scholarly journals Comparison of approximate solutions to the phonon Boltzmann transport equation with the relaxation time approximation: Spherical harmonics expansions and the discrete ordinates method

2018 ◽  
Vol 123 (17) ◽  
pp. 174304 ◽  
Author(s):  
J. G. Christenson ◽  
R. A. Austin ◽  
R. J. Phillips
Keyword(s):  
Relaxation Time ◽  
Transport Equation ◽  
Spherical Harmonics ◽  
Boltzmann Transport Equation ◽  
Approximate Solutions ◽  
Discrete Ordinates ◽  
Boltzmann Transport ◽  
Discrete Ordinates Method ◽  
Time Approximation ◽  
Relaxation Time Approximation

On Efficient Particle Methods for Solving the Boltzmann Transport Equation in the Relaxation-Time Approximation

2009 ◽  
Author(s):  
Nicolas G. Hadjiconstantinou ◽  
Gregg A. Radtke ◽  
Lowell L. Baker
Keyword(s):  
Relaxation Time ◽  
Transport Equation ◽  
Variance Reduction ◽  
Boltzmann Transport Equation ◽  
Computational Cost ◽  
Particle Methods ◽  
Boltzmann Transport ◽  
Time Approximation ◽  
Relaxation Time Approximation ◽  
Stochastic Particle

We present and discuss a variance-reduced stochastic particle simulation method for solving the relaxation-time model of the Boltzmann transport equation. The variance reduction, achieved by simulating only the deviation from equilibrium, results in a significant computational efficiency advantage compared to traditional stochastic particle methods in the limit of small deviation from equilibrium. More specifically, the proposed method can efficiently simulate arbitrarily small deviations from equilibrium at a computational cost that is independent of the deviation from equilibrium, which is in sharp contrast to traditional particle methods. The proposed method is developed and validated in the context of dilute gases; despite this, it is expected to directly extend to all fields (carriers) for which the relaxation-time approximation is applicable.

A Hybrid SN-PN Formulation for Solution of the Boltzmann Transport Equation for Phonons

2011 ◽  
Author(s):  
Arpit Mittal ◽  
Sandip Mazumder
Keyword(s):  
Transport Equation ◽  
Large Scale ◽  
Heat Conduction Problem ◽  
A Priori ◽  
Boltzmann Transport Equation ◽  
Transient Heat Conduction ◽  
Discrete Ordinates ◽  
Boltzmann Transport ◽  
Discrete Ordinates Method ◽  
Transient Heat Conduction Problem

A generalized form of the Ballistic-Diffusive Equations (BDE) for approximate solution of the Boltzmann Transport Equation (BTE) for phonons is formulated. The formulation presented here is new and general in the sense that, unlike previously published formulations of the BDE, it does not require a priori knowledge of the specific heat capacity of the material. Furthermore, it does not introduce artifacts such as media and ballistic temperatures. As a consequence, the boundary conditions have clear physical meaning. In formulating the BDE, the phonon intensity is split into two components: ballistic and diffusive. The ballistic component is traditionally determined using a viewfactor formulation, while the diffusive component is solved by invoking spherical harmonics expansions. Use of the viewfactor approach for the ballistic component is prohibitive for complex large-scale geometries. Instead, in this work, the ballistic equation is solved using two different established methods that are appropriate for use in complex geometries, namely the discrete ordinates method (DOM), and the control angle discrete ordinates method (CADOM). Results of each method for solving the BDE are compared against benchmark Monte Carlo results, as well as solutions of the BTE using standalone DOM and CADOM for a two-dimensional transient heat conduction problem at various Knudsen numbers. It is found that standalone CADOM (for BTE) and hybrid CADOM-P1 (for BDE) yield the best accuracy. The hybrid CADOM-P1 is found to be the best method in terms of computational efficiency.

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