Khalanchuk L. Structured discrete models for solving boundary value problems. - Qualification work on the rights of the manuscript.
The dissertation on competition of a scientific degree of the philosophy doctor on a specialty 113 Applied mathematics. – Zaporizhzhia National University, Zaporizhzhia, 2021.
The object of research is structured and block-structured grids for geometric models.
The subject of research – methods of generating structured and block-structured grids of geometric models.
The purpose of the dissertation is to develop a mathematical apparatus for the construction of structured and block-structured grids with specified parameters of thickening and quality assurance of the model. To achieve this goal in the work the following main tasks:
- to develop and test a method for obtaining structured discrete mathematical models performed on quadrilateral finite elements for two-dimensional geometric objects using the Poisson equation;
- to develop methods for optimizing the generation of structured discrete models of geometric objects by choosing the method of initial partitioning;
- to develop approaches to control the shape and intensity of grid lines thickening to a given area of the model of two-dimensional and three-dimensional geometric object.
The Introduction substantiates the topicality of the thesis, outlines its relationship to scientific and technical research projects. It formulates the research goal and objectives, specifies the object, subject, and methods of research, and highlights the scientific novelty and practical value of the obtained results. It sketches out how the research results were used in practical cases.
The first section reviews the current state of generation of structured discrete models, which is a continuation and addition to previous studies, namely: describes the basic steps for building a grid; algebraic, elliptic, variational, hyperbolic methods of generation of structured and block-structured grids are considered; analyzed special software tools that are used to generate computer grids; the numerical methods used at generation of grids are resulted; a wide range of objects and processes have been studied, for which structured grids are used. The generation of a structured grid of electron density in a given region of various quantum points with different wave numbers is also performed.
In the second section, a mathematical apparatus was developed for constructing structured grids in a freely distributed Scilab software package by the differential method on the example of the Poisson equation with given parameters of control function condensation and model quality assurance. The influence of the geometry of the region and the choice of the method of constructing the initial grid on the generation speed of a given structured model of this region by the elliptical method is investigated by empirical method.
In the third section, the influence of the parameters of the control functions of the Poisson equation on the thickening of the grid of surfaces of different types was investigated. Rotation surfaces are considered in more detail, as such surfaces are widely used in the modern technical industry, namely: aircraft construction, rocketry, etc. Therefore, the thickening of the grid on the surfaces of the cylinder, the cone, as well as at the junction of the cylindrical and conical, cylindrical and spherical, two conical surfaces was investigated. The compression of the grid on the surface is also investigated, which simulates the probability density of an electron in a given region of a quantum dot on the example of a cubic quantum dot.
The fourth section discusses the generation of structured discrete models for three-dimensional objects. The study was performed on the example of the cross section of the angular and direct connection of two beams. The fourth section shows the solution of the boundary value problem of bending a thin plate, solves the difference schemes of Sophie Germain's equation, condenses the grid to certain construction zones for different plate shapes.
The conclusions emphasize that the methods of generation of structured discrete models of geometric objects developed in the dissertation work allow to qualitatively increase the results of mathematical modeling, to use them in the analysis and optimization of engineering structures. A software product was created in the freely distributed Scilab engineering software package, which allows the elliptical method to automate the generation of structured discrete models of geometric objects such as two-dimensional curvilinear quadrilaterals, body surfaces, sections of three-dimensional objects. The obtained solutions of dissertation research problems can be used by design organizations and productions as applications of mathematical modeling of geometric objects.
