Method Of Analytical Regularization Research Articles

High-Precision Calculation Using the Method of Analytical Regularization for the Cut-Off Wave Numbers for Waveguides of Arbitrary Cross Sections with Inner Conductors

A method for the accurate calculation of the cut-off wavenumbers of a waveguide with an arbitrary cross section and a number of inner conductors is demonstrated. Concepts of integral and infinite-matrix (summation) operator-valued functions depending nonlinearly on the frequency spectral parameter provide a secure basis for formulating the spectral problem, and the Method of Analytical Regularization is employed to implement an effective algorithm. The algorithm is based on a mathematically rigorous solution of the homogeneous Dirichlet problem for the Helmholtz equation in the interior of the waveguide, excluding the regions occupied by the inner conductor boundaries. A highly efficient method of calculating the cut-off wavenumbers and the corresponding non-trivial solutions representing the modal distribution is developed. The mathematical correctness of the problem statement, the method, and the ability to calculate the cut-off wavenumbers with any prescribed and proven accuracy provide a secure basis for treating these as “benchmark solutions”. In this paper, we use this new approach to validate previously obtained results against our benchmark solutions. Furthermore, we demonstrate its universality in solving some new problems, which are barely accessible by existing methods.

A novel near field radiation shaping technique by using data driven surrogate based optimization for nondestructive hyperthermia

AbstractHerein, a novel and computationally efficient near field shaping technique for nondestructive hyperthermia applications is presented. The aim of nondestructive hyperthermia studies is to treat cancer cells in the body without damaging the healthy cells using the focused near‐field characteristic of the antenna array. However, optimal near field shaping requires an optimization process for having accurately focus the energy into the desired coordinates, which is, a computational expensive and inefficient task due to the nature of full‐wave electromagnetic (EM) analysis. To enable computationally efficient, and accurate near field shape optimization, a surrogate model is required. To achieve this, the near field radiation of the designed horn antenna array in free space is obtained by using a fast and reliable technique, analytical regularization method (ARM). After that, by using Artificial Neural Network based regression algorithm, a data driven surrogate model is created using the data samples generated by ARM. By this means, near field shaping optimization for focusing the energy to the region where the tumor is located can be achieved, as a multi‐objective optimization problem to be solved via data driven surrogate model assisted optimization techniques. The performance results of K‐fold validation and hold‐out data sets for proposed data driven surrogate model are calculated using Mean Absolute Error metric which are obtained as 0.9 × 10−3 and 1.4 × 10−3 respectively. By observing the presented study case scenario, after 10 min of illumination with antenna array with optimally selected configurations, the targeted area (position of tumor) had reached to the critical temperature of 43 Celsius. Thus, it can be said that the proposed method is an efficient method, both in terms of (1) destroying tumor cells by heating without damaging the healthy cells, and (2) being a computationally efficient method.

Q Factor Enhancement of Open 2D Resonators by Optimal Placement of a Thin Metallic Rod in Front of the Longitudinal Slot

A rigorous approach was employed for the accurate evaluation of the electromagnetic interaction between a thin metallic rod and a two-dimensional (2D) slotted cavity. The problem was posed as a classical boundary value problem for the Helmholtz equation in which a 2D slotted open cavity is bounded by an arbitrary but otherwise smooth contour with a longitudinal slit. Using the method of analytical regularization, the problem was transformed to well-conditioned coupled infinite systems of linear algebraic equations for the Fourier coefficients in the expansions of induced surface currents on the rod and slotted cavity. When truncated to finite size, their solutions exhibit fast convergence to the exact solution as the order is increased. This feature makes it possible to investigate the spectral and scattering characteristics of the coupled cavity and rod to within any desired accuracy. In this paper, the complex eigenvalues for a slotted cavity in the presence of a thin rod and the dependence upon their relative location were investigated, particularly to find where there is significant or optimal enhancement of the Q factor. Such optimisation may be exploited in the design of advanced slot antennas and slotted waveguides.

On an electrostatic micropump with a rigorous mathematical model

The novel electrostatic micropump model for applications such as in biomedical drug delivery is presented. The geometrical arrangement of the coupling rigid electrodes lets us exploit our mathematically rigorous boundary integral equation formulation and its solution. Thus, the charge densities involving the fringe effects on the plates are obtained by means of analytical regularization method (ARM) constructed for annular strips earlier. The efficiency of the constructed method is demonstrated with respect to the direct integral equation solvers implemented via the entire domain Galerkin method and point matching. The main physical characteristics of the suggested system and their deviation from that of infinitely large plane approximation are discussed.

Spoiling of tunability of on-substrate graphene strip grating due to lattice-mode-induced transparency.

We report a prediction of the optical effect apparently not discussed earlier. As known both from theory and experiment, the gratings of flat graphene strips lying on dielectric substrates display moderate-Q resonances on the strip plasmon modes in the H-polarization case. In the plasmon resonances, high reflectance and absorbance are observed. These characteristics are tunable with the aid of the graphene chemical potential, which controls the plasmon-mode frequency. However, if this frequency coincides with the high-Q lattice-mode frequency, a narrow-band regime of electromagnetically induced transparency (EIT) appears. A new point in our finding is that, in the EIT regime, the tunability of the reflectance and absorbance of a grating of narrow graphene strips get spoiled profoundly. This is established using a full-wave meshless code based on the method of analytical regularization, which leads to a Fredholm second-kind matrix equation that guarantees the code convergence. Numerical results are presented for the strip width and period, having the microsize dimensions so that all resonances lie in the THz range. However, the same effect takes place in the infrared range for narrower strips and smaller periods. The lattice modes are caused by the periodicity and can have ultra-high Q-factors; however, they do not exist if the substrate is absent. The loss of tunability at EIT is explained by the lattice-mode field pattern, which has deep E-field minima at the strips.

FMM Accelaration of 2D Well-Conditioned T-Matrix Method for Multiple Scatterers

The T-Matrix method for a multitude of two-dimensional dielectric targets can be well-conditioned based on the ideas on Analytical Regularization Method. A peculiar implementation of Fast Multipole Acceleration for this method is revisited. The advantages of working with linear algebraic equations of the second kind instead of the first kind are underlined in terms of reduction in the number of iterations for the iterative solver for a specific residual error.

A Rigorous Algorithm for the Boundary Value Problem of Dielectric Bodies Based on the Analytical Regularization Method

In this paper, the application of the Analytical Regularization Method (ARM) to a dielectric cylinder is presented. At first, the matrix of the algebraic system is constructed as a 2 × 2 block matrix where the off-diagonal blocks consist of the Fourier coefficients of non-singular operators and the entries of the diagonal blocks involve the associated singularities. Then the left- and right-hand side operators of the ARM built up on these singularities are applied to the algebraic system, transforming it to a second kind one in form of $(I + H)y=b$ where $I$ the identity operator and $H$ is a compact operator in space $l_{2}$ . The algorithm is applied to a dielectric cylinder illuminated by a TM -z polarized plane wave. The formulation of TE-z polarization is almost the same except the field functions will be in terms of magnetic field and its normal derivative. The numerical results are validated by the solution of separation of variables for a circular dielectric cylinder and then the superiority of the second kind system to the first kind one is shown by means of the condition number with respect to the truncation number of the algebraic system.

Electrostatic Problem for Toroid Surfaces: Analytical Regularization

The electrostatic problem described by the Laplace equation with Dirichlet boundary condition was rigorously and efficiently solved for closed and open arbitrarily shaped isolated toroid surface of revolution. Application of Orthogonal Polynomial Method and Analytical Regularization Method allowed the problem to be reduced to a well-conditioned infinite system of linear algebraic equations of the second kind. It possesses robust numerical solution with, in principle, any desired accuracy.

Complex TE-modes in Slotted Waveguides of Arbitrary Cross Section

Implementation of the rigorous Analytical Regularization Method is presented for plane wave scattering by arbitrary two-dimensional cavities with longitudinal slit. The general solution of the Neumann boundary value problem for Helmholtz equation is obtained in a form of two coupled infinite systems of linear algebraic equations of the second kind. Compactness of the matrix operator provides efficient numerical solution of the joint system by truncation method. Fast convergence of the truncated system of linear algebraic equations makes this method an effective instrument for accurate analysis of spectral and diffraction problems. Considering the homogeneous version of equations and finding the complex roots of their determinants, we calculate the complex eigenvalues of complex oscillations. Employing spline interpolation and mollifier function the problem geometry is generalized.

Efficient Modeling of Large Gratings

A robust, numerically stable and efficient method for simulation of the wave diffraction by flat finite gratings containing large number of perfectly conductive rods is presented. The approach is twofold. At first, we use further development of analytical regularization. At second, we select two domains of the gratings. One of them is the middle part of the grating, and another one - end-set rods - consists of the rods near the ends of the gratings. The currents on the rods in the middle set are nearly the same as ones for the infinite grating of the same structure. End-set rods can be managed with the grating of much smaller size. The solutions of both these auxiliary problems are constructed by means of Analytical Regularization Method.

Improving radiation performance of the cylindrical dielectric reflector sandwiched by thin resistive layer illuminated by a complex line source

We studied two-dimensional (2-D) thin dielectric parabolic reflector, sandwiched by thin resistive layer from both sides. It is illuminated by an E-polarized electromagnetic plane wave from front side. It is expected to observe the radiation performance of such a composite reflector depending on the electrical and geometrical parameters. We applied two-sided generalized boundary conditions which are modified for a thin multilayer structure. The electromagnetic boundary value problem is formulated and it finally produced a set of two coupled singular integral equations (SIEs). They are handled with the analytical regularization method based on the Riemann-Hilbert Problem solution. The final numerical results are in the Fredholm second-kind matrix equation form and any desired accuracy can possibly be obtained. Also the solution procedure guarantees the convergence. It is observed that the proper selection of the electrical resistivity of the thin resistive layer produces wider bandwidth in the aperture efficiency and the directivity variation approaches to the perfect electric conductor (PEC) reflector case for higher dielectric constants. This selection can increase the allowable frequency band in the applications by coating dielectric layer from both sides. We demonstrate these statements in the numerical results section in the microwave frequency range.

Integral transforms and the regularisation method in the time‐domain excitation of open PEC slotted cone scatterers

An initial-boundary value problem for the transient electromagnetic field in the presence of a complicated-shape scatterer is studied. The considered biconical zero-thickness and perfectly conducting surface with periodic longitudinal slots has many special cases, each of which is of separate practical interest. A new analytical-numerical technique is presented that is based on the Mehler–Fock integral transform and the method of analytical regularisation. The source of excitation of the slotted bicone is a radial electric dipole with an arbitrary time dependence on the field. The proposed method makes it possible to obtain an analytical and numerical solution of the problem to carry out a qualitative and numerical analysis, and to study the characteristic features of the main scattering. Due to the use of the method, the initial-boundary value problem is reduced to a Fredholm type of the second kind system of linear algebraic equations (SLAE-2). The dependences of the condition number on the problem parameters and the estimation of the SLAE-2 convergence with larger truncation orders are given. The far-field radiation patterns are plotted for various values of the problem parameters, the behaviour of the near field at the apex of the slotted cone is studied, and the regime of the slot wave propagation is analysed.

The Cauchy method of analytical regularisation in the modelling of plane wave scattering by a flat pre‐fractal system of impedance strips

The Cauchy method of analytical regularisation, first in full explanation, is used for the modelling of the plane wave scattering by a flat pre-fractal system of impedance strips. It is based on a modification of the singular integral equation technique and the correct type of inversion formula of the simplest Cauchy singular integral equation. The approach is effective for the plane H-polarised electromagnetic (or sonic) wave scattering problems. The principle of strict mathematical orderings of the pre-fractal strip grating deals with a five-parts class of perfect fractal sets, which is characterised by a variable positive fractal dimension. Considerable attention is devoted to the asymptotic model of the plane wave scattering by a sparsely filled grating, which has an explicit solution. This result is very important for verification of the mathematical models presented and testing of direct numerical methods. The scattered electromagnetic field in the far zone is successfully analysed with the aid of the asymptotic solution derived for electrically narrow strips.

Complex eigenvalues of natural TM ‐oscillations in an open resonator formed by two sinusoidally corrugated metallic strips

The diffraction of an E-polarized plane wave from two finite sinusoidal gratings is investigated by a rigorous approach, based on the Method of Analytical Regularization. This allows us to analyse the problem in a wide frequency band without limitations on the depth of the sinusoidal corrugation, the wave size of the gratings, or their relative location. The homogeneous version of the equations obtained is used for accurate computation of the complex eigenvalues in an open resonator of Fabry–Perot type with sinusoidally corrugated strips. In the calculations, two types of structures were used: one is formed by parallel translation of the sinusoidally corrugated strips and the other is obtained by rotation of one of the strips in the first structure. For each relative location of the corrugated strips, the resonance excitation of the structure by an obliquely incident E-polarized plane wave is considered. Based on the calculation of the surface current density for different incidence angles, their specific values for optimal excitation of the complex modes are established.

Scattering of an obliquely incident E − polarised plane wave from ensembles of slotted cylindrical cavities: A rigorous approach

A rigorous theory is developed for the multiple scattering analysis of an E − polarised plane wave from an ensemble of 2-D slotted arbitrary cylinders. The Method of Analytical Regularisation, previously applied to the solution of scattering wave problems for isolated slotted cylinders, is generalised to problems incorporating ensembles of closed or slotted cylinders. Numerical calculations of the complex eigenvalues, Q − factors and resonant current densities are performed for various configurations of two slotted circular or elliptical cavities, focussing on an accurate assessment of differing peculiarities of the mutual resonant electromagnetic coupling.

Integral equation modelling of unmanned aerial vehicle radar scattering characteristics in VHF to S frequency bands

Methods for modelling the radar scattering characteristics of resonance-size airborne objects containing metallic and dielectric design elements are discussed. The developed algorithms are based on solving Fredholm second-kind boundary integral equations (IEs) that place them within the context of the method of analytical regularisation. The magnetic field IE is used for metal elements, and the Müller IE set is used for dielectric elements. The proposed algorithms provide important novelties in calculating the radar scattering characteristics of objects with various curvature radii, including electrically thin scatterers. The description is accompanied by an analysis of developed algorithm convergence. For the validation, a comparison of results for model object obtained by different methods is presented. The results of modelling the radar scattering characteristics of an unmanned aerial vehicle and its separate metallic and dielectric elements within the VHF and S frequency bands are demonstrated and discussed.

The Carleman regularization technique in the modelling of the plane E ‐polarized electromagnetic wave scattering by a flat system of impedance strips

The Carleman method of analytical regularization is used to model the plane E-polarized electromagnetic wave scattering by flat grating of a finite number of impedance strips. It is based on a modification of the singular integral equation technique, the perturbation method and the Carleman inversion formula for the simplest logarithmically singular integral equation. Attention is focussed on the asymptotic model of the plane E-polarized electromagnetic wave scattering by a sparsely filled impedance strip grating. In this case, the corresponding singular integral equation system has an explicit solution. This result is important to verify direct numerical methods. The scattered electromagnetic field in the far zone is successfully analysed with the aid of the asymptotic solution derived for electrically narrow strips.

Complex WGM frequencies of gyroelectric cylindrical resonators

Asymptotic closed-form expressions for calculating complex transverse electric (TE)/transverse magnetic (TM) whispering gallery mode (WGM) frequencies in homogeneous gyroelectric circular cylindrical resonators of infinite length are derived. In addition, a volume integral equation (VIE)-cylindrical Dini series expansion method is extended to support the prediction of complex WGMs for continuously varying highly inhomogeneous gyroelectric circular cylindrical resonators. To this end, the entire domain orthogonal vectorial basis of VIE is extended to support very large indices of the involved Dini-type cylindrical vector wave functions via asymptotic closed-form expressions. This way, the eigenbasis required to solve the VIE becomes free of numerical instabilities arising when very large orders of the involved Bessel functions are employed. The complex frequencies obtained by the asymptotic closed-form expressions for the case of the homogeneous gyroelectric resonator, as well as those obtained by the VIE when the multilayered gyroelectric resonator is reduced to one layer, are validated by comparisons with the complex roots extracted by numerically solving the TE/TM characteristic equations obtained from the separation of variables method, using complex root finding techniques. We demonstrate the calculation of very high-order WGM frequencies for cylindrical resonators composed of homogeneous and highly inhomogeneous permittivity profiles. This asymptotic theory constitutes a rigorous tool that may serve for verifying method of analytical regularisation–based numerical solutions for other non-circular inhomogeneous cylinders, and for interpreting experimental data for applications such as WGM lasing, refractometric sensing, and magneto-optic coupling.

Axisymmetric acoustic oscillations of a rigid spherical shell with two symmetric circular apertures

The plane wave scattering problem for a rigid spherical shell with two coaxial circular holes has been solved rigorously by the Method of Analytical Regularization. An original approach to solving the triple series equations with Legendre polynomials as kernels that arise in the formulation of the problem led to its solution, which is uniformly valid with respect to aperture size. The solution presents two independent infinite systems of linear algebraic equations of the second kind for the even and odd Fourier coefficients respectively. The fast convergence of the systems when truncated is the key to accurate and comprehensive studies of scattering characteristics (total and sonar cross-sections, mechanical force on the walls of doubly-connected sphere) in a wide frequency range. The solution is valid for investigation of the Rayleigh scattering region (where the sphere radius a is much smaller that the wavelength λ), for the resonance region (a∼λ), and also for the deep quasi-optics region (a≫λ), where the radius a of the sphere may be thousands of wavelengths. The unique properties of the obtained solution have been used for calculation of the first 16 complex eigenvalues (in the rigid doubly-connected sphere) with a guaranteed accuracy of 8 significant decimal digits. The new phenomenon of a huge drop in the sonar cross-section σB near the Helmholtz mode (σB/πa2∼10−10) was identified and studied both analytically and numerically. The resonance behaviour of the reflectivity and mechanical force factor is studied in a wide frequency range.

Trade‐off between threshold gain and directionality of emission for modes of two‐dimensional eccentric microring lasers analysed using lasing eigenvalue problem

This article discusses how a trade-off between the high directionality of emissions and low threshold gain can be achieved in active eccentric microring cavities. Our findings are based on the lasing eigenvalue problem formalism, considered using the method of analytical regularisation, and an extremely fast and accurate dedicated Galerkin method, applied to a set of associated Muller boundary integral equations. This method allows us to investigate symmetric and antisymmetric modes separately, on the threshold of nonattenuation in time emission. Numerical results show that the directivities of emission of working modes in a given frequency range, together with their threshold values of gain, are controlled by the size and location of the air hole in the cavity. The high efficiency of the developed code allows us to make an elementary optimisation of the considered cavity; this code is a promising engineering tool in the design of microring lasers.