# Accurate multiscale solutions to quasi-stationary thermal problems involving a heterogeneous heat source

Isabelle Ramière, Louis Belgrand, Frédéric Lebon

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Isabelle Ramière, Louis Belgrand, Frédéric Lebon. Accurate multiscale solutions to quasi-stationary thermal problems involving a heterogeneous heat source. WCCM 2024 / PANACM 2024 - the 16th World Congress on Computational Mechanics and 4th Pan American Congress on Computational Mechanics, Jul 2024, Vancouver, Canada. cea-04911643

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Submitted on 24 Jan 2025

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# Accurate multiscale solutions to quasi-stationary thermal problems involving a heterogeneous heat source

Isabelle Ramière¹, Louis Belgrand¹,² and Frédéric Lebon²

¹ CEA, DES, IRESNE, DEC, SESC ; Cadarache, F-13108 Saint-Paul-Lez Durance

² Aix-Marseille Université, CNRS, Centrale Marseille, LMA ; F-13453 Marseille cedex 13

* corresponding author, e-mail: isabelle.ramiere@cea.fr

**Keywords:** Heterogeneous material, Computational Homogenization, Asymptotic expansion, Generalized Hill-Mandel principle, Finite element square

We address the multiscale resolution of a (quasi-)stationary thermal problem with source term on a heterogeneous material. This situation differs from the mechanical computational homogenization framework in that a precise primal solution (temperature) is sought on the Representative Volume Element (RVE) and that a source term (possible heterogeneous at the microscale) is considered. Various Finite Element square (FE²) approaches are reviewed, differing mainly in their treatment of the source on the microstructure. These approaches are analyzed, in particular through a possible parallel with methods based on double-scale asymptotic expansion.

For this problem, the classical FE² method [2] respects the Hill-Mandel principle and does not impose a source on the microstructure. This approach is of order 1. In the case of low or no insulated inclusions, these approaches are proving to be sufficiently precise. In counterpart, for insulating inclusions, it is essential to consider the local influence of the source to obtain accurate solutions on the microstructure [1]. Recently introduced, the generalized Hill-Mandel [3, 4] principle makes it possible to consider the source on the microstructure in a FE² framework. We show that, thanks to this strategy, methods of order greater than 1 can be constructed.

The accuracy of the numerical results obtained by these different approaches is assessed using a direct simulation of the structure discretizing the heterogeneities. This study allows us to conclude that it is essential to consider the complete heat source on the microstructure [4], and not just its fluctuations [3], to obtain a reliable estimate of temperature variations at this scale.

Finally, a quasi-stationary FE² calculation with temporal loading and nonlinear behavior has been successfully carried out. This simulation confirms the importance of considering the generalized Hill-Mandel principle for avoiding underestimating the temperature in the microstructure inclusions.

## REFERENCES

[1] J. F. Bourgat. Numerical experiments of the homogenization method. *Computing Methods in Applied Sciences and Engineering*, 1977.

[2] G. Chatzigeorgiou et al. Periodic homogenization for fully coupled thermomechanical modeling of dissipative generalized standard materials. *International Journal of Plasticity*, 2016.

[3] E. A. de Souza Neto et al. An RVE-based multiscale theory of solids with micro-scale inertia and body force effects. *Mechanics of Materials*, 2015.

[4] G. R. Ramos et al. An extension of the Hill-Mandel principle for transient heat conduction in heterogeneous media with heat generation incorporating finite RVE thermal inertia effects. *International Journal for Numerical Methods in Engineering*, 2017.