Incomplete Galerkin Method Research Articles

Excitation Equations for Longitudinally-Azimutally Irregular Waveguides Taking into Account the Finite of the Walls Conductivity

Excitation equations for longitudinal-azimuthally irregular waveguides are formulated taking into account losses in the walls. The inner surface of the waveguide walls is given by an arbitrary smooth function b(ϕ, z). The coordinate transformation method replaces the original cylindrical coordinate system r, ϕ, z with a new one ρ, ϕ, z, where ρ = r/(b(ϕ, z)). The new system defines the waveguide boundary as ρ = 1 = const, i. e. the waveguide geometry transforms as a regular cylinder. Taking these functions into account, the standard procedure of the incomplete Galerkin method is used to determine the amplitudes of partial waves. The resulting general equations can be used in the calculation and optimization of both microwave and EHF electronic devices of various types, as well as passive microwave devices of various applications.

Calculation of special functions arising in the problem of diffraction by a dielectric ball

To apply the incomplete Galerkin method to the problem of the scattering of electromagnetic waves by lenses, it is necessary to study the differential equations for the field amplitudes. These equations belong to the class of linear ordinary differential equations with Fuchsian singularities and, in the case of the Lneburg lens, are integrated in special functions of mathematical physics, namely, the Whittaker and Heun functions. The Maple computer algebra system has tools for working with Whittaker and Heun functions, but in some cases this system gives very large values for these functions, and their plots contain various kinds of artifacts. Therefore, the results of calculations in the Maple11 and Maple2019 systems of special functions related to the problem of scattering by a Lneburg lens need additional verification. For this purpose, an algorithm for finding solutions to linear ordinary differential equations with Fuchsian singular points by the method of Frobenius series was implemented, designed as a software package Fucsh for Sage. The problem of scattering by a Lneburg lens is used as a test case. The calculation results are compared with similar results obtained in different versions of CAS Maple. Fuchs for Sage allows computing solutions to other linear differential equations that cannot be expressed in terms of known special functions.

On the Calculation of Electromagnetic Fields in Closed Waveguides with Inhomogeneous Filling

The paper investigates waveguides of constant cross-section with ideally conducting walls and arbitrary filling. The problem of finding the normal modes of a waveguide in a full vector formulation has been set and discretized. In the framework of numerical experiments, the guiding and evanescent modes of the waveguide are calculated for several variants of the fillings. The problem of diffraction of the normal waveguide mode incident on the joint of two waveguides, the cross-sections of which coincide, and the filling at the junction varies abruptly, is set and discretized. The results of numerical experiments for specific configurations of waveguide joints are presented, and the transmission and reflection coefficients of the guided modes are calculated. The solution of the Maxwell equations system is based on the decomposition of fields with the help of four potentials, and in the present work a symbolic-numerical method is realized that uses this approach. The numerical experiments presented in this paper show that the proposed approach and the method on its basis allow the effective calculation of various characteristics of waveguide systems. The adequacy of the approach used is also evidenced by comparing the results obtained with the results of V.V. Shevchenko for the diffraction problem at the junction of two open waveguides The symbolic-numerical method used in the work is implemented in the computer algebra system Maple, in particular, the calculations of matrix elements in the framework of the incomplete Galerkin method are carried out in symbolic form to accelerate further calculations using numerical methods.

Analysis of the incomplete Galerkin method for modelling of smoothly-irregular transition between planar waveguides

The paper deals with a numerical solution of the problem of waveguide propagation of polarized light in smoothly-irregular transition between closed regular waveguides using the incomplete Galerkin method. This method consists in replacement of variables in the problem of reduction of the Helmholtz equation to the system of differential equations by the Kantorovich method and in formulation of the boundary conditions for the resulting system. The formulation of the boundary problem for the ODE system is realized in computer algebra system Maple. The stated boundary problem is solved using Maples libraries of numerical methods.

Modeling Open Transitions of the “Horn” Type between Open Planar Waveguides

The generalization of the incomplete Galerkin method to the description of the transitions of the “horn” type between open planar waveguides is discussed. The obtained result is characterized by a high degree of analyticity of the derived equations and this is expected to enhance the efficiency of the eigenwave modeling in open irregular waveguides.

Analysis and optimization of matching two-dimensionally periodic dielectric structures with cone-shaped cavities

The problem of scattering of plane electromagnetic waves by two-dimensionally periodic dielectric structures with cone-shaped cavities placed on a homogeneous dielectric substrate is considered. A modified hybrid projection method for solving this problem, which combines the coupled-wave method (incomplete Galerkin method), the projection field matching on domains’ boundaries, and the one-dimensional finite-element method, is proposed. The modified method enables one to avoid the use of the excessive number of unknown expansion coefficients, typical for the previous versions. The method is used for calculating the effective permittivity of a dielectric layer of finite thickness with cylindrical holes and for optimizing the parameters of conical cavities providing the matching of a dielectric half-space for various incidence angles.

Joint application of the incomplete Galerkin method and scattering matrix method for modeling multilayer diffraction gratings

A mathematical model of wave diffraction from a one-dimensional multilayer periodic grating is considered. A rigorous mathematical statement is given for the problem where partial radiation conditions are used for domain confinement. A complete formulation of the numerical algorithm based on a combination of the incomplete Galerkin method and the scattering-matrix method is presented. The algorithm’s convergence rate and computational complexity are analyzed and the optimal values of the computational parameters are recommended.

Modeling a waveguide with a nonlinear insert with a quadratic nonlinearity

The scalar problem of the scattering of a wave from a nonlinear insertion lying in the interior of a waveguide is reduced by the incomplete Galerkin method to the boundary value problem for a Hamiltonian system. The cases in which this problem admits a solution in finite terms are indicated. Examples are given to illustrate specific phenomena due to the nonlinearity of the problem.

Mathematical modeling of waveguides with fractal insets

The paper is devoted to mathematical modeling of rectangular dielectric waveguides with local periodic and fractal insets. A three-dimensional scalar boundary-value problem for the Helmholtz equation is considered. The solution is obtained numerically by means of incomplete Galerkin method and finite-difference techniques. The transmission spectra for one-, two- and three-dimensional periodic and fractal insets are computed and compared with each other. Practical applications are suggested.

Mathematical modeling of waveguide transitions

Results of simulation of waveguide transitions performed at the Department of Mathematics of the Faculty of Physics, Moscow State University, are presented. Synthesis of waveguide transitions is based on the incomplete Galerkin method and the finite-difference technique formulated in the direct and projection forms (the finite-element method) and involves the Tikhonov regularization method.

Application of the incomplete Galerkin method to solve problems of diffraction of electromagnetic waves by a non-uniform cylinder

The problem of the diffraction of a plane electromagnetic wave with an arbitrary angle of incidence by an infinite cylinder is solved using the incomplete Galerkin method. The functions describing the distribution of the permittivity and magnetic permeability over the cross-section of the cylinder are assumed to be piecewise continuous. A justification of the method is given and numerical results are presented.

Galerkin's method in non-selfadjoint boundary value problems of the theory of waveguides

A generalization of the incomplete Galerkin's method to complex spaces with a pseudo scalar product is given, in which the basis system of the complex eigenfunctions is orthogonal. Using this method a numerical algorithm is constructed, and its efficiency is demonstrated using the example of the solution of the non-selfadjoint boundary value problem of the theory of a plane impedance waveguide.

A method of solving two-dimensional problems of diffraction by periodic structures

A numerical method, which is a synthesis of the incomplete Galerkin method and the semi-inversion method, is described for the two-dimensional problem of diffraction by a periodic structure with a non-coordinate form of the period. A rigorous proof of this approach is given. The results of calculations for a model problem are presented, which prove its effectiveness.

The stability of the incomplete Galerkin method in problems of wave diffraction at a bounded body

The problem of wave diffraction at a bounded body is examined, and the stability of an approximate solution obtained by the incomplete Galerkin method is proved.

Stability to rounding errors of directed orthogonalization

We described earlier a method of directed orthogonalization for solving boundary value problems for systems of linear ordinary differential equations. The method may be used for the numerical solution of the boundary value problems resulting from application of the incomplete Galerkin method to problems of wave diffraction in a waveguide with inhomogeneous filling, and gives high efficiency. We show that, given a sufficiently small orthogonalization step, the method enables a numerical algorithm to be constructed which is stable to computer rounding errors.