The pattern equation method (PEM) has been used as the main method of studying the characteristics of diffraction. In this paper, the method is extended to solving three–dimensional diffraction problems for a group of solids and solids of a complex shape for which the Sommerfeld–Weyl integral representation (the key one for the PEM) turns out to be diverging towards parts of the diffractor boundaries.
The PEM was first offered by A.G. Kyukrchan for solving the two–dimensional scalar problem of wave diffraction by a compact obstacle. He later used this method to solve a wide range of problems in the theory of wave diffraction, scattering and propagation. It is about such problems as wave diffraction by a single solid and a two–dimensional group of solids both in a homogeneous medium and in a layered medium, wave diffraction on a mattress periodic array near the medium boundary surface, wave propagation in dielectric waveguides with a complex cross section and others.
The scientific novelty of this paper is the extension of the PEM to scalar and vector three-dimensional problems of wave diffraction by a group of solids and by solids of a complex shape with various variants of boundary conditions. This method has no analogs among both Russian and foreign methods. It has high convergence speed of the computational algorithm that weakly depends on the shape of diffractors and on the distance between them.
The method is based on reducing the initial boundary value problem for Maxwell's equations to a system of integral operator equations of the second type relative to scatter plots. To do it, the generalized Sommerfeld–Weyl representation of the diffraction field in the form of a plane wave integral whose spectral function is a scatter plot is used. After that this system of integral operator equations is algebraized through the series expansion of the function in question by vector angular spherical harmonics forming an orthogonal basis on a unit sphere and then through projecting the left and right parts of the equality onto the same basis. In case of certain restrictions on the geometry of the problem that can be strictly imposed, the resulting infinite algebraic system can be solved with the help of the reduction method.
The main idea of the pattern equation method used to get the system of integral operator equations comes down to the fact that it is possible to extend the wave field by the scatter plots up to the convex hulls singularities, which in its turn makes it possible to find these plots out of the corresponding boundary value problem.
It should be also mentioned that the pattern equation method is strongly valid and it has strict validity boundaries. It distinguishes the pattern equation method from other numerical methods that have no explicit strong substantiation.
The results of the performed research show that it is possible to use the pattern equation method to solve the problems of diffraction by groups of solids and by solids of a complex shape in various frequency ranges with the preciseness acceptable for practical purposes. Besides, it has been established that the pattern equation method can be applied to effectively model the characteristics of diffraction for various solids of a complex shape and randomly heterogeneous material parameters by replacing them with a set of small diffractors.
The numerical check of Ufimtsev's theorem was done during the consideration of the pattern equation method for a group of solids with impedance boundary conditions. According to the theorem, integral diffraction diameter of a black body is two times smaller than the integral diffraction diameter of an ideal waveguide body with the same shadow contour. The results of this check show hat it is valid for a group of solids in case their size is equal to several wavelengths of the active field. It is also shown that the obtained results correlate well with data for single diffractors.
The author has obtained and is ready to defend the following results:
| — | Extension of the pattern equation method to three–dimensional problems of wave diffraction by a group of solids and by solids of a complex shape with various boundary conditions. |
| — | It has been established that the convergence speed of the pattern equation method algorithm for groups of solids of various shapes weakly depends on the shape of diffractors and on the distance between them. |
| — | The algorithm of numerical solution for scalar and vector problems both for groups of solids of various shapes and for various boundary conditions has been offered and implemented on the basis of the pattern equation method. |
| — | The mutual influence of objects located at various distances from each other has been studied and the limits when this distance can be ignored without serious losses in preciseness have been established. |
| — | It has been shown that it is possible to model the characteristics of diffraction by solids of a complex shape and random material parameters by replacing them with a set of objects of a simpler shape. |
| — | It has been shown that the pattern equation method makes it possible not only to find scatter plots of solids, but also use them to restore the field practically in any point in space. It has been also shown that normalized scatter plot and field at distances meeting the far–field zone condition practically coincide. |
| — | The correctness of all numerical researches has been evaluated using both the check of the optical system that has showed good preciseness that does not depend on the distance between diffractors and the comparison with other numerical methods. |
The patter equation method is strictly valid. The correctness of the obtained results has been evaluated through checking the optical theorem. Whenever possible, the results of scatter plot calculation have been compared to the results obtained with the help of other numerical methods.
