Modeling Close Packing of 3D Objects
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MODELING CLOSE PACKING OF 3D OBJECTS Y. G. Stoyan,1† V. V. Semkin,1‡ and A. M. Chugay1††
UDC 519.85
Abstract. The paper represents the concept of F-functions and quasi F-functions as an efficient tool for mathematical modeling of three-dimensional packing problems for convex geometrical objects with continuous translations and rotations. A mathematical model of close packing of convex geometrical objects is formulated and its basic properties are considered. A solution method is proposed, which includes the following stages: constructing initial points, computing local extrema, and passing from one local minimum to another. The solution approach is efficient to solve optimization packing problems. Numerical examples are given. Keywords: packing, mathematical modeling, optimization, quasi F-function, convex three-dimensional objects, translation, rotation.
INTRODUCTION In many situations, contributors need to solve problems where the irregular space structures can be interpreted as systems of close-packed solid particles of different forms. The possibility of constructing models and applying computational methods for computer modeling of new materials is studied for a long time. Close packing of non-spherical particles is used to model the structure and analyze the properties of various composite and vitreous materials, colloid solutions, granulated and porous media, mixed fuels [1– 3]. For example, in [4] the authors have proposed a new method to model dense materials on the basis of spheropolyhedra (objects obtained with the use of the Minkowski sum of a full sphere and a parallelepiped) packing algorithm. The method proposed in the present paper can be applied to the virtual design of dense materials with non-spherical particles. In [5] the optimization approach to simulation modeling of various microstructures is considered. The authors solve optimization problems for the porosity function and find the optimal model of granulometric composition of the microstructure under study. The problem of 3D-reconstruction of kernel microstructure is an important application of the proposed approach. The paper [6] considers the procedure of generating close random packing of solid particles with the use of spherocylinders. The described method carries out packing of spherocylinders that models rather well the structure of anisotropic powders and colloid substances. The paper [7] represents a computer model intended to determine the density of granulated materials by packing three-dimensional ellipsoids. A special approach is proposed that prevents blocking of particles as they move inside the container. A process of sampling survey is implemented to determine average packing density either in the entire container or in any of its parts. The obtained results are compared with the distribution of sand grains in ordinary sand. The paper [8] reviews the studies in packing of spheroids, cylinders, and arbitrary bodies in powder metallurgy problems. The approaches described in the paper are based on various approximations of a
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