Topological asymptotic analysis of the Kirchhoff plate bending problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 705-721.

The topological asymptotic analysis provides the sensitivity of a given shape functional with respect to an infinitesimal domain perturbation, like the insertion of holes, inclusions, cracks. In this work we present the calculation of the topological derivative for a class of shape functionals associated to the Kirchhoff plate bending problem, when a circular inclusion is introduced at an arbitrary point of the domain. According to the literature, the topological derivative has been fully developed for a wide range of second-order differential operators. Since we are dealing here with a forth-order operator, we perform a complete mathematical analysis of the problem.

DOI : 10.1051/cocv/2010010
Classification : 35J30, 49Q10, 49Q12, 74K20, 74P15
Mots clés : topological sensitivity, topological derivative, topology optimization, Kirchhoff plates
@article{COCV_2011__17_3_705_0,
     author = {Amstutz, Samuel and Novotny, Antonio A.},
     title = {Topological asymptotic analysis of the {Kirchhoff} plate bending problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {705--721},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {3},
     year = {2011},
     doi = {10.1051/cocv/2010010},
     mrnumber = {2826976},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2010010/}
}
TY  - JOUR
AU  - Amstutz, Samuel
AU  - Novotny, Antonio A.
TI  - Topological asymptotic analysis of the Kirchhoff plate bending problem
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2011
SP  - 705
EP  - 721
VL  - 17
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2010010/
DO  - 10.1051/cocv/2010010
LA  - en
ID  - COCV_2011__17_3_705_0
ER  - 
%0 Journal Article
%A Amstutz, Samuel
%A Novotny, Antonio A.
%T Topological asymptotic analysis of the Kirchhoff plate bending problem
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2011
%P 705-721
%V 17
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2010010/
%R 10.1051/cocv/2010010
%G en
%F COCV_2011__17_3_705_0
Amstutz, Samuel; Novotny, Antonio A. Topological asymptotic analysis of the Kirchhoff plate bending problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 705-721. doi : 10.1051/cocv/2010010. http://archive.numdam.org/articles/10.1051/cocv/2010010/

[1] G. Allaire, F. De Gournay, F. Jouve and A. Toader, Structural optimization using topological and shape sensitivity via a level set method. Control Cybern. 34 (2005) 59-80. | MR | Zbl

[2] S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property. Asymptotic Anal. 49 (2006) 87-108. | MR | Zbl

[3] S. Amstutz and H. Andrä, A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216 (2006) 573-588. | MR | Zbl

[4] S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the topological gradient method. Control Cybern. 34 (2005) 81-101. | MR | Zbl

[5] D. Auroux, M. Masmoudi and L. Belaid, Image restoration and classification by topological asymptotic expansion, in Variational formulations in mechanics: theory and applications, Barcelona, Spain (2007).

[6] G.R. Feijóo, A new method in inverse scattering based on the topological derivative. Inv. Prob. 20 (2004) 1819-1840. | MR | Zbl

[7] S. Garreau, Ph. Guillaume and M. Masmoudi, The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756-1778. | MR | Zbl

[8] M. Hintermüller and A. Laurain, Electrical impedance tomography: from topology to shape. Control Cybern. 37 (2008) 913-933. | MR | Zbl

[9] M. Hintermüller and A. Laurain, Multiphase image segmentation and modulation recovery based on shape and topological sensitivity. J. Math. Imag. Vis. 35 (2009) 1-22. | MR

[10] I. Larrabide, R.A. Feijóo, A.A. Novotny and E. Taroco, Topological derivative: a tool for image processing. Comput. Struct. 86 (2008) 1386-1403.

[11] R.W. Little, Elasticity. Prentice-Hall, New Jersey (1973).

[12] S.A. Nazarov and J. Sokołowski, Asymptotic analysis of shape functionals. J. Math. Pures Appl. 82 (2003) 125-196. | Zbl

[13] A.A. Novotny, R.A. Feijóo, C. Padra and E. Taroco, Topological derivative for linear elastic plate bending problems. Control Cybern. 34 (2005) 339-361. | MR | Zbl

[14] A.A. Novotny, R.A. Feijóo, E. Taroco and C. Padra, Topological sensitivity analysis for three-dimensional linear elasticity problem. Comput. Methods Appl. Mech. Eng. 196 (2007) 4354-4364. | MR | Zbl

[15] J. Sokołowski and A. Żochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251-1272. | Zbl

Cité par Sources :