A comparative study of no-time-counter and majorant collision frequency numerical schemes in DSMC
27 Nov 2012-Vol. 1501, Iss: 1, pp 489-495
TL;DR: In this article, the authors compare and contrast two O(N) collision schemes -no-time-counter (NTC) and majorant collision frequency (MCF) -with the goal of identifying the fundamental differences.
Abstract: The direct simulation Monte Carlo (DSMC) method is a stochastic approach to solve the Boltzmann equation and is built on various numerical schemes for transport, collision and sampling. This work aims to compare and contrast two popular O(N) DSMC collision schemes - no-time-counter (NTC) and majorant collision frequency (MCF) - with the goal of identifying the fundamental differences. MCF and NTC schemes are used in DSMC simulations of a spatially homogeneous equilibrium gas to study convergence with respect to various collision parameters. While the MCF scheme forces the reproduction of the exponential distribution of time between collisions, the NTC scheme requires larger number of simulators per cell to reproduce this Poisson process. The two collision schemes are also applied to the spatially homogeneous relaxation from an isotropic non-Maxwellian given by the Bobylev exact solution to the Boltzmann equation. While the two schemes produce identical results at large times, the initial relaxation shows some differences during the first few timesteps.
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TL;DR: In this paper, the basic ideas underlying the Direct Simulation Monte Carlo (DSMC) method are examined and a novel nonhomogeneous N-particle kinetic equation describing the randomized mathematical model of DSMC is derived.
Abstract: In this paper, the basic ideas underlying the Direct Simulation Monte Carlo (DSMC) method are examined and a novel nonhomogeneous N-particle kinetic equation describing the randomized mathematical model of DSMC is derived. It is shown that different collision-partner selection schemes, including No-Time-Counter (NTC) and Bernoulli-trials schemes, are approximations of the general transition operator of the randomized model. The popular collision-partner selection schemes, represented by the standard NTC and Bernoulli-trials approximations of the general transition operator, represented by Simplified Bernoulli-trials and Generalized Bernoulli-trials schemes, are tested on the one-dimensional rarefied gas heat transfer problem against conditions of two approximation limits: first, leading to the Boltzmann equation and, second, leading to the novel N-particle kinetic one.In this paper, the basic ideas underlying the Direct Simulation Monte Carlo (DSMC) method are examined and a novel nonhomogeneous N-particle kinetic equation describing the randomized mathematical model of DSMC is derived. It is shown that different collision-partner selection schemes, including No-Time-Counter (NTC) and Bernoulli-trials schemes, are approximations of the general transition operator of the randomized model. The popular collision-partner selection schemes, represented by the standard NTC and Bernoulli-trials approximations of the general transition operator, represented by Simplified Bernoulli-trials and Generalized Bernoulli-trials schemes, are tested on the one-dimensional rarefied gas heat transfer problem against conditions of two approximation limits: first, leading to the Boltzmann equation and, second, leading to the novel N-particle kinetic one.
19 citations
TL;DR: Su et al. as discussed by the authors simulated the near nucleus coma of Comet 9P/Tempel 1 with the 3D Direct Simulation Monte Carlo (DSMC) code PDSC++.
17 citations
Cites methods from "A comparative study of no-time-coun..."
...To the best knowledge of the authors, Ivanov’s group has not used this idea to improve the collision quality; instead, this group used their own so-called Majorant Collision Frequency Scheme (Venkattraman et al., 2012) to improve the collision quality....
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TL;DR: In this paper, the authors compared the results of the SBT and GBT collision schemes in treating the higher-order moments of the velocity distribution function and comparison with theory and the solution of the standard No-Time-Counter (NTC) method and its new variant, nearest neighbor scheme, using the DSMC0F program.
Abstract: The impetus of this paper is to assess the newly suggested direct simulation Monte Carlo (DSMC) collision schemes, that is, the “Simplified Bernoulli Trails (SBT)” and “Generalized Bernoulli Trials (GBT)” schemes in the prediction of the higher-order moments of the velocity distribution function for both confined and non-confined gas flows. Two fundamental rarefied gas dynamics problems are considered: spatially homogeneous relaxation process of a gas flow from a non-Maxwellian condition given by Bobylev–Krook–Wu exact (analytical) solution of the Boltzmann equation and the stationary shock wave problem. To perform the relaxation test, SBT and GBT schemes were implemented in the DSMC0F program. For the shock wave test, changes were made in the DSMC1 code to include the SBT and GBT schemes. A detailed comparison of the SBT and GBT collision schemes in treating the higher-order moments of the velocity distribution function and comparison with theory and the solution of the standard No-Time-Counter (NTC) method and its new variant, nearest neighbor scheme, using the DS1 code, is reported. Some higher moments beyond the usual moments were computed. The results of the fourth moment of the velocity distribution function in the homogeneous relaxation problem show that while both collision schemes produce identical results at an ample time, the initial relaxation process indicates the difference between the schemes. Even though the NTC schemes required a large number of particles per cell to produce the same results as the theory, the SBT scheme successfully simulates the solution using a low number of particles per cell.
13 citations
10 Jul 2017
TL;DR: In this article, a physikalisch konsistente modellierung des Transports zerstaubten metalls durch eine inerte Gasphase mittels kinetischer Ansatze is presented.
Abstract: Oberflachenprozessierung mittels Zerstaubungsplasmen findet breite Anwendung in der Abscheidung dunner Schichten. Vereinfacht dargestellt werden Atome aus einem Festkorperprakursor herausgeschlagen und anschliesend durch die Prozesskammer transportiert, bis sie auf eine Wand treffen und abgeschieden werden. Fur eine prazise und verlassliche Vorhersage der intrinsischen, physikalischen Vorgange, ist die Kenntnis uber Dichten und Flusse aller relevanten Teilchenspezies unerlasslich. Eine theoretische Vorhersage dieser Kenngrosen ist unmittelbar mit einer konsistenten Beschreibung des Teilchentransports verknupft. Letztere gibt das Thema dieser Arbeit vor: Vorgestellt wird eine physikalisch konsistente Modellierung des Transports zerstaubten Metalls durch eine inerte Gasphase mittels kinetischer Ansatze. Die Monte Carlo Methode wird als ausgewahlter statistischer Simulationsansatz vorgestellt und zur Beschreibung des Teilchentransports innerhalb diverser Zerstaubungsprozesse herangezogen.
11 citations
Abstract: A fundamental and yet computationally feasible parameter based on the characteristic function of the velocity distribution function (VDF) is proposed for determining the deviation from near-equilibrium conditions in rarefied flow simulations using the direct simulation Monte Carlo (DSMC) method. The proposed parameter utilizes the one-to-one correspondence between the VDF and its characteristic function (or Fourier transform), thereby correlating the deviation of the VDF (from a Chapman-Enskog VDF) with the deviation of the characteristic function (also from that of a Chapman-Enskog VDF). The results are first presented for an unsteady Bobylev solution for approach to equilibrium in 0-D, free-molecular Fourier-Couette flow problem and the Mott-Smith solution for the shock wave all of which have analytical solutions for the VDF, thereby confirming that the proposed parameter indeed captures the deviation from near-equilibrium conditions accurately. The utility of the proposed parameter is then demonstrated using two benchmark problems—Couette flow (over a range of Knudsen numbers) and structure of a normal shock (for upstream Mach numbers of 1.5, 3, and 5)—solved using the DSMC method. While the current work only presents results for benchmark one-dimensional DSMC simulations, the approach can be extended easily to rarefied flows in higher dimensions. Therefore, the proposed parameter has the potential to be used for understanding the nature of VDF and its deviation from near-equilibrium conditions at all locations in a flow field without the need for explicitly sampling the VDF.A fundamental and yet computationally feasible parameter based on the characteristic function of the velocity distribution function (VDF) is proposed for determining the deviation from near-equilibrium conditions in rarefied flow simulations using the direct simulation Monte Carlo (DSMC) method. The proposed parameter utilizes the one-to-one correspondence between the VDF and its characteristic function (or Fourier transform), thereby correlating the deviation of the VDF (from a Chapman-Enskog VDF) with the deviation of the characteristic function (also from that of a Chapman-Enskog VDF). The results are first presented for an unsteady Bobylev solution for approach to equilibrium in 0-D, free-molecular Fourier-Couette flow problem and the Mott-Smith solution for the shock wave all of which have analytical solutions for the VDF, thereby confirming that the proposed parameter indeed captures the deviation from near-equilibrium conditions accurately. The utility of the proposed parameter is then demonstrated...
7 citations
References
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40,330 citations
"A comparative study of no-time-coun..." refers background in this paper
...The majorant collision frequency (MCF) scheme [7, 8, 9] computes the majorant frequency (νmax) as...
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Book•
16 Jun 1994
TL;DR: The direct simulation Monte Carlo (or DSMC) method has, in recent years, become widely used in engineering and scientific studies of gas flows that involve low densities or very small physical dimensions as mentioned in this paper.
Abstract: The direct simulation Monte Carlo (or DSMC) method has, in recent years, become widely used in engineering and scientific studies of gas flows that involve low densities or very small physical dimensions. This method is a direct physical simulation of the motion of representative molecules, rather than a numerical solution of the equations that provide a mathematical model of the flow. These computations are no longer expensive and the period since the 1976 publication of the original Molecular Gas Dynamics has seen enormous improvements in the molecular models, the procedures, and the implementation strategies for the DSMC method. The molecular theory of gas flows is developed from first principles and is extended to cover the new models and procedures. Note: The disk that originally came with this book is no longer available. However, the same information is available from the author's website (http://gab.com.au/)
5,311 citations
1,664 citations
15 Jun 1998
290 citations
01 Jan 1989
TL;DR: In this paper, the majorant frequency (MFF) method is proposed for the simulation of rarefied gas flows. But the main difference is that it requires a large number of sampling particles.
Abstract: In this work the direct statistical simulation method for spatially uniform relaxation in rarefied gas is developed on the basis of a probabilistic interpretation for the integral representation of the master kinetic equation (Kats equation). It is shown that under certain conditions the conventional schemes for the relaxation simulation follow directly from this equation. A new accurate simulation scheme is proposed, the majorant frequency scheme, which requires a computing capacity that scales directly with the number of sampling particles. The relation between the solutions of the master kinetic equation of rarefied gas and the Boltzmann equation is studied for the uniform case. It is shown that the correlations occurring in the finite-number particle system affect significantly the statistical simulation results. The criterion for estimating the influence of such correlations during the computation process is suggested. Introduction At present, the direct statistical simulation method, based on splitting up the evolution of a gas system in two stages, is widely used in the dynamics of rarefied gas. The method is realized in the following way. The flowfield computed is divided into cells of a finite size Ax and, according to the initial distribution function, N sampling particles are placed into each cell. Then the spatially-uniform relaxation stage and the stage of a free-molecular transition are successively carried out in all the cells. The free-molecular transition simulation can be performed without difficulties. In this case, the realization of the spatially-uniform relaxation stage is of primary importance. Copyright © 1989 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. * Institute of Theoretical and Applied Mechanics. t Computing Center, 171 172 M. S. IVANOV ET AL For the simulation of collisional relaxation, the numerical schemes suggested in Refs. 1-4 are used. All these schemes are derived from heuristic considerations on the basis of physical understanding of the relaxation process in an actual gas; as a result there is no direct relation with the Boltzmann kinetic equation. The heuristic character of these schemes allow comparative analysis only qualitatively, with the use of the "Boltzmannian" collision frequency as the main criterion." The stochastic process for an approximate solution of the Boltzmann equation is constructed in Ref. 8 using the Euler scheme for the Boltzmann spatiallyuniform equation with its further randomization. In such an approach, for colliding particles the conservation laws are not valid, and this is the basic difference from the schemes given in Refs. 1-4. It seems reasonable to consider the known numerical schemes for the statistical simulation of rarefied gas flows in light of a general theory of Monte Carlo techniques. Such a unified consideration enables one to carry out a comparative analysis to show the inner relation between these schemes and also justifies the use of various Monte Carlo weight techniques.^ Derivation of Simulation Technique from Master Kinetic Equation In the construction of the Monte Carlo numerical technique we shall directly proceed from the master kinetic equation for the N-particle distribution function, which in the spatially-uniform case has the following r°° {fN(t,cjj) fN(t,c)} | vi-vj | by dbij dey l C) K2(t'->t I c) cp (t,c) dc dt + cp0(t,c) /2) Jo J where 9 (t,c) = t) (c) f^ (t,c) is the collision density. DIRECT SIMULATION AND MASTER KINETIC EQUATION 173 9o(t,c)=fN(o,c)\)(c)exp{-\)(c)t} _ . _ • _ _ N _ _• '-* C) = -^Y D'^WCv^V;' I Vi,Vj) II 5(vm-Vm)
10 citations