Choporov S. Mathematical modeling of geometrical objects on the basis of R-functions

Object of research - is complex engineering constructions. The purpose of the research is to develop principles of mathematical modeling of complex geometrical objects on the basis of the theory of R-functions and to develop principles of discretization on the basis of quadrilateral and hexahedral elements. Research methods are based on methods of mathematical analysis, analytical geometry, mathematical modeling, computational mathematics, systems analysis. The theory of R-functions is used for description of the mathematical models of complex geometrical objects. Theoretical and practical results are principles and methods of mathematical modeling of complex geometrical objects on the basis of R-functions and principles of generation of quadrilateral and hexahedral finite element meshes. Developed principles and methods allow both to improve the quality results of mathematical modeling and to use them in the design, analysis and optimization of complex engineering constructions. The scientific novelty: the R-functions method for the mathematical modeling complex solids has been developed in the constructing, visualization and mathematical models for 3D geometrical objects analysis; first the method based on superposition method and isomorphism principle for constructing discrete mathematical models founded on finite quadrangular items for analytical mathematical models of complex 2D geometrical objects defined by R-functions was developed; first the method based on superposition method and isomorphism principle for constructing discrete mathematical models founded on finite hexagon items for analytical mathematical models of complex 3D geometrical objects defined by R-functions was developed; the rectification method for geometrical object discrete model and the creating of a universal template rectification with partial recursion obtain the further development by using the information about the curvature of the object boundary. The practical significance of research results confirmed their implementation: a method of sampling the three-dimensional structures of finite hexagon elements and the geometric modeling computer system for complex objects used as a preprocessor for finite element method to further analysis of the stress strain state of elements of industrial designs by LTD "Teсhelektropromremont"; formalization methods for mathematical modeling of complex geometric objects; methods of discretization of analytical mathematical models that described on the basis of R-functions, methods of optimization and refinement of discrete representation, the software for mathematical modeling and discretization on quadrilateral and hexahedral elements of complex solids which was elaborated in the dissertation is implemented in study process of Zaporizhzhya National University, Department of Mathematics.