II. Low-Frequency Formulations

The scattering problem of a perfectly electrically

conducting (PEC) body subject to illumination by a time

harmonic incident plane wave can be formulated via the

EFIE as

En() [JwA(r) + (7) on S (1)

where E¹ represents the incident electric field, S is the

surface of the scatterer, and the subscript tan denotes the

component of a quantity tangential to the surface S. A and

are the magnetic vector potential and the electric scalar

potential defined by

where

* *[ J{r"}G(r,r¹\d$",

() == [(ands.

A(r) =

G(r,r')=

on S

r' on S

(2)

N

J= 244,00

(3)

4|P-P

and where k = u(e)/2, and and are the permeability

and permittivity of the surrounding medium. The surface

charge density is related to the surface divergence of J

through the equation of continuity

v. J = -jud

(5)

Many method of moments [7] schemes have been

developed to obtain the numerical solution for equation (1).

One of these is the triangular patch model [6], which is

based on a method of moments solution of the EFIE in

conjunction with a planar triangular patch model of the

scatterer and a special set of basis functions. In this section,

for completeness, we first describe the basis function used

in the original patch code [6]; then the two other vector

basis function sets that are suitable for low-frequency use

are described.

In the method of moments solution procedure, the

surface current density J is approximated as

where N is the number of unknowns, I, is an unknown

coefficient to be determined, and #, is a vector basis

function. For the formulation described here, u, in (6) is

chosen from one of three different sets of basis functions:

U). (J. and J5), or (Jand J.). These three sets of

basis functions are briefly described in following.

The original vector basis function fr

As in [6], f, is a vector basis function defined on a pair

of adjacent triangles 7, associated with the non-

boundary edge of the model, as shown in Figure 1 and

defined by equation (7), where ,, is the length of edge

and A,* is the area of triangle 1².

1₂(1) -

.

4

24,

.

24,

n-th edge

Pa

-Pa

0

r in T

r in T

Figure 1. Local coordinates associated with an

edge.

otherwise

To extend the original patch code [6] to the low

frequency range, vector basis functions are presented based

on the work in [1-3]. These new vector basis functions J

are divided into two types, J and either J, or J,, with

the following properties which make them suitable for use

in the magnetic vector and electric scalar potentials at low

frequencies:

Jis associated with interior nodes and is

divergenceless;

.

J, is associated with faces and is curl-free;

J is equivalent to f, but is only associated with

the interior edges of the model that lie along a

tree structure connecting the centroids of the

triangular patches.