Maksymenko-Sheyko K. R-functions in mathematical modeling of geometric objects and physical fields

The research object is the analytical identification of geometrical objects of the nonclassical form, and fields of the various physical nature. The purpose of the work is the research and improvement of constructive tools of the R-functions theory at the solving of the analytical geometry inverse problem in 2D and 3D; the mathematical and computer modeling of fields of the various physical nature, including ones having screw type of symmetry with the R-functions method. Research methods: the R-functions method, variational methods, the tensor analysis. The basic R-operations systems and operations for building sites of boundary were investigated. The methods to build the normalized equations of geometrical objects with various types of symmetry and fractal geometry objects were developed. The equations of some structure elements in mechanical engineering, including the automobile frames were obtained in 3D. The algorithm of automatic formation of the predicate equation for composed geometrical object by common logic formula was developed. The technique of numerical experiments was developed for practical problems including nuclear power plant elements. The mathematical and computer models of incompressible viscous liquid flow and heat exchange were constructed in curvilinear non-orthogonal coordinates at motion of liquid along screw type symmetry channels. Scientific novelty: the developed methods and constructive means have allowed to realize for the first time the analytical identification of nonclassical geometrical objects in 3D, fractal geometry objects. The process of construction of symmetric functions for symmetric areas was simplified essentially and automated. The mathematical models of an incompressible viscous liquid current and heat exchange were constructed at flow of a liquid on channels with screw type of symmetry in curvilinear non-orthogonal coordinates, and influence of twisting parameter was investigated for the first time on hydrodynamical and temperature fields distribution character. The technique of computing experiments conducting has received the further development at the solving of research problems of the various nature physical fields with the R-functions method. The results of work are used in the Moscow state technological university "Stankin" as the theoretical and practical bases at mathematical modeling of geometrical objects for an accuracy parameters estimation of metal-cutting machines, for development of the computer-graphic method of the mathematical modelling problems solving and for the organization of prototyping process using R-functions, and also into educational process of the Kharkov and Zaporozhye national universities. The areas of application are power engineering, mechanical engineering.