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

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

Volume 7: Fluids and Heat Transfer, Parts A, B, C, and D ◽

10.1115/imece2012-89352 ◽

2012 ◽

Author(s):

James M. Loy ◽

Sanjay R. Mathur ◽

Jayathi Y. Murthy

Keyword(s):

Transport Equation ◽

Convergence Rates ◽

Control Volume ◽

Boltzmann Transport Equation ◽

Physical Space ◽

Convergence Acceleration ◽

Solution Procedure ◽

Boltzmann Transport ◽

Knudsen Numbers ◽

Sequential Solution

Sequential 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 3–4 orders of magnitude at room temperature, diffuse-limit solutions are not feasible for all K vectors. Consequently, non-gray solutions of the BTE almost always experience extremely slow convergence. In this paper, we develop a coupled-ordinates method for solving the phonon BTE in the relaxation time approximation. Here, inter-equation 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. The solution procedure is benchmarked against a traditional sequential solution procedure for thermal transport in silicon. Significant acceleration, between 10 to 300 times, over the sequential procedure is found for heat conduction problems.

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A Fast Hybrid Fourier–Boltzmann Transport Equation Solver for Nongray Phonon Transport

Journal of Heat Transfer ◽

10.1115/1.4007654 ◽

2012 ◽

Vol 135 (1) ◽

Cited By ~ 20

Author(s):

James M. Loy ◽

Jayathi Y. Murthy ◽

Dhruv Singh

Keyword(s):

Transport Equation ◽

Thermal Transport ◽

Boltzmann Transport Equation ◽

Iterative Solution ◽

Phonon Transport ◽

Computational Effort ◽

Boltzmann Transport ◽

Knudsen Numbers ◽

Sequential Solution ◽

Hybrid Solver

Nongray phonon transport solvers based on the Boltzmann transport equation (BTE) are being increasingly employed to simulate submicron thermal transport in semiconductors and dielectrics. Typical sequential solution schemes encounter numerical difficulties because of the large spread in scattering rates. For frequency bands with very low Knudsen numbers, strong coupling between other BTE bands result in slow convergence of sequential solution procedures. This is due to the explicit treatment of the scattering kernel. In this paper, we present a hybrid BTE-Fourier model which addresses this issue. By establishing a phonon group cutoff Knc, phonon bands with low Knudsen numbers are solved using a modified Fourier equation which includes a scattering term as well as corrections to account for boundary temperature slip. Phonon bands with high Knudsen numbers are solved using the BTE. A low-memory iterative solution procedure employing a block-coupled solution of the modified Fourier equations and a sequential solution of BTEs is developed. The hybrid solver is shown to produce solutions well within 1% of an all-BTE solver (using Knc = 0.1), but with far less computational effort. Speedup factors between 2 and 200 are obtained for a range of steady-state heat transfer problems. The hybrid solver enables efficient and accurate simulation of thermal transport in semiconductors and dielectrics across the range of length scales from submicron to the macroscale.

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Solving Nongray Boltzmann Transport Equation in Gallium Nitride

Journal of Heat Transfer ◽

10.1115/1.4036616 ◽

2017 ◽

Vol 139 (10) ◽

Cited By ~ 5

Author(s):

Ajit K. Vallabhaneni ◽

Liang Chen ◽

Man P. Gupta ◽

Satish Kumar

Keyword(s):

Gallium Nitride ◽

Transport Equation ◽

Thermal Transport ◽

Hot Spot ◽

Boltzmann Transport Equation ◽

Significant Loss ◽

Computational Time ◽

Boltzmann Transport ◽

Phonon Modes ◽

Fourier Model

Several studies have validated that diffusive Fourier model is inadequate to model thermal transport at submicron length scales. Hence, Boltzmann transport equation (BTE) is being utilized to improve thermal predictions in electronic devices, where ballistic effects dominate. In this work, we investigated the steady-state thermal transport in a gallium nitride (GaN) film using the BTE. The phonon properties of GaN for BTE simulations are calculated from first principles—density functional theory (DFT). Despite parallelization, solving the BTE is quite expensive and requires significant computational resources. Here, we propose two methods to accelerate the process of solving the BTE without significant loss of accuracy in temperature prediction. The first one is to use the Fourier model away from the hot-spot in the device where ballistic effects can be neglected and then couple it with a BTE model for the region close to hot-spot. The second method is to accelerate the BTE model itself by using an adaptive model which is faster to solve as BTE for phonon modes with low Knudsen number is replaced with a Fourier like equation. Both these methods involve choosing a cutoff parameter based on the phonon mean free path (mfp). For a GaN-based device considered in the present work, the first method decreases the computational time by about 70%, whereas the adaptive method reduces it by 60% compared to the case where full BTE is solved across the entire domain. Using both the methods together reduces the overall computational time by more than 85%. The methods proposed here are general and can be used for any material. These approaches are quite valuable for multiscale thermal modeling in solving device level problems at a faster pace without a significant loss of accuracy.

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Prediction of Non-Equilibrium Heat Conduction Using Parallel Computation of the Phonon Boltzmann Transport Equation

Volume 8B: Heat Transfer and Thermal Engineering ◽

10.1115/imece2014-36084 ◽

2014 ◽

Author(s):

Syed A. Ali ◽

Gautham Kollu ◽

Sandip Mazumder ◽

P. Sadayappan

Keyword(s):

Heat Conduction ◽

Transport Equation ◽

Parallel Computation ◽

Large Scale ◽

Phonon Scattering ◽

Boltzmann Transport Equation ◽

Computational Time ◽

Spectral Space ◽

Boltzmann Transport ◽

Non Equilibrium

Non-equilibrium heat conduction, as occurring in modern-day sub-micron semiconductor devices, can be predicted effectively using the Boltzmann Transport Equation (BTE) for phonons. In this article, strategies and algorithms for large-scale parallel computation of the phonon BTE are presented. An unstructured finite volume method for spatial discretization is coupled with the control angle discrete ordinates method for angular discretization. The single-time relaxation approximation is used to treat phonon-phonon scattering. Both dispersion and polarization of the phonons are accounted for. Three different parallelization strategies are explored: (a) band-based, (b) direction-based, and (c) hybrid band/cell-based. Subsequent to validation studies in which silicon thin-film thermal conductivity was successfully predicted, transient simulations of non-equilibrium thermal transport were conducted in a three-dimensional device-like silicon structure, discretized using 604,054 tetrahedral cells. The angular space was discretized using 400 angles, and the spectral space was discretized into 40 spectral intervals (bands). This resulted in ∼9.7×109 unknowns, which are approximately 3 orders of magnitude larger than previously reported computations in this area. Studies showed that direction-based and hybrid band/cell-based parallelization strategies resulted in similar total computational time. However, the parallel efficiency of the hybrid band/cell-based strategy — about 88% — was found to be superior to that of the direction-based strategy, and is recommended as the preferred strategy for even larger scale computations.

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Solution of the Phonon Boltzmann Transport Equation Employing Rigorous Implementation of Phonon Conservation Rules

Heat Transfer, Volume 3 ◽

10.1115/imece2006-14090 ◽

2006 ◽

Cited By ~ 3

Author(s):

Tianjiao Wang ◽

Jayathi Y. Murthy

Keyword(s):

Thin Film ◽

Energy Conservation ◽

Transport Equation ◽

Phonon Dispersion ◽

Boltzmann Transport Equation ◽

Physical Space ◽

Finite Volume Scheme ◽

Boltzmann Transport ◽

Bulk Silicon ◽

Silicon Thin Film

A finite volume scheme is developed to solve the phonon Boltzmann transport equation in an energy form accounting for phonon dispersion and polarization. The physical space and the first Brillouin zone are discretized into finite volumes and the phonon BTE is integrated over them. Second-order accurate differencing schemes are used for the discretization. The scattering term employs a rigorous implementation of phonon momentum and energy conservation laws in determining the rate of normal and Umklapp processes. The method is applied to a variety of bulk silicon and silicon thin-film conduction problems and shown to perform satisfactorily.

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