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Random sequential packing simulations in three dimensions for aligned cubes

Published online by Cambridge University Press:  14 July 2016

Douglas W. Cooper
Douglas W. Cooper*
Affiliation:
IBM Research Division
*
∗Postal address: IBM Research Division, Thomas J. Watson Research Center, Room 01–203, P.O. Box 218, Yorktown Heights, NY 10598, USA.
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Abstract

This particular three-dimensional random packing limit problem is to determine the mean fraction of a cubic space that would be occupied by aligned, fixed, equalsize cubes, placed at random locations sequentially until no more can be added. No analytical solution has yet been found for this problem. Simulation results for a finite region and finite number of attempts were extrapolated to an infinite number of attempts (N →∞) in an infinite region by multiple linear regression, using volume fraction occupied (F) as a linear combination of the ratio of the length of the small cube sides (S) to the length of the cubic region side (L) and the cube root of the ratio of the region volume to the total volume of cubes tried, (L3/NS3)⅓. These results for random packing in a volume with penetrable walls can be adjusted with a multiplicative correction factor to give the results for impenetrable walls. A total of N = 107 attempts at placement were made for L/S = 20/1 and N = 14 × 106 attempts were made for L/S = 10/1. The results for volume fraction packed are correlated by F = 0.430(±0.008) + 0.966(±0.072)(S/L) – 0.236(±0.029)(L3/NS)⅓. The numbers in parentheses are twice the standard errors of estimate of the coefficients, indicating the 95% confidence intervals due to random errors. This value for the packing density limit, 0.430 ± 0.008, is slightly larger than that given by a conjecture by Palásti [10], 0.4178. Our value is consistent with that obtained by rather different simulation methods by Jodrey and Tory [8], 0.4227 ± 0.0006, and by Blaisdell and Solomon [2], 0.4262.

Keywords

RANDOM SAMPLING WITHOUT REPLACEMENTRANDOM SEQUENTIAL SAMPLINGPACKING LIMITSIMULATION WITH REGRESSIONPALÁSTI CONJECTURE
Type
Short Communications
Information
Journal of Applied Probability , Volume 26 , Issue 3 , September 1989 , pp. 664 - 670
Copyright
Copyright © Applied Probability Trust 1989 

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References

[1] Blaisdell, B. E. and Solomon, H. (1970) On random sequential packing in the plane and a conjecture of Palásti. J. Appl. Prob. 7, 667–698.Google Scholar
[2] Blaisdell, B. E. and Solomon, H. (1982) Random sequential packing in Euclidean spaces of dimensions three and four and a conjecture of Palásti. J. Appl. Prob. 19, 382–390.Google Scholar
[3] Cooper, D. W. (1987) Parking problem (sequential packing) simulations in two and three dimensions. J. Colloid Interface Sci. 119, 442–450.CrossRefGoogle Scholar
[4] Cooper, D. W. (1988) Random-sequential-packing simulations in three dimensions for spheres. Phys. Rev. A38, 522–524.Google Scholar
[5] Feder, J. (1980) Random sequential absorption. J. Theoret. Biol. 87, 237–254.Google Scholar
[6] Gotoh, K., Jodrey, W. S. and Tory, E. M. (1978) Average nearest-neighbour spacing in a random dispersion of equal spheres. Powder Technol. 21, 285–287.CrossRefGoogle Scholar
[7] Haile, J. M., Massobrio, C. and Torquoto, S. (1985) Two-point matrix probability function for two-phase random media: computer simulation for impenetrable spheres. J. Chem. Phys. 83, 4075–4078.Google Scholar
[8] Jodrey, W. S. and Tory, E. M. (1980) Random sequential packing in Rn. J. Statist. Comput. Simul. 10, 87–93.Google Scholar
[9] Lotwick, H. W. (1982) Simulation of some spatial hard core models and the complete packing problem. J. Statist. Comput. Simul. 15, 295–314.CrossRefGoogle Scholar
[10] Palásti, I. (1960) On some random space filling problems. Publ. Math. Inst. Hung. Acad. Sci. 5, 353–360.Google Scholar
[11] Pomeau, Y. (1980) Some asymptotic estimates in the random parking problem J. Phys. A13, L193–L196.Google Scholar
[12] Rényi, A. (1958) On a one-dimensional problem concerning random space filling. Publ. Math. Inst. Hung. Acad. Sci. 3, 109–127.Google Scholar
[13] Swendsen, R. H. (1981) Dynamics of random sequential absorption. Phys. Rev. A24, 504–507.Google Scholar
[14] Tanemura, M. (1979) On random complete packing by discs. Ann. Inst. Statist. Math. 31, 351–365.Google Scholar
[15] Tory, E. M. and Jodrey, W. S. (1983) Comments on some types of random packing. In Advances in the Mechanics and the Flow of Granular Materials, ed. Shahinpoor, M., Vol. 1, 75, Trans. Tech., Clausthal-Zellerfeld, West Germany.Google Scholar
[16] Tory, E. M., Jodrey, W. S. and Pickard, D. K. (1983) Simulation of random sequential absorption: efficient methods and resolution of conflicting results. J. Theoret Biol. 102, 439–445.Google Scholar
[17] Weiner, H. J. et al. (1980) Letters to the editor: Further comments on a paper by H. J. Weiner. J. Appl. Prob. 17, 878–892.CrossRefGoogle Scholar