Curl Equations Research Articles

Analysis of singly and doubly periodic absorbers by frequency-domain finite-difference method

In this paper, the theory and numerical algorithms of a periodic finite-difference frequency domain (PFDFD) method are presented for the analysis of arbitrarily shaped inhomogeneously filled, singly and doubly periodic absorbers. The PFDFD method is based on a finite-difference solution of the Maxwell's curl equations plus an absorbing boundary condition, metallic backing, and Floquet's periodic condition to define the problem in a finite region. Compared to the integral equation (IE) and moment method techniques, the PFDFD algorithm is simpler to develop, more efficient to model electrically large geometries, and more flexible to analyze absorbers with both electric and magnetic loadings. The theory and algorithms proposed in the paper have been validated via numerous examples and agree very well with those of other techniques and measurements.

The finite element modeling of static and stationary electric and magnetic fields

A finite-element method is described for computing static and stationary electric and magnetic fields directly in terms of the electric or magnetic field strength. In this way the use of (vector) potentials is avoided and a much higher accuracy is achieved with the same computational effort. The curl equation is supplemented with the electromagnetic compatibility relations and a problem is defined which is known to have a unique solution. For satisfying the conditions at the interfaces between different media use is made of both edge and nodal elements. The validity of our approach is illustrated by solving a simple sample problem.

Influence of the RHS on the convergence behaviour of the curl-curl equation

The vector potential formulation is widely used in electromagnetic field computation due to its robustness. Numerical experiences on the convergence behaviour of the non-gauged vector potential formulation (curl curl equation) are reported. The convergence of the system depends on the discretisation of the source variable (the right hand side of the equation). The system converges if the matrix equation is compatible, i.e. if the RHS is in the range of the curl-curl matrix. The compatibility is ensured when the current density is expressed by the curl of a source field (vector potential) and when this source field is projected on the space curl W/sup 1/, where W/sup 1/ is the space of the Whitney edge element. An explanation of the convergence behaviour is given through the analysis of the matrix structure: under the condition of the compatibility, the curl-curl equation is implicitly ganged by an iterative solver.

The Origin of Spurious Solutions in Computational Electromagnetics

It is commonly believed that the divergence equations in the Maxwell equations are “redundant” for transient and timeharmonic problems, therefore most of the numerical methods in computational electromagnetics solve only two first-order curl equations or the second-order curl–curl equations. This misconception is the true origin of spurious modes and inaccurate solutions in computational electromagnetics. By studying the div–curl system this paper clarifies that the first-order Maxwell equations are not “overdetermined,” and the divergence equations must always be included to maintain the ellipticity of the system in the space domain, to guarantee the uniqueness of the solution and the accuracy of the numerical methods, and to eliminate the infinitely degenerate eigenvalue. This paper shows that the common derivation and usage of the second-order curl–curl equations are incorrect and that the solution of Helmholtz equations needs the divergence condition to be enforced on an associated part of the boundary. The div–curl method and the least-squares method introduced in this paper provide rigorous derivation of the equivalent second-order Maxwell equations and their boundary conditions. The node-based least-squares finite element method (LSFEM) is recommended for solving the first-order full Maxwell equations directly. Examples of the numerical solutions by LSFEM are given to demonstrate that the LSFEM is free of spurious solutions.

Divergence Preserving Discrete Surface Integral Methods for Maxwell's Curl Equations Using Non-orthogonal Unstructured Grids

Several new discrete surface integral methods for solving Maxwell's equations in the time-domain are presented. These methods, which allow the use of general non-orthogonal mixed-polyhedral unstructured grids, are direct generalizations of the canonical staggered-grid finite difference method. These methods are conservative in that they locally preserve "divergence" or charge. Employing mixed polyhedral cells (hexahedral, tetrahedral, etc.), these methods allow more accurate modeling of non-rectangular structures and objects because the traditional "stair stepped" boundary approximations associated with the orthogonal grid based finite difference methods can be avoided. Numerical results demonstrating the accuracy of these new methods are presented.

Hybrid finite element solutions of time dependent Maxwell's curl equations

Hybrid finite element (FE) solutions of transient electromagnetic field are proposed. Time dependent Maxwell's curl equations are solved separately using edge elements for the electric field approximation and nodal-based elements for the magnetic field approximation. Two different formulations are presented: the first adopts a unique mesh, the second two complementary FE meshes. The solution of the resulting equations systems is performed by explicit methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Field-to-wire coupling using the finite element-time domain (FE-TD) method

A new numerical method is proposed to analyze transient induced effects in thin wires excited by external fields. Maxwell's curl equations and transmission line equations are solved by the point-matching finite element-time domain (FE-TD) method. The method is based on a finite element discretization of space and a finite difference discretization of time. The leap-frog scheme is adopted and an explicit solution scheme is obtained.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

FD-TD analysis of Vivaldi flared horn antennas and arrays

This paper summarizes a detailed computational study of Vivaldi flared horn antenna designs including single element, double element crossed-pair subarray elements, and linear arrays using the finite-difference time-domain (FD-TD) method. FD-TD, which numerically solves Maxwell's time dependent curl equations, accounts for the complex geometrical and electrical characteristics associated with this antenna design and array implementation. Validations have been made relative to a moment method (MM) model of an electrically large linearly tapered slot antenna. Also, good correlation is shown to exist in the primary features of the antenna patterns between computed and measured data for all cases. This work has achieved a number firsts for the FD-TD methodology. It represents the first exact computational model of a single quad Vivaldi slot antenna; the first exact computational model of a phased array of Vivaldi quad elements; and the first FD-TD model to demonstrate grating lobes for a phased array antenna of any sort. Lastly, this research represents an extensive study of the largest grid-based antenna models conducted to date. >

On Two Numerical Techniques for Light Scattering by Dielectric Agglomerated Structures.

Smoke agglomerates are made of many soot sphcres, and their light scattering response is of interest in fire research. The numerical techniques chiefly used for theoretical scattering studies are the method of moments and the coupled dipole moment. The two methods have been obtained in this tutorial paper directly from the monochromatic Maxwell curl equations and shown to be equivalent. The effects of the finite size of the primary spheres have been numerically delineated.

Maxwell's Equations in the 21st Century

Maxwell's equations in the 21st Century The finite-difference time-domain method is a novel method for solving Maxwell's curl equations, especially when parallel-processing techniques are applied. The next generation of computers will bring a revolution by exploiting the use of parallel processing in computation to the maximum.

FDTD analysis of magnetized ferrites: an approach based on the rotated Richtmyer difference scheme

Electromagnetic wave propagation in magnetized ferrites is modeled by solving Maxwell's time-dependent curl equations coupled with the equation of motion of the magnetization vector. A discretization approach based on the rotated Richtmyer finite-difference scheme is proposed. The new approach has been used to calculate the phase constants of transversally magnetized ferrite-loaded waveguides. The numerical dispersion equation for TE/sub n0/ modes is derived. The results obtained with this approach for a ferrite-filled and a ferrite-slab loaded waveguide are compared with those obtained with Yee's scheme extended for the treatment of ferrites and with the exact results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Combined tangential-normal vector elements for computing electric and magnetic fields

A direct method for computing electric and magnetic fields in two dimensions is developed. This method determines both the fields and fluxes directly from Maxwell's curl and divergence equations without introducing potential functions. This allows both the curl and the divergence of the field to be set independently in all elements. The technique is based on a novel type of vector finite element that simultaneously interpolates to the tangential component of the electric or magnetic flux. Continuity conditions are imposed across element edges. This guarantees the continuity of the field and flux at the mid-point of each edge, and its also guarantees that for all edges the average values of the tangential component of the field and of the normal component of the flux are identical.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A class of symmetrical condensed node TLM methods derived directly from Maxwell's equations

A series of general transmission line matrix (TLM)-type methods, which include the symmetrical condensed node method, are derived directly from Maxwell's curl equations without recourse to transmission line models. Written as a system of conservation laws, Maxwell's equations in 3-D plus time are decomposed along the orthogonal characteristic directions of a rectangular grid. The Riemann invariants in this method correspond to the voltage pulses of the TLM method. A new method of handling inhomogeneous media is proposed based on a new transfer event. The dispersive nature of these schemes is also investigated.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Numerical dispersion and stability characteristics of time-domain methods on nonorthogonal meshes

The familiar finite-difference time-domain method for discretizing Maxwell's curl equations on orthogonal grids has been extended to nonorthogonal grids by a number of researchers. While it is difficult to determine the dispersion and stability characteristics of these methods when applied on arbitrary grids, analysis of the idealized but representative case of a uniform skewed mesh proves to be quite tractable in 2-D. This analysis demonstrates that numerical dispersion errors are small for well-resolved spatial wavelengths and that these methods converge to the continuous-space solution in the limit as the cell and time step sizes vanish. Grid anisotropy (variations in wave propagation speed as a function of the propagation angle relative to the mesh coordinates) increases as the mesh is skewed. In spite of this, there exist some angles where waves propagate through the skewed mesh with virtually no dispersion. This analysis also provides a stability limit for the time step size in terms of geometrical mesh quantities. >

A multigrid method for solution of vorticity—velocity form of 3‐D navier—stokes equations

AbstractA multigrid method has been developed and applied to the simulation of unsteady three‐dimensional incompressible flow, using the Navier—Stokes equations in terms of vorticity and velocity variables. A system of equations for the velocity problem, consisting of the continuity equation and three curl equations, is first formulated as a uniquely determined, non‐singular system. An efficient and robust technique, namely the multigrid distributive Gauss—Seidel scheme, is developed for solution of this system of algebraic equations. The method is applied to study 3‐D flow in a shear‐driven cavity.

Planetary influences on electrical engineering

The author describes how Maxwell's equations are indebted to observations of Mars, Jupiter, and Saturn. The electromagnetic relations were derived in terms of a medium with 'small but real density' and with the capability of yielding elastically, but the mechanical origin to which the curl and div equations trace back is a surprise to many engineering students. Maxwell developed his ability to deal with waves by starting his career with a four-year study of the rings of Saturn, which constitute a rare, elastic medium subject to internal inverse-square forces. Applying the laws of motion, Maxwell deduced that electricity and magnetism should combine to support what he called an electromagnetic wave, whose velocity in that medium he could calculate. Meanwhile, observations of Jupiter's satellites revealed that light has a finite velocity. Knowing this value, Maxwell was able to propose that light itself might be electromagnetic. These foundations of modern electrical engineering thus rest firmly on astronomical observation of Mars, Jupiter, and Saturn. Astronomy has continued to contribute to many branches of electrical engineering.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Solar wind-magnetosphere interaction as simulated by a 3-D EM particle code

Results are presented for simulating the solar-wind-magnetosphere interaction with a new three-dimensional electromagnetic particle code. Previously, such global simulations were done with MHD codes while lower-dimensional particle or hybrid codes served to account for microscopic processes and such transport parameters as have to be introduced ad hoc in MHD. The kinetic model presented attempts to combine the macroscopic and microscopic tasks. It relies only on the Maxwell curl equations and the Lorentz equation for particles, which are ideally suited for computers. The preliminary results shown are for an unmagnetized solar-wind plasma streaming past a dipolar magnetic field. The results show the formation of a bow shock and a magnetotail, the penetration of energetic particles into cusp and radiation belt regions, and dawn-dusk asymmetries. >

Application of time-domain finite-volume method to some radiation problems in two and three dimensions

Maxwell's curl equations in the time domain are cast into a conservation form and discretized using a finite-volume procedure. The Lax-Wendroff explicit scheme is then used to solve the discretized Maxwell's equations, resulting in second-order accuracy with respect to both time and space. This time-domain scheme does not require an interlaced mesh for electric and magnetic fields. Radiation pattern and input impedance of some typical parallel-plate waveguides and mutual coupling for arrays of parallel-plate waveguides are computed and, whenever possible, compared with closed form exact or approximate solutions. Radiation patterns for a two-dimensional horn and a rectangular waveguide are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Solving the helmholtz equation using multiply propagated fields

AbstractComputer models in electromagnetics are based primarily either on integral or on differential equations. The former arise from source integrals using some appropriate Green's function whereas the latter originate from the Maxwell curl equations. Although requiring volume rather than surface sampling even for spatially homogeneous problems, in contrast to integral‐equation (IE) models, differential‐equation (DE) models are geneally a better choice for problems involving spatial inhomogeneities. This is because such problems require volumetric sampling using either approach, but the DE model produces a sparse matrix rather than the full matrix of the IE formulation.In this paper we describe a new approach based on using multiply propagated fields for numerically solving the banded matrix that results from discretizing the Helmholtz equation. A computer‐time savings of N1/2 and N2/3 for two‐dimensional (2‐D) and three‐dimensional (3‐D) problems, respectively, is made possible, where N is the total number of field samples or unknowns. For even moderate‐size problems where 100 samples per linear dimension are used (N2 = 10,000 and N3 = 1,000,000), the time savings can be of the order of 100 and 10,000 respectively. Another advantage of this procedure, which we call Helmholtz equation multiple propagator (HEMP), is that the radiation or closure condition needed to terminate the spatial solution mesh for exterior problems can be enforced rigorously with essentially no additional computational cost.The method is illustrated for a 2‐D problem by application to plane‐wave scattering from an infinite, metal, circular cylinder. Results are presented for the mode amplitudes of the scattered field, the induced surface current, and the bistatic far field as obtained from HEMP, and shown to be in good agreement with the analytical results. Although limited here to the simplest possible application in order to establish its feasibility, the approach's advantage would be its applicability to 2‐D and 3‐D problems involving inhomogeneous, penetrable objects.

Time-explicit simulation of wave interaction in optical waveguide crossings at large angles

The time-explicit finite-difference time-domain method is used to simulate wave interaction in optical waveguide crossings at large angles. The wave propagation at the intersecting structure is simulated by time stepping the discretized form of the Maxwell's time dependent curl equations. The power distribution characteristics of the intersections are obtained by extracting the guided-mode amplitudes from these simulated total field data. A physical picture of power flow in the intersection is also obtained from the total field solution; this provides insights into the switching behavior and the origin of the radiations.