Donald R. Wilton

Incorporation of edge conditions in moment method solutions

The use of an edge condition in moment method solutions is investigated. Using as an example TM scattering by a strip, various numerical schemes for treating the edge behavior are examined for low frequencies where an analytical solution of the quasi-static integral equation is available for comparison. The same numerical techniques are then applied in the general dynamic case. It is found that failure to incorporate the correct edge behavior in the current expansion can resuit in erroneous currents and anomalous behavior of the solution near edges. The use of ordinary pulses or triangle expansion functions with appropriately singular pulses at the edges circumvents these difficulties. In the case of pulses, a simple correction factor is derived which can be used a posteriori to correct the calculated edge current.

On the existence of branch points in the eigenvalues of the electric field integral equation operator in the complex frequency plane

It is demonstrated by specific examples that eigenvalues of the electric field integral equation (EFIE) operator for finite extent objects in lossless media can have branch points in the complex frequency s -plane. Specifically, it is shown from the analysis of the spheroidal wave equation in the s -plane that branch points are present in the eigenvalues of the integral operator associated with the scalar scattering problem from a perfectly conducting spheroid. Similarly, it is concluded from the analysis of the Mathieus differential equation that branch points, besides that associated with the infinite extent of the object, exist in the s -plane behavior of the eigenvalues of the EFIE operator for a perfectly conducting infinite elliptic cylinder. A proof is given that branch point singularities cannot occur in the eigenvalues of EFIE operators for structures in which geometrical symmetry completely determines the eigenfunctions (and hence they are frequency independent). It is conjectured that branch points may always be present when sufficient object symmetry is lacking. This conjecture is supported by the fact that branch points appear when a sphere is deformed into a spheroid or when a circular cylinder is deformed into an elliptic one. An analogous phenomenon has been observed in circuit and transmission line problems. For example, it is shown that when the taper parameter of an exponential transmission line goes to zero (the line becomes uniform and thus a symmetrical structure), the branch points in the eigenvalues of the impedance matrix move away to infinity.

A new numerical approach to the calculation of electromagnetic scattering properties of two-dimensional bodies of arbitrary cross section

A new method is introduced for formulating the scattering problem in which the scattered fields (and the interior fields in the case of a dielectric scatterer) are represented in an expansion in terms of free-space modal wave functions in cylindrical coordinates, the coefficients of which are the unknowns. The boundary conditions are satisfied using either an analytic continuation procedure, in which the far-field pattern (in Fourier series form) is continued into the near field and the boundary conditions are applied at the surface of the scatterer; or the completeness of the modal wave functions, to approximately represent the fields in the interior and exterior regions of the scatterer directly. The methods were applied to the scattering of two-dimensional cylindrical scatterers of arbitrary cross section and only the TM polarization of the excitation is considered. The solution for the coefficients of the modal wave functions are obtained by inversion of a matrix which depends only on the shape and material of the scatterer. The methods are illustrated using perfectly conducting square and elliptic cylinders and elliptic dielectric cylinders. A solution to the problem of multiple scattering by two conducting scatterers is also obtained using only the matrices characterizing each of the single scatterers. As an example, the method is illustrated by application to a two-body configuration.

Efficient numerical techniques for solving Pocklington's equation and their relationships to other methods

It is shown that testing Pocklingtons equation with piecewise sinusoidal functions yields an integro-difference equation whose numerical solution is identical to that of the point-matched Hallens equation when a common set of basis functions is used with each. For any choice of basis functions, the integro-difference equation has the simple kernel, the fast convergence, the simplicity of point-matching, and the adequate treatment of rapidly varying incident fields, but none of the additional unknowns normally associated with Hallens equation. Furthermore, for the special choice of piecewise sinusoids as the basis functions, the method reduces to Richmonds piecewise sinusoidal reaction matching technique, or Galerkins method. It is also shown that testing with piecewise linear (triangle) functions yields an integro-difference equation whose solution converges asymptotically at the same rate as that of Hallens equation. The resulting equation is essentially that obtained by approximating the second derivative in Pocklingtons equation by its finite difference equivalent. The authors suggest a simple and highly efficient method for solving Pocklingtons equation. This approach is contrasted to the point-matched solution of Pocklingtons equation and the reasons for the poor convergence of the latter are examined.

EFFECTIVE METHODS FOR SOLVING INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS

ABSTRACT Effective numerical methods for solving simple integral and integro-differential equations with logarithmically singular kernels are presented tutorially in this paper. Fundamental features and interrelationships of the methods are discussed. Procedures for applying the techniques to practical equations of electromagnetics are suggested.

SOME IMPLICATIONS OF THE LAPLACE TRANSFORM INVERSION ON SEM COUPLING COEFFICIENTS IN THE TIME DOMAIN

ABSTRACT The issues associated with the choice of.coupling coefficient forms in the singularity expansion and the closely-related subject of whether expansions may be written without an entire function present have persisted as major points of concern and confusion in the development of singularity expansion method theory. In this paper we show that the variety of choices available to one in constructing a singularity expansion relates directly to the large-s asymptotic behavior of the resolvent kernel for the integral equation from which the expansion is derived. The choices range from a cautious extreme in which causality is enforced explicitly to a bold extreme where one depends upon the expansion to sum to a causal result well ahead of the time of arrival of the excitation. By appealing to recently-reported estimates of this asymptotic behavior we define what the acceptable constructions are and discuss them on a comparative basis. A geometrical interpretation of the domain of integration for specific...

LARGE FREQUENCY ASYMPTOTIC PROPERTIES OF RESOLVENT KERNELS

ABSTRACT ABSTRACT A conjecture on the large complex frequency asymptotic behaviour of the resolvent kernel of the electric field integral equation operator is presented. The conjecture is based on a detailed examination of the corresponding large frequency behaviour of a matrix approximant to the operator. From this analysis it is concluded that the resolvent decays exponentially on a sequence of concentric circular contours of increasing radius threading between poles in the left half plane. The decay rate is proportional to the distance between observation and source points.

A TM-TE decomposition of the electromagnetic field due to arbitrary sources radiating in unbounded regions

A decomposition of the electromagnetic field in unbounded homogeneous isotropic regions into transverse magnetic (TM) and transverse electric (TE) parts is presented. The form obtained is distinctive in that the potentials generating the decomposition are explicitly expressed in terms of source currents.