Andrew F. Peterson

Higher order interpolatory vector bases for computational electromagnetics

Low-order vector basis functions compatible with the Nedelec (1980) representations are widely used for electromagnetic field problems. Higher-order functions are receiving wider application, but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases. In this paper, fully interpolatory higher order vector basis functions of the Nedelec type are defined in a unified and consistent manner for the most common element shapes. It is shown that these functions can be obtained as the product of zeroth-order Nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points. The completeness properties of the vector functions are discussed, and expressions for the vector functions of arbitrary polynomial order are presented. Sample numerical results confirm the faster convergence of the higher order functions.

The “Interior Resonance” Problem Associated with Surface Integral Equations of Electromagnetics: Numerical Consequences and a Survey of Remedies

ABSTRACT It is well known that certain surface integral equations used to describe exterior electromagnetic scattering problems may not produce unique solutions if applied to closed geometries that also represent resonant cavities. This paper explores the numerical consequences of the uniqueness problem. The “interior resonance” problem and several proposed remedies are illustrated using the eigenvalues of the integral operators for circular cylinders. Although an eigenvalue of the continuous integral operator vanishes at a resonance frequency, discretization error may prevent the associated matrix eigenvalue from becoming appreciably small. Inaccurate results are due to an incorrect balance between the excitation and the small matrix eigenvalue near a resonance frequency.

Higher-order vector finite elements for tetrahedral cells

Edge-based vector finite elements are widely used for two-dimensional (2-D) and three-dimensional (3-D) electromagnetic modeling. This paper seeks to extend these low-order elements to higher orders to improve the accuracy of numerical solutions. These elements have relaxed normal-component continuity to prohibit spurious modes, and also satisfy Nedelecs constraints to eliminate unnecessary degrees of freedom while remaining entirely local in character. Element matrix derivations are given for the first two vector finite element sets. Also, results of the application of these basis functions to cavity resonators demonstrate the superiority of the higher-order elements.

Higher order interpolatory vector bases on pyramidal elements

In the numerical solution of three-dimensional (3-D) electromagnetic field problems, the regions of interest can be discretized by elements having tetrahedral, brick or prismatic shape. However, such different shape elements cannot be linked to form a conformal mesh; to this purpose pyramidal elements are required. In this paper, we define interpolatory higher order curl- and divergence-conforming vector basis functions on pyramidal elements, with extension to curved pyramids, and discuss their completeness properties. A general procedure to obtain vector bases of arbitrary polynomial order is given and bases up to second order are explicitly reported. These new elements ensure the continuity of the proper vector components across adjacent elements of equal order but different shape. Results to confirm the faster convergence of higher order functions on pyramids are presented.

A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields

An iterative method based on the conjugate gradient (CG) algorithm is developed for the efficient treatment of equations involving multiple excitations. Examples show that significant time savings can be obtained as compared to treating each excitation individually with the conjugate gradient algorithm. However, these savings are not obtained without the drawback of increased memory requirements to store the additional excitations, residuals, and solutions. The efficiency of this algorithm tends to increase as additional excitations are added. >

Application of the integral equation-asymptotic phase method to two-dimensional scattering

A hybrid-procedure called the integral equation-asymptotic phase (IE-AP) method is investigated for scattering from perfectly conducting cylinders of arbitrary cross-section shape. The IE-AP approach employs an asymptotic solution to predict the relatively rapid phase dependence of the unknown current distribution, to leave a slowly varying residual function that can be represented by a coarse density of unknowns. In the present investigation, the current density appearing within the combined-field integral equation is replaced by the product of a rapidly varying phase function obtained from the physical optics current and a residual function. The resulting equation is discretized by the method of moments, using subsectional quadratic polynomial basis functions defined on curved cells to represent the residual function. Results show that the required density of unknowns can often be as few as one per wavelength on average without a significant loss of accuracy in the computed current density, even for scatterers with corners. >

A comparison of acceleration procedures for the two-dimensional periodic Green's function

Alternate acceleration procedures are evaluated for the two-dimensional free-space periodic Greens function. Two of these techniques yield exponential convergence rates and are preferable if high accuracies are required. Numerical aspects of these procedures are described.

Vector finite element formulation for scattering from two-dimensional heterogeneous bodies

A formulation is proposed for electromagnetic scattering from two-dimensional heterogeneous structures that illustrates the combination of the curl-curl form of the vector Helmholtz equation with a local radiation boundary condition (RBC). To eliminate spurious nonzero eigenvalues in the spectrum of the matrix operator, vector basis functions incorporating the Nedelec constraints are employed. Basis functions of linear and quadratic order are presented, and approximations made necessary by the use of the local RBC are discussed. Results obtained with linear-tangential/quadratic normal vector basis functions exhibit excellent agreement with exact solutions for layered circular cylinder geometries, and demonstrate that abrupt jump discontinuities in the normal field components at material interfaces can be accurately modeled. The vector 2D formulation illustrates the features necessary for a general three-dimensional implementation. >

Variational nature of Galerkin and non-Galerkin moment method solutions

There is renewed interest in the use of variational methods in conjunction with numerical solutions of electromagnetic radiation and scattering problems. The variational aspects of secondary calculations based on a method of moments (MoM) solution are investigated. These calculations exhibit a type of second-order accuracy, regardless of whether or not the operator being discretized is self-adjoint, and regardless of whether or not testing functions are identical to the basis functions (Galerkins method). Numerical results support these conclusions and suggest that the advantage of Galerkins method in actual calculations is grossly overstated.

Analysis of heterogeneous electromagnetic scatterers: research progress of the past decade

A review of the current status of integral equation and finite element (FEM) (including time domain) modeling as applied particularly to penetrable bodies is provided. The direct numerical solution of the appropriate frequency-domain partial differential equations is an alternative that has recently been the focus of much effort by the research community and appears to offer computational advantages over the integral equation formulations. These frequency-domain approaches require the solution of a matrix equation; the direct time-domain solution of partial differential equations offers a third formulation that does not require a matrix solution and may be more efficient for electrically large structures. The author summarizes the development of these modeling procedures and attempts to identify some of the unresolved issues associated with their development. >