scholarly journals On Variance-Reduced Simulations of the Boltzmann Transport Equation for Small-Scale Heat Transfer Applications

2010 ◽  
Vol 132 (11) ◽  
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 Simulation ◽  
Particle Methods ◽  
Small Scale ◽  
Boltzmann Transport ◽  
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 with 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.

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.

Phonon Transport Modeling Using Boltzmann Transport Equation With Anisotropic Relaxation Times

2012 ◽  
Vol 134 (8) ◽  
Author(s):  
Chunjian Ni ◽  
Jayathi Y. Murthy
Keyword(s):  
Relaxation Time ◽  
Transport Equation ◽  
Thermal Transport ◽  
Relaxation Times ◽  
Boltzmann Transport Equation ◽  
Phonon Transport ◽  
Oxide Semiconductor ◽  
Boltzmann Transport ◽  
Scattering Model ◽  
Anisotropic Relaxation

A sub-micron thermal transport model based on the phonon Boltzmann transport equation (BTE) is developed using anisotropic relaxation times. A previously-published model, the full-scattering model, developed by Wang, directly computes three-phonon scattering interactions by enforcing energy and momentum conservation. However, it is computationally very expensive because it requires the evaluation of millions of scattering interactions during the iterative numerical solution procedure. The anisotropic relaxation time model employs a single-mode relaxation time, but the relaxation time is derived from detailed consideration of three-phonon interactions satisfying conservation rules, and is a function of wave vector. The resulting model is significantly less expensive than the full-scattering model, but incorporates directional and dispersion behavior. A critical issue in the model development is the role of three-phonon normal (N) scattering processes. Following Callaway, the overall relaxation rate is modified to include the shift in the phonon distribution function due to N processes. The relaxation times so obtained are compared with the data extracted from equilibrium molecular dynamics simulations by Henry and Chen. The anisotropic relaxation time phonon BTE model is validated by comparing the predicted thermal conductivities of bulk silicon and silicon thin films with experimental measurements. The model is then used for simulating thermal transport in a silicon metal-oxide-semiconductor field effect transistor (MOSFET) and leads to results close to the full-scattering model, but uses much less computation time.

DSMC for Phonon Transport in Solid Thin Films

2009 ◽  
Author(s):  
Mitsuhiro Matsumoto ◽  
Masaya Okano
Keyword(s):  
Heat Transfer ◽  
Thin Films ◽  
Relaxation Time ◽  
Transport Equation ◽  
Boltzmann Transport Equation ◽  
Thermal Design ◽  
Dynamics Simulation ◽  
Boltzmann Transport ◽  
Phonon Dynamics ◽  
Solid Thin Films

As the scale of electronic devices decreases, heat transfer analysis and thermal design becomes more important. In particular, heat transfer through various solid thin films is strongly affected by thickness dependence of thermal conductivity and interfacial thermal resistance. Analysis of phonon dynamics based on a linearized Boltzmann transport equation, or the so-called relaxation time approximation, has been widely used, but detailed analysis using molecular dynamics simulation reveals that couplings among various phonon modes can affect the energy transfer. In this study, we propose a DSMC scheme to simulate phonon dynamics starting from the original Boltzmann transport equation. In contrast to the linearized model, this scheme requires no relaxation time as an input parameter, and we can investigate the couplings among phonons with different modes, although we have to assume some appropriate model of phonon-phonon collisions. As a test calculation, energy flux was evaluated for model thin films of various thicknesses, and a phenomenon similar to the Casimir limit was retrieved. This scheme will enable us to include other factors, such as phonon-electron couplings.

Improved Phonon Transport Modeling Using Boltzmann Transport Equation With Anisotropic Relaxation Times

2009 ◽  
Author(s):  
Chunjian Ni ◽  
Jayathi Y. Murthy
Keyword(s):  
Relaxation Time ◽  
Transport Equation ◽  
Relaxation Times ◽  
Boltzmann Transport Equation ◽  
Phonon Transport ◽  
Dynamics Simulation ◽  
Boltzmann Transport ◽  
Scattering Model ◽  
Anisotropic Relaxation ◽  
Thermal Conductivities

A sub-micron thermal transport model based on the phonon Boltzmann transport equation (BTE) is developed using anisotropic relaxation times. A previously-published model, the full-scattering model, developed by Wang, directly computes three-phonon scattering interactions by enforcing energy and momentum conservation. However, it is computationally very expensive because it requires the evaluation of millions of scattering interactions during the iterative numerical solution procedure. The anisotropic relaxation time phonon BTE model employs a single-mode relaxation time idea, but the relaxation time is a function of wave-vector. The resulting model is significantly less expensive than the full-scattering model, but incorporates directional and dispersion behavior as well as relaxation times satisfying conservation rules. A critical issue in the model development is the accounting for the role of three-phonon N scattering processes. Direct inclusion of N processes into the anisotropic relaxation time model is not possible because such an inclusion would engender thermal resistance. Following Callaway, the overall relaxation rate is modified to include the shift in the phonon distribution function due to N processes. The relaxation times so obtained are compared with the data extracted from equilibrium molecular dynamics simulation by Henry and Chen. The anisotropic relaxation time phonon BTE model is validated by comparing the predicted bulk thermal conductivities of silicon and silicon thin-film thermal conductivities with experimental measurements.

scholarly journals On efficient simulations of multiscale kinetic transport

2013 ◽  
Vol 371 (1982) ◽  
pp. 20120182 ◽  
Author(s):  
Gregg A. Radtke ◽  
Jean-Philippe M. Péraud ◽  
Nicolas G. Hadjiconstantinou
Keyword(s):  
Variance Reduction ◽  
Evolution Equations ◽  
Computational Cost ◽  
Practical Interest ◽  
Simulation Method ◽  
Transport Processes ◽  
Particle Simulation ◽  
Small Scale ◽  
Control Variate ◽  
Equilibrium Distributions

We discuss a new class of approaches for simulating multiscale kinetic problems, with particular emphasis on applications related to small-scale transport. These approaches are based on a decomposition of the kinetic description into an equilibrium part, which is described deterministically (analytically or numerically), and the remainder, which is described using a particle simulation method. We show that it is possible to derive evolution equations for the two parts from the governing kinetic equation, leading to a decomposition that is dynamically and automatically adaptive, and a multiscale method that seamlessly bridges the two descriptions without introducing any approximation . Our discussion pays particular attention to stochastic particle simulation methods that are typically used to simulate kinetic phenomena; in this context, these decomposition approaches can be thought of as control-variate variance-reduction formulations, with the nearby equilibrium serving as the control. Such formulations can provide substantial computational benefits in a broad spectrum of applications because a number of transport processes and phenomena of practical interest correspond to perturbations from nearby equilibrium distributions. In many cases, the computational cost reduction is sufficiently large to enable otherwise intractable simulations.

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