Computational Shape Derivatives in Heat Conduction: an Optimization Approach for Enhanced Thermal Performance

Abstract

We analyze an optimization problem of the conductivity in a composite material arising in a heat conduction energy storage problem. The model is described by the heat equation that specifies the heat exchange between two types of materials with different conductive properties with Dirichlet-Neumann boundary conditions on the external part of the domain, and on the interface characterized by the resisting coefficient between the highly conductive material and the less conductive material. The main purpose of the paper is to compute a shape gradient of an optimization functional in order to accurately determine the optimal location of the conductive material using a classical shape optimization strategy. We also present some numerical experiments for the special case of a ball to illustrate the efficiency of the proposed method.

Keywords

Heat conduction / Shape derivatives / Optimization / Numerical simulations / 35Q93 / 35R30 / 49Q10

References

[1] Allaire, G., Dapogny, C., Delgado, G., Michailidis, G.: Multi-phase structural optimization via a level set method. ESAIM: Control. Opt. Calc. Variations 20(2), 576–611 (2014)

[2] Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. C.R. Math. 334(12), 1125–1130 (2002)

[3] Belgacem, F.B., Bernardi, C., Jelassi, F., Mint Brahim, M.: Finite element methods for the temperature in composite media with contact resistance. J. Sci. Comput. 63, 478–501 (2015)

[4] Bendsoe, M., Sigmund, O.: Topology Optimization. Theory, Method and Applications. Springer-Verlag, Berlin, Heidelberg (2003)

[5] Brahim, M.M.: Méthodes d’éléments finis pour le problème de changement de phase en milieux composites. Université de Bordeaux, Thèse de doctorat (2016)

[6] Brezis, H.: Analyse Fonctionnelle. Masson, Paris (1983)

[7] Delgado, G.: Optimization of composite structures: a shape and topology sensitivity analysis. PhD dissertation (2014)

[8] Evans, L.C.: Partial differential equations. In: Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (1998)

[9] Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3/4), 251–265 (2012)

[10] Henrot, A., Pierre, M.: Variation et optimisation de formes: une analyse géométrique. In: Mathématiques & Applications, vol. 48. Springer, Berlin (2005)

[11] Larnier, S., Masmoudi, M.: The extended adjoint method. ESAIM: Mathematical Modeling and Numerical Analysis, 47(1), 83–108 (2013)

[12] Pietrak, K., Wiśniewski, T.S.: A review of models for effective thermal conductivity of composite materials. Journal of Power Technologies 95, 14–24 (2015)

[13] Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for 2-nd order elliptic problems. In: Galligani, I., Magenes, E. (eds) Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, vol. 606, pp. 292–315. Springer, Berlin, Heidelberg (1977)

[14] Sai MKS. Review of composite materials and applications. Int. J. Latest Trends Eng. Technol., 2016, 6(3): 129-135

[15] Sokolowski, J., Zolésio, J.-P.: Introduction to Shape Optimization. Springer-Verlag, Berlin, Heidelberg (1992)

[16] Tartar, L.: The general theory of homogenization. In: Lecture Notes of the Unione Matematica Italiana. Springer-Verlag, Berlin (2009)

[17] Zhou, S., Li, Q.: Computational design of multi-phase microstructural materials for extremal conductivity. Comput. Mater. Sci. 43, 549–564 (2008)

Funding

Spanish National Plan for Scientific and Technical Research and Innovation(PID2020-114976GB-I00, PY20 01125)

Institut de Mathématiques de Bordeaux(UMR 5251)

ANR project(NumOpTes ANR-22-CE46-0005 (2023–2026))

ANR project(PHASEFIELD ANR-16–CE40-0026-01)

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Shanghai University

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