On Density Estimation in Digital Modeling of Random Packings

I. E. Naats, G. N. Parvatov

(Presented by the scientific seminar of the Department of Higher Mathematics)

In the theory of equilibrium statistical mechanics and in modeling problems, one of the basic conditions is the condition of statistical homogeneity of the structure under consideration.

In the case of infinitely large systems the homogeneity condition is satisfied. However, for actually existing systems the presence of surfaces bounding the volume under consideration disturbs the homogeneity of the system of particles. The condition of homogeneity in statistical mechanics is the invariance of the particle distribution functions with respect to the coordinates in the region under consideration [1].

In [2] an estimate was obtained for the distance from the boundaries beginning with which the boundary effect may be neglected in studying the distribution of particles in a one-dimensional gas model.

In the present work an estimate is given of the influence of the boundary effect on the density of random packing of space by a system of nonintersecting spheres in digital models used in problems of studying the structure of inhomogeneous materials [3].

Suppose that a random packing of the unit cube by a system of spheres in a digital experiment is carried out according to one of the algorithms described in [4, 5]. Having determined the packing density of this unit cube, it is required to estimate the packing density, according to the given algorithm, of three-dimensional space, i.e., to eliminate the influence of the cube boundaries in the digital experiment.

A sufficiently reliable and at the same time simple estimate can be obtained by the method of boundary transfer.

The essence of the method consists in constructing a new cube of smaller dimensions. In other words, from the cube given in the experiment an inner cube is “cut out,” on the packing of which the influence of the former boundaries is negligibly small. Let \(\rho_1\) denote the packing density of the unit cube by a system of nonintersecting spheres, and \(\rho_2\) the density of the inner cube. Then the ratio

\[ \frac{\rho_2}{\rho_1}=k \]

can characterize the influence of the boundaries on the density of random packing. We shall obtain the numerical characteristic \(k\) for random packing of the unit cube by algorithm [4].

To obtain estimates of the values of “$k$” on a digital computer, random fillings of the unit cube were carried out with density $\rho_1$ in the range $0.0104$—$0.229$ and with step $0.0104$. For each value of $\rho_1$, a cube of smaller dimensions was then obtained (with edge length $(1-2d)$, set back from each face by the diameter of a sphere, “$d$”), and the corresponding density value $\rho_2$ was then found. Repeating the random-filling procedure 10 times for each value of $\rho_1$, we obtain the corresponding mean value $\rho_2$, and the variance of the density of the small volume as a function of $\rho_1$.

The results of the digital experiment for estimating $\rho_2$ and $\sigma_2^2$, performed on the M-20 computer, are summarized in Table 1, from which it is seen that the values of “$k$” lie in the range $1.15$—$1.35$. Here the smallest value corresponds to a denser random filling of the unit cube. This is explained by the fact that, with a larger number of spheres packed in a cube of finite dimensions, one random filling differs little from another in the configuration of the spatial arrangement of the spheres. Therefore, the influence of the boundaries on the estimate of the density $\rho_2$ decreases as $\rho_2$ increases.

Using the results of the calculations in Table 1, it is possible to estimate the density of random filling of large regions of three-dimensional space according to the packing algorithm described, if the digital experiment was carried out only for packing a sample unit cube. For the density range under consideration, this correction is on average a factor of $1.2$.

Table 1

Figure 1 shows a block diagram of the algorithm for estimating the influence of the boundaries of the packed volume on the packing density. It consists of four basic blocks: a block for finding random coordinates $X, y, Z$, a block for packing the unit cube, a block for packing the small cube, and a block for computing $\bar{\rho}, \sigma^2$. The first two blocks completely coincide with the block diagram of the random-packing algorithm of [4], with the only difference that in the present case the packing of the large cube is carried out with a prescribed density from 20 to 440 spheres with a step of 20 spheres. The block for packing the small volume is a sequence of operations for checking

conditions for the entry of spheres into the small cube, previously packed in the large one, operations for counting the spheres of the small and large volumes, recording coordinates, and transmitting control.

Fig. 1. Flowchart of the program for random filling and computation of characteristics

The operation of checking the conditions for a sphere to enter the small cube consists in checking the coordinates of the center of the sphere for fulfillment of the conditions:

\[ X,\;Y,\;Z \geq d;\qquad X,\;Y,\;Z \leq 0.9, \tag{1} \]

where \(d\) is the “left” boundary,
\(0.9\) is the “right” boundary (or \(0.9 = 1 - d\)).

A sphere is considered packed into the small cube if it satisfies these conditions. A special feature of the algorithm is that both counters—the number of spheres \(N_k\) in the large cube and the counter of the number of spheres \(n_k\)—belong to the small-cube packing block and follow one after the other. This is due to the fact that every sphere entering the large cube and not intersecting the previously packed ones is immediately checked for the condition of its entry into the small volume. A sphere satisfying conditions (1) is recorded in the volu-

their counters. But if even one coordinate of the center of the sphere being packed does not satisfy condition (1), then the sphere is recorded only in the counter for the large cube, and then the coordinates of the spheres of the large cube are recorded. After the coordinates have been recorded, the contents of the sphere counter \(N_k\) of the large cube are compared with the number \(20i\), where \(i\) is the number of the digital experiment (the row number in Table 1). If \(N_k = 20i\), control is transferred to the computation block; if \(N_k < 20i\), control is transferred to the beginning of the search for random numbers \(X, Y, Z\) in order to continue the packing.

The computation block is intended for carrying out the computation of \(\rho_2\).
To run a program compiled according to this algorithm, 18 minutes of M-20 machine time are required.

REFERENCES

1. M. E. Fisher. J. Math. Phys., 6, 1943 (1965).

2. H. S. Ziff and H. H. Cooperswith, J. of Mathematical Physics, v. 8, No. 2, p. 306 (1967).

3. I. E. Naats, A. A. Vorob’ev, V. A. Vorob’ev. Application of probabilistic-geometric methods in radiation defectoscopy of bodies with a heterogeneous structure. Reports at the Third Interuniversity Conference on radiation methods and means of nondestructive testing of materials and products. Tomsk. TSU Publishing House, 1968. (In press).

4. V. A. Vorob’ev, V. K. Kivran, I. E. Naats. Digital modeling of random packings of spheres of equal diameters with low filling density. Present collection.

5. V. A. Vorob’ev, V. K. Kivran, I. E. Naats. Modeling of random dense packings of equal spheres on digital electronic computers. Present collection.