ИСТИНА

The method of boundary integral equations is based on the integral representation of the electric and magnetic fields through surface integrals.At that, the conditions at infinity are fulfilled automatically and the issue of enlargement of the domain, covered by the spatial grid, for the correct consideration of these conditions is eliminated. In the problems of diffraction the latter issue is particularly acute because the step of spatial discretization is de_ned not only by the complexity of geometry but by the length of wave as well (the step of discretization should be considerably less than the wave length). The limitation imposed by the wave length on the grid step is often determinative and does not allow employment of the non-uniform grids. In case of the numerical solution of boundary integral equations, the problem is reduced to the system of linear algebraic equations with dense matrix. A significant increase in the number of cells of the partition used for the solution can be achieved by applying special methods for compressing dense matrices and fast matrix algorithms. Such methods are widely used in problems of electrodynamics, allowing to significantly increase the computational complexity of the problems being solved. At the same time, the extension of the classes of solved wave scattering problems requires the progress in the of the boundary integral equations method. The problem of electromagnetic diffraction in a piecewise homogeneous medium that can consist of domains with different dielectric properties and can contain ideally conducting inclusions in the form of solid objects and screens is considered in the present work. This problem is reduced to a system of boundary integral equations with hypersingular integrals over surfaces separating the media with different dielectric properties and over surfaces of the ideally conducting objects. This approach is specified by the possibility to apply it both to diffraction problems on bodies and to diffraction problems on thin screens. The method of hypersingular integral equations has proved to be universal for the construction of efficient numerical algorithms in the modeling of a broad class of electrodynamic processes. An approach based on expanding a strong singularity in the sense of the Hadamard finite value with the use of methods of piecewise constant approximations and collocations for the discretization of these equations is used for the numerical solution of hypersingular integral equations. The proposed numerical method is based on ideas borrowed from the vortex frame method, widely used in computational aerodynamics.