The Singularity Expansion Method

The History of the SEM

The SEM method was first presented by Dr.Carl Baum [9] and later formalized by Marin et

al [40] in May 1972, who proposed a representation of transient scattered fields in terms of

free oscillations of bodies. This formulation of magnetic field intensity, in integral operator form, showed the operator inverse to be an analytic function of the complex wavenumber k, except at certain singularities, where it had poles. The approach found its origins in 2 places:

• The theoretical analyses of canonical problems, with the aim being to determine natural oscillations particular to the structures considered. Pocklington [41] calculated the

oscillations of a thin wire bent into a circular ring. Oseen [42] calculated oscillations of a straight thin wire, and H´allen [43] the fundamental oscillation of a thin wire represented by an integral equation of the first kind. Prolate spheroids and perfectly conducting spheres were solved analytically by Abraham [44] and Stratton [45], respectively. • Observations of induced currents and scattered fields seemingly describable by expo-

nentially damped sinusoidal oscillations. Experimental observations of the transient response of complicated scatterers, such as missiles and aircraft led to the develop- ment of the SEM formalism [9]. Experimental results to excitations of antennas were presented by Ross [46] and Schmitt et al [47].

Compounding analytical difficulties in finding these oscillations, the response to an arbitrary excitation had proved intractable.

Applications of the SEM

The first canonical problem used in the numerical validation of SEM, as presented in the open literature, was that of the thin wire, analyzed by Tesche [48]. He calculated the exterior natural resonant frequencies of the thin wire from an integral equation formulation of the scattering problem. Umashankar et al [49] extended this work to the case of an L-shaped wire, which was characterized by a H´allen-type integral. It was observed that the pole patterns for the L-shaped wire closely followed those of the original thin wire. It was also noted that the bend had relatively small effect on the distribution of the currents on the wire. To test the SEM in its application to image-type problems, Umashankar et al [50] considered the problem of scattering by a thin wire parallel to a ground plane. The natural frequencies were found to be very close to the natural frequencies of the same thin wire in free space. As the wire was moved closer to the ground plane, the poles were found to spiral around the free space pole.

SEM has been applied to many problems from as early as the 1970s till today. A very few of these, that were more significant are:

• The SEM applied to perpendicular crossed wires (a basic aircraft model) [51] • Thin wire transient analysis with types of coupling coefficients [52].

• The transient response of cylindrical dipole antennas [53]. • In the application to target detection [54].

• Far-field responses of step-excited linear antennas [56].

• Representation of the early and late time fields scattered from wire targets [57]. • Application to a matrix-pencil method for both time and frequency domain represen-

tations for antennas [58]

Critical Review and Observations

SEM builds on the concept of natural frequencies and natural modes corresponding to “free oscillations”. We consider therefore, the homogeneous equations for the scattering problem, defined in terms of an impedance (dyadic) operator, Γ(r, r0_{; s}

α) and mode vectors, Mα, Cα.

hΓ(r, r0; sα); Mα(r0)i = 0 (2.4.4)

and

hCα(r); Γ(r, r0; sα)i = 0 (2.4.5)

where Γ(r, r0_{; s}

α) is the kernel of the Fredholm integral equation defined in chapter 3, evaluated

at s = sα. h·, ·i is the usual inner product, as defined in chapter 4. This follows of course

from the general integral equation for the excitation V(r, s)

hΓ(r, r0; sα); I(r0, s)i = V(r, s) (2.4.6)

Expanding this integral equation near s = sα using the Taylor and Laurent series formula,

the required spatial current distribution is

I(r, s) =X

α

ηα(s)Mα(r)

(s − sα)mα

+ I0(r, s) (2.4.7)

The second term, I0_{(r, s), corresponds to the analytic power series expansion of (B.7.8), often}

referred to as the entire function contribution. η_α(s) is the coupling coefficient, defining the strength of the natural oscillation in terms of the object and incident wave parameters. Detailed derivations are provided in Appendix B.7. mα defines the multiplicity of the poles.

A number of key observations can be made and questions posed regarding the SEM in this electromagnetic application.

• Dolph [59] observed that many of the papers on SEM were difficult to interpret mathe- matically, since neither the properties of the integral operators, nor the space in which solutions are sought were specified.

• Solutions and SEM representations for integral equations of the first kind were purported to facilitate easier numerical solution, without considering the ill-posed nature of the equations of the first kind.

• There are questions regarding compactness of first-kind integral operators and therefore the applicability of solution methods. This suggests the possible need for regularization prior to analysis by Fredholm methods.

• Dolph [59] showed that the natural frequencies in the SEM consisted of 2 non-intersecting sets of “wavenumber parameters”; the interior resonant frequencies on the negative imaginary axis and a set in the left-hand half plane.

• Dolph and Ramm [37, 38] found solutions for current components tangential to the surface and in an L2 space, i.e. they are Lebesgue-square integrable. In other ap- plications, H¨older-continuous spaces are required. Do these requirements/spaces differ for the EFIE and MFIE? Furthermore, would these spaces lend themselves to a viable computation scheme?

• Is there a practical integration approach for applying an SEM approach within Su- perNEC, while still maintaining the integrity of the SuperNEC formulation?

• What is the relationship between SEM and other expansion methods in modes, eigen- functions and singular functions?

• Are there specific requirements on the integral operator for a valid SEM to apply? Does it need to be of the first kind, second kind, Hilbert-Schmidt, Carleman?

• Is the SEM approach a feasible method for complex structures, as opposed to the simple problems reviewed in the previous section? Does application of a Method of Moments discretization or a wire-grid segmentation corrupt the solution, or is this incidental to the underlying sound mathematical basis?

We can clearly conclude that attention to the mathematical basis of MBPE, SEM and EEM is required. The choice of functional space for analysis is important, as are the operator characteristics of the integral and differential operators. The characteristics of integral equa- tions with respect to uniqueness and ill-posedness have been considered in the literature and need to be addressed as part of the mathematical basis for integration of spectral expansion methods into SuperNEC.