Development of a System for the Computation of Electromagnetic Wave Scattering from Non-Penetrable Objects by Solving the Electric Field Integral Equation

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(1)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 NONPENETRABLE OBJECTS BY SOLVING THE ELECTRIC FIELD INTEGRAL EQUATION. ALBERTO MONJE REAL 2016.

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(3) GRADO EN TECNOLOGÍAS TELECOMUNICACIÓN. Y. SERVICIOS. DE. 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….

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(5) 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 NONPENETRABLE OBJECTS BY SOLVING THE ELECTRIC FIELD INTEGRAL EQUATION. ALBERTO MONJE REAL 2016.

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(7) 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..

(8) 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..

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(10) 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.

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(12) 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..

(13) 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.

(14) 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) + 7𝑓(𝑥) 𝑑𝑥. (3). ℒ [𝑓(𝑥)] = ∭ 4 𝑓(𝑥) 𝑑𝑉. (4). 𝑉. 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: 𝑁. 𝑓(𝑥) ≈ ∑ 𝛼𝑛 𝑓𝑛 (𝑥). (5). 𝑛=1. And, substituting (5) in (1), 𝑁. 𝑁. ℒ [𝑓(𝑥)] ≈ ℒ [∑ 𝛼𝑛 𝑓𝑛 (𝑥)] = ∑ 𝛼𝑛 ℒ[𝑓𝑛 (𝑥)] ≈ 𝑔(𝑥) 𝑛=1. (6). 𝑛=1. We define the residual r(x) as: 𝑁. 𝑁. 𝑟(𝑥) = ℒ [𝑓(𝑥)] − ℒ [∑ 𝛼𝑛 𝑓𝑛 (𝑥)] = 𝑔(𝑥) − ℒ [∑ 𝛼𝑛 𝑓𝑛 (𝑥)] 𝑛=1. (7). 𝑛=1. 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)..

(15) 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: 𝑁. 𝑁. < 𝑤𝑚 , ℒ [∑ 𝛼𝑛 𝑓𝑛 (𝑥)] > = ∑ 𝛼𝑛 < 𝑤𝑚 , ℒ[𝑓𝑛 (𝑥)] > ≈ < 𝑤𝑚 , 𝑔(𝑥) > 𝑛=1. (8). 𝑛=1. This yields a system that we can see in a matrix form: 𝑍11 𝑍21 ( ⋮ 𝑍𝑁1. 𝑍12 𝑍22 ⋮ 𝑍𝑁2. 𝛼1 𝑍1𝑁 𝑏1 𝛼2 𝑍2𝑁 𝑏2 )·( ⋮ )=( ) , ⋮ ⋮ 𝛼 𝑍𝑁𝑁 𝑏𝑁 𝑁. ⋯ ⋯ ⋱ ⋯. (9). where 𝑍𝑚𝑛 = < 𝑤𝑚 , ℒ[𝑓𝑛 (𝑥)] >. (10). 𝑏𝑚 = < 𝑤𝑚 , 𝑔(𝑥) > and where we define the dot product as: 𝑏. < 𝑓(𝑥), 𝑔(𝑥) > = ∫ 𝑓(𝑥)𝑔∗ (𝑥) 𝑑𝑥,. 𝑎≤𝑥≤𝑏. (11). 𝑎. 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: 𝑑𝑓(𝑥) ℒ [𝑓(𝑥)] = −2 = 𝑥3 + 2 , 𝑑𝑥 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.

(16) 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 Ndimensional 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..

(17) 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) = ∬ 𝑠. 𝜌(𝑥 ′ , 𝑦 ′ ) 𝑑𝑥 ′ 𝑑𝑦′ 4𝜋𝜀0 𝑅. In our particular situation, where the capacitor is formed by two parallel plates: 𝜌(𝑥 ′ , 𝑦 ′ ) 𝑑𝑥 ′ 𝑑𝑦 ′ , 𝑈𝑃+𝐵𝑃 4𝜋𝜀0 𝑅. 𝜙(𝑥, 𝑦, 𝑧) = ∬. (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: 𝑉 , 𝑔={ 2 𝑉 − , 2. (𝑥, 𝑦, 𝑧) ∈ 𝑈𝑃 (15) (𝑥, 𝑦, 𝑧) ∈ 𝐵𝑃. Continuing with the process, f(x) must be approximated as in (5), in this case by N squares of side length 2b (surface Δ𝑠 = 4𝑏 2 ), as it is described in Figure 2.1. Parallel Plate Capacitor Geometry and supposing that the charge surface density is constant in each square: 𝑁 ′. 𝜌(𝑥 , 𝑦. ′). = ∑ 𝛼𝑛 𝑓𝑛. (16). 𝑛=1. where we define fn constant in the surface of the nth square, i.e.: 𝑓𝑛 = {. 1, 0,. 𝑜𝑛 Δ𝑠𝑛 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒. (17). Finally, this is the resulting system: 𝑉 ′ ′ , ∑𝑁 𝑛=1 𝛼𝑛 𝑓𝑛 (𝑥 , 𝑦 ) ∬ 𝑑𝑥 ′ 𝑑𝑦 ′ = { 2 𝑉 4𝜋𝜀0 𝑅 𝑈𝑃+𝐵𝑃 − , 2. (𝑥, 𝑦, 𝑧) ∈ 𝑈𝑃 (18) (𝑥, 𝑦, 𝑧) ∈ 𝐵𝑃. And it can be rewritten as: 𝑉 , 𝑓𝑛 (𝑥 ′ , 𝑦 ′ ) 2 ′ ′ ∑ 𝛼𝑛 ∬ 𝑑𝑥 𝑑𝑦 = { 𝑉 𝑈𝑃+𝐵𝑃 4𝜋𝜀0 𝑅 𝑛=1 − , 2 𝑁. (𝑥, 𝑦, 𝑧) ∈ 𝑈𝑃 (19) (𝑥, 𝑦, 𝑧) ∈ 𝐵𝑃.

(18) 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 𝑉 𝑉 deltas in the center of each Δ𝑠𝑛 . This way, in those points, the potential will be or − depending on 2 2 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) 𝜋𝜀0. (20). Finally, when the system is solved and the 𝛼𝑛 coefficients 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: 𝑁 ′. 𝑄 = ∬ 𝜌(𝑥 , 𝑦. ′ )𝑑𝑥′𝑑𝑦′. 𝑈𝑃. ≈ ∑ 𝛼𝑛 𝑓𝑛 Δ𝑠𝑛. (21). 𝑛=1. 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.

(19) 8. RESULTS AND SIMULATIONS. Figure 2.2. Charge density distribution in the capacitor, for 20x20 and 60x60 mesh sizes..

(20) 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=. 𝐴𝜀 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 · 106 elements, 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..

(21) 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 + 𝑗𝜔𝜀𝐄. (25). ∇ ∙ 𝐃 = 𝑞𝑒. (26). ∇ ∙ 𝐁 = 𝑞𝑚. (27). Being 𝐃 = 𝜀𝐄 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 𝑒 𝑗𝜔𝑡 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 −𝐧. (28). ̂𝐧 × (𝐇2 − 𝐇1 ) = 𝐉s. (29). ̂𝐧 ∙ (𝐃2 − 𝐃1 ) = 𝑞𝑒. (30). ̂ ∙ (𝐁2 − 𝐁1 ) = 𝑞𝑚 𝐧. (31). ̂ is the normal vector to the surface boundary that points from region 2 to region 1. Where 𝐧 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:.

(22) 11. ̂ × 𝐄2 = 0 −𝐧. (32). ̂ × 𝐇 2 = 𝐉𝑠 𝐧. (33). ̂𝐧 ∙ 𝐃2 = 𝑞𝑒. (34). ̂ ∙ 𝐁2 = 0 𝐧. (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). ∇ × ∇ × 𝐄 = 𝜔2 𝜇𝜀𝐄 − 𝑗𝜔𝜇𝐉. (37). Substituting (25) in (36) yields:. And taking the first term in the right hand side of (37) to the left hand side, ∇ × ∇ × 𝐄 − 𝜔2 𝜇𝜀𝐄 = −𝑗𝜔𝜇𝐉. (38). ∇ × ∇ × 𝐄 = ∇(∇ ∙ 𝐄) − ∇2 𝐄. (39). ∇(∇ ∙ 𝐄) − ∇2 𝐄 − 𝑘 2 𝐄 = −𝑗𝜔𝜇𝐉. (40). Knowing the vector identity. we can rewrite (38) as:. where 𝑘 = 𝜔√𝜇𝜀 =. 2𝜋 𝜆. is the wavenumber.. Substituting (26) in the above yields ∇2 𝐄 + 𝑘 2 𝐄 = 𝑗𝜔𝜇𝐉 + ∇. 𝑞𝑒 𝜀. (41). The relationship between the electric surface current and the electric charge is the equation of continuity: ∇ ∙ 𝐉 = −𝑗𝜔𝑞𝑒. that in (41) yields:. (42).

(23) 12. ∇2 𝐄 + 𝑘 2 𝐄 = 𝑗𝜔𝜇𝐉 −. 1 1 ∇(∇ ∙ 𝐉) = 𝑗𝜔𝜇(𝐉 + 2 ∇(∇ ∙ 𝐉)) 𝑗𝜔𝜀 𝑘. (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: ∇2 𝐺(𝐫, 𝐫 ′ ) + 𝑘 2 𝐺(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 ′ ) + 𝑉. 1 ∇′∇′ ∙ 𝐉(𝐫 ′ )] 𝑑𝐫′ 𝑘2. (45). THE GREEN’S FUNCTION In order to complete equation (45), the Green’s function must be obtained via solving the threedimension 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: ∇2 𝐺 =. 1 𝑑 2 𝑑𝐺 𝑑 2 𝐺 2 𝑑𝐺 1 𝑑 2 (𝑟𝐺) (𝑟 ) = + = 𝑟 2 𝑑𝑟 𝑑𝑟 𝑑𝑟 2 𝑟 𝑑𝑟 𝑟 𝑑𝑟 2. (46). Substituting the last term in (44), 1 𝑑 2 (𝑟𝐺) 𝑑 2 (𝑟𝐺) 2 + 𝑘 𝐺 = 0 => + 𝑘 2 (𝑟𝐺) = 0 𝑟 𝑑𝑟 2 𝑑𝑟 2. (47). To solve this homogeneous system, we try a solution of the form of: 𝐴𝑒 𝑠𝑟 𝑟. (48). 𝐴𝑒 𝑠𝑟 𝑠𝑟 ) 𝑟 + 𝑘 2 (𝑟 𝐴𝑒 ) = 0 => 𝑠 2 𝐴𝑒 𝑠𝑟 + 𝑘 2 𝐴𝑒 𝑠𝑟 = 0 => 𝑠 = ±𝑗𝑘 𝑑𝑟 2 𝑟. (49). 𝐺=. Resulting in: 𝑑 2 (𝑟. So the solution for the homogeneous part is: 𝐺=𝐴. 𝑒 −𝑗𝑘𝑟 𝑒 𝑗𝑘𝑟 +𝐵 𝑟 𝑟. (50).

(24) 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=𝐴. 𝑒 −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: 𝑒 −𝑗𝑘𝑟 𝑒 −𝑗𝑘𝑟 ] 𝑑𝑉 = ∭ −𝛿(𝒓, 𝒓′ ) 𝑑𝑉 = − 1 𝐴 ∭ [∇ ∙ ∇ ( ) + 𝑘2 𝑟 𝑟 𝑉 𝑉. (52). Using the Gauss theorem to solve the first term in the integral: 𝑒 −𝑗𝑘𝑟 𝑒 −𝑗𝑘𝑟 ̂ ∙ ∇( ∭ ∇ ∙ ∇( ) 𝑑𝑉 = ∬ 𝒏 ) 𝑑𝑆 𝑟 𝑟 𝑉 𝑆. (53). ̂ = 𝒓̂, and substituting in the above equation: Since our surface is a sphere,𝐧 𝑒 −𝑗𝑘𝑟 𝜕 𝑒 −𝑗𝑘𝑟 ∬ 𝐫̂ ∙ ∇ ( ) 𝑑𝑆 = ∬ ( ) 𝑑𝑆 𝑟 𝑟 𝑆 𝑆 𝜕𝑟. (54). 𝜕 𝑒 −𝑗𝑘𝑟 4𝜋𝑎2 [ ( )] 𝜕𝑟 𝑟 𝑟=𝑎. (55). and integrating, it yields:. To solve this, in the limit when 𝑎 → 0, (55) results lim 4𝜋𝑎. 𝑎→0. 𝜕 𝑒 −𝑗𝑘𝑟 ( )] = −4𝜋 𝜕𝑟 𝑟 𝑟=𝑎. 2[. (56). Evaluating the second term of the integral in (52): ∭ [𝑘 2 𝑉. 𝑎 −𝑗𝑘𝑟 𝑎 𝑒 −𝑗𝑘𝑟 𝑒 ] 𝑑𝑉 = 𝑘 2 ∫ 4𝜋𝑟 2 𝑑𝑟 = 4𝜋𝑘 2 ∫ 𝑟𝑒 −𝑗𝑘𝑟 𝑑𝑟 𝑟 𝑟 0 0. (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).

(25) 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). ∇ × ∇ × 𝐀 = ∇(∇ ∙ 𝐀) − ∇2 𝐀. (61). With this vector calculus identity. and taking the curl of both sides of (60) μ ∇ × 𝐇 = ∇ × ∇ × 𝐀 = ∇(∇ ∙ 𝐀) − ∇2 𝐀. (62). We now substitute (62) in (25), that results 𝜇𝐉 + 𝑗𝜔𝜇𝜀𝐄 = ∇(∇ ∙ 𝐀) − ∇2 𝐀. (63). On the other hand, substituting (60) in (24): ∇ × 𝐄 = −𝑗𝜔∇ × 𝐀 => ∇ × (𝐄 + 𝑗𝜔𝐀) = 0. (64). ∇ × (−∇Ф𝑒 ) = 0. (65). and using the identity. where −∇Ф𝑒 is an arbitrary electric scalar potential, this yields: 𝐄 = −𝑗𝜔𝐀 − ∇Ф𝑒. (66). Substituting the above in (63), leads to 𝜇𝐉 + 𝑗𝜔𝜇𝜀(−𝑗𝜔𝐀 − ∇Ф𝑒 ) = ∇(∇ ∙ 𝐀) − ∇2 𝐀. (67). which with some rearrangements yields: ∇2 𝐀 + 𝑘 2 𝐀 = −𝜇𝐉 + ∇(∇ ∙ 𝐀 + 𝑗𝜔𝜇𝜀Ф𝑒 ). (68).

(26) 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 = −𝑗𝜔𝜇𝜀Ф𝑒. (69). This definition applied to (67) gets it simplified: ∇2 𝐀 + 𝑘 2 𝐀 = −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 𝐉(𝐫 𝑉. ′ )𝑑𝐫 ′. 𝑒 −𝑗𝑘|𝐫−𝐫 | = 𝜇 ∭ 𝐉(𝐫 ) 𝑑𝐫′ 4𝜋|𝐫 − 𝐫 ′ | 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 ∇(∇ ∙ 𝐀) 𝜔𝜇𝜖. (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: 𝐄 𝑆 (𝐫) = −𝑗𝜔𝜇 ∬ 𝐺(𝐫, 𝐫 ′ ) [𝐉(𝐫 ′ ) + 𝑆. 1 ′ ′ ∇ ∇ ∙ 𝐉(𝐫 ′ )] 𝑑𝐫′ 𝑘2. (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) × 𝐄 𝑇 (𝐫) = 𝐧 ̂(𝐫) × (𝐄 𝑆 (𝐫) + 𝐄 𝑖 (𝐫)) = 0 𝐧. (74). ̂(𝐫) × 𝐄 𝑆 (𝐫) = −𝐧 ̂(𝐫) × 𝐄 𝑖 (𝐫) 𝐧. (75). This yields to:. Therefore, taking the cross product at both sides of (73) and substituting the above: −. 𝑗 1 ̂(𝐫) × 𝐄 𝑖 (𝐫) = 𝐧 ̂(𝐫) × ∬ 𝐺(𝐫, 𝐫 ′ ) [𝐉(𝐫 ′ ) + 2 ∇′ ∇′ ∙ 𝐉(𝐫 ′ )] 𝑑𝐫′ 𝐧 𝜔𝜇 𝑘 𝑆. (76).

(27) 16 We can express this equation with the formerly explained magnetic vector potential, which is a common practice: −. 𝑗 1 ̂(𝐫) × 𝐄 𝑖 (𝐫) = 𝐧 ̂(𝐫) × [𝐀(𝐫) + 2 ∇∇ ∙ 𝐀(𝐫)] 𝐧 𝜔𝜇 𝑘. (77). We can finally make use of the tangent vector instead of the normal vector to the surface: −. 𝑗 1 [𝐭(𝐫) ∙ 𝐄 𝑖 (𝐫)] = 𝐭(𝐫) ∙ ∬ [1 + 2 ∇∇ ∙] 𝐉(𝐫 ′ )𝐺(𝐫, 𝐫 ′ )𝑑𝐫′ 𝜔𝜇 𝑘 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. 2𝐷 𝑅> or 𝑅 ≫ 𝜆, being R the distance of the object to the source, and D the maximum linear 𝜆 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′ = 𝒓̂ · 𝒓′ |𝑟|. (80).

(28) 17. Then, the far field approximation will be 𝑅 = 𝑟 − 𝒓̂ · 𝒓′. (81). 𝑅=𝑟. (82). for phase variations, and. for amplitude variations. 1. The radiated field expressed in (72) expresses fields that vary in the form in the first term of the 𝑟 right hand side, whilst the second term of the right hand side expresses fields that vary in the forms of 1 1 and 3 or superior. Given that the observation point is very far from the source, the attenuation of 𝑟2 𝑟 the second term of the right hand side of that expression will be much greater, and thus only the 1 contributions of the fields that attenuate in the form of will prevail significantly. Therefore, we can 𝑟 express the radiated field as: 𝐄(𝐫) = −𝑗𝜔𝐀(𝐫). (83). Then, using the approximations explained, and with equation (71), the radiated field finally is: 𝑗𝜔𝜇 𝑒 −𝑗𝑘(r − 𝐫 ∭ 𝐉(𝐫 ′ ) 𝐄(𝐫) = − 4𝜋 𝑟 𝑉. ′ ∙𝐫 ̂). 𝑑𝐫′. (84). 𝑗𝜔𝜇 𝑒 −𝑗𝑘𝑟 ′ ∭ 𝐉(𝐫 ′ )𝑒 𝑗𝑘𝐫 ∙𝐫̂ 𝑑𝐫′ 4𝜋 𝑟 𝑉. (85). which can be simplified to 𝐄(𝐫) = −. Finally, the RCS can be calculated, knowing the scattered and the incident field as: |𝐄 𝑆 |2 𝜎 = 4𝜋𝑟 = 4𝜋𝑟 2 |𝐄 𝑆 |2 |𝐄 𝑖 |2 2. for |𝐄 𝑖 | = 1, which is usually chosen for computation easiness.. (86).

(29) 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. 𝜆 Therefore, the surface shall be meshed in triangular elements (Recommended size of the mesh < ). 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)..

(30) 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: 𝑁. 𝐉(𝐫) = ∑ 𝑎𝑛 𝐟𝑛 (𝐫). (87). 𝑛=1. 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: 𝐿𝑛 + 𝝆𝑛 (𝐫) 2𝐴+ 𝑛 𝐿𝑛 (𝐫) 𝐟𝑛 (𝐫) = − 𝝆− 2𝐴𝑛 𝑛 𝐟𝑛 (𝐫) = 0. 𝐟𝑛 (𝐫) =. , 𝑓𝑜𝑟 𝒓 𝑖𝑛 𝑇𝑛 +. (88). , 𝑓𝑜𝑟 𝒓 𝑖𝑛 𝑇𝑛 −. (89). , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (90). − where 𝑇𝑛 + and 𝑇𝑛 − are the triangles (of area 𝐴+ 𝑛 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 , 𝐴+ 𝑛. 𝐿𝑛 , 𝐴− 𝑛 ∇𝑠 ∙ 𝐟𝑛 (𝐫) = 0,. ∇𝑠 ∙ 𝐟𝑛 (𝐫) =. 𝑓𝑜𝑟 𝒓 𝑖𝑛 𝑇𝑛 +. (91). 𝑓𝑜𝑟 𝒓 𝑖𝑛 𝑇𝑛 −. (92). 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒. (93).

(31) 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: 𝑁. 𝑗 1 𝑒 −𝑗𝑘𝑟 − 𝐭̂(𝐫) ∙ 𝐄 𝑖 (𝐫) = ∬ (1 + 2 ∇∇ ∙) ∑ 𝑎𝑛 𝐟𝑛 (𝐫 ′ ) 𝑑𝐫′ 𝜔𝜇 𝑘 4𝜋𝑟 𝑆. (94). 𝑛=1. 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 𝐭̂(𝐫) ∙ 𝐄 𝑖 (𝐫)) 𝑑𝒓 𝜔𝜇 𝑁. 1 𝑒 −𝑗𝑘𝑟 = ∬ 𝐟𝑚 (𝒓) · (∬ (1 + 2 ∇∇ ∙) ∑ 𝑎𝑛 𝐟𝑛 (𝐫 ′ ) 𝑑𝐫′) 𝑑𝒓 𝑘 4𝜋𝑟 𝑆 𝑆 𝑛=1. (95).

(32) 21 which, for just a 𝑚 and a 𝑛 element is: 𝑗 𝐭̂(𝐫) ∙ 𝐄 𝑖 (𝐫)) 𝑑𝒓 𝜔𝜇 𝑓𝑚 1 𝑒 −𝑗𝑘𝑟 = ∬ 𝐟𝑚 (𝒓) · (∬ (1 + 2 ∇∇ ∙)𝑎𝑛 𝐟𝑛 (𝐫 ′ ) 𝑑𝐫′) 𝑑𝒓 𝑘 4𝜋𝑟 𝑓𝑚 𝑓𝑛 ∬ 𝐟𝑚 (𝒓) · (−. (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) · (∬ (1 + 𝑓𝑚. 𝑓𝑛. 1 𝑒 −𝑗𝑘𝑟 ′ ∇∇ ∙) 𝐟 (𝐫 ) 𝑑𝐫′) 𝑑𝒓 𝑛 𝑘2 4𝜋𝑟. (97). can be rewritten as: 1 𝑒 −𝑗𝑘𝑟 𝑧𝑚𝑛 = [∬ ∬ 𝐟𝑛 (𝐫 ′ ) · 𝐟𝑚 (𝒓)𝑑𝒓′ 𝑑𝒓 + ∬ 𝐟𝑚 (𝐫) · ( 2 ∇∇ ∙ ∬ 𝐟𝑛 (𝐫 ′ ) 𝑑𝐫′) 𝑑𝒓] 𝑘 4𝜋𝑟 𝑓𝑚 𝑓𝑛 𝑓𝑚 𝑓𝑛. (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 ∇∇ ∙ ∬ 𝐟𝑛 (𝐫 ′ )𝐺(𝐫, 𝐫 ′ )𝑑𝐫′) 𝑑𝒓 𝑘2 𝑓𝑛. (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: 1 ∬ 𝐟𝑚 (𝐫) · ( 2 ∇ ∬ [∇𝐺(𝐫, 𝐫 ′ )] · 𝐟𝑛 (𝐫 ′ )𝑑𝐫′) 𝑑𝒓 𝑘 𝑓𝑚 𝑓𝑛. (101). Because of the symmetry of the Green’s function: ∇𝐺(𝐫, 𝐫 ′ ) = − ∇′𝐺(𝐫, 𝐫 ′ ). (102). Then, (101) is: ∬ 𝐟𝑚 (𝐫) · (− 𝑓𝑚. 1 ∇ ∬ [∇′𝐺(𝐫, 𝐫 ′ )] · 𝐟𝑛 (𝐫 ′ )𝑑𝐫′) 𝑑𝒓 𝑘2 𝑓𝑛. If the vector identity in (100) is used to solve the gradient in the above, it yields:. (103).

(33) 22 1 ∬ 𝐟𝑚 (𝐫) · ( 2 [∇ ∬ 𝐺(𝐫, 𝐫 ′ )∇′ · 𝐟𝑛 (𝐫 ′ )𝑑𝐫 ′ − ∇ ∬ ∇′ ∙ [ 𝐟𝑛 (𝐫 ′ )𝐺(𝐫, 𝐫 ′ )]𝑑𝐫 ′ ] ) 𝑑𝒓 𝑘 𝑓𝑚 𝑓𝑛 𝑓𝑛. (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 ∇ ∬ 𝐺(𝐫, 𝐫 ′ )∇′ · 𝐟𝑛 (𝐫 ′ )𝑑𝐫 ′ ) 𝑑𝒓 = ∬ 𝐟𝑚 (𝐫) · ∇F(𝐫)d𝐫 𝑘2 𝑓𝑛 𝑓𝑚. (105). Making use once more of the vector identity: 𝐟𝑚 (𝐫) · ∇F(𝐫) = ∇ · [ 𝐟𝑚 (𝐫)F(𝐫)] − [∇ · 𝐟𝑚 (𝐫)]F(𝐫). (106). ∬ 𝐟𝑚 (𝐫) · ∇F(𝐫)d𝐫 = ∬ ∇ · [ 𝐟𝑚 (𝐫)F(𝐫)] d𝐫 − ∬ [∇ · 𝐟𝑚 (𝐫)]F(𝐫) d𝐫. (107). 𝑓𝑚. 𝑓𝑚. 𝑓𝑚. 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) · ∇F(𝐫)d𝐫 = ∬ ∇ · 𝐟𝑚 (𝐫) · (− 𝑓𝑚. 𝑓𝑚. 1 ∬ 𝐺(𝐫, 𝐫 ′ )∇′ · 𝐟𝑛 (𝐫 ′ )𝑑𝐫 ′ ) 𝑑𝒓 𝑘 2 𝑓𝑛. (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 ′ ) · 𝐟𝑚 (𝒓)𝑑𝒓′𝑑𝒓 − 𝑓𝑚. 𝑓𝑛. 1 𝑒 −𝑗𝑘𝑟 ′ ′ )][ [ (𝐫 (𝐫)] ∇ · 𝐟 ∇ · 𝐟 𝑑𝒓′𝑑𝒓] 𝑛 𝑚 𝑘2 4𝜋𝑟. (109). Substituting the RWG functions and their divergences, as calculated in (88), (89), (91) and (92): 𝑧𝑚𝑛 =. 𝐿𝑚 𝐿𝑛 1 1 𝑒 −𝑗𝑘𝑟 ± ′ ∬ ∬ [ 𝝆± 𝑑𝐫′𝑑𝐫 𝑚 (𝐫) ∙ 𝝆𝑛 (𝐫 ) ± 2 ] 𝐴𝑚 𝐴𝑛 𝑇𝑚 𝑇𝑛 4 𝑘 4𝜋𝑟. (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: 𝑀. 𝑧𝑚𝑛. 𝑀. 𝐿𝑚 𝐿𝑛 1 1 𝑒 −𝑗𝑘𝑅𝑝𝑞 ∓ ′ ] ≈ ∑ ∑ 𝑤𝑝 𝑤𝑞 [ 𝝆± (𝐫 ) ∙ 𝝆 (𝐫 ) ± 𝑛 𝑞 4𝜋 4 𝑚 𝑝 𝑘 2 𝑅𝑝𝑞 𝑝=1 𝑞=1. (111).

(34) 23 It is important to notice that, since the divergence sign is different depending on whether the function 1 is defined on 𝑇 + or 𝑇 − as shown in (91) and (92), the sign of 2 will be negative if both testing and 𝑘 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. 1. To extract the singularity, adding and subtracting to the Green’s function yields: 𝑟. 𝑒 −𝑗𝑘𝑟 𝑒 −𝑗𝑘𝑟 1 1 =[ − ]+ 𝑟 𝑟 𝑟 𝑟. (113). Where the first term of the right hand side can be directly solved: 𝑒 −𝑗𝑘𝑟 1 lim [ − ] = −𝑗𝑘 𝑟→0 𝑟 𝑟. (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: ± ′ 1 𝐼1 = ∬ 𝝆± 𝑚 (𝐫) ∙ ∬ 𝛒𝑛 (𝐫 ) 𝑑𝐫′𝑑𝐫 𝑟 𝑇 𝑇′ 1 𝐼2 = ∬ ∬ 𝑑𝐫′𝑑𝐫 𝑇 𝑇′ 𝑟. (115) (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 1. The Duffy transform is a widely used method to solve integrals on a triangle with a singularity at its 𝑟 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..

(35) 24. Figure 2.7. Geometric explanation of the Duffy transform. The Duffy transform converts the integral in one of the kind: 1 1 𝑓(𝐫 ′ ) ′ 𝑓(𝑢, 𝛾) |𝐽(𝑢, 𝛾)| 𝑑𝑢 𝑑𝛾 ∫ ∫ 𝑑𝐫 = ′ 𝑇 𝑟(𝐫 ) 0 0 𝑟(𝑢, 𝛾). ∬. (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 1 |𝐽(𝑢, 1 𝛾)| 𝑑𝐫′ = ∫ ∫ 𝑑𝑢 𝑑𝛾 𝑇′ 𝑟 0 0 𝑟(𝑢, 𝛾). ∬. (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).

(36) 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: 2. 𝑎(𝑢, 𝛾) = (1 − 𝑢)2 [𝑥1 − [(1 − 𝛾)𝑥2 + 𝛾𝑥3 ]]. (127). Proceeding in a similar way: 2. 𝑏(𝑢, 𝛾) = (1 − 𝑢)2 [𝑦1 − [(1 − 𝛾)𝑦2 + 𝛾𝑦3 ]]. 2. 𝑐(𝑢, 𝛾) = (1 − 𝑢)2 [𝑧1 − [(1 − 𝛾)𝑧2 + 𝛾𝑧3 ]]. (128) (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: 1 1 ± 1 𝛒𝑛 (u, γ) ′ |𝐽(𝑢, 𝛾)|𝑑𝑢 𝑑𝛾 ∬ 𝛒± ∫ ∫ (𝐫 ) 𝑑𝐫′ = 𝑛 𝑟 𝑇′ 0 0 𝑟(𝑢, 𝛾). (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 𝛒± 𝑛 (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.

(37) 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 𝐭̂(𝐫) ∙ 𝐄 𝑖 (𝐫) 𝜔𝜇. (131). This equation, when dot multiplied with the test functions yields, for each test function: 𝑏𝑚 = −. 𝑗 ∬ 𝐟 (𝐫) ∙ 𝐄 𝑖 (𝐫)𝑑𝐫 𝜔𝜇 𝐟𝑚 𝑚. (132). that, when extracting the constants, yields: 𝑏𝑚 = −. 𝑗 𝐿𝑚 ∬ 𝝆± (𝐫) ∙ 𝐄 𝒊 (𝐫)𝑑𝐫 𝜔𝜇 2𝐴𝑚 𝑇𝑚 𝑚. (133).

(38) 27. This integral, as explained in the former point, can be numerically approximated using M Gauss quadrature points, resulting in the following: 𝑀. 𝑗 𝐿𝑚 𝑖 𝑏𝑚 = − ∑ 𝑤𝑝 𝝆± 𝑚 (𝐫𝑝 ) ∙ 𝐄 (𝐫𝑝 ) 𝜔𝜇 2. (134). 𝑝=1. where 𝐄 𝑖 (𝐫𝑝 ) is the incident electric field to the surface, and so it has a polarization, that has to be scalar multiplied with 𝝆± 𝑚 (𝐫𝑝 ), and a phase depending of the position of 𝐫𝑝 . The total electric field of the plane wave can be described as: 𝐄 𝑖 (𝐫𝑝 ) = 𝐸0 𝑒̂ (𝜃, 𝜙)𝑒 −𝑗𝑘𝒓̂·𝐫𝑝. (135). So the final excitation element is obtained as: 𝑀. 𝑗 𝐿𝑚 −𝑗𝑘𝒓̂·𝐫𝑝 𝑏𝑚 = − 𝐸 ∑ 𝑤𝑝 𝝆± 𝑚 (𝐫𝑝 ) ∙ 𝑒̂ (𝜃, 𝜙)𝑒 𝜔𝜇 2 0. (136). 𝑝=1. 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 𝑒 −𝑗𝑘𝑟 𝑎𝑚 𝐿𝑚 ′ 𝑗𝑘𝐫 ′ ∙𝐫̂ ∬ 𝝆± 𝑑𝐫′ 𝑚 (𝐫 )𝑒 4𝜋 𝑟 2 𝑇𝑚. (137). that, once more, can be computed numerically: 𝑀. 𝑗𝜔𝜇 𝑒 −𝑗𝑘𝑟 𝑎𝑚 𝐿𝑚 𝑗𝑘𝒓′𝒑 ∙𝐫̂ ′ 𝐄𝑚 (𝐫) ≈ − ∑ 𝑤𝑝 𝝆± 𝑚 (𝐫𝑝 )𝑒 4𝜋 𝑟 2. (138). 𝑝=1. 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..

(39) 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: 𝐴1 𝐴 𝐴2 𝛽= 𝐴 𝐴3 𝛾= 𝐴 and where 𝐴1 , 𝐴2 and 𝐴3 are the areas of the three triangles shown in Figure 2.9. 𝛼=. Figure 2.9. Components of the simplex coordinates. (140) (141) (142).

(40) 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 𝐴2 𝐴3 𝐴 + + = =1 𝐴 𝐴 𝐴 𝐴. (143). so, one component is usually expressed in terms of the other two: 𝛾 =1−𝛼−𝛽. (144). 𝐫 = (1 − 𝛼 − 𝛽)𝐯1 + 𝛼𝐯2 + 𝛽𝐯3. (145). Finally,. 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: 1. 1−𝛼. ∬ 𝑓(𝒓) 𝑑𝒓 = ∬ 𝑓(𝛼, 𝛽)|J(𝛼, 𝛽)| 𝑑𝛼 𝑑𝛽 = 2𝐴 ∫ ∫ 𝑇. 𝑇𝑐. 0. 𝑓(𝛼, 𝛽) 𝑑𝛼 𝑑𝛽. (146). 0. 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.

(41) 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: 𝑀. ∬ 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦 ≈ 𝐴 ∑ 𝑤(𝛼𝑞 , 𝛽𝑞 )𝑓(𝛼𝑞 , 𝛽𝑞 ) 𝑆. q. 𝛼. 1. 0.33333333. 2. (147). 𝑞=1. 𝛽. 𝛾. 𝑤. 0.33333333. 0.33333333. 0.225. 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.

(42) 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..

(43) 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 𝜆 the sphere was 0.006m and the frequency measures were at 5GHz (radius = ) and at 30GHz (radius =. 6𝜆 10. 10. ). The respective meshes can be seen in Figure 3.1.. Figure 3.1. Sphere geometry and meshes for 5GHz (left) and 30GHz (right) 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..

(44) 33. Figure 3.2. Bistatic radar cross section of a sphere of r =. Figure 3.3. Bistatic radar cross section of a sphere of r =. 𝝀 𝟏𝟎. 𝟔𝝀 𝟏𝟎.

(45) 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 VV 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.