2d electromagnetic field mom calculations using well...

Andries Denturck

well-conditioned higher order polynomials.2D Electromagnetic field MoM calculations using

Academiejaar 2012-2013Faculteit Ingenieurswetenschappen en ArchitectuurVoorzitter: prof. dr. ir. Daniël De ZutterVakgroep Informatietechnologie

Master in de ingenieurswetenschappen: toegepaste natuurkundeMasterproef ingediend tot het behalen van de academische graad van

Begeleider: Ignace BogaertPromotor: prof. dr. ir. Daniël De Zutter

Preface

Dear reader,

Before we get started with the serious part of this thesis, I would first like to thank you forinvesting some of your precious time in the lecture of this thesis. In this day and age, time isthe most valuable asset we hold, and it is an honour to have some of yours. However, beforewe really get started, I would like thank the following people who helped me in writing thisthesis.

First of all, I would like to thank Professor Daniel De Zutter, promotor of this thesis and headof the INTEC department of Ghent University. It is through his support, and the support ofINTEC’s Electromagnetics Group that this work ever saw the light of dawn.

Next and foremost, I would like to thank dr. ir. Ignace Bogaert. His supervising role wasmost crucial in the creation of this work and without him. What started out as weekly meet-ings on Tuesday mornings at 10 am, later emerged into sheer daily meetings, e-mails, phonecalls, Skype calls, etc... Time and again, he managed to stun me with his deep understandingof the matter. Tirelessly he tried to transfer to me his knowledge filling dozens of papers withhis idiosyncratic red scribblings. His office was always open to me, and I am ever grateful forall the discussions that took place there.

In view of this, I would also like to thank Dieter Dobbelaere and Yves Beghein. They occupiedthe same office, and soldiered on through all these at times heated discussions. Furthermore,they provided me with lots of useful tips and tricks, Octave-commandos and KDevelop sup-port.

Apart from all these hard working people at the INTEC, I would like to thank those hardworking people which are dearest to me: my parents. For 23 years they have cared for me,fed me, housed me, stimulated me and kept up with all my shenanigans; even when we werethousand miles apart. Special thanks go to my father, who helped editing this thesis.

Finally, I would like to thank the ravishing Marina Borovkova, who made sure I always keptsmiling.

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Permission to loan

The author gives permission to make this master dissertation available for consultation andto copy parts of this master dissertation for personal use. In the case of any other use, thelimitations of the copyright have to be respected, in particular with regard to the obligationto state expressly the source when quoting results from this master dissertation.

Andries Denturck June 3, 2013

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Faculty of Engineering & Architecture

Department of Information Technology

Chairman: Prof. dr. ir. D. De Zutter

2D Electromagnetic field MoM

calculations using well-conditioned

higher order polynomials.

Andries Denturck

Promotor: Prof. dr. ir. D. De Zutter

Supervisor: dr. ir. I. Bogaert

Master Thesis presented to achieve the academic degree of:

Master of Engineering: Applied Physics

Academic year 2012–2013

Keywords: 2DMethod of Moments, higher-order, preconditioning, Field

Integral Equations

2D Electromagnetic field MoMcalculations using well-conditoned

higher order polynomials.

Andries Denturck

Promotor: Prof. dr. ir. D. De ZutterSupervisor: dr. ir. I. Bogaert

Abstract

In this article we will present the results of a new, higher-order MoM solver, capable of handling2D Field Integral Equations in the frequency domain. We will especially dedicate our attention to thewell-conditioned bases which were used to reduce the influence of mesh refinement and polynomial erroron the EFIE condition number.Keywords: 2D Method of Moments, Field Integral Equations, higher-order, preconditioning.

I. Introduction

Traditionally, electromagnetic field Method ofMoments calculations are performed usingonly the lowest order polynomials. This resultsin equations that are easy to implement andlead to reasonable condition numbers. Improv-ing the accuracy of this method is done byrefining the mesh. Moreover, using FMM andMLFMA techniques, computational complex-ity can be reduced significantly, resulting in adecrease in run time.

The downside of this method is the rapidlyincreasing amount of unknowns when a highaccuracy is wished for. In those cases, onewould be better off using higher-order schemes,as these are more efficient in reducing theamount of unknowns in the MoM system.However, using traditional sets of polynomials,like the Legendre or Lagrange Polynomials, re-sults in high condition numbers, especially forthe EFIE. In this article we will show that aset of polynomials can be found for which thecondition number is well-behaved under the in-fluence of mesh refinement and/or polynomial

order.

II. Higher-Order solver.

A lot of time went first into programming ahigher-order solver using Legendre Polynomi-als. To this extent, the TM-EFIE, and MFIEequations were implemented. The solver wastested on numerical accuracy and used to solvescattering problems for a variety of geome-tries. Figure 1 shows the absolute value ofHz for a plane wave impinging on one half ofBernouilli’s Lemniscate.

Figure 1: Lemniscate: Total Field |Hz| (k=2)

1

III. Polynomial sets

The Legendre Polynomials are a decent basisfor Method of Moments calculations, especiallyfor calculating the MFIE. However, in somecases they are insufficient as in the exampleof the TE-EFIE. In that case, we prefer a basiswhich vanishes on segment edges. To this ex-tent, we used the following basis based on theAssociated Legendre Polynomials:

qn =P2

n+2(2t− 1)

‖P2n+2(2t− 1)‖ (1)

This basis was completed with an expan-sion on the mesh-nodes:

hn∧n+1(ρ) =

{hn/(ρ(t)) = c+g+(t),h\n+1(ρ(t)) = c−g−(t),

(2)

With:

g± =d

dx(N + 2)PN ± NPN+1√

4N(N + 1)2(N + 2)(3)

The Associate Legendre Polynomials to-gether with the nodal function were calledthe Continuous Base Line Functions (CBLF).They were used with succes as test and basisfunctions in a Galerkin-scheme for the TE-EFIEand TE-CFIE.

Higher order schemes, as we already men-tioned before, tend to suffer from large con-dition numbers. This was especially true forthe EFIE. It turns out that the TM-EFIE ker-nel is too smooth, whereas the TE-EFIE isnot smooth enough. The way we solved thisfor the TM-EFIE, is to test the smooth kernelwith non-smooth polynomials which were thederivative of a continuous basis, the CBLF. Inthis way, the Derived CBLF or DCBLF was con-structed. Because of a null-space in this basis,we had to further expand the DCBLF with aconstant function. Moreover, this testing was

actually done implicitly using a preconditioner.

For the TE-EFIE, we looked for testing func-tions which were smoother than the CBLF.Eventually, for n > 0, a basis of IntegratedLegendre polynomials was chosen:

φn(t) =∫ t

0

Pn(2x− 1)‖Pn(2x− 1)‖dx (4)

This basis was extended with linear hatfunctions spanning several segments, as seenon figure 2.

Figure 2: Hat ExpansionThey are given by the following equation:

hn =

{i− 1 + t t ∈ [0, 1] on segment in(1− t) t ∈ [0, 1] on segment n+1

(5)

IV. Results

We calculated impedance matrices for theLemniscate, at 10 different frequencies for 5levels of refinement and polynomial order

from 1 to 10. On Figure 3 and 4 the conditionnumbers of the TE and TM EFIE were plottedas a function of the spatial frequency k. Thered lines are the condition numbers of the

non-preconditioned matrices, whereas the redlines are from the preconditioned ones. What

we notice is that for the preconditioner, the

2

lines cluster in function of mesh refinement.This is most obvious for the TM case, wherewe see 5 thick lines, corresponding to the 5

levels of refinement. These are by themselvesthe superposition of 10 lines corresponding tothe polynomial order. Thus, compared to the

non-preconditioned system, thepreconditioner stabilizes the influence of mesh

refinement and polynomial order on thecondition number. However, the effectiveness

of the preconditioner is still frequencydependent. At high frequencies, the condition

number of the preconditoned curve evenexceeds that of the non-preconditioned one.

This frequency dependency can be the subjectof further research.

Figure 3: Lemniscate TM: κ

Figure 4: Lemniscate TE: κ

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2D Elektromagnetische MoMveldberekeningen met behulp vangoed geconditioneerde hogere orde

polynomen

Andries Denturck

Promotor: Prof. dr. ir. D. De ZutterBegeleider: dr. ir. I. Bogaert

Abstract

In dit artikel presenteren we de resultaten bekomen met een hogere orde solver die de momenten-methode gebruikt voor het oplossen van 2D integraalvergelijkingen voor electromagnetische velden inhet frequentiedomein. We richten onze aandacht specifiek op goed geconditioneerde basissen die gebruiktzullen worden om de invloed van rasterverfijning en polynomiale order op het EFIE conditie nummerte verminderen. Sleutelwoorden: 2D Momentenmethode, veldintegraalvergelijkingen, hogere orde,pre-conditionering.

I. Inleiding

Traditioneel worden veldberekeningen met demomenten methode beperkt tot lagere ordeveeltermen. Dit resulteert in vergelijkingen diegemakkelijk te implementeren zijn en tot re-delijke conditiegetallen leiden. Verbetering vande nauwkeurigheid van deze methode wordtbekomen door het raster te verfijnen. Daarbovenop worden FMM en MLFMA techniekengebruik om de resulterende computationelecomplexiteit te reduceren.

De voorgaande techniek is het nadeel vande snelle toename van het aantal onbekendenwanneer een betere nauwkeurigheid vereistwordt. In die gevallen zou men beter af zijnmet het gebruik van hogere orde schema’s,gezien die beter geschikt zijn om het aantalvariabelen in de hand te houden. Het gebruikvan traditionele sets van veeltermen zoals deLegendre veeltermen of de Lagrange veelter-men resulteert echter in hogere conditiege-

tallen, in het bijzonder in het geval van de EFIE.In dit artikel zullen we aantonen dat we eenset veeltermen kunnen vinden waarvoor hetconditiegetal onder controle blijft bij het opdri-jven van de raster verfijning en de verhogingvan de orde van de veeltermen.

II. Hogere orde solver.

Er werd veel tijd besteed aan het program-meren van een hogere orde solver die ge-bruik maakt van Legendre veeltermen. DeTM-EFIE en MFIE vergelijkingen werden indie mate in de solver geïmplementeerd. Ditprogramma werd getest op haar numeriekenauwkeurigheid en werd daarna gebruikt omeen aantal verstrooiingsproblemen te bereke-nen op een aantal geometrieën. Figuur 1 bi-jvoorbeeld toont de absolute waarde van Hzvoor een vlakke golf die invalt op een lichaamdat de vorm heeft van de helft van het lemnis-caat van Bernouilli.

1

Figure 1: Lemniscaat: Totaal veld |Hz| (k=2)

III. Veelterm sets

In het algemeen vormen de Legendre veelter-men een degelijke basis voor de momenten-methode, in het bijzonder voor het berekenenvan de MFIE. In sommige gevallen volstaan zeechter niet, zoals bij de TE-EFIE. In dat gevalgeven we de voorkeur aan een basis die verd-wijnt op de rasterpunten. Om dit te bekomengebruiken we de geassocieerde Legendre veel-termen:

qn =P2

n+2(2t− 1)

‖P2n+2(2t− 1)‖ (1)

Deze basis wordt vervolledigd met eennodale functie:

hn∧n+1(ρ) =

{hn/(ρ(t)) = c+g+(t),h\n+1(ρ(t)) = c−g−(t),

(2)

Met:

g± =d

dx(N + 2)PN ± NPN+1√

4N(N + 1)2(N + 2)(3)

De geassocieerde Legendre veeltermensamen met de nodale functies noemen wede Continous Base Line Functions (CBLF).Deze werden met succes gebruikt als test- enbasisfuncties in een Galerkinschema voor deTE-EFIE en de TE-CFIE.

Zoals we reeds vermeldden hebbben hogereorde schema’s het nadeel van hoge conditiege-tallen te hebben. Dit is in het bijzonder hetgeval voor de EFIE. Het blijkt dat de TM-EFIEkern te glad is, terwijl de TE-EFIE juist nietglad genoeg is. Voor de TM-EFIE hebben wedit opgelost door de kern te testen met mindergladde functies dan de Legendre veeltermen.Daartoe werd de afgeleide van de CBLF gecon-strueerd, de DCBLF. Om het probleem van denull-space in deze basis te omzeilen werd dezetevens uitgebreid met een constante functieover de gehele rand.

Voor de TE-EFIE zochten we test functiesdie gladder waren dan de CBLF. Uiteindelijkhebben we gekozen, voor N > 0, om een ba-sis van geïntegreerde Legendre veeltermen tegebruiken:

φn(t) =∫ t

0

Pn(2x− 1)‖Pn(2x− 1)‖dx (4)

Deze basis werd uitgebreid met lineaire hoe-denfuncties die verscheidene segmenten over-spannen zoals hieronder te zien is:

Figure 2: Hat Expansion

Ze worden bekomen aan de hand van de

2

volgende vergelijkingen:

hn =

{i− 1 + t t ∈ [0, 1] on segment in(1− t) t ∈ [0, 1] on segment n+1

(5)

IV. Resultaten

We hebben impedantie-matrices berekend voorhet lemniscaat bij tien frequenties en vijfniveaus van verfijning op het raster, dit in com-binatie met veelterm ordes gaande van 1 tot10. In figuur 3 en 4 geven we de conditiege-tallen van de TE en TM EFIE weer als functievan de spatiale frequentie k. De rode lijnengeven de conditiegetallen van de matrices zon-der preconditioner, terwijl de blauwe lijnen deresultaten geven bij het gebruik van een precon-ditioner. Wat we opmerken is dat bij gebruikvan een preconditioner de lijnen zich groeperenals functie van de rasterverfijning. Dit springthet meest uit het oog in het TM geval waar weslechts vijf lijnen zien die corresponderen metde resultaten van de vijf niveaus van rasterver-fijning. Deze lijnen zijn op zich al de samen-vallende resultaten voor de polynomiale ordegaande van 1 tot 10. We kunnen dus stellendat het gebruik van de preconditioner de in-vloed van de rasterverfijing en de veeltermordeop het conditiegetal nagenoeg nivelleert. Deefficiëntie van de preconditioner is echter welnog frequentie-afhankelijk. Bij hogere frequen-ties loopt het conditiegetal op, en wordt het

zelfs hoger dan de niveaus die bekomen wor-den zonder het gebruik ervan. Deze frequentieafhankelijkheid kan het onderwerp zijn voorverder onderzoek.

Figure 3: Lemniscaat TM: κ

Figure 4: Lemniscaat TE: κ

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Contents

1 Introduction 11.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theoretical Background 32.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Constitutive equations & Frequency domain . . . . . . . . . . . . . . . 42.1.2 Interface conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 3D Representation formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.1 The magnetic monopole . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Dyadic Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.3 Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Representation formulas in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.1 TM Representation Formulas . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 TE Representation Formulas . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Integral Equations for 2D PEC-objects . . . . . . . . . . . . . . . . . . . . . . 132.4.1 Magnetic Field Integral Equation (MFIE) . . . . . . . . . . . . . . . . 132.4.2 Electric Field Integral Equation (EFIE) . . . . . . . . . . . . . . . . . 152.4.3 Combined Field Integral Equation . . . . . . . . . . . . . . . . . . . . 16

3 Method of Moments 173.1 Introduction to the Method of Moments . . . . . . . . . . . . . . . . . . . . . 173.2 Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 TM-EFIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.2 MFIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.3 TE-EFIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Test and Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.1 TM-EFIE, TM-MFIE and TM-CFIE . . . . . . . . . . . . . . . . . . . 213.3.2 TE-MFIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.3 TE EFIE and TE-CFIE . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5.1 Far-patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5.2 Self-patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5.3 Neighbour-patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.6 Incoming Field Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

vii

CONTENTS viii

3.7 Current and Field Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 263.7.1 Solving the linear system . . . . . . . . . . . . . . . . . . . . . . . . . 263.7.2 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.7.3 Field-values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Scattering Solutions 284.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.1 Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1.2 Flat strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1.3 Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1.4 Lemniscate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Condition numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Preconditioner 355.1 Introduction to preconditioners . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.1.1 Kernel dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.1.2 Preconditioners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.1.3 MFIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.1.4 TM EFIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.1.5 TE EFIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Conclusion 43

A The infinite PEC cylinder: analytical solution 45A.1 TM-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.2 TE-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

B Polynomial Basis 50B.1 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50B.2 Integrated Legendre Polynomials + Hat expansion . . . . . . . . . . . . . . . 51

B.2.1 Integrated Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . 51B.2.2 Hat expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

B.3 Continuous Base Line Functions (CBLF) . . . . . . . . . . . . . . . . . . . . . 52B.3.1 Associated Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . 52B.3.2 Nodal function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

B.4 Derived Continuous Base Line Function (DCBLF) . . . . . . . . . . . . . . . 56B.4.1 Derivative of the Associated Legendre Polynomials . . . . . . . . . . . 56B.4.2 Derivative of the nodal function . . . . . . . . . . . . . . . . . . . . . . 57

Chapter 1

Introduction

1.1 Context

Ever since computers were developed in the 1960’s, scientists have tried to figure out waysin which to exploit this new computational resource to solve differential equations. Overthe years, various methods have been developed that take Maxwell’s equations, be it in dif-ferential or integral formulation, and transform them in numerical equations which can besolved by machines. The three main examples of these techniques are the Finite DifferenceTime Domain (FDTD) method, Finite Element Method (FEM), and the Method of Moments(MoM); the latter of these being the subject of this work.

As outlined in [1], the main advantage of MoM lies within the fact that for a a given object,only the boundary needs to be meshed. In the example of a 2D object, the mesh is 1D, whichmeans a drastical reduction in the number of unknowns. This is especially useful when study-ing scattering problems. Whereas FEM and FDTD require the discretization of large areasin spaces, the MoM only requires the discretization of the boundaries of the scattering objects.

However, there are also a few downsides to the method of moments. For one, the impedancematrices which are calculated are usually dense matrices. Moreover, they tend to be ill-conditioned as well. Both these aspects make that run-times for the MoM can be very long.Another downside is that, while the method delivers very good results for homogeneous ob-jects, inhomogeneous objects cannot be handled as well as for example in FEM. This is oneof the reasons why in this work we shall focus mainly on homogeneous objects.

Traditionally, the MoM is performed using only the lowest order polynomials. This results inequations which are easy to implement and lead to reasonable condition numbers. Improvingthe accuracy of this method is done by refining the mesh. Moreover, using FMM and MLFMAtechniques, computational complexity can be reduced significantly, resulting in a decrease inrun time [2].

Despite this, low-order discretization also has its drawbacks. As a rule of thumb, mesh re-

1

CHAPTER 1. INTRODUCTION 2

finement should be lower than λ10 to reach a certain level of accuracy. This means that for

high-accuracy, high-frequency simulations, the number of unknowns can become impracticallylarge. Luckily, this problem can be solved by using higher order polynomials. Indeed, higherorder polynomials are superior in their ability to store information, in such way that using anth order discretization on a certain segment can prove to be vastly superior to a low-orderscheme on n subsegments of said segment. Superior in error that is. Condition number wise,higher order schemes tend to be a far worse option, which is one of the main reasons whythey are relatively unpopular.

1.2 Goals

The goal of this work was to program a higher-order solver, called HO2D, capable of simu-lating 2D electromagnetic field problems in the frequency domain. Building on this solver,the goal was to investigate whether certain polynomial sets could be used to improve thecondition number.

Chapter 2

Theoretical Background

In this chapter a short overview will be given of the theoretical background of boundaryintegral equations. First off, a general introduction to Maxwell’s equations is given, detailingsome of the assumptions that were made while programming the solver. This is followed by ageneral derivation of the representation formulas for 3D problems. Next, using the results forthe 3D problem, the more relevant case of 2D problems will be derived. Finally, a transitionto Perfectly Electrical Conductor (PEC) objects will be made, obtaining the EFIE, MFIEand CFIE equations.

2.1 Maxwell’s Equations

At the core of probably every electromagnetic simulation are of course some form of Maxwell’sequations. Here, we will start from the non-relativistic formulation using vector-calculus [3]:

∇×E(r, t) = −∂B(r, t)

∂t(2.1a)

∇×H(r, t) = J(r, t) +∂D(r, t)

∂t(2.1b)

∇ ·D(r, t) = ρ (2.1c)

∇ ·B(r, t) = 0 (2.1d)

In words:

• Faraday’s Law: the curl of the electric field E (V/m) equals minus the time derivativeof the magnetic flux density B (Wb/m2)

• Ampere’s Law: the curl of the magnetic field H (A/m) equals the sum of the totalcurrent density J A/m and the time derivative of the electric flux density D (C/m2)

• Gauss’ Law for Electricity: the divergence of the electric flux density D (C/m2) equalsthe total charge density ρ (C/m3)

• Gauss’ Law for Magnetism: the divergence of the magnetic flux density B (Wb/m2)equals zero.

3

CHAPTER 2. THEORETICAL BACKGROUND 4

To be complete, we also want to mention the Law of Conservation of Charge, which can beobtained by combining the divergence of Ampere’s Law 2.1b with Gauss’ Law for Electricity2.1c.

∂ρ(r, t)

∂t+∇ · J(r, t) = 0. (2.2)

2.1.1 Constitutive equations & Frequency domain

Maxwell’s equations as listed in 2.1 do not form a closed set of equations, even if one assumesthe current and charge densities to be known quantities. If we want to perform field calcula-tions, we need a set of constitutive equations, the nature of which will strongly influence thecomplexity of the problem. In the most general case, the field components of D and H in rat a time t can depend on the field components of both E and H in all of time and space.However, in this work, we will limit ourselves to linear, isotropic, local media, so that theconstitutive equations take the following form:

D(r, t) = ε(r, t) ?E(r, t) (2.3)

B(r, t) = µ(r, t) ?H(r, t) (2.4)

ε is called the permittivity of the medium, whereas µ is called the permeability. The ?symbolizes the time-convolution operator. This convolution operator is hard to implement inthe time-domain, which is why we will opt for calculations in the frequency domain. In thatcase, we can write the constitutive equations as follows:

D(r) = ε(r)E(r) (2.5)

B(r) = µ(r)H(r) (2.6)

If we finally choose the engineering convention of ejwt for the frequency domain representation,all of the above boils down to the following formulation of Maxwell’s equations:

∇×E(r) = −jωµH(r) (2.7a)

∇×H(r) = J(r) + jωεE(r) (2.7b)

∇ · εE(r) = ρ(r) (2.7c)

∇ · µH(r) = 0 (2.7d)

2.1.2 Interface conditions

Maxwell’s equation as listed above can only be solved in media where µ and ε are continuousfunctions. In fact, in this work we will exclusively work with media of constant µ and ε. Thismeans that on the boundary of two such media, we can’t calculate a solution. Instead, we


need to impose additional interface conditions. If we take n1 to be the outer normal of thefirst medium (pointing into the second), these interface conditions take the following form:

n1 × (E2(r)−E1(r)) = 0 (2.8)

n1 × (H2(r)−H1(r)) = Js(r) (2.9)

n1 · (D2(r)−D1(r)) = ρs (2.10)

n1 · (B2(r)−B1(r)) = 0 (2.11)

Therefore, at a boundary, both the tangential component of the electric field and the normalcomponent of the magnetic flux density are continuous, while this can not be said for thetangential magnetic field and the normal magnetic flux density. The field-difference betweenthe two media in these latter cases define the surface current density Js (A/m) and surfacecharge density ρs (C/m2)

In this work, we will only work with PEC-objects embedded in a single medium. As insidesuch objects, all fields vanish, what is left to compute is the field in the embedding medium.Using the interface conditions listed above, the interface conditions on the object boundariesreduce to:

n×E(r) = 0 (2.12)

n×H(r) = Js(r) (2.13)

n ·D(r) = ρs(r) (2.14)

n ·B(r) = 0 (2.15)

2.2 3D Representation formulas

Within the above constraints, we will now make the transition from a vector-calculus formu-lation, to the integral representation formulas in three dimensions. A thorough derivationof these 3D-representation formulas can be found in [4]. Here, we will also start from thefictitious magnetic monopole, but take a somewhat different path, ultimately obtaining thesame result. In this discussion, we will limit ourselves to the case of the electric field, sincethe magnetic case can be derived easily using the electromagnetic principle of duality.

2.2.1 The magnetic monopole

We spent the whole previous section outlining the exact form of Maxwell’s equations wewere going to use in our field equations. However, in order to derive we will now shortlydigress from this path, by temporarily introducing the theoretical magnetic monopole. Thismonopole, although it hasn’t been observed, would be the theoretical analogon of the electron.As such, it would give rise to a magnetic current density M Wb/m2s and magnetic charge


density π Wb/m3. Part of its appeal lies in the way it would completely symmetrize Maxwell’sequations:

∇×E(r) = −M(r)− jωµH(r) (2.16a)

∇×H(r) = J(r) + jωεE(r) (2.16b)

∇ · εE(r) = ρ(r) (2.16c)

∇ · µH(r) = π(r) (2.16d)

By introducing a magnetic surface current density M s Wb/ms and magnetic charge densityπs Wb/m2, the interface conditions in 2.8 are then extended to:

n1 × (E2 −E1)(r) = −M s(r) (2.17)

n1 × (H2 −H1)(r) = Js(r) (2.18)

n1 · (D2 −D1)(r) = ρs(r) (2.19)

n1 · (B2 −B1)(r) = πs(r) (2.20)

2.2.2 Dyadic Green’s Function

So far, we are still dealing with purely differential. To make the transition to an integralformulation, we start by taking the curl of 2.16a and substituting 2.16b into it. This way weobtain the following equation:

∇×∇×E(r)− k2E(r) = −∇×M(r)− jωµJ(r) (2.21)

We note that in the case of insufficiently smooth currents, this equation needs to be interpretedin a distributional sense. To derive an integral formulation from it, we first need to obtain asolution to the following equation:

∇×∇× Gk(|r − r′|)− k2Gk(|r − r′|) = Iδ(r − r′) (2.22)

The solution is the dyadic Green’s function, which has the following expression:

Gk(|r − r′|) =

(1+∇∇k2

)Gk(|r − r′|) (2.23)

Gk itself is the more well-known Green’s function of the 3D Helmholtz equation:

∇2Gk(|r − r′|) + k2Gk(|r − r′|) = −δ(r − r′) (2.24)

The result is a symmetrical plane wave modulated by a hyperbola:


Gk(|r − r′|) =ejk|r−r

′|

4π|r − r′| (2.25)

Now that we have an expression for the dyadic Green’s function, the field can be found usingthe superposition principle:

E(r) =

∫

R3

dV ′[−∇×M(r′)− jωµJ(r′)

]· Gk(|r′ − r|) (2.26)

The first term of this expression contains the curl of the magnetic current, which is ratherinconvenient in further calculations. Therefore, we will end this subsection by transferringthe curl to the dyadic Green’s function. We proceed, by remembering the following identityfrom vector calculus: ∇ · (A ·B) = B · (∇ ×A) −A · (∇ ×B) Extending this to the casetensors, we can write the above integral as:

E(r) = −∫

R3

[∇× Gk(|r − r′|)

]·M(r′) + jωµGk(|r − r′|) · J(r′) dV ′

−∫

R3

∇ ·[M(r′)× Gk(|r − r′|)

]dV ′ (2.27)

Applying the divergence theorem, assuming there are no currents at infinity, we can easilyconclude that the second term of the previous expression vanishes. Thus the final integralbecomes:

E(r) = −∫

R3

[∇× Gk(|r − r′|)

]·M(r′) + jωµGk(|r − r′|) · J(r′) dV ′ (2.28)

2.2.3 Equivalence Principle

Let us now consider two regions in 3D space, V1 and V2, with the former region enveloping thelatter one. Furthermore, in both regions we define current densities J1 and J2. The surfacebetween both regions will be named S. We can now use the equivalence principle as outlinedin [5].

Suppose we know the fields in all points of R3. Let us then propose a new configuration, inwhich we remove J1. Then, we introduce the following surface currents on the boundary S:

M s(r) = −n×E(r) (2.29)

Js(r) = n×H(r) (2.30)


We now propose a solution to this new problem. First off, we take the fields inside V1 to bezero. As there are no currents anymore, this zero field is a possible solution to Maxwell’sequations. In V2, we assume the field to be the same as for the original problem. Becausethese fields obeyed Maxwell’s equations in the original problem, they still do now. As for thesurface S, the currents obey the interface conditions for the new problem. Therefore we mustconclude that the solution we constructed is the one and only solution to the new problem.

Inside V2 the solution to the new problem is exactly the same as the solution for the originalone. If we introduce the incoming field as the field generated by the sources in V2 we canwrite for the fields inside that region:

E(r) = Ei(r) +

∮

S

(∇× Gk(|r − r′|)

)· (n×E(r′))− jωµGk(|r − r′|) · n×H(r′) dS′

(2.31)

Ei(r) = −jωµ∫

V2

Gk(|r − r′|) · J(r′) dS′ (2.32)

As mentioned before, we can easily obtain the representations formulas for the magnetic fieldusing duality. We will only mention the result:

H(r) = H i(r) +

∮

S

(∇× Gk(|r − r′|)

)· (n×H(r′)) + jωεGk(|r − r′|) · n×E(r′) dS′

(2.33)

H i(r) = −∫

V2

∇× Gk(|r − r′|) · J(r′) dS′ (2.34)

2.3 Representation formulas in 2D

So far, we have derived all our equations in three dimensions. However, as the title of thiswork hints at, it was rather the goal to limit ourselves to two dimensional problems. Ofcourse, two dimensional problems are merely three dimensional problems in which all mediaare invariant in a certain direction. We will now first list some of the changes which thetransition to 2D problems brings about.

2D vs 3D integration

In the limit to infinity, volume integrals can be replaced by surface integrals in a planeperpendicular to the invariant direction, whereas surface integrals reduce to contour-integralsin this same plane. In general, we will calculate our integrals in the xy-plane. We can thenreplace r = xx + yy + zz by ρr = xx + yy + 0z


Nature of the nabla-operator

If we choose the z-axis to be the invariant direction, this means that all derivatives in thez-direction need to vanish. This means we can without any loss of generality replace thenabla-operator by the following 2D variant:

∇T =∂

∂xx +

∂

∂yy + 0 z (2.35)

TE and TM decomposition

A rather interesting feature of this new nabla, is that Maxwell’s equations can now be de-coupled into two separate problems. In the Transverse Magnetic (TM) problem, the electricfield lays along the invariant direction:

∇T × Ez(ρ)z = −jωµHT (ρ) (2.36)

∇T ×HT (ρ) = (Jz(ρ) + jωεEz(ρ))z (2.37)

While in the Transverse Electrical (TE) problem, it is the magnetic field which only has az-component.

∇T ×Hz(ρ)z = jωεET (ρ) + JT (ρ) (2.38)

∇T ×ET (ρ) = −jωµHz(ρ)z (2.39)

∇T · εET (ρ) = ρ(ρ) (2.40)

2D dyadic Green’s function

Substituting ∇T for ∇ in 2.23 gives us the two dimensional dyadic Green’s function:

Gk(ρ,ρ′) =

(1+∇T∇T

k2

)Gk(|ρ− ρ′|) (2.41)

However, the Green’s function Gk which appears in this expression is not the same as for the3D-case. Indeed, a new Green’s function needs to be found, this time as a solution for the2D Helmholtz equation:

∇2TGk(|ρ− ρ′|) + k2Gk(|ρ− ρ′|) = −δ(ρ− ρ′) (2.42)


The Green’s function which is the solution to this problem is the 0th order Hankel-functionof the second kind (multiplied with a constant)

Gk(|ρ− ρ′|) =j

4H

(2)0 (k|ρ− ρ′|) (2.43)

2.3.1 TM Representation Formulas

We will now derive the representation formulas for the TM-problem. We will limit ourselvesstrictly to the representation formulas for the electric field, as the derivations for the magneticfield are strictly identical. Moreover, to make matters simpler, we will omit the incoming fields,since they do not play any role in the following discussion. For the TM case, the electric fieldonly has a z-component, and the integral equation for the field outside of a boundary Cbecomes:

Ez(ρ) = z ·Ei(ρ) = z ·∮

C−∇T × Gk(|ρ− ρ′|) · (n′ × z′)Ez(ρ′)dc′

−∮

CjωµGk(|ρ− ρ′|) · (n′ × t)Ht(ρ

′) dc′ (2.44)

The of all, it is quite straightforward to see that the second term, reduces to:

−jωµ∮

CGk(|ρ− ρ′|)Ht(ρ

′) dc′

The first term is some more problematic though, and requires some calculus. Here we willwork out the curl and the dot-products step by step. Starting with the curl, we can make thefollowing derivation:

∇T × Gk|r − r′|) = ∇T ×[(1+∇T∇Tk2

)Gk(|ρ− ρ′|)

](2.45)

= ∇T ×(1Gk(|ρ− ρ′|)

)(2.46)

= −1×∇Gk(|ρ− ρ′|) (2.47)

= 1×∇′Gk(|ρ− ρ′|) (2.48)

Perhaps the most crucial step is the first one, where we have used the fact that the curlof any divergence vanishes. In the second step, we have used the fact that ∇ × (fA) =f∇×A−A×∇f We can continue now by bringing dotproduct with the vector z into theequation:

z ·(1×∇′Gk(|ρ− ρ′|)

)= −1 ·

(∇′Gk(|ρ− ρ′|)× z

)(2.49)

= −∇′Gk(|ρ− ρ′|)× z (2.50)


Finally, we can introduce (n′ × z′) = t′. Knowing that a · (b× c) = c · (a× b) we can write:

(∇′Gk(|ρ− ρ′|)× z

)· t′ = ∇′Gk(|ρ− ρ′|) ·

(z× t′

)(2.51)

=∂Gk(|ρ− ρ′|)

∂n′(2.52)

To conclude, we can now write down the representation formula for the Ez field in 2D:

Ez,i(ρ) =

∮ [Ez(ρ

′)∂Gk(|ρ− ρ′|)

∂n′− jk2

ωεGk(|ρ− ρ′|)Ht(ρ

′)

]dc′i (2.53)

2.3.2 TE Representation Formulas

For the electric field in the TE case, we can follow the same reasoning as in the previoussection. Introducing a local coordinate system t, n, z in the point ρ, the representation formulabecomes:

Et,i(ρ) = t ·Ei(ρ) = t ·∮

C∇× Gk(|ρ,ρ′|) · (n′ × t′)Et(ρ′)dc′

+

∮

CjωµGk(|ρ,ρ′|) · (n′ × z′)Hz(ρ

′) dc′ (2.54)

The first term strongly resembles the first term in the TM problem, and the derivation canbe treated analogously. For the second term, we will now prove that the following identityholds:

t · Gk(|ρ,ρ′|) · t = t ·(1+∇T∇Tk2

)Gk(|ρ− ρ′|) · t′ =

∂2

∂n∂n′Gk(ρ− ρ′)−

t · t′k2

δ(ρ− ρ′)(2.55)

We will start here by examining the nature of the dyadic Green’s function itself:

(1+∇T∇Tk2

)Gk(|ρ− ρ′|) =

1

k2

k2 + ∂2x ∂x∂y 0

∂y∂x k2 + ∂2y 0

0 0 k2

Gk(|ρ− ρ′|) (2.56)

Keeping in mind that Gk is the solution of 2.42 we can write this as:


(1+∇T∇Tk2

)Gk(|ρ− ρ′|) =

1

k2

−∂2

y ∂x∂y 0

∂y∂x −∂2x 0

0 0 1

Gk(|ρ− ρ′))

−

1 0 00 1 00 0 0

δ(ρ− ρ′)

(2.57)

Before we progress, we will point out that: (z×∇T ) = −∂yx+∂xy. Moreover, we remind thereader that in the dyadic Green’s function we can interchange ∇T and −∇′T . We can thenwrite

t · Gk(|ρ− ρ′|) · t = t ·(−(z×∇′T )(∇T × z)

k2Gk(|ρ− ρ′|)

)· t′ − t · t′

k2δ(ρ− ρ′) (2.58)

All that is left now is a bit of calculus

t · Gk(|ρ,ρ′|) · t = t ·(−(z×∇′T )(∇T × z)

k2Gk(|ρ− ρ′|)

)· t′ − t · t′

k2δ(ρ− ρ′) (2.59)

= n ·(∇′T∇T

k2Gk(|ρ− ρ′|)

)· n′ − t · t′

k2δ(ρ− ρ′) (2.60)

=∂2

∂n∂n′Gk(ρ− ρ′)−

t · t′k2

δ(ρ− ρ′) (2.61)

In practical calculations, the delta-function always vanishes, which is why it is usually notmentioned. We can then finally write the result for the TE electric field as:

Et,i(ρ) =

∮ [jωµ

k2Hz(ρ

′)∂2Gk(|ρ− ρ′|)

∂n′∂n− ∂Gk(|ρ− ρ′|)

∂nEt(ρ

′)

]dc′ (2.62)

As mentioned before, this whole exercise can be redone for the magnetic field. In fact, theyalso follow from the duality principle. We will only mention the results

Hst (ρ) =

∮ [−jωεk2

Ez(ρ′)∂2Gk(|ρ− ρ′|)

∂n′∂n− ∂Gk(|ρ− ρ′|)

∂nHt(ρ

′)

]dc′ (2.63)

Hsz (ρ) =

∮ [Hz(ρ

′)∂Gk(|ρ− ρ′|)

∂n′+jk2

ωµGk(|ρ− ρ′|)Et(ρ′)

]dc′ (2.64)


2.4 Integral Equations for 2D PEC-objects

We will now show how the representation formulas can be used to derive the Electric andMagnetic Field Integral Equations for 2D objects. We will start with the magnetic field,as this is the most interesting one. At the end, the Combined Field Integral Equations arediscussed.

2.4.1 Magnetic Field Integral Equation (MFIE)

Let us start by considering a PEC-object with a boundary C, and let us furthermore considera boundary C+ which encircles this object. Using the results obtained above, while keepingin mind the incoming field, we obtain that in every point ρ of C, the representation formulasyield:

Ht(ρ) = H it(ρ) +

∮

C+

[−jωεk2

Ez(ρ′)∂2Gk(|ρ− ρ′|)

∂n′∂n− ∂Gk(|ρ− ρ′|)

∂nHt(ρ

′)

]dc′ = 0 (2.65)

Hz(ρ) = H iz(ρ) +

∮

C+

[Hz(ρ

′)∂Gk(|ρ− ρ′|)

∂n′+j2

ωµGk(|ρ− ρ′|)Et(ρ′)

]dc′ = 0 (2.66)

If we now let C+ approach C, remembering that the electric field must vanish on the objectboundary, we have in the limit:

H it(ρ)− lim

C+→C

∮

C+

∂Gk(|ρ− ρ′|)∂n

Ht(ρ′)dci = lim

C+→CHt(ρ

+) (2.67)

H iz(ρ) + lim

C+→C

∮

C+

∂Gk(|ρ− ρ′|)∂n′

Hz(ρ′)dc′i = lim

C+→CHz(ρ

+) (2.68)

(2.69)

There are two issues with these equations that need to be addressed. First of all, we need toask ourselves how the second term on the left hand side behaves in the limit when ρ′ → ρ.Next to that, we will need to check under which circumstances the limit on the right handside is meaningful. Let us first discuss the limit on the left. As we will derive in the nextchapter, we can write the normal derivatives of the Green’s function as follows:

∂

∂n′Gk(ρ− ρ′) =

jk

4H

(2)1 (k|ρ− ρ′|) n′ · (ρ− ρ′)

|ρ− ρ′| (2.70)

∂

∂nGk(ρ− ρ′) = −jk

4H

(2)1 (k|ρ− ρ′|) n · (ρ− ρ′)

|ρ− ρ′| (2.71)

These functions are also called the kernels of the integral equation. On [6] we find that thefirst order Hankel function of the second kind has the following limiting form:


limkz→0

H(2)1 (kz) ∼ 2j

π

1

kz(2.72)

There is no problem when ρ′ approaches ρ tangentially. In that case, the kernel vanishes,and the contour integral can be performed without any difficulties. However, when we let C+

approach C, the approach will be, in one way or another, along the normal, in which case thekernel does not vanish. Fortunately, this is not a big problem, as we will now prove for theTE-case.

We will start the discussion, which we borrowed heavily from [7], with the TE-kernel (2.70).Let us then take the two boundaries C and C+. Since we are free to let C+ approach as welike, we will let it approach in such way that the limit is reached everywhere on C except onan infinitesimal small patch dC around the point ρ. In the case of a smooth curve, this patchcan be taken small enough as to yield a straight line, which we will do now.

C = C+dC

C+

ρ′

ρ

~n′

a

Figure 2.1: Singularity at ρ− ρ′

Let us then approach dC in such way that C+ forms a half-circular arc over this patch, withcenter point ρ and radius a. Assuming that Hz(ρ

′) is constant over this arc we can write theabove TE-MFIE as follows:

H iz(ρ) = lim

C+→CHz(ρ

+)−∮

C−dC

∂Gk(|ρ− ρ′|)∂n′

Hz(ρ′)dc′i − lim

C+→CHz(ρ

+)

2π

∫ π

0

n · (ρ− ρ′)|ρ− ρ′|2 adθ

(2.73)

Since n · (ρ− ρ′) = |ρ− ρ′| and |ρ− ρ′| = a on our tiny circular arc, the last integral simplyequals π. Furthermore, since the kernel vanishes in a tangential approach of ρ, the secondterm can be calculated as a full contour integral. Thus we can write our equation as follows:

H iz(ρ) = lim

C+→CHz(ρ

+)− limC+→C

Hz(ρ+)

2−∮

C

∂Gk(|ρ− ρ′|)∂n′

Hz(ρ′)dc′i (2.74)


In the case of a non-smooth boundary, for example the tip of a wedge, it is not possible toapproximate the function as a straight patch. Instead, we need to replace the integral overπ in equation 2.73 by an integral over the total outer angle Θ of the wedge. In this way, thesecond term becomes

Θ

2πlim

C+→CHz(ρ

+)

Using the fact that in the limit limρ→ρ′ n′ = n, we can extend the previous result to the caseof the TM-problem (for a smooth scatter):

H it(ρ) = lim

C+→CHt(ρ)

2−∮

C

∂Gk(|ρ− ρ′|)∂n′

Ht(ρ′)dc′i (2.75)

Now all that is left is to discuss the limit of the H-field which is still present in 2.74 and 2.75.For objects with a closed geometry, (e.g, cylinders) there is no problem, as each point of C+

can be mapped onto C. However for open boundaries, like a flat strip, this is not the case.Here, the left limit and the right limit do not correspond and the limit is undetermined. Aquite drastic result of this is that the MFIE will fail for open geometries, and thus in thosecases one needs to resort to the EFIE.

Assuming we can write down the MFIE, we can do so more elegantly using operator notation.Reminding ourselves of the interface conditions for PEC objects 2.13, we get:

(1

2+MTM

)Jz = H i

t (2.76)

(1

2+MTE

)Jt = H i

z (2.77)

We end by noting that the above results can easily be extended to the case of a cavity. Here,a vacuum ”object” is completely surrounded by PEC material, and the previous discoursecan be repeated by taking the contour C− inside the cavity.

2.4.2 Electric Field Integral Equation (EFIE)

In the case of the EFIE, we can proceed analogously to the MFIE. This time the equationsare a bit simpler though:

Eiz(ρ) = limC+→C

−jωµ∮

C+

Gk(|ρ− ρ′|)Ht(ρ′)dc′i (2.78)

Eit(ρ) = limC+→C

j

ωε

∮

C+

∂2Gk(|ρ− ρ′|)∂n′∂n

Hz(ρ′)dc′i (2.79)


Furthermore, these limits are significantly more well-behaved than their MFIE counterparts.First of all, the TM kernel doesn’t contain any normal derivative, so the direction of approachdoesn’t matter for the kernel singularity. The same case can be made for the TE kernel, ifwe consider that the original form of this kernel 2.55 contains only tangential components.Thus, we can remove the limits, and state:

Eiz(ρ) = −jωµ∫

C+

Gk(|ρ− ρ′|)Ht(ρ′)dc′i (2.80)

Eit(ρ) =j

ωε

∫

C+

∂2Gk(|ρ− ρ′|)∂n′∂n

Hz(ρ′)dc′i (2.81)

In operator notation, one usually writes:

L Jz = Eiz (2.82)

N Jt = Etz (2.83)

2.4.3 Combined Field Integral Equation

Even though the EFIE and the MFIE have proven their merit, there are cases in which theyfail due to internal resonances. For the EFIE, these resonances are due to the existence ofa non-trivial solution at resonance frequencies. These effectively create a null-space solution,which make the problem ill-conditioned. However, as the currents at the resonance frequencyare non radiating, these don’t give rise to much problems. Some error will be caused bythe fact that numerical integration is only approximative, so that there will always be someelectromagnetic energy leaking from the object. However, problems are more severe in theMFIE case. Here, resonances will gives rise to radiating currents which strongly distort theresults.

The solution to these problems is the Combined Field Equation (CFIE). This equation takesa linear combination of the EFIE and MFIE, using the impedance Z to bridge the differencein dimensions between electric and magnetic fields:

[αL+ Z(1− α)

(1

2+MTM

)]Jz = αEiz + Z(1− α)H i

t (2.84)

[αN + Z(1− α)

(1

2+MTE

)]Jt = αEit + Z(1− α)H i

z (2.85)

It can be shown that this type of equation effectively avoids the internal resonance problem.The ideal value of α, which has a value in the interval [0, 1] is a matter of some debate. [8] citesvarious sources claiming that a value of α = 0.2 delivers the best results. Other authors [9] [4]stick to a value of 0.5.

Chapter 3

Method of Moments

A short introduction to the method of moments is given, followed by a discussion of thecomputational implementation. This latter part contains a discussion of the kernels, the testand basis functions, and meshing schemes. Furthermore, the numerical integration of the far-,neighbour- and self-patches are handled, followed by a brief discussion of the incoming fieldvector. We end with the procedure for obtaining boundary currents and fields.

3.1 Introduction to the Method of Moments

We will start the discussion by introducing the pseudo-inner product of 2 functions on acertain object boundary C:

〈f, g〉 =

∮

Cf(ρ)g(ρ)dc (3.1)

This product differs from the regular inner product, as it contains f instead of the complexconjugate f∗. In some texts it is also called the reaction between the two functions. Withthis in mind, let us now consider the following equation for a linear operator Z:

ZI = V (3.2)

Clearly, by filling in the right currents, incoming fields and integral operators, we can obtainthe EFIE, MFIE and CFIE equations derived in the previous chapter. However, in theircurrent form, they’re almost impossible to compute. A first step is to expand the unknownvector I using a set of basis functions:

Z∑

i

Iiφi = V (3.3)

17

CHAPTER 3. METHOD OF MOMENTS 18

Using the linearity of the Z-operator, this can rewritten in the following way:

∑

i

FiZ φi = V (3.4)

This result is only marginally better than before. To come to an equation which we canactually implement computationally, we now take the pseudo inner product of both sideswith another set of test functions ψ:

∑

i

Ii〈ψj ,Z φi〉 = 〈ψjV 〉 (3.5)

In this way, we can finally rewrite our original equation as follows:

∑

i

ZjiIi = Vj (3.6)

Zij = 〈ψj ,Z φi〉 =

∮ψj(ρ)Z(ρ,ρ′)φi(ρ′)dc (3.7)

Gj = 〈ψj , V 〉 =

∮ψjρ)V (ρ)dc (3.8)

Therefore, we have transformed the original integral equation to a set of linear equationswhich could be arbitrarily large, depending on the desired accuracy. Yet, to implement this,a few assumptions and approximations need to be made. One issue is the number and natureof the basis and test functions. Next to that, there is the problem of calculating the matrixelements. These are all line integrals which we will not be able to calculate exactly. Bothissues will be addressed in the next sections of this chapter. First however, we will discussthe nature of the kernels which occur in the field integral equations.

3.2 Kernels

3.2.1 TM-EFIE

As mentioned before, the 2D-Green function corresponding to (2.42) is the 0th order Hankelfunction of the second kind (with a multiplication factor):

Gk(|ρ− ρ′|) =j

4H

(2)0 (k|ρ− ρ′|) (3.9)

This Hankel-function was obtained using the AmostoC package, which translates the Amosfunction library [10] to C-language. The computation of Hankel-functions is computationallyexpensive, and it represents the main bottle-neck in the computation of the matrix elements.


3.2.2 MFIE

The next kernels we will discuss are the ∂∂nGk and ∂

∂n′Gk. We will concentrate ourselves onthe former, but it is obvious that the treatment for the latter kernel will be very similar.Let us again consider Green’s function:

Gk(|ρ− ρ′|) =j

4H

(2)0 (k|ρ− ρ′|) (3.10)

In [6] we find the following expression for the derivative of the Hankel function of the secondkind.

d

dzH(2)n (z)

1

2

[H

(2)n−1(z)−H(2)

n+1(z)]

(3.11)

H(2)−n(z) = (−1)nH(2)

n (z) (3.12)

Thus our kernel becomes:

∂

∂nGk(|ρ− ρ′|) = −jk

4H

(2)1 (k|ρ− ρ′|)n ·∇T |ρ− ρ′| (3.13)

(3.14)

Finally, since ∇T |ρ− ρ′| = (ρ−ρ′)|ρ−ρ′| we can write for this kernel:

∂

∂nGk(|ρ− ρ′|) = −jk

4H

(2)1 (k|ρ− ρ′|)n · (ρ− ρ

′)|ρ− ρ′| (3.15)

Keeping in mind that ∇′T will merely causes a minus sign to appear, we get for the otherkernel:

∂

∂n′Gk(|ρ− ρ′|) =

jk

4H

(2)1 (k|ρ− ρ′|)n

′ · (ρ− ρ′)|ρ− ρ′| (3.16)

3.2.3 TE-EFIE

Let us now examine the last kernel, which has contains a double derivative. As we have shownin the previous chapter, this kernel can be written in two ways

∂2

∂n∂n′Gk(|ρ− ρ′|) +

t · t′k2

δ(ρ− ρ′) = k2ut · u′tGk −∂2

∂ut∂u′tGk (3.17)

We will use the right-hand side, consisting of two terms. The first term constitutes gives riseis an extra kernel that needs to be evaluated, as it contains the dot product of the tangentialderivatives along the two patches for which the interaction is evaluated. However, we willnow show how the kernel in the second term, can be reduced to Gk, provided we use adequate


testing and basis functions.

Let us start by considering the interaction integral of this second term:

Gn,m =

∮ ∮ψ(sn)

∂2Gk(|ρn(sn)− ρm(sm)|)∂sn∂sm

φ(sm)dsndsm (3.18)

Integrate by parts along tn, we get the following:

Gn,m =

∮ψ(sn)

∂Gk(|ρn(sn)− ρm(sm)|)∂sm

φ(sm)dsndsm

]sn=s

sn=s

−∮ ∮

∂ψ(sn)

∂sn

∂Gk(|ρn(sn)− ρm(sm)|)∂sm

φ(sm)dsndsm (3.19)

If now we choose the test functions continuous on the whole boundary, then the first term in(3.19) vanishes, and we have transferred the derivative from the kernel to the testing function.We then repeat this procedure, starting from (3.19), and make the further assumption thatthe basis functions also be continuous along the object-boundary. In this way, we effectivelytransfer the double derivative to the test and basis function, so that the kernel is Gk Thetotal matrix element can then be written as follows:

Zn,m = k2

∮ ∮ψ(tn)Gk(|ρn(tn)− ρm(tm)|)∂ρ(tn)

∂tn· ∂ρ(tm)

∂tmφ(tm)dtndtm

−∮ ∮

∂ψ(tn)

∂tnGk(|ρn(tn)− ρm(tm)|)∂φ(tm)

∂tmdtndtm (3.20)

3.3 Test and Basis Functions

In an ideal world, we would work with infinite sets of test and basis functions. Of course, inreality we will have to approximate by limiting ourselves to sets of only a few functions. Wedistinguish two strategies in choosing these functions:

For one, we might define a set of functions over the whole object boundary. In this case,accuracy can be improved by increasing the amount of functions defined on this boundary.Thus, in the case of polynomials, we might take a function which is constant over the wholeboundary, one which is linear,quadratic,cubic, etc...

Another, more popular strategy starts by dividing the boundary in segments, a process calledmeshing. We can then define functions which are non-zero on only one or a few segments.


Thus, instead of increasing the polynomial order, we might as well divide the object into seg-ments, and limit ourselves to functions which are constants on these segments. This approachis actually the most common, and the majority of MoM-solvers are limited to polynomial or-ders upto 1.

In this work we use both strategies, and study the interplay of higher polynomial order withmesh refinement. We will now discuss the polynomials which were used in the different fieldintegral equations.

3.3.1 TM-EFIE, TM-MFIE and TM-CFIE

Both test and basis functions for these cases were based on the shifted orthonormal Legendrepolynomials as defined in section B.1. To elaborate on this, we take as an example theZ-matrix elements for the TM-EFIE:

Zi,j =

∮ ∮‖∂ρ(t)

∂t‖ψi(ρ(t))Gk(|ρ(t)− ρ(t′)|)φj(ρ(t′))‖∂ρ(t′)

∂t‖dtdt′ (3.21)

In order not to overload the notation we will from now on write the Jacobian as:

‖∂ρ(t)

∂t‖ = J(t)

We now cut up the boundary in different segments, and let us look at a random segment n.We will take the boundary integral such that on this segment t ∈ [0, 1]. We then define abasis function and a testing function of order p on this segment:

ψn,p = ψn,p

{Pp(2t−1)

J(t)‖Pp(2t−1)‖ , t ∈ [0, 1]

0, elsewhere(3.22)

This scheme, in which both basis and testing functions is also called a Galerkin-weighing. Ifwe plug these functions into the expression for the Z-matrix element, we get the followingelement:

Znp,mq =

∫ 1

0

∫ 1

0

Pp(2tn − 1)

‖Pp(2tn − 1)‖Gk(|ρn(tn)− ρm(tm)|) Pp(2tm − 1)

‖Pp(2tm − 1)‖dtndtm (3.23)

These functions, which we will call ”adapted Legendre polynomials” were also used to calcu-late the TM-MFIE matrix elements. Moreover, combining the MFIE and EFIE, the TM-CFIEalso uses this scheme.


3.3.2 TE-MFIE

For the TE-MFIE, we initially used the same polynomials as for the TM-MFIE. However, aswe will see in the discussion of the TE-EFIE, it the purely normalized Legendre polynomialsturned out to be a better choice. The testing and basis functions then are:

ψn,p = ψn,p

{Pp(2t−1)‖Pp(2t−1)‖ , t ∈ [0, 1]

0, elsewhere(3.24)

The matrix elements are given by:

Znp,mq =

∫ 1

0

∫ 1

0‖∂ρ(tn)

∂tn‖ Pp(2tn − 1)

‖Pp(2tn − 1)‖∂

∂nGk(|ρn(tn)− ρm(tm)|) Pp(2tm − 1)

‖Pp(2tm − 1)‖‖∂ρ(tm)

∂tm‖dtndtm

(3.25)

3.3.3 TE EFIE and TE-CFIE

The case of the TE-EFIE is the most complicated one. As we saw in the section on kernels,our implementation crucially depends on the usage of continuous test and basis functions(and their derivatives). The functions (3.22) and (3.24) do not fit this definition, since theyare discontinuous on the segment edges. Thus, a different breed of polynomials is needed.Initially, our eye fell on polynomials based on the Jacobi polynomials:

qn(t) = Ant (1− t)P 2,2n (2t− 1) (3.26)

An being a normalization factor. The occurence of x(1− x) makes sure that the polynomialsvanish at the edges x = 0 and x = 1, whereas the orthogonality follows from the followingcharacteristic of the Jacobi polynomials:

∫ 1

−1(1− t)α(1 + t)βP (α,β)

m (t)P (α,β)n (t) dt =

2α+β+1

2n+ α+ β + 1

Γ(n+ α+ 1)Γ(n+ β + 1)

Γ(n+ α+ β + 1)n!δnm

(3.27)

As outlined in section B.3.1, it can be easily shown that these polynomials are nothing lessthan the normalized second order associated Legendre polynomials

qn(t) =P 2n+2(2t− 1)

‖P 2n+2(2t− 1)‖ (3.28)

Unfortunately, they can only model phenomena which are zero on segment edges, which isfairly useless in practical applications. That’s why, this basis needs to be expanded with a


nodal function. This the higher order analogon of the hat-functions which are used in lowerorder calculations, and would ideally resemble the behaviour of a delta distribution centredon a node. Of course, using polynomials, we can only approximate this peaked behaviour.We chose the following functions to simulate this behaviour:

g±(x) =d

dx

(N + 2)PN (2x− 1)±NPN+1(2x− 1)√4N(N + 1)2(N + 2)

(3.29)

The idea here is to have one function span two segments. So on one segment, the functionrises like g+, whereas on the next segment, it descends like g−. Of course, this is in the casewhere the polynomial order on both segments is equal. When the polynomial is different ontwo adjacent segments, coupling constants need to be introduced to ensure that the nodalfunction be continuous. A derivation of these constants can be found in B.3.2. From nowon we will refer to this basis of associated Legendre Polynomials and nodal function as theContinuous Base Line Functions (CBLF). Their derivatives are the Derived CBLF (DCBLF).

If we now take a look at (3.20), the question arises how to calculate the matrix elements us-ing (D)CBLF. We could theoretically implement more computational functions to calculatethem. However, this would be rather impractical, as all the other matrix elements are cal-culated using Legendre polynomials. Moreover, the TM-EFIE kernel which returns in (3.20)was already computed using the Legendre polynomials; it would be convenient if we couldreuse these matrix elements. Luckily, the Legendre polynomials form a polynomial basis,which means we can expand (D)CBLF using the normalized Legendre polynomials. Thismeans that we can, by making a basis-transformation matrix, transform any Z-matrix cal-culated with the (adapted) normalized Legendre Polynomials, to a Z-matrix calculated with(D)CBLF. This is the reason why we chose to evaluate all kernels using the same polyno-mials. A justification for our choice of the Legendre Polynomials over all other orthonormalpolynomial bases can be found in [11].

In (B.3.1), the expansion of CBLF in normalized Legendre polynomials is described. In(B.4.1) the same is done for the DCBLF. Notice that we could not expand the CBLF in theadapted Legendre polynomials because the Jacobian would cause troubles while deriving theDCBLF. This also explains why we had to use the normalized Legendre polynomials andnot the adapted Legendre polynomials in the TE-MFIE. Indeed, thanks to the TE-EFIE,the CFIE has to be calculated using the CBLF as well. The only way we can transform theTE-MFIE matrix using this basis is when the matrix elements were already calculated usingthe ordinary normalized Legendre polynomials. Luckily, the Jacobian doesn’t occur at all inthe Gk-part of (3.20), so we can still reuse the same matrix elements.

3.4 Meshing

In the previous section we introduced the concept of meshing. We will now elaborate on howthis was done in our solver. One option could be hard-programming the different segment


geometries in the solver itself. However, this would be highly impractical, as it would requireus to compile different versions of the program for each geometry. Instead we chose to firstapproximate the segments using Legendre polynomials:

ρn(t) =

N∑

i

ρiPi(2t− 1) (0 < t < 1) (3.30)

ρi = (2i+ 1)

∫ 1

0ρn(t)Pi(2t− 1)dt (3.31)

This meshing can be done in an easy, high level programming environment; in our case Octavewas chosen. The output of our Octave-scripts was then written to a .txt-file, which served asthe input for the HO2D-solver.

This is how it was done from a technical point of view. A more interesting question is how toactually mesh a boundary. For this, we have used two different schemes: constant meshingand exponential meshing. Constant meshing is the simplest kind of meshing, in which a pa-rameter, preferably the perimeter of the object, is divided in N equal parts. This is a goodtechnique for sufficiently smooth boundaries, like ellipses and circles. One way this techniquecould be extended to non-smooth objects is by increasing the polynomial order near a wedge.This will type of refinement will be discussed in this work.

Indeed, for non-smooth objects, we used a different scheme of exponential meshing. Insteadof segments which have the same length over the whole boundary, segments were chosen sothat their length decreased exponentially when approaching the corners. We will explainthis method using the example of a flat strip with length L. Let us than assume we have areference value L0 for a segment length, and start from the example of even N . In that case,starting from the middle of the segment (i=0), we can take our segments to be of length:

Li = L0f|i| (3.32)

Here, the factor f has to be chosen so that the length-difference between two adjacent patchesis not too large. For the flat strip, 0.5 is a good value. The value of L0 should be , so thatthe total sum of segment lengths equals L:

L0 = L1

N mod 2 + 21−fbN/2c

1−f(3.33)

For uneven N an extra segment of length L0 is included in the middle of the line. This is thereason for the modulo in the expression 3.33.


3.5 Numerical integration

3.5.1 Far-patch

The far patch is the integral between test and basis functions on two different patches sepa-rated by at least another patch. For the far-patch interaction with the TM-MFIE kernel, thematrix elements are defined as follows:

Znp,mq =

∫ 1

0

∫ 1

0Pp(2tn − 1)Gk(|ρn(tn),ρm(tm)|)Pq(2tm − 1)dtndtm (3.34)

Again, in an ideal world, we can evaluate this integral exactly. In numerical computinghowever, integrals are computed using quadrature-rules.

∫ 1

0f(x)dx =≈

n∑

i=1

wif(xi) (3.35)

In the case of the far-patch, in which the singularity of the Hankel-function is neatly avoided,we can make use of the ordinary Gauss-Legendre quadrature.The points xi and weights wi forthis quadrature were taken from a pre-computed library provided by Ignace Bogaert, KristofCools, Jan Fostier and Joris Peeters. This quadrature rule was able to calculate integrals onthe interval [0, 1], and as such, we can write our numerically obtained matrix elements as:

Znp,mq =N∑

i=0

M∑

j=0

Pp(2xi − 1)Zk(ρn(xi),ρm(xj))Pq(2xj − 1)wiwj (3.36)

3.5.2 Self-patch

For the self patch, where both test and basis function are non-zero on the same patch, thingsare somewhat more difficult. In this case the singularity of the kernels is approached everytime ρ(tn) and ρ(tm) coincide. In theory, we could still use the Gauss-Legendre quadraturewe used for the far-patch, but the rate of convergence is terribly low. Instead a GeneralizedQuadrature rule was used. This quadrature, computed by Ignace Bogaert (inspired by [12]was designed to exactly integrate the functions having the following form:

f(t) = P (t) ln(t) +Q(t)

With P and Q polynomials of order N ,(t ∈ [0, 1]). As the limiting behaviour of H(2)0 is indeedlogarithmic [6], we can use this quadrature rule to calculate matrix elements for the EFIE.For the MFIE, we recall from the previous chapter that no singularity appears, although wecan still use this quadrature rule.

The singularity only appears when ρ(tn) = ρ(tm). This means that to optimally use ourquadrature rule, we can first write the integrals as follows:


Zni,mj =1

2

∫ 1

0

∫ 2−v

vfnp,mq

(u+ v

2,u− v

2

)dudv +

1

2

∫ 1

0

∫ 2−v

vfnp,mq

(u− v

2,u+ v

2

)dudv

(3.37)

One can see that the singularity now lies exclusively at v = 0, so that we can use theGeneralized Gaussian quadrature to integrate over v. The inner integral does not need sucha quadrature rule. However, this integral is still dependent on v, which would be highlyimpractical to implement. Thus, the integration boundaries were adapted using the transform:u→ v+(2−2v)u to enable us to use the Gauss-Legendre Quadrature on the interval 0 ≤ u ≤ 1

Zni,mj =1

2

∫ 1

0

∫ 1

0fnp,mq (v + (1− v)u, (1− v)u) (1− v)dudv +

1

2

∫ 1

0

∫ 1

0fnp,mq

((1− v)u′), v + (1− v)u′

)(1− v)du′dv

(3.38)

3.5.3 Neighbour-patch

Here, much of the reasoning from the previous paragraph can be repeated. However, nowthe singularity merely occurs when the boundary point between two segments is approached.This means that using 3.38, the singularity would be at v = 1. Thus, the transformationv → 1− v

Zni,mj = −1

2

∫ 1

0

∫ 1

0fnp,mq (1− (1− u)v, uv) vdudv − 1

2

∫ 1

0

∫ 1

0fnp,mq (uv, 1− (1− u)v) vdudv

(3.39)

3.6 Incoming Field Vector

We will now shortly discuss the calculation of the incoming field vector. Hereto, the incomingfield was assumed known, and constructed in a class, in a specific class containing functions forthe transversal and z-components of the field. In this way, the code was able to generate anystrictly 2D incoming field through class inheritance. The field vector was then computed bytesting with the Legendre functions which were also used for the computation of the Z-matrixelements.

3.7 Current and Field Calculations

3.7.1 Solving the linear system

We have now discussed the calculation of the Z-matrix and V-Matrix elements. The questionis now how we can solve the linear system ZI = V for I, as this is the quantity of interest.Two paths can be taken to solve this problem. The first one, limited to square-invertiblematrices, is by calculating the inverse of Z, so that the current-vector follows simply from:


I = Z−1V (3.40)

Different schemes exist for inverting the Z-matrix. A common method for the computationof this inverse is based on LU-decomposition. This is the method used by LAPACK libraries,and this path was taken in the computation of the current vector in C++.

Another option is the use of iterative solvers to obtain a solution for the current vector. Alot of different techniques exist, but most of them follow the same scheme. First of all, anestimation is made for I, let us call it I0. Then ZI0 = V0 is calculated. The main differencein iterative methods is now to compare V0 with V and to conjecture a new estimate I1 forthe current vector. This goes on until a certain In is found for which ‖Vn − V ‖ < ε, with εthe desired tolerance. Of course, not all iterative methods are suited for the electromagneticmethod of moment. This mainly depends on the nature of the Z-matrix. For example, oneof the most basic methods, the conjugate gradient method, is only applicable for symmetricpositive definite matrices, which is not the case here.

3.7.2 Current

The above solution renders the current vector, that is, the coefficients of the current in termsof the basis functions. To get the actual current, an inverse transformation needs to be done.The resulting current is the most crucial parameter to solve, and serves as the main outputof the program. In the case of a cylinder, we define the relative RMS current-error as follows:

∆ =

√ΣNn=1|Jn − Jn|

2(3.41)

where Jn denotes the numerical value and Jn the exact value in a point n of the boundary.

3.7.3 Field-values

Using the current we obtained above, we can calculate the fields in any point of the 2D plane.We chose to limit ourselves to evaluating the z-components of the fields, since the tangentialcomponents are dependent on the local coordinate system one uses. Based on our discussionof representation formulas in chapter 2 we can then state:

Eiz(ρ) = Eiz(ρ)− jωµ∮

CGk(|ρ− ρ′|)Jz(ρ′)Jzdc′i (3.42)

Hz(ρ) = H iz(ρ) +

∮

C(ρ′)

∂Gk(|ρ− ρ′|)∂n′

Jtdc′ = 0 (3.43)

Chapter 4

Scattering Solutions

In this discussion, we will briefly discuss some numerical results we obtained using the higher-order solver. We will start with the case of the infinite cylinder, focusing on the current error.After that, we look into some interesting non-smooth geometries.

4.1 Numerical results

4.1.1 Circle

The first geometry which was extensively studied was a simple cylindrical scatterer. Thisgeometry is of great importance, as it is the only one we tested which has an actual analyticalsolution, see also appendix A. As this geometry is perfectly smooth and has rotationalsymmetry, we could use a constant meshing scheme. Since the arc length on a circle isproportional to the angle, we get for ith segment:

{x = cos ti

y = sin titi ∈

[2π

n(i− 1),

2π

ni

](4.1)

While the existence of an analytical solution makes it a preferred way to benchmark scatteringsolvers, it has the downside of being too symmetrical for debugging purposes. Indeed, thehigh degree of symmetry can make for unpleasant surprises when one switches to less regulargeometries. We will now first take a look at the scattered fields of these elements. Figures 4.1and 4.2 show absolute values of the total fields Ez and Hz. The cylinder had a radius a = 1and was impinged by a plane-wave e−jkx, with k = 2. Notice how inside the cylinder, bothfields vanish, which is the way it should be for PEC-objects.

Of course, these plots only give a qualitative indication of the solver-performance. For aquantitative approach, we can for example take a look at the values for:

28

CHAPTER 4. SCATTERING SOLUTIONS 29

Figure 4.1: Circle: Total Field |Ez| (k=2) Figure 4.2: Circle: Total Field |Hz| (k=2)

δ =|Jn − Jn||Jn|

(4.2)

This value is related to ∆ defined in (3.41). Here however, the relative error in every pointon the scatterer is considered. For example, for the same problem, meshed with 10 segmentsand polynomial order 10, we get the following semi-log plots 4.4 and 4.3 for the EFIE current.

Figure 4.3: Circle: Current error Jz (k=2) Figure 4.4: Circle: Current error Jt (k=2)

We see that the TM-MFIE seems to be more accurate in describing the boundary currentthan the TM-EFIE. However, this is hardly conclusive evidence to suggest that this wouldbe a general trait. In the TE-case, these roles are actually switched and ∆TM = 9.4810−11 >∆TE = 7.3610−11. It is actually more interesting to investigate the general behaviour of ∆in function of both mesh refinement and polynomial order. To this extent we made a plotof the EFIE, MFIE and CFIE ∆ for the TM case. What you see is the course of the dis-crete RMS error in function of the amount of segments. For the CFIE, α was chosen to be 0.5.


Figure 4.5: Circle: TM Current error (k=2)

What we actually observe is that for a given polynomial order P , all three errors lie relativelyclose to each other. In fact, the lower of the two lines is actually more or less a superposition ofthe MFIE and CFIE lines, whereas the other line is the EFIE line. In general, the EFIE showsa larger error than both CFIE and the MFIE. When interpolating, we see that especially forhigher orders, the error behaves like hp+1, as predicted by [8]. The exact nature of this currenterror will not be studied any further though.

4.1.2 Flat strip

We already discussed the meshing of the flat strip in the previous chapter, when we introducedthe concept of exponential meshing. Here, we used exactly that scheme, using a factor of f =0.5 We can now look at what happens when the same plane-wave impinges perpendicularlyon the broadside of a flat strip, as shown on figure 4.6 and 4.7

As mentioned before in the derivation of the MFIE, this flat strip is an example of an openobject. As such, the MFIE (and consequently the CFIE) is not applicable here, and the aboveresults were obtained purely using the EFIE.

4.1.3 Square

The square is defined by having 4 equal sides and 4 right angles, and as such it is mostcertainly non-smooth. However, exponential meshing of the square was fairly easy, as onecould use the routines used for the flat-strip to construct the four sides of the square. Thisway, the figures of 4.8 and 4.9 were obtained.


Figure 4.6: Flat Strip: Total Field |Ez| (k=2)Figure 4.7: Flat Strip: Total Field |Hz| (k=2)

Figure 4.8: Square: Total Field |Ez| (k=2) Figure 4.9: Square: Total Field |Hz| (k=2)


We can now take a look at the current density on the boundary of the square. On figure 4.10,the current density Jz is plotted of the TM problem. This current clearly shows singularitieson the edges, whereas the current Jt on 4.11 doesn’t. This difference in current smoothnesscan be explained using the theory of Sobolev spaces [8].

Figure 4.10: Square: |Jz| (k=2) Figure 4.11: Square: |Jt| (k=2)

4.1.4 Lemniscate

Another Geometry which we testes is one half of Bernouilli’s Lemniscate, which was plottedon figure 4.12. This contour has the following equation [13]

Figure 4.12: Lemniscate: Meshing (f=0.9)

{x = a cos t

1+sin t2

y = a sin t cos t1+sin t2

t ∈[π

2,−π

2

](4.3)

With a perimeter s = La, where L =Γ2( 1

4)√(2π)≈ 2.62 is the lemniscate constant.


The lemniscate is an example of a convex wedge-geometry, since in the point t = π2 ↔ t = −π

2the curve is convex, yet it is non smooth as the angle is π

2 . This singularity, like in the case ofthe square, gives rise to singular currents for the TM-current. This means that here as wellan exponential meshing scheme needs to be used in the vicinity of the wedge-tip. However,in contrast to the case of the square, where a simple expression for the arc-length is available,here the expression for the arc length is s(t) = aF (t, i), F(t,i) an elliptic integral of the firstkind. Since this would be rather hard to implement, we chose to mesh t on the interval [−π

2 ,π2 ]

instead. Of course, the f -factor then had to be chosen so that the lengths of two neighbouringsegments didn’t differ too much from eachother. This factor was determined rather on sight,the resulting mesh is also visible on 4.12 for f = 0.9 Of course, this mesh is invisible on thefield plots shown on figures 4.13 and

Figure 4.13: Lemniscate: Total Field |Ez|(k=2)

Figure 4.14: Lemniscate: Total Field |Hz|(k=2)

4.2 Condition numbers

In the introduction we already mentioned concepts like well-conditioned basis and ill-conditionedsystems. We will now elaborate on this by explaining the concept of condition numbers. Inshort, condition numbers are used in numerical analysis to determine the sensitivity of asystem to small errors in the input. In our example, we take a linear equation Ax = y, anddefine for a certain operator norm, the condition number κ ≥ 1 is given by:

κ = ‖A‖‖A−1‖ (4.4)

For matrix equations like the ones we solve, we can write this more clearly using SVD. Indeed,κ then reduces to the ratio of the largest to the smallest singular value.

κ =σminσmax

(4.5)


If we now take the example from [14], and take the matrix:

(1.168 0.9332.172 1.735

)(4.6)

The singular values of this matrix are 3.15 and 1.2610−6, therefore the condition numberbecomes 2.49106 How thus this manifest itself? Well, let us take the equation Ax = y. Ify1 = (2.101, 3.907), the solution is x1 = (1, 1) whereas if we slightly change y, e.g. y2 =(2.100, 3.907, the exact solution is x2 = (432.75, 544.00). Needless to say that this is quiteproblematic. Let us take the example of y2 and try to solve for x using an iterative solver.If we would start the calculations with the educated guess of x = x1 = (1, 1), it would onlyseem that we are quite close to the solution, while in reality, we are miles away. Therefore,if our tolerance is too high, we will obtain wildly erroneous results, whereas if we don’t, theamount of iterations rise uncontrollably.

Figure 4.15: κ EFIE (k=2,P=10) Figure 4.16: κ MFIE (k=2, P=10)

Let us consider three geometries, Circle, Square and Lemniscate, and see how the conditionnumber varies as a function of the mesh refinement. On 4.15 we see a plot of the conditionnumbers for the EFIE in function of the discretization, whereas 4.16 shows the conditionnumbers for the MFIE . We see that in general, the EFIE has a much higher condition numberthan the MFIE. In the next section our main concern will be controlling the condition numberof this problem.

Chapter 5

Preconditioner

In the previous chapter we obtained numerical solutions for some reference problems. Whilethese results can be interesting by themselves, we are more interested in the condition number,and how to improve it. We will now first explain how it comes that the condition numbershows the behaviour we saw earlier. We will then propose our solution to this probleminvolving preconditioners. At the end, we discuss the results of this operation.

5.1 Introduction to preconditioners

5.1.1 Kernel dynamics

Let’s go back to the beginning of chapter 3, and look at the integral equations. For a generalkernel Z they look like:

ZI = V (5.1)∮Z(|ρ(t′))− ρ(t)|)I(ρ(t))‖∂ρ(tm)

∂t‖dtdt′ = V (ρ(t)) (5.2)

With a bit of imagination and on a sufficiently smooth boundary, we can recognize a circularconvolution in the left-hand side of this equation.

[Z ? I](ρ(t)) ≈ V (ρ(t)) (5.3)

If we now take the a Fourier transform along t, we get the following equation:

V (ω) ≈ Z(ω)I(ω) (5.4)

35

CHAPTER 5. PRECONDITIONER 36

Indeed, it is a well known property of the Fourier transform that it transforms a convolutioninto a product. To better illustrate our next point, we will write the previous equation inmatrix notation.

V (ω1)

V (ω2)...

=

Z(ω1) 0 · · ·0 Z(ω2) · · ·...

.... . .

I(ω1)

I(ω2)...

(5.5)

The Fourier transform ”diagonalizes” the operator Z. We can now ask ourselves how thebehaviour of the Fourier-transformed kernel will affect the condition number in this matrixequation. This then will give us an idea of the real condition number, since the Fouriertransform is a unitary transformation that leaves the condition number invariant. Let usthen, for the sake of simplicity, take as a first example the Dirac-delta distribution

Z(x) = δ(x) (5.6)

The Fourier-transform of the Dirac-delta is a constant function:

F(δ) = 1 (5.7)

This leads to

V (ω1)

V (ω2)...

=

1 0 · · ·0 1 · · ·...

.... . .

I(ω1)

I(ω2)...

(5.8)

The condition number here becomes:

κ =‖Z‖‖Z−1‖ = 1 (5.9)

This is of course the best result one can possibly hope for, as all the spectral componentsare being perfectly mapped. If on the contrary, one takes a smoother kernel, like extremeexample of a constant function, the Fourier-transform becomes:

F(1) = 2πδ(ω) (5.10)

This is a lot more problematic. Only the Fourier component at ω = 0 is left unchanged, whileall other spectral components are reduced to zero. This means that the condition number


becomes infinite, and the problem is practically unsolvable. However, the discussion doesn’tend here. Let us now go to the other end of the spectrum, and take a hypersingular kernel,such as the distribution

Z(x) = δ′(x) (5.11)

As the behaviour of this distribution is defined by partial integration, we can easily take itsFourier transformation:

F(δ′(x)) = −jω (5.12)

It is clear that low spatial frequencies will be reduced in magnitude, while the opposite istrue at high frequencies. This as well will lead to high condition numbers. And thus, wesee a pattern arise. When a kernel is too smooth, it will act as a low-pass filter and resultin an ill-conditioned problem. For the same reasons, a kernel which is too singular can actas a high-pass filter. The perfect kernel it seems, would strongly resemble the Dirac-deltadistribution, which is unfortunately not (always) the case in electromagnetic field calculations.

However, there is hope. As we have seen in the last example, we can influence the singularbehaviour by taking derivatives of and/or integrating the kernel. Indeed, the perfect behaviourof the delta-function was ruined after taking its derivative, but the same is true in the oppositedirection: By integrating the hypersingular δ′ we get the nice delta-function. In the nextsection we will discuss how these operations can be performed virtually using preconditioners.First however, we we will elaborate on the principles of preconditioners.

5.1.2 Preconditioners

Preconditioning is a common technique in computational mathematics by which the conditionnumber of a problem can be improved. Let us consider the following linear system:

ZI = V (5.13)

For the above linear system, a left-preconditioner M would be any matrix for which thesystem

MZI = MV (5.14)

while obtaining the same result, has a better condition number. Analogously, M is a right-preconditioner if it improves the condition number for


ZMI ′ = V I = MI ′ (5.15)

Needless to say that both right and left preconditioners can be combined; the main idea isthat the preconditioner leaves the result itself unchanged (up to rounding errors caused bythe limited machine precision). In fact, it the operation of the preconditioner merely boilsdown to a basis-transformation. This insight now explains the quest for a preconditioner. Weare looking for a basis, other than the Legendre polynomials which were used to calculatethe matrix elements, which will improve the condition number. We will now elaborate thissearch for the kernels that appear in the PEC integral equations.

5.1.3 MFIE

We will start this discussion with the TM-MFIE, noting that the discussion for TE-MFIEkernel is completely identical. Let us then write down the complete kernel, keeping in mindthe extra term, which can be considered as a delta distribution:

Z(ρn(sn),ρm(sm)) =1

2δ(sn − sm)− ∂

∂nGk(|ρn(sn)− ρm(sm)|) (5.16)

This kernel consists of 2 parts: The delta part, which as we have mentioned before exhibits theperfect behaviour, and ∂

∂nGk. This second part, thanks to the derivative, will be less smooththan the Hankel-function. Furthermore, its relative importance to the overall kernel is rathersmall. These two facts combined make for a near perfect behaviour of this kernel (and byextention the TE-EFIE kernel). The condition number for these problems is in general verygood, and these problems do not need preconditioning. Thus the Legendre polynomials werefor the calculation of these problems.

5.1.4 TM EFIE

Here the kernel is the 2D-Green function, and the matrix elements take the following form:

Znp,mq =

∫ s

0

∫ 1

0

fp(sn)

J(sn)Gk(|ρn(sn)− ρm(sm)|)gq(sm)

J(sm)dsndsm (5.17)

The 2D-Green function, which is the Hankel function, turns out to be too smooth, (ω →∞G(ω) ∼ 1

ω ). which as we have seen turns the integral operator into a a low-pass filter.The goal is now to take the derivative of this kernel, in order to reduce the smoothness andimprove the condition number. The most straightforward tactic to do so is of course by partialintegration. In a similar manner to the discussion of the ∂2

∂n∂n′Gk-kernel, we can take eithertest or basis functions to be the derivative of another basis.


Znp,mq =

∫ s

0

∫ s

0

∂f(sn)

∂snGk(|ρn(sn)− ρm(sm)|)g(sm)dsndsm (5.18)

Partial integration then gives us the following result:

Znp,mq =

∫ s

0f(sn)Gk(|ρn(sn)− ρm(sm)|)g(sm)dsm

]sn=s

sn=0

−∫ s

0

∫ s

0f(sn)

∂

∂snGk(|ρn(sn)− ρm(sm)|)g(sm)dsndsm (5.19)

If we now make the critical assumption that the function f is continuous on the boundary,the first term vanishes. In this way, we have effectively transferred the derivative to theGreen function. We are left with a kernel which is less smooth, and we can reasonably ex-pect the condition number to improve. Thus the basis we are looking for is the derivativeof a basis which is by itself continuous. Moreover, this original basis should by itself be awell-conditioned basis. Finally, the basis should be polynomial, such that we can expand thisbasis and its derivatives in terms of Legendre Polynomials.

Although there are many possible answers, we opted to use the DCBLF which were introducedfor the TE-EFIE as testing functions. As we have seen, the DCBLF is indeed the derivativeof a well-conditioned, continuous base. As basis function, the adapted Legendre polynomialswere used. But, there is a catch. Because of the derivation, the DCBLF is not full rankanymore. Indeed, the constant function, which is constant over all segments, is mapped tozero by the derivation. This null-space gives rise to a singular value of zero, and the conditionnumber becomes infinite.

The solution to this problem is to extend the basis of derivatives with its null space. Inthis way, the null space is contained in the basis, and the problem is solved. Partially. Itturns out that this artifice has two downsides. For one, the preconditioner which is formedout of this newly extended basis will not be square anymore. The nth order basis has n + 1members, which can be expanded in n + 1 Legendre Polynomials. This means the resultingpreconditioner, will be square. However, by adding an extra function to the total basis, weadd one row to the preconditioner M , resulting in a matrix M ′ which is not square anymore.One effect of this is that the preconditioned problem is not invertable anymore. The way tosolve this issue is to take the preconditioned equation:

M ′ZI = M ′V (5.20)

If we now postmultiply both sides with the transpose (M ′Z)†, the resulting matrix (M ′Z)†(M ′Z)is square, and with the null-space gone, the matrix will also be invertible. The downside ishowever that by postmultiplying, we have also squared the condition number. Indeed, ifκ(M ′Z) = λ, the condition number of the postmultiplied system will be:


κ(

(M ′Z)†(M ′Z))

= λ2 (5.21)

Thus, we have squared the condition number which we could have theoretically had. Thisproblem is quite minor in comparison with another, more fundamental issue we encounterwhen we use this technique. The second problem lies within the nature of the constant termitself. We recall that the set of CBLF are all normalized to 1. However, there exists nocontinuous function which has this extra constant term as its derivative. Needless to saythat this constant could greatly influence the behaviour of the preconditioner M ′, and thecondition number of M ′Z.

To solve this problem, we will start by looking at the spectral behaviour of the new kernel∂∂sGk at high frequencies

F[∂

∂sGk

](ω) ≈ 1

2

This means that our Z-matrix can be approximated by:

Zi,j =< fi,Zgj >≈1

2< fi, gj >

If we now first take the original DCBLF, then < fi, gj > is almost a unitary matrix, exceptfor the sad fact that the constant function is mapped onto zero. Thus, if we expand theDCBLF with the constant function, we only require that the corresponding singular value be12 as well.

The way this is done in practice, is by taking the matrix M ′Z and norming the row cor-responding to the constant function to 1

2 . Of course, the same needs to be done with thecorresponding row in the incoming field vector M ′V .

5.1.5 TE EFIE

Finally we can look at the case of the TE-EFIE. Here, the kernel has a double derivative, andas one might expect, this results in a hypersingular kernel. However, as we have discussedbefore, the integral can be split in two parts:

Znp,mq = k2

∫ 1

0

∫ 1

0f(2tn − 1)Gk(|ρn(tn)− ρm(tm)|)∂ρ(tn)

∂tn· ∂ρ(tm)

∂tmg(2tm − 1)dtndtm

−∫ 1

0

∫ 1

0

∂f(2tn − 1)

∂tnGk(|ρn(tn)− ρm(tm)|)∂f(2tm − 1)

∂tmdtndtm (5.22)

We see that especially for low-frequency problems, where k is small, the second term becomesdominant, which is why we will now focus mainly on this term.


The kernel for this second term is the same as for the TM-EFIE. Thus, we should be able touse the same reasoning to precondition this part of the equation. There, the testing functionsconsisted of the DCBLF, while the basis functions consisted of Legendre Polynomials. Incontrast, in the second term of the non-preconditioned TE-EFIE, we have both the basis andtesting functions being DCBLF. Thus to obtain the same effect as for the TM-EFIE, we wouldlike to change either test or basis functions of this second term into Legendre polynomials.

The way to do this, is to use for example a basis of integrated Legendre polynomials as abasis for total the TE-EFIE. In that case, the first term is calculated using this basis, while,the second term containing Gk will consist of ordinary Legendre polynomials. Here as wellthere is a catch. The functions which return in this second term must be the derivatives ofanother, continuous basis (this continuity was used to derive the above expression for thematrix element). Thus, while such an integrated function can be found for all LegendrePolynomials of order n ≥ 1 (see B.2.1), no such solution exists for P0. The solution is to takehat-functions with the following expression (see also figure 5.1):

hn =

{i− 1 + t t ∈ [0, 1] on segment i

n(1− t) t ∈ [0, 1] on segment n+1(5.23)

Figure 5.1: Hat Expansion

For n segments on a closed geometry, n − 1 such functions can be constructed. It turns outthat in order to obtain a full set of functions, we need to expand this basis with a constantfunction, like in the TM-EFIE. The value of this constant function is obtained analogouslyto the TM case.

5.2 Results

To investigate the effectiveness of the preconditioner, we chose the Lemniscate, because ithas the least regular shape of them all. Therefore we expect the most interesting behaviour.For this geometry, we calculated 5 exponential meshes, using f = 0.9 and a ”radius” a = 2∗π

L


(L the Lemniscate constant). We made meshes of 10, 20, 30, 40 and 50 segments. For thesemeshes, simulations were run for a polynomial P = 1→ 10 and for frequencies of k = 2i withi an integer between -4 and 5. As we calculated all matrices (EFIE,MFIE and CFIE, TMand TE)we accumulated quite some data this way, 3.9GB to be exact.

We can now look at the results on figures 5.2 and 5.3

Figure 5.2: Lemniscate TM: κ Figure 5.3: Lemniscate TE: κ

This plots shows the evolution of the condition number as a function of k. Here, the redresults are the non-preconditioned result, whereas the blue results are those from the precon-ditioned matrices. The results are remarkable. In both TE and TM, the polynomial orderbarely influences the condition number of the preconditioned system. Hence, the very thicklines, which are actually a superposition of 10 individual lines. Only at large frequencies,this breaks down and κ starts to show a more disturbed behaviour. Another feature of thesepreconditioned systems is that the condition number rises slightly with the mesh refinementin the TM case, hence the spacing between the fat lines. In the TE-case, this is less the case.

From the previous, we can conclude that our preconditioner is good at stabilizing the con-dition number in function of mesh refinement and polynomial order. The question is nowif it also serves it’s purpose of decreasing the condition number. Here, we can’ answer with”yes” or ”no”, as it all depends on the frequency. What we see is that for low frequencies, upto k=1, our preconditioner indeed does its job and we get a significant reduction of the con-dition number. However, at high frequencies, the opposite is true, especially for the TM-EFIE.

What is now the origin of this behaviour? We have two hypotheses. One is concerned withthe constant factor which we discussed earlier. As an alternative hypothesis, the smoothnessof the polynomials themselves might start to influence the condition number itself. In anycase, this matter should be investigated more thoroughly, which can be the subject of furtherresearch.

Chapter 6

Conclusion

We started this work with a discussion on the merits and limitations of the Method of Mo-ments calculations. We situated the goal for this work in the need for well-conditioned poly-nomial bases for these Method of Moment calculations. Practically, the goal was to programa higher-order MoM solver capable of solving 2D scattering problems, using preconditioners.

After the introduction, we progressed by giving an outline of the theoretical framework inwhich this work is embedded. First of all, Maxwell’s equations were discussed, where wemade some assumptions regarding the media to be handled by our solver. This discussionwas then followed by an original derivation of the representation formulas in three dimensions.Using these results, the representation formulas in two dimensions were derived. Finally, thistheoretical chapter was concluded with the derivation of the MFIE and the EFIE. Focus wasdirected to the MFIE because of its elusive singular behavior. The subject of the CFIE wasalso briefly touched upon.

Next, we discussed the Method of Moments. Here, after a brief general introduction, wewent on with a more thorough discussion of the different aspects of our computational imple-mentation. For one, we discussed the different kernels and how to compute them. Then weintroduced the testing and basis functions based on Legendre Polynomials, which were usedto calculate the matrix-elements. Furthermore, we addressed the need for a continuous basisfor the TE-EFIE. After a short intermezzo about meshing, we also discussed the numericalintegration of the matrix elements. Finally, we discussed the incoming field vector, and theexact scheme which was used to obtain the results which were discussed in the next chapter.

The code that was generated in this work is able to solve problems involving PEC-scatters,it could be further extended to more general problems but this was considered out of thescope. In the chapter 4 then, the code was used to calculate field problems for differentscatterer geometries. We saw how on circular scatterers, convergence of the error in functionof discretization was as expected. Furthermore, we discussed the currents and fields of theflat strip, square and lemniscate. We for example noticed that the MFIE could not be usedfor the flat strip, as was predicted in chapter 2. We concluded this chapter by introducingthe concept of condition numbers, and by addressing the need for preconditioning which was

43

CHAPTER 6. CONCLUSION 44

handled in the next chapter.

Indeed, the next and final chapter was dedicated to a discussion on preconditioners. Here,we started by discussing how kernel-smoothness influences the condition number of the FieldIntegral Equations. Indeed, using Fourier-analysis, we showed how a kernel can filter outcertain spectral components depending on its smoothness. We explained how the use of apreconditioner can counteract these detrimental effects on the condition number. This wasthen followed by a discussion of the polynomial bases on which our preconditioners were based.Here, we saw that there was still an unresolved issue regarding the nature of one of the basisfunction. The chapter was concluded with a discussion of the results of the preconditioner.

In general, we could conclude that the proposed basis was efficient in stabilizing the infuenceof mesh density and polynomial order on the condition number. So far we were not ableto resolve the frequency dependency. Two possible reasons are given for this. For one, thenature of the polynomials might be at the root of these problems. Furthermore, we are quitecertain that the proposed polynomials are not suited for high-frequency applications. As anextension to this work one could look into better ways to solve the frequency dependency andthe work could also be extended to non-PEC structures.

Appendix A

The infinite PEC cylinder:analytical solution

In this section we shall treat the analytical solution of what might be the simplest of scatteringproblems: a plane wave impinging on a an infinite circular PEC cylinder. As luck would haveit, this is one of the few problems for which an analytical solution can be found. This can beexplained through the fact that the cylindrical coordinate system is one of the few coordinatesystems for which that allows separation of variables for the Helmholtz equation. It was usedextensively in the testing of the solver, even though its circular symmetry, implied insensitivityto certain classes of bugs. Especially the MFIE and TE-EFIE suffered from this trait, thanksto the occurrence of tangent vectors.

A.1 TM-Polarization

The first problem we will discuss is the TM problem, following the derivation in [5]. Theincoming electromagnetic wave with amplitude E0 and wave vector k has the following form(all other field components obviously being zero):

Eiz(ρ) = E0e−jk·ρ (A.1)

H it(ρ) =

1

Z(ek × ez)E0e

−jk·ρ (A.2)

The approach we will follow here, is to calculate the electric field first. The expressions for Ht

are then simply derived using Ampere’s law. To this extent, we first examine the incomingelectric field.

Eiz(ρ) = E0e−jkρ cos(φ) (A.3)

As this expression is periodic in φ, we can take its Fourier-series with regard to that variable,while treating ρ as a constant. It turns out that the integral rendering the nmathdrmth

45

APPENDIX A. THE INFINITE PEC CYLINDER: ANALYTICAL SOLUTION 46

Fourier-coefficient strongly resembles a modified form of Bessel’s first integral, which generatesthe n-th order Bessel function of the first kind:

Jn(kρ) =1

2πjn

∫ 2π

0eikρ cos(φ)+nφdφ (A.4)

This means we can write the incoming field as follows:

Eiz(ρ) = E0

+∞∑

n=−∞(−j)nJn(kρ)ejnφ (A.5)

Let us now consider the scattered field, which is the matter at interest. Here again, we cancompute the Fourier-series, although this time, the coefficients themselves are unknown.

Esz(ρ) =+∞∑

n=−∞Esn(ρ)ejnφ (A.6)

However, as the scattered field itself needs to be a solution of Maxwell’s equations, we can tryto obtain an expression for these coefficients by inserting (A.6) into the source-free Helmholtzequation for the electric field:

∇2Ez + k2Ez = 0 (A.7)

The result is Bessel’s differential equation:

1

ρ

d

dρρ

d

dρEsn(ρ) +

(k2 − n2

ρ2

)Esn(ρ) = 0 (A.8)

The solutions to this equations are linear combinations of Bessel function of the first andsecond kind.

Esn(ρ) = AsnJn(kρ) +BsnYn(kρ) (A.9)

This is proof that our Fourier expansion of the plane wave is indeed a solution to the afore-mentioned Helmholtz equation. Moreover, we can now look for solutions that satisfy theradiation condition. The first and second kind Bessel functions are real functions, which havethe following behaviour

limkρ→∞

Jn(kρ) =

√2

πkρcos(kρ− 2n+ 1

4π) (A.10)

limkρ→∞

Yn(kρ) =

√2

πkρsin(kρ− 2n+ 1

4π) (A.11)

These functions by themselves, since they are real, are unable to simulate the correct be-haviour for a travelling wave. Instead, a complex linear combination needs to be taken, asfollows:

H(2)n (ρ) = Jn(kρ)− jYn(kρ) (A.12)


These are the Hankel Functions of the second kind. With this in mind, we can write thescattered field as follows:

Esz(ρ) = −E0

+∞∑

n=−∞cnH

(2)n (kρ)ejnφ (A.13)

The last step is to impose the interface conditions on the cylinder (2.12). On any PEC surface,the tangential component of the electric field vanishes Ez = Eiz +Esz = 0 This means that foreach n:

cn = −E0(−j)n Jn(ka)

H(2)n (ka)

(A.14)

Using these values for cn, we finally obtain the following expression for the scattered field:

Esz(ρ) = −E0

+∞∑

n=−∞(−j)n

[Jn(ka)

H(2)n (ka)

H(2)n (kρ)

]ejnφ (A.15)

The total field then follows simply after the addition of (A.5) and (A.15)

Esz(ρ) = E0

+∞∑

n=−∞(−j)n

[Jn(kρ)− Jn(ka)

H(2)n (ka)

H(2)n (kρ)

]ejnφ (A.16)

If we now look at the expressions for the magnetic field components, we can use the followingexpression:

Ht(ρ) =1

−jωµ∇× Ezz =j

kZ

(1

ρ

∂Ez∂φ

ρ− ∂Ez∂ρ

φ

)(A.17)

This gives us the expressions for the magnetic field components:

Hρ(ρ) = −E01

Zkρ

+∞∑

n=−∞(−j)nn

[Jn(kρ)− Jn(ka)

H(2)n (ka)

H(2)n (kρ)

]ejnφ (A.18)

Hφ(ρ) = −E0j

Z

+∞∑

n=−∞(−j)n

[J ′n(kρ)− Jn(ka)

H(2)n (ka)

H(2)′n (kρ)

]ejnφ (A.19)

Conveniently, we have also obtained the surface current density this way. Indeed, becauseof (2.13), it follows that the current density equals Js = zHφ We will end this section bytransforming this expression for the surface current into a more aesthetical one. We will startby rewriting it in the following way:

Hφ(ρ) = −E0j

Z

+∞∑

n=−∞

(−j)n

H(2)n (ka)

[Jn(kρ)H(2)

n (ka)− Jn(ka)H(2)′n (kρ)

]ejnφ (A.20)


On the cylinder surface, ρ = a, so we can now invoke the Wronskian identity [6]

J ′n(ka)H(2)n (ka)− Jn(ka)H(2)′

n (ka) =2j

πka(A.21)

We can write the surface current density then as follows:

Js(ρ) = E02

πkZa

+∞∑

n=−∞(−j)n 1

H(2)n (ka)

ejnφ (A.22)

A.2 TE-Polarization

For the TE polarization, we can start in the same way as the TE-polarization. First of all,we have similar expression for the incoming fields:

H iz(ρ) = H0e

−jk·ρ (A.23)

Eit(ρ) = −Z(ek × ez)H0e

−jk·ρ (A.24)

Much in the same way, we can write the Fourier expansions for the incoming and scatteredmagnetic field using Bessel functions and Hankel functions.

H iz(ρ) = H0

+∞∑

n=−∞(−j)nJn(kρ)ejnφ (A.25)

Hsz (ρ) = H0

+∞∑

n=−∞cnH

(2)n (kρ)ejnφ (A.26)

At this point, our discussion digresses from the previous section, as the tangential componentof the magnetic field does not vanish on the boundary of the cylinder. However, the tangentialelectric field of course still vanishes, so in order to impose these interface conditions, we firstneed to establish expressions for the incoming and scattering electric field. However, insteadof using (A.24), we will explicitly use Ampere’s law for the tangential field:

Eφ(ρ) =jZ

k

∂Hz

∂ρ(A.27)

For the scattered and incoming field components we get:

Eiφ(ρ) =jZ

k

∂H iz

∂ρ= jH0Z

+∞∑

n=−∞(−j)nJ ′n(kρ)ejnφ (A.28)

Esφ(ρ) =jZ

k

∂Hsz

∂ρ= jH0Z

+∞∑

n=−∞(−j)ncnH(2)′

n (kρ)ejnφ (A.29)


It is immediately clear that this time, the coefficients cn become:

cn = −H0(−j)n J ′n(ka)

H(2)′n (ka)

(A.30)

From here, we can immediately go to the results for the total field components:

Hz(ρ) = H0

+∞∑

n=−∞(−j)n

[Jn(kρ)− J ′n(ka)

H(2)′n (ka)

H(2)n (kρ)

]ejnφ (A.31)

Eρ(ρ) = −H0Z

kρ

+∞∑

n=−∞(−j)nn

[Jn(kρ)− Jn(ka)′

H(2)′n (ka)

H(2)n (kρ)

]ejnφ (A.32)

Eφ(ρ) = −jZH0

+∞∑

n=−∞(−j)n

[J ′n(kρ)− J ′n(ka)

H(2)′n (ka)

H(2)′n (kρ)

]ejnφ (A.33)

Using the same Wronskian identity as before, we get the current density:

Jt(ρ) =2jH0

πka

+∞∑

n=−∞(−j)n ejnφ

H(2)′n (ka)

(A.34)

Appendix B

Polynomial Basis

B.1 Legendre Polynomials

The polynomials which were used to calculate the matrix elements, and which were effectivelyused as testing and basis functions are the Legendre polynomials for some problems. Theyare generated by Legendre’s differential equation:

d

dx

[(1− x2)

d

dxPn(x)

]+ n(n+ 1)Pn(x) = 0 (B.1)

Usually, these are normalized such that Pn(1) = 1, rendering an orthogonal basis on theinterval −1 ≤ x ≤ 1:

∫ 1

−1Pm(x)Pn(x) dx =

2

2n+ 1δmn (B.2)

The polynomials used in our case differed in two ways. First of all, they were shifted so thatthe interval of orthogonality was 0 ≤ t ≤ 1, with the inner product:

∫ 1

−1Pm(2t− 1)Pn(2t− 1) dx =

1

2n+ 1δmn (B.3)

Then, they were normalized in order to obtain an orthonormal basis.

pn(x) =Pn(2x− 1)

‖Pn(2x− 1)‖ (B.4)

50

APPENDIX B. POLYNOMIAL BASIS 51

B.2 Integrated Legendre Polynomials + Hat expansion

B.2.1 Integrated Legendre Polynomials

φn(t) =

∫ t

0

Pn(2x− 1)

‖Pn(2x− 1)‖dx, for n > 0; t ∈ [0, 1] (B.5)

It is easy to see thatφn(0) = φn(1) = 0

The latter identity arises from the fact that φn(1) =< Pn, P0 >= 0 for n > 0

Now, to expand these functions in Legendre Polynomials, we can simply use the followingexpression

2Pn(2x− 1)

‖Pn(2x− 1)‖2=

d

dx[Pn+1(2x− 1)− Pn−1(2x− 1)] (B.6)

Thus,

φn(t) =

∫ t

0

Pn(2x− 1)

‖Pn(2x− 1)‖dx =‖Pn(2x− 1)‖

2[Pn+1(2x− 1)− Pn−1(2x− 1)] (B.7)

B.2.2 Hat expansion

The hats for this set of basis functions are linear hat functions spanning multiple segments.

hn =

{i− 1 + t t ∈ [0, 1] on segment i

n(1− t) t ∈ [0, 1] on segment n+1(B.8)

In terms of Legendre Polynomials, this becomes:

hn =

{(i− 1

2)P0 + P12 t ∈ [0, 1] on segment i

nP02 − nP1

2 t ∈ [0, 1] on segment n+1(B.9)

Of course, these should be normalized such that the L2 norm of their derivative be unity.Thus, in the end the expansion functions take the following form:

hn =

{(2i−1

4n )P0 + P14n t ∈ [0, 1] on segment i

P04 − P1

4 t ∈ [0, 1] on segment n+1(B.10)


B.3 Continuous Base Line Functions (CBLF)

B.3.1 Associated Legendre Polynomials

We will now prove that the functions defined 3.26 are equivalent to the associated Legendrepolynomials. Thus we start from:

qn(x) = Anx (1− x)P 2,2n (2x− 1) (B.11)

An =

√Γ(n+ 5)Γ(n+ 1)(2n+ 5)

Γ(n+ 3)2(B.12)

To make things easier, there exists an expression relating the derivatives of Jacobi Polynomi-als:

dkPα,βn (x)

dxk=

Γ(α+ β + n+ 1 + k)

2kΓ(α+ β + n+ 1)Pα+k,β+kn−k (x) (B.13)

Substituting this in B.11for k = −2 renders the following equation:

qn(x) = AnΓ(n+ 3)

Γ(n+ 5)x (1− x)

d2

dx2P 0,0n+2(2x− 1) (B.14)

qn(x) = AnΓ(n+ 3)

Γ(n+ 5)x (1− x)

d2

dx2Pn+2(2x− 1) (B.15)

In the last step, we used the fact that the Jacobi Polynomial P 0,0n+2 is identical to the ordinary

Legendre Polynomial Pn+2. Furthermore, we can now use the expression for the AssociatedLegendre Polynomials to further reduce the above expressions.

Pmn (x) = (−1)m(1− x2)m2

dm

dxmPn(x) (B.16)

qn(x) = AnΓ(n+ 3)

Γ(n+ 5)P 2n+2(2x− 1) (B.17)

To conclude, we check whether the polynomials are still normed to unity.

‖qn(x)‖ =

√Γ(n+ 5)Γ(n+ 1)(2n+ 5)

Γ(n+ 3)2

Γ(n+ 3)

Γ(n+ 5)‖P 2

n+2(2x− 1)‖ (B.18)

=

√Γ(n+ 1)(2n+ 5)

Γ(n+ 5)‖P 2

n+2(2x− 1)‖ (B.19)

= 1 (B.20)


In what follows we will consider:

Bn =

√Γ(n+ 1)(2n+ 5)

Γ(n+ 5)(B.21)

(B.22)

Expanding Associated Legendre Polynomials in Legendre Polynomials

Now the question arises how these Associated Legendres can be expressed as a linear combi-nation of Ordinary Legendres. Here to we first consider the following:

qn(x) = Bn(x− x2)d2

dx2Pn+2(2x− 1) (B.23)

= Bn

[d

dx

((x− x2)

d

dxPn+2(2x− 1)

)+ (2x− 1)

d

dxPn+2(2x− 1)

](B.24)

If we now apply the following identity:

d

dx

[(x− x2)

d

dxPn(2x− 1)

]= −n(n+ 1)Pn(2x− 1) (B.25)

This yields:

qn(x) = Bn

[−(n+ 2)(n+ 3)Pn+2(2x− 1) + (2x− 1)

d

dxPn+2(2x− 1)

](B.26)

We continue by eliminating the derivatives of the Legendres using the following property:

d

dxPn+2(2x− 1) = 2

bn+12c∑

i=0

(2(n+ 1− 2i) + 1)Pn+1−2i (B.27)

This leads to:

qn(x) = Bn [−(n+ 2)(n+ 3)Pn+2(2x− 1)

+ 2

bn+12c∑

i=0

(2(n+ 1− 2i) + 1)(2x− 1)Pn+1−2i(2x− 1)

(B.28)

Now all that’s left is convert the summation terms in (2x− 1)P (2x− 1) to terms containingpurely Legendre Polynomials. Hereto, we use the following identity.

(2x− 1)Pn(2x− 1) =(n+ 1)Pn+1(2x− 1) + nPn−1(2x− 1)

2n+ 1(B.29)


Which gives us:

qn(x) = Bn [−(n+ 2)(n+ 3)Pn+2(2x− 1)

+ 2

bn+12c∑

i=0

((n− 2i+ 2)Pn−2i+2(2x− 1) + (n+ 1− 2i)Pn−2i(2x− 1))2(n− 2i+ 2) + 1

2(n+ 1− 2i) + 1

(B.30)

If we now shuffle things up a bit:

qn(x) = Bn [−(n+ 2)(n+ 3)Pn+2(2x− 1)

+ 2

bn+12c∑

i=0

(n− 2i+ 2)Pn−2i+2(2x− 1) + (n+ 1− 2i)Pn−2i(2x− 1)

(B.31)

qn(x) = Bn

−(n+ 2)(n+ 1)Pn+2(2x− 1) + 2

bn2c∑

i=0

(2(n− 2i) + 1)Pn−2i(2x− 1)

(B.32)

We finally get the following expansion of associated Legendre Polynomials:

qn(x) = Bn

−(n+ 2)(n+ 1)Pn+2(2x− 1) + 2

bn2c∑

i=0

Pn−2i(2x− 1)

‖Pn−2i(2x− 1)‖2

(B.33)

Another useful expression which we shall use in the next section is the following one:

qn(x) = Bn

[−(n+ 2)(n+ 1)Pn+2(2x− 1) +

d

dxPn+1(2x− 1)

](B.34)

B.3.2 Nodal function

Expansion of the nodal function in Legendre polynomials

In this section we will discuss the rising and descending part of the nodal-function:

g±(x) =d

dx

(N + 2)PN (2x− 1)±NPN+1(2x− 1)√4N(N + 1)2(N + 2)

(B.35)

(B.36)


We will first set the constant:

Cn =1√

4N(N + 1)2(N + 2)

this becomes:

g±(x) = Cnd

dx((N + 2)PN (2x− 1)±NPN+1(2x− 1)) (B.37)

g±(x) = 2Cn

(N + 2)

bN−12c∑

i=0

PN−1−2i(2x− 1)

‖PN−1−2i(2x− 1)‖2±N

bN2c∑

i=0

PN−2i(2x− 1)

‖PN−2i(2x− 1)‖2

(B.38)

Coupling constants

The nodal functions are defined as follows:

hn∧n+1(ρ) =

hn/(ρ(t)) = c+g+(t), on segment n

h\n+1(ρ(t)) = c−g−(t), on segment n+1

0, elsewhere

(B.39)

There are two constraints on the nodal functions, Indeed they must be both normalized tounity and continuous. This leads to two conditions:

{‖hn∧n+1‖ = 1 Normalization

c+g+(1) = c−g−(0) Continuity(B.40)

This leads to the following system of equations:{‖c+g+(t)‖+ ‖c−g−(t)‖ = 1

c+g+(1) = c−g−(0)(B.41)

Taking into account the fact that the ascending and descending part of the hat functions arealready normalized, we can now determine the values of both c+ and c− as follows:

{|c+|+ |c−| = 1

|c+| = | c−g−(0)g+(1) |

(B.42)

{|c−| = |g+(1)|

|g−(0)|+|g+(0)|c+ = c−g−(0)

g+(1)

(B.43)

There are two distinct solutions to this problem, differing only in the sign.


B.4 Derived Continuous Base Line Function (DCBLF)

Here we use the derivative of the previously defined basis.

B.4.1 Derivative of the Associated Legendre Polynomials

We start by taking the derivative of equation B.34 derived earlier.

d

dxqn(x) = Bn

[−(n+ 2)(n+ 1)

d

dxPn+2(2x− 1) +

d2

dx2Pn+1(2x− 1)

](B.44)

In order to obtain an expansion in Legendre polynomials, it is useful to derive an expressionfor the second derivative of the Legendre Polynomials first.

d2

dx2Pn+1(2x− 1) = 2

bn2c∑

i=0

(2(n− 2i) + 1)d

dxPn−2i(2x− 1) (B.45)

= 2(2n+ 1)

bn2c∑

i=0

d

dxPn−2i(2x− 1) +

bn2c∑

i=0

(−8i)d

dxPn−2i(2x− 1) (B.46)

Now, we want to transform the latter two terms into an expression which consists purely outof Legendres. This can be done using (B.47):

2(2n+ 1)Pn(2x− 1) =d

dx[Pn+1(2x− 1)− Pn−1(2x− 1)] (B.47)

d2

dx2Pn+1(2x− 1) = 4(2n+ 1)

bn2c∑

i=0

(i+ 1)Pn−1−2i(2x− 1)

‖Pn−1−2i(2x− 1)‖2

− 8

bn2c∑

i=0

i(i+ 1)Pn−1−2i(2x− 1)

‖Pn−1−2i(2x− 1)‖2(B.48)

If we shift with the summation coefficient, i→ i− 1, this becomes:

= 4(2n+ 1)

bn+12c∑

i=0

iPn+1−2i(2x− 1)

‖Pn+1−2i(2x− 1)‖2− 8

bn+12c∑

i=0

i(i− 1)Pn−1−2i(2x− 1)

‖Pn−1−2i(2x− 1)‖2(B.49)

= 4

bn+12c∑

i=0

(2(n+ 1− i) + 1)iPn+1−2i(2x− 1)

‖Pn+1−2i(2x− 1)‖2(B.50)


Using (B.27), we can transform the total expression for de derivative of qn into:

d

dxqn(x) = −4Bn

bn+1

2c∑

i=0

((n+ 2)(n+ 1)

2− (2(n+ 1− i) + 1)i

)Pn+1−2i(2x− 1)

‖Pn+1−2i(2x− 1)‖2

(B.51)

The coefficients which finally turn up, are actually the triangular numbers, so we can aes-thetically write down the previous equation as:

d

dxqn(x) = −4Bn

bn+1

2c∑

i=0

Tn+1−2iPn+1−2i(2x− 1)

‖Pn+1−2i(2x− 1)‖2

(B.52)

Tn = 1 + 2 + 3 + ...n =m∑

i=1

i =n(n+ 1)

2(B.53)

B.4.2 Derivative of the nodal function

The derivative of the nodal parts is the sum of 2 double derivatives of Legendre polynomials.Using (B.27), we can simply state that the expansion in Legendres can be written as follows:

d

dxg±(x) = Cn

d2

dx2((N + 2)PN (2x− 1)±NPN+1(2x− 1)) (B.54)

= 4Cn

(N + 2)

bN2c∑

i=0

(2(N − i) + 1)iPN−2i(2x− 1)

‖PN−2i(2x− 1)‖2(B.55)

±NbN+1

2c∑

i=0

(2(N + 1− i) + 1)iPN+1−2i(2x− 1)

‖PN+1−2i(2x− 1)‖2

(B.56)

We can now reuse the same constants for the nodal function, and the derivative becomes:

d′n∧n+1(ρ) =

d′n/(ρ(t)) = c+ddtg+(t), on segment n

d′\n+1(ρ(t)) = c− ddtg−(t), on segment n+1

0, elsewhere

(B.57)

Bibliography

[1] D. B. Davidson, Computational electromagnetics for RF and microwave engineering.Cambridge University Press, 2005.

[2] I. Bogaert and J. De Zaeytijd, Elektromagnetische veldberekeningen van extreem grote2D-problemen. 2004.

[3] J. C. Maxwell, A treatise on electricity and magnetism, vol. 1. Clarendon press, 1881.

[4] W. C. Chew, M. S. Tong, and B. Hu, Integral equation methods for electromagnetic andelastic waves, vol. 12. Morgan & Claypool Publishers, 2009.

[5] D. De Zutter, Elektromagnetisme I. 2010.

[6] “NIST Digital Library of Mathematical Functions.” http://dlmf.nist.gov/, Release 1.0.5of 2012-10-01, 2012.

[7] J. Volakis, Integral Equation Methods for Electromagnetics. Institution of Engineeringand Technology, 2012.

[8] K. F. Warnick, Numerical analysis for electromagnetic integral equations. Artech HousePublishers, 2008.

[9] E. Jørgensen, Higher-order integral equation methods in computational electromagnetics.Ørsted-DTU, 2003.

[10] D. Amos, “A subroutine package for bessel and hankel functions of complex arguments,”1996.

[11] J. Hesthaven, “From electrostatics to almost optimal nodal sets for polynomial interpo-lation in a simplex,” SIAM Journal on Numerical Analysis, vol. 35, no. 2, pp. 655–676,1998.

[12] J. Ma, V. Rokhlin, and S. Wandzura, “Generalized gaussian quadrature rules for systemsof arbitrary functions,” SIAM Journal on Numerical Analysis, vol. 33, no. 3, pp. 971–996,1996.

[13] E. W. Weisstein, “Lemniscate. From MathWorld—A Wolfram Web Resource,” 2013.Last visited on 14/5/2013.

[14] N. Van den Bergh, Wiskundige Ingenieurstechnieken. 2010.

58

List of Figures

2.1 Singularity at ρ− ρ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1 Circle: Total Field |Ez| (k=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Circle: Total Field |Hz| (k=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Circle: Current error Jz (k=2) . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 Circle: Current error Jt (k=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.5 Circle: TM Current error (k=2) . . . . . . . . . . . . . . . . . . . . . . . . . . 304.6 Flat Strip: Total Field |Ez| (k=2) . . . . . . . . . . . . . . . . . . . . . . . . 314.7 Flat Strip: Total Field |Hz| (k=2) . . . . . . . . . . . . . . . . . . . . . . . . 314.8 Square: Total Field |Ez| (k=2) . . . . . . . . . . . . . . . . . . . . . . . . . . 314.9 Square: Total Field |Hz| (k=2) . . . . . . . . . . . . . . . . . . . . . . . . . . 314.10 Square: |Jz| (k=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.11 Square: |Jt| (k=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.12 Lemniscate: Meshing (f=0.9) . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.13 Lemniscate: Total Field |Ez| (k=2) . . . . . . . . . . . . . . . . . . . . . . . . 334.14 Lemniscate: Total Field |Hz| (k=2) . . . . . . . . . . . . . . . . . . . . . . . . 334.15 κ EFIE (k=2,P=10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.16 κ MFIE (k=2, P=10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.1 Hat Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Lemniscate TM: κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3 Lemniscate TE: κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

59

2d electromagnetic field mom calculations using well...

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