Modeling Close Packing of 3D Objects

The paper represents the concept of Φ-functions and quasi Φ-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.