In this section, the magnetic field integral equation (MFIE) is derived for a conducting scatterer. It is well-known that the MFIE applies only to closed bodies, so that throughout this section we assume that the object has no boundary edges. The vector basis functions given in Section 2.1, are used for both the expansion and testing functions in the numerical solution of the MFIE.
Let denote the surface of a perfectly conducting scatterer with unit
normal vector
. The incident magnetic field
is due
to an impressed source in the absence of the scatterer. The scatterer is
in a homogeneous space characterized by a pair of parameters (
), where
is the inductivity or permeability and
is
the capacitivity or permittivity.
The result of enforcing
the boundary condition on the magnetic field is given by
where
is an outward
unit normal vector on
,
is the surface is just inside of
,
and
and
are the incident and scattered magnetic fields,
respectively.
The tangential component of the scattered magnetic field can be expressed
as a limit for observation points
not on an edge, (see
[48] and [51])
where is the induced electric surface current on
, and the
integral on the right hand side in (
), with the field point
exactly on
, is interpreted as the
Cauchy principal value.
is free
space Green's function, and
is the gradient operator on the primed
coordinates. The Green's function and its gradient are given below as
and
where is the wave number as defined in the previous section,
is the distance between the source point and the observation
point which is
, and
approaches
from the interior. Substituting (
)
into (
),
we obtain the
magnetic field integral equation:
In order to apply the method of moments, the surface of the scatterer is
decomposed into a set of curved triangular patches using a parametric
description of
the surface. The procedure of the parametric surface model generation has
been described in Section 2.1. The next step after surface modeling is
to define a set of basis functions which are used to
approximate the surface current. Here, the basis functions defined
in ()
are to be used as expansion functions on the parametric surface.
Then, the electric surface current can be
approximated as a linear combination of the expansion functions with a set of
unknown coefficients as in (
). Substituting
(
) into (
) gives an integral equation with
unknown coefficients. The method of moments allows one to select a set of
testing functions which are used to test (
) with a
symmetric product which is defined in (
).
When the testing functions are the same as the expansion functions, the procedure is called the Galerkin procedure. This procedure gives
where is the
testing function and
.
is the unknown coefficient associated with the
basis
function
.
The linear equations for
unknowns in (
)
can be rewritten in a matrix
equation as
where and
are column vectors and
is an
square matrix. The
element of
is the unknown
coefficient associated with the
expansion function. The element
of the
row and the
column of the matrix
and the element of the
row of the vector
are given by
and
From the definition of the symmetric product in (),
(
) can
be rewritten as
Note that when the testing function and the
expansion function reside on the same
patch the singularity contribution is the same for both the flat patch and
the curved patch. In other words, there is a term when
for both the flat patch and the curved patch. However,
the contribution for the principal value integral
is different. In the flat patch case, the current vector
is
always on the same plane as
, so that
(see [16]).
However, it is not the case for a curved patch. Due to the
surface curvature the current vector
on the patch is not
always in the same plane as
, so that
,
as shown in Figure 2.4. Thus, the principal
value integral has a singular integrand when the
testing patch is also the source patch.
That may be the only disadvantage of this model for MFIE.
Fortunately, this integral is easily evaluated..
Following the procedure in the previous section, we intend to compute the elements of
sequentially by source-field patch pairs for all
integrals. It will avoid the costly and inefficient recomputation
of an identical integral up to nine times which would result if the elements of
were computed sequentially by basis functions.
As with the EFIE,
it is convenient to write all the required integrals in
terms of integrals over a source-field patch pair, as
where is the computation from testing over
the magnetic field due to the part of the electric current
on
,
and
are either + or
signs, and
is given by
For convenience, let be the inner integral
of the second term on the right-hand side of (
).
It can be expressed as
where the vector function is defined after
substituting (
) and (
) into the
integral, as
Here , and
is
when
is
+ and
when
is
. We transform the parametric coordinates to a local
system of area coordinates
within
, so
(
) can be rewritten as
where and
are given by
(
).
Directly applying the numerical integration technique for a triangular
region in [49] to (
) gives
where is the number of points where the integrand is sampled
on the triangle
,
and
is the weight corresponding to the integration point (
).
, and
are given in (Table 8.2,
[49]).
Similarly, the testing integral in () can also be treated in the
same way as above. The final numerical equation for
when
is then given by
where
is given in (
)
and
is given in (
).
,
,
, and
have the same meanings
as in (
), and
and
are given in (
).
When , the integrand of the integral over
in
(
) is
singular at the testing point. There are two ways to treat this
singularity. One is, as with EFIE, to subtract and add an analytically integrable function with the
same singularity as the integrand.
Another is to divide the
patch into three sub-patches for which the testing point is a common vertex and then apply numerical integration directly
on these sub-patches.
Numerically integrating over these sub-patches, we obtain
where ,
, and
are shown in Figure 2.5.
Since the testing point is a vertex of these three sub-patches and
vertices
are never picked as integration points,
the result of numerical integration over the sub-patches will be finite and, hopefully, accurate. The
numerical integration technique used in the previous section and this
section is applied to (
).
To avoid repeating work, the same notation as in the previous
section will be used. Thus, for
, the
is given by
where is defined on
in the same way that
is defined on
.
Because
,
is
of
(
) with the vertex
replaced by the vertex of
opposite the common edge of the two triangles where
exists.
For
,
is the contribution to
of (
)
due to integration over
.