DIGITAL MODELING IN PROBLEMS OF RADIATION FLAW DETECTION OF BODIES WITH A HETEROGENEOUS STRUCTURE

V. K. Kivran, I. E. Naats, G. N. Parvatov, V. I. Ses

(Presented by the scientific seminar of the sector for flaw detection of construction materials of the Scientific Research Institute of Electronic Introscopy)

Problems of radiation flaw detection of products with a heterogeneous structure of the concrete type are distinguished by a certain peculiarity caused by the heterogeneity of the flux behind such bodies [1]. For a theoretical study of questions connected with examining the influence of structural heterogeneity on various factors of radiation flaw detection, there arises the need for mathematical modeling of such structures as applied to calculations on digital computers. The present work considers questions of creating such models using a concrete-type structure as an example.

The structure of concrete, from the standpoint of the distribution of coarse aggregate, may be represented by a model of random filling of space by a system of convex bodies having random sizes, shape, and orientation. Taking into account that in concrete with gravel aggregate the particles have a more or less rounded surface, in the model of such a structure the packing of coarse aggregate is compared with a packing of spheres with a corresponding size distribution.

Generally speaking, all existing types of packings, according to the density of filling, may conventionally be divided into three groups: a) sparsely filled, with low filling density, in fact having no points of contact between elements; b) dense; and c) densest, with limiting filling density. Such a division of packings is conventional, all the more so because, when the density changes, these packings transform into one another with a number of intermediate states.

The densest packings are in general not random and, from the standpoint of modeling, are of no interest. Sparse and dense packings, as a rule, are random and can be modeled by Monte Carlo methods.

When modeling random packings of spheres, it is necessary to store in the memory of the digital computer the sizes and coordinates of the spheres. If one takes into account that, usually, in digital computers of the M-20, BESM, and similar types, the random-access memory consists of 4095 cells and that part of them must be allocated for the packing program and for calculating the necessary parameters, then it is obvious that only a limited number of packed spheres, on the order of a thousand, can be stored in random-access memory. The use of external memory in this case is inadvisable because of the sharp increase in machine-time costs. However, such a number of spheres is quite sufficient for studying various dependencies.

The second essential aspect of digital modeling of a packing is the packing sequence; that is, when each successive sphere is packed, the position of the spheres already packed is taken to be fixed. The use of iterative methods to refine the positions of the preceding spheres is at present a more complicated problem, owing to the sharp increase in machine-time costs.

In the model of a sparsely filled packing, the coordinates of the centers of each successive sphere being packed are generated by special programs for pseudorandom numbers with a uniform distribution law [2]. In this case, the next sphere to be packed is checked for the possibility that it intersects the previously packed spheres and the faces bounding the packing space. If there are no such intersections, the sphere is considered packed; if there is at least one intersection—which should not occur in real packings—all three coordinates of the sphere are generated again.

Using programs written to implement this algorithm, one can construct models of packings with a filling density of up to 0.25 for equal spheres; in this case, no more than two minutes of machine time on an M-20-type computer are spent for each 1000 spheres. The maximum filling density (at algorithm failure) reaches 0.3, with machine-time costs of up to 8 minutes.

To construct a model of dense packings, we developed three algorithms.

One of them is based on the fact that, near two generated coordinates of the center of each successive sphere being packed, with the aid of the above-mentioned pseudorandom-number programs, a position of the sphere is sought such that its center has the minimum value of the third coordinate, while not intersecting the spheres previously packed. This position of the sphere is found by successively making trial steps of the center of the sphere in the plane of the axes of the generated coordinates; in this plane the center of the sphere is displaced by the magnitude of the trial step parallel to these coordinate axes.

The filling density of such a packing is 0.45 for equal spheres; in this case, about 35–40 minutes of machine time on an M-20-type computer are spent for each 800–900 spheres.

The second algorithm, which differs from the first in that the trial steps are made in the direction of steepest descent, makes it possible to obtain packings with a filling density, for equal spheres, of 0.55; in this case the machine-time costs amount to 50–55 minutes for each 900–1000 spheres.

According to the third algorithm, an exact calculation is made of the coordinates of the sphere being packed when it is packed into the recess formed by three randomly selected previously packed spheres. With this algorithm, the filling density for equal spheres reaches 0.60; in this case the machine-time costs are 40–45 minutes for 600 packed spheres.

To compare the developed mathematical models of sphere packings in terms of filling density, experimental investigations were carried out on physical models of the structure—packings of balls. The filling density of packings of equal balls without preliminary compaction reached 0.62–0.63, while the filling density of a gravel mixture of one fraction was 0.64–0.67. Thus, the developed algorithms for modeling the structure make it possible to obtain mathematical models of the structure with a density sufficiently close to that of real structures.

A fairly good correspondence of the structure models to real structures is also found in determining the average number of contacts of each

particles in the structure. Thus, the average number of contacts in a model with a packing density of 0.45 is 4.6, and with a packing density of 0.60 it is, respectively, 7.9. In a packing of equal spheres this number, within a packing density of 0.6–0.7, is 6.3–7.9 [3]. The average number of contacts per particle in a gravel mixture, determined by us in concrete with a filler of one fraction, is 7.6–7.9.

The good agreement, in terms of the main characteristics, of the developed digital models of the structure of granular heterogeneous structures with their real representations makes it possible to carry out the necessary calculations of defect detectability in bodies with such a structure by radiation methods quite fully and with good agreement with experimental data. With the aid of these models one can calculate the shadow image, the sensitivity of radiation methods, and also study various structural properties.

REFERENCES

1. V. A. Vorob’ev. Radiation flaw detection of building materials and structures using bremsstrahlung radiation. Defektoskopiya, 5 (1965), 33.

2. D. I. Golenko. Modeling and statistical analysis of pseudorandom numbers on electronic computers. Moscow, “Nauka,” 1965.

3. G. D. Scott. Packing of Spheres. J. Nature, 188 (1960).