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

Hadjiconstantinou, Nicolas G.; Radtke, Gregg Arthur; Baker, Lowell L.

doi:10.1115/1.4002028

Cited by 35 publications

(33 citation statements)

References 27 publications

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“…However, as stated above, for calculating the thermal conductivity of nanostructures, unless symmetry allows, deviational particles need to be tracked well beyond their first free-path termination. This is also true for solving the Boltzmann equation under general conditions, since only then the correct non-equilibrium distribution of deviational carriers is obtained 2,12,14,15 . We conclude by emphasizing that the only approximation introduced in this work comes from the assumption that the governing BTE can be linearized.…”

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confidence: 99%

An alternative approach to efficient simulation of micro/nanoscale phonon transport

Péraud¹,

Hadjiconstantinou²

2012

Applied Physics Letters

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139

163

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Starting from the recently proposed energy-based deviational formulation for solving the Boltzmann equation [J.-P. Péraud and N. G. Hadjiconstantinou, Phys. Rev. B 84, 2011], which provides significant computational speedup compared to standard Monte Carlo methods for small deviations from equilibrium, we show that additional computational benefits are possible in the limit that the governing equation can be linearized. The proposed method exploits the observation that under linearized conditions (small temperature differences) the trajectories of individual deviational particles can be decoupled and thus simulated independently; this leads to a particularly simple and efficient algorithm for simulating steady and transient problems in arbitrary three-dimensional geometries, without introducing any additional approximation.In a previous paper 1 , we presented a low variance Monte Carlo method for solving the Boltzmann transport equation (BTE) for phonons in the relaxation-time approximation whereby computational particles simulate only the deviation from an equilibrium distribution. The benefits of such control-variate formulations 2 , which we will refer to as deviational, are twofold: first, in the limit of small temperature differences, deviational methods exhibit substantial computational speedup compared to traditional Monte Carlo methods 1 ; this speedup increases quadratically as the characteristic temperature difference goes to zero. Second, by simulating only the deviation from equilibrium, deviational methods seamlessly and automatically focus the computational effort on regions where it is needed and can thus be used for solving otherwise intractable multiscale problems. In the present article, we show that for problems exhibiting sufficiently small temperature differences such that the BTE can be linearized, deviational computational particles may be treated independently, thus lending themselves to a simulation algorithm that is simpler, does not use any approximation in space or time, and, depending on the application of interest, can be several orders of magnitude faster than the one presented in Ref. 1.The deviational approach can be introduced by writing the governing equation (with no approximation) in the formwhereis the relaxation time (ω, p and T respectively referring to the angular frequency, the polarization and the temperature), f = f (x, ω, p, θ, φ) is the occupation number of phonon states, V g is the phonon-bundle group velocity and f−1 is a Bose-Einstein distribution at the "control" temperature T eq (k b denotes Boltzmann's constant). Since this distribution does not depend on (T loc − T eq ) once normalized, a particle undergoing a scattering event can be drawn from (the normalized form of) (2) without knowledge of T loc ; energy conservation is simply ensured by conserving the particle. Although this formulation was originally introduced 1 as a means of truncating the discretization of a semi-infinite simulation domain (by limiting the region where computational cells were used), ...

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confidence: 99%

An alternative approach to efficient simulation of micro/nanoscale phonon transport

Péraud¹,

Hadjiconstantinou²

2012

Applied Physics Letters

Self Cite

139

163

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“…The BTE has been solved using various numerical techniques in recent years, including with discrete ordinates 2,16 , Monte Carlo techniques [17][18][19] , and a finite volume approach. 20 However, many studies used frequency averaged properties.…”

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Quasiballistic heat transfer studied using the frequency-dependent Boltzmann transport equation

et al. 2011

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Quasiballistic heat transfer occurs when there is a temperature gradient over length scales comparable to phonon MFPs. This regime has been of interest recently because observing quasiballistic transport can lead to useful information about phonon mean free paths (MFPs), knowledge of which is essential for engineering nanoscale thermal effects. Here, we use the Boltzmann transport equation (BTE) to understand how observations of quasiballistic transport can yield information about MFPs. We solve the transient, one-dimensional, frequency-dependent BTE for a double layer structure of a metal film on a substrate, the same geometry that is used in transient thermoreflectance experiments, using a frequency-dependent interface condition. Our results indicate that phonons with MFPs longer than the thermal penetration depth do not contribute to the measured thermal conductivity, providing a means to probe the MFP distribution. We discuss discrepancies between our simulation and experimental observations which offer opportunities for future investigation.

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“…However, we note that the assumption of small deviation from equilibrium is not required for the LVDSMC methodology which is valid for arbitrary deviations from equilibrium (e.g see Ref. [13]); simulations of the nonlinear operator (2) will be considered in the future.…”

Section: Simulation Methodsmentioning

confidence: 99%

Simulation of Heat Transport in Graphene Nanoribbons Using the Ab-Initio Scattering Operator

Landon

Hadjiconstantinou

2014

Volume 8A: Heat Transfer and Thermal Engineering

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We present a deviational Monte Carlo method for simulating phonon transport in graphene using the ab initio 3-phonon scattering operator. This operator replaces the commonly used relaxation-time approximation, which is known to neglect, among other things, coupling between out of equilibrium states that are particularly important in graphene. Phonon dispersion relations and transition rates are obtained from density functional theory calculations. The proposed method provides, for the first time, means for obtaining solutions of the Boltzmann transport equation with ab initio scattering for time-and spatially-dependent problems. The deviational formulation ensures that simulations are computationally feasible for arbitrarily small temperature differences; within this formulation, the ab initio scattering operator is treated using an efficient stochastic algorithm which, in the limit of large number of states, outperforms the more traditional deterministic methods used in solutions of the homogeneous Boltzmann equation. We use the proposed method to study heat transport in graphene ribbons. INTRODUCTIONTransport in graphene-based devices with length scales in the 10nm-10µm range is expected to be an area of considerable interest for some time, both because of the difficulty associated with manufacturing large area graphene sheets, but also due to the interest associated with the development of small scale electronic devices. Due to the relatively long phonon mean free path, Fourier-based descriptions cannot be used [1] to describe

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

Abstract: We present and discuss a variance-reduced stochastic particle

Cited by 35 publications

References 27 publications

An alternative approach to efficient simulation of micro/nanoscale phonon transport

An alternative approach to efficient simulation of micro/nanoscale phonon transport

Quasiballistic heat transfer studied using the frequency-dependent Boltzmann transport equation

Simulation of Heat Transport in Graphene Nanoribbons Using the Ab-Initio Scattering Operator

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