The object of the study is the process of optimization of arbitrary polyhedral packing in the container, taking into account the equilibrium constraints and placement constraints, including the minimum allowable distances. The purpose of the work is to increase the efficiency of solving optimization problems of optimal packing of arbitrary polyhedra by developing constructive tools of mathematical and computer modeling, new mathematical models and effective methods of local optimization with the use of modern solvers (NLP-solvers). Research methods: in the work analytical geometry and functional analysis are used for constructing phi-functions, pseudonormalized phi-functions, quasi phi-functions and pseudonormalized quasi phi-functions; geometrical design methods for constructing mathematical models and developing methods of searching for feasible starting points and methods of local optimization for the OPP problem. Practical results - the scientific results of the dissertation work are the further development of mathematical modeling and computational methods in geometric design: new mathematical models are created and effective methods for solving optimization problems of optimal packing of arbitrary polyhedra with a wide range of applications in the priority fields of science and technology (including additive technologies, materials science, logistics, mineralogy, medicine, nanotechnology, robotics, image recognition systems, control systems, space launch systems, energy, engineering, aerospace, construction). The scientific novelty of the results obtained is that the method of phi-functions has been further developed: firstly constructed phi-functions, pseudonormalized phi-functions, quasi phi-functions and pseudonormalized quasi phi-functions as tools of mathematical modeling of placement constraints for the OPP problem, which allows to describe in an analytical form: non-overlapping of arbitrary polyhedra; the containment of polyhedra in a convex container; minimum allowable distances between arbitrary polyhedra and between polyhedra and the boundary of the container; For the first time, a mathematical model of the OPP problem is built in the form of a nonlinear programming problem (including all globally optimal solutions) for convex polyhedra in a convex container, whose boundary is formed by spherical, cylindrical, elliptic surfaces and a plane, taking into account the placement constraints and the equilibrium constraints, which allows to use modern NLP-solvers; For the first time, a mathematical model of Optimal Polytopes Clustering (OPC) problem is constructed in spherical, cuboidal and cylindrical areas of minimal volume, which allows to generate effective feasible starting points to search for local extremes of the OPP problem; methods for solving geometric design problems have been further developed: the strategy for solving the OPP problem and effective methods for its main implementations were proposed, which, in contrast to existing approaches: take into account simultaneously continuous translations and rotations of objects, minimum allowable distances and equilibrium constraints; allow to obtain locally optimal solutions for OPP problems that are better than the target function (compared to benchmark instances known for published results). As a result of the dissertation, there were received: an act on the implementation of scientific results in the educational process of the Kharkiv National University of Radio Electronics; a reference on the use of the developed software module for solving the problem of optimal filling of a given volume by particles of non-spherical form in material science; a letter of support from G. Fasano - leading scientist and specialist in the field of mathematical modeling and optimization of the systems of the European company "Thales Alenia Space"; an act about the use of the results of the dissertation work in the IT company "Cloud Works" to solve the problems of optimizing the process of 3D printing, which uses SLS technology and problems of optimal packaging of cargoes in arbitrary containers in the field of logistics. The obtained results can be used for further fundamental research on the development of methods for solving the optimization problems of packing of non-oriented three-dimensional objects of arbitrary spatial shape.
