A three-dimensional time-domain finite-element formulation for periodic structures

A formulation is presented for a three-dimensional time-domain finite-element method that can be used to analyze the scattering of a plane wave obliquely incident on a (doubly) infinite periodic structure using one unit cell. A broadband frequency response can be obtained in a single execution. The specifics of the method are shown for scattering problems, but it should be straightforward to extend it to radiation problems. The method solves for a transformed field variable (instead of solving directly for the electric field) in order to easily enable periodic boundary conditions in the time domain. The accuracy and stability of the method is demonstrated by a series of examples where the new formulation is compared with reference solutions. Accurate results are obtained when the excitation (frequency range) and the geometry are such that no higher order propagating Floquet modes are present. The accuracy is degraded in the presence of higher order propagating modes due to the rather simple absorbing boundary condition that is used with the present formulation. The method is found to be stable even for angles of incidence close to grazing.     

Impedance boundary conditions for finite planar and curved frequency selective surfaces

We consider the time-harmonic electromagnetic scattering problem from a finite planar or curved, infinitesimally thin, frequency selective surface (FSS), the periodic unit cells of which are constituted, exclusively, by electric conductors and free-space. In order to avoid the meshing of these cells, the problem is solved by employing an integral equation formulation in conjunction with approximate impedance boundary conditions (IBC) prescribed on the sheet that models the FSS. The impedance in the IBC is derived from the exact reflection coefficient calculated, for the fundamental Floquet mode, on the infinite planar FSS illuminated by a plane-wave at a given incidence. When the FSS is curved, and/or the direction of the incident wave is unknown, higher order IBCs are proposed that are valid in a large angular range and can be implemented in a standard method of moments formulation. Also, a simple technique is presented that allows to reproduce the radiating Floquet modes in the scattered field even though those are not accounted for in these IBCs. Their numerical efficiencies are evaluated for a curved strip grating translationally invariant along one direction. Finally, we present an alternative approach where the impedance is approximated by its truncated Fourier series, that considerably enhances the accuracy of the results at the cost, however, of a denser mesh of the sheet.     

Analysis of periodic structures via a time-domain finite-element formulation with a Floquet ABC

An accurate absorbing boundary condition (ABC) is presented for three-dimensional time-domain finite-element (FEM-TD) analysis of infinitely periodic structures. The proposed ABC serves to truncate the computational domain in the nonperiodic directions and it is highly effective to absorb both the fundamental and the higher order Floquet modes. The formulation is applicable both for scattering and radiation analysis. Validation results are presented that demonstrate the ability of the FEM-TD to obtain highly accurate results over a broad frequency band that includes frequencies where a higher order Floquet mode is propagating (nonevanescent).     

A numerical absorbing boundary condition for finite difference andfinite element analysis of open periodic structures

In this paper we present a novel approach to deriving local boundary conditions, that can be employed in conjunction with the Finite Difference/Finite Element Methods (FD/FEM) to solve electromagnetic scattering and radiation problems involving periodic structures. The key step in this approach is to derive linear relationships that link the value of the field at a boundary grid point to those at the neighboring points. These linear relationships are identically satisfied not only by all of the propagating Floquet modes but by a few of the leading evanescent ones as well. They can thus be used in lieu of absorbing boundary conditions (ABCs) in place of the usual FD/FEM equations for the boundary points. Guidelines for selecting the orders of the evanescent Floquet modes to be absorbed are given in the paper. The present approach not only provides a simple way to derive an accurate boundary condition for mesh truncation, but also preserves the banded structure of the FD/FEM matrices. The accuracy of the proposed method is verified by using an internal check and by comparing the numerical results with the analytic solution for perfectly conducting strip gratings     

Numerical Simulation of BOR scattering and radiation using a higher order FEM

Numerical simulations of body-of-revolution geometries for scattering and radiation problems are presented. The formulation consists of a finite element-boundary integral (FE-BI) method which is based on a finite element method that uses higher order nodal-based scalar basis functions for the azimuthal field component and higher order edge-based vector basis functions for the transverse field. This formulation, when combined with a symmetric FE-BI hybridization scheme, yields a final system of equations that is more accurate than earlier first-order formulations. Numerical examples are given to demonstrate the accuracy and capabilities of the higher order solution.     

A numerically efficient finite-element formulation for the generalwaveguide problem without spurious modes

A numerically efficient finite-element procedure without spurious modes is presented for the analysis of propagation characteristics in arbitrarily shaped metal waveguides loaded with linear materials of arbitrary complex tensor permittivity and permeability. The method is straightforwardly derived from the first-order Maxwell curl equations and comprises both the transversal and longitudinal components of the electric and magnetic fields. Hence, all necessary boundary conditions on the tangential field components are a priori satisfied by the trial functions. With this formulation, an absence of spurious modes has been found. Furthermore, by imposing the additional boundary conditions on the normal components of the magnetic induction and electric displacement fields, the dimension of the resulting matrix equation can be significantly reduced. For the fundamental modes, both the convergence order and the accuracy of the presented method are found to be significantly higher than those of comparable methods when applied to some numerical examples     

Regular boundary integral formulation for the analysis of opendielectric/optical waveguides

A regular boundary element method is employed for the variational formulation of the Helmholtz equation that governs waveguiding problems. The problems are defined on the boundary as usual, but as in the charge simulation method, the source points associated with the fundamental solutions are allocated outside the domain so that the singular integrals which occur in the standard boundary element procedure can be avoided. First, the formulation is developed for the two-dimensional (2-D) scalar Helmholtz problem, solving for the axial components of either electric or magnetic fields. Then the formulation is extended for the analysis of dielectric waveguides of the open type incorporating axial components of both electric and magnetic fields, for the solution of the propagating modes which are generally of hybrid types. Very close agreements have been found when the solutions obtained by the present formulation are compared with the ones obtained by different methods. One merit of the extended formulation is that it has been fixed to suppress the spurious solutions     

Synthesis of absorbing boundary conditions for the FDTD method:numerical results

Presents numerical results in the form of a comparison of predictions due to a numerical algorithm using a previously published exact nonlocal absorbing boundary condition (ABC) and analytical results for the field radiated by an infinitely small dipole. The numerical approximations are derived, and application to the FDTD method is straightforward. The method is compared to a related approach due to De Moerlose et al., (see ibid., vol.41, no.7, p. 890-896, 1993) and it is shown that the method yields better accuracy since higher order interpolation and numerical integration schemes may be used with this formulation     

On combined source solutions for bodies with impedance boundary conditions

Numerical solutions to the impedance boundary condition (IBC) combined source integral equation (CSIE) for scattering from impedance spheres are presented. The CSIE formulation is a well-posed alternative to the IBC electric and magnetic field integral equations which can be contaminated by spurious resonant modes. Compared with the IBC combined field integral equation (CFIE), CSIE solutions have the same accuracy when the combined source coupling admittance is chosen to be the same value as the combined field coupling admittance. However, the CSIE formulation is better suited than the CFIE for creating a general purpose computer code capable of handling aperture radiation problems and/or a scatterer which has a spatially varying surface impedance.     

Fractal surfaces and electromagnetic extended boundary conditions

In this paper, we employ the extended boundary condition method with the Weierstrass-Mandelbrot (WM) fractal function to model and solve a relevant electromagnetic scattering problem. The key point of the procedure is the property of the WM to be an almost periodic function. This allows to generalize techniques employed for periodic problems and to express the field by means of a superposition of Floquet modes. The procedure is devised for the general case of dielectric surfaces. Criteria for assessing the validity of the method are discussed and provided. Validity of the method is confirmed by numerical results     

计算周期性导波结构的时域有限差分方法

本文提出一种计算周期性导波结构的时域有限差分方法.由Floquet定理建立边界条件,在电场边界和磁场边界上两次使用Floquet定理,从而将计算域限制在一个周期结构内,并且在导波结构侧面引入吸收边界条件,保证了计算精度.通过预先给出传播常数,经FDTD迭代计算,其谐振频率就是该传播常数所对应的工作频率.     

Integral equation modeling of multilayered doubly-periodic lossy structures using periodic boundary condition and a connection scheme

This paper presents a boundary integral formulation to analyze multilayered doubly-periodic lossy structures with arbitrary geometry. The formulation is based on the moment method using first-order triangular patch basis functions. Each individual layer is analyzed separately using the simple free-space Green's function. After discretization, periodic boundary conditions are imposed on each region and a connection scheme is used to connect the regions. Metallic patches between layers or on the periodic boundary are also included in the model. Several examples are presented showing both the flexibility and the accuracy of the method.     

Electromagnetic fields near a concave perfectly conducting cylindrical surface

Although no shadowing or diffraction effects occur, the surface fields excited by a high frequency source located on a perfectly conducting concave cylindrical boundary cannot be analyzed by geometrical optics since the caustics for rays, which have experienced many reflections, accumulate. In a previous study, alternative field representations in terms of whispering gallery (WG) modes, canonical integrals, and hybrid ray-mode combinations have been explored to compensate for the failure of geometrical optics. As the source and/or observation points move off the boundary, the number of relevant multiply reflected rays decreases, and the caustics eventually become separated sufficiently to be treated as isolated. Ray optics is then expected to apply provided that uniform corrections near caustics and their endpoints are included. This conjecture is confirmed in the present investigation, which tracks the field continuously from the "boundary layer" near the concave surface, where ray optics is invalid, to off-surface points where it applies, by generalizing the alternative field representations used previously. A rich variety of hybrid ray-mode combinations exists for off-surface source and observation points. Especially intriguing is the possibility of choosing a hybrid mix that completely avoids the need for the caustic (and endpoint) correction functions in a purely ray-optical formulation. The utility, accuracy, and range of validity of the various field representations is assessed by numerical comparison with a reference solution in terms of WG modes plus a continuous spectrum.     

A fast higher-order time-domain finite element-boundary integral method for 3-D electromagnetic scattering analysis

A novel hybrid time-domain finite element-boundary integral method for analyzing three-dimensional (3-D) electromagnetic scattering phenomena is presented. The method couples finite element and boundary integral field representations in a way that results in a sparse system matrix and solutions that are devoid of spurious modes. To accurately represent the unknown fields, the scheme employs higher-order vector basis functions defined on curvilinear tetrahedral elements. To handle problems involving electrically large objects, the multilevel plane-wave time-domain algorithm is used to accelerate the evaluation of the boundary integrals. Numerical results demonstrate the accuracy and versatility of the proposed scheme.     

Calculation of Cutoff Wavenumbers for TE and TM Modes in Tubular Lines with Offset Center Conductor (Short Paper)

The cutoff wavenumbers of TE and TM modes (higher order modes) in a tubular line having an offset center conductor have heen calculated. Whereas most previous methods used to study this structure were of an approximate nature, the analytical method developed by Singh and Kothari leads to a rigorous analytical formulation. The boundary conditions on both conductor boundaries, assumed to be perfectly conducting, are satisfied exactly. The cutoff values calculated show that some results previously reported are inaccurate.     

Finite periodic structure approach to large scanning array problems

There are two conventional techniques dealing with mutual coupling problems for antenna arrays. The "element-by-element" method is useful for small to moderate size arrays. The "infinite periodic structure" method deals with one cell of infinite periodic structures, including all the mutual coupling effects. It cannot, however, include edge effects, current tapers, and nonuniform spacings. A new technique called the "finite periodic structure" method, is presented and applied to represent the active impedance of an array, it involves two operations. The first is to convert the discrete array problem into a series of continuous aperture problems by the use of Poisson's sum formula. The second is to use spatial Fourier transforms to represent the impedance in a form similar to the infinite periodic structure approach. The active impedance is then given by a convolution integral involving the infinite periodic structure solution and the Fourier transform of the equivalent aperture distribution of the current over the entire area of the array. The formulation is particularly useful for large finite arrays, and edge effects, current tapers, and nonuniform spacings can also be included in the general formulation. Although the general formulation is valid for both the free and forced modes of excitation, the forced excitation problem is discussed to illustrate the method.     

Retarded time absorbing boundary conditions

Over the past few years simulations of electromagnetic problems in three dimensions using the finite difference time domain (FDTD) method have become increasingly popular. A major problem in such simulations is the truncation of the computational domain. A formulation of this boundary problem using retarded time values of the field inside the computational domain is suggested, and hence the name retarded time absorbing boundary condition (RT-ABC). This formulation allows the boundary to be situated in the near field of the problem and thereby reduces the necessary computational domain, and the present formulation allows error estimates for the numerically calculated fields     

A Lossy Dielectric-Ring Loaded Waveguide With Suppressed Periodicity for Gyro-TWTs Applications

A dielectric-loaded (DL) waveguide is an attractive possibility for interaction circuits with high-power sources in the millimeter-wave regime down to tenths of millimeters, particularly for gyrotron-traveling-wave-tube amplifiers (gyro-TWTs). We present results on a systematic investigation of the influence of the periodically loaded lossy dielectric on the propagation characteristics of the operating modes, which reveals that a complex mode in the periodic system can be mapped to a corresponding mode in an empty waveguide or a uniform DL waveguide. Dielectric losses not only induce modal transitions between different modes with similar field structures and close phase velocities in the uniform system but also unify the discrete mode spectrum into a continuous spectrum in the periodic system. Since the lossy dielectric functions as a power sink, the higher order Bloch harmonic components arising from the structural periodicity are suppressed, and the mode spectrum of the lossy periodic system degenerates into that of an empty waveguide. This alleviates the potential danger of spurious oscillations induced by the higher order harmonic components, making the periodic lossy DL waveguide promising in a high-power millimeter-wave gyro-TWT.     

An application of the boundary element method to the magnetic fieldintegral equation

A boundary element method (BEM) for the solution of electromagnetic scattering problems using the magnetic field integral equation (MFIE) is discussed. The discretized form of the MFIE is written in indicial notation with no limitations placed on the order of either the geometric or functional approximation. By considering several different types of boundary elements, it is determined that geometric errors can be significant and degrade the accuracy of the numerical solution. It is shown that a higher-order approximation for the current could significantly improve the accuracy of the numerical solution. The superparametric boundary element in which the geometry was given quadratic approximation and the current was given linear approximation was more efficient than elements using lower-order approximations. The BEM results are compared to the results obtained using the dielectric bodies of revolution (DBR) code     

Higher order impedance and absorbing boundary conditions

Traditionally, generalized impedance boundary conditions (GIBCs) have been used to model dielectrics and coated surfaces, and absorbing boundary conditions (ABCs) have been used to simulate nonreflecting surfaces. The two types have the same mathematical form and, in most instances, a higher order condition involving higher order field derivatives has a better accuracy. We demonstrate that there is a close connection between the two and this enables us to use a systematic method which is available for generating GIBCs of any desired order to derive new two- and three-dimensional ABCs. The method is applicable to curvilinear/doubly-curved surfaces and examples are given. Finally, curves are presented that quantify the accuracy of two-dimensional ABCs up to the fourth order, and show how higher order ABCs can improve the efficiency of large scale partial differential equation (PDE) solutions