Semerich Y. Mathematical modeling of electromagnetic processes for domains with a cyclical symmetry and geometric singularities by the R-functions method

The object of research is electromagnetic processes, which are described by the Maxwell equations in differential form. The purpose of work is development, proving and program realization of new effective methods for numerical analysis of electromagnetic fields in domain possessing a cyclical symmetry and containing geometric singularities. Methods of researches are mathematical tools of the R-functions theory for building normalized boundary equations of geometric objects with non-classical form; the R-functions method for building completeness structure of solutions for boundary value problems; polynomial and spline-approximation methods for approaching undefined components of structure of solutions; variation methods of Ritz and least square for functional minimization and determination of unknown coefficients of structure of solutions; methods of differential corteges algebra for differentiation; Gauss quadrature formulas for numerical integrations; linear algebra methods for solution systems of linear algebraic equations. Theoretical results of work are development new methods for boundary value problems solution in domains possessing a cyclical symmetry and containing geometric singularities. Practical results of work are elaborating mathematical and program tools for numerical investigation of physical-mechanical fields in domains possessing a translation or cyclical symmetry and containing geometric singularities. Scientific newness of developed results is for the first time new methods for solution boundary value problems of electromagnetic fields investigation in domain possessing cyclical symmetry and containing geometric singularities. Method for building of normalized boundary equations for geometric objects with translation and cyclical symmetry elaborated in the work allows to decrease number of base functions and R-operations for expressions of boundary equations, and also to automatize of process for building normalized boundary equations indicated geometric objects and to conduct multivariants numerical experiments for computer modelling of different physical-mechanical fields for symmetry domains. Results of scientific researches are implemented in training process Kharkiv national university named V.N.Karazin at the reading of lection courses. The implementation is planning for conducting of applied investigations in Institute of apply mathematics and mechanics of Ukrainian NAS (Donetsk), Institute of electrodynamics of Ukrainian NAS (Kiev), Institute of strength problem of Ukrainian NAS (Kiev), Institute of technical mechanics of Ukrainian NAS (Dnipropetrovs'k), Institute of mechanics named S.P.Timoshenko of Ukrainian NAS (Kiev) and other. Results of investigations can be used at the practice for computer modelling different physical-mechanical processes in elements and equipments possessing geometric symmetry of translation and cyclical type and containing geometric singularities, for example, in radio engineering, aircraft building, machinery, building and other.