Current Integral Equation Research Articles

Simple and Efficient Computational Method to Analyze Cylindrical Plasmonic Nanoantennas

We present in this work a simple and efficient technique to analyze cylindrical plasmonic nanoantennas. In this method, we take into account only longitudinal current inside cylindrical structures and use 1D integral equation for the electric field with a given surface impedance of metal. The solution of this integral equation is obtained by the Method of Moments with sinusoidal basis functions. Some examples of calculations of nanoantennas with different geometries and sources are presented and compared with the commercial software Comsol 3D simulations. The results show that the proposed technique provides a good precision in the near-infrared and lower optical frequencies 100–400 THz.

Split Formulation of the Charge and Current Integral Equations for Arbitrarily Shaped Dielectrics

The low-frequency instability in the method of moments can be resolved by reformulating the underlying integral equations to explicitly include charge as an unknown. A new technique based on this charge and current formulation that reduces the computational complexity, both in time and memory, will be introduced. Considerations when using a charge and current technique include the practical problems of basis functions and the evaluation of the singular surface integrals, well as more theoretical problems such as whether the continuity equation still holds. The technique is implemented using a simple set of basis and testing functions and has been verified quantitatively, checking the charge and current distributions against analytic solutions where existing, as well as qualitatively against the boundary conditions for the scattered field. These results show the method to be both stable and accurate.

A New Derivation of TGR with Naked Antenna in Lossy Medium

A new derivation of Transient Grounding Resistance (TGR) has been proposed in this paper. Take grounding system as naked antenna in lossy medium, and then get the input resistance of naked antenna in a given frequency based on Helen current integral equations. With inverse Fourier transform, the resistance calculation equations can be translated from frequency domain to time domain. The result can be applied to design a new grounding system and to calculate its TGR .

ChemInform Abstract: Integral Equations in the Study of Polar and Ionic Interaction Site Fluids

AbstractReview: 114 refs.

Numerical analysis of small parabolic antennas using the method of current and change integral equations

The method of current and charge integral equations [1] is applied to numerical electrodynamic analysis of radiation and impedance matching characteristics of parabolic antennas with reflector diameter from 0.5 to 10λ. As a result as opposed to the current method relying on physical optics approximation the influence of feed on reflector’s radiation pattern, the influence of reflector on feed’s impedance matching and the contribution of feed’s rear radiation into reflector’s radiation pattern are all accounted for. A new model of representing a parabolic surface in the form of its square approximation is suggested, which provides almost uniform partitioning grid and has at least 2.5 times less boundary elements that the common revolution surface representation while having the same sampling coefficient. Dependences of antenna’s directivity on reflector dimensions (0.5–10λ) are calculated for six different focus distance to reflector’s diameter ratios using the developed by the authors crystal_U software package. The calculated results are confirmed by good matching with well-known experimental results.

Extension to Nonconforming Meshes of the Combined Current and Charge Integral Equation

We bring out some mathematical properties of the current and charge boundary integral equation when it is posed on a surface without geometrical singularities. This enables us to show that it is then possible to solve this equation by a boundary element method that requires no interelement continuity. In particular, this property allows the use of meshes on various parts of the surface obtained independently of each other. The extension to surfaces with geometrical singularities showed that acute dihedral angles can lead to inaccuracies in the results. We built a two-dimensional version of this equation which brought out that the wrong results are due to spurious oscillations concentrating around the singular points of the geometry. Noticing that the system linking the current and the charge is a saddle-point problem, we have tried augmenting the approximation of the charge to stabilize the numerical scheme. We show that this stabilization procedure, when coupled with a refinement of the mesh in the proximity of the geometrical singularities, obtained by a simple subdivision of the triangles, greatly reduces the effect of these instabilities.

Integral Equations in the Study of Polar and Ionic Interaction Site Fluids

In this review article we consider some of the current integral equation approaches and application to model polar liquid mixtures. We consider the use of multidimensional integral equations and in particular progress on the theory and applications of three dimensional integral equations. The IEs we consider may be derived from equilibrium statistical mechanical expressions incorporating a classical Hamiltonian description of the system. We give example including salt solutions, inhomogeneous solutions and systems including proteins and nucleic acids.

Field Analysis for Thin Shields in the Presence of Ferromagnetic Bodies

Application of the current sheet integral equation for an efficient analysis of the electromagnetic field in the presence of thin shields is limited to the case of nonferromagnetic and homogeneous media. An extension of this method to thin shields in the vicinity of ferromagnetic bodies is proposed in this paper for time-harmonic fields. The fixed-point polarization technique is used, where the nonlinear material is replaced by a free space and a distribution of fictitious polarization whose magnitude is corrected iteratively in terms of the magnetic induction. A Fourier decomposition of the polarization is employed and the integral equation is solved for each harmonic. The procedure has a guaranteed convergence and is more and more rapid as the number of harmonics retained decreases. In the beginning only the fundamental harmonic is considered and, afterward, higher order harmonics are added to increase the computation accuracy.

Current and Charge Integral Equation Formulations and Picard's Extended Maxwell System

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Important connection between computational and mathematical electromagnetics is presented. The newly developed well-conditioned electromagnetic frequency domain surface integral equation formulations, the current and charge integral equations, are shown to be related to Picard's extended Maxwell system, an extended partial differential equation system that has the correct static behavior. Electromagnetic surface integral representations are derived in this paper for traditional surface integral equation formulations and for the Picard system using the fundamental solution approach, i.e., from the definition of Dirac's delta function. The surface integral representations are constructed with proper solid angle coefficients starting from the scalar Helmholtz equation. The traditional surface integral equation formulations are shown to be derived from Maxwell's curl equations and are thus lacking the contribution of the divergence equations at zero frequency. It is shown that the new current and charge formulations can be derived from the surface integral representation of the Picard system. </para>

Application of extended boundary conditions and the Haar wavelets in the analysis of wave scattering by thin screens

A recent approach to solution of 2D scattering problems for electromagnetic waves scattered by thin screens is analyzed. With the use of examples of scattering by a strip and an unclosed cylindrical surface, it is shown that the proposed approach has no advantages in terms of the efficiency of numerical solution over a well-known approach based on exact integral equations for currents that have singular kernels and that are solved with the Krylov-Bogoliubov method.

Current and Charge Integral Equation Formulation

A new stable frequency domain surface integral equation formulation is proposed for the three dimensional electromagnetic scattering of composite metallic and dielectric objects. The developed formulation does not suffer from the low frequency breakdown and leads to a well balanced and stable system on a wide frequency band. Surface charge densities are used as unknowns in addition to the traditional surface current densities. The balance of the system is achieved by using normalized field quantities and by enforcing the continuity of the fields across the boundaries with carefully chosen scaling factors. The linear dependence between the currents and charges is taken into account with an integral operator, and the linear dependence in charges is removed with the deflation method. A combined field integral equation form of the formulation is proposed to remove the internal resonance problem associated to the closed metallic objects. Due to the good balance in the new formulation, fast converging iterative solutions on a very wide frequency band can be obtained. The new formulation and its convergence is verified with numerical examples.

Three-dimensional (3D) eddy current time-domain integral equation with Coulomb gauge condition

A method is proposed to find eddy currents in massive bodies with relative motion. An integral equation solution in the time domain is proposed in this paper. This method can also be applied to magnetic-resonance imaging magnets to study transient effects occurring due to the sudden application of current. These effects may include perturbations of the homogeneity of the main field or the far field.

Using of the wavelet technique for the solution of the wave diffraction problems

The basic ideas of the wavelet technique used to solve boundary value problems of the diffraction theory are presented. The wavelet technique is combined with the methods of auxiliary currents, discrete sources and continued boundary conditions. The proposed approach can be applied to other methods, such as the current integral equation method. The corresponding numerical algorithms are developed.

Analytical comparison between the surface current integral equation and the second-order small-slope approximation

This paper is the third in a series discussing a new approximate bistatic model for electromagnetic scattering from perfectly conducting rough surfaces. Our previous approach supplemented the Kirchhoff model through the addition of new terms involving linear orders in slope and surface elevation differences that arise naturally from a second iteration of the surface current integral equation. This completion of the Kirchhoff was shown to provide the correct first-order small perturbation method (SPM-1) in the general bistatic context. The agreement with SPM-1 was achieved because differences of surface heights are no longer expanded in powers of surface slope. While consistent with SPM, our previous formulation fails to reconverge toward the Kirchhoff model, at some incidence and scattered angles, when the illuminated surface satisfies the high frequency roughness condition. This weakness is also shared with the first-order small slope approximation (SSA-1) which is structurally equivalent to our previous formulation where the polarization is independent of surface roughness. The second-order small slope approximation (SSA-2), which satisfies the SPM-1 and second-order small perturbation method (SPM-2) limits by construction, was shown by Voronovich to converge toward the tangent plane approximation of the Kirchhoff model under high frequency conditions. In the present paper, we show that, in addition to the linear orders in our previous model, one must now include cross-terms between slope and surface elevation to ensure convergence toward both high frequency and small perturbation limits. With the inclusion of these terms, our new formulation becomes comparable to the SSA-2 (second-order kernel) without the need to evaluate all the quadratic order slope and elevations terms. SSA-2 is more complete, however, in the sense that it guarantees convergence toward the second-order Bragg limit (SPM-2) in the fully dielectric case in addition to both SPM-1 and Kirchhoff. Our new generalization is shown to explain correctly extra depolarization in specular conditions to be caused by surface curvature and surface autocorrelation for incoherent and coherent scattering, respectively. This result will have large repercussions on the interpretation of bistatically reflected signals such as those from GPS.

Detection of a buried wire with two resistively loaded wire antennas

The use of two identical straight thin‐wire antennas for the detection of a buried wire is analyzed with the aid of numerical calculations. The buried wire is located below an interface between two homogeneous half‐spaces. The detection setup, which is formed by a transmitting and a receiving wire, is located above the interface. The transmitter is excited by a pulsed voltage in a small gap. A resistance profile according to Wu and King [1965] has been used to suppress the reflections at the end of the wires. This enhances the visibility of properties of the lower half‐space directly from the time response of the current along the receiving antenna. The problem is solved in several steps. First, the electric field integral equation for the total current along a single thin‐wire antenna in a homogeneous space is formulated. The result is then used to construct a set of coupled integral equations to describe the currents along all three wires without the resistive load. The integral equations contain the transmitted and reflected fields due to the interface. Next, the set of integral equations is adapted for the resistance profile along the wires of the detection setup. The reflected and transmitted fields in both half‐spaces are treated as secondary incident fields in the integral equation for the currents along the wires. In these equations, the response from a pulsed dipole source in the same configuration occurs as a Green's function. The inverse spatial Fourier transformation that occurs in the transmitted and reflected fields is carried out with the aid of a frequency independent, composite Gaussian quadrature rule. The set of coupled integral equations is solved by using the continuous‐time discretized‐space approach, where the space discretization is kept fixed for all frequencies. This results in a linear system of equations with a fixed dimension which is solved by the conjugate gradient‐fast Fourier transform (CG‐FFT) method. With the aid of a marching‐on‐in‐frequency scheme, the system is solved for a number of frequencies. Time domain results are obtained by applying an inverse Fourier transformation. Representative numerical results are presented and discussed.

Phase factors representation for electromagnetic scattering from arough-surface perfect conductor

The surface magnetic field (current) integral equation for a plane in an average rough interface between a vacuum and a perfect conductor is considered. This equation is presented in a form that contains the random elevations only as exponential functions with purely imaginary exponents. Iterations of this equation lead to a solution in powers of multiplicity of the Kirchhoff scattering, and each iterative term contains the random elevations only in exponents. Because the modulus of such functions is equal to unity, we may expect that convergence of such series is independent of the Rayleigh parameter. All mean values can be expressed directly in terms of joint characteristic functions of elevations in different points of a rough surface. To develop this representation of the scattering problem we use the three-dimensional Green function in terms of the one-dimensional Fourier integral with respect to z.

Numerical analysis of probe-excited multiple-arm printed wire antennas

A unified current integral equation is formulated for N-arm printed antennas with an arbitrary arm shape. The parametric moment method is employed for the current distribution. A succinct expression for the radiation fields is obtained with the help of the saddle point method. The validity of the formulation is verified by comparing the computed current distributions and radiation patterns for two-arm and four-arm Archimedian spiral antennas with published results. Also, the numerical results for a two-arm circular open-loop and a small four-arm Archimedian spiral are given to show potential applications of the present formulation.

Three‐dimensional nonlinear inversion in cross‐well electrode logging

Electrode logging as known in the oil industry is a method for determining the electrical conductivity distribution around a borehole or between two boreholes from the static field (dc) measurements in the borehole. In this paper we discuss the reconstruction of the conductivity in a three‐dimensional domain between two boreholes. The electric current density integral equation will be taken as point of departure to develop a nonlinear inversion scheme based on a new inversion method, the so‐called contrast source inversion method (CSI). The CSI method considers the inverse scattering problem as an inverse source problem in which the unknown contrast sources in the object domain are reconstructed by minimizing the object and data error using a conjugate gradient step, after which the contrast is updated by minimizing only the error in the object. The robustness and the abilities of this full‐vector three‐dimensional inversion method have been demonstrated with a number of numerical examples where the synthetic “measured” data used were generated by solving a forward scattering problem using the conjugate gradient fast Fourier transform method.

Performance of a numerical iterative solution of the surface current integral equation for surfaces containing small radii of curvature

An iterative method for numerically solving the surface current integral equation is applied to wedge‐like surfaces in order evaluate its applicability to surfaces that give rise to substantial edge diffraction. The convergence properties and accuracy of this approach are examined by comparing its results with the analytic Sommerfeld solution for incidence angles of 45° and 85° and scattering angles from −90° to 90°. The accuracy and computational efficiency of these results suggest that this method may be suitable for studying scattering from a variety of one‐ and two‐dimensional surfaces that contain regions of large surface curvature.

Current integral equations of boundary-penetrating antenna embedded in three-layer lossy medium

The Hallen-type current integral equations of a boundary-penetrating antenna embedded in a three-layer dissipative medium are deduced using the Hertz vector and the principle of superposition.