On Determining the Structural Properties of Random Fillings in Digital Models
Abstract Generated abstract
This paper examines how the radial density function of a random packing of equal, nonintersecting spheres can be determined using digital simulation rather than physical models. A unit cube is filled according to a previously described packing algorithm at discrete densities from 0.0104 to 0.229, and the radial function is computed by counting sphere centers within successively enlarged spheres around a centrally located particle, with repeated trials averaged for each density. The resulting program, implemented on an M-20 computer, produced radial function curves that agree with those reported for physical models, supporting digital modeling as an effective and comparatively economical approach for studying the structure of random sphere packings.
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ON DETERMINING THE STRUCTURAL PROPERTIES OF RANDOM PACKINGS USING DIGITAL MODELS
I. E. Naats, G. N. Parvatov
(Presented by the scientific seminar of the UVL)
The structural properties of materials can be assessed by means of physical models, which are random ensembles of a finite number of spheres [1].
One of the important parameters associated with the random structure of a model is the radial function \(g(r)\) of a random packing. We define the radial function as the packing density by a random ensemble of spheres of a sphere of radius \(r\). For small values of \(r\) \((2R \leq r \leq 4R)\), the behavior of \(g(r)\) is irregular and to a certain extent describes the local properties of the packing. For \(r=\infty\), \(g(\infty)=\rho_0\), i.e., the packing density of space.
In [2, 3] radial functions are presented that were obtained from experiments with physical models of random packings of hard spheres.
The purpose of the present work is to calculate the radial function by digital simulation of a random packing of nonintersecting spheres in a sample unit cube.
Let us consider the procedure for calculating the radial function for a random packing implemented by digital simulation according to the algorithm [4]. Let the packing density of the unit cube \(\rho_1\) take discrete values in the interval \(0.0104\)—\(0.229\) with step \(0.0104\). First, in the packed cube with value \(\rho_1\), we find a sphere as close as possible to the center of the cube. Then, from the center of this sphere, we draw a sphere of radius \(R=d\), where \(d\) is the diameter of a sphere, and find all coordinates of the centers of spheres falling on this sphere. The numerical values of the radial function are determined by computing the expression
\[ g(r)=N\cdot \frac{v}{V}, \]
where \(N\) is the number of spheres falling into the sphere of radius \(r\);
\(v\) is the volume of a sphere;
\(V\) is the volume of the sphere of radius.
Next we increase the radius of the sphere by \(0.25d\) and repeat the procedure for determining the number of spheres whose centers have fallen into the enlarged sphere. The radius of the sphere is increased until the sphere touches some face of the cube. At this point, the calculation of the radial function \(g(r)\) for the given cube packed with spheres to density \(\rho_1\) ends. The computations of \(g(r)\) were carried out ten times.
for each value of \(\rho_1\) in the interval from 0.0104 to 0.229 with a step of 0.0104. Then the mean value of this function corresponding to the given packing density was determined.
Fig. 1 shows a block diagram of the program. The block diagram consists of the unit-cube packing program, the program for finding the center
Text in Fig. 1:
- Search \(XYZ\)
- Cube packing
- \(N_k = 20\)
- \(N_k < 20\)
- \(20 + 20 + \cdots = N\)
- Search for the center of the sphere and “min” to the boundary
- Formation of the sphere \(D\)
- Beginning of \(g(r)\)
- Condition for a sphere to contain the ball
- \(\rho > \dfrac{\Delta x}{\Delta y\,\Delta z}\)
- \(\rho < \dfrac{\Delta x}{\Delta y\,\Delta z}\)
- \(\Delta x_i^2 + \Delta y_i^2 + \Delta z_i^2 = z_i^2\)
- \(D^2 > z_i^2\)
- \(D^2 < z_i^2\)
- Count of balls in the \(i\)-th sphere
- BPU
- Increase of the sphere \(D\)
- \(\min > D\)
- \(\min < D\)
- Preparation of data for \(g(r)\)
- Count of packed balls \(=N\)
- \(N \le 440\)
- \(N > 440\)
- Count of cycles “\(u\)”
- \(u \ge 1\)
- \(u < 1\)
Fig. 1. Block diagram of the program for computing the radial function
of the spheres, the program for forming the condition for a ball to fall into a sphere, the program for forming the radius of the sphere, the program for computing \(g(r)\), and several counters.
The unit-cube packing program fully corresponds to the packing algorithm [4].
It should be noted that there are certain features in the operation of the block diagram as a whole. The operation of finding “min” consists in finding the minimum radius at which a sphere touches some face. The entire program is enclosed in a loop over the number of trials “Ц”.
According to this block diagram, a program was written for the M-20 digital computer and a computation was carried out. The computation time was 18 min. The results of the digital experiment are presented in the form of graphs of \(g(r)\) (Fig. 2) for various values of \(\rho_1\).
Fig. 2. Dependence of density on distance
The curves in Fig. 2 agree completely with the curves obtained for radial functions on physical models [5]. This confirms the effectiveness of investigating the structure of random packings by digital modeling of random fillings on a digital computer, which makes it possible to obtain more accurate results at lower cost.
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