014), pp. 22-53 | DOI | MR

[2] Allaire, Grégoire; De Gournay, Frédéric; Jouve, François; Toader, Anca-Maria Structural optimization using topological and shape sensitivity via a level set method, Control Cybern., Volume 34 (2005) no. 1, p. 59 | MR | Zbl

[3] Allaire, Grégoire; Jouve, François; Toader, Anca-Maria 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] Beretta, Elena; Capdeboscq, Yves; De Gournay, Frédéric; Francini, Elisa 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] Beretta, Elena; Francini, Elisa 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] Beretta, Elena; Francini, Elisa; Vogelius, Michael S 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] Capdeboscq, Yves; Vogelius, Michael S 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] Ciarlet, Philippe G The finite element method for elliptic problems, 40, Society for Industrial and Applied Mathematics, 2002 | MR | Zbl

[9] Dapogny, Charles The topological ligament in shape optimization: an approach based on thin tubular inhomogeneities asymptotics (2020) (in preparation)

[10] Dapogny, Charles; Vogelius, Michael S 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] Feppon, Florian; Allaire, Grégoire; Dapogny, Charles Null space gradient flows for constrained optimization with applications to shape optimization (2019) (submitted, https://hal.archives-ouvertes.fr/hal-01972915/)

[12] Henrot, Antoine; Pierre, Michel Shape variation and optimization. A geometrical analysis, EMS Tracts in Mathematics, 28, European Mathematical Society, 2018 | Zbl

[13] Nazarov, Sergei; Slutskij, A.; Sokołowski, Jan 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] Nazarov, Sergei; Sokołowski, Jan 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] Nazarov, Sergei; Sokołowski, Jan Self-adjoint extensions of differential operators and exterior topological derivatives in shape optimization, Control Cybern., Volume 34 (2005), pp. 903-925 | MR | Zbl

[16] Nguyen, Hoai-Minh; Vogelius, Michael S 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] Pironneau, Olivier Optimal shape design for elliptic systems, Springer, 1982 | Zbl

[18] Sokołowski, Jan; Zolésio, Jean-Paul Introduction to shape optimization, Springer, 1992 | Zbl

Cité par Sources :