Mostovenko O. Geometric models of physical fields

The dissertation is devoted to the development of the methodology of applied geometry in the directions of expanding the tools of geometric modeling of physical processes and phenomena on the basis of creating a new generalized geometric model of physical fields, in particular the formation of energy fields, research of varieties of physical nature, which cannot be clearly imagined without a geometric model, which are current problems of both applied geometry and other scientific and technical fields. The proposed geometric models are based on a new method of interpolation of points taking into account the influence of distances from current interpolation points to given points on the interpolation result. In the geometric modeling of energy fields there are two main tasks: 1) for the known potentials of individual points of the field to restore the potentials of other points of the field; 2) for given energy sources to determine the potential of an arbitrary point of the field, taking into account the distances from the points of the field to energy sources. The solution of the first problem is proposed as a discrete interpolation of given points in four-dimensional space on a grid with a uniform step. The solution of the second problem is based on the method of continuous interpolation of points, taking into account the influence of distances from the current interpolation points to the given points on the interpolation result. This effect is taken into account due to a special parameter t, which can be determined by the two proposed schemes. This parameter is also taken into account when modeling energy fields. The potential of an arbitrary point of the energy field is defined as the sum of the products of the capacities of point energy sources for the corresponding parameters t. Linear and planar energy sources are considered in discrete form as sets of point sources. Visual representation of energy fields is considered as a one-parameter set of isosurfaces of equal potentials, on which a four-dimensional variety of energy field stratifies. It is shown how in the geometric model of the energy field to take into account the reflected and absorbed energy in the presence of flat screens. The methods of solving inverse problems in modeling energy fields are given, when the parameters of energy sources are determined by the given parameters of individual points of the field. A number of problems of optimization of energy field parameters in relation to the problem of energy saving in architectural design are formulated and solved. Key words: geometric model, interpolation, energy field, physical field, energy potential, power, point energy source, linear energy source, planar source, influence of distance, function, experimental data, parabolic dependence.