Choporov S. Mathematical Modeling and Analysis of Objects’ Geometric Shapes for Computer-Aided Design in Mechanical Engineering

The object of the research is mechanical engineering objects. The subject of the research is mathematical models of geometric shapes of mechanical engineering objects. This research uses methods of analytical geometry, mathematical analysis and mathematical modelling: the hybrid function representation of geometric shapes uses the theory of R-functions; the meshing methods employ algorithms of analytical geometry and the binary search method; the methods for nodes smoothing use the coordinate descent and conjugate gradient methods; the adaptive remeshing method employs the finite element analysis; the CAD software uses the object-oriented approach. In the process of the research, CAD software for mechanical engineering objects’ simulation has been developed. Finally, the following new scientific results have been obtained: the hybrid function representation of geometric shapes has been developed; this representation allows to combine implicitly and boundary defined shapes in one model of a domain; the projection method of boundary meshing for function represented objects has been developed; the method for boundary meshes smoothing has been developed; the local refinement method for boundary meshes has been improved; this method unified for both triangular and quadrilateral surface meshes; the method for meshing of two- and three-dimensional function represented objects into solid elements has been developed; the method of inner node’s smoothing has been developed; this method employs minimization of an area functional (in 2D) or a volume functional (in 3D); this method allows to remove degenerated elements and to improve a mesh quality; the method for adaptive meshing using solid elements for function represented objects has been developed; this method uses the finite element method and error estimation to indicate elements for subdivision; the parallel meshing and finite element analysis techniques have been developed; these techniques allow to improve performance of the simulation; the domain-specific language for formalization of function represented geometric shapes and finite elements analysis problems has been developed. Scientific results obtained in the dissertation is the solution of the scientific and technical problem of increasing efficiency of mechanical engineering objects’ design by improving an accuracy of mathematical modelling and reducing time of the simulation. The following scientific and technical effects are possible: improving of the mechanical engineering objects visualization; cost reduction replacing physical experiments by numerical simulations based on discrete models with improved accuracy. The methods obtained in the dissertation could be employed by design departments and scientific organizations for the CAD software development. Results of the dissertation have been used in the educational process of the software engineering department of Zaporizhzhia National University in the following courses: «Parallel and Distributed Computing» and «Software for Scientific Researches». Results of the dissertation have been used in the researches, which are funded by Ministry of Education and Science of Ukraine and by the Grant of the President of Ukraine for young scientists. Results of the dissertation also have been employed in the researches, which are funded Yuzhnoye State Design Office.