Coupling Mesoscopic Boltzmann Transport Equation and Macroscopic Heat Diffusion Equation for Multiscale Phonon Heat Conduction

ABSTRACT

Phonon heat conduction has to be described by the Boltzmann transport equation (BTE) when sizes or sources are comparable to or smaller than the phonon mean free paths (MFPs). When domains much larger than MFPs are to be treated or when regions with large and small MFPs coexist, the computation time associated with full BTE treatment becomes large, calling for a multiscale strategy to describe the total domain and decreasing the computation time. Here, we describe an iterative method to couple the BTE, under the Equation of Phonon Radiative Transfer approximation solved by means of the deterministic Discrete Ordinate Method, to a Finite-Element Modeling commercial solver of the heat equation. Small-size elements are embedded in domains where the BTE is solved, and the BTE domains are connected to a domain where large-size elements are located and where the heat equation is applied. It is found that an overlapping zone between the two types of domains is required for convergence, and the accuracy is analyzed as a function of the size of the BTE domain. Conditions for fast convergence are discussed, leading to the computation time being divided by more than five on a study case in 2D Cartesian geometry. The simple method could be generalized to other types of solvers of the Boltzmann and heat equations.

KEYWORDS:

• Multiscale
• phonon heat conduction
• Boltzmann equation
• discrete ordinate method

Acknowledgments

Early discussions with T. Nghiem and J. Randrianalisoa is acknowledged. We thank E. Nefzaoui and D. Lemmonier for discussions and providing numerical codes. We thank also anonymous referees for useful suggestions.

Additional information

Funding

This work was supported by the EU FP7 [QuantiHeat (FP7-NMP-2013-LARGE-7)]; EU H2020 [EFINED (FETOPEN-1-2016-2017/FETOPEN-01-2016-2017)]; ANR [TIPTOP (ANR-16-CE09-0023)].