Variance‐reduced Monte Carlo solutions of the Boltzmann equation for low‐speed gas flows: A discontinuous Galerkin formulation

Baker, Lowell L.; Hadjiconstantinou, Nicolas G.

doi:10.1002/fld.1724

Cited by 24 publications

(22 citation statements)

References 30 publications

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“…It originates from the LVDSMC class of methods 30,31,[48][49][50][51][52] for simulating the Boltzmann transport equation. LVDSMC methods achieve computational efficiency 26,27,53 by simulating only the deviation from a suitably chosen equilibrium distribution. For phonon systems, this is given by n…”

Section: Introductionmentioning

confidence: 99%

Deviational simulation of phonon transport in graphene ribbons with ab initio scattering

Landon

Hadjiconstantinou

2014

Journal of Applied Physics

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We present a deviational Monte Carlo method for solving the Boltzmann-Peierls equation with ab initio 3-phonon scattering, for temporally and spatially dependent thermal transport problems in arbitrary geometries. Phonon dispersion relations and transition rates for graphene are obtained from density functional theory calculations. The ab initio scattering operator is simulated by an energy-conserving stochastic algorithm embedded within a deviational, low-variance Monte Carlo formulation. The deviational formulation ensures that simulations are computationally feasible for arbitrarily small temperature differences, while the stochastic treatment of the scattering operator is both efficient and exhibits no timestep error. The proposed method, in which geometry and phonon-boundary scattering are explicitly treated, is extensively validated by comparison to analytical results, previous numerical solutions and experiments. It is subsequently used to generate solutions for heat transport in graphene ribbons of various geometries and evaluate the validity of some common approximations found in the literature. Our results show that modeling transport in long ribbons of finite width using the homogeneous Boltzmann equation and approximating phonon-boundary scattering using an additional homogeneous scattering rate introduces an error on the order of 10% at room temperature, with the maximum deviation reaching 30% in the middle of the transition regime.

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Section: Introductionmentioning

confidence: 99%

Deviational simulation of phonon transport in graphene ribbons with ab initio scattering

Landon

Hadjiconstantinou

2014

Journal of Applied Physics

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“…Recently, the lattice Boltzmann method (LBM) has experienced remarkable advances and has been developed into a widely used method for simulation of various fluid dynamics phenomena such as, open channel flows [14], low speed gas flows [15], shallow water flows [16], nanofluids [17], solidification [18,19], and so on [20,21]. Unlike conventional CFD methods based on the discretization of macroscopic continuum equations, the main goal of LBM is based on the microscopic models and mesoscopic kinetic equations in which the combined behavior of the particles is applied to simulate the physical mechanism of the systems.…”

Section: Introductionmentioning

confidence: 99%

Simulation of natural convection melting in a cavity with fin using lattice Boltzmann method

Jourabian

Farhadi

Sedighi

et al. 2011

Numerical Methods in Fluids

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SUMMARY In this study, a numerical investigation of melting phenomenon with natural convection in a cavity with fin has been performed using enthalpy‐based lattice Boltzmann method. The lattice D2Q9 model was applied to determine the density and velocity fields, and the D2Q5 model for the temperature field. The effect of vertical position and length of the fin on the melting rate was studied. The simulations were carried out for Stefan number of 10, Rayleigh number of 10 5 and relative thermal conductivity (kfin∕kfluid) ranging from 5 to 30. The obtained results show that the rate of melting increases when the relative thermal conductivity and the length of the fin become greater. We also found that the variation of vertical position of the fin from bottom to middle has an insignificant effect on melting while it causes the increase of full melting time when the fin is mounted on the top of the cavity. Copyright © 2011 John Wiley & Sons, Ltd.

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“…As a result of this limitation, variance reduction approaches have recently received considerable attention: Baker and Hadjiconstantinou [4] recently showed that solving for the deviation from equilibrium drastically reduces statistical uncertainty and thus enables the simulation of arbitrarily small deviations from equilibrium. The same authors also showed that variance-reduced formulations can be developed for both particle methods [5] and PDE-type approaches [4,6]; a particle method that is equivalent to the one detailed in [5] has also been proposed by Chun and Koch [11]. Unfortunately, in these particle methods, particle cancellation in the collision dominated regime is required for stability [5,11]; this adversely affects both accuracy and efficiency.…”

Section: Introductionmentioning

confidence: 99%

Low-variance direct Monte Carlo simulations using importance weights

Al-Mohssen¹,

Hadjiconstantinou²

2010

ESAIM: M2AN

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Abstract. We present an efficient approach for reducing the statistical uncertainty associated with direct Monte Carlo simulations of the Boltzmann equation. As with previous variance-reduction approaches, the resulting relative statistical uncertainty in hydrodynamic quantities (statistical uncertainty normalized by the characteristic value of quantity of interest) is small and independent of the magnitude of the deviation from equilibrium, making the simulation of arbitrarily small deviations from equilibrium possible. In contrast to previous variance-reduction methods, the method presented here is able to substantially reduce variance with very little modification to the standard DSMC algorithm. This is achieved by introducing an auxiliary equilibrium simulation which, via an importance weight formulation, uses the same particle data as the non-equilibrium (DSMC) calculation; subtracting the equilibrium from the non-equilibrium hydrodynamic fields drastically reduces the statistical uncertainty of the latter because the two fields are correlated. The resulting formulation is simple to code and provides considerable computational savings for a wide range of problems of practical interest. It is validated by comparing our results with DSMC solutions for steady and unsteady, isothermal and non-isothermal problems; in all cases very good agreement between the two methods is found.Mathematics Subject Classification. 60H30, 76P05.

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Variance‐reduced Monte Carlo solutions of the Boltzmann equation for low‐speed gas flows: A discontinuous Galerkin formulation

Cited by 24 publications

References 30 publications

Deviational simulation of phonon transport in graphene ribbons with ab initio scattering

Deviational simulation of phonon transport in graphene ribbons with ab initio scattering

Simulation of natural convection melting in a cavity with fin using lattice Boltzmann method

Low-variance direct Monte Carlo simulations using importance weights

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