N. S. Arkhipov - Dr.Sc. (Eng.), Associate Professor, Deputy Head of Research and Testing Institute of the Academy FSS of Russia
I. S. Polyanskij - Ph.D. (Eng.), Research Scientist, the Academy FSS of Russia. E-mail: van341@mail.ru
D. E. Stepanov - Junior Research Scientist, the Academy FSS of Russia

The method of analysis of the field distribution in the aperture area of a regular waveguide with an arbitrary cross section is proposed in the article based on the barycentric approach. Expanding waveguide proposed solution is defined as a polygon. The method has been developed with the use of the barycentric approach with the objective of improving the accuracy of the determination of the critical frequencies for various natural waves (modes) and the field distribution in the aperture of the waveguide at comparable with the finite element method of first and higher order capacitive and computational cost. In accordance with the well- known principle of minimum potential energy determination of the field distribution in the aperture of the waveguide is made on the basis of numerical solutions formed from the scalar (or vector ) of the Helmholtz equation system of linear equations. The novelty of the solution lies in the potential approximation in the basis of N-dimensional (the number of vertices of a convex polygon that defines the aperture of a regular waveguide with arbitrary cross-section) orthogonal basis functions. Formation of basis functions is invited to serve on the basis of a combination of polynomials of Lagrange type in N-dimensional system of barycentric coordinates. The barycentric method application doesn-t require the implementation of the sampling procedures followed by crosslinking surface symmetric matrix and Dirichlet matrix calculation for the case of the field distribution in the aperture of the waveguide with a cross section given convex polyhedron. If the aperture of the waveguide is presented by an arbitrary (non-convex) polygon the applicability of the method allows sampling of the aperture by a minimal set of convex polygons with an arbitrary number of vertices, and further combining (stitching) metric matrices and Dirichlet matrices of formed domains. Numerical evaluation of the proposed method is carried out in comparison with the finite element method of first and higher orders during the analysis of the field in the aperture of rectangular and U-shaped waveguides. In terms of these results, a computational experiment showed preference to the developed method.