# Topological derivatives for thermo-mechanical semi-coupled system

### Esparta, J.E.; Giusti, S.M.;

The topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation. According to the literature, the topological derivative has been fully developed for a wide range of one single physical phenomenon modeled by partial differential equations. In addition, up to our knowledge, the topological asymptotic analysis associated to multi-physics problems has so far not been re- ported in the literature. In this work, we present the topological asymptotic analysis for the total potential mechanical energy associated to a thermo-mechanical system, when a small circular inclusion is introduced at an arbitrary point of the domain. In particular, we con- sider the linear elasticity system (modeled by the Navier equation) coupled with the steady- state heat conduction problem (modeled by the Laplace equation). The mechanical coupling term comes out from the thermal stress induced by the temperature field. Since this term is non-local, we introduce a non-standard adjoint state, which allows to obtain a closed form for the topological derivative. Finally, we provide a full mathematical justification for the de- rived formulas and develop precise estimates for the remainders of the topological asymptotic expansion.

Palavras-chave: Topological derivative, thermo-mechanical system, multi-physic topology opti- mization, asymptotic analysis.,

Palavras-chave:

DOI: 10.5151/meceng-wccm2012-19623

##### Referências bibliográficas
•  G. Allaire, F. Jouve, and N. Van Goethem. Damage and fracture evolution in brittle materials by shape optimization methods. Journal of Computational Physics, 230:5010– 5044, 201
•  S. Amstutz. Sensitivity analysis with respect to a local perturbation of the material property. Asymptotic Analysis, 49(1-2):87–108, 2006.
•  S. Amstutz, S.M. Giusti, A.A. Novotny, and E.A. de Souza Neto. Topological derivative in multi-scale linear elasticity models applied to the synthesis of microstructures. International Journal for Numerical Methods in Engineering, 84(6):733–756, 2010.
•  S. Amstutz, I. Horchani, and M. Masmoudi. Crack detection by the topological gradient method. Control and Cybernetics, 34(1):81–101, 2005.
•  S. Amstutz and A.A. Novotny. Topological optimization of structures subject to von mises stress constraints. Structural and Multidisciplinary Optimization, 41(3):407–420, 2010.
•  M.C. Delfour and J.P. Zol´esio. Shapes and Geometries. Advances in Design and Control, vol. 4. SIAM, Philadelphia, 2001.
•  H.A. Eschenauer, V.V. Kobelev, and A. Schumacher. Bubble method for topology and shape optmization of structures. Structural Optimization, 8(1):42–51, 1994.
•  J.D. Eshelby. The elastic energy-momentum tensor. Journal of Elasticity, 5(3-4):321– 335, 1975.
•  S.M. Giusti, A.A. Novotny, E.A. de Souza Neto, and R.A. Feijóo. Sensitivity of the macroscopic elasticity tensor to topological microstructural changes. Journal of the Mechanics and Physics of Solids, 57(3):555–570, 200
•  N. Van Goethem and A.A. Novotny. Crack nucleation sensitivity analysis. Mathematical Methods in the Applied Sciences, 33(16):1978–1994, 20
•  M.E. Gurtin. An introduction to continuum mechanics. Mathematics in Science and Engineering vol. 158. Academic Press, New York, 1981.
•  M.E. Gurtin. Configurational forces as basic concept of continuum physics. Applied Mathematical Sciences vol. 137. Springer-Verlag, New York, 2000.
•  M. Hintermüller and A. Laurain. Multiphase image segmentation and modulation recovery based on shape and topological sensitivity. Journal on Mathematical Imaging and Vision, 35:1–22, 2009.
•  M. Hintermüller, A. Laurain, and A.A. Novotny. Second-order topological expansion for electrical impedance tomography. Advances in Computational Mathematics, 36(2):235– 265, 2012.
•  I. Larrabide, R.A. Feijóo, A.A. Novotny, and E. Taroco. Topological derivative: a tool for image processing. Computers Andamp; Structures, 86(13-14):1386–1403, 2008.
•  S.A. Nazarov and J. Sokolowski. Asymptotic analysis of shape functionals. Journal de Math´ematiques Pures et Appliqu´ees, 82(2):125–196, 2003.
•  A.A. Novotny. Análise de sensibilidade topológica. Phd thesis, Laboratório Nacional de Computac¸ ão Cient´ifica, Petrópolis, Brazil, 2003.
•  A.A. Novotny, R.A. Feijóo, E. Taroco, and C. Padra. Topological sensitivity analysis for three-dimensional linear elasticity problem. Computer Methods in Applied Mechanics and Engineering, 196(41-44):4354–4364, 2007.
•  J. Sokolowski and A. ? Zochowski. On the topological derivative in shape optimization. SIAM Journal on Control and Optimization, 37(4):1251–1272, 1999.
•  J. Sokolowski and J.P. Zol´esio. Introduction to shape optimization - shape sensitivity analysis. Springer Series in Computational Mathematics, vol. 16. Springer-Verlag, Berlin, 1992.
##### Como citar:

Esparta, J.E.; Giusti, S.M.; "Topological derivatives for thermo-mechanical semi-coupled system", p. 3888-3905 . In: In Proceedings of the 10th World Congress on Computational Mechanics [= Blucher Mechanical Engineering Proceedings, v. 1, n. 1]. São Paulo: Blucher, 2014.
ISSN 2358-0828, DOI 10.5151/meceng-wccm2012-19623

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