Analytical Regularization Procedure Research Articles

NATURAL ELECTROMAGNETIC MODES OF A COMPOSITE OPEN STRUCTURE INVOLVING A PERFECTLY CONDUCTING STRIP GRATING, AN INHOMOGENEOUS FERRITE LAYER, AND A MONOLAYER OF GRAPHENE

Subject and Purpose. Сonsidered are the natural modes and their correspondent eigenfrequencies of a composite structure which is nonuniform along one of the coordinates and consists of a lossy ferromagnetic layer placed in a static magnetic field. The layer involves a perfectly conducting strip grating at one of its boundaries and a graphene monolayer at the other. Methods and Methodology. The above stated problem can be approached within the analytical regularization procedure developed for dual series equations. The latter concern a broad class of diffraction problems which include, in particular, the diffraction of monochromatic plane waves on strip gratings placed at the boundary of a gyromagnetic medium. The amplitudes of the electromagnetic eigenmodes can be obtained from the infinite set of homogeneous linear algebraic equations solvable within a truncation technique. The roots of the system’s determinant represent complex-valued eigenfrequencies of the system under investigation. The material parameters adopted in our computations for the ferromagnetic layer correspond to such of yttrium iron garnet. Results. A number of numerical programs have been developed which permit analyzing the dependences of wave field eigenfunctions and complex eigenfrequencies upon geometrical parameters of the structure (such as grating slot width and period, and thickness of the lossy layer), as well as on electrodynamic parameters of the ferromagnet and graphene characteristics, specifically the chemical potential and relaxation energy of electrons. A number of behavioral regularities have been established, as well as the effect of non-uniformity of ferrite layer parameters upon the structure’s eigenfrequencies and wave field eigenfunctions. Conclusions. The structure under study has been shown to be is an open oscillatory system with a set of complex-valued natural frequencies demonstrating finite points of accumulation. The real parts of these eigenfrequencies lie in a certain interval determined by characteristic frequencies of the ferrite layer, while the imaginary parts are negative, such that the correspondent natural modes show an exponential decay with time. The grating edges represent the mirrors which the natural surface oscillations are reflected from, being supported at that by the ferromagnetic medium. The results obtained in this paper can be useful for creating the elemental base for microwave devices and the devices themselves.

Diffraction by semi-infinite cone formed with electric and magnetic surfaces: analytical regularization and Wiener–Hopf techniques

The problem of axially symmetric E-polarized electromagnetic waves diffraction from the circular cone formed by the junction of the perfectly electro-conducting (PEC) semi-infinite truncated cone and the perfectly magneto-conducting (PMC) finite cone is studied rigorously by applying the Kontorovich–Lebedev integral transformation technique, the mode-matching method, and the analytical regularization procedure. The convolution-type operator and its inverse operator are used to conduct the regularization procedure and to reduce the problem to the infinite systems of linear algebraic equations of the second kind. The left side and the right side of the regularization procedures are considered. Transition from the cone to the PEC plane with the circular magnetic window is analysed. The exact solution of the problem is obtained for the static limit, and the low-frequency approximate is derived. The Wiener–Hopf technique is also applied to obtain the solution. Both solutions obtained using the analytical regularization method and by the Wiener–Hopf technique are compared, and the relation between them is derived. Within the framework of the Macdonald’s model, the solution is applied to analyse the far field scattered from the two partially absorbing conical scatterers; (1) the PEC semi-infinite cone with the “black” vertex; and (2) the semi-infinite PMC cone with the “black” truncated conical surface.

Scattering of the plane acoustic wave from a finite hollow rigid cone at oblique incidence

AbstractThe paper deals with solving of the diffraction plane wave problem from a finite rigid hollow cone at oblique incidence. The diffraction problem is treated in terms of the scattering velocity potential by means of the mode matching technique and the analytical regularization procedure. The unknown expansion coefficients are found from an infinite system of linear algebraic equations of the second kind which allows us to obtain a solution with a desired accuracy. The influence of the cone parameters and angles of plane wave incidence on scattering characteristics of the cone is discussed. The obtained numerical results are verified by testing the satisfaction of the mode matching conditions and by comparing our calculations with those known for the rigid disc.

Axially symmetric TM-wave diffraction from bi-cone formed by semi-infinite and truncated semi-infinite cones

The problem of axially symmetric TM-wave diffraction from the perfectly conducting bi-cone formed by the semi-infinite and truncated semi-infinite cones is considered. The bi-cone is excited by the ring magnetic source. The problem is formulated in spherical coordinate system as a mixed boundary value problem for Helmholtz equation with respect of the unknown diffracted field. The solution of the problem is reduced to the infinite set of linear algebraic equations (ISLAE) of the second kind by means of the mode matching technique and the analytical regularization procedure. Two sorts of the second kind ISLAE are obtained based on the left- and right-side regularizations. The unknown expansion coefficients can be determined from each of the ISLAE with the given accuracy by the reduction method. The correlations between the ISLAE of our problem and of the supplementary problem of wave diffraction by the bi-cone formed with finite and semi-infinite shoulders are found. The approximate solution is obtained in the analytical form for the small aperture of the truncated shoulder. The numerically obtained results are focused on the practically important diffraction from the bi-cone formed by an infinite plane with circular hole and the semi-infinite conical probe.

RESONANCE WAVE SCATTERING BY A STRIP GRATING ATTACHED TO A FERROMAGNETIC MEDIUM

The difiraction of a uniform unit-amplitude E-polarized plane wave is considered in the case of its normal incidence on a strip periodic metal grating placed on the anisotropic hyrotropic ferromagnetic half-space boundary. The Dirichlet boundary conditions on the grating strips, the medium interface conjugation conditions, the Meixner condition that the energy is flnite in any conflned volume and the radiation condition are applied, and the boundary value difiraction problem in terms of Maxwell's (Helmholtz) equations is equivalently reduced to the dual system of functional equations with exponential kernel. The system is shown to be the Riemann-Hilbert problem in analytic function theory with the conjugation coe-cient difiering, in general, from \i1 and dependent on the incident wave frequency. An analytical regularization procedure based on the Riemann-Hilbert boundary value problem solution with the following use of the Plemelle- Sokhotsky formulas is suggested, resulting in the system of linear algebraic equations of the second kind with a compact operator. For these systems, the truncation technique possibility has been shown.

Elliptical to Linear Polarization Transformation by a Grating on a Chiral Medium

The phenomenon of wave polarization conversion occurs when an electromagnetic wave of elliptical polarization is incident upon a layered structure comprising a strip grating, magnetodielectric layer, chiral layer, and screen. For suitable choice of structural parameters, we discovered regimes where the specularly reflected wave is nearly totally transformed into a linearly polarized wave. Due to the presence of a chiral medium, the corresponding diffraction problem is an intrinsically vectorial problem.It is solved using an analytical regularization procedure based on the Riemann-Hilbert problem method and the obtained solution thus admits an effective numerical treatment.It is shown that a nearly total polarization transformation can be realized for different elliptical polarisation parameters of the incident wave over a rather wide range of incident wave angles. The exhibition of polarization transformation is affected by the resonance properties of the grating-screen volume and by the grating. The purity of polarization conversion may be effectively controlled by careful choice of the structure parameters.

Highly efficient full-vectorial integral equation solution for the bound, leaky, and complex modes of dielectric waveguides

A full-vectorial contour integral equation analysis of the natural modes of dielectric waveguides (DW) of arbitrary cross section is presented. The Galerkin method, together with the Analytical Regularization procedure, is applied to discretizing and solving the eigenvalue problem. This ensures the fast convergence and superior accuracy of the numerical algorithms. The waveguide cross section is characterized by a parametrical curve defining its contour, with a limited curvature at each point. This avoids the singularity points at corner regions and provides accurate results, even for waveguides with virtually sharp corners. Both fundamental and higher order mode propagation characteristics are studied in the bound, leaky, and complex regimes. Numerical results consistent with other theories and experimental data are presented for a wide range of practical dielectric waveguides that demonstrate the efficiency, accuracy, and versatility of the method developed. Finally, the technique is applied to model a fused fiber coupler.

E-Polarized Scalar Wave Diffraction by Perfectly Conductive Arbitrary Shaped Cylindrical Obstacles with Finite Thickness

The Analytical Regularization Method is applied to the problem of E-polarized wave diffraction by arbitrary shaped cylindrical obstacle which is perfectly conductive and homogeneous in longitudional direction. The initial electromagnetic boundary value problem is reduced to the infinite algebraic system of the second kind by means of constructed analytical regularization procedure. As it is well-known, an equation of such a kind, in principal, can be solved with any predetermined accuracy by means of truncation procedure. Numerical implementation of corresponding analytical regularization procedure is suggested. Numerical results thus obtained show high quality of the algorithm, including relatively small values and uniform boundness of condition number of truncated algebraic systems when their size tends to infinity. By the qualitative property of infinite algebraic system of the second kind, it guarantees the stability of numerical process of truncated solving as well as the convergence of solution of truncated system to the solution of infinite system. Relevant numerical results, including condition number behaviour, current density, field space distribution and far field pattern for a few different resonant and non-resonant obstacles are presented.

Elastodynamics by BEM: a new direct formulation

A new direct BE formulation is proposed for the solution of elastodynamic problems. An analytical regularization procedure is devised via an integration-by-parts technique without introducing any hypothesis about either the discretization of the boundary geometry or the space–time interpolation of the elastic fields involved. The only requirement is for continuity in the displacement field. The regularization of the integral equations for static elasticity using the same approach, already available in the literature, is presented as a particular case of the general procedure introduced herein. The numerical implementation of this technique is discussed and two-dimensional examples are presented. Copyright © 1999 John Wiley & Sons, Ltd.

Efficient analytical integration of symmetric Galerkin boundary integrals over curved elements; elasticity formulation

Abstract Substantial improvements are reported in the computational efficiency of Galerkin boundary element analysis (BEA) 1employing curved continuous boundary elements for elasticity problems. A direct analytical treatment of the singular and hypersingular double integrations, employing a limit to the boundary definition is adapted to gain computational economy. Symbolic manipulation has been employed to derive the analytical expressions for the singular contributions to these double integrals. The analytical regularization procedure separates the potentially singular Galerkin integrands into an essentially singular but simple part, plus a regular remainder that can be integrated numerically. The exact finite contributions from the simplified singular terms are then computed analytically. It is shown that the key to containing the explosive growth in the length of the formulae associated with such a hybrid analytical/numerical integration scheme is the strategic timing of when to take the limit to the boundary. This regularization also isolates the contribution from the curvature of the boundary element, thus facilitating enhanced computational efficiency in problems with many straight elements. Although the evaluation of the integrals of the Kelvin fundamental solution essentially follows the procedures described earlier for the scalar Laplace equation, the analytical treatment, is now more involved, in terms of the complexity and number of terms present in each of these second rank tensor kernel functions. The complete analytical formulation is explicitly provided in a series of tables. Example problems indicate that, with these techniques, Galerkin symmetric BEA can be more efficient than collocation for moderate or large scale problems.

Efficient analytical integration of symmetric Galerkin boundary integrals over curved elements: Thermal conduction formulation

Substantial improvements are reported in the computational efficiency of Galerkin boundary element analysis (BEA) employing curved continuous boundary elements. A direct analytical treatment of the singular double integrations involved in Galerkin BEA, adapting a limit to the boundary concept used successfully in collocation BEA, is used to obviate significant computation in the determination of the Galerkin coefficient matrices. Symbolic manipulation has been strategically employed to aid in the analytical evaluation of the singular contributions to these double integrals. The analytical regularization procedure separates the potentially singular Galerkin integrands into an essentially singular but simple part, plus a regular remainder that can be integrated numerically. The finite contribution from the simplified singular term is then computed analytically. It is shown that the key to containing the explosive growth in the length of the formulae associated with such a hybrid analytical/numerical integration scheme is the strategic timing of when to take the limit to the boundary. This regularization also isolates the contribution from the curvature of the boundary element, thus facilitating enhanced computational efficiency in problems with many straight elements. Example problems are presented to quantify the performance of this approach. It is concluded that with these techniques, Galerkin symmetric BEA can be more efficient than its collocation-based counterpart.