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GRADO DE INGENIERÍA DE TECNOLOGÍAS Y SERVICIOS DE
TELECOMUNICACIÓN
TRABAJO FIN DE GRADO
DEVELOPMENT OF A SYSTEM FOR THE COMPUTATION OF ELECTROMAGNETIC
WAVE SCATTERING FROM NON-PENETRABLE OBJECTS BY SOLVING THE
ELECTRIC FIELD INTEGRAL EQUATION
ALBERTO MONJE REAL
2016
GRADO EN TECNOLOGÍAS Y SERVICIOS DE
TELECOMUNICACIÓN
TRABAJO FIN DE GRADO
Título: Development of a System for the Computation of Electromagnetic Wave
Scattering from Non-Penetrable Objects by Solving the Electric Field Integral Equation
Autor: D. Alberto Monje Real
Tutor: D. Valentín de la Rubia Hernández
Ponente: D. Alberto Monje Real
Departamento: Departamento de Matemática Aplicada a las TIC
MIEMBROS DEL TRIBUNAL
Presidente: D. Ricardo Riaza Rodríguez
Vocal: D. Javier Jesús Lapazaran Izargain
Secretario: D. José Manuel Fernández González
Suplente: D. Francisco José Navarro Valero
Los miembros del tribunal arriba nombrados acuerdan otorgar la calificación de:
………
Madrid, a de de 20…
UNIVERSIDAD POLITÉCNICA DE MADRID
ESCUELA TÉCNICA SUPERIOR DE INGENIEROS DE TELECOMUNICACIÓN
GRADO DE INGENIERÍA DE TECNOLOGÍAS Y SERVICIOS DE TELECOMUNICACIÓN
TRABAJO FIN DE GRADO
DEVELOPMENT OF A SYSTEM FOR THE COMPUTATION OF ELECTROMAGNETIC
WAVE SCATTERING FROM NON-PENETRABLE OBJECTS BY SOLVING THE
ELECTRIC FIELD INTEGRAL EQUATION
ALBERTO MONJE REAL
2016
SUMMARY
In this Trabajo de Fin de Grado (TFG, from now on), the main objective is to determine the
radar cross section of perfect electrical conducting objects by numerically solving the Electric
Field Integral Equation, using the Method of Moments. A numerical code implementation of
the MoM is carried out in this TFG, taking into account all implementation details for an
accurate solution to EFIE.
The Method of Moments is a numerical analysis technique that is used to solve the Maxwell
Equations, like other numerical methods such as the Finite Element Method but, unlike this
last one, which determines the electric field in volumetric elements solving the Electric Field
Differential Equation, the Method of Moments solves the Electric Field Integral Equation
obtaining the surface current in triangular elements, and when the current distribution in the
object is known, then the total electric field in any point of the space can be obtained. In this
TFG it will be explained in detail how to get to the Electric Field Integral Equation, starting
from Maxwell’s Electromagnetic Equations in the frequency domain, as well as how to
numerically solve the problem to sufficiently well approximated results. Since a direct
approach to the topic might be of high complexity, a brief introduction of the Method of
Moments is given, and an electrostatic problem is also solved, as a demonstration.
The project focuses on the study of the frequency behavior of the radar cross section, both
monostatic (where the reception part of the radar is placed in the same location as the
transmission one) and bistatic (where the transmission and reception parts of the system are
in different locations), of metallic objects of moderate electric size. The radar cross section of
an object measures the strength of the scattering that the object produces when an
electromagnetic plane wave impacts on it.
Finally, the influence of the geometry in the electrical behavior of the different perfect
electrical conducting objects to the incidence of an electromagnetic plane wave is detailed.
This is an important factor since the direction of the incident plane wave determines whether
the object will be more or less visible to the radar.
KEYWORDS
Method of Moments (MoM), Radar Cross Section (RCS), Numerical Analysis, Electric Field
Integral Equation (EFIE), Scattering, Back-scattering, Finite Element Analysis, Boundary
Element Method, Green’s Function, Radar.
RESUMEN
En este Trabajo de Fin de Grado (TFG en adelante), el objetivo principal es determinar la
sección radar de objetos conductores eléctricos perfectos resolviendo numéricamente la
Ecuación Integral de Campo Eléctrico (EFIE por sus siglas en inglés), empleando el Método
de los Momentos, y programar un código que sirva para obtener esa sección radar tanto en un
caso electrostático, como en un caso electrodinámico.
El Método de los Momentos es una técnica de análisis numérico que se emplea para resolver
las ecuaciones de Maxwell, de la misma forma que se emplean otros métodos numéricos
como por ejemplo el Método de Elementos Finitos. Este último método determina el campo
eléctrico en elementos volumétricos resolviendo la Ecuación Diferencial de Campo Eléctrico,
aunque al contrario que éste, el Método de los Momentos resuelve la Ecuación Integral de
Campo Eléctrico obteniendo la corriente superficial en elementos triangulares. Cuando se
conoce la distribución de corriente en el objeto, el campo eléctrico total en cualquier punto
del espacio se puede obtener. En este TFG se explicará en detalle cómo llegar a la Ecuación
Integral de Campo Eléctrico, empezando desde las Ecuaciones de Maxwell en el dominio de
la frecuencia, y cómo resolver numéricamente el problema para obtener resultados
suficientemente aproximados. Dado que abordar el tema directamente puede ser de gran
complejidad, se proporciona una breve introducción del Método de los Momentos, y un
ejemplo electrostático se resolverá tanto teóricamente como programándolo, como
demostración. El proyecto se centra en el estudio del comportamiento en frecuencia de la
sección radar (parámetro de un objeto que mide la intensidad de la dispersión que se produce
cuando una onda plana incide sobre él) tanto monoestática (donde el sistema receptor del
radar está ubicado en el mismo lugar que la transmisión), como biestática (donde los sistemas
de transmisión y recepción del sistema están en diferentes ubicaciones), de objetos metálicos
de tamaño eléctrico moderado.
Finalmente, se detalla la influencia de la geometría en el comportamiento eléctrico de los
diferentes objetos conductores perfectos cuando incide una onda plana electromagnética. Este
es un factor importante dado que la dirección de la onda plana incidente determina si el
objeto será más o menos visible al radar.
PALABRAS CLAVE
Método de los Momentos, Sección Radar, Análisis Numérico, Ecuación Integral de Campo
Eléctrico, Dispersión, Método de Elementos Finitos, Método de Elementos Frontera,
Función de Green, Radar.
CONTENT INDEX
1. INTRODUCTION AND OBJECTIVES ..................................................... 1
1.1. Introduction ............................................................................................................................. 1
1.2. Objetives ................................................................................................................................. 2
2. DEVELOPMENT ................................................................................... 3
2.1. Introduction to the Method of Moments ................................................................................. 3
2.1.1. The Method of Moments ................................................................................................. 3
2.1.2. Application of the MoM to an electrostatic problem ...................................................... 5
2.2. Application of the Method of Moments to EFIE .................................................................. 10
2.2.1. The Electric Field Integral Equation ............................................................................. 10
2.2.2. Using the MoM to solve the 3D EFIE .......................................................................... 18
2.2.3. Appendix ....................................................................................................................... 28
3. RESULTS ........................................................................................... 32
3.1. Testing simulations ............................................................................................................... 32
3.1.1. Sphere ........................................................................................................................... 32
3.1.2. Cube .............................................................................................................................. 34
3.1.3. Nasa almond .................................................................................................................. 35
3.1.4. Ogive ............................................................................................................................. 36
3.1.5. Double ogive ................................................................................................................. 37
3.2. Other simulations .................................................................................................................. 38
3.2.1. Torus ............................................................................................................................. 38
3.2.2. Destroyer ....................................................................................................................... 40
4. CONCLUSIONS AND FUTURE LINES .................................................. 42
4.1. Conclusions ........................................................................................................................... 42
4.2. Future Lines .......................................................................................................................... 43
5. BIBLIOGRAPHY ................................................................................ 44
1
1. INTRODUCTION AND OBJECTIVES
1.1. INTRODUCTION
The determination of the way an object appears in the radar has been a subject of great interest since
the radar technologies were initially developed and even though it has usually been related to the
military industry, it can be of use for civil engineering. With respect to the military industry, we can
see it from two points of view: supposing we are air surveillance radar designers, it is very important
to know what kind of object our radar is detecting, since it can be dangerous not to identify a possible
threat, mistaking it with an animal or with a civil aircraft and, in the same way it is also important not
to mistake a civil aircraft with a fighter or a bomber, for obvious reasons. Taking a look from the
aircrafts designer’s perspective, it is our goal to make our fighters and bombers as stealthy as possible,
so that they cannot be easily detected by the enemies’ radar systems, while in civil aircraft
engineering, we will try to design airplanes that are very detectable to radar systems, making our
planes easy to be tracked from land and airports, and increasing the safety of people and goods
transportation.
The parameter that was described before is the Radar Cross Section (RCS) of a given object, and it
gives a measurement of how detectable is that object by a radar. In the situations described before, we
would like to design military aircrafts with the lowest RCS possible, so that they are harder to detect,
and civil airplanes with high RCS, making them easier to be detected. Being the radar designer, we
would like to know what RCS will a given aircraft have, so that we can identify that airplane
immediately and know if it is a threat or not. The RCS depends on many different factors such as:
The size and shape (i.e. the geometry) of the object we are trying to detect.
The material of that object.
The frequency which the system is working on.
The direction the radar wave impacts the object and the direction the wave is scattered (very
important in bistatic radar systems).
The polarization we are employing.
However, the RCS does not depend on the transmitted power or the distance to the target.
Now that we are aware of the importance of the RCS of an object, there are several ways to calculate
it. The first and easiest solution is to simply measure it: we just have to illuminate the object with our
radar, detect the backscattering (monostatic radar) or other direction scattering (bistatic radar) and we
will know what its RCS is. However, there is a great problem with this: we need to have the object
physically, and thus, it must have been designed and built beforehand without knowing what its RCS
will be. A more difficult but with great advantages solution will be simulating the RCS. This way, we
would know the RCS of the object before building it and if it doesn’t match our requirements, we can
re-design the object to match the specifications. It is in this point where this project focuses.
Using numerical solutions to the Electric Field Integral Equation (EFIE), we can calculate the
backscattering an arbitrarily shaped object will have, and then what its radar cross section will be.
The method we are using in this project to solve the Maxwell equations is the Method of Moments
(MoM), also known as the Boundary Element Method (BEM), with which we can calculate the
surface current distribution of an object of arbitrary shape given the incident electric field. Likewise,
we can obtain the scattered field if we know the surface current distribution. In this project, a detailed
explanation of the whole Method of Moments is given and a numerical code implementation is
carried out, providing final results and tests of the method over different shapes.
2
1.2. OBJETIVES
As mentioned before, the main objective of this project is the determination of the radar cross section
of perfect electrical conducting objects of arbitrary shape, but in order to achieve this main goal, we
must accomplish some intermediate objectives:
Explain the Method of Moments.
o Explain the Method of Moments, in a general way.
o Apply the Method of Moments to an electrostatic example (charge and capacity of a
capacitor).
Apply the Method of Moments to the EFIE.
o Analytically determine the Electric Field Integral Equation (EFIE).
o Obtain the three dimensional Green’s Function.
o Use the Method of Moments to solve the EFIE.
Obtain the monostatic and bistatic scattering.
Test the obtained results and compare them to real measurements or analytic solutions
3
2. DEVELOPMENT
2.1. INTRODUCTION TO THE METHOD OF MOMENTS
2.1.1. THE METHOD OF MOMENTS
As explained in [1], the Method of Moments (MoM) is a mathematical algorithm that allows us to
solve equations of the form:
ℒ[𝑓(𝑥)] = 𝑔(𝑥) (1)
Where g(x) is a known function, ℒ is a linear operator, and our objective is to calculate f(x).
The fact that ℒ is a linear operator means that it must satisfy that:
ℒ[ 𝛼𝑠(𝑥) + 𝛽 𝑟(𝑥)] = 𝛼ℒ[𝑠(𝑥)] + 𝛽ℒ[𝑟(𝑥)] (2)
Some examples of ℒ can be a differential operator, an integral one, a multiplication by a constant...:
ℒ[𝑓(𝑥)] = −2 𝑑𝑓(𝑥)
𝑑𝑥+ 7𝑓(𝑥) (3)
ℒ[𝑓(𝑥)] = ∭ 4 𝑓(𝑥)
𝑉
𝑑𝑉
(4)
Now, since the linear operator ℒ might be very hard to solve, finding a solution for equation (1)
analytically will only be possible in the easiest cases. What we can do for those difficult situations,
taking advantage of computation, is to approximate f(x) by some basis or expansion functions, fn(x) of
our choice (carefully chosen, but known), weighted by some coefficients 𝛼𝑛, 𝑛 = 1, 2, … , 𝑁, that will
be the unknowns that we want to calculate in order to solve the problem:
𝑓(𝑥) ≈ ∑ 𝛼𝑛𝑓𝑛(𝑥)
𝑁
𝑛=1
(5)
And, substituting (5) in (1),
ℒ[𝑓(𝑥)] ≈ ℒ [∑ 𝛼𝑛𝑓𝑛(𝑥)
𝑁
𝑛=1
] = ∑ 𝛼𝑛 ℒ[𝑓𝑛(𝑥)]
𝑁
𝑛=1
≈ 𝑔(𝑥) (6)
We define the residual r(x) as:
𝑟(𝑥) = ℒ[𝑓(𝑥)] − ℒ [∑ 𝛼𝑛𝑓𝑛(𝑥)
𝑁
𝑛=1
] = 𝑔(𝑥) − ℒ [∑ 𝛼𝑛𝑓𝑛(𝑥)
𝑁
𝑛=1
] (7)
Our goal is to make the residual r(x) as little as possible, because this would mean that our
approximation of f(x) will be closer to the exact value of f(x).
4
The problem that we have now is that we have N unknowns,𝛼1, 𝛼2, … , 𝛼𝑁, and just one equation. In
order to make the system solvable, we will use some testing or weighting functions that we are free to
choose, 𝑤𝑚, 𝑚 = 1, 2, … , 𝑁, as many testing functions as basis functions there are, and we are going
to dot product each side of (6) by each testing function:
< 𝑤𝑚 , ℒ [∑ 𝛼𝑛𝑓𝑛(𝑥)
𝑁
𝑛=1
] > = ∑ 𝛼𝑛 < 𝑤𝑚 , ℒ[𝑓𝑛(𝑥)] >
𝑁
𝑛=1
≈ < 𝑤𝑚 , 𝑔(𝑥) > (8)
This yields a system that we can see in a matrix form:
(
𝑍11 𝑍12 ⋯ 𝑍1𝑁
𝑍21 𝑍22 ⋯ 𝑍2𝑁
⋮ ⋮ ⋱ ⋮𝑍𝑁1 𝑍𝑁2 ⋯ 𝑍𝑁𝑁
) · (
𝛼1
𝛼2
⋮𝛼𝑁
) = (
𝑏1
𝑏2
⋮𝑏𝑁
) , (9)
where
𝑍𝑚𝑛 = < 𝑤𝑚 , ℒ[𝑓𝑛(𝑥)] >
𝑏𝑚 = < 𝑤𝑚 , 𝑔(𝑥) > (10)
and where we define the dot product as:
< 𝑓(𝑥), 𝑔(𝑥) > = ∫ 𝑓(𝑥)𝑔∗(𝑥)𝑏
𝑎
𝑑𝑥, 𝑎 ≤ 𝑥 ≤ 𝑏 (11)
Once we have solved the system in (9), written abbreviated as follows:
𝑍 · 𝛼 = 𝑏 → 𝛼 = 𝑍−1 · 𝑏 , (12)
we have the values of 𝛼𝑛 for every n, and thus, we have solved the problem.
It is important to notice a few aspects of this method:
The first one is to realize that not every arbitrary set of basis functions 𝑓𝑛 will be suitable for
our problem. For example, if our aim is to solve a problem of the form:
ℒ[𝑓(𝑥)] = −2 𝑑𝑓(𝑥)
𝑑𝑥= 𝑥3 + 2 ,
we cannot choose a set of basis functions that are not differentiable since the system will have
singularities. Apart from these situations, we should choose a set of functions that will be
similar to the function f(x) that we are trying to approximate, so that we can approximate
better our function by using less coefficients (less equations) and making our program more
efficient.
Not just the choice of the basis functions is crucial; the choice of the testing functions is
equally important, because it will determine the weights of our approximation, and will
eventually affect to the final result. The simplest weighting function, and generally a bad one,
is Dirac’s delta: 𝛿(𝑚 − 𝑚0), where we force our approximation to be equal to the function
g(x) in one point (m0). This solution is called Point Matching, and we will use it in the
5
electrostatic example. A common way of choosing the testing functions is choosing wm(x) =
fn(x). This is called the Galerkin’s Method, which we will use for solving the EFIE.
The second aspect to take into account is that if the basis functions are properly chosen, the
more we increase 𝑛, the better our approximation will be. A good way to see this, which is an
alternative way to understand the method, is that we approximate the function f(x) by an N-
dimensional vector space, with the fn(x) functions being a base of the space, and the
coefficients being the weights of the vectors. This way, we can approximate any vector (i.e.
function) of the space with these N vectors (functions) and the appropriate weights. What we
are doing when we force the residual r(x) to be equal to zero, is simply forcing the projection
of the error to be zero, which implies that the error is forced to be orthogonal to our vector
space. Therefore, each time that we increase the dimension of the space, we force the error to
be orthogonal to another dimension, and thus, it must be smaller than before. Then, the bigger
the dimension of our space is, the smaller our error will be and for an infinite dimension
space, our approximation will be perfect, and the residual will be equal to zero.
It is also important to realize the complexity that the MoM has. Since we have to fill matrix Z
which is of size NxN, we see that the complexity of the fill is O(N2), while the inversion of
the matrix using Gaussian elimination or LU decomposition is of complexity O(N3). It makes
this method a very complex one computationally speaking. This is why the system in (10) is
not solved using the methods written above, but by iterative solutions.
2.1.2. APPLICATION OF THE MOM TO AN ELECTROSTATIC PROBLEM
THEORY
Now that the theory of the Method of Moments has been exposed, we are going to use the MoM to
solve an easy electrostatic problem: Calculating the charge and the capacity of a parallel plate
capacitor connected to a difference of potential of V volts and with a space between plates of h. The
geometry of the problem is graphically described for better understanding:
Figure 2.1. Parallel Plate Capacitor Geometry with h=0.01cm.
6
First, we will get to expression (1). The potential generated in any point of the space by a charge
surface distribution is:
𝜙(𝑥, 𝑦, 𝑧) = ∬𝜌(𝑥′, 𝑦′)
4𝜋휀0𝑅
𝑠
𝑑𝑥′𝑑𝑦′
In our particular situation, where the capacitor is formed by two parallel plates:
𝜙(𝑥, 𝑦, 𝑧) = ∬𝜌(𝑥′, 𝑦′)
4𝜋휀0𝑅
𝑈𝑃+𝐵𝑃
𝑑𝑥′𝑑𝑦′, (13)
where UP and BP are the surface of the upper plate and the one of the bottom plate respectively, and
R is defined as the Euclidean distance from the point where the charge is located to the point of
observation, i.e.
𝑅 = √(𝑥 − 𝑥′)2 + (𝑦 − 𝑦′)2 + (𝑧 − 𝑧′)2 (14)
Equation (13) will be the left hand side of expression (1), and the right hand side of (1) will be the
potential in each plate, which is known:
𝑔 = {
𝑉
2, (𝑥, 𝑦, 𝑧) ∈ 𝑈𝑃
−𝑉
2, (𝑥, 𝑦, 𝑧) ∈ 𝐵𝑃
(15)
Continuing with the process, f(x) must be approximated as in (5), in this case by N squares of side
length 2b (surface Δ𝑠 = 4𝑏2), as it is described in Figure 2.1. Parallel Plate Capacitor Geometry and
supposing that the charge surface density is constant in each square:
𝜌(𝑥′, 𝑦′) = ∑ 𝛼𝑛
𝑁
𝑛=1
𝑓𝑛 (16)
where we define fn constant in the surface of the nth square, i.e.:
𝑓𝑛 = {1, 𝑜𝑛 Δ𝑠𝑛
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 (17)
Finally, this is the resulting system:
∬∑ 𝛼𝑛
𝑁𝑛=1 𝑓𝑛(𝑥′, 𝑦′)
4𝜋휀0𝑅𝑈𝑃+𝐵𝑃
𝑑𝑥′𝑑𝑦′ = {
𝑉
2, (𝑥, 𝑦, 𝑧) ∈ 𝑈𝑃
−𝑉
2, (𝑥, 𝑦, 𝑧) ∈ 𝐵𝑃
(18)
And it can be rewritten as:
∑ 𝛼𝑛
𝑁
𝑛=1
∬𝑓𝑛(𝑥′, 𝑦′)
4𝜋휀0𝑅𝑈𝑃+𝐵𝑃
𝑑𝑥′𝑑𝑦′ = {
𝑉
2, (𝑥, 𝑦, 𝑧) ∈ 𝑈𝑃
−𝑉
2, (𝑥, 𝑦, 𝑧) ∈ 𝐵𝑃
(19)
7
which is an equation in the form of (6), and has to be evaluated with N testing functions. Carrying the
formerly explained solution of point matching, the testing functions wm will be defined as Dirac’s
deltas in the center of each Δ𝑠𝑛. This way, in those points, the potential will be 𝑉
2 or −
𝑉
2 depending on
which plate (upper or bottom respectively) the weighting function is located.
When seeing (19) in the form of (9), it is important to realize that the coefficients of the Z matrix
(Znm), that can be also called the impedance matrix, represent the potential that a density charge of
1C/m2 in the nth element would cause in the mth element. Since the coefficients of the main diagonal
of the matrix (elements Zmn with m=n) are the contribution of a charge density over itself, it yields a
singularity, and thus, the elements on the main diagonal have to be evaluated analytically as explained
in [2], equation(2-31), resulting
𝑍𝑛𝑛 =0.8814(2𝑏)
𝜋휀0 (20)
Finally, when the system is solved and the 𝛼𝑛coefficients are determined, the charge in one plate of
the capacitor can be calculated as the sum of all the basis functions weighted by the coefficients:
𝑄 = ∬ 𝜌(𝑥′, 𝑦′)𝑑𝑥′𝑑𝑦′𝑈𝑃
≈ ∑ 𝛼𝑛
𝑁
𝑛=1
𝑓𝑛Δ𝑠𝑛 (21)
And the capacity of a capacitor is, by definition:
𝐶 =𝑄
𝑉 (22)
Now that all the steps have been described and explained, some results are provided in Figure 2.2 and
Figure 2.3
8
RESULTS AND SIMULATIONS
Figure 2.2. Charge density distribution in the capacitor, for 20x20 and 60x60 mesh sizes.
9
Figure 2.3 Curve of convergence of the capacity.
As it can be seen in Figure 2.3, when the size of the mesh (i.e. the number of mesh elements)
increases, the MoM approximation fits better the theoretic calculations, asymptotically tending to the
analytic parallel plate capacitor capacity as explained in [3]:
𝐶 =𝐴휀
ℎ (23)
This simple simulations show the importance of choosing a sufficiently large size of the mesh in order
to get accurate results. To further illustrate this point, in Figure 2.2, it can be seen how the size of the
mesh is specially crucial when the function that is being calculated (the charge density in this case)
varies rapidly: The charge distribution is almost constant in the center of the capacitor, but in the
edges of the capacitor it shows strong variations, so a thinner mesh is needed in order to get accurate
results. Nevertheless, as it can be seen in the simulated capacity, it tends to the analytic one but there
is always an error. This can be because of many factors; the main one probably is that the point
matching is not a very accurate technique despite its simplicity and, while Galerkin’s method would
have been a better choice, this is an illustrative example not only of the point matching but also of a
different basis and testing functions choice.
However, as discussed in the last point of 2.1.1, increasing the number of unknowns leads to an even
higher complexity proportional to the square of the number of elements in the mesh, requiring more
memory and computing time. In this example provided, no more than 120 elements per side could be
calculated (as shown in Figure 2.3) because this yielded to 14400 elements per plate, 28800 elements
in the capacitor, and a Z matrix of 829.44 · 106elements, and even more memory and computing time
is required to solve the system inverting the matrix. This shows the compromise between accuracy
and complexity that has to be satisfied.
10
2.2. APPLICATION OF THE METHOD OF MOMENTS TO EFIE
In this chapter, the Method of Moments will be applied in order to solve real electromagnetic
problems, particularly, as it is the objective of this project, the determination of the radar cross section
of Perfect Electrical Conducting (PEC) arbitrarily shaped objects. In order to do so, and following the
explanations given in [4], a brief introduction to electromagnetics will be presented, as well as the
process to get to the Electric Field Integral Equation (EFIE). Later on, following the steps in chapter
2.1.1, a computational solution of that equation will be shown.
2.2.1. THE ELECTRIC FIELD INTEGRAL EQUATION
INTRODUCTION TO ELECTROMAGNETICS
In this point a development to get to the final form of the Electric Field Integral Equation will be
presented, starting from Maxwell’s Equations, and discussing some important aspects such as
boundary conditions, Green’s Function or far field approximations.
MAXWELL’S EQUATIONS
The Equations that every electromagnetic field must verify, in the frequency domain are the ones
exposed by Maxwell:
∇ × 𝐄 = −𝐌 − 𝑗𝜔𝜇𝐇 (24)
∇ × 𝐇 = 𝐉 + 𝑗𝜔휀𝐄 (25)
∇ ∙ 𝐃 = 𝑞𝑒 (26)
∇ ∙ 𝐁 = 𝑞𝑚 (27)
Being 𝐃 = 휀𝐄 and 𝐁 = 𝜇𝐇. The magnetic current M, and the magnetic charge 𝑞𝑚 are two magnitudes
that don’t exist physically, but they are used for mathematical purposes. The phase because of time
propagation is of the form of 𝑒𝑗𝜔𝑡 and is assumed, and thus, not written. From now on, vectors will be
noted in bold.
BOUNDARY CONDITIONS
In the boundary between two generic surfaces, the electromagnetic fields have to verify these four
equations.
−�̂� × (𝐄2 − 𝐄1) = 𝐌s (28)
�̂� × (𝐇2 − 𝐇1) = 𝐉s (29)
�̂� ∙ (𝐃2 − 𝐃1) = 𝑞𝑒 (30)
�̂� ∙ (𝐁2 − 𝐁1) = 𝑞𝑚 (31)
Where �̂� is the normal vector to the surface boundary that points from region 2 to region 1.
In the case that is being studied in this project, one of the regions (region number 1) will be formed by
a Perfect Electrical Conductor (PEC) and the other one (region number 2), will be a dielectric. The
above boundary conditions can be rewritten to this particular situation as follows since the Electric
and Magnetic fields are both zero in the PEC, and no magnetic current or charge will be present:
11
−�̂� × 𝐄2 = 0 (32)
�̂� × 𝐇2 = 𝐉𝑠 (33)
�̂� ∙ 𝐃2 = 𝑞𝑒 (34)
�̂� ∙ 𝐁2 = 0 (35)
DERIVATION TO EFIE
The way of solving the electric field scattering in this project will be considering that a radiated field
is originated by a surface electric current distribution, while this surface current distribution is
originated by another electric field, the incident one. The objective is to get to a formulation that
allows the problem to be solved this way.
Taking the curl of equation (24),
∇ × ∇ × 𝐄 = −𝑗𝜔𝜇∇ × 𝐇 (36)
Substituting (25) in (36) yields:
∇ × ∇ × 𝐄 = 𝜔2𝜇휀𝐄 − 𝑗𝜔𝜇𝐉 (37)
And taking the first term in the right hand side of (37) to the left hand side,
∇ × ∇ × 𝐄 − 𝜔2𝜇휀𝐄 = −𝑗𝜔𝜇𝐉 (38)
Knowing the vector identity
∇ × ∇ × 𝐄 = ∇(∇ ∙ 𝐄) − ∇2𝐄 (39)
we can rewrite (38) as:
∇(∇ ∙ 𝐄) − ∇2𝐄 − 𝑘2𝐄 = −𝑗𝜔𝜇𝐉 (40)
where 𝑘 = 𝜔√𝜇휀 =2𝜋
𝜆 is the wavenumber.
Substituting (26) in the above yields
∇2𝐄 + 𝑘2𝐄 = 𝑗𝜔𝜇𝐉 + ∇𝑞𝑒
휀 (41)
The relationship between the electric surface current and the electric charge is the equation of
continuity:
∇ ∙ 𝐉 = −𝑗𝜔𝑞𝑒 (42)
that in (41) yields:
12
∇2𝐄 + 𝑘2𝐄 = 𝑗𝜔𝜇𝐉 −1
𝑗𝜔휀∇(∇ ∙ 𝐉) = 𝑗𝜔𝜇(𝐉 +
1
𝑘2∇(∇ ∙ 𝐉)) (43)
With this equation, and the linearity of Maxwell’s Equations, it is possible to calculate the electric
field integrating the contribution of each current distribution in the volume where they are located.
In order to calculate the electric field, the Helmholtz scalar equation is of the form:
∇2𝐺(𝐫, 𝐫′) + 𝑘2𝐺(𝐫, 𝐫′) = −𝛿(𝐫, 𝐫′) (44)
where 𝐺(𝐫, 𝐫′) is the Green’s function and it is assumed to be known (it will be discussed and
obtained later). Now, with the Green’s function, and through the superposition principle, the electric
field can be calculated integrating in the volume all the current contributions:
𝐄(𝐫) = −𝑗𝜔𝜇 ∭ 𝐺(𝐫, 𝐫′) [𝐉(𝐫′) +1
𝑘2∇′∇′ ∙ 𝐉(𝐫′)] 𝑑𝐫′
𝑉
(45)
THE GREEN’S FUNCTION
In order to complete equation (45), the Green’s function must be obtained via solving the three-
dimension Helmholtz scalar equation (44). Since it is a differential equation, the first thing to
calculate is the solution to the homogeneous differential equation, and afterwards, the solution to the
inhomogeneous case to obtain a unique solution.
Since the Green’s function is the solution of a point of source, it must be spherically symmetric, and
then, only the radial component will be considered:
∇2𝐺 =1
𝑟2
𝑑
𝑑𝑟(𝑟2
𝑑𝐺
𝑑𝑟) =
𝑑2𝐺
𝑑𝑟2+
2
𝑟
𝑑𝐺
𝑑𝑟=
1
𝑟
𝑑2(𝑟𝐺)
𝑑𝑟2 (46)
Substituting the last term in (44),
1
𝑟
𝑑2(𝑟𝐺)
𝑑𝑟2+ 𝑘2𝐺 = 0 =>
𝑑2(𝑟𝐺)
𝑑𝑟2+ 𝑘2(𝑟𝐺) = 0 (47)
To solve this homogeneous system, we try a solution of the form of:
𝐺 =𝐴𝑒𝑠𝑟
𝑟 (48)
Resulting in:
𝑑2(𝑟𝐴𝑒𝑠𝑟
𝑟)
𝑑𝑟2+ 𝑘2 (𝑟
𝐴𝑒𝑠𝑟
𝑟) = 0 => 𝑠2𝐴𝑒𝑠𝑟 + 𝑘2𝐴𝑒𝑠𝑟 = 0 => 𝑠 = ±𝑗𝑘 (49)
So the solution for the homogeneous part is:
𝐺 = 𝐴𝑒−𝑗𝑘𝑟
𝑟+ 𝐵
𝑒𝑗𝑘𝑟
𝑟 (50)
13
This solution includes outgoing and incoming waves, and for the solution of this problem, only
outgoing waves will be taken into consideration. Then:
𝐺 = 𝐴𝑒−𝑗𝑘𝑟
𝑟 (51)
where 𝑟 is the relative distance from the observation for the source: 𝑟 = |𝒓 − 𝒓′|
Now, in order to solve the inhomogeneous part of the equation to get a unique solution, and to
determine A, the integration of (44) over a sphere of radius 𝑎 around the source yields:
𝐴 ∭ [∇ ∙ ∇ (𝑒−𝑗𝑘𝑟
𝑟) + 𝑘2
𝑒−𝑗𝑘𝑟
𝑟] 𝑑𝑉
𝑉
= ∭ −𝛿(𝒓, 𝒓′) 𝑑𝑉 =
𝑉
− 1 (52)
Using the Gauss theorem to solve the first term in the integral:
∭ ∇ ∙ ∇ (𝑒−𝑗𝑘𝑟
𝑟) 𝑑𝑉
𝑉
= ∬ �̂� ∙ ∇ (𝑒−𝑗𝑘𝑟
𝑟) 𝑑𝑆
𝑆
(53)
Since our surface is a sphere,�̂� = �̂�, and substituting in the above equation:
∬ �̂�
𝑆
∙ ∇ (𝑒−𝑗𝑘𝑟
𝑟) 𝑑𝑆 = ∬
𝜕
𝜕𝑟
𝑆
(𝑒−𝑗𝑘𝑟
𝑟) 𝑑𝑆 (54)
and integrating, it yields:
4𝜋𝑎2 [𝜕
𝜕𝑟(
𝑒−𝑗𝑘𝑟
𝑟)]
𝑟=𝑎
(55)
To solve this, in the limit when 𝑎 → 0, (55) results
lim𝑎→0
4𝜋𝑎2 [𝜕
𝜕𝑟(
𝑒−𝑗𝑘𝑟
𝑟)]
𝑟=𝑎
= −4𝜋 (56)
Evaluating the second term of the integral in (52):
∭ [𝑘2𝑒−𝑗𝑘𝑟
𝑟] 𝑑𝑉
𝑉
= 𝑘2 ∫𝑒−𝑗𝑘𝑟
𝑟
𝑎
0
4𝜋𝑟2𝑑𝑟 = 4𝜋𝑘2 ∫ 𝑟𝑒−𝑗𝑘𝑟𝑎
0
𝑑𝑟 (57)
It can be easily seen that in the limit when 𝑎 → 0, the result of that integral is zero. Then, finally
𝐴 =1
4𝜋 (58)
14
and
𝐺(𝐫, 𝐫′) =𝑒−𝑗𝑘|𝐫−𝐫′|
4𝜋|𝐫 − 𝐫′| (59)
This is the Green’s function in three dimensions.
MAGNETIC VECTOR POTENTIAL
Since it may be useful in future derivations, a short explanation of the magnetic vector potential will
be given in this point. Some vector calculus identities will be used here.
Since the magnetic vector 𝜇𝑯 is always solenoidal (i.e. ∇ ∙ 𝜇𝐇 = 0 , eq (27), for qm=0), by the
fundamental theorem of vector calculus, it can be expressed as the curl of an arbitrary vector A:
μ𝐇 = ∇ × 𝐀 (60)
With this vector calculus identity
∇ × ∇ × 𝐀 = ∇(∇ ∙ 𝐀) − ∇2𝐀 (61)
and taking the curl of both sides of (60)
μ ∇ × 𝐇 = ∇ × ∇ × 𝐀 = ∇(∇ ∙ 𝐀) − ∇2𝐀 (62)
We now substitute (62) in (25), that results
𝜇𝐉 + 𝑗𝜔𝜇휀𝐄 = ∇(∇ ∙ 𝐀) − ∇2𝐀 (63)
On the other hand, substituting (60) in (24):
∇ × 𝐄 = −𝑗𝜔∇ × 𝐀 => ∇ × (𝐄 + 𝑗𝜔𝐀) = 0 (64)
and using the identity
∇ × (−∇Ф𝑒) = 0 (65)
where −∇Ф𝑒 is an arbitrary electric scalar potential, this yields:
𝐄 = −𝑗𝜔𝐀 − ∇Ф𝑒 (66)
Substituting the above in (63), leads to
𝜇𝐉 + 𝑗𝜔𝜇휀(−𝑗𝜔𝐀 − ∇Ф𝑒) = ∇(∇ ∙ 𝐀) − ∇2𝐀 (67)
which with some rearrangements yields:
∇2𝐀 + 𝑘2𝐀 = −𝜇𝐉 + ∇(∇ ∙ 𝐀 + 𝑗𝜔𝜇휀Ф𝑒) (68)
15
Because of the fact that the divergence of A has not been defined yet, it can be freely set to a value if
everything remains consistent with that definition. Then, the divergence for A is chosen to be:
∇ ∙ 𝐀 = −𝑗𝜔𝜇휀Ф𝑒 (69)
This definition applied to (67) gets it simplified:
∇2𝐀 + 𝑘2𝐀 = −𝜇𝐉 (70)
This expression is similar to (44), it is a Helmholtz scalar equation. Thus, A can be known by
knowing the current distribution and the Green’s function, as done in (45):
𝐀(𝐫) = 𝜇 ∭ 𝐺(𝐫, 𝐫′)
𝑉
𝐉(𝐫′)𝑑𝐫′ = 𝜇 ∭ 𝐉(𝐫′)𝑒−𝑗𝑘|𝐫−𝐫′|
4𝜋|𝐫 − 𝐫′|𝑑𝐫′
𝑉
(71)
Finally, the electric field in any point of space can be computed from (6666), when substituting Ф𝑒
with its value in (69):
𝐄 = −𝑗𝜔𝐀 − ∇Ф𝑒 = −𝑗𝜔𝐀 −𝑗
𝜔𝜇𝜖∇(∇ ∙ 𝐀) (72)
EFIE
Having explained all these derivations and reviewed briefly the fundamentals of electromagnetics,
obtaining of the Electric Field Integral Equation can be dealt with now. This equation will allow the
calculation of the surface currents knowing the incident electric field, and once those currents are
known, they can be used to compute the scattered field in any point of space. In this TFG, only far
field approximations will be explained, even though this could be used to solve near field situations if
the correct changes were made.
For the derivation of EFIE, the starting point is (45), where a linear equation related the surface
current distribution in an object with the scattered field generated by these currents in any point of
space. Apart from this, the volume equation doesn’t have to be calculated, since the surface currents
are distributed along a surface, so that volume integral may be substituted with a surface integral.
Then, the scattered field is calculated as:
𝐄𝑆(𝐫) = −𝑗𝜔𝜇 ∬ 𝐺(𝐫, 𝐫′) [𝐉(𝐫′) +1
𝑘2∇′∇′ ∙ 𝐉(𝐫′)]
𝑆
𝑑𝐫′ (73)
To be able to calculate the surface currents on a PEC body knowing the incident electric field, the
boundary condition imposed by (32) allows relating the incident field with the scattered field:
�̂�(𝐫) × 𝐄𝑇(𝐫) = �̂�(𝐫) × (𝐄𝑆(𝐫) + 𝐄𝑖(𝐫)) = 0 (74)
This yields to:
�̂�(𝐫) × 𝐄𝑆(𝐫) = −�̂�(𝐫) × 𝐄𝑖(𝐫) (75)
Therefore, taking the cross product at both sides of (73) and substituting the above:
−𝑗
𝜔𝜇�̂�(𝐫) × 𝐄𝑖(𝐫) = �̂�(𝐫) × ∬ 𝐺(𝐫, 𝐫′) [𝐉(𝐫′) +
1
𝑘2∇′∇′ ∙ 𝐉(𝐫′)] 𝑑𝐫′
𝑆
(76)
16
We can express this equation with the formerly explained magnetic vector potential, which is a
common practice:
−𝑗
𝜔𝜇�̂�(𝐫) × 𝐄𝑖(𝐫) = �̂�(𝐫) × [𝐀(𝐫) +
1
𝑘2∇∇ ∙ 𝐀(𝐫)] (77)
We can finally make use of the tangent vector instead of the normal vector to the surface:
−𝑗
𝜔𝜇[𝐭(𝐫) ∙ 𝐄𝑖(𝐫)] = 𝐭(𝐫) ∙ ∬ [1 +
1
𝑘2∇∇ ∙] 𝐉(𝐫′)𝐺(𝐫, 𝐫′)𝑑𝐫′
𝑆
(78)
This is the Electric Field Integral Equation. As it can be seen, with this equation, we can determine the
inducted current in a PEC object when illuminating it with an incident electric field.
FAR FIELD AND OBTENTION OF THE RCS
As the main goal of this project is the determination of the Radar Cross Section of objects of arbitrary
shape, the calculation of the scattered far field once the currents are known must be explained. The
RCS is a measured employed by radar operators; therefore, it is defined in the far field which greatly
simplifies the calculations.
Some approximations can be made when we are referring to objects located far from the source (i.e.
𝑅 >2𝐷
𝜆 or 𝑅 ≫ 𝜆 , being R the distance of the object to the source, and D the maximum linear
dimension of the antenna, if it is the case. In this region, the radiated field propagates as a plane wave.
This situation is exposed in Figure 2.4.
Figure 2.4. Far field situation.
As it can be seen, for an object situated at a great distance from the source, the attenuation will be
approximately the same as if the source was located in the origin, since 𝑟′ ≪ 𝑅. It can also be noticed
that the difference of phase from the source to the object compared with the one from the origin to the
object is 𝑟′𝑐𝑜𝑠𝛼. Then, in terms of phase,
𝑅 = 𝑟 − 𝑟′𝑐𝑜𝑠𝛼 (79)
Knowing that the dot product of two vectors is:
𝒓 · 𝒓′ = |𝒓||𝒓′| 𝑐𝑜𝑠𝛼 => 𝑟′𝑐𝑜𝑠𝛼 =𝒓 · 𝒓′
|𝑟|= �̂� · 𝒓′ (80)
17
Then, the far field approximation will be
𝑅 = 𝑟 − �̂� · 𝒓′ (81)
for phase variations, and
𝑅 = 𝑟 (82)
for amplitude variations.
The radiated field expressed in (72) expresses fields that vary in the form 1
𝑟 in the first term of the
right hand side, whilst the second term of the right hand side expresses fields that vary in the forms of 1
𝑟2 and 1
𝑟3 or superior. Given that the observation point is very far from the source, the attenuation of
the second term of the right hand side of that expression will be much greater, and thus only the
contributions of the fields that attenuate in the form of 1
𝑟 will prevail significantly. Therefore, we can
express the radiated field as:
𝐄(𝐫) = −𝑗𝜔𝐀(𝐫) (83)
Then, using the approximations explained, and with equation (71), the radiated field finally is:
𝐄(𝐫) = −𝑗𝜔𝜇
4𝜋∭ 𝐉(𝐫′)
𝑒−𝑗𝑘(r − 𝐫′∙�̂�)
𝑟𝑑𝐫′
𝑉
(84)
which can be simplified to
𝐄(𝐫) = −𝑗𝜔𝜇
4𝜋
𝑒−𝑗𝑘𝑟
𝑟∭ 𝐉(𝐫′)𝑒𝑗𝑘𝐫′∙�̂�𝑑𝐫′
𝑉
(85)
Finally, the RCS can be calculated, knowing the scattered and the incident field as:
𝜎 = 4𝜋𝑟2|𝐄𝑆|2
|𝐄𝑖|2= 4𝜋𝑟2|𝐄𝑆|2 (86)
for |𝐄𝑖| = 1, which is usually chosen for computation easiness.
18
2.2.2. USING THE MOM TO SOLVE THE 3D EFIE
Now that the Electric Field Integral Equation derivation has been detailed, the next step in order to
determine the scattered electric field is the determination of the surface current distribution. To do so,
the Method of Moments will be used, following the same steps as in 2.1.1, and deriving to the
numerical computed solution of the currents. This generalized method will allow the computation of
the current distribution on a PEC surface of any shape, so the solution from the simplest of the shapes
(a conducting sphere) to the hardest can be obtained, provided that the system in which the
computation is made has enough memory as well as computing power. Although this arbitrary 3D
shape solver might be harder to program than a specific situation, such as thin wires, a plane surface
(2D solution) or bodies of revolution, it will be able to solve them, in addition to shapes that are not
included in one of those. The advantages greatly overcome the disadvantages if the programmer’s will
is to have a polyvalent solver, hence the importance of the generalization.
MESHING THE SURFACE AND PREPROCESSING
To be able to solve arbitrarily shaped PEC objects, their surface must be reduced to a simpler one.
Therefore, the surface shall be meshed in triangular elements (Recommended size of the mesh <𝜆
10).
The program chosen for the realization of this project is gmsh [5], which can be downloaded for free
under a GNU License. This mesher was chosen because of its capability, the fact that it is free
software, allowing the programmer to build this program from scratch and without cost. Once the
surface is meshed, the file that gmsh writes is of the kind:
Figure 2.5. Mesh output format and corresponding figure.
The output numbers in Figure 2.5 express different characteristics of the elements they are noting. For
example, the second column after the $Elements tag notes if the element of the mesh is a Point (15,
where the last number of the row expresses the node in the $Nodes list), a line (1, where the last two
numbers of the row express the nodes that make that line), or a triangle (2, where, as it can be
inferred, the last three numbers in the row represent the three nodes that form that triangle).
19
PROCESSING THE IMPEDANCE MATRIX
BASIS FUNCTIONS: RWG
Following the process that was described in 2.1.1, the function to determine must be approximated by
a set of basis functions, just like in equation (5). In this case the function we want to determine is the
current distribution J, so the approximation will be:
𝐉(𝐫) = ∑ 𝑎𝑛𝐟𝑛(𝐫)
𝑁
𝑛=1
(87)
As it was formerly discussed, the basis functions have to resemble the behavior of the original
function as much as possible. For this kind of electrodynamic problems, one of the most commonly
used basis functions are the ones described in [6] by Rao, Wilton and Glisson which are known as
RWG functions. The form of those functions is illustrated in Figure 2.6.
Figure 2.6. RWG functions for an edge.
These functions are defined as:
𝐟𝑛(𝐫) =𝐿𝑛
2𝐴𝑛+ 𝝆𝑛
+(𝐫) , 𝑓𝑜𝑟 𝒓 𝑖𝑛 𝑇𝑛+ (88)
𝐟𝑛(𝐫) =𝐿𝑛
2𝐴𝑛− 𝝆𝑛
−(𝐫) , 𝑓𝑜𝑟 𝒓 𝑖𝑛 𝑇𝑛− (89)
𝐟𝑛(𝐫) = 0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (90)
where 𝑇𝑛+ and 𝑇𝑛
− are the triangles (of area 𝐴𝑛+ and 𝐴𝑛
− respectively) that share edge 𝑛, which length
is 𝐿𝑛.
Since the tangential component of the surface current distribution can be discontinuous, charge
density may be accumulated in the edges. The charge density can be related to the current using (42),
so taking the divergence of the RWG functions:
∇𝑠 ∙ 𝐟𝑛(𝐫) = −𝐿𝑛
𝐴𝑛+ , 𝑓𝑜𝑟 𝒓 𝑖𝑛 𝑇𝑛
+ (91)
∇𝑠 ∙ 𝐟𝑛(𝐫) =𝐿𝑛
𝐴𝑛− , 𝑓𝑜𝑟 𝒓 𝑖𝑛 𝑇𝑛
− (92)
∇𝑠 ∙ 𝐟𝑛(𝐫) = 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 (93)
20
As it can be seen, the divergence is not zero only in those triangles that share edge 𝑛. These basis
functions imposing normal continuity across edges but allowing for tangential discontinuity, are said
to be div-conforming basis functions.
The proceeding way with the elements of the mesh is to loop around the edges that define the
triangles, but only in those who are part of two triangles. This is because the basis and testing
functions that will be used will be assigned to one edge each, but not on the boundary edges. A good
preprocessing of the problem will make the computations much faster. The solution that was
developed to make the list of the edges was the following one:
1. After reading all the triangles, they are stored in an array.
2. For the first triangle, the edge formed by the first two vertexes is stored in another array (an
edge array), with an attribute of the identification number of the triangle it belongs (the index
of the triangle in its array). The same is done for the other two edges of the first triangle.
3. For the following triangles, the array of the edges must be looped, looking for coincidence
with the nodes that form that edge. If there is a match with a previously stored edge, it means
that the edge found is the same as this one. In this case, the edge stored in first place gets also
the identification number of the triangle this new edge belongs to. In case there is not a match,
the new edge is stored in a new position of the array. Repeat this step until finish
With this procedure, what will be achieved is that every edge has the identification number of the two
triangles that it shares, and that identification number will be the index of the triangle in its array, so
direct access to that triangle is granted, saving a great amount of time in the processing steps. In case
that an edge only belongs to one triangle (i.e. it is a boundary edge), it will only have one
identification number for a triangle, and this way it will be known that no function must be defined for
that edge.
BUILDING OF THE SYSTEM
Now that we have defined the basis functions by which we are going to approximate the EFIE, we
substitute the currents in (78) by their RWG approximation, yielding:
−𝑗
𝜔𝜇�̂�(𝐫) ∙ 𝐄𝑖(𝐫) = ∬ (1 +
1
𝑘2∇∇ ∙) ∑ 𝑎𝑛𝐟𝑛(𝐫′)
𝑒−𝑗𝑘𝑟
4𝜋𝑟𝑑𝐫′
𝑁
𝑛=1
𝑆
(94)
where 𝑟 = |𝒓 − 𝒓′|
Now that the unknown function has been approximated, the next step is to apply the testing or
weighting functions to this equation to have a complete system of equations. In this case, Galerkin’s
Method is the chosen to select the weighting functions, so they will be the same as the basis functions,
the RWG ones. This results in:
∬ 𝐟𝑚(𝒓) ·
𝑆
(−𝑗
𝜔𝜇�̂�(𝐫) ∙ 𝐄𝑖(𝐫)) 𝑑𝒓
= ∬ 𝐟𝑚(𝒓) ·
𝑆
(∬ (1 +1
𝑘2∇∇ ∙) ∑ 𝑎𝑛𝐟𝑛(𝐫′)
𝑒−𝑗𝑘𝑟
4𝜋𝑟𝑑𝐫′
𝑁
𝑛=1
𝑆
) 𝑑𝒓
(95)
21
which, for just a 𝑚 and a 𝑛 element is:
∬ 𝐟𝑚(𝒓) ·
𝑓𝑚
(−𝑗
𝜔𝜇�̂�(𝐫) ∙ 𝐄𝑖(𝐫)) 𝑑𝒓
= ∬ 𝐟𝑚(𝒓) ·
𝑓𝑚
(∬ (1 +1
𝑘2∇∇ ∙)𝑎𝑛𝐟𝑛(𝐫′)
𝑒−𝑗𝑘𝑟
4𝜋𝑟𝑑𝐫′
𝑓𝑛
) 𝑑𝒓
(96)
REDISTRIBUTION OF THE DIFFERENTIAL OPERATOR
To evaluate the left hand side of the equation in an easier way to compute it, some vector calculus
must be used. First, the right hand side for just the 𝑧𝑚𝑛 element of the impedance matrix:
𝑧𝑚𝑛 = ∬ 𝐟𝑚(𝒓) ·
𝑓𝑚
(∬ (1 +1
𝑘2∇∇ ∙) 𝐟𝑛(𝐫′)
𝑒−𝑗𝑘𝑟
4𝜋𝑟𝑑𝐫′
𝑓𝑛
) 𝑑𝒓 (97)
can be rewritten as:
𝑧𝑚𝑛 = [∬ ∬ 𝐟𝑛(𝐫′)
𝑓𝑛
· 𝐟𝑚(𝒓)𝑑𝒓′𝑑𝒓 + ∬ 𝐟𝑚(𝐫)
𝑓𝑚
𝑓𝑚
· (1
𝑘2∇∇ ∙ ∬ 𝐟𝑛(𝐫′)
𝑒−𝑗𝑘𝑟
4𝜋𝑟𝑑𝐫′
𝑓𝑛
) 𝑑𝒓] (98)
The first term of the right hand side is not problematic, whereas the second term is, so it has to be
dealt with:
∬ 𝐟𝑚(𝐫)
𝑓𝑚
· (1
𝑘2∇∇ ∙ ∬ 𝐟𝑛(𝐫′)𝐺(𝐫, 𝐫′)𝑑𝐫′
𝑓𝑛
) 𝑑𝒓 (99)
Using the vector identity:
∇ ∙ 𝐟𝑛(𝐫′)𝐺(𝐫, 𝐫′) = [∇𝐺(𝐫, 𝐫′)] · 𝐟𝑛(𝐫′) + [∇ · 𝐟𝑛(𝐫′)]𝐺(𝐫, 𝐫′) (100)
where the second term in the right hand side is zero since 𝐟𝑛(𝐫′) is not a function of 𝒓 (unprimed).
This way, (99) yields:
∬ 𝐟𝑚(𝐫)
𝑓𝑚
· (1
𝑘2∇ ∬ [∇𝐺(𝐫, 𝐫′)] · 𝐟𝑛(𝐫′)𝑑𝐫′
𝑓𝑛
) 𝑑𝒓 (101)
Because of the symmetry of the Green’s function:
∇𝐺(𝐫, 𝐫′) = − ∇′𝐺(𝐫, 𝐫′) (102)
Then, (101) is:
∬ 𝐟𝑚(𝐫)
𝑓𝑚
· (−1
𝑘2∇ ∬ [∇′𝐺(𝐫, 𝐫′)] · 𝐟𝑛(𝐫′)𝑑𝐫′
𝑓𝑛
) 𝑑𝒓 (103)
If the vector identity in (100) is used to solve the gradient in the above, it yields:
22
∬ 𝐟𝑚(𝐫)
𝑓𝑚
· (1
𝑘2[∇ ∬ 𝐺(𝐫, 𝐫′)∇′ · 𝐟𝑛(𝐫′)𝑑𝐫′
𝑓𝑛
− ∇ ∬ ∇′ ∙ [ 𝐟𝑛(𝐫′)𝐺(𝐫, 𝐫′)]𝑑𝐫′
𝑓𝑛
] ) 𝑑𝒓 (104)
Since the basis and test functions are distributed over a surface, we can use the Gauss divergence
theorem in the second term of the above equation, converting it in an integral over its boundary. Since
the surface can be chosen as big as wanted, in the boundary of that surface, there will be no current.
Then the integral over the surface must be equal to zero, so finally it yields:
∬ 𝐟𝑚(𝐫)
𝑓𝑚
· (−1
𝑘2∇ ∬ 𝐺(𝐫, 𝐫′)∇′ · 𝐟𝑛(𝐫′)𝑑𝐫′
𝑓𝑛
) 𝑑𝒓 = ∬ 𝐟𝑚(𝐫)
𝑓𝑚
· ∇F(𝐫)d𝐫 (105)
Making use once more of the vector identity:
𝐟𝑚(𝐫) · ∇F(𝐫) = ∇ · [ 𝐟𝑚(𝐫)F(𝐫)] − [∇ · 𝐟𝑚(𝐫)]F(𝐫) (106)
∬ 𝐟𝑚(𝐫)
𝑓𝑚
· ∇F(𝐫)d𝐫 = ∬ ∇ · [ 𝐟𝑚(𝐫)F(𝐫)]
𝑓𝑚
d𝐫 − ∬ [∇ · 𝐟𝑚(𝐫)]F(𝐫)
𝑓𝑚
d𝐫 (107)
Here, using the same procedure with the Gauss divergence theorem as before, the first term is equal to
zero and the second term is the only one prevailing so, finally:
∬ 𝐟𝑚(𝐫)
𝑓𝑚
· ∇F(𝐫)d𝐫 = ∬ ∇ · 𝐟𝑚(𝐫)
𝑓𝑚
· (−1
𝑘2∬ 𝐺(𝐫, 𝐫′)∇′ · 𝐟𝑛(𝐫′)𝑑𝐫′
𝑓𝑛
) 𝑑𝒓 (108)
MAKING THE SYSTEM COMPUTABLE
MATRIX ELEMENTS
After distributing the differential operators as explained before, the 𝑧𝑚𝑛 element of the impedance
matrix yields:
𝑧𝑚𝑛 = [∬ ∬ 𝐟𝑛(𝐫′)
𝑓𝑛
· 𝐟𝑚(𝒓)𝑑𝒓′𝑑𝒓
𝑓𝑚
−1
𝑘2[ ∇′ · 𝐟𝑛(𝐫′)][ ∇ · 𝐟𝑚(𝐫)]
𝑒−𝑗𝑘𝑟
4𝜋𝑟𝑑𝒓′𝑑𝒓] (109)
Substituting the RWG functions and their divergences, as calculated in (88), (89), (91) and (92):
𝑧𝑚𝑛 =𝐿𝑚𝐿𝑛
𝐴𝑚𝐴𝑛∬ ∬ [
1
4𝝆𝑚
± (𝐫) ∙ 𝝆𝑛±(𝐫′) ±
1
𝑘2]
𝑒−𝑗𝑘𝑟
4𝜋𝑟𝑑𝐫′𝑑𝐫
𝑇𝑛
𝑇𝑚
(110)
These integrals, in order to be processed by a computer, have to be numerically calculated. This is
achieved by approximating them by an M-point numerical Gauss quadrature which will be explained
in the appendix:
𝑧𝑚𝑛 ≈𝐿𝑚𝐿𝑛
4𝜋∑ ∑ 𝑤𝑝𝑤𝑞 [
1
4𝝆𝑚
± (𝐫𝑝) ∙ 𝝆𝑛∓(𝐫𝑞
′ ) ±1
𝑘2]
𝑒−𝑗𝑘𝑅𝑝𝑞
𝑅𝑝𝑞
𝑀
𝑞=1
𝑀
𝑝=1
(111)
23
It is important to notice that, since the divergence sign is different depending on whether the function
is defined on 𝑇+ or 𝑇−as shown in (91) and (92), the sign of 1
𝑘2 will be negative if both testing and
basis function are in a triangle with the same sign or positive if they are in triangles of different sign.
In the former equation, 𝑅𝑝𝑞 is the distance from 𝐫𝑝 to 𝐫𝑞′ :
𝑅𝑝𝑞 = √(𝑥𝑝 − 𝑥𝑞)2 + (𝑦𝑝 − 𝑦𝑞)2 + (𝑧𝑝 − 𝑧𝑞)2 (112)
For points that are far enough, the expression (111) can be used as a good approximation, but for pairs
of basis and testing functions with overlapping triangles, in which some RWG are in the same triangle
as others, and 𝑅𝑝𝑞 tends to zero, the integrals in (110) have to be performed analytically. In these
cases the integrand is singular and a proper treatment of this integral has to be carried out. To do so,
the singularity must be extracted, and then a Duffy transform will be applied.
To extract the singularity, adding and subtracting 1
𝑟 to the Green’s function yields:
𝑒−𝑗𝑘𝑟
𝑟= [
𝑒−𝑗𝑘𝑟
𝑟−
1
𝑟] +
1
𝑟 (113)
Where the first term of the right hand side can be directly solved:
lim𝑟→0
[𝑒−𝑗𝑘𝑟
𝑟−
1
𝑟] = −𝑗𝑘 (114)
This part of the singularity can be numerically calculated over the same quadrature points as in (111).
The second term leads to integrals of the form:
𝐼1 = ∬ 𝝆𝑚± (𝐫) ∙ ∬ 𝛒𝑛
±(𝐫′)1
𝑟𝑑𝐫′𝑑𝐫
𝑇′
𝑇
(115)
𝐼2 = ∬ ∬1
𝑟𝑑𝐫′𝑑𝐫
𝑇′
𝑇
(116)
where 𝑇 and 𝑇′ are overlapping. Although analytical integration or a combination of analytical and
numerical integration might be used, in this project, the method chosen to solve these integrals is the
Duffy transform, which is one of the most commonly used.
DUFFY TRANSFORM
The Duffy transform is a widely used method to solve integrals on a triangle with a 1
𝑟 singularity at its
vertex. This transform reduces the singularity at the vertex of the triangle allowing a numerical
integration. Therefore, these integrals will also be performed numerically. The Duffy transform is
graphically explained in Figure 2.7, where it is shown that any point in the triangle (in the blue
vertical lined area) is transformed in another one in the rectangle (the orange horizontal lined area). In
the picture, it can be appreciated that the singularity is reduced to a softer one by “expanding” the
singularity in vertex 𝐯1, from just one point, to a whole side of the rectangle so that the integral can be
performed.
24
Figure 2.7. Geometric explanation of the Duffy transform.
The Duffy transform converts the integral in one of the kind:
∬𝑓(𝐫′)
𝑟(𝐫′)𝑑𝐫′ = ∫ ∫
𝑓(𝑢, 𝛾)
𝑟(𝑢, 𝛾)|𝐽(𝑢, 𝛾)| 𝑑𝑢 𝑑𝛾
1
0
1
0
𝑇
(117)
being 𝐽(𝜐, 𝛾) the Jacobian of the transform, which will be calculated later. In the Duffy transform, a
point 𝒓′ becomes:
𝐫′ = 𝑢𝐯1 + (1 − 𝑢)(1 − 𝛾)𝐯2 + 𝛾(1 − 𝑢)𝐯3 (118)
where 𝐯1, 𝐯2 and 𝐯3 are the triangle vertexes, for 0 ≤ 𝑢, 𝛾 ≤ 1, and being the singularity located in the
vertex 𝐯1.
Now that the transform has been described, the solution of the inner integral in (116) is:
∬1
𝑟𝑑𝐫′ = ∫ ∫
|𝐽(𝑢, 𝛾)|
𝑟(𝑢, 𝛾)𝑑𝑢 𝑑𝛾
1
0
1
0
𝑇′ (119)
The Jacobian of this transformation is easier to calculate when realizing that a change from the
original triangle to simplex coordinates as explained in the appendix simplifies the calculations.
A point from the original triangle expressed in simplex coordinates is represented as:
𝐫′ = 𝛾𝐯1 + 𝛼𝐯2 + 𝛽𝐯3 = (1 − 𝛼 − 𝛽)𝐯1 + 𝛼𝐯2 + 𝛽𝐯3 (120)
as it is explained in 2.2.3. And the Jacobian of this transformation is
|𝐽(𝛼, 𝛽, 𝛾)| = 2𝐴 (121)
25
where A is the area of the original triangle. Now, identifying terms from (118) and (120), it can be
seen that:
{𝛼 = (1 − 𝑢)(1 − 𝛾)
𝛽 = 𝛾(1 − 𝑢) (122)
The Jacobian of this second transform is:
|𝐽(𝑢, 𝛾)| = (1 − 𝑢) (123)
Finally, multiplying (121) and (123), the total Jacobian of the transformation is:
|𝐽(𝑢, 𝛾)| = (1 − 𝑢)2𝐴 (124)
Now the next step to solve (119) is determining the distance:
𝑟(𝑥, 𝑦) = √(𝑥1 − 𝑥)2 + (𝑦1 − 𝑦)2 + (𝑧1 − 𝑧)2 = √𝑎(𝑢, 𝛾) + 𝑏(𝑢, 𝛾) + 𝑐(𝑢, 𝛾) = 𝑟(𝑢, 𝛾) (125)
where
𝑎(𝑢, 𝛾) = (𝑥1 − 𝑥)2 = [𝑥1 − (𝑢𝑥1 + (1 − 𝑢)(1 − 𝛾)𝑥2 + 𝛾(1 − 𝑢)𝑥3)]2 (126)
After operating, it yields:
𝑎(𝑢, 𝛾) = (1 − 𝑢)2[𝑥1 − [(1 − 𝛾)𝑥2 + 𝛾𝑥3]]2 (127)
Proceeding in a similar way:
𝑏(𝑢, 𝛾) = (1 − 𝑢)2[𝑦1 − [(1 − 𝛾)𝑦2 + 𝛾𝑦3]]2 (128)
𝑐(𝑢, 𝛾) = (1 − 𝑢)2[𝑧1 − [(1 − 𝛾)𝑧2 + 𝛾𝑧3]]2 (129)
After getting to this point, it is noticeable that the singularity, which was present in the denominator of
the integrand as (1 − 𝑢) is cancelled with the same factor in the Jacobian. Therefore the integral is no
longer singular and it can be calculated with a numerical Q-point Gauss approximation for a
rectangle, which are provided in the appendix. The outer integral in (116) can be performed
numerically with the M-point Gauss approximation for a triangle.
To solve equations of the form of (115), the same procedure is used:
∬ 𝛒𝑛±(𝐫′)
1
𝑟𝑑𝐫′
𝑇′
= ∫ ∫𝛒𝑛
±(u, γ)
𝑟(𝑢, 𝛾)|𝐽(𝑢, 𝛾)|𝑑𝑢 𝑑𝛾
1
0
1
0
(130)
In this case, the calculations of the Jacobian and the distance remain the same, cancelling the
singularity at the vertex of the triangle and, thus, allowing a numerical integration. Then, taking each
of the quadrature points in the rectangle and, translating them to the Cartesian plane, the vector
𝛒𝑛±(u, γ) can be obtained numerically. Just like before, the outer integral can be performed
numerically over triangle T.
The Duffy transform that has been explained is only valid for triangles with a singularity at their
vertex. Bearing in mind the integrals over overlapping triangles that are to be performed, integrating
26
over an M-point quadrature will lead to singularities within the triangles. The solution in this case is
to split each triangle in three, having each triangle the split (and singularity) point at its vertex as
described in Figure 2.8.
Figure 2.8. Process of splitting
Despite the difficulty that the singular integration involves, it is this part the one that comprises the
strongest contribution to the determination of the currents, so special attention is required in this point.
Once these singular integrations can be solved, the entire impedance matrix can be filled. The
impedance matrix depends only on the geometry of the object and the frequency at which the RCS is
to be calculated. Therefore, it has to be computed only once, for both monostatic and bistatic radar
systems.
Another consideration to take into account about the impedance matrix is that it is symmetric (i.e. the
interaction of the RWG belonging to edge 𝑚 with the one belonging to edge 𝑛 is the same as the
interaction of the RWG of edge 𝑛 with that of edge 𝑚). Thus, 𝑧𝑚𝑛 = 𝑧𝑛𝑚 and only a triangular half
of the matrix has to be calculated, reducing considerably the computing time.
EXCITATION VECTOR
After filling the impedance matrix, the following step to solve equation (94) is to calculate the
excitation vector. Once this has been done, the system can be solved and the currents can be obtained.
As shown in that equation, the excitation is the left hand side of the equation:
−𝑗
𝜔𝜇�̂�(𝐫) ∙ 𝐄𝑖(𝐫) (131)
This equation, when dot multiplied with the test functions yields, for each test function:
𝑏𝑚 = −𝑗
𝜔𝜇∬ 𝐟𝑚(𝐫) ∙ 𝐄𝑖(𝐫)𝑑𝐫
𝐟𝑚
(132)
that, when extracting the constants, yields:
𝑏𝑚 = −𝑗
𝜔𝜇
𝐿𝑚
2𝐴𝑚∬ 𝝆𝑚
± (𝐫) ∙ 𝐄𝒊(𝐫)𝑑𝐫
𝑇𝑚
(133)
27
This integral, as explained in the former point, can be numerically approximated using M Gauss
quadrature points, resulting in the following:
𝑏𝑚 = −𝑗
𝜔𝜇
𝐿𝑚
2∑ 𝑤𝑝𝝆𝑚
± (𝐫𝑝) ∙ 𝐄𝑖(𝐫𝑝)
𝑀
𝑝=1
(134)
where 𝐄𝑖(𝐫𝑝) is the incident electric field to the surface, and so it has a polarization, that has to be
scalar multiplied with 𝝆𝑚± (𝐫𝑝), and a phase depending of the position of 𝐫𝑝. The total electric field of
the plane wave can be described as:
𝐄𝑖(𝐫𝑝) = 𝐸0�̂�(𝜃, 𝜙)𝑒−𝑗𝑘�̂�·𝐫𝑝 (135)
So the final excitation element is obtained as:
𝑏𝑚 = −𝑗
𝜔𝜇
𝐿𝑚
2𝐸0 ∑ 𝑤𝑝𝝆𝑚
± (𝐫𝑝) ∙ �̂�(𝜃, 𝜙)𝑒−𝑗𝑘�̂�·𝐫𝑝
𝑀
𝑝=1
(136)
Up to this point, the system of algebraic equations is completed and can be solved. The current vector
can be obtained, and only the far field computation is needed.
SCATTERED FIELD
Once the currents are known, the last step in order to calculate the radar cross section of the object is
obtaining the scattered field. The radiated field is described as a function of the currents in (85), and it
can be approximated for each triangle as:
𝐄𝑚(𝐫) = −𝑗𝜔𝜇
4𝜋
𝑒−𝑗𝑘𝑟
𝑟
𝑎𝑚𝐿𝑚
2∬ 𝝆𝑚
± (𝐫′)𝑒𝑗𝑘𝐫′∙�̂�𝑑𝐫′
𝑇𝑚
(137)
that, once more, can be computed numerically:
𝐄𝑚(𝐫) ≈ −𝑗𝜔𝜇
4𝜋
𝑒−𝑗𝑘𝑟
𝑟
𝑎𝑚𝐿𝑚
2∑ 𝑤𝑝𝝆𝑚
± (𝐫𝑝′ )𝑒𝑗𝑘𝒓𝒑
′ ∙�̂�
𝑀
𝑝=1
(138)
and the total scattered electric field will be the sum of the contributions of all the triangles. However,
it is important to underline that the scattered field is often calculated with a polarization, i.e. that only
the electric field that is polarized in a certain direction will be of interest. Thus, the total scattered
field must be scalar multiplied with the polarization vector in which direction the measure is wanted.
The RCS can then be obtained for any incident polarization and propagation direction and for any
scattered polarization.
Depending on the problem, monostatic or bistatic RCS may be calculated. The difference between
them is that while for monostatic RCS, the system and the scattered field have to be solved as many
times as measuring points are looked for, for bistatic RCS, the system has to be solved only once, and
the scattered field for that current distribution is the one that has to be calculated as many times as
desired. This difference results in execution times much lower for bistatic than for monostatic
scattering, since the solution of a large system of equations takes large time to be computed, and has
to be done many times.
28
2.2.3. APPENDIX
It is the objective of this appendix to provide some useful extra information about certain aspects of
the project, such as the code that has been implemented, the compiler and libraries that have been
used, and quadrature points tables.
QUADRATURE POINTS
In many occasions throughout the text, a Gaussian M-point quadrature approximation has been used,
so a description of the mentioned quadrature over triangles seems important. Simplex coordinates are
often chosen to represent those points and, thus, they deserve a brief explanation.
SIMPLEX COORDINATES
In a similar way as it was explained in the Duffy transform, any point within a triangle can be
expressed as a weighted sum of the position vector of its vertexes:
𝐫 = 𝛾𝐯1 + 𝛼𝐯2 + 𝛽𝐯3 (139)
where 𝛼, 𝛽and 𝛾 are defined as:
𝛼 =𝐴1
𝐴 (140)
𝛽 =𝐴2
𝐴 (141)
𝛾 =𝐴3
𝐴 (142)
and where 𝐴1, 𝐴2 and 𝐴3 are the areas of the three triangles shown in Figure 2.9.
Figure 2.9. Components of the simplex coordinates
29
It can be easily noticed that since the point has to be located in the triangle (i.e. a surface), only two
components are needed to described its position, and as it can be also seen:
𝛼 + 𝛽 + 𝛾 = 𝐴1
𝐴+
𝐴2
𝐴+
𝐴3
𝐴=
𝐴
𝐴= 1 (143)
so, one component is usually expressed in terms of the other two:
𝛾 = 1 − 𝛼 − 𝛽 (144)
Finally,
𝐫 = (1 − 𝛼 − 𝛽)𝐯1 + 𝛼𝐯2 + 𝛽𝐯3 (145)
where 0 ≤ 𝛼, 𝛽 ≤ 1
This transformation converts one arbitrary triangle in a canonical one as shown in Figure 2.10. The
integrations performed under this transformation result:
∬ 𝑓(𝒓) 𝑑𝒓
𝑇
= ∬ 𝑓(𝛼, 𝛽)|J(𝛼, 𝛽)| 𝑑𝛼 𝑑𝛽
𝑇𝑐
= 2𝐴 ∫ ∫ 𝑓(𝛼, 𝛽) 𝑑𝛼 𝑑𝛽1−𝛼
0
1
0
(146)
This Jacobian is the one used in the Duffy’s transform section to calculate the Duffy’s transform
Jacobian.
Figure 2.10. Simplex coordinates geometrical description
30
QUADRATURE POINTS TABLES
Now that simplex coordinates have been introduced, the tables used for the determination of the
Gauss-Legendre quadrature points of the triangles of this project will be given. A seven point
approximation for the triangles was used in this project, because a seven point approximation can
estimate with exact result polynomials of degree up to 2𝑛 − 1 where 𝑛 = 5 for this number of points.
Exact solution for polynomials of degree = 9 has been considered a good approximation for this
project. The quadrature points are expressed in terms of the upper coordinates, and are normalized to
the original triangle’s area, and will allow the approximation of the integral by a sum:
∬ 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦
𝑆
≈ 𝐴 ∑ 𝑤(𝛼𝑞 , 𝛽𝑞)𝑓(𝛼𝑞 , 𝛽𝑞)
𝑀
𝑞=1
(147)
q 𝛼 𝛽 𝛾 𝑤
1 0.33333333 0.33333333 0.33333333 0.225
2 0.05971587 0.47014206 0.47014206 0.13239415
3 0.47014206 0.05971587 0.47014206 0.13239415
4 0.47014206 0.47014206 0.05971587 0.13239415
5 0.79742698 0.10128650 0.10128650 0.12593918
6 0.10128650 0.79742698 0.10128650 0.12593918
7 0.10128650 0.10128650 0.79742698 0.12593918
Table 2-1. Gauss quadrature points for a triangle
Likewise, for the Duffy transform, Gauss quadrature points had to be used but this time over a
rectangle. Given the delicate situation of a singular integration, nine points of quadrature have been
chosen to have an even better approximation. These points are for a rectangle in the first quadrant and
are also normalized to the rectangle area.
q 𝑢 𝛾 𝑤
1 0.8872983346 0.8872983346 0.07716049383
2 0.5 0.8872983346 0.12345678901
3 0.1127016645 0.8872983346 0.07716049383
4 0.8872983346 0.5 0.12345678901
5 0.5 0.5 0.1975308642
6 0.1127016645 0.5 0.12345678901
7 0.8872983346 0.1127016645 0.07716049383
8 0.5 0.1127016645 0.12345678901
9 0.1127016645 0.1127016645 0.07716049383
Table 2-2. Quadrature points for a rectangle
31
SOFTWARE
It is this author’s opinion that knowing on which software this code has been implemented may be
useful for programming a method like this one, as well as to give credit for the programs and libraries
used.
The capacitor example in 2.1.2 was entirely computed using MATLAB, as well as the results
were presented using this same tool.
The program used to obtain the mesh from the arbitrary shapes, as formerly said, is gmsh, a
free license software [5].
All the program processing of 2.2 was built on C++, reading the input from gmsh and
providing the data output. The compiler used is the GNU GCC compiler for 32 bits windows.
The Armadillo library [7] was used for the solution of the system (i.e. solving eq. (12)). This
library works on BLAS and LAPACK (libraries made to work with linear algebra), so these
libraries had to be also installed. The solver used for the system is an iterative one, i.e.
Armadillo does not invert the impedance matrix and then multiply it by the excitation vector,
Armadillo guesses a solution for the current vector and performs a multiplication by the
impedance matrix; then it calculates the residual trying to make it tend to zero as fast as
possible with an algorithm that may be a gradient one. This leads to much faster solution
times. A compiled object-oriented language was chosen because of its scalable possibilities
against other non-object-oriented, and the fact that it is compiled makes it more efficient,
since large and complex problems were intended to be solved.
The results were plotted using MATLAB once again for its simplicity. Data was imported
from the output of the processing program in C++ and it was plotted.
While hoping that these suggestions may be of use for future developments of a similar system, they
are not the only options for the development of a system of these characteristics.
32
3. RESULTS After having implemented the code, some tests are required to check the accurate performance of the
program. Some examples were contrasted such as a sphere, measured against the Mie series, a cube,
and some other bodies of revolution such as the NASA almond, an ogive, a double ogive, and others.
Other objects were simulated to show the results, after knowing that the program worked correctly.
3.1. TESTING SIMULATIONS
3.1.1. SPHERE
The solution of the sphere is a good experiment to check the accuracy of the MoM code implemented
of the frequency is known. For this reason, the plots extracted from the code can be contrasted with a
known to be exact plot, and no doubts about whether the original measure was right or wrong will
occur. The Method of Moments programmed in this project was compared with another code in
MATLAB that generated the Mie series result for the bistatic scattering of the sphere. The radius of
the sphere was 0.006m and the frequency measures were at 5GHz (radius = 𝜆
10) and at 30GHz (radius
= 6𝜆
10 ). The respective meshes can be seen in Figure 3.1.
The computation results are shown in Figure 3.2 and Figure 3.3 for both frequencies. These results are
compared to the analytical Mie series expected result so they can provide a trustworthy verification of
a correct behavior of the code. Both HH (in blue) and VV (in red) polarizations are studied and as it
can be appreciated, the maximum deviations are lower than 0.5dB. While more tests will have to be
studied to be sure that the program is working as expected since the sphere has no edges that may
have some influence in the results, this is the only test where the accurate solution is known, and is
not the result of measurements.
Figure 3.1. Sphere geometry and meshes for 5GHz (left) and 30GHz (right)
33
Figure 3.2. Bistatic radar cross section of a sphere of r = 𝝀
𝟏𝟎
Figure 3.3. Bistatic radar cross section of a sphere of r = 𝟔𝝀
𝟏𝟎
34
3.1.2. CUBE
The analysis of the monostatic RCS of a cube has also been performed. The side of the cube was 1m
in length and the frequency used was 0.43GHz. Due to very high computation time, only RCS with V-
V polarization was calculated, and only one plot per degree was calculated. The mesh is shown in
Figure 3.4 and the results can be seen in Figure 3.5. In this last figure, drawn in red is this program’s
simulation results, the continuous black line is the measurements for this cube, and the discontinuous
black line are the computation results that were obtained in [8]. As it can be seen the simulation fits
the expected results with a reasonable accuracy.
Figure 3.4. Cube of side length = 1m geometry and mesh
Figure 3.5. Monostatic RCS of the cube for f=0.43GHz, V-V polarization
35
3.1.3. NASA ALMOND
Beyond this point some famous bodies will be simulated and their RCS will be obtained and
compared with the expected results that were provided in [9]. The first of these shapes is the famous
NASA almond, a body of revolution designed to have very low RCS. In this simulation, RCS in both
H-H and V-V polarizations is measured, and compared with the expected results. The geometry and
mesh of the almond can be seen in Figure 3.6.
The frequency for the simulation of this figure is 1.19 GHz, since at this frequency the length of the
almond is approximately the wavelength. The results are shown in Figure 3.7. The continuous red and
blue lines are this program’s results, the continuous black line is the measurements done to the
almond, and the discontinuous lines are other simulations made in [9]. The simulated curves fit
correctly between the simulation and the measurements taken as a reference.
Figure 3.7. NASA almond RCS, for V-V and H-H polarizations, at f=1.19GHz
Figure 3.6. The NASA almond geometry and mesh
36
3.1.4. OGIVE
The ogive is a body of revolution that has been widely used to test the code. It has a symmetry edge
and two tips, as shown in Figure 3.8.
For this simulation, the mesh chosen is less dense than for other shapes, a mesh of element side size of 𝜆
10 has been chosen, which is often considered enough. The simulation is not as accurate as the former
ones, but this can also allow noticing that even though the mesh is not as dense as in former examples,
the results given are a good approximation, with maximum deviations of around 1dB, as it can be seen
in, where the blue and red lines are once again, this simulation results. As it can be appreciated, the
simulations are a good approximation for the measured and computed results provided.
Figure 3.9. Ogive RCS, for H-H and V-V polarizations, at f=1.18GHz
Figure 3.8. Geometry and mesh of the ogive
37
3.1.5. DOUBLE OGIVE
When putting together two different half ogives, the result is a double ogive (Figure 3.10), which will
be analyzed and compared with the expected values at a frequency of 1.57GHz (Figure 3.11). For this
simulation, a thinner mesh was chosen than for the ogive, so better results were obtained, but taking
much longer to simulate (as mentioned, the complexity increases not linearly, but as 𝑂(𝑁2)).
Figure 3.11. Double ogive RCS, for H-H and V-V polarizations, at f=1.57GHz
Figure 3.10. Double ogive geometry and mesh
38
3.2. OTHER SIMULATIONS
Once the code has been tested and works properly, two more objects were simulated in monostatic
and bistatic radar configurations, for both HH and VV polarizations.
3.2.1. TORUS
The torus is another body of revolution and it has been simulated because it is an object different from
the above because there is no surface in the center of it. Its geometry is shown in Figure 3.12, where
the bigger radius is 𝑅 = 0.1 𝑚 = 𝜆 and the smaller one is 𝑟 = 0.03 𝑚. The torus has been simulated
at f=3GHz, and the bistatic and monostatic results are shown in Figure 3.13 and Figure 3.14,
respectively. In the bistatic case, the incident field propagates in − �̂� direction.
Figure 3.13. Bistatic RCS of the torus, at f=3 GHz
Figure 3.12. Torus geometry and mesh
39
Figure 3.14. Monostatic RCS of the torus, at f=3 GHz
40
3.2.2. DESTROYER
The final object to be analyzed is a destroyer ship of approximately 700ft = 213m. The geometry of
the problem is illustrated in Figure 3.15. Since it is a very large problem, the frequency of analysis is
3.3 MHz, a really low one (no radar system operates at this frequencies) but it is useful to illustrate the
behavior of the ship at this frequency. It is important to notice the high RCS values, measured for all
the angles and in both bistatic and monostatic configurations, as it can be seen in Figure 3.16 and
Figure 3.17, respectively. The plane wave is propagating in the −�̂� direction in the bistatic situation,
and the RCS is calculated in azimuth (situation of detection from one ship to another).
Figure 3.15. Destroyer geometry and mesh
41
Figure 3.16. Bistatic RCS of the destroyer at f=3.3 MHz
Figure 3.17. Monostatic RCS of the destroyer at f=3.3 MHz
42
4. CONCLUSIONS AND FUTURE LINES
4.1. CONCLUSIONS
In this project, the Method of Moments has been thoroughly reviewed, starting from the mathematical
base, then applying the method to an electrostatic problem (the capacitor, where two of its
characteristics were studied: capacity and charge distribution), which was programmed using
MATLAB. Up to this point, the potential of a program with these characteristics can be noticed. It is a
program that can provide accurate results and plenty information about the behavior of the object once
it has been solved. Finally, the MoM has been programmed for the solution of electromagnetic wave
scattering in non-penetrable PEC objects of arbitrary shape, allowing the determination of the radar
cross section of those objects.
The main code was developed in C++ and the results were represented exporting the data from C++
and plotting it using MATLAB. This means that C++ programming language had to be learnt from
scratch, and no other tools apart from one library to operate with matrix were used. In the
development of the final solution of the code, some concepts such as the Duffy transform, some
vector calculus operations, quadrature points and others that were considered to be important enough
were explained in detail. As it has been shown, many concepts from mathematics, physics, and
programming languages get involved in this project, converting it in a very transversal one, and
requiring large amounts of time not only in its programming but also in the understanding of the
theory lying behind it, as well as some mathematical tools that were essential throughout the whole
process.
One of the most difficult parts of the development process was the debugging of the code. Since the
program is like a black box, where, as a programmer, one only knows the inputs and outputs, but not
the intermediate values the program should produce. Thus, finding out the mistakes was really
tedious. Nevertheless, learning how to proceed with problems of this kind was useful, and surely will
be in the future. An additional drawback for the debugging part were the high times of computing the
program took to execute, that forced to make some blind changes before executing the program and
checking if they were correct. In this regard, having programmed the electrostatic problem first
provided some clues for the future development of the electrodynamic problem to understand why
and/or where a programmed Method of Moments may fail, and speeding up the debugging process.
In the results section, many figures with different characteristics (soft edged, hard edged, symmetric,
non-symmetric, etc.) were simulated and compared with the expected values obtained analytically or
measured. These comparisons showed accurate results and can be taken as good approximations to the
RCS of the object that is being simulated, as long as a thin enough mesh is selected. With these
results, the RCS of an object of a desired shape can be simulated with the guarantee of accurate results
before its building, saving great costs and time in the process of design.
To summarize, although it took a long time and effort to build and debug a program of these
characteristics, the results are by far satisfactory because the fact that the objective was to solve a
generic problem, not just 2D or body of revolution problems, makes this program a very powerful
one, with many future improvements that can expand its possibilities even further as it is explained in
4.2.
43
4.2. FUTURE LINES
There are many future lines that can be developed to improve this program, expanding its applications
or getting more accurate results:
One future line can be the implementation of the Magnetic Field Integral Equation (MFIE),
which can be used to analyze cavities, while the EFIE is useful for calculation of thin or open
surfaces. Once the MFIE has been implemented, in order to obtain better results, a weighted
combination of both can be used, yielding the Combined Field Integral Equation (CFIE)
𝐶𝐹𝐼𝐸 = 𝛼 𝐸𝐹𝐼𝐸 +𝑗
𝑘(1 − 𝛼)𝑀𝐹𝐼𝐸 (148)
with 𝛼 chosen between 0.2 and 0.5 to eliminate spurious solutions.
Another interesting future line to expand the applications of the program beyond the
determination of the RCS of an object could be the implementation of a delta-gap model for
the excitation of planar antennas, such as Vivaldi antennas, or Dipole antennas. The delta-gap
model is based on a feed generating an electric field in a small space (edge 𝑚). The excitation
vector in (132) is for this case reduced to:
𝑏𝑚 = −𝑗
𝜔𝜇𝐿𝑚𝑉𝑖𝑛 (149)
where 𝐿𝑚 is the length of edge 𝑚 and 𝑉𝑖𝑛 is the input voltage of the source. Once the currents
are obtained, the radiation for any point of the space can be obtained, as well as the radiation
pattern. Another interesting use for this implementation is the knowledge of the input
impedance, which can be obtained as:
𝑍𝑖𝑛 =𝑉𝑖𝑛
𝐼𝑖𝑛 (150)
where 𝐼𝑖𝑛 is known once the current distribution is solved since
𝐼𝑖𝑛 = 𝐿𝑚𝛼𝑚 (151)
being 𝛼𝑚 the current density across edge 𝑚.
One first development for enhancing performance of the program might be to optimize the
speed of the program with multithread programming, allowing to fill the impedance matrix
much faster (in a quad core processor, four times faster), as well as calculating the scattering
for multiple space points at the same time. The next step to improve greatly the size of the
objects that are being computed is the implementation of the Fast Multipole Method (FMM),
which treats far objects clusters as if they were a single particle, filling the matrix faster and
requiring less memory to be used, allowing shapes of larger size to be analyzed.
44
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