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Research Papers: Micro/Nanoscale Heat Transfer

A Coupled Ordinates Method for Convergence Acceleration of the Phonon Boltzmann Transport Equation

[+] Author and Article Information
James M. Loy, Sanjay R. Mathur

Department of Mechanical Engineering,
The University of Texas at Austin,
Austin, TX 78712-0209

Jayathi Y. Murthy

Department of Mechanical Engineering,
The University of Texas at Austin,
Austin, TX 78712-0209
e-mail: jmurthy@me.utexas.edu

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 18, 2013; final manuscript received August 12, 2014; published online November 11, 2014. Assoc. Editor: Zhuomin Zhang.

J. Heat Transfer 137(1), 012402 (Jan 01, 2015) (10 pages) Paper No: HT-13-1026; doi: 10.1115/1.4028806 History: Received January 18, 2013; Revised August 12, 2014; Online November 11, 2014

Sequential numerical solution methods are commonly used for solving the phonon Boltzmann transport equation (BTE) because of simplicity of implementation and low storage requirements. However, they exhibit poor convergence for low Knudsen numbers. This is because sequential solution procedures couple the phonon BTEs in physical space efficiently but the coupling is inefficient in wave vector (K) space. As the Knudsen number decreases, coupling in K space becomes dominant and convergence rates fall. Since materials like silicon have K-resolved Knudsen numbers that span two to five orders of magnitude at room temperature, diffuse-limit solutions are not feasible for all K vectors. Consequently, nongray solutions of the BTE experience extremely slow convergence. In this paper, we develop a coupled-ordinates method for numerically solving the phonon BTE in the relaxation time approximation. Here, interequation coupling is treated implicitly through a point-coupled direct solution of the K-resolved BTEs at each control volume. This implicit solution is used as a relaxation sweep in a geometric multigrid method which promotes coupling in physical space. The solution procedure is benchmarked against a traditional sequential solution procedure for thermal transport in silicon. Significant acceleration in computational time, between 10 and 300 times, over the sequential procedure is found for heat conduction problems.

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Figures

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Fig. 1

Discretized control volume in physical space

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Fig. 2

Schematic of a control volume in wave vector space. The Brillouin zone shown here is for a face centered cubic lattice, adapted from Ref. [15].

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Fig. 3

Flow chart for sequential solution procedure

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Fig. 4

Flow chart for one relaxation sweep for COMET

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Fig. 5

Convergence rates for the angular and spatial discretization. On the abscissa, N refers to Nx for the spatial mesh, and Nθ for the angular mesh. The error is defined as the average deviation from Ref. [42].

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Fig. 6

Computational domain used for benchmarking COMET

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Fig. 7

Ratio of the iteration count for the sequential and COMET procedures. The tabulation below the graph shows the values.

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Fig. 8

Ratio of the total time taken for sequential and COMET procedures. The tabulation below the graph shows the values.

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Fig. 9

Dispersion relation for silicon in the [100] direction at 300 K using EDIP [45]

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Fig. 10

Discretization of the Brillouin zone for a nongray dispersion relation

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Fig. 11

Phonon mean free path as a function of polarization and frequency for bulk silicon

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Fig. 12

Ratio of the iteration count for the sequential and COMET procedures. The spatial and angular discretizations used are Nx × Ny = 50 × 50 and Nθ × Nϕ = 2 × 2 in the octant, respectively.

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Fig. 13

Ratio of the total time for the sequential and COMET procedures. The spatial and angular discretizations used are Nx × Ny = 50 × 50 and Nθ × Nϕ = 2 × 2 in the octant, respectively.

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