The problem of diffraction of a TE-polarized electromagnetic wave by a circular slotted cylinder is investigated. The boundary value problem in question for the Helmholtz equation is reduced to an infinite system of linear algebraic equations of the second kind (SLAE-II) using integral summation identities (ISI). A detailed study of the matrix operator of the problem is performed and its Fredholm property in the weighted Hilbert space of infinite sequences is proven. The convergence of the truncation method constructed in the paper for the numerical solution of SLAE-II is justified and the results of computations are presented and discussed, specifically considering the determination of resonance modes.
In this paper we study inverse problems of electric conductivity that arise in the design of spherical shielding or cloaking shells and other functional devices used to control static electric fields. The shells are considered consisting of a finite number of layers filled with homogeneous isotropic or anisotropic medium. The inverse problems under study are reduced to control problems with the layers electric conductivities taken as controls. A different choice of minimized functionals allows us to solve a wide range of design problems using one approach. A numerical algorithm to solve these problems is based on particle swarm optimization. Various results of numerical experiments are discussed in order to find the most effective designs. The findings obtained in this study describe a broad set of specific easy-to-manufacture structures that have the highest cloaking or shielding performance in the class of layered shells.
We investigate a generalization of one-parameter eigenvalue problems arising in the theory of wave propagation in waveguides filled with nonlinear media to more general nonlinear multi-parameter eigenvalue problems for a nonlinear operator. Using an integral equation approach, we derive functional dispersion equations (DEs) whose roots yield the desired eigenvalues. The existence of the roots of DEs is proved and their distribution is described.
The paper deals with different formulations of mathematical models for the analysis of processes of resonance scattering and generation of plane wave packets on isotropic, nonmagnetic, linearly polarised media with a nonlinear, layered dielectric structure of cubic polarisability. For each formulation, sufficient conditions for the existence and, partially, uniqueness of the corresponding solution are derived.
In the frequency domain, the resonant properties of nonlinear structures are determined by the proximity of the scattering/generation frequencies of the nonlinear structures to the complex eigenfrequencies of the corresponding homogeneous linear spectral problems with the induced nonlinear permeability of the medium. Here the case of cubically polarisable, canalising, and decanalising layers is considered.
Metasurfaces have been extensively exploited in recent years for mantle cloaking applications. In this type of problems it is of fundamental importance to determine the connection between the metasurface geometrical parameters and the realised value of surface impedance, in order to properly design the metasurface. In this paper the surface impedance of a non homogeneous metasurface, based on a sinusoidally modulated metallic pattern is analysed.
Reconstruction of electromagnetic parameters of artificial materials is an urgent problem [1], [2]. In practice, as a rule, these parameters cannot be directly measured. Therefore, methods of mathematical modeling must be applied. Conventional waveguide measurement equipment employs network analyzers and dielectric samples having the simplest shape of a parallelepiped adjusted to the walls of a standard rectangular waveguide. Such setups open the way to use well-developed analytical-numerical techniques [3], [4] of determining permittivity and other parameters of the inclusion using accurate closed-form solutions and asymptotic approximations. This work is a continuation of the series of papers [3], [4] devoted to the analysis of inverse problems of finding permittivity and permeability of layered materials. We apply a specially developed mathematical and numerical technique that combines analytical and numerical approaches providing the results with guaranteed accuracy. Particularly, we develop the methods [3] of determining permittivity and permeability tensors of anisotropic materials in a rectangular waveguide.
A great variety of new and artificial materials have been created during the last decades, such as metamaterials, various composites and many others. To analyze and determine the characteristics of such materials advanced mathematical methods are applied. This work is devoted to the developing the technique for reconstruction of electromagnetic parameters of multi-sectional anisotropic diaphragms in a waveguide of rectangular cross section. To solve this inverse problem we make use of the method of rotation applied for anisotropic materials and reconstruct the quantities to be determined, including the diaphragm positions, from the values of the transmission coefficient measured at different frequencies. The efficiency and accuracy of computations are validated in the course of analytical-numerical solution to the inverse problem. The developed techniques and obtained results can be implemented in practical measurements when anisotropic materials and media with unknown properties are investigated with the aid of commonly used waveguide devices and analyzers.
The results are presented of the numerical solution to the forward problem of the electromagnetic wave scattering by inhomogeneous dielectric bodies in a waveguide and inverse problem of determining parameters of dielectric inclusions from the values of the transmission coefficient of the scattered electromagnetic wave. A comparison with the results of mesurements and analysis of the achieved threshold computational accuracy levels enable one to estimate and validate the simulation results and formulate the ways of improving the model and codes.
The combined system with microstrip antennas (f1=1.85GHz; f2=3.7GHz, f3=1.75GHz, f4= 3.5GHz) is presented. The measurements of the prototype characteristics have shown that elevation angle of peak directivity of the antennas is oriented to zenith. Simulated S11 at stated frequencies with the following minimum values of S11 = -34dB, f=1.85GHz; S11=-21dB, f=3.7GHz; S11 = -22dB, f=1.75GHz; S11=-19dB, f=3.5GHz are observed.
A numerical-analytical method of reconstructing complex frequency-dependent permittivities is proposed. Measurements of the transmission coefficient for three typical metamaterial antenna units are performed and the frequency-dependent effective complex permittivity of the units is reconstructed in a wide frequency range.
A novel resonance method for determining the complex permittivity of materials in the X-band is proposed. A distinctive feature of this method is that the sample under test, located in a radio-transparent isotropic and homogeneous dielectric matrix, is both a local inclusion and the excitation element of the resonance in the microwave module 'waveguide–dielectric matrix–inclusion'. The complex permittivity of an inclusion is determined by comparing the measured resonance frequency and quality factor with the computed database value. The permittivity and loss factor of microwave ceramics, natural granite and ferrite are investigated, as well as crystals of lead fluoride PbF2 doped with holmium trifluoride HoF3 and erbium fluoride ErF3 of different concentrations. The method allows the permittivity and loss tangent to be determined with uncertainties of 0.1% and 5%, respectively.
In this paper a mathematical technique is developed to find the parameters of a medium in terms of its scattered electromagnetic fields. Optical nonlinearity plays an important role in finding the scattering parameters of a medium. Using perturbation theory and nonlinear inverse scattering techniques with first order, second order and third order optical nonlinearity we find scattered electromagnetic fields. Using error minimization techniques parameters are estimated in term of permittivity and permeability up to second order. © 2019 IEEE.
A novel mechanically tunable waveguide dielectric filter is presented in this paper. The resonant structure of the filter is made of a rectangular waveguide cavity and an H-plane dielectric cylinder cut in half. The resonating frequency of the proposed structure can be significantly changed by moving two split cylinder parts. The coupling matrix synthesis technique is used to create the required bandpass filter. A drawback of the proposed structure is that the tuning process invokes a mismatch between the feeding structure and the filter. To overcome this drawback, the tuning screws between the coupling irises are used. A filter with the proposed geometry has a large frequency tuning range as well as relatively constant passband bandwidth.
This paper proposes a novel mechanically tunable waveguide filter design, based on a coupled resonator topology. The resonator geometry consists of a rectangular waveguide cavity and a dielectric cylinder. Unlike conventional cylindrical resonators which are made out of a single part, here the cylinders are cut in half. By changing the distance between these two parts, the resonating frequency of the resonator can be changed, which allows for a tunable filter design. A filter using the proposed geometry has a large tuning bandwidth as well as relatively constant passband frequencies. The performance of the proposed filter is demonstrated by evaluating the filter using 3D simulation software CST Studio Suite.
In the present paper, the high-sensitivity technique for reducing the uncertainty of the free-space measurements of the real part of the permittivity of dielectric slab is proposed. The real part of the permittivity is extracted from the absolute value of the reflection coefficient measured by means of horn antenna. It is shown that making measurements for the slab under test (SUT) alone (original experimental model), in many cases results in large uncertainty because of low sensitivity of the model in the range of possible values of the real part of permittivity. The total measurement uncertainty depends on the slope of the measurement curve that can be used as a measure of the sensitivity of the model; consequently, to reduce it one has to increase the slope of the measurement curve that can be achieved by varying parameters of the model, such as the operating frequency or thickness of SUT. However, when the measurements are to be made in a non-destructive manner at a given frequency (in case of highly dispersive materials), neither frequency, nor thickness can be changed thus making it impossible to increase the slope of the measurement curve in the region of interest and, therefore, reduce the uncertainty. To overcome the problem we propose to extend the original model by introducing into it one or two auxiliary slabs (extended model) with known constitutive properties thereby extending the set of model parameters affecting the shape of the curve. Alternatively, with extended models one can achieve higher sensitivity without having to vary the frequency and thickness. Optimal choice of dimensions of auxiliary slab(s) is discussed in detail herein. The validity of the proposed method is verified through numerical experiments.
An accelerated boundary integral method for the analysis of scattering of the dominant mode by multiple multi-layered full-height circular cylindrical posts in a rectangular waveguide is presented. After some transformations the surface integral equation is converted to a system of equations whose entries can be evaluated analytically yielding Schlömilch series. Slow convergence of these series is accelerated using the Ewald technique. The proposed method gives results that are comparable in terms of accuracy with other approaches, including those incorporated in solvers HFSS and CST Studio and outperforms them in terms of computation time, especially for posts with large electrical sizes. The efficiency of the proposed method is confirmed by the examples of H-plane cylinder bandpass filters design.
A novel and efficient methodology is proposed for reducing measurement uncertainty associated with the dielectric constant of low-loss materials. The experimental setup consists of two H-plane cylindrical rods located in a rectangular waveguide. One of the rods has the unknown dielectric constant to be extracted, while the other dielectric rod whose dielectric constant is known in advance and serves as a tool for altering the shape of the dependence of the absolute value of the reflection coefficient on the dielectric constant in such a way that the resulting curve exhibits more rapid variations in the range of possible values of the dielectric constant of the post being characterized. The distance between the rods, radius of the optimizing rod, and position offset of the optimizing post serve as optimization parameters to be adjusted so that measurement uncertainty could be reduced which is confirmed by the results of numerical modeling. However, in the vicinity of the resonances, the method cannot be applied. © 2018 Electromagnetics Academy. All rights reserved.
In the present paper an approach to reduce uncertainty in the measurement of the dielectric constant of a rectangular dielectric slab situated in a rectangular waveguide is discussed. The experimental model under consideration consists of two rectangular full height and full width slabs located in an otherwise empty section of a rectangular waveguide. The dielectric constant of one of these slabs is to be measured, while the dielectric constant of the other slab is known (measured with high accuracy in advance). The slab with known constitutive properties is introduced for the purpose of altering the shape of the curve representing the relationship between the absolute value of the reflection coefficient and the dielectric constant to be measured, since curves having larger steepness in the neighborhood of the actual value of the dielectric constant result in smaller values of uncertainty. Although it is possible to change the shape of the curve by varying sample parameters, an increase in steepness obtained this way is not always sufficient. Furthermore, it is not always possible or convenient to alter dimensions of the sample under test and/or frequency. The results of the present study show that this issue can be overcome by extending the experimental model, i.e., by introducing an auxiliary dielectric slab with known constitutive properties. Additionally, it is shown that under certain conditions it is always possible to increase the steepness of the curve in the range of values in which the value of the dielectric constant is expected to fall, by varying the thickness of the auxiliary slab and distance between the slabs. The efficiency of the proposed approach is confirmed by results of numerical modeling.
We consider normal waves in multi-layered dielectric rod and Goubau line that are basic types of open metal-dielectric waveguides. The dispersion equations are analyzed using the notion of generalized cylindrical polynomials and methods of calculating determinants of block-diagonal matrices. Sufficient conditions of the existence of symmetric waves are established.
Existence of complex TM and TE waves in a dielectric waveguide of circular cross section and a Goubau line is proved by analyzing functional properties of the dispersion equations (DEs) using the theory of functions of several complex variables and validating the existence of complex roots of DE. The method proceeds from the analysis of general setting involving hybrid nonsymmetric azimuthally dependent real and complex waves.
Existence of the higher-order surface waves of the Goubau line is proved and their structure is analyzed. An efficient computational approach is proposed based on numerical solution to initial-value problems obtained by the parameter differentiation. Several applications and further research directions are discussed.
Existence of symmetric surface complex waves in a Goubau line—a perfectly conducting cylinder of circular cross-section covered by a concentric dielectric layer—is proved by constructing perturbation of the spectrum of symmetric real waves with respect to the imaginary part of the permittivity of the dielectric cover. Closed-form iteration procedures for calculating the roots of the dispersion equation (DE) in the complex domain supplied with efficient choice of initial approximation are developed. Numerical modeling is performed with the help of a parameter-differentiation method applied to the analytical and numerical solution of DEs.
We consider a homogeneous Goubau line (GL) with a lossy cover. We analyze the dispersion equation for symmetric waves with respect to the problem parameters, find radially symmetric complex waves, and examine their behavior. We show that longitudinal wavenumbers of complex waves are regular perturbations of the propagation constants of eigenwaves of lossless GL, weakly depend om the imaginary part of permittivity of the cover, and that attenuation in GL is low at higher losses.
A new method of digital radar signal processing for remote sensing is proposed. The technique allows one to obtain two-dimensional images of objects with super resolution. The method is based on solving a convolution-type two- dimensional linear integral equation of the first kind by algebraic methods.
We show that dispersion characteristics of antenna patterns should be taken into account when developing systems using UWB signals. The accuracy of measurement of angular coordinates of objects using UWB pulses is increased by optimizing their shape based on known characteristics of the antenna system and at least partially known characteristics of the investigated signal source. Optimization algorithms allow to minimize the width of the directional pattern for a given level of the useful signal.
Dispersion properties of the mutual impedance of emitters significantly alter the shape and spectrum of ultra-wideband (UWB) signals which must be taken into account when forming the signal. In order to take into consideration mutual coupling of elements we propose to use pulse characteristics in the antenna performance analysis and calculations. This approach allows us to simplify calculations of UWB antenna systems and improve their accuracy.
New methods of signal processing based on nonlinear regression methods are presented. They allow us to restore images of individual objects of group targets with superresolution at signal-to-noise ratios that are significantly lower than those provided by the known methods.
Resolution of goniometric systems on the basis of antenna arrays can be increased due to the secondary digital processing of the accepted signals. Necessary algorithms are created on the basis of solution to inverse problems.
A new method of signal processing by smart antennas is proposed and justified. It allows to improve the accuracy of angle measurement and to restore the image of the object with superresolution. The method is based on the extrapolation of the signals received by each element of the antenna array, outside the aperture. This allows introducing new virtual elements and thus synthesizing significantly larger antenna array. The method is tested in numerical experiments using a mathematical model and the maximum effective angular resolution is found for different cases and objects. Algorithms based on the method of digital aperture synthesis provide angular superresolution 3-7 times greater than that according to the Rayleigh criterion for a signal/noise ratio of 12-13 dB.
The results of theoretical studies and mathematical modeling indicate the possibility of obtaining the angular superresolution and its limits by using algebraic methods of imaging. The necessary algorithms are created on the basis of solution to inverse problems. The limiting levels of the achieved angular superresolution are found depending on the signal‐to‐noise ratio for objects of various types. A number of specific advantages of the method are shown as applied to the solution of two‐dimensional problems. Capabilities of the method are illustrated by examples. High performance of the proposed approach allows one to use it in real time when scanning and tracking targets.
We consider the problem of estimation of the right-hand sides of the Helmholtz equation that models electromagnetic and acoustic wave fields when that initial data is uncertain. In the case when measurement errors depend on solutions, we construct algorithms of computation of the optimal estimates which are compatible with the measurement data. It is shown that approximate a posteriori estimates of the right-hand sides are expressed via solutions of linear algebraic equations.
In this work we describe a method of obtaining guaranteed a posteriori estimates of unknown right-hand sides of the Helmholtz transmission problems from indirect measurements of a solution to this problem. The obtained results can be applied in various models of electromagnetics and acoustics that describe excitation of transparent bodies by sources of different kinds.
The analysis is performed and theorems are formulated concerning general form of guaranteed estimates of linear functionals from unknown data of Helmholtz transmission problems arising in electromagnetics and acoustics of inhomogeneous dielectric media.
We consider the problem of guaranteed estimation of unknown right-hand sides of the equations entering the statement of the exterior Neumann problems for the Helmholtz equation from indirect observations of their solutions. A method is developed for the determination of guaranteed a posteriori estimates of this right-hand sides which are compatible with measurement data. It is shown that such estimates can be expressed via solutions of a uniquely solvable system of the Helmholtz equations.
The paper deals with Cauchy problems for first-order systems of linear ordinary differential equations with unknown data. It is assumed that the right-hand sides of equations belong to certain bounded sets in the space of square-integrable vector-functions, and the information about the initial conditions is absent. From indirect noisy observations of solutions to the Cauchy problems on a finite system of points and intervals, the guaranteed mean square estimates of linear functionals on unknown solutions of the problems under consideration are obtained. Under an assumption that the statistical characteristics of noise in observations are not known exactly, it is proved that such estimates can be expressed in terms of solutions to well-defined boundary value problems for linear systems of impulsive ordinary differential equations.
We present complete mathematical statements and perform detailed investigations of the minimax estimation problems of unknown data for the Helmholtz transmission problems from indirect noisy observations of their solutions. We construct optimal, in certain sense, estimates, which are called minimax mean-square estimates, of the values of linear functionals from unknown data. It is established that when unknown data and correlation functions of errors in observations belong to special sets, the minimax mean square estimates are expressed via solutions to certain transmission problems for systems of Helmholtz equations. We prove that these systems are uniquely solvable. Several possible generalizations of the techniques and results are proposed including applications to the problems with incomplete data and pointwise observations.
The theorems on a general form of guaranteed estimates of linear functionals from unknown solutions of Helmholtz transmission problems are formulated in the case of pointwise noisy observations of these solutions.
We investigate the estimation problems of linear functionals from solutions to transmission problems for Helmholtz equation with inexact data. The right-hand sides of equations entering the statements of transmission problems and the statistical characteristics of observations errors are supposed to be unknown and belonging to the certain sets. It is shown that the linear mean square estimates of the above-mentioned functionals and estimation errors are expressed via solutions to the systems of transmission problems of the special type.
We are looking for linear with respect to observations optimal estimates of solutions and right-hand sides of Maxwell equations called minimax or guaranteed estimates. We develop constructive methods for finding these estimates and estimation errors which are expressed in terms of solutions to special variational equations and prove that Galerkin approximations of the obtained variational equations converge to their exact solutions.
We investigate the guaranteed estimation problem of linear functionals from solutions to transmission problems for the Helmholtz equation with inexact data. The right-hand sides of equations entering the statements of transmission problems and the statistical characteristics of observation errors are supposed to be unknown and belonging to certain sets. It is shown that the optimal linear mean square estimates of the above mentioned functionals and estimation errors are expressed via solutions to the systems of transmission problems of the special type. The results and techniques can be applied in the analysis and estimation of solution to forward and inverse electromagnetic and acoustic problems with uncertain data that arise in mathematical models of the wave diffraction on transparent bodies.
We investigate the problem of guaranteed estimation of values of linear continuous functionals defined on solutions to mixed variational equations generated by linear elliptic problems from indirect noisy observations of these solutions. We assume that right-hand sides of the equations, as well as the second moments of noises in observations are not known; the only available information is that they belong to given bounded sets in the appropriate functional spaces. We are looking for linear with respect to observations optimal estimates of solutions of aforementioned equations called minimax or guaranteed estimates. We develop constructive methods for finding these estimates and estimation errors which are expressed in terms ofsolutions to special mixed variational equations and prove that Galerkin approximations of the obtained variational equations converge to their exact solutions. We study also the problem of guaranteed estimation of right-hand sides of mixed variational equations.
Fast algorithms for the solution of volume singular integral equations of electromagnetics are presented. Numerical results are demonstrated that confirm efficiency of the developed fast computational algorithms.
Problems of electromagnetic wave scattering on 3D dielectric structures in the presence of bounded perfectly conducting surfaces are reduced to a system of singular integral equations. We study this system mathematically and suggest a numerical solution method.
Theorems providing the convergence of approximate solutions of linear operator equations to the solution of the original equation are proved. The obtained theorems are used to rigorously mathematically justify the possibility of numerical solution of the 3D singular integral equations of electromagnetism by the Galerkin method and the collocation method.
An introductory course in numerical analysis and computational mathematics that covers numerical methods of linear algebra; numerical solution to equations; a general approach to numerical solution of ordinary differential equations; basic approximation theory and interpolation techniques; and fundamentals of iteration methods. The book is equipped with a great number of solved exercises and problems and can be used for self-studies
Generalized Chebyshev iteration (GCI) applied for solving linear equations with nonselfadjoint operators is considered. Sufficient conditions providing the convergence of iterations imposed on the domain of localization of the spectrum on the complex plane are obtained. A minimax problem for the determination of optimal complex iteration parameters is formulated. An algorithm of finding an optimal iteration parameter in the case of arbitrary location of the operator spectrum on the complex plane is constructed for the generalized simple iteration method. The results are applied to numerical solution of volume singular integral equations (VSIEs) associated with the problems of the mathematical theory of wave diffraction by 3D dielectric bodies. In particular, the domain of the spectrum location is described explicitly for low-frequency scattering problems and in the general case. The obtained results are discussed and recommendations concerning their applications are given.
It has always been an urgent issue for the oil and gas industry to improve oil, gas, and condensate recovery at liquid and gaseous hydrocarbon fields developed with the use of artificial formation pressure maintenance techniques that involve injection of water or water combined with other displacement agents. Therefore, due to the aforesaid issues, permanent attention should still be paid to the practical problem of optimizing the non-stationary hydrodynamic pressure applied to a reservoir by regulating the operating conditions of the production and injection wells, development process optimization in general, and water flooding in particular. The theory of Buckley and Leverett, does not take into account the loss of stability of the displacement front, which provokes a stepwise change and the triple value of water saturation. Traditionally a mathematically simplified approach was proposed-a repeatedly differentiable approximation to eliminate the “jump” in water saturation. Such a simplified solution led to negative consequences well-known from the water flooding practice, recognized by experts as “viscous instability of the displacement front” and “fractal geometry of displacement front”. The core of the issue is an attempt to predict the beginning of the stability loss of the front of oil displacement by water and to prevent its negative consequences on the water flooding process under difficult conditions of interaction of hydro-thermodynamics, capillary, molecular, inertial, and gravitational forces. In this study, catastrophe theory methods applied for the analysis of nonlinear polynomial dynamical systems are used as a novel approach. Namely, a mathematical growth model is developed and an inverse problem is formulated so that the initial coefficients of the system of differential equations for a two-phase flow can be deter mined using this model. A unified control parameter has been selected, which enables one to propose and validate a discriminant criterion for oil and water growth models for monitoring and optimizing.
Non-stationary flooding of oil-saturated reservoirs has a long-standing and durable place as the main secondary method of oil production and maintenance of reservoir pressure in the development of most oil reservoirs. The water injection into the reservoir creates a delayed problem - the inevitable, often catastrophic flooding of oil production wells, provoked by a sudden and irreversible change in water saturation. The theory of two-phase flow filtration created by Buckley and Leverett does not take into account the loss of stability of the displacement front, which provokes an abrupt change and a triplicity of the water saturation value. Therefore, a mathematically simplified approach was proposed at one time, a repeatedly differentiable approximation to exclude a “jump” in water saturation. Such a simplified solution led to well-known negative consequences of the waterflooding practice, which experts call the “viscous instability of the displacement front”, the “fingering displacement front”. This work has presented a novel approach to formulation decisive rules for the first time allowing timely detection and prevention of the consequences of loss of stability of the displacement front and targeted control of the flooding system by stopping, forcing, limiting operating modes, assigning workover solutions of producing and injection wells. It is possible to quickly solve important short-term practical tasks passing traditional labor- intensive incorrect deterministic tasks and complex methods of solution mobilizing the injected water and controlling the fluid production rate, more precisely water and oil on the basis of the discriminant criterion.