Paired Pulse Basis Functions and Triangular Patch Modeling for the
Method of Moments Calculation of Electromagnetic Scattering
from ThreeDimensional, ArbitrarilyShaped Bodies
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classified information.
Anne I. Mackenzie
Certificate of Approval:
Lloyd S. Riggs
Professor
Electrical and Computer Engineering
Sadasiva M. Rao, Chair
Professor
Electrical and Computer Engineering
Stuart M. Wentworth
Associate Professor
Electrical and Computer Engineering
Hulya Kirkici
Associate Professor
Electrical and Computer Engineering
George T. Flowers
Dean
Graduate School
Paired Pulse Basis Functions and Triangular Patch Modeling for the
Method of Moments Calculation of Electromagnetic Scattering
from ThreeDimensional, ArbitrarilyShaped Bodies
Anne I. Mackenzie
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
December 19, 2008
Paired Pulse Basis Functions and Triangular Patch Modeling for the
Method of Moments Calculation of Electromagnetic Scattering
from ThreeDimensional, ArbitrarilyShaped Bodies
Anne I. Mackenzie
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Anne Isobel Mackenzie was born in Cottingham, Yorkshire, England on March 16,
1953. Her parents, James Ross and Veronica Mackenzie, who were medical doctors, emi
grated with Anne in 1956 to Canada. After attending grade school in Woodstock, Ontario
and secondary schools in Corpus Christi, Texas and Indianapolis, Indiana, Anne attended
Purdue University in West Lafayette, Indiana, where she obtained a B.S. degree in General
Biology (1974). While working for the Biology Department at the Indiana University
Purdue University campus at Indianapolis, she completed an A.A.S. degree in Electrical
Engineering Technology (1980). She then worked for five years as an electronics repair
technician at Wavetek Electronics in Beech Grove, Indiana. Following this period, Anne
obtained a B.S. degree in Electrical Engineering (1986) from Purdue University in Indi
anapolis and moved to Virginia to work at the NASA Langley Research Center, where she
is still employed as a researcher in the Electromagnetics & Sensors Branch. Anne obtained
an M.S. degree in Electrical Engineering from Virginia Tech in 1992. Her work at NASA has
included weather radar analysis and design, radiometry, materials measurements, and elec
tromagnetic modeling. Anne?s parents and her siblings, Judith Marie Mackenzie, Veronica
Jane Martin, and James Willis Mackenzie, now reside in Indianapolis, Indiana.
iv
Dissertation Abstract
Paired Pulse Basis Functions and Triangular Patch Modeling for the
Method of Moments Calculation of Electromagnetic Scattering
from ThreeDimensional, ArbitrarilyShaped Bodies
Anne I. Mackenzie
Doctor of Philosophy, December 19, 2008
(M.S.E.E., Virginia Polytechnic Institute and State University, 1992)
(B.S.E., Purdue University, 1986)
106 Typed Pages
Directed by Sadasiva M. Rao
Due to an increasing emphasis on fabrication with composite materials, it is important
to be able to model accurately the electromagnetic properties of composite structures. In
this work, we demonstrate a new pair of orthogonal pulse vector basis functions for the
calculation of electromagnetic scattering from arbitrarilyshaped material bodies. These
subdomain basis functions are intended for use with triangular surface patch modeling ap
plied to a method of moments (MoM) solution. For modeling the behavior of dielectric
materials, several authors have used the same set of basis functions to represent equiva
lent electric and magnetic surface currents. This practice can result in zerovalued or very
small diagonal terms in the moment matrix and an unstable numerical solution. To pro
vide a more stable solution, we have developed orthogonally placed, pulse basis vectors:
one for the electric surface current and one for the magnetic surface current. The basis
function for the electric surface current is placed perpendicular to each patch edge, while
the basis function for the magnetic surface current is placed parallel to each patch edge.
This combination, together with appropriate testing functions, ensures strongly diagonal
moment matrices. The basis functions are suitable for implementing solutions using the
v
electric field integral equation (EFIE) or the magnetic field integral equation (HFIE.) To
obtain unique solutions at all frequencies, including characteristic frequencies for closed
bodies, the EFIE and HFIE may be expanded with paired pulse vector basis functions
and then arithmetically combined by any of the combinedfield methods such as combined
field integral equation (CFIE), PoggioMillerChangHarringtonWuTsai (PMCHWT), or
M?uller formulations. In this work, we describe the numerical implementations of EFIE and
HFIE solutions and show example results for threedimensional, canonical figures. Those
scattering results obtained by using pulse vector basis functions are compared to results
obtained from an exact method or a more accurate numerical method specialized for a par
ticular type of geometry, such as a body of revolution. In successive chapters, the numerical
procedures and solutions are shown for perfect conductors (PEC?s), dielectric bodies, and
PEC/dielectric composites. The composite scatterers may contain multiple dielectric and
PEC parts, either touching or nontouching.
vi
Acknowledgments
I wish to acknowledge my major professor, Dr. S. M. Rao, for sharing his insights into
electromagnetics during the past four years. Dr. Mike Baginski has provided much practical
help and encouragement, and without him I may not have embarked on this most recent
adventure. My undergraduate teachers Drs. Ahmet Fer and Han Paik inspired me to start
o? in this direction many years ago. Thanks to the faculty and sta? of Auburn University
who did more than they had to. Thanks to NASA Langley Research Center for giving me
the time to work on this graduate degree. Thanks to all of my family, friends, and coworkers,
who were surprisingly positive about the whole thing. Early in my mathematical career,
Dad recognized the importance of the directed segment. And thanks to Kitty Columbus,
who was a constant friend for as long as he could be.
vii
Style manual or journal used IEEE Editorial Style Manual (together with the Auburn
dissertation style known as ?auphd.?)
Computer software used The document preparation package LaTeX2e together with
the style file auphd.sty. Figures were prepared with MATLAB, PowerPoint, and Acrobat.
viii
Table of Contents
List of Figures xi
List of Tables xiv
1 Introduction 1
1.1 Overview ..................................... 1
1.2 Background.................................... 1
1.2.1 ScatteringSolutions ........................... 1
1.2.2 MethodofMoments ........................... 3
1.3 StatementoftheProblem............................ 6
1.4 TheProposedMethod.............................. 7
1.5 Scope ....................................... 9
1.6 Organization ................................... 10
2 New Basis Functionsforthe ElectromagneticSolutionof Arbitrarily
shaped, Threedimensional Conducting Bodies Using Method of Mo
ments 11
2.1 Overview ..................................... 11
2.2 Introduction.................................... 11
2.3 DescriptionoftheProblem ........................... 12
2.4 DescriptionofBasisFunctions ......................... 13
2.5 NumericalSolutionProcedure.......................... 15
2.6 NumericalResults ................................ 18
2.7 Summary ..................................... 18
3 An Alternate Set of Basis Functions for the Electromagnetic So
lution of ArbitrarilyShaped, ThreeDimensional, Closed Conducting
Bodies Using Method of Moments 22
3.1 Overview ..................................... 22
3.2 Introduction.................................... 22
3.3 DescriptionoftheProblem ........................... 23
3.4 DescriptionofBasisFunctions ......................... 24
3.5 NumericalSolutionProcedure.......................... 26
3.6 NumericalResults ................................ 30
3.7 Summary ..................................... 31
ix
4 Electromagnetic Scattering from ArbitrarilyShaped Dielectric Bod
ies Using Paired Pulse Vector Basis Functions and Method of Moments 34
4.1 Overview ..................................... 34
4.2 Introduction.................................... 34
4.3 IntegralEquations ................................ 36
4.4 BasisandTestingFunctions........................... 39
4.5 NumericalSolutionProcedure.......................... 40
4.5.1 Calculation of A and F ......................... 45
4.5.2 Calculation of ??F and ??A .................... 46
4.5.3 Calculation of ? and ? ......................... 46
4.5.4 TestingtheIncidentFields ....................... 51
4.6 NumericalExamples............................... 51
4.7 Summary ..................................... 56
5 Electromagnetic Scattering from Arbitrarily Shaped Composites Us
ing Paired Pulse Vector Basis Functions and Method of Moments 57
5.1 Overview ..................................... 57
5.2 Introduction.................................... 57
5.3 IntegralEquations ................................ 59
5.4 BasisandTestingFunctions........................... 63
5.5 NumericalSolutionProcedure.......................... 65
5.5.1 Calculation of A and F ......................... 68
5.5.2 Calculation of ??F and ??A .................... 69
5.5.3 Calculation of ? and ? ......................... 69
5.5.4 TestingtheIncidentFields ....................... 74
5.6 NumericalExamples............................... 74
5.7 Summary ..................................... 80
6Conclusion 81
Bibliography 83
Appendices 86
A Derivation of Dielectric Field Equations 87
B Pulse Basis Functions in EFIE and HFIE Solutions 91
x
List of Figures
1.1 Basis functions f
n
and g
n
associated with the n
th
edge. ........... 7
1.2 Testing functions t
m
and lscript
m
associated with the m
th
edge. ......... 9
2.1 Arbitrarilyshaped conducting body excited by an incident electromagnetic
planewave. .................................... 12
2.2 Basisfunctiondescription............................. 14
2.3 Testing paths associated with the m
th
edge................... 15
2.4 Electric charge patch within the T
th
i
triangle. ................. 17
2.5 Bistatic RCS of a square plate of length 0.15? excited by a plane wave
travelinginthezdirection............................ 19
2.6 Bistatic RCS of a circular disk of diameter 0.15? excited by a plane wave
travelinginthezdirection............................ 20
2.7 Bistatic RCS of a sphere of diameter 0.15? excited by a plane wave traveling
inthezdirection. ................................ 20
2.8 Bistatic RCS of a circular cylinder of diameter 0.15? and height 0.15? excited
byaplanewavetravelinginthezdirection. ................. 21
3.1 Arbitrarilyshaped conducting body excited by an incident electromagnetic
planewave. .................................... 23
3.2 Basisfunctiondescription............................. 25
3.3 Testing path associated with the m
th
edge. .................. 26
3.4 Electric charge patch around the i
th
node.................... 28
3.5 Electric source patches S
n1
and S
n2
for ?
mn
calculation............ 31
3.6 Bistatic RCS of a sphere of diameter 0.15 ? excited by a plane wave traveling
inthezdirection. ................................ 32
xi
3.7 Bistatic RCS of a cube of length 0.15 ? excited by a plane wave traveling in
thezdirection................................... 32
4.1 An arbitrarilyshaped dielectric body with surface S excited by an external
source........................................ 36
4.2 Basis functions f
n
and g
n
associated with the n
th
edge. ........... 41
4.3 Testing functions t
m
and lscript
m
associated with the m
th
edge. ......... 41
4.4 Observation points for mn
th
vector(o)andscalar(x)potentials. ...... 45
4.5 Normal electric current components for ? calculation. ............ 47
4.6 Magnetic charge source area for ? calculation. ................ 49
4.7 Magnetic source patches S
n1
and S
n2
for ?
mn
calculation........... 50
4.8 Orientation of dielectric sphere and cube to the incident plane wave. Sphere
radius = 0.1 ?; epsilon1
R
= 4. Cube length = 0.2 ?; epsilon1
R
=4. ............ 52
4.9 Orientation of dielectric cone to the incident plane wave. Cone
radius = 0.1 ?; apex halfangle = 30
?
; epsilon1
R
=3. ................ 52
4.10 Bistatic RCS for a dielectric sphere at ? =0
?
, radius = 0.1 ?, epsilon1
R
=4. .. . 53
4.11 Bistatic RCS for a dielectric sphere at ? =90
?
, radius = 0.1 ?, epsilon1
R
=4. . . 53
4.12 Bistatic RCS for a dielectric cube at ? =0
?
, length = 0.2 ?, epsilon1
R
=4. . .. . 54
4.13 Bistatic RCS for a dielectric cube at ? =90
?
, length = 0.2 ?, epsilon1
R
=4. .. . 54
4.14 Bistatic RCS for a dielectric cone at ? =0
?
, radius = 0.1 ?,apex
halfangle = 30
?
, epsilon1
R
=3,incidentwavetravelingtowardapex. ....... 55
4.15 Bistatic RCS for a dielectric cone at ? =90
?
, radius = 0.1 ?,apex
halfangle = 30
?
, epsilon1
R
=3,incidentwavetravelingtowardapex. ....... 55
5.1 Arbitrarilyshaped PEC and dielectric bodies with surfaces C, D1,andD2
excitedbyanexternalsource........................... 59
5.2 Basis functions f
n
and g
n
associated with the n
th
edge. ........... 63
5.3 Testing functions t
m
and lscript
m
associated with the m
th
edge. ......... 64
5.4 Observation points for mn
th
vector(o)andscalar(x)potentials. ...... 68
xii
5.5 Normal electric current components for ? calculation. ............ 70
5.6 Magnetic charge source area for ? calculation. ................ 71
5.7 Magnetic source patches S
n1
and S
n2
for ?
mn
calculation........... 73
5.8 Geometries for which bistatic RCS was calculated, including a) two spheres,
b)adisk/cone,andc)amissile.......................... 75
5.9 Bistatic RCS for two nontouching spheres, one dielectric, epsilon1
R
=4,andone
PEC......................................... 76
5.10 Bistatic RCS for a composite disk/cone, cone epsilon1
R
=2,PECdisk. ...... 76
5.11 Bistatic RCS for a composite missile, nose cone epsilon1
R
= 7.5, PEC cylinder,
incidentwaveapproachingthenoseofthemissile. .............. 77
5.12 Bistatic RCS for a composite missile, nose cone epsilon1
R
= 7.5, PEC cylinder,
incidentwaveapproachingthetailofthemissile. ............... 77
5.13 Dielectric cube of length 0.1? capped with PEC plates, epsilon1
R
=4........ 78
5.14 Bistatic RCS at ? =0
?
for a dielectric cube of length 0.1? capped with PEC
plates, epsilon1
R
=4. .................................. 79
5.15 Bistatic RCS at ? =90
?
for a dielectric cube of length 0.1? capped with
PEC plates, epsilon1
R
=4. ............................... 79
B.1 Bistatic RCS for a PEC sphere, diameter = 0.18 ?............... 91
B.2 Bistatic RCS for a PEC sphere, diameter = 0.18 ?............... 92
xiii
List of Tables
5.1 IntegralEquationSurfaces............................ 60
xiv
Chapter 1
Introduction
1.1 Overview
In this work we present a new method for calculating the electromagnetic scattering
from arbitrarily shaped, threedimensional objects in the resonant size [1] range up to a
few wavelengths; the objects may be electrical conductors, dielectrics, or a composite of
materials having di?erent conducting properties. The novelty of the method lies in the
application of new basis functions specially designed for electromagnetic field problems. Two
spatially orthogonal, unit pulse basis vectors are defined in conjunction with flat, triangular
patches on the scattering surfaces. The pulse basis functions represent the unknown electric
and magnetic equivalent surface currents [2] that will be determined by application of
the method of moments (MoM) [3]. The geometric properties of the new basis functions
allow unique and accurate solutions to be obtained for any geometry/material configuration
using any standard combined field formulation such as Combined Field Integral Equation
(CFIE) [4], PoggioMillerChangHarringtonWuTsai (PMCHWT) [5], [6], or the M?uller [7]
formulation.
1.2 Background
1.2.1 Scattering Solutions
How can we calculate the radiation from the surface of an object that has been electri
fied, either by the attachment of a currentcarrying wire, or by irradiation from an exterior
or interior source? If the scatterer?s surface geometry conforms to the coordinate surfaces in
one of eleven known orthogonal coordinate systems [8], the Helmholtz equation variables in
each coordinate may be separated and solved for analytically to produce an exact scattering
1
solution. This is the case for the sphere [2], the infinite cylinder, the prolate and oblate
spheroids, and a variety of conic sections. However, most scattering objects do not conform
to these geometries, and for other shapes, approximations must su?ce.
Balanis [9] and Richmond [10] have given informative reviews of calculation techniques
for PEC and dielectric scatterers, respectively. For electrically large, smooth objects longer
than ten wavelengths, simple analytic solutions are available. The rayoptics, or geometric
optics (GO) method, for example, has been used for dielectric cylinders and spheres [11].
Biggs [12] applied GO to calculate the RCS for a PEC prolate spheroid. According to
Richmond, this method ?often provides reasonably accurate results for slightly curved di
electric shells but is inaccurate for rapidly curving shells and the edge region of a truncated
shell.? The geometric theory of di?raction (GTD), originated by Keller [13], and the phys
ical theory of di?raction (PTD), originated by Ufimtsev [14], are two further refinements
that incorporate edge di?raction into the rayoptics solutions. Molinet [15] observes that
ray tracing methods have been greatly improved for more complex geometries, and eventu
ally accurate results can be found for scatterers of about one wavelength if special attention
and methods are applied to all discontinuities such as corners, vertices, edges, and curve
discontinuities.
For scattering problems with no exact solution, perturbation theory and variational
methods are two mathematical approximation methods [2]. Perturbation methods start
from a geometrically similar problem that does have an exact solution and calculate the
change in the solution; Eftimiu [16], [17] used this method to compute scattering from corru
gated PEC cylinders. Variational methods approximate the desired quantities themselves;
Cohen [18] employed a variational approach for a circular dielectric cylinder. However,
Richmond states that this approach becomes complicated and lengthy for dielectric bodies
of arbitrary shape.
For layered dielectric bodies of revolution (BOR?s) of up to a few wavelengths, iter
ative methods have been used. In the 1950?s, Rhodes [19] and Andreasen [20] calculated
the scattering from thin dielectric shells. Govind, Wilton, and Glisson [21] later modeled
2
inhomogeneous missile plumes by dividing the plumes into piecewise homogeneous layers.
Their method was an extension of Mei?s unimoment method for BOR?s [22].
For arbitrary geometries in the resonant size range, numerical methods provide a prac
tical means of obtaining a very good approximation for scattering. Numerical methods
may be classified into integral equation (IE) and di?erential equation (DE) methods. Using
integral equation methods such as MoM, we solve first for equivalent currents and second
for the scattered fields resulting from those currents. In contrast, using di?erential equation
methods such as finite element analysis and the finite di?erence time domain method, we
solve directly for the scattered fields. In our opinion, integral equations provide the most
accurate solutions because they incorporate Green?s functions that enforce the radiation
condition: the scattered fields diminish to zero at infinite distance. If the scattering object
contains regions of homogeneous permittivity epsilon1 and permeability ?, it is su?cient to write
boundary equations to express the continuity of the tangential electric and magnetic fields
across the region boundaries. Our work is based on such a technique, the MoM surface
integral equation technique.
1.2.2 Method of Moments
Harrington [23] has written a history of the development of MoM, beginning with
Galerkin?s work with linear matrix equations circa 1915. In his development, Harrington
wove Galerkin?s method together with Rumsey?s reaction concept [24] and the Rayleigh
Ritz variational method; in a departure from Galerkin?s method, he decided that weighting
and testing functions could be made di?erent from each other in order to facilitate the
speed of calculation. Harrington chose the name ?method of moments? to describe his
method because it most closely followed the work of Kantorovich and Akilov [25], who
used that name. Some of the earliest MoM solutions were published in 1963 by Mei and
Van Bladel [26], who calculated scattering from PEC rectangular cylinders. Richmond [10]
in 1965 calculated the fields from a thin dielectric cylinder of arbitrary shape after first
using surface integrals and MoM to find the polarization currents. In 1968, Harrington [3]
3
popularized MoM with his text book, Field Computation by Moment Methods. Since then,
continuous increases in digital computer processing speed and memory size have served to
make numerical methods commonplace tools for electromagnetic analysis and design. Due
to their faster computational speed, surface MoM techniques have remained more popular
than volume integral MoM techniques for solving threedimensional problems.
Chang and Harrington used fictitious equivalent electric and magnetic surface currents,
a concept described by Schelkuno? [27], to calculate the characteristic modes for two
dimensional dielectric bodies. We will use the electric field integral equation (EFIE) and
magnetic field integral equation (HFIE) in the form given in their 1977 paper [5] to calculate
the scattered fields from arbitrarilyshaped, threedimensional dielectric scatterers. For each
region of interest:
EFIE :
bracketleftbigg
j?A + ?? +
1
?
??F
bracketrightbigg
tan
=
bracketleftbig
E
i
bracketrightbig
tan
(1.1)
HFIE :
bracketleftbigg
j?F + ?? ?
1
?
??A
bracketrightbigg
tan
=
bracketleftbig
H
i
bracketrightbig
tan
(1.2)
where the magnetic and electric vector potentials, Aand F, respectively, are defined in
terms of the equivalent electric and magnetic surface currents J
S
and M
S
as
A = ?
integraldisplayintegraldisplay
S
J
S
GdS
prime
(1.3)
F = epsilon1
integraldisplayintegraldisplay
S
M
S
GdS
prime
, (1.4)
the electric scalar potential ? is defined as
? =
1
epsilon1
integraldisplayintegraldisplay
S
q
e
S
GdS
prime
(1.5)
=
j
?epsilon1
integraldisplayintegraldisplay
S
??J
S
GdS
prime
, (1.6)
4
the magnetic scalar potential ? is defined as
? =
1
?
integraldisplayintegraldisplay
S
q
m
S
GdS
prime
(1.7)
=
j
??
integraldisplayintegraldisplay
S
??M
S
GdS
prime
, (1.8)
q
e
S
is the electric surface charge density related to the electric current density by the equation
??J
S
= ?j?q
e
S
, (1.9)
q
m
S
is the magnetic surface charge density related to the magnetic current density by the
equation
??M
S
= ?j?q
m
S
, (1.10)
the Green?s function G is defined as
G =
e
?jkR
4?R
(1.11)
R = r ? r
prime
, (1.12)
? and epsilon1 are the permeability and permittivity constants of the surrounding medium, S
prime
denotes the source surface, and k is the wave number. The vectors r and r
prime
are position
vectors to observation and source points, respectively, from a global coordinate origin. The
left sides of (1.1) and (1.2) represent tangential reflected fields, while the right sides represent
tangential incident fields. The derivation of (1.1) and (1.2) is given in Appendix A.
How can the surface integral method equate the incident and scattered fields at the
surface of a dielectric body? Chang and Harrington write separate equations for the inner
and outer sides of the surface. The total solution is viewed as the superposition of two cases;
in the first case, the inner region has zero field and in the second case, the outer region has
zero field. When both cases are added together, the tangential fields have a nonzero sum in
both inner and outer regions. The equivalent surface currents J
S
and M
S
are expressions
5
of the actual fields ?n ? H
tan
and E
tan
? ?n, rather than actual currents, where ?n is a unit
vector normal to the surface and pointing into the region of scattering. It is assumed that
the outer and inner equivalent surface currents have opposite direction, and therefore, sum
to zero.
1.3 Statement of the Problem
Maue?s integral equation [29] is commonly employed to expand the portions of (1.1)
and (1.2) containing
1
?
??F and
1
?
??A. Most often, this equation is seen in the form:
H
s
(J)=?
1
?
??A = ???
integraldisplayintegraldisplay
S
J
S
GdS
prime
= ??n ?
J
S
2
?
integraldisplayintegraldisplay
S
a96
a96
?G ? J
S
dS
prime
(1.13)
where the deleted integral symbol
integraltextintegraltext
a96
indicates the principal value. By duality [2],
E
s
(M)=
1
?
??F = ??
integraldisplayintegraldisplay
S
M
S
GdS
prime
= ?n ?
M
S
2
+
integraldisplayintegraldisplay
S
a96a96
?G ? M
S
dS
prime
. (1.14)
In MoM scattering solutions, (1.13) and (1.14) are usually treated with RaoWilton
Glisson (RWG) [28] basis functions together with triangular patch modeling. In the RWG
method, the testing functions are the same as the basis functions. It may be noted that the
RWG basis functions were originally defined for PEC MoM problems. The application of
these basis functions for dielectric/composite body problems is not entirely satisfactory for
the following reason.
When (1.13) and (1.14) are expanded and tested, unstable and incorrect solutions can
occur if the basis and testing functions are not carefully chosen. For example, if we call
the testing vectors t and lscript, the expressions lscript ? (??n ?
J
S
2
)ort ? (?n ?
M
S
2
) will tend to
be insignificantly small if, in either expression, the testing vector and the current basis
vector are parallel. Unfortunately, use of the RWG basis functions for both currents J
S
6
and M
S
tends to create this scenario. In the MoM matrix equation, the terms ??n?
J
S
2
and
?n?
M
S
2
are moment matrix self terms, which lie on the diagonal and should be dominant.
We implement basis and testing functions that will preserve these self term portions as well
as the principal values of the curl terms in (1.1) and (1.2).
At certain frequencies, known as characteristic frequencies, false or spurious mathemat
ical solutions exist for the EFIE and HFIE for a closed body, whether the body is made of
PEC, dielectric or composite material. Yaghjian [30] has discussed this problem thoroughly.
In order to eliminate these spurious solutions, a number of combined field methods have
been devised, such as CFIE, PMCHWT, and M?uller. Each of these methods combines the
EFIE and HFIE in a particular way that obtains correct solutions at all frequencies includ
ing the characteristic frequencies. The paired pulse basis functions presented in this work
allow the implementation of any of these formulations accurately and e?ciently. Use of the
basis and testing functions presented here represents a relatively straightforward approach
and a simpler composite solution, when compared to the methods proposed by Sheng, Jin,
Song et al. [31] or Kishk and Shafai [32].
1.4 The Proposed Method
n
th
edge
midpoint
centroid
edge node
?
n
f
+
n
f
n
g
+
n
T
?
n
T
S
n
Figure 1.1: Basis functions f
n
and g
n
associated with the n
th
edge.
7
At the outset, we assume a triangular patch model for the given object. Within the
triangular surface mesh, T
+
n
and T
?
n
represent two triangles connected to the n
th
edge as
shown in Fig. 1.1. The edges of each triangle other than the n
th
edge we will call free edges.
Within each triangle, the surface is planar. We define two mutually orthogonal vector basis
functions associated with the n
th
edge as
f
n
(r)=
?
?
?
?
?
?n
?
? g
n
, r ? S
n
,
0, otherwise
(1.15)
and
g
n
(r)=
?
?
?
?
?
unit vector bardbl n
th
edge, r ? S
n
,
0, otherwise
(1.16)
where ?n
?
represents the unit vector normal to the plane of the triangle T
?
n
. The domain of
the basis functions is S
n
, the region whose perimeter is drawn by connecting the midpoints
of the free edges to the centroids of triangles T
?
n
and to the nodes of edge n. Shown as a
shaded area in Fig. 1.1, S
n
is 2/3 of the total triangular patch area. The basis functions
defined in (1.15) and (1.16) are unit pulse functions that are orthogonal to each other. In
dielectric scattering problem solutions, we will use f
n
to expand J
S
and g
n
to expand M
S
in the integral equations.
We further define the testing functions associated with edge m as vectors t
?
m
and lscript
m
,as
shown in Fig. 1.2. Vector t
+
m
extends from the triangle T
+
m
centroid to the edge m midpoint;
t
?
m
extends from the edge m midpoint to the triangle T
?
m
centroid. Vector lscript
m
extends from
the beginning to the end of edge m, in the direction of g
m
. The testing vector t is used
in conjunction with the EFIE, while the testing vector lscript is used in conjunction with the
HFIE to solve dielectric problems. Testing the expanded integral equations then results in
the nonzero products t ? f, t ? (?n ?
g
2
), lscript ? g, and lscript ? (??n ?
f
2
). With this arrangement,
all of the currents are welltested.
8
m
th
edge
centroid
+
m
t
?
m
t
m
T
+
m
T
?
m
l
edge node
Figure 1.2: Testing functions t
m
and lscript
m
associated with the m
th
edge.
1.5 Scope
The example problems we address are frequency domain problems concerning scattering
bodies that may be decomposed into homogeneous regions having real or complex epsilon1. While
the accuracy of a given solution may be somewhat a?ected by the meshing and integral
evaluation techniques chosen, such questions are not the focus of the paper and are not
dwelt upon. The triangular meshes have been drawn fine enough to produce solutions that
appear to the eye to be reasonably well converged when plotted. Meshes in the example
problems follow the rule of thumb of at least 300 unknowns per square wavelength of
scattering surface area. Also, the meshes have been drawn somewhat irregularly to avoid
a particular type of modelinduced error called grid error. The intent of the dissertation
is to make a mathematical argument for the pulse basis pair method and to demonstrate
correct scattering solutions for a number of mostly canonical geometries that can be solved
by another trusted method for comparison. Numerical integrations have been performed
using the Gaussian quadrature method for triangles.
9
1.6 Organization
The body of this document contains four chapters that were initially written to be pub
lished as individual journal articles. They have been reformatted and slightly expanded for
inclusion here. Each chapter is selfcontained and may contain some repetition of previous
text. Chapter 2 discusses the f basis function for equivalent electric surface currents in
PEC bodies. Chapter 3 discusses the g basis function, also for equivalent electric surface
currents in PEC bodies. While either basis function may be employed for EFIE or HFIE
solutions for PEC bodies, chapters 2 and 3 present EFIE solutions only. Chapter 4 describes
the use of the f and g basis functions for equivalent electric and magnetic surface currents,
respectively, in dielectric bodies. In chapter 4, EFIE and HFIE solutions are compared for
the same problems; the two solutions can be combined in the appropriate manner to obtain
any of the combined field formulations. In chapter 5, the use of f and g basis functions is
described for the solution of composite scattering bodies. For those examples, EFIE solu
tions are shown for composites modeled as combinations of open and closed bodies, and an
HFIE solution is shown for a composite modeled as a combination of two closed bodies.
A derivation of the dielectric integral equations may be found in Appendix A. Appendix
B contains an example of the flexibility of the pulse vector basis functions; a PEC sphere
scattering problem is solved four times consecutively by using either f or g basis functions
for the EFIE or HFIE method.
10
Chapter 2
New Basis Functions for the Electromagnetic Solution of
Arbitrarilyshaped, Threedimensional Conducting Bodies
Using Method of Moments
2.1 Overview
In this chapter, we present a new set of basis functions, defined over a pair of planar
triangular patches, for the solution of electromagnetic scattering and radiation problems as
sociated with arbitrarilyshaped surfaces using the method of moments solution procedure.
The basis functions are constant over the function subdomain and resemble pulse functions
for one and twodimensional problems. Further, another set of basis functions, orthogonal
to the first set, is also defined over the same physical space. The primary objective of
developing these basis functions is to utilize them for electromagnetic solutions involving
conducting, dielectric, and composite bodies. The present chapter, however, involves only
conducting bodies along with several numerical results.
2.2 Introduction
The solution of electromagnetic scattering/radiation problems involving arbitrary shapes
and material composition is of much interest to commercial as well as defense industries.
The method of moments (MoM) [2] solutions to these problems generally involve triangular
patch modeling, utilizing RaoWiltonGlisson (RWG) [28] basis functions. It may be noted
that the RWG basis functions have been defined for the solution of conducting bodies and
the utilization of the same basis functions for dielectric/composite bodies is less than satis
factory. The primary di?culty associated with a material body solution is the requirement
of two orthogonal basis functions to express unknown electric and magnetic surface currents
11
J
S
and M
S
. In our opinion, using the same basis functions for both J
S
and M
S
is not a
good idea and invariably results in numerical di?culties. Consequently, a host of techniques
has been developed which involve either tinkering with the basis functions or modifying the
testing procedures to apply for material bodies [31, 33, 34]. Keeping these di?culties in
perspective, in this work, we present two orthogonal sets of basis functions that can be used
for conducting as well as material bodies. The solution of the material body problem will
be presented in due course.
2.3 Description of the Problem
x
y
z
S
PEC
?
?
E
i
Figure 2.1: Arbitrarilyshaped conducting body excited by an incident electromagnetic
plane wave.
Let S denote the surface of an arbitrarilyshaped perfectly conducting body illuminated
by an incident electromagnetic plane wave E
i
as shown in Fig. 2.1. We assume S to be open
or closed and orientable, possessing a piecewise continuous normal. S may be composed of
intersecting surfaces. Using the equivalence principle, potential theory, and the freespace
12
Green?s function [2], the electric field integral equation (EFIE) is given by
[j?A + ??]
tan
= E
i
tan
(2.1)
where the subscript tan refers to the tangential component. In (2.1),
A = ?
integraldisplayintegraldisplay
S
J
s
GdS
prime
(2.2)
? = epsilon1
?1
integraldisplayintegraldisplay
S
q
s
GdS
prime
(2.3)
G =
e
?jkR
4?R
(2.4)
R = r ? r
prime
 , (2.5)
epsilon1 and ? are the permittivity and permeability constants of the surrounding medium, k is
the wave number, and r and r
prime
represent the position vectors to observation and source
points, respectively, from a global coordinate origin. The unknown surface current density
J
s
is related to the charge density q
s
by the continuity equation, given by
??J
s
= ?j? q
s
. (2.6)
For the numerical solution of (2.1), we apply the method of moments formulation
using planar triangular patch modeling and the basis functions as described in the following
section:
2.4 Description of Basis Functions
Let T
+
n
and T
?
n
represent two triangles connected to the edge n of the triangulated
surface model as shown in Fig. 2.2. We define two mutually orthogonal vector basis
13
n
th
edge
midpoint
centroid
edge node
?
n
f
+
n
f
n
g
+
n
T
?
n
T
S
n
Figure 2.2: Basis function description.
functions associated with the n
th
edge as
f
n
(r)=
?
?
?
?
?
?n
?
?
?
lscript, r ? S
n
,
0, otherwise
(2.7)
and
g
n
(r)=
?
?
?
?
?
?
lscript, r ? S
n
,
0, otherwise
(2.8)
where S
n
represents the region obtained by connecting the midpoints of the free edges to
the centroids of triangles T
?
n
, and to the nodes of edge n. This is shown shaded in Fig. 2.2.
Also,
?
lscript and ?n
?
represent the unit vector along the n
th
edge and the unit vector normal to
the plane of the triangle T
?
n
, respectively. Note that the basis functions defined in (2.7) and
(2.8) are actually the pulse functions defined over the region S
n
. It is wellknown that the
pulse functions do not have continuous derivatives but result in delta distributions along
the boundary. This point is crucial in modeling the charge density and the calculation of
scalar potential, which may be accomplished as described in the following section. Also,
note that in this chapter, only perfect electric conductor (PEC) bodies are analyzed and
hence only f
n
?s are used in the method of moments solution.
14
2.5 Numerical Solution Procedure
m
th
edge
centroid
+c
m
r
?c
m
r
m
T
+ m
T
?
m
r
edge midpoint
Figure 2.3: Testing paths associated with the m
th
edge.
As a first step, we consider the testing procedure. Consider the m
th
interior edge,
associated with triangles T
?
m
, as shown in Fig. 2.3. We integrate the vector component of
(2.1) parallel to the path from the centroid r
c+
m
of T
+
m
to the midpoint of the edge r
m
and
thence from r
m
to the centroid of T
?
m
given by r
c?
m
. For each section of the path integration,
we approximate A and E
i
by their respective values at the midpoints of the path. Thus,
we have,
j?A
parenleftbigg
r
m
+ r
c+
m
2
parenrightbigg
? (r
m
? r
c+
m
)+j?A
parenleftbigg
r
m
+ r
c?
m
2
parenrightbigg
? (r
c?
m
? r
m
)+
bracketleftbig
?(r
c?
m
) ? ?(r
c+
m
)
bracketrightbig
= E
i
parenleftbigg
r
m
+ r
c+
m
2
parenrightbigg
? (r
m
? r
c+
m
)+
E
i
parenleftbigg
r
m
+ r
c?
m
2
parenrightbigg
? (r
c?
m
? r
m
) (2.9)
for m =1,2,3,???,N,whereN represents the total number of interior edges in the trian
gulation scheme, i.e., excluding the edges on the boundary for an open body.
Next, we consider the expansion procedure. Using the basis functions f
n
defined in
(2.7), we approximate the unknown current J
S
as
J
S
=
N
summationdisplay
n=1
I
n
f
n
. (2.10)
15
Next, substituting the current expansion (2.10) into (2.9) yields an N ?N system of linear
equations which may be written in matrix form as ZI = V ,whereZ =[Z
mn
]isanN ?N
matrix and I =[I
n
]andV =[V
m
] are column vectors of length N. The elements of Z and
V are given by
Z
mn
= j?
bracketleftbig
A
+
mn
? (r
m
? r
c+
m
)+A
?
mn
? (r
c?
m
? r
m
)
bracketrightbig
+ ?
?
mn
? ?
+
mn
(2.11)
V
m
= E
+
m
? (r
m
? r
c+
m
)+E
?
m
? (r
c?
m
? r
m
) (2.12)
where
A
?
mn
= ?
integraldisplayintegraldisplay
S
f
n
e
?jkR
?
m
4?R
?
m
dS
prime
(2.13)
?
?
mn
=
?1
j?epsilon1
integraldisplayintegraldisplay
S
?
s
? f
n
e
?jkR
c?
m
4?R
c?
m
dS
prime
(2.14)
R
?
m
=
vextendsingle
vextendsingle
vextendsingle
vextendsingle
r
m
+ r
c?
m
2
? r
prime
vextendsingle
vextendsingle
vextendsingle
vextendsingle
(2.15)
R
c?
m
= r
c?
m
? r
prime
 (2.16)
E
?
m
= E
i
parenleftbigg
r
m
+ r
c?
m
2
parenrightbigg
. (2.17)
The numerical evaluation of the vector potential, shown in (2.13), is straightforward
and may be accomplished by the procedure described in [35]. However, the numerical
evaluation of the scalar potential term, described in (2.14), may be carried out as follows:
Let us define the unknown charge density q
S
in (2.3) as
q
S
=
N
p
summationdisplay
i=1
?
i
P
i
(2.18)
where N
P
represents the number of triangular patches in the model, ?
i
is the unknown
coe?cient, and
P
i
(r)=
?
?
?
?
?
1, r ? T
i
,
0, otherwise.
(2.19)
16
I
i2
I
i1
I
i3
T
i
i1
l
2i
l
3i
l
Figure 2.4: Electric charge patch within the T
th
i
triangle.
Next, consider a triangular patch T
i
with associated nonboundary edges, i
1
, i
2
,andi
3
.
Then, using (2.6), the wellknown divergence theorem, and simple vector calculus, we have
integraldisplayintegraldisplay
T
i
q
S
dS =
integraldisplayintegraldisplay
T
i
?
S
? J
S
?j?
dS
=
j
?
contintegraldisplay
C
i
J
S
? ?n
lscript
=
j
?
[I
i
1
lscript
i
1
+ I
i
2
lscript
i
2
+ I
i
3
lscript
i
3
] (2.20)
where C
i
is the contour bounding the triangle T
i
, ?n
lscript
is the unit vector normal to the contour
C
i
in the plane of T
i
,andlscript
i
j
,j=1,2,3 represent the edge lengths. This scheme is shown
in Fig. 2.4. Also, note that
integraldisplayintegraldisplay
T
i
q
S
dS =
integraldisplayintegraldisplay
T
i
?
i
dS
= ?
i
A
i
(2.21)
17
where A
i
represents the area of the triangle T
i
. Lastly, using (2.20) and (2.21), we have
?
i
=
j
?
bracketleftbigg
I
i
1
lscript
i
1
+ I
i
2
lscript
i
2
+ I
i
3
lscript
i
3
A
i
bracketrightbigg
. (2.22)
Thus, we can write the scalar potential term in (2.14) as
?
?
mn
=
jlscript
n
?epsilon1
bracketleftBigg
1
A
n
+
integraldisplayintegraldisplay
T
+
n
e
?jkR
c?
m
4?R
c?
m
dS
prime
+
1
A
n
?
integraldisplayintegraldisplay
T
?
n
e
?jkR
c?
m
4?R
c?
m
dS
prime
bracketrightBigg
. (2.23)
Finally, once the matrices Z and V are determined, we may easily solve the system of
linear equations to obtain I.
2.6 Numerical Results
In this section, we present numerical results for a square plate (length = 0.15?), a
circular disk (diameter = 0.15?), a sphere (diameter = 0.15?) and a circular cylinder
(diameter = 0.15?,length=0.15?), and compare with the solutions obtained using the
procedure presented in [28]. Also, for the case of the sphere, the results are compared with
the exact solution. The plate, the disk, the sphere, and the cylinder are modeled with
312, 258, 500, and 320 triangles, respectively. In every case, the body is placed at the
center of the coordinate system and illuminated by an xpolarized plane wave traveling in
the negative direction along the zaxis. Further, the square plate and the circular disk are
oriented parallel to the xyplane. The bistatic radar cross sections (RCS?s) are presented in
Figs. 2.5?2.8. We note that the results compare well with the other numerical results.
2.7 Summary
In this chapter, we have presented a new set of basis functions, which we called the f
basis functions, for the method of moments solution of electromagnetic scattering by bodies
of arbitrary shape. The f basis functions are vectors perpendicular to the mesh edges.
Another set of basis functions, orthogonal to the first set, has also been presented; these are
18
the g basis functions, vectors parallel to the mesh edges. Both of these new basis functions
are pulse vectors defined over adjacent pairs of triangular patches. It is hoped that these
two sets of basis functions, in conjunction with the method of moments solution procedure,
will provide a more stable solution to material problems. However, in the present chapter,
only conducting scatterers were analyzed with the new basis function f and the results were
compared with those from other solution methods. The new basis functions will be applied
to material bodies in chapters 4 and 5.
0 30 60 90 120 150 180
0
0.5
1
1.5
2
x 10
3
Theta [degrees]
RCS
/
????
2
Pulse
Pulse
RWG
RWG
E
i X
Y
Z
phi = 0?
phi = 90
?
Figure 2.5: Bistatic RCS of a square plate of length 0.15? excited by a plane wave traveling
in the z direction.
19
0 30 60 90 120 150 180
0
2
4
6
8
x 10
4
Theta [degrees]
RC
S
/
????
2
Pulse
Pulse
RWG
RWG
E
i
phi = 0 ?
phi = 90 ? Z
Y
X
Figure 2.6: Bistatic RCS of a circular disk of diameter 0.15? excited by a plane wave
traveling in the z direction.
0 30 60 90 120 150 180
0
2
4
6
8
x 10
3
Theta [degrees]
RC
S
/
????
2
Pulse
Pulse
RWG
Exac t
Exac t
Exac t
phi = 90 ?
phi = 0 ?
Z
Y
X
E
i
Figure 2.7: Bistatic RCS of a sphere of diameter 0.15? excited by a plane wave traveling
in the z direction.
20
0 30 60 90 120 150 180
0
0.004
0.008
0.012
0.016
Theta [degrees]
RC
S
/
????
2
Pulse
Pulse
RWG
RWG
phi = 90 ?
phi = 0 ?
X
Y
Z
E
i
Figure 2.8: Bistatic RCS of a circular cylinder of diameter 0.15? and height 0.15? excited
by a plane wave traveling in the z direction.
21
Chapter 3
An Alternate Set of Basis Functions for the Electromagnetic
Solution of ArbitrarilyShaped, ThreeDimensional,
Closed Conducting Bodies Using Method of Moments
3.1 Overview
In chapter 2, we introduced two new sets of pulsetype basis functions, each defined
over adjacent pairs of planar triangular patches, to calculate the electromagnetic scatter
ing/radiation associated with threedimensional, arbitrarilyshaped material bodies. We
then explored in detail the suitability of one set of basis functions, the f pulse basis vec
tors, to calculate electromagnetic scattering from arbitrarilyshaped conducting bodies, ei
ther open or closed. In this chapter, we explore the use of an alternate set of basis functions,
the g pulse basis vectors, which are orthogonal to the f pulse basis functions previously
defined. We describe the numerical solution scheme using g basis vectors and calculate the
perfect electric conductor (PEC) scattering for two canonical closed geometries. The pulse
basis results are then compared with the results calculated by other means.
3.2 Introduction
The primary motivation for this work is to develop an e?cient and wellconditioned
method of moments (MoM) [3] solution for dielectric material bodies via a surface integral
equation (SIE) approach [36]. It may be noted that since the SIE approach involves both
electric and magnetic currents as unknowns in the MoM formulation, it is necessary to define
two mutually orthogonal sets of basis functions to generate a wellconditioned moment
matrix. We emphasize here that the mathematical equations appearing in the dielectric
22
body SIE formulation are very similar to the equations we encounter in the PEC body
problem; therefore, solving the PEC case provides confidence in the solution methodology.
3.3 Description of the Problem
x
y
z
S
PEC
?
?
E
i
Figure 3.1: Arbitrarilyshaped conducting body excited by an incident electromagnetic
plane wave.
Let S denote the surface of an arbitrarilyshaped, perfectly conducting body illumi
nated by an incident electromagnetic plane wave E
i
as shown in Fig. 3.1. We assume S to
be closed and orientable, possessing a piecewise continuous normal. S may be composed of
intersecting surfaces. Using the equivalence principle, potential theory, and the freespace
Green?s function [2], the electric field integral equation (EFIE) is given by
[j?A + ??]
tan
= E
i
tan
(3.1)
23
where the subscript tan refers to the tangential component. In (3.1),
A = ?
integraldisplayintegraldisplay
S
J
S
GdS
prime
(3.2)
? = epsilon1
?1
integraldisplayintegraldisplay
S
q
S
GdS
prime
(3.3)
G =
e
?jkR
4?R
(3.4)
R = r ? r
prime
, (3.5)
epsilon1 and ? are the permittivity and permeability constants of the surrounding medium, k is
the wave number, and r and r
prime
represent the position vectors to observation and source
points, respectively, from a global coordinate origin. The unknown surface current J
S
is
related to the charge density q
S
by the continuity equation, given by
??J
S
= ?j?q
S
. (3.6)
For the numerical solution of (3.1), we apply the method of moments formulation using
planar triangular patch modeling and the basis functions as described in the following:
3.4 Description of Basis Functions
Let T
+
n
and T
?
n
represent two triangles connected to the edge n of the triangulated
surface model as shown in Fig. 3.2. We define two mutually orthogonal vector basis
functions associated with the n
th
edge as
f
n
(r)=
?
?
?
?
?
?n
?
?
?
lscript, r ? S
n
,
0, otherwise
(3.7)
and
g
n
(r)=
?
?
?
?
?
?
lscript, r ? S
n
,
0, otherwise
(3.8)
24
n
th
edge
midpoint
centroid
edge node
?
n
f
+
n
f
n
g
+
n
T
?
n
T
S
n
Figure 3.2: Basis function description.
where S
n
represents the region obtained by connecting the midpoints of the free edges
to the centroids of triangles T
?
n
and to the nodes of edge n. This area is shown shaded
in Fig. 3.2. Also,
?
lscript and ?n
?
represent the unit vector along the n
th
edge and the unit
vector normal to the plane of the triangle T
?
n
, respectively. Note that the basis functions
defined in (3.7) and (3.8) are actually pulse functions, orthogonal to each other, defined
over the region S
n
. The electromagnetic solution procedure using the basis functions f
n
was described in the previous chapter, and in this chapter we present a similar solution
using the basis functions g
n
.
It may also be noted here that the basis functions g
n
are less versatile than the basis
functions f
n
. Unlike the basis functions f
n
, which are applicable to both open and closed
bodies as demonstrated in chapter 2, the basis functions g
n
are applicable to closed bodies
only. The main reason for this restriction is that the functions g
n
are defined parallel to
the edges of the planar triangular patches and represent the tangential component of the
current. It is wellknown that the tangential component of the surface current is undefined
for an open surface at the boundary. However, our primary motivation to use these functions
is their applicability in the solution of the dielectric body problem, which is always posed
for a closed body. Hence, the basis functions g
n
do not hinder our purpose.
25
3.5 Numerical Solution Procedure
m
th
edge
m
T
+ m
T
?
m
l
edge node m
1
edge node m
2
Figure 3.3: Testing path associated with the m
th
edge.
As a first step, we consider the testing procedure. Consider the m
th
interior edge,
associated with triangles T
?
m
. We integrate the vector component of (3.1) along the m
th
edge, shown in Fig. 3.3, to obtain
integraldisplay
lscript
m
j?A(r) ? dlscript +
integraldisplay
lscript
m
??(r) ? dlscript =
integraldisplay
lscript
m
E
i
(r) ? dlscript (3.9)
which may be rewritten as
integraldisplay
lscript
m
j?A(r) ? dlscript + ?
m2
n
? ?
m1
n
=
integraldisplay
lscript
m
E
i
(r) ? dlscript (3.10)
for n =1,2,3,???,N,wheren represents the source charge region and N represents the
total number of edges in the triangulation scheme. Note that in (3.10), ?
m1
n
and ?
m2
n
represent the scalar potentials evaluated at the nodes connected to the m
th
edge. Further,
the integrals appearing in (3.10) may be easily evaluated using any accurate numerical
integration algorithm such as the onepoint, twopoint or fourpoint trapezoidal rule. In
thiswork,wechoosetouseatwopointmethod.
26
Next, we consider the expansion procedure. Using the basis functions g
n
defined in
(3.8), we approximate the unknown current J
S
as
J
S
=
N
summationdisplay
n=1
I
n
g
n
. (3.11)
This is followed by a substitution of the current expansion (3.11) into (3.10), yielding an
N ?N system of linear equations which may be written in matrix form as ZI = V ,where
Z =[Z
mn
]isanN ?N matrix and I =[I
n
]andV =[V
m
] are column vectors of length N.
The elements of the Z and V are given by
Z
mn
= j?
integraldisplay
lscript
m
A
n
? dlscript + ?
m2
n
? ?
m1
n
(3.12)
V
m
=
integraldisplay
lscript
m
E
i
? dlscript (3.13)
where
A
n
= ?
integraldisplayintegraldisplay
S
g
n
e
?jkR
4?R
dS
prime
(3.14)
?
m1
n
=
?1
j?epsilon1
integraldisplayintegraldisplay
S
?
s
? g
n
e
?jkR
m1
n
4?R
m1
n
dS
prime
(3.15)
?
m2
n
=
?1
j?epsilon1
integraldisplayintegraldisplay
S
?
s
? g
n
e
?jkR
m2
n
4?R
m2
n
dS
prime
(3.16)
R =
vextendsingle
vextendsingle
r ? r
prime
vextendsingle
vextendsingle
(3.17)
R
m1
n
= r
m1
? r
prime
 (3.18)
R
m2
n
= r
m2
? r
prime
 (3.19)
and r
m1
and r
m2
are the position vectors to the nodes m1andm2, respectively, connected
to the m
th
edge.
The numerical evaluation of the vector potential, shown in (3.14), is straightforward
and may be accomplished by the procedure described by Wilton et al. [35]. However, the
27
S
i
T
1
T
2
T
3
T
4
T
5
T
6
i
th
node
E
2
E
3
E
4
E
5
E
1
E
6
?
ij
lll
c+
ij
r
c
ij
r
j
th
edge
Figure 3.4: Electric charge patch around the i
th
node.
numerical evaluation of the scalar potential terms described in (3.15) and (3.16) may be
carried out as follows:
Let us define the unknown charge density q
S
in (3.3) as
q
S
=
N
n
summationdisplay
i=1
?
i
P
i
(3.20)
where N
n
represents the total number of nodes (vertices) in the model, ?
i
is an unknown
coe?cient, and
P
i
(r)=
?
?
?
?
?
1, r ? S
i
,
0, otherwise.
(3.21)
In (3.21), S
i
is the i
th
charge patch, formed by connecting the centers of the edges and
the centroids of the triangles associated with the node i, as shown by the shaded area in
Fig. 3.4.
28
Now, let us consider the i
th
charge patch. Using (3.6), the wellknown divergence
theorem, and simple vector calculus, we have
integraldisplayintegraldisplay
S
i
q
S
dS =
integraldisplay
S
i
?
s
? J
S
?j?
dS
=
j
?
contintegraldisplay
C
i
J
S
? ?n
lscript
dlscript (3.22)
where C
i
is the contour bounding the charge patch S
i
,and?n
lscript
is the unit vector normal to
the contour C
i
in the plane of the triangle containing the contour segment. By considering
the left hand side of (3.22), we have
integraldisplayintegraldisplay
S
i
q
S
dS =
integraldisplayintegraldisplay
S
i
?
i
dS
= ?
i
A
Si
(3.23)
where A
Si
represents the area of the charge patch S
i
. Considering the right hand side of
(3.22), we have
j
?
contintegraldisplay
C
i
J
S
? ?n
lscript
dlscript =
j
?
E
K
summationdisplay
j=1
I
ij
[
?
lscript
ij
? (?n
+
ij
? r
c+
ij
+ ?n
?
ij
? r
c?
ij
)] (3.24)
where
?
lscript
ij
and ?n
?
ij
, respectively, represent the unit vector along the j
th
edge connected to
node i and the outward unit vector normal to the plane of the T
?
ij
triangle associated with
the j
th
edge connected to node i. As shown in Fig. 3.4, the vector r
c?
ij
extends from the
centroid of the T
?
ij
triangle to the center of the j
th
edge, while r
c+
ij
extends from the edge
center to the centroid of the T
+
ij
triangle. Also, E
K
represents the total number of edges
connected to node i. Using (3.23) and (3.24), we have
?
i
=
j
?A
Si
E
K
summationdisplay
j=1
I
ij
[
?
lscript
ij
? (?n
+
ij
? r
c+
ij
+ ?n
?
ij
? r
c?
ij
)]. (3.25)
29
Substituting the basis function g for
?
lscript, we can write the scalar potential terms in (3.15)
and (3.16) as
?
m1
n
=
j
?epsilon1
bracketleftBigg
g
n
? (?n
+
n
? r
c+
n
+ ?n
?
n
? r
c?
n
)
A
Sn1
integraldisplayintegraldisplay
S
n1
e
?jkR
m1
n1
4?R
m1
n1
dS
prime
?
g
n
? (?n
+
n
? r
c+
n
+ ?n
?
n
? r
c?
n
)
A
Sn2
integraldisplayintegraldisplay
S
n2
e
?jkR
m1
n2
4?R
m1
n2
dS
prime
bracketrightBigg
=
j
?epsilon1
g
n
?
parenleftbig
?n
+
n
? r
c+
n
+ ?n
?
n
? r
c?
n
parenrightbig
parenleftbigg
1
A
S
n1
integraldisplayintegraldisplay
S
n1
e
?jkR
m1
n1
4?R
m1
n1
dS
prime
?
1
A
S
n2
integraldisplayintegraldisplay
S
n2
e
?jkR
m1
n2
4?R
m1
n2
dS
prime
parenrightbigg
(3.26)
?
m2
n
=
j
?epsilon1
bracketleftBigg
g
n
? (?n
+
n
? r
c+
n
+ ?n
?
n
? r
c?
n
)
A
Sn1
integraldisplayintegraldisplay
S
n1
e
?jkR
m2
n1
4?R
m2
n1
dS
prime
?
g
n
? (?n
+
n
? r
c+
n
+ ?n
?
n
? r
c?
n
)
A
Sn2
integraldisplayintegraldisplay
S
n2
e
?jkR
m2
n2
4?R
m2
n2
dS
prime
bracketrightBigg
=
j
?epsilon1
g
n
?
parenleftbig
?n
+
n
? r
c+
n
+ ?n
?
n
? r
c?
n
parenrightbig
parenleftbigg
1
A
S
n1
integraldisplayintegraldisplay
S
n1
e
?jkR
m2
n1
4?R
m2
n1
dS
prime
?
1
A
S
n2
integraldisplayintegraldisplay
S
n2
e
?jkR
m2
n2
4?R
m2
n2
dS
prime
parenrightbigg
(3.27)
where r
m1
, r
m2
, r
n1
,andr
n2
are the position vectors to nodes m1andm2 connected to
the edge m, and nodes n1andn2 connected to the edge n, respectively. Further, S
n1
and
S
n2
are the charge patches associated with the nodes n1andn2, respectively, as depicted in
Fig. 3.5. The integrals in (3.26) and (3.27) may be evaluated with the procedures described
in [35] .
Finally, once the matrices Z and V are determined, we may easily solve the system of
linear equations to obtain I.
3.6 Numerical Results
In this section, we present numerical results for a PEC sphere (diameter = 0.15?)and
a cube (length = 0.15?), and we compare the results with the solutions obtained using the
procedure presented in chapter 2. Also, for the case of the sphere, the results are compared
with the exact solution. The sphere and the cube are modeled with 500 and 960 triangles,
30
S
n1
S
n2
g
n
T
n
+
T
n

r
n
c+
r
n
c
Figure 3.5: Electric source patches S
n1
and S
n2
for ?
mn
calculation.
respectively. For both examples, the body is placed symmetrically at the center of the
coordinate system and illuminated by an xpolarized plane wave traveling in the negative
direction along the zaxis. The bistatic radar cross sections are presented in Figs. 3.6 and
3.7. We note that the new basis function results compare well with those determined by
other methods.
3.7 Summary
In this chapter, we have applied a new set of basis functions, which we called the g basis
functions, for the method of moments solution of electromagnetic scattering by conducting
bodies of arbitrary shape. The new basis functions are pulse basis vectors defined over a
pair of triangular patches and tangential to the common edge. We have shown that similar
numerical results are obtained by using the f basis function perpendicular to the edge or
the g basis function tangential to the edge; in order to do this we changed the testing
function as well as the basis function. The present set of basis functions along with the
functions discussed in our previous chapter should prove helpful in obtaining a stable and
31
0 30 60 90 120 150 180
0
2
4
6
8
x 10
?3
Theta [degrees]
RCS/
?
2
Pulse f
Pulse g
Exact
? = 0?
x
y
z
? = 90?
?
?E
i
Figure 3.6: Bistatic RCS of a sphere of diameter 0.15 ? excited by a plane wave traveling
in the z direction.
0 30 60 90 120 150 180
0
0.01
0.02
0.03
0.04
Theta [degrees]
RCS/
?
2
Pulse f
Pulse g
? = 90?
? = 0?
z
y
x
?
?
E
i
Figure 3.7: Bistatic RCS of a cube of length 0.15 ? excited by a plane wave traveling in the
z direction.
32
wellconditioned solution to the material body problem. The new basis function pair will
be applied to material bodies in chapters 4 and 5.
33
Chapter 4
Electromagnetic Scattering from ArbitrarilyShaped Dielectric Bodies
Using Paired Pulse Vector Basis Functions and Method of Moments
4.1 Overview
In the previous two chapters, we demonstrated a pair of orthogonal pulse vector basis
functions for the calculation of electromagnetic scattering from arbitrarilyshaped conduct
ing bodies. In this chapter, we extend the use of the same two basis vectors, f and g,
to solve dielectric body problems. Further, the numerical solution procedures to calculate
scalar and vector potentials for the dielectric case are based upon procedures shown in the
last two chapters. The reason to use an orthogonal basis function pair is now made apparent
by the calculations required for either the electric field integral equation (EFIE) or the mag
netic field integral equation (HFIE) dielectric solution. The importance of demonstrating
both EFIE and HFIE solutions is to show the suitability of the basis functions for combined
field formulations in order to guarantee unique solutions at all frequencies. In this chapter,
we detail the implementations for EFIE and HFIE formulations and show example results
for canonical figures.
4.2 Introduction
For scattering problems concerning perfect electric conductors (PEC?s) of arbitrary
shape and of electrical size in the resonance region, the method of moments (MoM) has
provided a practical means of solution using surface integral equations [3]. MoM solutions
are particularly advantageous for calculating radar cross sections (RCS?s) when compared to
di?erential methods such as the finite element method, because the MoM solution incorpo
rates a Green?s function that, by definition, reduces the scattered field strengths to zero at
34
infinite distance. When the scattering objects are PEC or homogeneous dielectric bodies,
the boundary problem can be solved by surface integral equations, resulting in a com
putational savings compared to volume integral equations. Numerous successful meshing
schemes and basis functions have been employed for surface integral PEC MoM problems.
For example, a popular combination has been triangular patch modeling in combination
with RaoWiltonGlisson (RWG) [28] vector basis functions to expand the unknown equiva
lent electric surface currents [2]. For the more complex case of dielectric scattering, several
authors [37], [38] have represented equivalent electric and magnetic surface currents with a
single basis function. A computational di?culty then arises because the testing functions
and the basis functions are all vectors having the same direction; implementation of the curl
operation in the integral equations results in zerovalued or very small diagonal impedance
matrix terms and an unstable numerical solution.
To provide a more stable solution for dielectric bodies, we use orthogonally placed,
pulse basis vectors defined over each contiguous pair of triangular patches: one for the
equivalent electric surface current J
S
and one for the equivalent magnetic surface current
M
S
. This combination ensures strongly diagonal MoM matrices. The basis functions for
M
S
and J
S
are placed parallel and perpendicular, respectively, to each edge in the triangu
lar patch scheme, providing a smooth transition from dielectric areas to conducting areas.
Consequently, the proposed pulse vector basis functions may be used to solve composite
problems. The surface integral technique described here is suitable for dielectric regions
of homogeneous composition and requires the scattering bodies to be modeled as closed
surfaces.
To avoid the illconditioned problem associated with characteristic frequencies, the
paired pulse vector basis functions may be used in integral equations for any of the com
bined field methods. In the past, the combined field integral equation (CFIE) technique
has been used to model conductors [4], while the PoggioMillerChangHarringtonWuTsai
(PMCHWT) [5], [6] technique or the M?uller [39] technique has been used to model dielectric
35
bodies. Pulse pair vector basis functions allow the use of any combined field method for
dielectric bodies.
In the following sections, we discuss the integral equations for dielectric scatterers. The
basis and testing functions are defined and implemented for EFIE and HFIE formulations.
Numerical examples are presented to show the calculated RCS for a number of canonical
geometries.
4.3 Integral Equations
E
1
,H
1
?
1
,?
1
E
2
,H
2
?
2
,?
2
E
i
S
1
?n
2
?n
Figure 4.1: An arbitrarilyshaped dielectric body with surface S excited by an external
source.
Figure 4.1 shows an arbitrarilyshaped, closed dielectric body with surface S.An
unseen source in region 1 outside the body is radiating at a frequency of ?; the incident
electric field is labeled E
i
. For this problem, there is no source inside the body in region 2.
The media in regions 1 and 2 are characterized by ?
1
and epsilon1
1
, ?
2
and epsilon1
2
, respectively. We
write the resulting fields, equivalent surface currents, and their associated potentials as
phasor quantities that are understood to vary at the same frequency ?. Our objective will
be to calculate J
S
and M
S
, fictitious surface currents on S that would produce the same
scattered E and H as the actual source. By applying the equivalence principle [2], [5], we
36
write the EFIE?s for the dielectric body:
bracketleftbigg
j?A
1
+ ??
1
+
1
?
1
??F
1
bracketrightbigg
tan
=
bracketleftbig
E
i
1
bracketrightbig
tan
(4.1)
bracketleftbigg
j?A
2
+ ??
2
+
1
?
2
??F
2
bracketrightbigg
tan
=0 (4.2)
where the magnetic and electric vector potentials, A
i
and F
i
for i =1,2, are defined in
terms of the equivalent currents as
A
i
= ?
i
integraldisplayintegraldisplay
S
J
S
G
i
dS
prime
(4.3)
F
i
= epsilon1
i
integraldisplayintegraldisplay
S
M
S
G
i
dS
prime
(4.4)
and the electric scalar potential ?
i
is defined as
?
i
=
1
epsilon1
i
integraldisplayintegraldisplay
S
q
e
S
G
i
dS
prime
(4.5)
=
j
?epsilon1
i
integraldisplayintegraldisplay
S
??J
S
G
i
dS
prime
. (4.6)
The Green?s function G
i
is defined as
G
i
=
e
?jk
i
R
4?R
(4.7)
R = r ? r
prime
, (4.8)
?
i
and epsilon1
i
are the permeability and permittivity constants of the surrounding medium, S
prime
denotes the source surface, and k
i
is the wave number for each region. The vectors r and r
prime
are position vectors to observation and source points, respectively, from a global coordinate
origin. The electric charge density q
e
S
is related to the unknown surface current J
S
by the
continuity equation, given by
??J
S
= ?j?q
e
S
. (4.9)
37
By Maue?s integral [29],
??n
1
???
integraldisplayintegraldisplay
S
M
S
G
1
dS
prime
=
M
S
2
? ?n
1
?
integraldisplayintegraldisplay
S
a96a96
?G
1
? M
S
dS
prime
(4.10)
where ?n
1
is the unit vector normal to S pointing away from the surface into region 1 and the
deleted integral symbol
integraltextintegraltext
a96
indicates the principal value. The normal ?n
1
and its opposite,
?n
2
, are shown in Fig. 4.1. Taking the cross product of ?n
1
with each side of (4.10), we may
write the curl operation in (4.1) as
1
?
1
??F
1
= ??
integraldisplayintegraldisplay
S
M
S
G
1
dS
prime
= ?n
1
?
M
S
2
+
integraldisplayintegraldisplay
S
a96a96
?G
1
? M
S
dS
prime
. (4.11)
Similarly, for the surface in region 2, we obtain
??
integraldisplayintegraldisplay
S
M
S
G
2
dS
prime
= ?n
2
?
M
S
2
+
integraldisplayintegraldisplay
S
a96a96
?G
2
? M
S
dS
prime
. (4.12)
Thus, using the modifications presented in (4.11) and (4.12), the EFIE?s for the dielectric
body may be written as
bracketleftbigg
j?A
1
+ ??
1
+ ?n
1
?
M
S
2
+
integraldisplayintegraldisplay
S
a96a96
?G
1
? M
S
dS
prime
bracketrightbigg
tan
=
bracketleftbig
E
i
1
bracketrightbig
tan
(4.13)
bracketleftbigg
j?A
2
+ ??
2
+ ?n
2
?
M
S
2
+
integraldisplayintegraldisplay
S
a96a96
?G
2
? M
S
dS
prime
bracketrightbigg
tan
=0. (4.14)
Next, again invoking the equivalence principle, we write the HFIE?s for the dielectric
body:
bracketleftbigg
j?F
1
+ ??
1
?
1
?
1
??A
1
bracketrightbigg
tan
=
bracketleftbig
H
i
1
bracketrightbig
tan
(4.15)
bracketleftbigg
j?F
2
+ ??
2
?
1
?
2
??A
2
bracketrightbigg
tan
= 0 (4.16)
38
where the magnetic scalar potential ?
i
for i =1,2 is defined as
?
i
=
1
?
i
integraldisplayintegraldisplay
S
q
m
S
G
i
dS
prime
(4.17)
=
j
??
i
integraldisplayintegraldisplay
S
??M
S
G
i
dS
prime
(4.18)
and q
m
S
is the magnetic charge density related to the fictitious magnetic current density by
the equation
??M
S
= ?j?q
m
S
. (4.19)
By reasoning analogous to that used in expanding the EFIE?s, we may rewrite the HFIE?s
for the dielectric body as
bracketleftbigg
j?F
1
+ ??
1
? ?n
1
?
J
S
2
?
integraldisplayintegraldisplay
S
a96a96
?G
1
? J
S
dS
prime
bracketrightbigg
tan
=
bracketleftbig
H
i
1
bracketrightbig
tan
(4.20)
bracketleftbigg
j?F
2
+ ??
2
? ?n
2
?
J
S
2
?
integraldisplayintegraldisplay
S
a96
a96
?G
2
? J
S
dS
prime
bracketrightbigg
tan
=0. (4.21)
Note that for either the EFIE or the HFIE method, the region 1 and region 2 equations
look similar.
When the MoM matrix is calculated for (4.13) and (4.14), the M
S
/2 terms will be
the dominant terms in the submatrix dealing with the E(M
S
) portion of the evaluation.
Similarly, when the MoM matrix is calculated for (4.20) and (4.21), the J
S
/2 terms will
dominate the H(J
S
) portion of the evaluation. It is important that we choose the proper
basis and testing functions so that this dominance is preserved in the numerical solution.
4.4 Basis and Testing Functions
Let us assume that the surface is modeled by a triangular mesh. T
+
n
and T
?
n
repre
sent two triangles connected to the n
th
edge of the triangulated surface model as shown
in Fig. 4.2. The edges of each triangle other than the n
th
edge we will call free edges.
Within each triangle, the surface is planar. We define two mutually orthogonal vector basis
39
functions associated with the n
th
edge as
f
n
(r)=
?
?
?
?
?
?n
?
? g
n
, r ? S
n
,
0, otherwise
(4.22)
and
g
n
(r)=
?
?
?
?
?
unit vector bardbl n
th
edge, r ? S
n
,
0, otherwise
(4.23)
where ?n
?
represents the unit vector normal to the plane of the triangle T
?
n
. S
n
represents
the domain of the basis functions: the region whose perimeter is drawn by connecting the
midpoints of the free edges to the centroids of triangles T
?
n
and to the nodes of edge n.
Shown as a shaded area in Fig. 4.2, S
n
is 2/3 of the total triangular patch area. Note that
the basis functions defined in (4.22) and (4.23) are unit pulse functions orthogonal to each
other. Throughout the problem solution, we will use f
n
to expand J
S
and g
n
to expand
M
S
.
The testing functions associated with edge m are vectors t
?
m
and lscript
m
, for EFIE and
HFIE solutions, respectively, as shown in Fig. 4.3. Vector t
+
m
extends from the triangle T
+
m
centroid to the edge m midpoint; t
?
m
extends from the edge m midpoint to the triangle T
?
m
centroid. Vector lscript
m
extends from the beginning to the end of edge m, in the direction of
g
m
.
4.5 Numerical Solution Procedure
We have discretized the surface of interest using a triangular mesh containing a total
of N edges. The MoM solution procedure results in 2N linear equations, written as
?
?
?
?
?
[Z
1
(J
S
)] [Z
1
(M
S
)]
[Z
2
(J
S
)] [Z
2
(M
S
)]
?
?
?
?
?
?
?
?
?
?
[J
S
]
[M
S
]
?
?
?
?
?
=
?
?
?
?
?
[V
1
]
[V
2
]
?
?
?
?
?
(4.24)
40
n
th
edge
midpoint
centroid
edge node
?
n
f
+
n
f
n
g
+
n
T
?
n
T
S
n
Figure 4.2: Basis functions f
n
and g
n
associated with the n
th
edge.
m
th
edge
centroid
+
m
t
?
m
t
m
T
+
m
T
?
m
l
edge node
Figure 4.3: Testing functions t
m
and lscript
m
associated with the m
th
edge.
41
where [Z
1
(J
S
)], [Z
1
(M
S
)], [Z
2
(J
S
)], and [Z
2
(M
S
)] are N?N matrices and the numerical
subscripts refer to the medium in which the matrix elements are evaluated. [J
S
]and[M
S
]
are column vectors of length N. For the EFIE solution,
[V
1
]=
bracketleftbig
E
i
1
E
i
2
??? E
i
N
bracketrightbig
T
, (4.25)
for the HFIE solution,
[V
1
]=
bracketleftbig
H
i
1
H
i
2
??? H
i
N
bracketrightbig
T
, (4.26)
and for either solution,
[V
2
]=[0
1
0
2
??? 0
N
]
T
. (4.27)
The four Z matrices represent the influence of the incident E and Hfields on edge currents
in regions 1 and 2.
In order to write Z
mn
in scalar terms, we integrate each of the equations (4.13), (4.14),
(4.20), and (4.21) along the appropriate m
th
testing vector. Letting ?n
1
= ??n
2
,weobtain
for the EFIE in region 1:
j?A
1
? t
m
+ ?
?
n
? ?
+
n
+
parenleftbigg
?n
1
?
M
S
2
parenrightbigg
? t
m
+
parenleftbiggintegraldisplayintegraldisplay
S
a96
a96
?G
1
? M
S
dS
prime
parenrightbigg
? t
m
= E
i
1
? t
m
(4.28)
for the EFIE in region 2:
j?A
2
? t
m
+ ?
?
n
? ?
+
n
?
parenleftbigg
?n
1
?
M
S
2
parenrightbigg
? t
m
+
parenleftbiggintegraldisplayintegraldisplay
S
a96a96
?G
2
? M
S
dS
prime
parenrightbigg
? t
m
= 0 (4.29)
42
for the HFIE in region 1:
j?F
1
? lscript
m
+ ?
2
n
? ?
1
n
?
parenleftbigg
?n
1
?
J
S
2
parenrightbigg
? lscript
m
?
parenleftbiggintegraldisplayintegraldisplay
S
a96a96
?G
1
? J
S
dS
prime
parenrightbigg
? lscript
m
= H
i
1
? lscript
m
(4.30)
and for the HFIE in region 2:
j?F
2
? lscript
m
+ ?
2
n
? ?
1
n
+
parenleftbigg
?n
1
?
J
S
2
parenrightbigg
? lscript
m
?
parenleftbiggintegraldisplayintegraldisplay
S
a96
a96
?G
2
? J
S
dS
prime
parenrightbigg
? lscript
m
=0. (4.31)
In (4.28?4.31), n = 1, 2, 3, ..., N identifies the location of the source charge. Expressions of
the form
parenleftBig
integraltext
t
m
?? ? t
m
parenrightBig
have been simplified to the form (?
?
n
? ?
+
n
), where ?
?
n
and ?
+
n
are
the scalar potentials due to charges near the n
th
edge evaluated at the minus and plus ends
of the testing vector, as defined by the assigned current direction [28]. Areas of magnetic
positive and negative charge have been designated by superscripts 1 and 2, respectively.
Next, we expand the currents J
S
and M
S
as
J
S
=
N
summationdisplay
n=1
I
n
f
n
(4.32)
and
M
S
=
N
summationdisplay
n=1
I
N+n
g
n
(4.33)
where [I] is the column matrix of complex scalar coe?cients. Substituting (4.32) and
(4.33) into (4.284.31) yields a 2N ?2N system of linear equations which may be written in
matrix form as [Z][I]=[V ], corresponding to the elements of (4.24). For the EFIE solution,
43
region 1, the elements Z
mn
are given by
Z
mn
(J
S
)=j?A
n
? t
m
+ ?
?
n
??
+
n
(4.34)
Z
mn
(M
S
)=
parenleftbigg
?n
1
?
g
n
2
parenrightbigg
? t
m
+
parenleftbiggintegraldisplayintegraldisplay
S
a96
a96
?G
1
? g
n
dS
prime
parenrightbigg
? t
m
(4.35)
where A
n
and ?
n
aregivenbythedefinitionsofA and ?, (4.3) and (4.6), respectively,
except that f
n
has replaced J
S
and the source areas are restricted to the n
th
source regions.
The elements Z
mn
for the EFIE solution, region 2, are found in a similar manner. For the
HFIE solution, region 1, the elements Z
mn
are given by
Z
mn
(J
S
)=?
parenleftbigg
?n
1
?
f
n
2
parenrightbigg
? lscript
m
?
parenleftbiggintegraldisplayintegraldisplay
S
a96
a96
?G
1
? f
n
dS
prime
parenrightbigg
? lscript
m
(4.36)
Z
mn
(M
S
)=j?F
n
? lscript
m
+ ?
2
n
??
1
n
(4.37)
where F
n
and ?
n
are given by the definitions of F and ?, (4.4) and (4.18), respectively,
except that g
n
has replaced M
S
and the source areas are restricted to the n
th
source
regions. The elements Z
mn
for the HFIE solution, region 2, are found in a similar manner.
The calculation of the vector and scalar potentials is detailed in the following three sections.
The elements V
m
are given by
V
1m
= E
i
m
? t
m
, EFIE solution, region 1 (4.38)
V
2m
=0, EFIE solution, region 2 (4.39)
V
1m
= H
i
m
? lscript
m
, HFIE solution, region 1 (4.40)
V
2m
=0, HFIE solution, region 2 . (4.41)
Once the matrices [Z]and[V ] have been determined, the unknowns in [I] may be calculated
by matrix algebra. The equivalent surface currents so determined may be used to calculate
fields inside or outside the scattering body, as desired.
44
4.5.1 Calculation of A and F
The following discussion applies to both regions 1 and 2, and the subscripts have been
dropped for A, F, epsilon1, ?,and?n. The vector potentials A
mn
and F
mn
are found by numerical
integration of the Green?s function over the n
th
source region shown shaded in Fig. 4.2.
The observation points r in the Green?s function definition are points chosen on or near
the testing vector; for this work, we obtained good EFIE and HFIE results by using one
T
+
m
point and one T
?
m
point. Each test point was the centroid of the smaller triangle whose
nodes were the n
th
edge nodes and the T triangle centroid. These observation points are
sketched in Fig. 4.4, in which the vector potential observation points are marked by o?s,
the scalar potential observation points by x?s. The final testing equations are written to
incorporate this segmentation of the vector potential, e.g.,
j?A
mn
? t
m
? j?
parenleftbig
A
+
mn
? t
+
m
+ A
?
mn
? t
?
m
parenrightbig
(4.42)
j?F
mn
? lscript
m
?
j?
2
parenleftbig
F
+
mn
+ F
?
mn
parenrightbig
? lscript
m
(4.43)
where A
+
mn
and F
+
mn
are the vector potentials observed at t
+
m
due to the n
th
source region,
and A
?
mn
and F
?
mn
are the vector potentials observed at t
?
m
due to the n
th
source region.
m
th
edge
+
m
t
?
m
t
m
T
+ m
T
?
m
l
x
x
x
x
o
o
edge node m
1
edge node m
2
Figure 4.4: Observation points for mn
th
vector (o) and scalar (x) potentials.
45
4.5.2 Calculation of ??F and ??A
The dielectric EFIE contains the term ??F, as shown in (4.1) and (4.2). In the
corresponding testing equation,
parenleftbigg
1
epsilon1
??F
m,n
parenrightbigg
? t
m
=
parenleftBig
?n
+
?
g
n
2
parenrightBig
? t
+
m
+
parenleftBig
?n
?
?
g
n
2
parenrightBig
? t
?
m
+
parenleftbiggintegraldisplayintegraldisplay
S
a96a96
?G
+
? g
n
dS
prime
parenrightbigg
? t
+
m
+
parenleftbiggintegraldisplayintegraldisplay
S
a96a96
?G
?
? g
n
dS
prime
parenrightbigg
? t
?
m
(4.44)
where ?n
?
denotes the normal to the T
?
m
field patch and the G superscript also refers to
the T
?
m
field patch associated with the Green?s function. Because the basis function g
n
is
constant over the n
th
source region, it may be moved outside the integral before the cross
product is calculated. The calculation of ??A
mn
is analogous. Thus,
parenleftbigg
?
1
?
??A
mn
parenrightbigg
? lscript
m
= ?
parenleftbigg
?n
+
?
f
+
n
2
+ ?n
?
?
f
?
n
2
parenrightbigg
?
lscript
m
2
?
parenleftbiggintegraldisplayintegraldisplay
S
?
a96
a96
?G
+
? f
?
n
dS
prime
+
integraldisplayintegraldisplay
S
?
a96
a96
?G
?
? f
?
n
dS
prime
parenrightbigg
?
lscript
m
2
. (4.45)
4.5.3 Calculation of ? and ?
We have stated in (4.6) that the electric scalar potential ? is defined as
? =
j
?epsilon1
integraldisplayintegraldisplay
S
??J
S
GdS
prime
.
Because the basis function f
n
is a pulse function, direct calculation of ??f
n
would produce
impulse functions at the edges of the source charge region. Rather than integrating impulse
functions, we will use the divergence theorem to calculate
integraltextintegraltext
S
??J
S
dS
prime
directly. Assuming
that ??J
S
is constant over a triangular source area and given that
integraldisplayintegraldisplay
S
??J
S
dS
prime
=
contintegraldisplay
C
J
S
? ?n
C
dC (4.46)
46
L
1
I
2
I
1
I
3
L
2
L
3
T
n
+
n
th
edge
Figure 4.5: Normal electric current components for ? calculation.
where ?n
C
is the unit vector normal to the contour in the plane of surface S
prime
,wemaywrite
? for a source triangle (shown in Fig. 4.5) as
? =
j (I
1
L
1
+ I
2
L
2
+ I
3
L
3
)
?epsilon1A
integraldisplayintegraldisplay
S
GdS
prime
(4.47)
where I
1
,I
2
,andI
3
are the current components of J
S
normal to the three sides; L
1
, L
2
,
and L
3
are the side lengths; and A is the triangle area.
The unknown electric charge density q
e
S
may be defined as
q
e
S
=
N
T
summationdisplay
i=1
?
i
P
i
(4.48)
where N
T
is the number of triangular patches in the model,
?
i
=
j
?
bracketleftbigg
I
i
1
lscript
i
1
+ I
i
2
lscript
i
2
+ I
i
3
lscript
i
3
A
i
bracketrightbigg
, (4.49)
and
P
i
(r)=
?
?
?
?
?
1, r ? T
i
,
0, otherwise.
(4.50)
47
In order to calculate ?
mn
, we assume that the electric charge associated with J
n
is now
spread out from S
n
(in Fig. 4.2) over the two larger, triangular regions T
?
n
.Forthemn
th
scalar potential term,
?
+
mn
=
j
?epsilon1
parenleftbigg
I
n
L
n
A
+
f
+
n
integraldisplayintegraldisplay
T
+
n
G
+
dS
prime
?
I
n
L
n
A
?
f
?
n
integraldisplayintegraldisplay
T
?
n
G
+
dS
prime
parenrightbigg
(4.51)
where the superscript on G indicates that the observation point lies on T
+
m
,andthepoten
tials associated with T
+
n
and T
?
n
have been di?erenced to obtain a result for the n
th
edge,
observed from the m
th
edge. ?
?
mn
is similarly calculated with the observation point on T
?
m
.
Equation (4.51) applies to both regions 1 and 2.
To calculate the scalar magnetic potential, we likewise start by defining areas of mag
netic charge associated with each edge current M
n
. Let us define the unknown charge
density q
m
S
in (4.17) as
q
m
S
=
N
N
summationdisplay
i=1
?
i
P
i
(4.52)
where N
N
represents the total number of nodes (vertices) in the model, ?
i
is a scalar to be
determined, and
P
i
(r)=
?
?
?
?
?
1, r ? S
i
,
0, otherwise.
(4.53)
In (4.53), S
i
is the i
th
charge patch, formed by connecting the centers of the edges and
the centroids of the triangles associated with the i
th
node, as shown by the shaded area in
Fig. 4.6. Again making use of the divergence theorem, we can write ? for the i
th
source
patch as
48
S
i
T
1
T
2
T
3
T
4
T
5
T
6
i
th
node
E
2
E
3
E
4
E
5
E
1
E
6
?
ij
lll
c+
ij
r
c
ij
r
j
th
edge
Figure 4.6: Magnetic charge source area for ? calculation.
? =
j
??
integraldisplayintegraldisplay
S
??M
S
GdS
prime
(4.54)
=
j
??
contintegraldisplay
C
M
S
? ?n
C
dC
integraldisplayintegraldisplay
S
GdS
prime
(4.55)
=
j
??A
i
braceleftbigg
N
E
summationdisplay
j=1
I
ij
bracketleftBig
?
lscript
ij
? (?n
+
ij
? r
c+
ij
+ ?n
?
ij
? r
c?
ij
)
bracketrightBig
bracerightbiggbraceleftbiggintegraldisplayintegraldisplay
S
GdS
prime
bracerightbigg
(4.56)
where
?
lscript
ij
and ?n
?
ij
, respectively, represent the unit vector along the j
th
edge connected to
node i and the outward unit vector normal to the plane of the T
?
ij
triangle associated with
the j
th
edge connected to node i. The vector r
c?
ij
extends from the centroid of the T
?
ij
triangle to the center of the j
th
edge, while r
c+
ij
extends from the edge center to the centroid
of the T
+
ij
triangle. N
E
represents the total number of edges connected to node i and A
i
is
the area of S
i
. From (4.56), we see that ?
i
in (4.52) is
?
i
=
j
?A
i
N
E
summationdisplay
j=1
I
ij
bracketleftBig
?
lscript
ij
? (?n
+
ij
? r
c+
ij
+ ?n
?
ij
? r
c?
ij
)
bracketrightBig
. (4.57)
49
In order to calculate ?
mn
, we will use the positive and negative magnetic charge
patches, designated S
n1
and S
n2
respectively, associated with the n
th
edge and shown in
Fig. 4.7. We will find the normal components of g
n
flowing across the mutual boundary,
designated r
c+
n
and r
c?
n
in Fig. 4.7. The value of ?
1
mn
, the scalar potential at the n
th
edge
as observed from end node 1 on the m
th
edge, is
?
1
mn
=
j
??A
S
n1
g
n
?
parenleftbig
?n
+
n
? r
c+
n
+ ?n
?
n
? r
c?
n
parenrightbig
parenleftbiggintegraldisplayintegraldisplay
S
n1
G
1
dS
prime
parenrightbigg
?
j
??A
S
n2
g
n
?
parenleftbig
?n
+
n
? r
c+
n
+ ?n
?
n
? r
c?
n
parenrightbig
parenleftbiggintegraldisplayintegraldisplay
S
n2
G
1
dS
prime
parenrightbigg
=
j
??
g
n
?
parenleftbig
?n
+
n
? r
c+
n
+ ?n
?
n
? r
c?
n
parenrightbig
parenleftbigg
1
A
S
n1
integraldisplayintegraldisplay
S
n1
G
1
dS
prime
?
1
A
S
n2
integraldisplayintegraldisplay
S
n2
G
1
dS
prime
parenrightbigg
. (4.58)
In (4.58), the superscript on G refers to the end of the m
th
edge where the observation is
made. A similar calculation is done to find ?
2
mn
. The equations are the same for regions 1
and 2.
S
n1
S
n2
g
n
T
n
+
T
n

r
n
c+
r
n
c
Figure 4.7: Magnetic source patches S
n1
and S
n2
for ?
mn
calculation.
50
4.5.4 Testing the Incident Fields
We test the V matrix as follows:
V
m
= t
+
m
? E
i+
+ t
?
m
? E
i?
, EFIE solution (4.59)
where E
i
is evaluated at the vector potential test points, near the midpoints of the test
vectors t.
V
m
= lscript
m
? H
i
, HFIE solution (4.60)
where H
i
is evaluated at the midpoint of the test vector lscript.
4.6 Numerical Examples
For three dielectric scatterers, a sphere, a cube, and a cone, the bistatic RCS has
been calculated by both EFIE and HFIE solution methods, incorporating pulse basis vector
functions. The MoM solutions are compared to solutions from at least one other calculation
method.
A sphere of radius 0.1 ? and epsilon1
R
= 4 is irradiated by a plane wave traveling in the +z
direction. Its mesh contains 500 patches and 750 edges. The MoM RCS is calculated by
using pulse basis functions and is compared to the Mie series analytic result. The orientation
of the sphere and the cube to the incident wave are shown in Fig. 4.8; the sphere RCS plots
are shown in Figs. 4.10 and 4.11.
Acubeoflength0.2? and epsilon1
R
= 4 is irradiated by a plane wave traveling in the +z
direction. Its mesh contains 480 patches and 720 edges. The MoM RCS is calculated by
using pulse basis functions and is compared to surface and volume integral results reported
by Sarkar, Arvas, and Ponapalli [40] for the same case. The cube RCS plots are shown in
Figs. 4.12 and 4.13.
A cone of radius 0.1 ?, apex halfangle = 30
?
,andepsilon1
R
= 3 is irradiated by a plane wave
traveling in the z direction. Its mesh contains 606 patches and 909 edges. The MoM RCS
51
is calculated by using pulse basis functions and is compared to MoM body of revolution
(BOR) results. The orientation of the cone to the incident wave is shown in Fig. 4.9; the
RCS plots are shown in Figs. 4.14 and 4.15.
?a
i
E
x
1
0.5
0
0.5
1
1
0.5
0
0.5
1
1
0.5
0
0.5
1
0.5
0
0.5
x
y
z
?
?
Figure 4.8: Orientation of dielectric sphere and cube to the incident plane wave. Sphere
radius = 0.1 ?; epsilon1
R
= 4. Cube length = 0.2 ?; epsilon1
R
=4.
0.5
0
0.5
0.5
0
.5
?a
i
E
x
x
y
z
?
?
Figure 4.9: Orientation of dielectric cone to the incident plane wave. Cone radius = 0.1 ?;
apex halfangle = 30
?
; epsilon1
R
=3.
The RCS plots show good agreement between the pulse vector basis results and the
standards of comparison. There is some di?erence between the EFIE and HFIE results;
at this time, no generalizations can be made concerning the conditions under which either
52
0 30 60 90 120 150 180
50
40
30
20
10
0
Theta [degrees]
No
r
m
a
l
i
z
e
d
RCS
[
d
B
]
Analytic
Pulse EFIE
Pulse HFIE
Figure 4.10: Bistatic RCS for a dielectric sphere at ? =0
?
, radius = 0.1 ?, epsilon1
R
=4.
0 30 60 90 120 150 180
4
3
2
1
0
Theta [degrees]
No
r
m
a
l
i
z
e
d
RCS
[
d
B
]
Analytic
Pulse EFIE
Pulse HFIE
Figure 4.11: Bistatic RCS for a dielectric sphere at ? =90
?
, radius = 0.1 ?, epsilon1
R
=4.
53
0 30 60 90 120 150 180
50
40
30
20
10
0
Theta [degrees]
No
r
m
a
l
i
z
e
d
RCS
[
d
B
]
Sarkar SIE
Sarkar VIE
Pulse EFIE
Pulse HFIE
Figure 4.12: Bistatic RCS for a dielectric cube at ? =0
?
, length = 0.2 ?, epsilon1
R
=4.
0 30 60 90 120 150 180
4
3
2
1
0
Theta [degrees]
No
r
m
a
l
i
z
e
d
RCS
[
d
B
]
Sarkar SIE
Sarkar VIE
Pulse EFIE
Pulse HFIE
Figure 4.13: Bistatic RCS for a dielectric cube at ? =90
?
, length = 0.2 ?, epsilon1
R
=4.
54
0 30 60 90 120 150 180
40
30
20
10
0
Theta [degrees]
N
o
rm
al
i
zed
R
C
S
[
d
B
]
BOR
Pulse EFIE
Pulse HFIE
Figure 4.14: Bistatic RCS for a dielectric cone at ? =0
?
, radius = 0.1 ?,apex
halfangle = 30
?
, epsilon1
R
= 3, incident wave traveling toward apex.
0 30 60 90 120 150 180
4
3
2
1
0
Theta [degrees]
N
o
rm
al
i
zed
R
C
S
[
d
B
]
BOR
Pulse EFIE
Pulse HFIE
Figure 4.15: Bistatic RCS for a dielectric cone at ? =90
?
, radius = 0.1 ?,apex
halfangle = 30
?
, epsilon1
R
= 3, incident wave traveling toward apex.
55
method is superior. Scattering from smoother objects, as expected, is more easily modeled
than from ones with sharp corners or points.
4.7 Summary
In this chapter we have demonstrated the use of a pair of orthogonal pulse vector basis
functions to solve dielectric MoM surface integral problems for closed bodies. We used the
f basis function for electric surface currents and the g basis function for magnetic surface
currents. Together with the pair of pulse basis vectors, we used testing vectors t and lscript
for EFIE and HFIE solutions, respectively. An important part of the numerical solution
procedure was the expression of the electric and magnetic scalar and vector potentials, which
themselves were expanded in terms of the basis functions. The pulse basis functions may
be used in combined field methods to insure unique solutions at resonant frequencies. Use
of these basis functions will allow reliable and accurate scattering solutions for conductors,
dielectric bodies, or composites of arbitrary shape.
56
Chapter 5
Electromagnetic Scattering from Arbitrarily Shaped Composites Using
Paired Pulse Vector Basis Functions and Method of Moments
5.1 Overview
In previous chapters, a pair of orthogonal pulse vector basis functions was demonstrated
for the calculation of electromagnetic scattering from arbitrarilyshaped perfect electric
conductors (PEC?s) (chapters 2 and 3) or dielectric bodies (chapter 4). In this chapter,
the basis functions are applied to dielectric/PEC composites. For the general case, i.e.,
perfect or lossy dielectric surfaces, the f and g pulse vector basis function pair is used to
represent equivalent electric and magnetic surface currents. For the special case of PEC
surfaces, only the f pulse vector basis function is needed to represent the equivalent electric
surface current. The composite scatterer may contain multiple dielectric and PEC parts,
either touching or nontouching. We describe here the scattering solution for a composite
structure and we show example electric field integral equation (EFIE) and magnetic field
integral equation (HFIE) results for several two or threepart figures.
5.2 Introduction
Due to the increasing development and use of a variety of building materials, metallic
and nonmetallic, for all structures large and small, it is very important to be able to
model correctly the scattering behavior of composite structures. For homogeneous regions
in dielectric bodies, we may often calculate electromagnetic scattering more e?ciently by
using equivalent surface currents [2] rather than the polarization volume currents. PEC
bodies naturally lend themselves to surface current modeling for another reason: the actual
currents occur in a thin layer near the surface. In the previous chapters, we described a
57
surface integral method using orthogonally placed, pulse basis vectors for the method of
moments (MoM) solution of scattering problems involving dielectric bodies [3]. The surface
integral technique described here is suitable for dielectric and PEC regions of homogeneous
composition; a triangular patch scheme is used for the surface mesh [28].
We wish to be able to use a common formulation for both the dielectric and the PEC
components of a closed body composite structure in order to calculate scattering accurately
and e?ciently while avoiding the illconditioned problem associated with characteristic fre
quencies. Kishk and Shafai [32] have reviewed a number of available composite formulations.
These include the combined field integral equation (CFIE) [4], the PoggioMillerChang
HarringtonWuTsai (PMCHWT) [5], [6], and the M?uller [7] formulations. For dielectric
problems, the paired pulse basis functions may be used with any of these integral equation
formulations to guarantee unique solutions. However, for a composite problem, only the
CFIE can be used throughout. The paired pulse basis functions allow use of the CFIE for
composite scattering solutions. But, if a portion of the composite structure is described
as an open body, the HFIE cannot be written; consequently, a combined field formulation
cannot be used.
To provide stable EFIE or HFIE solutions for dielectric bodies, we use orthogonally
placed, pulse basis vectors defined over each contiguous pair of triangular patches: one
for the equivalent electric surface current J
S
and one for the equivalent magnetic surface
current M
S
. This combination allows correct calculation of the curl terms in the EFIE
and HFIE, ensuring strongly diagonal moment matrices. To solve a composite problem, we
model all surface currents on both the dielectric and the PEC elements with pulse vector
basis functions. For the dielectric elements, we employ the orthogonal basis vector pair; for
the PEC elements we retain only the electric current basis vector.
In the following sections, we discuss solutions for composite dielectric/PEC scatterers.
The basis and testing functions are defined and the matrix equation is explained. Numerical
examples of EFIE and HFIE solutions are presented to show the calculated radar cross
sections (RCS?s) for several canonical geometries, including two nontouching spheres, a
58
disk/cone structure, a simplified missile shape, and a cube capped with PEC plates at two
ends.
5.3 Integral Equations
E
1
,H
1
?
1
, ?
1
E
2
,H
2
?
2
, ?
2
E
i
D1
1
?n
2
?n
C
PEC
D2
Figure 5.1: Arbitrarilyshaped PEC and dielectric bodies with surfaces C, D1,andD2
excited by an external source.
Figure 5.1 shows two arbitrarilyshaped bodies, one PEC and one dielectric. An unseen
source in region 1 outside the bodies is radiating at a frequency of ?. In region 1, fields
E
1
and H
1
exist as a result of the incident energy combined with scattering from the
two bodies. Surface C on the conductor exterior marks the boundary between region 1,
characterized by ?
1
and epsilon1
1
, and the interior of the conductor, where no fields exist. Surfaces
D1 and D2 are the two sides of the dielectric body surface separating region 1 from region
2. Region 2 inside the dielectric is characterized by ?
2
and epsilon1
2
and contains fields E
2
and
H
2
. By applying the equivalence principle [2], [5], we will calculate J
C
, J
D
,andM
D
,
fictitious surface currents on the surfaces that would produce the same scattered E and H
as the actual volume sources. We write the fields, equivalent surface currents, and their
associated potentials as phasor quantities that are understood to vary at the frequency ?.
To solve the scattering problem by either the EFIE or the HFIE surface integral method,
three equations are required. Table I lists the surfaces where the field points and current
sources are located for each equation.
59
Table 5.1: Integral Equation Surfaces
Equation Field Point Location Source Current Location
I C C, D1
II D1 C, D1
III D2 D2
The EFIE?s to be solved simultaneously are of the form
?[E
s
(J
C
,J
D
,M
D
)]
tan
=
bracketleftbig
E
i
bracketrightbig
tan
(5.1)
where s denotes scattered and i denotes incident.
EFIE I:
bracketleftbigg
j?A(J
C
,?
1
)+??(J
C
,?
1
,?
1
)+j?A(J
D1
,?
1
)+??(J
D1
,?
1
,?
1
)
+
1
?
1
??F (M
D1
,?
1
)
bracketrightbigg
tan
=
bracketleftbig
E
i
C
bracketrightbig
tan
. (5.2)
EFIE II:
bracketleftbigg
j?A(J
C
,?
1
)+??(J
C
,?
1
,?
1
)+j?A(J
D1
,?
1
)+??(J
D1
,?
1
,?
1
)
+
1
?
1
??F(M
D1
,?
1
)
bracketrightbigg
tan
=
bracketleftbig
E
i
D1
bracketrightbig
tan
. (5.3)
EFIE III:
bracketleftbigg
j?A(J
D2
,?
2
)+??(J
D2
,?
2
,?
2
)+
1
?
2
??F (M
D2
,?
2
)
bracketrightbigg
tan
=
bracketleftbig
E
i
D2
bracketrightbig
tan
. (5.4)
60
The magnetic and electric vector potentials, A and F, respectively, are defined in terms of
the equivalent currents as
A = ?
integraldisplayintegraldisplay
S
J
S
GdS
prime
(5.5)
F = epsilon1
integraldisplayintegraldisplay
S
M
S
GdS
prime
(5.6)
where S represents the source surface of interest. The electric scalar potential ? is defined
as
? =
1
epsilon1
integraldisplayintegraldisplay
S
q
e
S
GdS
prime
(5.7)
=
j
?epsilon1
integraldisplayintegraldisplay
S
??J
S
GdS
prime
(5.8)
where q
e
S
is the electric charge density related to the fictitious electric current density by
the equation
??J
S
= ?j?q
e
S
. (5.9)
The Green?s function G is defined as
G =
e
?jkR
4?R
(5.10)
R = r ? r
prime
, (5.11)
? and epsilon1 are the permeability and permittivity constants of the surrounding medium, S
prime
denotes the source surface, and k is the wave number. The vectors r and r
prime
are position
vectors to observation and source points, respectively, from a global coordinate origin.
Using Maue?s integral [29], we express the
1
?
??F portion of the EFIE?s as
1
?
i
??F
i
= ??
integraldisplayintegraldisplay
S
M
S
G
i
dS
prime
= ?n
i
?
M
S
2
+
integraldisplayintegraldisplay
S
a96a96
?G
i
? M
S
dS
prime
(5.12)
61
for i =1,2, where ?n
i
is the unit vector normal to S pointing away from the surface into
region i and the deleted integral symbol
integraltextintegraltext
a96
indicates the principal value. The normal ?n
1
and its opposite, ?n
2
,areshowninFig.5.1.
Referring to Table I again, for the same surfaces we may write three simultaneous
HFIE?s of the form
?[H
s
(J
C
,J
D
,M
D
)]
tan
=[H
i
]
tan
. (5.13)
HFIE I:
bracketleftbigg
?
1
?
1
??A(J
C
,?
1
) ?
1
?
1
??A(J
D1
,?
1
)
+ j?F(M
D1
,?
1
)+??(M
D1
,?
1
,?
1
)
bracketrightbigg
tan
=
bracketleftbig
H
i
C
bracketrightbig
tan
. (5.14)
HFIE II:
bracketleftbigg
?
1
?
1
??A(J
C
,?
1
) ?
1
?
1
??A(J
D1
,?
1
)
+ j?F(M
D1
,?
1
)+??(M
D1
,?
1
,?
1
)
bracketrightbigg
tan
=
bracketleftbig
H
i
D1
bracketrightbig
tan
. (5.15)
HFIE III:
bracketleftbigg
?
1
?
2
??A(J
D2
,?
2
)+j?F(M
D2
,?
2
)+??(M
D2
,?
2
,?
2
)
bracketrightbigg
tan
=
bracketleftbig
H
i
D2
bracketrightbig
tan
. (5.16)
The magnetic scalar potential ? is defined as
? =
1
?
integraldisplayintegraldisplay
S
q
m
S
GdS
prime
(5.17)
=
j
??
integraldisplayintegraldisplay
S
??M
S
GdS
prime
(5.18)
62
and q
m
S
is the magnetic charge density related to the unknown magnetic current density by
the equation
??M
S
= ?j?q
m
S
. (5.19)
Using Maue?s integral, we express the ?
1
?
??A portion of the HFIE?s as
?
1
?
i
??A
i
= ???
integraldisplayintegraldisplay
S
J
S
G
i
dS
prime
= ??n
i
?
J
S
2
?
integraldisplayintegraldisplay
S
a96
a96
?G
i
? J
S
dS
prime
(5.20)
for i =1,2.
5.4 Basis and Testing Functions
n
th
edge
midpoint
centroid
edge node
?
n
f
+
n
f
n
g
+
n
T
?
n
T
S
n
Figure 5.2: Basis functions f
n
and g
n
associated with the n
th
edge.
Let us assume that the surface is modeled by a triangular mesh. T
+
n
and T
?
n
repre
sent two triangles connected to the n
th
edge of the triangulated surface model as shown
in Fig. 5.2. The edges of each triangle other than the n
th
edge we will call free edges.
Within each triangle, the surface is planar. We define two mutually orthogonal vector basis
63
functions associated with the n
th
edge as
f
n
(r)=
?
?
?
?
?
?n
?
? g
n
, r ? S
n
,
0, otherwise
(5.21)
and
g
n
(r)=
?
?
?
?
?
unit vector bardbl n
th
edge, r ? S
n
,
0, otherwise
(5.22)
where ?n
?
represents the unit vector normal to the plane of the triangle T
?
n
. S
n
represents
the domain of the basis functions: the region whose perimeter is drawn by connecting the
midpoints of the free edges to the centroids of triangles T
?
n
and to the nodes of edge n.
Shown as a shaded area in Fig. 5.2, S
n
is 2/3 of the total triangular patch area. Note that
the basis functions defined in (5.21) and (5.22) are unit pulse functions, orthogonal to each
other. Throughout the problem solution, we will use f
n
to expand J
S
and g
n
to expand
M
S
.
m
th
edge
centroid
+
m
t
?
m
t
m
T
+
m
T
?
m
l
edge node
Figure 5.3: Testing functions t
m
and lscript
m
associated with the m
th
edge.
The testing functions associated with edge m are vectors t
?
m
and lscript
m
,fortheEFIEand
HFIE solutions, respectively, as shown in Fig. 5.3. Vector t
+
m
extends from the triangle T
+
m
centroid to the edge m midpoint; t
?
m
extends from the edge m midpoint to the triangle T
?
m
64
centroid. Vector lscript
m
extends from the beginning to the end of edge m, in the direction of
g
m
.
5.5 Numerical Solution Procedure
In this section, the EFIE and HFIE solution methods are shown for one PEC body
and one dielectric body. After meshing each surface of interest, we obtain N
C
edges for the
conductor and N
D
edges for the dielectric body. The MoM solution procedure results in
N
C
+2N
D
linear equations, written as
?
?
?
?
?
?
?
?
?
?
?
[ Z
C
(J
C
)] [Z
C
(J
D
)] [Z
C
(M
D
)]
[Z
D1
(J
C
)][Z
D1
(J
D
)] [Z
D1
(M
D
)]
[0] [Z
D2
(J
D
)] [Z
D2
(M
D
)]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
[J
C
]
[J
D
]
[M
D
]
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
[V
C
]
[V
D1
]
[0]
?
?
?
?
?
?
?
?
?
?
?
. (5.23)
The Z subscripts C, D1, and D2 indicate the conductor, outer dielectric, and inner di
electric surfaces, respectively, where the field points reside. J and M subscripts C and
D, respectively, indicate the conductor and dielectric surfaces where the source currents
reside. The (3,1) block of the moment matrix contains only zeros because, in the equiva
lent problem, at the inner dielectric surface, no fields are observed due to currents on the
conductor.
For the EFIE solution, [V
C
]and[V
D1
] are written
[V ]=
bracketleftbig
E
i
1
E
i
2
??? E
i
N
bracketrightbig
T
, (5.24)
for the HFIE solution, [V
C
]and[V
D1
] are written
[V ]=
bracketleftbig
H
i
1
H
i
2
??? H
i
N
bracketrightbig
T
, (5.25)
and for either solution
[V
D2
]=[0
1
0
2
??? 0
N
]
T
. (5.26)
65
The [V
D2
] column of the incident field matrix contains only zeros because the incident fields
from the external source in this problem strike only the outer surfaces of each body.
In order to write Z
mn
in scalar terms, we integrate each of the EFIE?s or HFIE?s along
the appropriate m
th
testing vector. Thus, we obtain for the EFIE?s:
j?A
i
? t
m
+ ?
?
n
? ?
+
n
+
parenleftbigg
?n
i
?
M
S
2
parenrightbigg
? t
m
+
parenleftbiggintegraldisplayintegraldisplay
S
a96a96
?G
i
? M
S
dS
prime
parenrightbigg
? t
m
= E
i
i
? t
m
(5.27)
where the subscripts on A, ?n, G,andE
i
denote the region into which the source cur
rent is radiating. Expressions of the form
parenleftBig
integraltext
t
m
?? ? t
m
parenrightBig
have been simplified to the form
(?
?
n
? ?
+
n
), where ?
?
n
and ?
+
n
are the scalar potentials due to charges near the n
th
edge
evaluated at the minus and plus ends of the testing vector, as defined by the assigned current
direction [28].
Similarly, we obtain for the HFIE?s:
j?F
i
? lscript
m
+ ?
2
n
? ?
1
n
?
parenleftbigg
?n
i
?
J
S
2
parenrightbigg
? lscript
m
?
parenleftbiggintegraldisplayintegraldisplay
S
a96a96
?G
i
? J
S
dS
prime
parenrightbigg
? lscript
m
= H
i
i
? lscript
m
. (5.28)
Next, we expand the currents J
S
and M
S
as
J
S
=
N
summationdisplay
n=1
I
n
f
n
(5.29)
and
M
S
=
N
summationdisplay
n=1
I
n
g
n
(5.30)
where [I] is a column matrix of complex scalar coe?cients. Substituting (5.29) and (5.30)
into equations of the form (5.27) and (5.28) yields a (2N
C
+ N
D
) ? (2N
C
+ N
D
)systemof
66
linear equations which may be written in matrix form as [Z][I]=[V ], corresponding to the
elements of (5.23). For the EFIE solution, the elements Z
mn
are of the form
Z
mn
(J
S
)=j?A
n
? t
m
+ ?
?
n
? ?
+
n
(5.31)
Z
mn
(M
S
)=
parenleftbigg
?n
i
?
g
n
2
parenrightbigg
? t
m
+
parenleftbiggintegraldisplayintegraldisplay
S
a96
a96
?G
i
? g
n
dS
prime
parenrightbigg
? t
m
(5.32)
where A
n
and ?
n
aregivenbythedefinitionsofA and ?, (5.5) and (5.8), respectively,
except that f
n
has replaced J
S
and the source areas are restricted to the n
th
source regions.
For the HFIE solution, the elements Z
mn
are of the form
Z
mn
(J
S
)=?
parenleftbigg
?n
i
?
f
n
2
parenrightbigg
? lscript
m
?
parenleftbiggintegraldisplayintegraldisplay
S
a96a96
?G
i
? f
n
dS
prime
parenrightbigg
? lscript
m
(5.33)
Z
mn
(M
S
)=j?F
n
? lscript
m
+ ?
2
n
? ?
1
n
(5.34)
where F
n
and ?
n
are given by the definitions of F and ?, (5.6) and (5.18), respectively,
except that g
n
has replaced M
S
and the source areas are restricted to the n
th
source regions.
The calculation of the vector and scalar potentials is detailed in the following three sections.
The elements V
m
are given by
V
m
= E
i
m
? t
m
, EFIE solution (5.35)
V
m
= H
i
m
? lscript
m
, HFIE solution . (5.36)
Once the matrices [Z]and[V ] have been determined, the unknowns in [I] may be calculated
by matrix algebra. The equivalent surface currents so determined may be used to calculate
fields inside or outside the scattering body, as desired.
67
5.5.1 Calculation of A and F
The following discussion applies to both regions 1 and 2, and the subscripts have been
dropped for A, F, epsilon1, ?,and?n. The vector potentials A
mn
and F
mn
are found by numerical
integration of the Green?s function over the n
th
source region shown shaded in Fig. 5.2.
The observation points r in the Green?s function definition are points chosen on or near
the testing vector; for this work, we obtained good EFIE and HFIE results by using one
T
+
m
point and one T
?
m
point. Each test point was the centroid of the smaller triangle whose
nodes were the n
th
edge nodes and the T triangle centroid. These observation points are
sketched in Fig. 5.4, in which the vector potential observation points are marked by o?s,
the scalar potential observation points by x?s.
m
th
edge
+
m
t
?
m
t
m
T
+ m
T
?
m
l
x
x
x
x
o
o
edge node m
1
edge node m
2
Figure 5.4: Observation points for mn
th
vector (o) and scalar (x) potentials.
The final testing equations are written to incorporate this segmentation of the vector
potential, e.g.,
j?A
mn
? t
m
? j?
parenleftbig
A
+
mn
? t
+
m
+ A
?
mn
? t
?
m
parenrightbig
(5.37)
j?F
mn
? lscript
m
?
j?
2
parenleftbig
F
+
mn
+ F
?
mn
parenrightbig
? lscript
m
(5.38)
where A
+
mn
and F
+
mn
are the vector potentials observed at t
+
m
due to the n
th
source region,
and A
?
mn
and F
?
mn
are the vector potentials observed at t
?
m
due to the n
th
source region.
68
5.5.2 Calculation of ??F and ??A
The dielectric EFIE contains the term ??F, as shown in (5.1) and (5.2). In the
corresponding testing equation,
parenleftbigg
1
epsilon1
??F
m,n
parenrightbigg
? t
m
=
parenleftBig
?n
+
?
g
n
2
parenrightBig
? t
+
m
+
parenleftBig
?n
?
?
g
n
2
parenrightBig
? t
?
m
+
parenleftbiggintegraldisplayintegraldisplay
S
a96a96
?G
+
? g
n
dS
prime
parenrightbigg
? t
+
m
+
parenleftbiggintegraldisplayintegraldisplay
S
a96a96
?G
?
? g
n
dS
prime
parenrightbigg
? t
?
m
(5.39)
where ?n
?
denotes the normal to the T
?
m
field patch and the G superscript also refers to
the T
?
m
field patch associated with the Green?s function. Because the basis function g
n
is
constant over the n
th
source region, it may be moved outside the integral before the cross
product is calculated. The calculation of ??A
mn
is analogous. Thus,
parenleftbigg
?
1
?
??A
mn
parenrightbigg
? lscript
m
= ?
parenleftbigg
?n
+
?
f
+
n
2
+ ?n
?
?
f
?
n
2
parenrightbigg
?
lscript
m
2
?
parenleftbiggintegraldisplayintegraldisplay
S
?
a96
a96
?G
+
? f
?
n
dS
prime
+
integraldisplayintegraldisplay
S
?
a96
a96
?G
?
? f
?
n
dS
prime
parenrightbigg
?
lscript
m
2
. (5.40)
5.5.3 Calculation of ? and ?
We have stated in (5.8) that the electric scalar potential ? is defined as
? =
j
?epsilon1
integraldisplayintegraldisplay
S
??J
S
GdS
prime
.
Because the basis function f
n
is a pulse function, direct calculation of the ??f
n
would
produce impulse functions at the edges of the source charge region. Rather than integrating
impulse functions, we will use the divergence theorem to calculate
integraltextintegraltext
S
??J
S
dS
prime
directly.
Assuming that ??J
S
is constant over a triangular source area and given that
integraldisplayintegraldisplay
S
??J
S
dS
prime
=
contintegraldisplay
C
J
S
? ?n
C
dC (5.41)
69
where ?n
C
is the unit vector normal to the contour in the plane of surface S
prime
,wecanwrite
? for a source triangle (shown in Fig. 5.5) as
L
1
I
2
I
1
I
3
L
2
L
3
T
n
+
n
th
edge
Figure 5.5: Normal electric current components for ? calculation.
? =
j (I
1
L
1
+ I
2
L
2
+ I
3
L
3
)
?epsilon1A
integraldisplayintegraldisplay
S
GdS
prime
(5.42)
where I
1
,I
2
,andI
3
are the current components of J
S
normal to the three sides; L
1
, L
2
,
and L
3
are the side lengths; and A is the triangle area. Let us define the unknown electric
charge density q
e
S
in (5.7) as
q
e
S
=
N
T
summationdisplay
i=1
?
i
P
i
(5.43)
where N
T
is the number of triangular patches in the model,
?
i
=
j
?
bracketleftbigg
I
i
1
lscript
i
1
+ I
i
2
lscript
i
2
+ I
i
3
lscript
i
3
A
i
bracketrightbigg
, (5.44)
and
P
i
(r)=
?
?
?
?
?
1, r ? T
i
,
0, otherwise.
(5.45)
70
The electric charge associated with J
n
is now spread out from S
n
(in Fig. 5.2) over the two
larger, triangular regions T
?
n
; T
+
n
is drawn in Fig. 5.5. For the mn
th
scalar potential term,
?
+
mn
=
j
?epsilon1
parenleftbigg
I
n
L
n
A
+
f
+
n
integraldisplayintegraldisplay
T
+
n
G
+
dS
prime
?
I
n
L
n
A
?
f
?
n
integraldisplayintegraldisplay
T
?
n
G
+
dS
prime
parenrightbigg
(5.46)
where the superscript on G indicates that the observation point lies on T
+
m
,andthepoten
tials associated with T
+
n
and T
?
n
have been di?erenced to obtain a result for the n
th
edge,
observed from the m
th
edge. ?
?
mn
is similarly calculated with the observation point on T
?
m
.
Equation (5.46) applies to both regions 1 and 2.
S
i
T
1
T
2
T
3
T
4
T
5
T
6
i
th
node
E
2
E
3
E
4
E
5
E
1
E
6
?
ij
lll
c+
ij
r
c
ij
r
j
th
edge
Figure 5.6: Magnetic charge source area for ? calculation.
To calculate the scalar magnetic potential, we likewise start by defining areas of mag
netic charge associated with each edge current M
n
. Let us define the unknown charge
density q
m
S
in (5.17) as
q
m
S
=
N
N
summationdisplay
i=1
?
i
P
i
(5.47)
71
where N
N
represents the total number of nodes (vertices) in the model, ?
i
is a scalar to be
determined, and
P
i
(r)=
?
?
?
?
?
1, r ? S
i
,
0, otherwise.
(5.48)
In (5.48), S
i
is the i
th
charge patch, formed by connecting the centers of the edges and
the centroids of the triangles associated with the i
th
node, as shown by the shaded area in
Fig. 5.6. Again making use of the divergence theorem, we can write ? for the i
th
source
patch as
? =
j
??
integraldisplayintegraldisplay
S
??M
S
GdS
prime
(5.49)
=
j
??
contintegraldisplay
C
M
S
? ?n
C
dC
integraldisplayintegraldisplay
S
GdS
prime
(5.50)
=
j
??A
i
braceleftbigg
N
E
summationdisplay
j=1
I
ij
bracketleftBig
?
lscript
ij
? (?n
+
ij
? r
c+
ij
+ ?n
?
ij
? r
c?
ij
)
bracketrightBig
bracerightbigg
braceleftbiggintegraldisplayintegraldisplay
S
GdS
prime
bracerightbigg
(5.51)
where
?
lscript
ij
and ?n
?
ij
, respectively, represent the unit vector along the j
th
edge connected to
node i and the outward unit vector normal to the plane of the T
?
ij
triangle associated with
the j
th
edge connected to node i. The vector r
c?
ij
extends from the centroid of the T
?
ij
triangle to the center of the j
th
edge, while r
c+
ij
extends from the edge center to the centroid
of the T
+
ij
triangle. N
E
represents the total number of edges connected to node i and A
i
is
the area of S
i
. From (5.51), we see that ?
i
in (5.47) is
?
i
=
j
?A
i
N
E
summationdisplay
j=1
I
ij
bracketleftBig
?
lscript
ij
? (?n
+
ij
? r
c+
ij
+ ?n
?
ij
? r
c?
ij
)
bracketrightBig
. (5.52)
In order to calculate ?
mn
, we will use the positive and negative magnetic charge
patches, designated S
n1
and S
n2
respectively, associated with the n
th
edge and shown in
72
Fig. 5.7. We will find the normal components of g
n
flowing across the mutual boundary,
designated r
c+
n
and r
c?
n
in Fig. 5.7. The value of ?
1
mn
, the scalar potential at the n
th
edge
as observed from node 1 on the m
th
edge, is
?
1
mn
=
j
??A
S
n1
g
n
?
parenleftbig
?n
+
n
? r
c+
n
+ ?n
?
n
? r
c?
n
parenrightbig
parenleftbiggintegraldisplayintegraldisplay
S
n1
G
1
dS
prime
parenrightbigg
?
j
??A
S
n2
g
n
?
parenleftbig
?n
+
n
? r
c+
n
+ ?n
?
n
? r
c?
n
parenrightbig
parenleftbiggintegraldisplayintegraldisplay
S
n2
G
1
dS
prime
parenrightbigg
=
j
??
g
n
?
parenleftbig
?n
+
n
? r
c+
n
+ ?n
?
n
? r
c?
n
parenrightbig
parenleftbigg
1
A
S
n1
integraldisplayintegraldisplay
S
n1
G
1
dS
prime
?
1
A
S
n2
integraldisplayintegraldisplay
S
n2
G
1
dS
prime
parenrightbigg
. (5.53)
In (5.53), the superscript on G refers to the end of the m
th
edge where the observation is
made. A similar calculation is done to find ?
2
mn
. The equations are the same for regions 1
and 2.
S
n1
S
n2
g
n
T
n
+
T
n

r
n
c+
r
n
c
Figure 5.7: Magnetic source patches S
n1
and S
n2
for ?
mn
calculation.
73
5.5.4 Testing the Incident Fields
We test the V matrix as follows:
V
m
= t
+
m
? E
i+
+ t
?
m
? E
i?
, EFIE solution (5.54)
where E
i
is evaluated at the vector potential test points, near the midpoints of the test
vectors t.
V
m
= lscript
m
? H
i
, HFIE solution (5.55)
where H
i
is evaluated at the midpoint of the test vector lscript.
5.6 Numerical Examples
The scattering solution using an orthogonal pair of pulse basis vectors is demonstrated
by calculating the bistatic RCS for four composite cases: a dielectric sphere close to a
PEC sphere, a dielectric cone capped with a PEC disk, a missile composed of a dielectric
nose cone and a PEC cylinder, and a cube capped with PEC plates at opposite ends. The
geometries are shown in Figs. 5.8 and 5.13; the EFIE solution method was employed for all
four cases. In addition, the twospheres problem was also solved by using the HFIE.
In the first problem, a dielectric sphere of radius 0.2 ? and epsilon1
R
= 4 and a PEC sphere
of radius 0.3 ? are situated on the zaxis. There is a gap of 0.1 ? between them, and a
plane wave traveling in the z direction impinges on the dielectric sphere first. The dielectric
sphere mesh has 324 edges, or N
D
= 324, while the conducting sphere mesh has 750 edges,
or N
C
= 750. The bistatic RCS results are shown for the pulse basis MoM EFIE and HFIE
solutions and compared to a body of revolution (BOR) MoM solution in Fig. 5.9. The
vertical axis represents RCS normalized by the region 1 wavelength.
In the second problem, a dielectric cone has height = 0.6 ?, radius = 0.3 ?,andepsilon1
R
=2.
The circular end of the cone is covered by a PEC disk. Because the EFIE solution was
chosen and the disk was PEC, it was allowable to model the disk as an open body in
74
contact with the closed dielectric cone portion of the figure, thus reducing the size of the
required PEC mesh. The results are shown for N
D
= 741 and N
C
= 205. For an HFIE
solution, the disk would be modeled as a closed body having a larger mesh. The plane wave
was assumed to be traveling in the z direction, and the bistatic RCS is shown in Fig. 5.10
for the pulse basis MoM solution, again compared to a BOR MoM solution.
E
x
i
,H
y
i
?
R
=4
PEC
r = 0.2 ?
r = 0.3 ?
0.1 ?
?
R
= 7.5
PEC
0.9 ?
5.58 ?
r = 0.18 ?
PEC
?
R
=2
r = 0.3 ?
0.6 ?
a) = Two spheres b) Disk/cone c) Missile
Figure 5.8: Geometries for which bistatic RCS was calculated, including a) two spheres, b)
a disk/cone, and c) a missile.
In the third problem, an airtoair missile shape was selected having a curved dielectric
nose cone with length = 0.9 ? and epsilon1
R
=7.5. The PEC cylinder has length = 5.58 ? and
radius = 0.18 ?. N
D
= 483, while N
C
= 3681. A plane wave was assumed to be traveling
in the z direction, toward the nose. The bistatic RCS is shown for the pulse basis MoM
EFIE solution in Fig. 5.11. Additional results are shown in Fig. 5.12 for the case where
the incident wave is traveling in the +z direction, toward the tail.
In the fourth problem, a dielectric cube having length = 0.1? and epsilon1
R
= 4 is sandwiched
between two PEC plates and the structure is irradiated from below as shown in Fig. 5.13.
75
0 30 60 90 120 150 180
0
1
2
3
4
5
Theta [degrees]
RC
S
/
????
2
Pulse Basis EFIE
Pulse Basis HFIE
BOR
? = 0
?
? = 90
?
Figure 5.9: Bistatic RCS for two nontouching spheres, one dielectric, epsilon1
R
= 4, and one PEC.
0 30 60 90 120 150 180
0
1
2
3
Theta [degrees]
RC
S
/
????
2
Pulse Basis EFIE
BOR
? = 90
?
? = 0
?
Figure 5.10: Bistatic RCS for a composite disk/cone, cone epsilon1
R
= 2, PEC disk.
76
0 30 60 90 120 150 180
0
1
2
3
4
5
6
7
Theta [degrees]
RC
S
/
????
2
Pulse Basis EFIE ? = 0?
Pulse Basis EFIE ? = 90?
Figure 5.11: Bistatic RCS for a composite missile, nose cone epsilon1
R
= 7.5, PEC cylinder,
incident wave approaching the nose of the missile.
0 30 60 90 120 150 180
0
1
2
3
4
5
6
7
Theta [degrees]
RC
S
/
????
2
Pulse Basis EFIE ? = 0?
Pulse Basis EFIE ? = 90?
Figure 5.12: Bistatic RCS for a composite missile, nose cone epsilon1
R
= 7.5, PEC cylinder,
incident wave approaching the tail of the missile.
77
Two pulse basis EFIE results are shown for the structure, one obtained by using a coarse
mesh and one using a finer mesh. For the coarse mesh, N
D
= 144 and for each plate,
N
C
= 20. For the finer mesh, N
D
= 909 and for each plate, N
C
= 136. The normalized
bistatic RCS in dB is shown in Figs. 5.14 and 5.15 and compared with combination volume
integral equation (VIE) and surface integral equation (SIE) results from Sarkar et al. [41].
Their formulation used 192 unknowns for the dielectric volume currents and 32 unknowns
for the PEC plate currents.
0.1 ?
?a
i
E
x
PEC
PEC
x
y
z
?
? ?
R
= 4
Figure 5.13: Dielectric cube of length 0.1? capped with PEC plates, epsilon1
R
=4.
The graphical results show very good agreement between RCS plots calculated with
orthogonal pulse basis vectors and their corresponding BOR plots. In the example of the
dielectric cube with PEC plates at top and bottom, the pulse basis and VIE/SIE methods
similarly indicate a deep null at ? =90
?
in the ? =0
?
RCS curve (Fig. 5.14). Compared
to the finely meshed pulse basis RCS curve (Fig. 5.15), the coarsely meshed pulse basis
RCS curve at ? =90
?
more closely resembles the VIE/SIE results of Sarkar et al., who also
used a coarse mesh for the PEC plates. Finer meshes show convergence to an almost flat
? =90
?
RCS curve.
78
0 30 60 90 120 150 180
60
50
40
30
20
10
0
Theta [degrees]
N
o
rm
al
i
zed
R
C
S
[
d
B
]
Pulse EFIE, N = 136
Pulse EFIE, N = 20
VIE/ S IE, N = 3 2
C
C
C
Figure 5.14: Bistatic RCS at ? =0
?
for a dielectric cube of length 0.1? capped with PEC
plates, epsilon1
R
=4.
0 30 60 90 120 150 180
2
1.5
1
0.5
0
0.5
Theta [degrees]
No
r
m
a
l
i
z
e
d
RCS
[
d
B
]
Pulse EFIE, N = 136
Pulse EFIE, N = 20
VIE/ S IE, N = 3 2
C
C
C
Figure 5.15: Bistatic RCS at ? =90
?
for a dielectric cube of length 0.1? capped with PEC
plates, epsilon1
R
=4.
79
5.7 Summary
We have demonstrated the solution of PEC/dielectric composite scattering problems by
using a pair of orthogonally placed pulse basis vectors. These basis functions, which model
equivalent electric and magnetic surface currents, allow for the correct implementation of
the EFIE and HFIE for dielectric bodies. In addition, the electric current pulse basis vector
allows the implementation of the EFIE and HFIE for PEC bodies. For the example com
posite EFIE solutions, the EFIE has been implemented with f and ?n?g basis vectors and
t testing vectors. The corresponding composite HFIE solutions for closed bodies required
?n?f and g basis vectors and lscript testing vectors. By arithmetically combining the EFIE and
the HFIE expanded with pulse basis vectors, the CFIE may be used to guarantee unique
solutions for composite scattering problems.
80
Chapter 6
Conclusion
We have demonstrated a new pair of basis functions for the solution of dielectric scat
tering problems by the method of moments (MoM) surface integral method. These basis
functions may be used for perfect electric conductor (PEC), dielectric, or composite struc
tures. They are particularly advantageous for PEC/dielectric composites, o?ering a simpler
solution method than has been previously published. The basis functions are designed for
use with triangular meshing, which is a convenient way to mesh threedimensional objects
of arbitrary shape. While there are MoM techniques that o?er more accurate solutions
for specialized geometries, we expect that the numerical procure developed here will allow
quite accurate solutions for most shapes of objects. It is therefore a solution of very wide
applicability.
In the surface integral problem solution, an electric surface current J and a magnetic
surface current M are determined for each mesh edge. For the n
th
mesh edge, J
n
is
expanded by the associated f
+
n
and f
?
n
basis vectors over a portion of the adjacent triangles,
and M
n
is expanded by the associated g
n
basis vector over the same area. The vectors
f
+
n
and f
?
n
are perpendicular to the edge, while g
n
is parallel to the edge; each basis
function is a unit vector. The orthogonality of the basis function pair within each triangle
and their relationship to the testing vectors is the key to the correct numerical solution
of the dielectric electric field integral equation (EFIE) and magnetic field integral equation
(HFIE). We showed that the pulse basis vectors will correctly solve dielectric and composite
problems in a straightforward manner.
An important part of the development of the numerical procedure was the represen
tation of the electric and magnetic vector and scalar potentials in the expanded matrix
equations. Each vector potential component was expressed as a surface integral of the basis
81
functions over a current source patch; the components were then summed. Each scalar
potential component was expressed as the sum of contour integrals of the normal compo
nents of the basis functions out of two charge source patches; the components were then
di?erenced.
Using canonical figures, we showed the numerical method and scattering results for PEC
bodies using f or g basis functions for an EFIE solution. For dielectric bodies, we showed
f and g basis functions used in EFIE and HFIE solutions. Finally, for composite bodies,
we showed f and g basis functions used in EFIE and HFIE solutions. Both methods were
detailed so that the combined field integral equation (CFIE) solution could be implemented
if desired for closed bodies. Application of this method for composites should be more
e?cient and accurate than previous surface integral methods.
A future problem to solve with paired pulse basis functions would be a composite
problem where one body is partly embedded in the other. Then, layered or coated bodies
would be a further application of interest.
82
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85
Appendices
86
Appendix A
Derivation of Dielectric Field Equations
We start with Maxwell?s equations in phasor form. Here, a steadily oscillating source
having frequency ? is assumed and each time varying quantity is written as its complex
peak value, which is understood to be multiplied by e
j?t
.
??E = ?M
i
? j??H (A1)
??H = J
i
+ J
c
+ J
d
= J
i
+ j??E (A2)
??E =
q
e
?
(A3)
??H =
q
m
?
(A4)
where E and H, J and M are the electric and magnetic fields and electric and magnetic
currents, respectively, i, c,andd denote impressed, conduction,anddisplacement, respec
tively, q
e
is the timevarying electric charge density, q
m
is the timevarying magnetic charge
density, ? is the complex permittivity, and ? is the complex permeability. In order to de
scribe currents within a general material that may be a perfect dielectric, a lossy dielectric,
or a perfect conductor, J
c
and J
d
have been combined as follows:
J
c
+ J
d
= j??E (A5)
= j?(?
prime
+
?
j?
)E (A6)
= j?(?
prime
? j?
primeprime
)E (A7)
where ?
prime
and ?
primeprime
are the real and imaginary parts of ?, respectively, ?
primeprime
triangle
= ?/?,and? is
the conductivity. For the problems concerned in this work, the source is distant from the
87
scatterer. Therefore, J
i
= M
i
= 0, and (A1) and (A2) may be simplified to
??E = ?j??H (A8)
??H = j??E . (A9)
We define the equivalent surface currents J
S
and M
S
on surface S as
J
S
= ?n ? H
tan
(A10)
M
S
= E
tan
? ?n (A11)
where ?n is a unit vector normal to the surface, pointing into the region on the side of S
where the field is tangent. We further define magnetic vector potential A and electric vector
potential F in relation to the scattered fields such that
H
s
(J
S
)=
1
?
??A (A12)
E
s
(M
S
)=?
1
?
??F (A13)
where s denotes scattered. Applying (A8) to scattered fields, substituting for H
s
,and
rearranging terms, we obtain:
??E
s
(J
S
)=?j???A (A14)
??E
s
(J
S
)+??j?A = 0 (A15)
??[E
s
(J
S
)+j?A]=0. (A16)
For the problems treated in this work, we use the vector identity that says that the curl
of a gradient equals 0 and equate E
s
(J
S
)+j?A toagradient???. We now invoke the
Lorenz gauge condition to define the scalar electric potential ? such that
? =
?1
j???
??A. (A17)
88
Given that
E
s
(J
S
)=?j?A ??? , (A18)
we combine (A18) with (A13) to obtain the total scattered E:
E
s
= E
s
(J
S
)+E
s
(M
S
) (A19)
= ?j?A ??? ?
1
?
??F . (A20)
Similarly, we apply (A9) to scattered fields and substitute for E
s
to obtain
??H
s
(M
S
)=?j???F (A21)
??H
s
(M
S
)+j???F = 0 (A22)
??[H
s
(M
S
)+j?F ]=0. (A23)
Defining the scalar magnetic potential ? such that
? =
?1
j???
??F , (A24)
we equate H
s
(M
S
)+j?F to the gradient ???.Now,
H
s
(M
S
)=?j?F ??? . (A25)
Combining (A25) with (A12), we obtain the total scattered H:
H
s
= H
s
(M
S
)+H
s
(J
S
) (A26)
= ?j?F ??? +
1
?
??A. (A27)
89
Solving for the vector potentials yields the following:
A = ?
integraldisplayintegraldisplay
S
J
S
e
?jkR
4?R
dS
prime
(A28)
F = ?
integraldisplayintegraldisplay
S
M
S
e
?jkR
4?R
dS
prime
(A29)
where S
prime
is the source surface, e
?jkR
/(4?R) is the Green?s function, k is the wave number
2?/?
0
,andR is the distance from the potential evaluation point to a source point on S.
Equations (A20) and (A27) will be implemented at the boundary surface of the scat
terer. If the scatterer is a perfect electric conductor, then M= 0 and the field equations
simplify to
E
s
= E
s
(J
S
)=?j?A ??? (A30)
H
s
= H
s
(J
S
)=
1
?
??A. (A31)
In surface integral problems, the boundary conditions equate the tangential components
of the scattered and incident fields:
bracketleftbigg
j?A + ?? +
1
?
??F
bracketrightbigg
tan
=
bracketleftbig
E
i
bracketrightbig
tan
(A32)
bracketleftbigg
j?F + ?? ?
1
?
??A
bracketrightbigg
tan
=
bracketleftbig
H
i
bracketrightbig
tan
. (A33)
Equations of the form (A32) and (A33) are written for each side of the boundary surface,
except in the case of a PEC, where only one side need be considered. More description of
these derivations is given in [2] and [9].
90
Appendix B
Pulse Basis Functions in EFIE and HFIE Solutions
Chapters 2 and 3 demonstrated the use of f and g basis vectors to expand the electric
surface currents J
S
for EFIE PEC solutions. As a point of interest, however, either set of
basis functions could be also used to expand J
S
for the HFIE solution. Figure B.1 shows
a PEC sphere of diameter 0.18 ? illuminated by a plane wave traveling in the z direction.
The scattering problem was worked four times using either f or g basis functions in the
EFIE or HFIE solution. The bistatic RCS results are shown in Fig. B.2.
?a
i
E
x
0.5
0
0.5
x
y
z
?
?
Figure B.1: Bistatic RCS for a PEC sphere, diameter = 0.18 ?.
91
30 60 90 120 150 180
0
0.005
0.010
0.015
0.020
0.025
Theta [degrees]
RCS
/
????
2
exact
f basis EFIE
g basis EFIE
f basis HFIE
g basis HFIE
Figure B.2: Bistatic RCS for a PEC sphere, diameter = 0.18 ?.
92