1. 398 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 47, NO. 4, APRIL 1999
On the Fully Numerical Evaluation of the
Linear-Shape Function Times the 3-D
Green’s Function on a Plane Triangle
Luca Rossi and Peter J. Cullen, Member, IEEE
Abstract—A numerical technique is reported for the evaluation
of improper integrals associated with the self-interaction terms
arising in the application of linear (Rao–Wilton–Glisson) current
basis functions, defined on planar triangular patches, to three-
dimensional electromagnetic surface scattering problems. The
two-dimensional numerical integration arising in more conven-
tional approaches, which follow Graglia [3], is replaced by a
one-dimensional integration by means of a suitable change of
the local coordinate system, and analytical expressions for the
functions to be numerically integrated are derived. Numerical
results obtained using Graglia’s method, our alternative method,
and a reliable reference solution are compared for accuracy and
computational complexity. The alternative technique appears to
be conceptually simpler than the conventional method, is easier
to implement, and causes no degradation in accuracy; in fact,
it seems to more efficiently achieve a slightly specified level of
accuracy.
Index Terms— Electromagnetic scattering, moment methods,
numerical analysis.
I. INTRODUCTION
THE numerical evaluation of multiple integrals with sin-
gular kernels is a necessary part of the moment-method
solution of electromagnetic scattering problems formulated
as surface integral equations. Typically, the scatterer surface
is constructed using a set of joined triangular patches or
facets. Given an incident field, we may compute the surface
current using, for example, the electric-field integral equation
(EFIE) formulation. The EFIE may be solved numerically
when the unknown surface current on the patches can be
represented using a suitable basis. A particularly useful basis
set is described in [1], where the basis functions are linear
functions of the local patch coordinates.
The first step in any solution of the EFIE (either direct or
iterative) is the calculation of the elements of the impedance
matrix (operator). The self-term elements describe the self
interaction (the scattered field on a patch due to the current on
that patch). Accurate computation of the self terms is important
and involves the evaluation of multiple integrals with singular
kernels, and is the subject of this paper.
Important results for the numerical computation of multi-
ple integrals involving the three-dimensional (3-D) Green’s
function and its gradient have already been published [2], [3].
Manuscript received November 30, 1997.
The authors are with the Department of Electronic and Electrical Engineer-
ing, Trinity College, Dublin 2, Ireland.
Publisher Item Identifier S 0018-9480(99)02993-2.
Specifically, the calculation of the self-term integral, namely,
the integral over a planar triangle of a linear function (defined
on the triangle) times the 3-D Green’s function, can be carried
out following the general method presented by Graglia [3],
which involves both analytical and numerical integrations.
In this paper, we develop an alternative fully numerical
method, which is suitable for the specific case of the evaluation
of the self-term integral, and compare it to the method reported
by Graglia [3]. The method makes use of a polar coordinate
system , and, to the best of our knowledge, has not been
proposed yet in electromagnetic literature. A similar approach
is described in [4] as a general numerical integration rule to
evaluate weakly singular integrals arising in many practical
computations in geodesy. However, unlike the general case,
our singular kernel is such that can be analytically integrated
over the domain and, due to its smoothness, the resulting
integral of a single variable ( ) function is evaluated through
a simple integration rule.
II. FORMULATION
Fig. 1 illustrates the planar triangle representing the
surface domain of the integrals that we wish to evaluate. We
adopt the same geometry and notation as in [3]. The following
improper integrals need to be calculated:
(1)
(2)
(3)
with
where is the observation point and the integration point.
where is the wavelength of the electromagnetic
radiation.
Referring to Fig. 1 in the plane containing the triangle ,
every point is associated with a pair of coordinates , .
Hence,
(4)
(5)
0018–9480/99$10.00 © 1999 IEEE

2. ROSSI AND CULLEN: LINEAR-SHAPE FUNCTION TIMES 3-D GREEN’S FUNCTION ON PLANE TRIANGLE 399
Fig. 1. Plane triangle T, local coordinate system (u, v), and auxiliary
systems (ua, va) and (, ).
where and are orthonormal vectors. For convenience,
(1)–(3) can be grouped in a more compact form as follows:
(6)
For completeness, we observe that the following series of
inequalities hold:
where is the circular domain centered at , and
having radius equal to the maximum distance between any
of the three vertices of and the observation point itself
( ). Thus, exists and is finite; however, care must
be taken with its numerical evaluation due to the presence of
the singularity in the integrand.
A. Conventional Method
The typical approach [3] to the calculation of is to
separately evaluate the integrals and as follows:
(7)
(8)
The integral (7), namely, , can be calculated analytically.
The resulting expression, see for instance [3], is
(9)
where is the index associated with the side of and
the functions , , and depend on geometrical
parameters, which are illustrated in Fig. 1.
In fact, a simpler expression of can be obtained by
manipulating some fundamental results extracted from [5].
Since we have not seen this formulation published, we present
it here for convenience. We refer to the coordinate system
, centered at , . Let us define for each side of
the characteristic coefficients , , and of the line
containing the side itself so that either the equation
(10)
or the equation (in the case of a “vertical” line )
(11)
is satisfied by the coordinate pairs of the endpoints of .
Furthermore, referring to Fig. 1, let and be the angles
associated with the endpoints of . The reference axis for
measuring the angles is chosen to be with angles increasing
in the counterclockwise direction. As can be seen in Fig. 1,
, , , , , . The subscripts and
are related to the and endpoint of , which is defined as
follows: connects the points (subscript ) and
(subscript ), , , where denotes
the remainder of , where , are integers. It can then
be shown that
(12)
where we have introduced the following triplet of functions
of the angle :
with

3. 400 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 47, NO. 4, APRIL 1999
when (10) is satisfied or
with
when (11) is satisfied. We notice that when , the th
term of the sum given in (12) is zero by definition. The results
achieved using (12) are identical to those obtained using (9).
The integral (8), i.e., , can be evaluated numerically
without difficulties, the integrand being bounded over the
integration domain . As described in [3], the numerical
multiple integration of a bounded function over a triangular
domain can always be deduced from an integral of the
following type:
(13)
where are the so-called triangle area coordinates, which
are related to the coordinates , by a linear operator. The
numerical integration of (13) can be performed by several
methods. For example, a generalized product rule (for instance,
combination of two Gaussian rules) can be applied as follows:
(14)
or, alternatively, an -point quadrature formula, as reported in
[6], can be considered as follows:
(15)
Thus, the final result is given by
(16)
In summary, to evaluate the integral it is first split into two
parts. The first part is performed analytically using either
(9) or alternatively (12) and the second is evaluated
numerically.
B. Alternative Method
In the previous section, we outlined the Graglia’s procedure
for the evaluation of and presented a modified formulation
of the analytical part of that method which we find useful.
In this section, we develop an alternative approach for
the evaluation of the integral (6), which we feel is both
conceptually simpler and also easier to apply. We also begin
by splitting the integral into two parts, but in our case, the
analytical part is zero. Observe that
with
(17)
(18)
is the intersection between a disk of radius centered at
the observation point , and . It is straightforward to
obtain
where the function , is defined as
and if is not located at a vertex; otherwise, is
the angle between the two sides of meeting at the vertex
where falls.
The integral (17) may now be expressed as
(19)
where the sum over extends to the three triangles
formed by the observation point and the endpoints of :
(see Fig. 1). is the domain given by
the intersection between and . We can now write
(20)
where
(21)
where is the distance between the integration and observation
points, which is considered as the origin of a polar coordinate
system ( is the reference axis, as previously stated).
is the distance of any point of from the observation point
and is a function of the characteristic coefficients and
in (10) as follows:
or the coefficient in (11) as follows:

4. ROSSI AND CULLEN: LINEAR-SHAPE FUNCTION TIMES 3-D GREEN’S FUNCTION ON PLANE TRIANGLE 401
TABLE I
NUMERICAL AND REFERENCE RESULTS FOR FIVE DIFFERENT OBSERVATION POINTS
with and defined in the previous section. Once again,
we observe that if , then the terms associated with
in (21) are zero. The integrals in (21) are easily evaluated.
The results are
where
Now, the final step in the solution of the original problem is the
evaluation of the three integrals involving the functions ,
, and over the domains .
This can be achieved numerically employing, for example, a
Gaussian quadrature formula
(22)
where and are, respectively, the
sets of weights and abscissas adopted for each .
The first remark about the alternative approach presented
here relates to the integrand functions , , and .
We observe that the longer side of a triangular patch, where
a basis current function shall be defined, is smaller than ,
with for an accurate implementation of the moment
method. Thus, we get
which shows that the domains
correspond to a relatively small portion of the period
of the function. This guarantees a sufficiently smooth
behavior of the functions to be integrated numerically, which,
in turns, provides a closer approximation.
Essentially, the alternative method presented in this section
is fully numerical since the analytical part evaluates to zero.
The problem of calculating is now reduced to the problem
of evaluating three triplets of integrals (one triplet for each
side of ) of functions of one variable .
In the conventional approach outlined in the previous sec-
tions, is evaluated as the sum of three triplets of integrals
calculated analytically ( ) and one triplet of multiple in-
tegrals carried out numerically ( ). Thus, the alternative
technique is certainly simpler than the usual one (the difficulty
in handling numerical integrations decreases in passing from
two dimensions to one dimension).
Another factor to be taken into account is the accuracy of
the two methods. In the conventional approach, the evaluation
of can be done by employing a product rule (
points considered) or a simpler quadrature formula ( points
sampled), with , which is obviously faster as well
as less accurate than the product rule. However, with our
alternative method, is calculated straightforwardly by an
-point Gaussian scheme, with satisfactory results.
It is also worth noting that in the implementation of the
moment method, one faces the double integration of the type
where is the vector belonging to given by (5) and,
similarly, is the vector belonging to given by (4). is
the vector associated with one of the vertices of as follows:
Thus, introducing the vector defined as
we can write the following equation:

5. 402 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 47, NO. 4, APRIL 1999
Recalling that can be calculated as described above for every
couple of coordinates , , it is now clear that the integral
given in (23) can be easily evaluated via a numerical scheme
of the following type:
Finally, it is straightforward to generalize the method ex-
posed here in the case of planar polygonal domains. In fact,
a planar polygon having sides can always be represented
as the union of disjointed plane triangles.
C. Numerical Results
The two techniques described in the previous section have
been compared for a triangle having the following features:
, , (see Fig. 1 and
m). Table I displays five different results obtained for
five different observation points belonging to the line
. The first two rows refer to the values and ,
which correspond to two different numerical evaluation of
the integral defined in (22) with and ,
respectively. In the successive couple of rows, results returned
implementing (16) are reported. and are the
numerical integrals defined in (8) and evaluated according to
two different algorithms: a seven-point rule described in [6]
and a product Gaussian rule with [see (14)]. Each
algorithm has been implemented using C programming
language. Finally, the last row shows the reference results
returned by the well-known Mathematica 3.0 software
package.
III. CONCLUSIONS
An alternative method for the numerical integration of
a linear-shape function times a 3-D Green’s function on a
planar triangle has been presented. The alternative method is
carefully compared to the popular approach, as described by
Graglia, and is demonstrated to possess certain advantages.
The technique presented is suitable for an accurate evaluation
of the impedance matrix elements in 3-D electromagnetic
scattering problems.
REFERENCES
[1] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering
by surfaces of arbitrary shape,” IEEE Trans. Antennas Propagat., vol.
AP-30, pp. 409–418, May 1982.
[2] D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-
Bundak, and C. M. Butler, “Potential integrals for uniform and linear
source distributions on polygonal and polyhedral domains,” IEEE Trans.
Antennas Propagat., vol. AP-32, pp. 276–281, Mar. 1984.
[3] R. D. Graglia, “On the numerical integration of the linear shape
functions times the 3-D Green’s function or its gradient on a plane
triangle,” IEEE Trans. Antennas Propagat., vol. 41, pp. 1448–1455,
Oct. 1993.
[4] R. Klees, “Numerical calculation of weakly singular surface integrals,”
J. Geodesy, vol. 70, pp. 781–797, 1996.
[5] I. S. Gradshsteyn and I. M. Ryzhik, Tables of Integrals, Series and
Products. New York: Academic, 1980.
[6] C. T. Reddy and D. J. Shippy, “Alternative integration formulae for
triangular finite elements,” Int. J. Numer. Methods Eng., vol. 17, pp.
133–139, 1981.
Luca Rossi was born in Carrara, Italy, in 1971. He
received the Laurea (Doctor) degree in telecommu-
nications engineering from the University of Pisa,
Pisa, Italy, in 1996, and is currently working toward
the Ph.D. degree at Trinity College, Dublin, Ireland.
Since September 1996, he has been with the De-
partment of Electrical and Electronic Engineering,
Trinity College. His research interests include com-
putational electromagnetics and its application to
high-frequency radio-wave propagation predictions.
Peter J. Cullen (M’95) has been a Lecturer of engi-
neering science at Trinity College, Dublin, Ireland,
since 1990. His research interests are mainly in
the field of electromagnetic-wave propagation and
scattering applied to radio communications. He is
a Director of Teltec Ireland, Trinity College, which
is an Irish Government program in advanced com-
munications technology. (Further details regarding
Teltec Ireland may be obtained from the URL:
http://www.mee.tcd.ie/mobile_radio.) He is on the
management committee of the European research
initiatives Cost 259 (wireless flexible personalized communications) and Cost
255 (satellite propagation at Ku-band and above).