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A Probabilistic Automatic Steady State Detection Method for the Direct Simulation Monte Carlo

Part of: Rarefied gas flows, Boltzmann equation Equilibrium statistical mechanics

Published online by Cambridge University Press:  02 November 2016

A. Karchani ,
O. Ejtehadi  and
R. S. Myong
A. Karchani*
Affiliation:
School of Mechanical and Aerospace Engineering, Research Center for Aircraft Parts Technology, Gyeongsang National University, Jinju, Gyeongnam 52828, South Korea
O. Ejtehadi*
Affiliation:
School of Mechanical and Aerospace Engineering, Research Center for Aircraft Parts Technology, Gyeongsang National University, Jinju, Gyeongnam 52828, South Korea
R. S. Myong*
Affiliation:
School of Mechanical and Aerospace Engineering, Research Center for Aircraft Parts Technology, Gyeongsang National University, Jinju, Gyeongnam 52828, South Korea
*
*Corresponding author. Email addresses:karchani@gnu.ac.kr (A. Karchani), o_ejtehadi@gnu.ac.kr (O. Ejtehadi), myong@gnu.ac.kr (R. S. Myong)
*Corresponding author. Email addresses:karchani@gnu.ac.kr (A. Karchani), o_ejtehadi@gnu.ac.kr (O. Ejtehadi), myong@gnu.ac.kr (R. S. Myong)
*Corresponding author. Email addresses:karchani@gnu.ac.kr (A. Karchani), o_ejtehadi@gnu.ac.kr (O. Ejtehadi), myong@gnu.ac.kr (R. S. Myong)
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Abstract

The statistical error associated with sampling in the DSMC method can be categorized as type I and II, which are caused by the incorrect rejection and acceptance of the null hypothesis, respectively. In this study, robust global and local automatic steady state detection methods were developed based on an ingenious method based purely on the statistics and kinetics of particles. The key concept is built upon probabilistic automatic reset sampling (PARS) to minimize the type II error caused by incorrect acceptance of the samples that do not belong to the steady state. The global steady state method is based on a relative standard variation of collisional invariants, while the local steady state method is based on local variations in the distribution function of particles at each cell. In order to verify the capability of the new methods, two benchmark cases — the one-dimensional shear-driven Couette flow and the two-dimensional high speed flow past a vertical wall—were extensively investigated. Owing to the combined effects of the automatic detection and local reset sampling, the local steady state detection method yielded a substantial gain of 30-36% in computational cost for the problem studied. Moreover, the local reset feature outperformed the automatic detection feature in overall computational savings.

Keywords

Direct simulation Monte Carlostatistical errortype II errorsteady state detectionreset sampling

MSC classification

Secondary: 82B80: Numerical methods (Monte Carlo, series resummation, etc.) 76P05: Rarefied gas flows, Boltzmann equation
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 5 , November 2016 , pp. 1183 - 1209
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Bird, G.A., Molecular Gas Dynamics and the Direct Simulation Monte Carlo of Gas Flows, Oxford Engineering Science Series, 1994.CrossRefGoogle Scholar
[2] Oran, E., Oh, C., Cybyk, B., Direct simulation Monte Carlo: Recent advances and applications, Annual Review of Fluid Mechanics, 30 (1998) 403–441.Google Scholar
[3] Wagner, W., A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation, Journal of Statistical Physics, 66 (1992) 1011–1044.Google Scholar
[4] Karchani, A., Myong, R.S., Convergence analysis of the direct simulation Monte Carlo based on the physical laws of conservation, Computers & Fluids, 115 (2015) 98–114.CrossRefGoogle Scholar
[5] Wu, J.S., Tseng, K.C., Analysis of micro-scale gas flows with pressure boundaries using direct simulation Monte Carlo method, Computers & Fluids, 30 (2001) 711–735.Google Scholar
[6] Sun, Q., Boyd, I.D., A direct simulation method for subsonic, microscale gas flows, Journal of Computational Physics, 179 (2002) 400–425.Google Scholar
[7] Wu, J.S., Tseng, K.C., Yang, T.J., Parallel implementation of DSMC using unstructured mesh, International Journal of Computational Fluid Dynamics, 17 (2003) 405–422.Google Scholar
[8] Darbandi, M., Karchani, A., Schneider, G., The study of rarefied gas flow through microfilters with different openings using MONIR-DSMC, in: ICNMM2011, ASME, Alberta, Canada, (2011) 215–221.Google Scholar
[9] Plotnikov, M.Y., Shkarupa, E., Theoretical and numerical analysis of approaches to evaluation of statistical error of the DSMC method, Computers & Fluids, 105 (2014) 251–261.Google Scholar
[10] Plotnikov, M.Y., Shkarupa, E., Estimation of the statistical error of the direct simulation Monte Carlo method, Computational Mathematics and Mathematical Physics, 50 (2010) 335–344.Google Scholar
[11] Karchani, A., Ejtehadi, O., Myong, R.S., A steady-state convergence detection method for Monte Carlo simulation, AIP Conference Proceedings, 1628 (2014) 313–317.Google Scholar
[12] Cramér, H., Mathematical Methods of Statistics, Princeton University Press, 1999.Google Scholar
[13] Chen, G., Boyd, I.D., Statistical error analysis for the direct simulation Monte Carlo technique, Journal of Computational Physics, 126 (1996) 434–448.Google Scholar
[14] Schwartzentruber, T.E., Scalabrin, L.C., Boyd, I.D., A modular particle-continuum numerical method for hypersonic non-equilibrium gas flows, Journal of Computational Physics, 225 (2007) 1159–1174.Google Scholar
[15] Gallis, M.A., Torczynski, J., Rader, D., Bird, G.A., Convergence behavior of a new DSMC algorithm, Journal of Computational Physics, 228 (2009) 4532–4548.Google Scholar
[16] Sun, Q., Fan, J., Boyd, I.D., Improved sampling techniques for the direct simulation Monte Carlo method, Computers & Fluids, 38 (2009) 475–479.Google Scholar
[17] Hadjiconstantinou, N.G., Garcia, A.L., Bazant, M.Z., He, G., Statistical error in particle simulations of hydrodynamic phenomena, Journal of Computational Physics, 187 (2003) 274–297.Google Scholar
[18] Cave, H.M., Tseng, K.C., Wu, J.S., Jermy, M.C., Huang, J.C., Krumdieck, S.P., Implementation of unsteady sampling procedures for the parallel direct simulation Monte Carlo method, Journal of Computational Physics, 227 (2008) 6249–6271.Google Scholar
[19] Bird, G.A., Molecular Gas Dynamics, Oxford Engineering Science Series, 1976.Google Scholar
[20] Rieffel, M.A., A method for estimating the computational requirements of DSMC simulations, Journal of Computational Physics, 149 (1999) 95–113.Google Scholar
[21] Bird, G.A., Sophisticated DSMC, Notes from DSMC07, Santa Fe, 2007.Google Scholar
[22] Bird, G.A., The DSMC Method, CreateSpace Independent Publishing Platform, 2013.Google Scholar
[23] Burt, J.M., Boyd, I.D., Convergence detection in direct simulation Monte Carlo calculations for steady state flows, Communications in Computational Physics, 10 (2011) 807.Google Scholar
[24] Shreider, Y.A., Method of Statistical Testing: Monte Carlo Method, Elsevier Publishing Company, 1964.Google Scholar
[25] Parzen, E., Modern Probability Theory and Its Applications, John Wiley & Sons, 1960.Google Scholar
[26] Ermakov, S.M., Mikhailov, G.A., Statistical Modeling, Nauka, Moscow, 1982.Google Scholar
[27] Vincenti, W.G., Kruger, C.H., Introduction to Physical Gas Dynamics, New York, Wiley, 1965.Google Scholar
[28] Ochi, M.K., Applied Probability and Stochastic Processes in Engineering and Physical Sciences, Wiley, New York, 1990.Google Scholar
[29] Kostinski, A., Jameson, A., On the spatial distribution of cloud particles, Journal of the Atmospheric Sciences, 57 (2000) 901–915.Google Scholar
[30] MacDonald, I., A relation between the Maxwell-Boltzmann and chi-squared distributions, Journal of Chemical Education, 63 (1986) 575.Google Scholar
[31] Garcia, A.L., Alder, B.J., Generation of the Chapman-Enskog distribution, Journal of Computational Physics, 140 (1998) 66–70.Google Scholar
[32] Garwood, F., Fiducial limits for the Poisson distribution, Biometrika, (1936) 437–442.Google Scholar
[33] Johnson, N.L., Kemp, A.W., Kotz, S., Univariate Discrete Distributions, John Wiley & Sons, 2005.Google Scholar
[34] Montgomery, D.C., Runger, G.C., Hubele, N.F., Engineering Statistics, John Wiley & Sons, 2009.Google Scholar
[35] Forbes, C., Evans, M., Hastings, N., Peacock, B., Statistical Distributions, John Wiley & Sons, 2011.Google Scholar
[36] Breslow, N.E., N.E. Day, , Statistical Methods in Cancer Research, International Agency for Research on Cancer, 1987.Google Scholar