Let a basis function as defined in the previous section be
associated
with the
non-boundary edge of the curved triangulated structure.
Surface current exists on both sides of the structure for an open
surface. The unknown current
, being solved for in the integral
equation, is the vector sum of the currents on
opposite sides of
. At boundaries of
, the
component of this vector sum normal to the boundary must vanish due to
continuity of the current; therefore, we need not define basis functions
associated with boundary edges. The current on
can be approximated as a linear combination of the basis functions with
a set of unknown coefficients.
where is the total number of non-boundary edges, and
is the
unknown coefficient associated with the
basis function
.
Since there may be up
to three non-boundary edges in a triangle, so there will be up to three
non-zero basis functions within each triangular face.
To convert (
) into a matrix equation, we choose the expansion
functions as testing functions. The symmetric product for
any two vector functions
and
as given by Harrington [3] is
where is defined in (
),
.
Utilizing a surface calculus identity [48], the last term of left
hand side in (
)
can be rewritten as
The continuity equation () is substituted into
(
) to make the current be the only unknown for the
scattered field. Inserting the current expansion (
) into
eqs (
) and (
) then into (
) yields N linear
equations for
unknown coefficients. It may be written in a matrix form as:
where is an
square matrix which is called the ``moment
matrix" or the ``generalized impedance" matrix.
and
are column vectors.
is called the ``generalized
current" vector, and
is named the ``generalized voltage" vector.
The
element of
is
. The
element of
is given by:
The evaluation of the generalized impedance matrix elements follows from
equations () through (
). Thus, utilizing the approximation (
),
where denotes the domain of
, and
and
are given by:
and
where and
are the magnetic vector potential and
the scalar potential due to the current
on the
edge,
Substituting the
expansion function into (
) and
(
), (
) and (
) can be
rewritten as:
and
where denotes the domain of
and
is the
surface divergence operator on the primed variables.
We note that each matrix element of is associated with a pair of
non-boundary edges
and
. However, the domains of the
integrals and locations of the observation points are associated with the
faces attached to these edges. For each pair of triangular patches,
contributions to the interactions between up to nine different combinations of source and field basis functions must be computed.
Each source-field basis function interaction corresponds to a
single matrix element. Much of the information required to compute the
interaction between the source and field basis functions is only related to
geometry. This information is the same regardless of which basis function
is currently considered.
Once computed, the geometry information between a pair of patches may be used
to obtain the contributions to a maximum of nine different matrix elements.
To evaluate these surface integrals, we first transfer the curved triangular
domain to a flat triangular parametric domain. Secondly, we transform the (u,
v) parametric space to
a local system of area coordinates (see Appendix ) with the
corresponding triangle. Finally, the numerical integration formula for
a triangular
region in (see Chapter 8, [49]) are used to evaluate
these surface integrals. Implementing this idea,
we can rewrite
in terms of each pair of faces as:
where is the contribution from testing over
on
the electric field due to the electric current
on
,
and p and q are
either + or
signs.
where and
are given by:
and
where and
are, respectively,
the magnetic vector potential and the electric scalar potential produced by
the part of
on
the patch
.
Both
and
can be numerically computed
after mapping the curved physical
space to the
parametric space with parametric description
in (
) and Jacobian in (
).
Then, we transform the
space to a local coordinate system.
After substituting the basis
function and the expression for the surface element,
is given by
where is
if
is + and
if
is
(see Fig.
),
and
is the area of the triangle
.
Since
and
are, respectively,
and
where
is given by (
),
can be rewritten as:
where , a vector function, is defined by
where () is used to replace
by
and
by
.
Similarly, can be written as
where , a scalar function of
, is defined by
Equations () and (
) can be evaluated
by Gaussian quadrature after transforming the
coordinates to a local system
of area coordinates
within
.
The details of the local system definition
are given in Appendix A. Then,
and
are
given by:
where only and
appear in the above equations
because
and
are
given in (
) where
is a linear combination of
and
.
Note that direct application
of a technique for numerical technique over a triangular
region [49] allows equations (
) and
(
) to
be evaluated as:
where is the total number of integration points,
is the
weight for the
integration point
, where
and
are
given in
Table 8.2, [49], and
and
are
given in (
).
The testing integrals over
in (
) can also be evaluated numerically in
the same way.
For
, both
and
are well behaved. Substituting (
) and (
) into
(
) and evaluating the testing integrals using the same
procedure as for the potential integrals,
the numerical representation of
is given,
for
, by
where is the total number of integration points on
chosen
according to the accuracy requirement, and
is the weight associated with the
integration point at
. The quantities
and
are given in
Table 8.2, [49],
and
are given in (
),
is either
+
or
, and
is the area of the triangle
.
The vector function
and the function
are defined by
For , the integrands of
both the
vector and scalar potentials integrals are singular. A term will be added and
subtracted from each integrand.
The term to be added and subtracted
must have the same
singular behavior when
approaches to
and can be
analytically evaluated.
Thus, the result of the magnetic vector potential
in (
) is presented
after adding and subtracting a selected term.
Adding and subtracting a term which has a
singularity when
approaches zero in the integraand of the scalar potential integral
, one has
where is either + or
, and
is the distance
between the testing point
and the source point
.
is an approximation to
when
is very close to
,
which can be
expressed in terms of a Taylor series approximation. Here,
and
are
given by
Now, has the same behavior as
near the singularity.
Hence, the integrand of the first integral on the right-hand side
of each of (
) and
(
) is well-behaved on entire
region, so these
integrals can be evaluated numerically with the technique in [49].
The second
integral on the right-hand
side of each of (
) and (
) can be evaluated
analytically,
and is discussed in Appendix C of [16] and [50]. Therefore,
for
is given by
where is either + or
depending on whether the patch of the testing
function
that contributes to
resides on
or
. The vector
function
and the scalar function
are defined
in (
).
,
,
,
and
are given by
and
where is the total number of integration points and
is the
weight for the
integration point
. Here
and
are
given in
(Table 8.2 [49]). Furthermore,
and
can be found in (
).
The vector functions and
in (
) and (
) are defined by
and
where is
if
is + and
if
is -.
And the scalar functions in (
) and (
) are
defined by
and
The analytical evaluations of the integrals in and
are given in [50].