Chugay A. Mathematical modeling and methods for solving problems of optimal package of three-dimevsional objiects

The thesis is devoted to the solution of optimization problems of packing threedimensional bodies by construction exact mathematical models and developing approaches based on the use of optimization methods of non-linear programming and modern solvers. Constructive tools of mathematical modeling and computer modeling of the relationship between oriented and non-oriented three-dimensional bodies which the boundary is formed by cylindrical, conical, spherical surfaces and planes in the form of new classes of Ф-functions and quasi-Ф-functions are developed. Based on the tools of mathematical modeling the basic mathematical model of the problem of optimal packing of three-dimensional bodies which the boundary is formed by cylindrical, conical, spherical surfaces and planes is constructed and investigated. Also various implementations that cover a wide class of scientific and applied problems of packing three-dimensional bodies are constructed. A general methodology for solving the problems of packing three-dimensional bodies that simultaneously allow continuous rotations and translations are developed. Strategies, methods and algorithms for solving optimization problems of packing threedimensional bodies with account for technological constraints (minimum permissible distances, prohibition zones, the possibility of continuous translations and rotations, etc.) are proposed. Based on the proposed tools of mathematical modeling, mathematical models, methods and algorithms, software using parallel computing technology for automatically solving the optimization problems of packing three-dimensional bodies is created. The results obtained can be used to solve problems of optimization of layout solutions, computer modeling in materials science, powder metallurgy and nanotechnologies, optimization of the 3D printing process for SLS additive production technology, and in information and logistics systems that optimize transportation and storage of goods. Keywords: packing, three-dimensional bodies, geometric design, Ф-functions, mathematical modeling, continuous rotations, nonlinear optimization.