Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-19T13:33:36.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological asymptotic analysis of the Kirchhoff plate bending problem

Published online by Cambridge University Press:  31 March 2010

Samuel Amstutz  and
Antonio A. Novotny
Samuel Amstutz
Affiliation:
Laboratoire d'analyse non linéaire et géométrie, Faculté des Sciences, 33 rue Louis Pasteur, 84000 Avignon, France. [email protected]
Antonio A. Novotny
Affiliation:
Laboratório Nacional de Computação Científica LNCC/MCT, Coordenação de Matemática Aplicada e Computacional, Av. Getúlio Vargas 333, 25651-075 Petrópolis – RJ, Brasil. [email protected]
Get access

Abstract

The topological asymptotic analysis provides the sensitivity of a givenshape functional with respect to an infinitesimal domain perturbation, likethe insertion of holes, inclusions, cracks. In this work we present thecalculation of the topological derivative for a class of shape functionalsassociated to the Kirchhoff plate bending problem, when a circular inclusionis introduced at an arbitrary point of the domain. According to theliterature, the topological derivative has been fully developed for a widerange of second-order differential operators. Since we are dealing here witha forth-order operator, we perform a complete mathematicalanalysis of the problem.

Keywords

Topological sensitivitytopological derivativetopologyoptimizationKirchhoff plates
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 3 , July 2011 , pp. 705 - 721
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allaire, G., de Gournay, F., Jouve, F. and Toader, A., Structural optimization using topological and shape sensitivity via a level set method. Control Cybern. 34 (2005) 59–80.
Amstutz, S., Sensitivity analysis with respect to a local perturbation of the material property. Asymptotic Anal. 49 (2006) 87–108.
Amstutz, S. and Andrä, H., A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216 (2006) 573–588. CrossRef
Amstutz, S., Horchani, I. and Masmoudi, M., Crack detection by the topological gradient method. Control Cybern. 34 (2005) 81–101.
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).
Feijóo, G.R., A new method in inverse scattering based on the topological derivative. Inv. Prob. 20 (2004) 1819–1840. CrossRef
Garreau, S., Guillaume, Ph. and Masmoudi, M., The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim. 39 (2001) 1756–1778. CrossRef
Hintermüller, M. and Laurain, A., Electrical impedance tomography: from topology to shape. Control Cybern. 37 (2008) 913–933.
Hintermüller, M. and Laurain, A., Multiphase image segmentation and modulation recovery based on shape and topological sensitivity. J. Math. Imag. Vis. 35 (2009) 1–22. CrossRef
Larrabide, I., Feijóo, R.A., Novotny, A.A. and Taroco, E., Topological derivative: a tool for image processing. Comput. Struct. 86 (2008) 1386–1403. CrossRef
R.W. Little, Elasticity. Prentice-Hall, New Jersey (1973).
Nazarov, S.A. and Sokołowski, J., Asymptotic analysis of shape functionals. J. Math. Pures Appl. 82 (2003) 125–196. CrossRef
Novotny, A.A., Feijóo, R.A., Padra, C. and Taroco, E., Topological derivative for linear elastic plate bending problems. Control Cybern. 34 (2005) 339–361.
Novotny, A.A., Feijóo, R.A., Taroco, E. and Padra, C., Topological sensitivity analysis for three-dimensional linear elasticity problem. Comput. Methods Appl. Mech. Eng. 196 (2007) 4354–4364. CrossRef
Sokołowski, J. and A. Żochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251–1272. CrossRef