Development of a Domain Decomposition Method for the...

Wouter Nys

for the Modelling of Through-Silicon ViasDevelopment of a Domain Decomposition Method

Academic year 2016-2017Faculty of Engineering and ArchitectureChair: Prof. dr. ir. Bart DhoedtDepartment of Information Technology

Master of Science in Engineering PhysicsMaster's dissertation submitted in order to obtain the academic degree of

Counsellors: Ir. Martijn Huynen, Ir. Michiel GossyeSupervisors: Prof. dr. ir. Dries Vande Ginste, Prof. dr. ir. Hendrik Rogier

Preface

Sunday 23 September 2012 – I remember as if it were only yesterday – I was waiting on the

railway platform in Kortrijk, West-Flanders. Young, self-confident, maybe a bit naive, I didn’t

know exactly what I was up to, except for the fact that this railway trip would at least take five

years. Today, almost five years later, the major part of this journey is already behind me. It

sure has been an adventurous trip, sometimes smooth, sometimes bumpy. But luckily, despite

some delays and strikes, the train has always arrived in time for the past four summer vacations.

Especially the last part of the journey has been extraordinary. Therefore, I would like to

express my sincerest gratitude to a few people who made that possible. First of all, I would

like to thank prof. dr. ir. Dries Vande Ginste and prof. dr. ir. Hendrik Rogier, for selling me the

ticket and for giving me the opportunity to immerse myself in the interesting research domain

of this master’s dissertation at the Electromagnetics Group of the Department of Information

Technology. Thanks to the progress meetings, feedback and words of encouragement, the train

has always remained on track.

Furthermore, this last trip would be unthinkable without the train attendants, my two wise

counsellors, ir. Martijn Huynen and ir. Michiel Gossye, who were always available to answer my

numerous questions and offered me the most profound guidance and feedback. Thanks to them,

I was able to not only learn a lot more about computational electromagnetics, but also set some

important steps towards a master’s degree in English. I appreciate all the patient help with the

HYBRID-code and the thorough corrections of the various versions of this master’s dissertation.

I would like to give all the other members of the EM group a sincere word of gratitude as well,

for creating a great atmosphere and always being interested in my progress.

Some of my fellow travellers deserve to be mentioned as well. Firstly, all the Engineering Physics

students, for the hundreds of classes we spent together. Without their presence and the funny

moments during breaks and lunchtimes, I would not form such a broad smile when recalling the

journey. In addition, I would like to thank my fellow thesis students Robbe Riem, Robin Sercu

and Stijn Meersman, for their empathy and understanding during the last ride. Last but not

least, words are not adequate to express my deepest gratitude to my beloved family and friends,

who have not only supported me relentlessly during this complete railway trip, but foremost put

me on that platform on 23 September 2012, with the necessary luggage.

Finally, I would like to thank the train driver. Whoever he or she is, without him or her, I would

still stand on that platform, waiting for the train to arrive.

Now that the destiny is within reach, I am looking forward to arrive at the terminus of this

five-year railway trip, and I am convinced that it will not be the end of the journey – only the

beginning.

Wouter Nys, June 2017

Copyright Agreement

The author gives permission to make this master’s dissertation available for consultation and

to copy parts of this master’s dissertation for personal use. In the case of any other use,

the copyright terms have to be respected, in particular with regard to the obligation to state

expressly the source when quoting results from this master’s dissertation.

Wouter Nys, June 2017

Development of a Domain Decomposition Methodfor the Modelling of Through-Silicon Vias

by

Wouter Nys

Master’s Dissertation submitted to obtain the academic degree of

Master of Science in Engineering Physics

Academic 2016–2017

Promoters: Prof. dr. ir. Dries Vande Ginste, Prof. dr. ir. Hendrik Rogier

Supervisors: Ir. M. Huynen, Ir. M. Gossye

Faculty of Engineering and Architecture

Ghent University

Department of Information Technology

Chairman: Prof. dr. ir. Bart Dhoedt

Summary

To keep up with Moore’s law, the semiconductor industry is making a transition from 2-D to 3-D

ICs. This encompasses stacking of several layers of wafers, which are electrically interconnected

by through-silicon vias (TSVs). The modelling and prediction of the electrical performance

of these components is of great practical interest. When operating at high frequencies (of

the order of GHz), full-wave electromagnetic simulation techniques have to be employed to

obtain an accurate solution. Full-wave domain decomposition methods (DDMs) subdivide the

computational domain into clearly defined regions and describe each region with the preferred

solution technique. These methods have gained considerable attention in the past decades, due

to ongoing research that allows their application in complex and practically relevant problems.

In this master’s dissertation, a novel DDM is designed, in which each of the N subdomains,

embedded in a homogeneous background medium, is solved with the boundary element method.

The unknowns are the currents on the interface between the regions, which are coupled with

Robin transmission conditions, resulting in a solvable set of linear equations.

With this matrix equation at hand, the developed method is implemented and thoroughly tested.

First, the formalism is validated using the analytical solution for the scattering at a sphere, for

a range of different materials. Subsequently, the same method is applied to the scattering at a

cube and a cylinder. Further, we broaden our scope and test the formalism for two objects, first

separated, then brought together to form a junction. For all those cases, validation with in-house

data is provided. Finally, a new formalism to describe junctions is proposed, implemented and

validated, which lays the foundation for the full-wave modelling of TSVs.

Keywords

Through-silicon vias; full-wave electromagnetic simulation; domain decomposition methods;

boundary integral equations; method of moments; Robin transmission conditions; junctions

Development of a Domain Decomposition Methodfor the Modelling of Through-Silicon Vias

Wouter Nys

Supervisors: prof. dr. ir. D. Vande Ginste, prof. dr. ir. H. Rogier, ir. M. Huynen and ir. M. Gossye

Abstract— The goal of this master’s dissertation is to develop a domaindecomposition method to model through-silicon vias. Therefore, a matrixequation to solve the scattering problem at a piecewise homogeneous objectin a homogeneous background medium is constructed. All subdomains arehandled with the boundary integral equation method and the respective setsof unknowns are coupled with Robin transmission conditions. Rao-Wilton-Glisson basis and test functions are utilised in our discretisation scheme,making it compatible with available method of moments software. Numer-ical results for several examples of a single scatter and multiple scatterersare provided for validation purposes.

Keywords— through-silicon vias; full-wave electromagnetic simulation;domain decomposition methods; boundary integral equations; method ofmoments; Robin transmission conditions; junctions

I. INTRODUCTION

ONE of the recent developments in the semiconductor in-dustry is a transition from 2-D to 3-D integrated circuits(ICs) [1]. This encompasses stacking of several layers of sub-strates on top of each other. As such, complex devices can bemade substantially smaller, and a lot of functionality is imple-mented on a limited footprint. Furthermore, due to shorter in-terconnection lengths, total power consumption, wire delay andparasitic effects are reduced. Moreover, ICs with different tech-nologies and functionalities can be stacked on top of each other.All these advantages illustrate the potential of 3-D ICs to extendMoore’s law.

Electrical interconnections between the several stacked lay-ers are ensured by through-silicon vias (TSV), i.e. cylindricalmetal lines (usually copper or tungsten), coated with an oxidelayer, embedded in a silicon substrate. Models to predict theperformance of TSVs are of utmost interest to the semiconduc-tor industry, since they reduce the time-to-market and produc-tion costs. Contemporary devices operate at high clock speeds(of the order of GHz), so full-wave electromagnetic simulationtechniques have to be employed to achieve an accurate solu-tion [2].

In the context of modelling electrically large objects withcomplex geometries and materials, the attention of the compu-tational electromagnetics (CEM) community was drawn to do-main decomposition methods (DDMs) during the past decades,due to ongoing research that allows their application in complexand practically relevant problems. These DDMs subdivide thecomputational domain in distinct regions and solve each subdo-main with the preferred solution technique [3]. Heterogeneousbounded regions are handled with 3-D volume methods, suchas the finite element (FE) method, which adequately deal withvariations in the material parameters. When homogeneous re-gions are considered, the problem can be projected onto a set ofunknowns on the interface; this is called the boundary integralequation (BIE) method.

As we are interested in high-frequency applications, the prob-lem of modelling TSVs should be treated with a full-wave sim-

ulation method. The TSV structure shows some clearly distin-guishable homogeneous subdomains. Therefore, we construct anovel DDM based on BIEs.

In Section II, a DDM based on Robin transmission conditions(RTCs) is developed. This framework is validated for a singlescatterer in Section III-A and for two scatterers in Section III-B.Finally, an alternative method to model junctions is proposed.

II. DOMAIN DECOMPOSITION METHOD

We consider a general 3-D set-up in which N homogeneousregions Ω1, Ω2, ..., ΩN , characterised by the permittivity εi andpermeability µi (i ∈ {0, ..., N}), are situated in a homogeneousbackground medium Ω0. The interface between two adjacentregions Ωp and Ωq is denoted as Γpq . On the surface of Ωi, n̂iis the normal vector, pointing inwards. The ejωt convention isemployed for the incident plane waves. This set-up is depictedin Fig. 1.

Fig. 1. General 3-D volume with N+1 regions.

A BIE method can be employed to express the problem interms of the magnetic and electric currents mp and jp, residingon each boundary of Ωp. This is mathematically expressed bythe Stratton-Chu representation theorem [4]. In order to find aunique solution to the global problem, boundary conditions haveto be imposed. In this master’s dissertation, RTCs are employed.On Γpq , the following conditions are imposed:

jp + n̂p ×mp = −jq + n̂q ×mq , (1)n̂p × jp −mp = n̂q × jq + mq . (2)

After expanding the unknowns in basis functions and passingto the weak formulation, the following system matrix equationis obtained:

[Si][D]i{x}+∑

q∈S(i)[Ciq][D]q{x} = [D]i{b}, (3)

with b the known excitation vector, x the unknown solution vec-tor, S(i) the set of neighbours of Ωi, [D]i a matrix selecting theelements belonging to Ωi and [Si], [Ciq] the subdomain matrix

of Ωi and the coupling matrix between Ωi and Ωq , respectively,defined by:

[Si] =

( −Ki ηiTi+ 12G′i− 1ηiTi −

12G′i −Ki

), (4)

[Ciq] =

(− 12Giq − 12G′iq12G′iq − 12Giq

), (5)

with the interaction and projection matrices given by:

[Ki]kl = 〈n̂i ×wk,Ki[f l]〉Γi , (6)[Ti]kl = 〈n̂i ×wk, Ti[f l]〉Γi , (7)[G′i]kl = 〈n̂i ×wk, n̂i × f l〉Γi , (8)

[Giq]kl = 〈n̂i ×wk,f l〉Γiq , (9)[G′iq]kl = 〈n̂i ×wk, n̂q × f l〉Γiq , (10)

where Ki, Ti are the magnetic and electric field integral opera-tors [5] and f l and wk general basis and test functions, respec-tively (k, l ∈ {1, ..., Li} and Li the number of edge elementsof Γi).

III. NUMERICAL RESULTS

A. Application to a single scatterer

Using this system matrix, the developed DDM is imple-mented and tested for the simple case of a single scattererfor validation purposes. Starting with a sphere, an analyti-cal validation is provided by the Mie series. Three differentmeshes are employed, with 192 (Mesh A), 720 (Mesh B) and1152 (Mesh C) edge elements. The accuracy of the devel-oped DDM, with Rao-Wilton-Glisson functions (RWGs) as ba-sis and test functions [6], is further evaluated by comparisonwith two other popular numerical full-wave methods, viz. theCalderón preconditioned Poggio-Miller-Chew-Harrington-Wu-Tsai (CP-PMCHWT) [5] and the N-Müller equations [7]. Theresults for a sphere with radius 1 m, εr = 4, at a frequency of100 MHz, are depicted in Table I.

TABLE IRMS ERROR ON THE RADAR CROSS SECTION (RCS) FOR THE SCATTERING

AT A DIELECTRIC SPHERE WITH εr = 4 FOR THREE DIFFERENT MESHES

AND METHODS, AT 100 MHZ.

Method Mesh A Mesh B Mesh CDeveloped DDM 0.64% 0.25% 0.11%CP-PMCHWT 0.95% 0.26% 0.10%N-Müller 1.33% 0.39% 0.16%

The method developed in this master’s dissertation is clearlysuperior to the N-Müller equation and equivalent to theCP-PMCHWT equation in terms of accuracy. However, condi-tion numbers of the order of 106 are recorded, resulting in a highnumber of iterations and long computation times, making thismethod less time-efficient than the other two. Efforts are madeto lower these condition numbers, by inspecting different test-ing schemes on the one hand, viz. employing Buffa-Christiansen(BC) functions [8] and a mixed testing scheme, and by applyinga technique dubbed current rescaling, on the other hand. Thelatter attempt reduces the condition number to an order of 104.

B. Application to multiple scatterers

First, a linearly polarised plane wave with a frequency of100 MHz is scattered at two non-adjacent cubes with side 1 m,εr,1 = εr,2 = 4 and separation 1 m. The correctness of ourmethod is demonstrated by validation with two in-house BIEmethods [9], [10]. The radar cross section (RCS) in the xz-planefor the three methods is depicted in Fig. 2.

0 30 60 90 120 150 180−10

−5

0

5

10

15

20

θ [◦]

RC

S[d

B]

Validation 1Validation 2Results

Fig. 2. The RCS in the xz-plane for the scattering at two cubes withεr,1 = εr,2 = 4, side 1m and separation 1m, at a frequency of 100MHz.

Further, we consider the more interesting case of two adjacentregions. This configuration of a junction, which will be exam-ined in the remainder of this section, is illustrated in Fig. 3.

Fig. 3. Schematic illustration of the scattering of a linearly polarised plane waveat a junction of two blocks of material.

An intuitive course of action is considered, where a junctionis formed by bringing two cubes together step by step. To val-idate the results, the RCS curves for the scattering at two iden-tical cubes for decreasing separations are compared to the RCSplot of the scattered field at a cuboid of the same material withcombined dimensions. This comparison is depicted in Fig. 4.

0 30 60 90 120 150 180

0

5

10

15

20

θ [◦]

RC

S[d

B]

Cuboid1µm1 mm1 cm5 cm10 cm20 cm

Fig. 4. RCS in the xz-plane for the scattering at two cubes withεr,1 = εr,2 = 4 and side 1m, for different separations, at a frequency of100MHz, in comparison with the scattering at a cuboid of 1m ×1m ×2m.

One observes a continuous evolution of the RCS curves to-wards the one of the cuboid. The large discrepancy at separa-

tions of 20 cm, 10 cm and 5 cm is easily explained by the pres-ence of the relatively large gap between the cubes. Nevertheless,when the separation equals 1 cm, this discrepancy becomes quitesmall, and a quasi perfect agreement is observed for 1 mm and1µm, as expected.

Next a similar methodology is applied for a junction of twocubes consisting of different materials. A validation with theframework implemented for a single scatterer is impossible dueto the nature of the problem. However, we can have a look at theRCS curves of two approaching cubes with side 1 m, εr,1 = 4and εr,2 = 15, as depicted in Fig. 5.

0 30 60 90 120 150 1802

4

6

8

10

θ [◦]

RC

S[d

B]

3 cm2 cm1 cm5 mm2.5 mm500 µm1 µm

Fig. 5. The RCS in the xz-plane for the scattering at two cubes with εr,1 = 4,εr,2 = 15 and side 1m, for different separations, at a frequency of100MHz.

A smooth transition is visible in the RCS curves from twonon-adjacent cubes up till the point they touch. The above obser-vations demonstrate the developed method to function properlyfor the case of a junction.

So far, a junction was constructed by bringing two cubes to-gether. This approach is very intuitive and yields correct re-sults, but is not perfectly in line with the discussion of Sec-tion II, since Γ0 is seen as Γ1 ∪ Γ2, while in fact, the inter-face Γ12 /∈ Γ0. Based on this observation, an alternative imple-mentation for junctions is developed, by introducing new basisfunctions, spanning over the two cubes. The RCS curves forthe examples of a junction with εr,1 = εr,2 = 4 and a junctionwith εr,1 = 4, εr,2 = 15, respectively, are compared in Fig. 6for the two methods. RMS errors of 0.07% and 0.77% are ob-tained for junction 1 and junction 2, respectively, which provesthe alternative implementation to be correct.

0 30 60 90 120 150 1802

4

6

8

10

12

14

16

18

θ [◦]

RC

S[d

B]

junction 1, v. 1junction 1, v. 2junction 2, v. 1junction 2, v. 2

Fig. 6. A comparison between the two approaches to model junctions, denotedv. 1 and v. 2, respectively. The RCS in the xz-plane for the scattering attwo junctions, with εr,1 = 2 and εr,2 = 4 (junction 1) or εr,2 = 15(junction 2) at a frequency of 100MHz.

IV. CONCLUSIONS

The goal of this master’s dissertation was to develop a do-main decomposition method to model the electrical performanceof TSVs. First, the method was developed theoretically, and asystem matrix based on a BIE solution for each of the distinctsubdomains and RTCs to couple the unknowns at the interfaces,was obtained.

This framework was then successfully applied to a single scat-terer. Validations with an analytical solution for a sphere andwith in-house simulation software, proved our method to func-tion properly and to deliver an excellent accuracy. However, infuture research, some fundamental efforts have to be performedto make the system better conditioned.

Finally, the developed DDM was implemented for the case oftwo objects. Starting from two non-adjacent cubes, the separa-tion distance was decreased step by step, which provides an in-tuitive plan to construct a junction. Furthermore, an alternativeapproach to model junctions was proposed. For both methods,numerical validations show that our novel domain decomposi-tion method for junctions of materials yields excellent results.The final step, viz. the modelling of more complex, TSV-likegeometries is hence a straightforward extension of our research.

REFERENCES[1] E. Sicard, W. Jianfei, R. J. Shen, E. P. Li, E. X. Liu, J. Kim, J. Cho and

M. Swaminathan, “Recent Advances in Electromagnetic Compatibility of3D-ICs – Part I”, IEEE Electromagnetic Compatibility Magazine, vol. 4,no. 4, pp. 79-89, April 2015.

[2] X. Gu, B. Wu, M. Ritter and L. Tsang, “Efficient Full-Wave Modelingof High Density TSVs for 3D Integration”, 2010 IEEE 60th ElectronicComponents and Technology Conference (ETCT), pp. 663-666, June 2010.

[3] J. Guan, S. Yan and J. M. Jin, “A Multi-Solver Scheme Based on RobinTransmission Conditions for Electromagnetic Modeling of Highly ComplexObjects”, IEEE Transactions on Antennas and Propagation, vol. 64, no. 12,pp. 5346-5358, December 2016.

[4] D. Colton and R. Kress, “Integral Equation Methods in Scattering Theory”,Society for Industrial and Applied Mathematics, 2013.

[5] K. Cools, F. P. Andriulli and E. Michielssen, “A Calderón MultiplicativePreconditioner for the PMCHWT Integral Equation”, IEEE Transactions onAntennas and Propagation, vol. 59, no. 12, pp 4579-4587, December 2011.

[6] S. Rao, D. Wilton and A. Glisson, “Electromagnetic Scattering by Surfacesof Arbitrary Shape” IEEE Transactions on Antennas and Propagation, vol.30, no. 3, pp. 409-418, May 1982.

[7] S. Yan, J. M. Jin and Z. Nie, “Calderón Preconditioner: from EFIE andMFIE to n-Müller Equations”, IEEE Transactions on Antennas and Propa-gation, vol. 58, no. 12, pp. 4105-4110, December 2010.

[8] A. Buffa and S. H. Christiansen, “A Dual Finite Element Complex on theBarycentric Refinement”, Mathematics of Computation, vol. 76, no. 260,pp. 1743-1769, May 2007.

[9] M. Gossye, M. Huynen, D. Vande Ginste, D. De Zutter and H. Rogier,“A Calderón Preconditioner for High Dielectric Contrast Media”, IEEETransactions on Antennas and Propagation, Submitted.

[10] M. Huynen, M. Gossye, D. De Zutter and D. Vande Ginste, “A 3-D Dif-ferential Surface Admittance Operator for Lossy Dipole Antenna Analysis”,IEEE Antennas and Wireless Propagation Letters, vol. 16, pp. 1052-1055,December 2017.

CONTENTS i

Contents

List of Figures iii

List of Tables v

List of Abbreviations and Symbols vi

1 Introduction 1

1.1 Through-Silicon Vias in Recent Semiconductor Technology . . . . . . . . . . . . 1

1.2 Full-wave EM modelling of electrically large objects . . . . . . . . . . . . . . . . 2

1.3 Goal and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Full-wave Simulation Methods 4

2.1 Finite Element Method and Method of Moments . . . . . . . . . . . . . . . . . . 5

2.2 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Domain Decomposition Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Domain Decomposition Method for the scattering at N objects 17

3.1 Definitions and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Huygens equivalence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Scattering at a set of objects using RTCs . . . . . . . . . . . . . . . . . . . . . . 20

3.3.1 Preliminary case: one scatterer . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.2 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4.1 Junction of two dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4.2 Junction of one dielectric and one PEC . . . . . . . . . . . . . . . . . . . 32

4 Application to single scatterers 36

4.1 Scattering at a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.1 Meshing of the sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.2 Numerical solution of scattering problems . . . . . . . . . . . . . . . . . . 38

4.1.3 Scattering at a dielectric sphere . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.4 Validation with other computational methods . . . . . . . . . . . . . . . . 41

4.1.5 Efforts to lower the condition number . . . . . . . . . . . . . . . . . . . . 42

4.1.6 A note on test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

CONTENTS ii

4.1.7 Scattering at lossy media . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Scattering at a cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 Meshing of the cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.2 Scattering at a dielectric cube . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Scattering at a cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Application to multiple scatterers 54

5.1 Scattering at two separate objects . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1.1 Constructing the system matrix . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1.2 Scattering at two cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Scattering at a junction of two materials . . . . . . . . . . . . . . . . . . . . . . . 61

5.2.1 Junction as the limit of two approaching objects . . . . . . . . . . . . . . 61

5.2.2 Alternative junction formalism . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Conclusions and future research 68

Bibliography 71

LIST OF FIGURES iii

List of Figures

1.1 Examples of recent advances in IC technologies. . . . . . . . . . . . . . . . . . . . 1

2.1 Schematic overview of full-wave simulation techniques. . . . . . . . . . . . . . . . 4

2.2 Schematic 3-D configuration of a general volume Ω in which a source g(r) is

located, with boundary Γ and normal n̂, pointing inwards. . . . . . . . . . . . . . 7

2.3 Treatment of the singular behaviour of the integrals. . . . . . . . . . . . . . . . . 8

2.4 Schematic illustration of the first two steps of FMMs. . . . . . . . . . . . . . . . 11

2.5 Two examples of div-conforming basis functions. . . . . . . . . . . . . . . . . . . 15

3.1 General 3-D volume with N+1 regions. . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Illustration of Huygens equivalence theorem in a single volume Ω1 surrounded by

free space or any combination of other volumes. . . . . . . . . . . . . . . . . . . . 19

3.3 Configuration of one object in 3-D free space. . . . . . . . . . . . . . . . . . . . . 21

3.4 Configuration of one dielectric object forming a junction with one perfectly con-

ducting region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1 Schematic illustration of the scattering of an incident electromagnetic field at a

sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Meshes of a sphere created with GMSH. . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Scattering at a sphere with εr,1 = 4 and radius 1 m, at a frequency of 100 MHz,

compared to the analytical results (Mie series). . . . . . . . . . . . . . . . . . . . 39

4.4 Scattering at a sphere with εr,1 = 4, at a frequency of 100 MHz, compared to the

analytical results for a sphere with an effective radius reff < 1 m. . . . . . . . . . 40

4.5 Numerical results of the combined testing scheme for different values of f , eval-

uated on Mesh A (192 edge elements) and Mesh B (720 edge elements), at a

frequency of 100 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.6 Scattering at a sphere of radius 1 m (reff = 0.991 m), at a frequency of 100 MHz

for different lossy materials, calculated with Mesh B (720 edge elements). . . . . 46

4.7 Scattering at a sphere of radius 1 m, at a frequency of 100 MHz, comparison

between copper and a PEC, calculated with Mesh B (720 edge elements). . . . . 47

4.8 Scattering at a sphere of radius 1 m, at a frequency of 100 MHz, for different

values of σε0ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48


cube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

LIST OF FIGURES iv

4.10 Meshes of a cube with side 1 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.11 Scattering at a cube of side 1 m, at a frequency of 100 MHz, for different computa-

tional methods. We look at the complete far field, and zoom in on the interesting

region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.12 Scattering at a cube with εr,1 = 4 and side 1 m, at a frequency of 100 MHz. A

comparison among three different meshes. . . . . . . . . . . . . . . . . . . . . . . 51


cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.14 Meshes of a cylinder with radius λ/3 and height 2λ. . . . . . . . . . . . . . . . . 52

4.15 Scattering at a cylinder with radius λ/3 and height 2λ, at a frequency of 1 GHz. 53

5.1 Schematic illustration of the scattering of an incident electromagnetic field at two

separated cubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Employed mesh for the scattering at two cubes separated by a given distance,

with S = 5 subdivisions along the side of the cube. . . . . . . . . . . . . . . . . . 58

5.3 Scattering at two cubes with εr,1 = εr,2 = 4, side 1 m and separation 1 m, at a

frequency of 100 MHz. A comparison between the two described formulations. . . 58

5.4 Scattering at two cubes with εr,1 = εr,2 = 4, side 1 m and separation 1 m, at a

frequency of 100 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.5 Scattering at two cubes with εr,1 = 4 and εr,2 = 15, side 1 m and separation 1 m,

at a frequency of 100 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.6 Scattered electric field in the plane between the two scattering cubes at a fre-

quency of 100 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.7 Real part of the x- and z-component of the scattered electric field in the plane

between the two scattering cubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61


junction of two blocks of material. . . . . . . . . . . . . . . . . . . . . . . . . . . 62


cuboid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.10 Scattering at two cubes with εr,1 = εr,2 = 4, side 1 m and different separations, at

a frequency of 100 MHz, in comparison with the scattering at their wraparound

cuboid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.11 Scattering at two cubes with εr,1 = 4 and εr,2 = 15 and side 1 m, for different

separations, at a frequency of 100 MHz. . . . . . . . . . . . . . . . . . . . . . . . 64

5.12 Illustration of the employed meshes for a junction composed of two blocks of

material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.13 Comparison of the intuitive approach (v. 1) and the alternative formalism (v. 2) to

model the scattering at a junction, with εr,1 = εr,2 = 4 (junction 1) and εr,1 = 4,

εr,2 = 15 (junction 2), respectively, at a frequency of 100 MHz. . . . . . . . . . . 67

LIST OF TABLES v

List of Tables

2.1 Advantages and disadvantages of the FE method and the MoM. . . . . . . . . . 12

4.1 Numerical results of the scattering at a dielectric sphere for three different meshes. 41

4.2 Numerical results of the N-Müller equations applied to the scattering at a dielec-

tric sphere for three different meshes. . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Numerical results of the CP-PMCHWT equations applied to the scattering at a

dielectric sphere for three different meshes. . . . . . . . . . . . . . . . . . . . . . 42

4.4 Numerical results of current rescaling applied to the scattering at a dielectric

sphere for Mesh B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.5 Numerical results of the scattering at different lossy materials. . . . . . . . . . . 47

4.6 Numerical results of the scattering at a cube for different methods. . . . . . . . . 50

4.7 Comparison of the numerical results of our system matrix equation working on

three different cubic meshes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1 Definitions of the different subsets of edge elements in the mesh. . . . . . . . . . 65

LIST OF TABLES vi

List of Abbreviations and Symbols

Abbreviations2-D two-dimensional

3-D three-dimensional

BC Buffa-Christiansen

BEM boundary element method

BIE boundary integral equation

CEM computational electromagnetics

CP-PMCHWT Calderón Preconditioned Poggio-Miller-Chang-Harrington-Wu-Tsai

DDM domain decomposition methods

EM electromagnetic

FDTD finite-difference time-domain

FE finite element

FMM fast multipole method

GMRES Generalized Minimal Residual method

GO geometrical optics

HDC high dielectric contrast

IC integrated circuit

MEMS micro-electromechanical system

MLFMM multilevel fast multipole method

MoM method of moments

PEC perfect electric conductor

PMCHWT Poggio-Miller-Chang-Harrington-Wu-Tsai

PML perfectly matched layer

PO physical optics

RCS radar cross section

RMS root mean square

RTC Robin transmission condition

RWG Rao-Wilton-Glisson

TSV through-silicon via

UTD uniform theory of diffraction

LIST OF TABLES vii

Symbols

Mathematical symbols

j imaginary unit

∇ nabla operator·T matrix transpose< real part= imaginary part〈·, ·〉X inner product of two functions over Xx vector

n̂i normal on the surface of Ωi, pointing inwards

Geometrical domains

R3 threedimensional spaceΩ general 3-D region

Γpq interface between Ωp and Ωq

Electromagnetic symbols

b magnetic induction

d dielectric displacement

e electric field

h magnetic field

m magnetic current

j electric current

k wave vector

λ wavelength

ω angular frequency

ε0 vacuum permittivity

εr relative permittivity

µr relative permeability

η wave impedance

σ conductivity

Operators

D general linear operator

γt tangential trace operator

K magnetic field integral operatorT electric field integral operatorX general operator

LIST OF TABLES viii

Functions

G Green’s function

f l general basis function

wk general test function

Vector spaces

D(D) domain of a linear operator DHk Sobolev space with p = 2L2(Ω) Hilbert space of square integrable functions on ΩWk,p Sobolev space

Miscellaneous

R distance between source and observation point

T± bottom or upper triangle

A± area of the bottom or upper triangle

l length of an edge element

r far-field radius

reff effective radius

S subdivision, number of edges along the side of a cube or cuboid

ξi subset of edge elements

INTRODUCTION 1

Chapter 1

Introduction

1.1 Through-Silicon Vias in Recent Semiconductor Technology

One of the most recent advances in semiconductor technology involves a shift from two-dimensio-

nal (2-D) integrated circuits (ICs) to three-dimensional (3-D) ICs [1], consisting of a stack of

horizontal layers of silicon wafers. By exploring the third dimension, devices can be made

substantially smaller, and highly integrated systems are formed. Many authors believe 3-D

integration to provide an extension of Moore’s law [2–4], stating that the number of transistors

on a dense IC doubles every two years [5].

This transition from 2-D to 3-D brings along certain advantages. First of all, devices with a

smaller footprint can be fabricated, delivering an increased functionality. Besides, since the

components can be placed closer together, the interconnection length is strongly reduced, re-

sulting in a reduced power consumption, lower wire delay, less parasitic effects and a higher

clock frequency [6]. Furthermore, heterogeneous integration provides the possibility to stack

ICs with different technologies and functionalities, e.g. memory, micro-electromechanical sys-

tems (MEMS), antennas, sensor chips, etc. [4, 6]. This is depicted in Fig. 1.1a. Summarised,

greater interconnection density leads to an improved overall system performance. The other side

of the coin is a set of problems and challenges involved with the introduction of the third di-

mension. 3-D systems encounter increased couplings, a heterogeneous temperature distribution,

complex power delivery paths, etc. [1, 2].

(a) Schematic illustration of heterogeneous in-

tegration in a 3-D IC [6].

(b) Schematic cross-section of a TSV.

Figure 1.1: Examples of recent advances in IC technologies.

INTRODUCTION 2

In order to electrically interconnect several, vertically stacked layers, a through-silicon via (TSV)

is introduced. TSVs are the pillars holding up the mansion of the 3-D IC [7]. A schematic cross-

section of a TSV is depicted in Fig. 1.1b. As the name indicates, a TSV consists of a metal

core, usually made out of copper (superior conductivity) or tungsten (lower cost), coated with

an oxide on the cylindrical walls, residing in a silicon substrate. This description is the most

common configuration of a TSV. Nevertheless, other shapes are proposed in literature, such as

tapered, rectangular, coaxial, etc. [8]. In order to obtain the desired performance, TSVs are

often placed in a via array, with densities of the order of 105 − 108/cm2 [9].

Because they cut down time-to-market and production costs, models to predict the performance

of TSVs, including crosstalk, interference and attenuation, are of utmost interest to the semicon-

ductor industry. Up till now, most efforts in this direction include equivalent-circuit models by

introducing lumped circuit elements [1, 3, 4, 10]. However, 3-D electromagnetic (EM) full-wave

analysis does not rely on approximative assumptions and is hence superior in accuracy, particu-

larly in capturing nonlinear effects such as biasing, or when the frequencies of interest grow too

large (i.e. of the order GHz). In literature, some ideas and methods have been proposed. They

range from the use of commercial full-wave solvers [11, 12], over the introduction of cylindrical

wave expansions [2] and cylindrical modal basis functions [6,9], to the addition of a non-uniform

diameter, based on conical modal basis functions [13]. This describes the context of this master’s

dissertation, i.e. the full-wave EM modelling of TSVs for the accurate prediction of electrical

performance of 3-D ICs.

1.2 Full-wave EM modelling of electrically large objects

The modelling of electrically large objects with complex geometries and materials has been

widely explored by the computational electromagnetics (CEM) community. Next to the EM

group of the Department of Information Technology of Ghent University (INTEC), i.a. the

Center for Computational Electromagnetics of the University of Illinois, USA, has done some

splendid work in this research field. They premise a hybrid finite element–boundary integral

method as a powerful tool to solve such problems accurately and efficiently [14]. The strength

of the method relies on partitioning the computational domain, i.e. a popular technique dubbed

the domain decomposition method (DDM, see Section 2.3).

With DDM, each region is solved individually – with the optimal technique – as a function of

unknowns defined on the interfaces between the domains. Complex, heterogeneous materials

can thus be tackled with the finite element (FE) method, while for homogeneous inner regions

and the infinite background medium, the boundary element method (BEM) can be used. A

more detailed discussion on these methods is provided in Section 2.1. To relate these sets of

unknowns on the boundaries of the different domains, certain boundary conditions are applied.

The authors of [14] propose Robin transmission conditions (RTCs) to couple the unknowns of

the distinct domains into a global system equation. These conditions enforce continuity of a

certain linear combination of the electromagnetic fields on the interface between two regions.

INTRODUCTION 3

1.3 Goal and outline

In Section 1.1, the concept of a TSV was introduced and the relevance of this structure for

both the recent semiconductor technology in general, and the development of 3-D ICs more

specifically, was reviewed. Further, in Section 1.2, a popular full-wave EM simulation method

for electrically large objects with multiple distinguishable subdomains, DDM, was introduced,

employing a particular kind of boundary conditions. As we are interested in modelling the

behaviour of TSVs for high frequencies, the aim of this master’s dissertation is to bring these

two stories together.

The goal is to develop a DDM to model TSV-like structures. The set-up consists of a metal core

with an oxide cladding, embedded in a silicon substrate, and can be reduced to a cylindrical metal

core inside a silicon layer, as a first step. This problem consists of two homogeneous subdomains,

i.e. the metal cylinder and the silicon substrate. These regions have a fixed value for the material

parameters, viz. the relative permittivity εr and permeability µr, hence the method of moments

(MoM) can be employed (see Chapter 2 for an extensive explanation). The fundamental issue is

thus to develop a DDM for a junction of two blocks of different materials, in which the individual

regions are solved with the MoM. This bottom-line problem can subsequently be translated into

the TSV-configuration, which is a purely geometrical issue.

As a starting point, we take a step back in Chapter 2, where an overview of the established

full-wave EM simulation techniques is given. Two of them, the FE method and the BEM are

analysed thoroughly. Moreover, some notes on discretisation are proposed, in order to pass from

a continuous to a discretised – and thus numerically solvable – formulation. Eventually, the

concept of domain decomposition techniques is elaborated.

Next, the central DDM of this master’s dissertation is developed in Chapter 3, starting from

the Huygens equivalence principle [15] and the Stratton-Chu representation theorem [16]. The

elaborations are initially valid for the simple case of a single scatterer, and then generalised

to the case of N scatterers and the background medium. Finally, the general formulation is

specified for a junction of two materials in a background medium.

In Chapter 4, the dedicated DDM is applied to a single scatterer. This rather simple case is

examined in order to familiarise ourselves with the employed software, to validate the method

w.r.t. an analytical solution (for the case of a sphere), to characterise the implementation of the

formalism and to write a meshing code for cubic structures.

This dedicated meshing code is then employed in Chapter 5, where the matrix formalism of

Chapter 3 to multiple scatters. Two approaches to treat a junction of two materials are pre-

sented. First, the two constituent blocks are brought closer together step by step, until they

touch and a junction is established. Second, we develop an alternative formalism to handle

junctions, taking into account overlapping edge elements.

This master’s dissertation is completed with an overview of conclusions and subjects for future

research in Chapter 6.

FULL-WAVE SIMULATION METHODS 4

Chapter 2

Full-wave Simulation Methods

Only a few electromagnetic problems can be solved analytically. Most realistic EM issues, con-

cerning scattering, radiation, modelling of waveguides, etc., rely on numerical solution methods.

There is a wide gamma of full-wave simulation techniques available to solve Maxwell’s equations

and tackle 3-D EM problems, both in time and frequency domain [17]. A schematic overview is

depicted in Fig. 2.1.

Figure 2.1: Schematic overview of full-wave simulation techniques [17].

In this chart, several full-wave EM simulation techniques are plotted against two quantities,

viz. the complexity of the materials and the electrical size of the objects to be simulated. Com-

plexity of materials is defined in terms of variations in material properties, i.e. the relative

permittivity εr and relative permeability µr, while electrical size is defined as the geometrical

size divided by the wavelength (λ). When dealing with electrically very large objects, the scale

of λ becomes negligible and ray optics methods can be utilized, such as the uniform theory of

diffraction (UTD), physical optics (PO) and geometrical optics (GO). When solving for rela-

tively simple materials, at a broad range of dimensions, the method of moments (MoM) and

the multilevel fast multipole method (MLFMM) are used. For handling complex configurations,

e.g. inhomogeneous media, the finite element (FE) method and the finite-difference time-domain

method (FDTD) are applicable.


2.1 Finite Element Method and Method of Moments

Two methods are discussed in more detail here, the finite element (FE) method and the method

of moments (MoM), since they are of uppermost relevance at the contemplated length scale [18].

At the end of this section, a few notes on the MLFMM are made, as an extension of MoM.

The FE method is the oldest method of the two, described by Courant in 1943 [19]. It was

explored initially for static field problems, and has become one of the most successful techniques

for solving engineering problems. The computational domain is subdivided into a large number of

small cells; the full 3-D space is thus discretised. Local interactions are then calculated between

these cells or elements. We concisely elaborate on this technique, starting from a general scalar1

linear operator equation [20]:

Dp(r) = g(r), (2.1)

with D a general linear operator, g(r) a known excitation vector, p(r) the unknown response in

the domain D(D) and r ∈ Ω, the computational 3-D domain. As described in the introductionof this chapter, analytical solutions for (2.1) are often non-existing. Instead of demanding a

pointwise correct solution, we impose the equation in an average, weighted way. This is dubbed

as the weak form of (2.1):

〈w(r), Dp(r)〉Ω = 〈w(r), g(r)〉Ω , (2.2)

with w(r) a general test function and the inner product 〈·, ·〉Ω defined as:

〈a(r), b(r)〉Ω =∫

Ωa(r)b(r) dΩ. (2.3)

The aim is now to find an approximate solution for the problem in the weak sense, by looking in

an L-dimensional subspace, spanned by a finite set of basis functions fl(r), with l ∈ {1, ..., L}:

p(r) ≈L∑

l=1

plfl(r), (2.4)

with pl the so-called expansion coefficients. We now apply the weak formulation, with L weight-

ing functions wk(r), k ∈ {1, ..., L}, which span an L-dimensional space as well. The aboveprocedure results in an (L× L)-dimensional matrix system:

L∑

l=1

pl 〈wk(r), Dfl(r)〉Ω = 〈wk(r), g(r)〉 , ∀ k ∈ {1, ..., L}, (2.5)

which can be solved numerically with a wide spectrum of direct or iterative techniques. This

general work-flow can be applied to 1-D, 2-D or 3-D wave equations, e.g. Maxwell’s equations,

which are the most interesting for this master’s dissertation. Different sets of boundary condi-

tions complement the wave equation. A problem arises when considering unbounded regions.

It is impossible to extend the 3-D mesh up to infinity, so at some point, the computational

1We choose a scalar quantity here in order to make the FE technique clear. The extension to vector functions

can be made easily, but this would distract the reader from the relevant concepts.


domain has to be cut off. This is accomplished by extending the cut-off domain by an absorbing

medium; the perfectly matched layer (PML) [21]. In this concept, a balance is found between a

small number of absorbing layers on the one hand, and a minimal contrast between the medium

of the original computational domain and the absorber on the other hand. As such, the number

of extra unknowns and non-physical reflections are minimised.

In practice, the weighting functions wk(r) and basis functions fl(r) are mesh-based2. As the

expansion functions need to follow the field variations, the mesh should be fine enough. A well-

known rule of thumb for the FE method is to make the length of the mesh elements of the order

of λ/20 – λ/30. Even with such a relatively fine grid, a non-physical phenomenon called grid

dispersion will occur, when the mesh density is dependent on the direction. This anisotropy

affects the speed of waves in different directions; a non-physical effect.

Since the whole computational domain Ω is discretised using a 3-D volume mesh, which has to be

fine enough w.r.t. λ in order to obtain accurate results and minimise grid dispersion, it is clear to

the reader that the number of unknowns and thus the dimension of the system matrix becomes

huge when the computational domain is electrically large. This is one of the main drawbacks

of the FE method. Nevertheless, as only local interactions occur, most inner products of test

and basis functions vanish, resulting in a sparse matrix. Therefore, a benefit can be derived in

terms of storage efficiency. Besides, one could reduce the number of unknowns by employing

a coarse mesh in the regions of low complexity and refining it locally for more complex areas.

However, non-physical reflections would occur at the transition from a coarse to a finer mesh,

dubbed spurious reflections.

It will be clear that – though the FE method has won its spurs in various engineering disciplines

over time and it is the preferred technique in EM problems with inhomogeneous objects – the

method has some major drawbacks. Many efforts have been devoted to overcome these prob-

lems (PMLs, refining the mesh up to λ/30, memory efficiency, variations in mesh parameter).

However, these efforts barely suppress the symptoms. A fundamentally different approach is

needed, and offered by introducing boundary integral equations (BIE), also called boundary ele-

ment methods (BEM). Once the integrals are drafted, they can be discretised following a similar

procedure as described above. In the context of this BIE method, the discretisation procedure

is called the MoM. This name is often employed for the BIE method as a whole.

In contrast to the 3-D volume mesh of the complete computational domain as used in the FE

method, the unknowns of the problem, i.e. currents and fields, should only be determined on

the boundary. The strength of the BIE method lies in the fact that it takes the analytical

discussion one step further than the FE method, by treating the unknowns on the boundary

as sources generating a field, which is calculated via a convolution with the Green’s function

(G). This function is the fundamental solution of the equation under consideration when an

impulse source is applied. Green’s functions can be calculated analytically for homogeneous and

2Common test and basis functions are piecewise constant, piecewise linear or continuous over the complete

computational domain. More on this in Section 2.2.


multilayered background media. This directly yields one of the limitations of the BIE method,

as only piecewise homogeneous materials can be tackled.

The work-flow of the BIE method is elaborated on for the case of the scalar 3-D wave equation:

∇2p(r) + k2p(r) = g(r), with r ∈ Ω. (2.6)This equation has to be met for all points in the region of interest, Ω, bounded by Γ, containing

a known excitation source g(r). The normal vector on Γ is denoted by n̂ and points inwards.

This general configuration is depicted in Fig. 2.2.

Figure 2.2: Schematic 3-D configuration of a general volume Ω in which a source g(r) is located, with

boundary Γ and normal n̂, pointing inwards.

As we explained, G is fundamental solution of (2.6), when replacing the general source term

g(r) by an impulse source:

∇2G(r, r′) + k2G(r, r′) = −δ(r − r′), (2.7)

with r ∈ Ω the source point, and r′ ∈ Ω the observation point. Since this equation is of thesecond order, two linearly independent solutions emerge, of which only the one obeying the

radiation condition is appropriate. The Green’s function can easily be derived for the 3-D case:

G(r, r′) =e−jk|r−r

′|

4π |r − r′| , (2.8)

which is 1/r-singular if the source and observation points coincide. Starting from the wave

equation and this Green’s function, some mathematical manipulations are performed, in order

to arrive at the general BIE. We subtract p(r)·(2.7) from G(r, r′)·(2.6), yielding the followingexpression:

G(r, r′)∇2p(r)− p(r)∇2G(r, r′) = G(r, r′)g(r) + p(r)δ(r − r′), (2.9)

which we integrate over Ω:∫

Ω

(G(r, r′)∇2p(r)− p(r)∇2G(r, r′)

)dΩ =

∫

ΩG(r, r′)g(r) dΩ + p(r′). (2.10)

In the last term of the right hand side, the unknown response function p(r′) emerges, to be

determined in each point r′ ∈ Ω. We employ the Gauss divergence theorem to convert thevolume integral over Ω of the left hand site to a surface integral over Γ:

∮

Γ

(−G(r, r′)∂p(r)

∂n+ p(r)

∂G(r, r′)∂n

)dΓ =

∫

ΩG(r, r′)g(r) dΩ + p(r′). (2.11)


This means p(r′) can be computed everywhere in Ω when the source term g(r) and the values

of p(r) and its normal derivatives on Γ are known. In order to determine p(r), source and

observation point need to be interchanged. Therefore, we write the arguments of G as |r − r′|(see (2.8)), define ∂n′ =

∂·∂n′ = n̂

′ · ∇′ and introduce pinc:

pinc(r) = −∫

ΩG(∣∣r − r′

∣∣)g(r′) dΩ′. (2.12)

This finally leads to the following representation formulas for p(r) and ∇p(r):

p(r) =

∮

Γ∂n′G(

∣∣r − r′∣∣)p(r′) dΓ′ −

∮

ΓG(∣∣r − r′

∣∣)∂n′f(r′) dΓ′ + pinc(r), (2.13)

∇p(r) =∮

Γ∇(∂n′G(

∣∣r − r′∣∣))p(r′) dΓ′ −

∮

Γ∇G(

∣∣r − r′∣∣)∂n′f(r′) dΓ′ +∇pinc(r). (2.14)

It will be clear to the reader that the discussion does not end with the above representation

formulas, since the integrals contain singularities, due to the behaviour of the Green’s function

when r → r′. The next step in our journey towards computationally solvable boundary integralequations, is thus to examine these integrals when the observation point r approaches Γ:

limr→Γ

p(r) = limr→Γ

∮

Γ∂n′G(

∣∣r − r′∣∣)p(r′) dΓ′ − lim

r→Γ

∮

ΓG(∣∣r − r′

∣∣)∂n′f(r′) dΓ′ + pinc(r),

(2.15)

limr→Γ

n̂ · ∇p(r) = limr→Γ

∂np(r)

= limr→Γ

∮

Γ∂n∂n′G(

∣∣r − r′∣∣)p(r′) dΓ′− lim

r→Γ

∮

Γ∂nG(

∣∣r − r′∣∣)∂n′f(r′) dΓ′+∂npinc(r).

(2.16)

(a) Schematic 3-D configuration of a general

volume Ω with a small sphere ΩR isolating

the singularity.

(b) Detail of the relevant part of Γ with ΩR,

a sphere with radius R and normals n̂ and

boudary Σ = ΓR ∪ ΣR.

Figure 2.3: Treatment of the singular behaviour of the integrals.

For the second term in the right hand side of (2.15), the singularity is integrable, no matter

where the observation point is located. We can thus omit the limit. The first term needs to be

treated more carefully. The singularity at r = r′ is isolated by introducing a small half sphere

ΩR with radius R and and surface Σ = ΓR ∪ ΣR, with ΓR the part of Γ inside the sphere, and


ΣR the surface of the sphere at the inside of Ω. Our set-up with this newly introduced sphere

centred around r is depicted in Fig. 2.3.

With this configuration, we can split the first integral of (2.15) over Γ in a regular part over

Γ \ ΓR and a singular part over Σ:

limr→Γ

∮

Γ∂n′G(

∣∣r−r′∣∣)p(r′) dΓ′= lim

R→0

∫

Γ\ΓR∂n′G(

∣∣r−r′∣∣)p(r′) dΓ′+ lim

R→0

∫

Σ∂n′G(

∣∣r−r′∣∣)p(r′) dΓ′,

(2.17)

of which the first integral can be calculated regularly and the second can be further elaborated

on as follows:

limR→0

∫

Σ∂n′G(

∣∣r − r′∣∣)p(r′) dΓ′ = lim

R→0

∫

ΣR+ΓR

n̂′ ·∇′G(∣∣r − r′

∣∣)p(r′) dΓ′

= p(r) limR→0

∫

ΣR+ΓR

n̂′ · uR(−jk e

−jkR

4πR− e−jkR

4πR2

)dΓ′

= p(r)

(limR→0

∫

ΓR

n̂′ ·uR(−jk e

−jkR

4πR− e−jkR

4πR2

)dΓ′+ lim

R→0

∫

ΣR

n̂′ ·uR(−jk e

−jkR

4πR− e−jkR

4πR2

)R2 dΦ

)

=p(r)

4πlimR→0

∫

ΣR

dΦ =p(r)

2. (2.18)

In the first line of (2.18), the two parts of Σ and the definition of ∂n′ are introduced. Since we

are looking at the case of r′ approaching r, the regular function p can be brought out of the

integral in the second line. Furthermore, the gradient of the Green’s function is calculated, with

uR =r′−r|r′−r| . When taking the scalar product with n̂

′, it will be clear to the reader that this

equals zero on ΓR if the surface is sufficiently flat around r, such that ΓR is a disk with radius

R. Only the second term on the third line survives and can be simplified because n̂′ and uRare parallel on the surface of a sphere. The integral is then evaluated over the solid angle Φ of

a half sphere, which results in a simple expression. In the end, the first limit of (2.15) yields:

limr→Γ

∮

Γ∂n′G(

∣∣r − r′∣∣)p(r′) dΓ′ =

∮

Γ∂n′G(

∣∣r − r′∣∣)p(r′) dΓ′ + p(r)

2. (2.19)

A similar procedure can be pursued for the second limit in (2.16), which results in:

limr→Γ

∮

Γ∂nG(

∣∣r − r′∣∣)∂n′f(r′) dΓ′ =

∮

Γ∂nG(

∣∣r − r′∣∣)∂n′f(r′) dΓ′ −

p(r)

2, (2.20)

where the minus sign stems from the fact that the derivative of p is calculated with respect to

r instead of r′. Finally, we discuss the first limit of (2.16). Due to the double derivative of

G, this term has to be handled with even more care. Without going into detail, by imposing

smoothness conditions on p(r), it can be derived that the limit can be omitted. We now have

come to the point where all integrals are calculated and thus both representation formulations

can be expressed safely. The two formulas are cast in a matrix format:

(p(r)

∂np(r)

)=

(12 + F

′ −GD 12 − F

)(p(r)

∂np(r)

)+

(pinc(r)

∂npinc(r)

), (2.21)


written in operator notation, with:

G∂np(r) =

∮

ΓG(∣∣r − r′

∣∣)∂n′p(r′) dΓ′,

F′p(r) =∮

Γ∂n′G(

∣∣r − r′∣∣)p(r′) dΓ′,

F∂np(r) =

∮

Γ∂nG(

∣∣r − r′∣∣)∂n′p(r′) dΓ′,

Dp(r) =

∮

Γ∂n∂n′G(

∣∣r − r′∣∣)p(r′) dΓ′. (2.22)

A similar system can be found for the region outside Γ. These results will be employed for an

extension of the scalar 3-D wave equation (2.6), namely the vectorial Helmholtz equation which

follows from Maxwell’s equations. These representation formulas will be the starting point of

the domain decomposition technique presented in Chapter 3 of this master’s dissertation.

The representation formulas (2.21) do not have a unique solution. The following step in the

general work-flow is thus to impose certain boundary conditions on Γ. This transforms the

equations to the (continuous) BIEs. Since the relevant boundary conditions will be introduced

in Chapter 3, we do not go in further detail here.

In order to obtain a numerically solvable system, the final step in the procedure involves dis-

cretising the integral equations. First, the scatterer is meshed. In case of the FE method,

this corresponds to a 3-D volume mesh of the complete computational domain Ω and the near-

field background medium. However, since in the BIE method the unknowns of the problem

are currents and fields on the boundary, a 3-D surface mesh is sufficient to solve the prob-

lem. Next, a set of basis functions from the correct function space (Section 2.2) are premised:

{f1(r), f2(r), ..., fL(r)}. A candidate solution is expressed as a linear combination of these basisfunctions:

p(r) ≈L∑

l=1

plfl(r), (2.23)

with pl the expansion coefficients and l ∈ {1, ..., L}. This candidate solution is plugged in theBIE. Further, the equation is weighted with a set of test functions with the same cardinality L

as the set of basis functions. This yields a square system of linear equations:

L∑

l=1

Mk,lpl = gk, ∀k ∈ {1, ..., L}, (2.24)

with M the (L × L) system matrix and gk the result of the integration of the excitation termweighted by the test functions. This system matrix equation can be solved directly or iteratively.

Contemplating the above discretisation procedure, the reader will have noticed the announced

similarities with the procedure of the FE method. Despite these resemblances, there are some

major differences between (2.5) and (2.24). In the discussion of the FE method, the sparseness

of the system matrix was mentioned, as a result of the limited amount of near-field interactions.

Given the Green’s function kernel in the expression for Mkl, a source at a given point r′ ∈ Γ


radiates in every observation point r ∈ R3. The result is a dense square system matrix, whichis generally ill-conditioned. However, the number of unknowns, the cardinality of the set of

basis functions, is reduced from O(M3) to O(M2), with M the number of mesh elements in onedimension, due to the fact that only Γ needs to be meshed, instead of Ω as a whole.

Practically all other drawbacks of the FE method are discarded in the MoM. As the Green’s

function respects the radiation condition, no PMLs have to be introduced to cope with the

truncation of the simulation domain. The phenomenon of grid dispersion does not occur, so

a mesh parameter of λ/10 is sufficient to fully capture the wave-like nature of the unknown

quantities in the absence of fine geometrical details. As a final remark, we explore the fast

multipole method (FMM), as a way to circumvent the problem of the dense system matrix-

vector product, lowering calculation times. Many algorithms have been developed to reduce the

computation time of the matrix elements, store the system matrix in a format requiring less

memory than O(L2) and finally solve the system faster than the typical O(L3), with L followingthe definition above.

When the number of unknowns L becomes large, it is advantageous to solve the system iter-

atively. This reduces the complexity from O(L3) to O(Niter · Nmv · C), with Niter the numberof iterations, Nmv the number of matrix-vector products and C the cost of these products. In

order to optimize the computation of the matrix-vector products, the FMM was developed. In

each iteration, the product

M·p =(

L∑

l=1

M1lpl,L∑

l=1

M2lpl, ...,L∑

l=1

MLlpl

)T, (2.25)

needs to be evaluated, with M the system matrix and p the vector of coefficients of the linear

combination of basis functions. The main idea of all FMMs is to lump the edge elements

together into groups ξi of neighbouring segments. Then, the edge elements in the system matrix

are rearranged to form the corresponding block matrices. This two-step process is depicted

schematically in Fig. 2.4. For clearness, the elements are drawn as 2-D elements, but this should

be interpreted as a 3-D object Ω.

(a) The edge elements are lumped together. (b) The edge elements are rearranged such

that M consists of block matrices represent-

ing the interaction of the clusters.

Figure 2.4: Schematic illustration of the first two steps of FMMs.


After this reorganisation, the system matrix of the configuration of Fig. 2.4 looks as follows:

M =

M11 M12 M13 M14 M15 M16

M21 M22 M23 M24 M25 M26

M31 M32 M33 M34 M35 M36

M41 M42 M43 M44 M45 M46

M51 M52 M53 M54 M55 M56

M61 M62 M63 M64 M65 M66

, (2.26)

in which Mij (i, j ∈ {1, ..., 6}) is the interaction matrix between ξi and ξj , representing thefield as observed in cluster ξi, excited by the sources in cluster ξj . The next step in the FMM

procedure depends on the electrical size of the object. If the structure exhibits small geometrical

details, the edge elements will be much smaller than λ, and the problem is located in the low

frequency regime. When the structure is large w.r.t. λ, the length of the edges is chosen to be

of the order of λ/10 and the problem is situated in the high frequency regime. Both regimes

require a different variant of the FMM, but both employ (2.26) as a starting point to perform

some smart manipulations and approximations. For a detailed derivation of the Low-Frequency

and High-Frequency FMM, the reader is referred to specialised literature [22]. Nevertheless,

even more progress can be made in terms of complexity. Extending the philosophy of Fig. 2.4,

a multi-level scheme can be proposed, in which the groups ξi can be clustered in parent groups,

etc., up to the final level of one single box. It can be proved that this approach leads to a

complexity of O(L log(L)), an enormous improvement compared to the initial O(L2).

To end this section, we give a recapitulatory overview of the advantages and disadvantages of

the FE method and the MoM, which should be kept in mind when choosing a numerical solution

method to solve a problem.

Table 2.1: Advantages and disadvantages of the FE method and the MoM.

FE method MoM

+ Heterogeneous objects can be handled

System matrix is sparse

Small number of unknowns

No grid dispersion

Discretisation of λ/10 sufficient3

Radiation condition fulfilled

Possibility to employ MLFMM

-

Huge number of unknowns

Grid dispersion

Discretisation of λ/20 required

Introduction of PML needed

MLFMM not possible

Only homogeneous objects

System matrix is dense

3Except for the case of fine geometrical details, w.r.t. λ.


2.2 Discretisation

In the previous section, we described the FE method and the MoM. At a particular point in

the discussion, we left the continuous description by meshing the proper part of the domain –

resulting in a volume mesh over Ω for the FE method and a surface mesh over Γ for the MoM.

Based on that particular mesh, a set of basis functions were premised. First of all, the employed

basis functions should lead to computationally calculable integrals. Besides, since in most cases,

the unknowns of the problem always have a physical meaning, it is of utmost importance that

the proposed basis functions exhibit the proper (dis)continuity characteristics. Last but not

least, the choice of basis functions should bring along a good computational efficiency. These

three requirements are elaborated on in this section.

The regularity of the integrals is translated into the concept of a Sobolev space, Wk,p; i.e. avector space of functions equipped with a special norm, which is a combination of Lp-norms of

the function and some of its weak derivatives, up to the kth order. This encompasses a rather

complicated mathematical theory, which falls outside the scope of this master’s dissertation [23].

We restrict the parameter p to p = 2, and introduce the more familiar notation Hk = Wk,p=2.The expansion functions have to belong to the correct space of quadratic integrable functions

in the Lebesgue sense4. Depending on the method, the boundary conditions and the choice of

weak formulation (which testing scheme is utilised), some of the derivatives of the functions

need to be sufficiently smooth as well. For the 1-D case, the Sobolev space is given by one of

the following two expressions, depending on demands on the first weak derivative:

H0([0, L]) ={f ∈ L2([0, L])

}, H1([0, L]) =

{f ∈ L2([0, L])

∣∣∣∣df

dx∈ L2([0, L])

}. (2.27)

The remaining of the discussion is restricted to the 3-D case and a high-level overview of the

important spaces is provided. For the FE method, the relevant quantities are considered in the

complete 3-D volume Ω. As such, (2.27) can simply be extrapolated to three dimensions and a

general vector function x. The index (1) representing the first weak derivative, is replaced by

an extra argument rot or div, respectively, both to be interpreted in the distributional sense.

The Sobolev spaces of the curl- and div-conforming volume vector functions are given by:

H(rot; Ω) ={x ∈ (L2(Ω))3

∣∣∣∣∇× x ∈ (L2(Ω))3}

, (2.28)

H(div; Ω) ={x ∈ (L2(Ω))3

∣∣∣∣∇ · x ∈ (L2(Ω))3}

. (2.29)

Depending on the integrals (and the physical nature of the unknowns, see further), the expansion

functions must belong to one of the two vector spaces. However, for the MoM, we are interested

in quantities defined on the boundary Γ. If certain technical conditions are met, a continuous

trace operator γt exists that maps x ∈ H(rot; Ω) on n̂ × x∣∣Γ ∈ H−1/2(div; Γ), the tangential

trace space [24], without elaborating on this remarkable index -1/2.

4The Hilbert space of square quadratic integrable functions in the Lebesgue sense in one dimension over an

interval [0, L], is given by L2([0, L]) ={f : [0, L] → C

∣∣∣∣ ∫ L0 |f(x)|2 dx


The physical (dis)continuity relations of the unknowns (i.e. field components, currents) need

to be reflected in a natural way by the choice of basis functions. For the FE method, the

electromagnetic fields in the bulk of Ω are employed as unknowns of the problem. At the

interfaces between the volume elements of the mesh, Maxwell’s equations impose continuity of

the tangential components of the electric and magnetic fields e, h and discontinuity of their

normal components. Hence, the expansion functions for e and h should belong to H(rot; Ω).When choosing the dielectric displacement d and the magnetic induction b, or the magnetic and

electric currents m and j as unknowns, Maxwell’s equations impose continuity of their normal

components and discontinuity of their tangential counterparts. The expansion functions are

then taken from H(div; Ω).

In the MoM formalism, m and j on Γ – which are in fact nothing else then −γte and γth,respectively – are often employed and will be the unknowns for the central problem of this

master’s dissertation (see Chapter 3). These currents should be divergence conforming as the

law of charge conservation provides a physical meaning for the divergence of the current. Since

singular charge distributions on boundaries radiate infinite fields, they should be avoided, and the

normal components of the currents should be continuous. The tangential components, however,

do not have to obey a boundary condition and can be discontinuous. It becomes clear that the

basis functions for the currents should be extracted from H−1/2(div; Γ).

The last requirement for the expansion functions is a good balance between accuracy – which

is guaranteed by the above two remarks – and efficiency. For the FE method, using simple

basis functions brings along simple interaction integrals, which can be calculated analytically

or exactly by means of quadrature rules, and thus take little CPU time. Subdomain functions,

i.e. basis functions with limited support (occupying only a few cells of the mesh), result in local

interactions, giving a sparse interaction matrix. Since the matrix for the MoM is dense anyway,

a choice can be made between subdomain and entire domain functions. Nevertheless, the latter

ones are not widely utilised as they impose restrictions on the geometry.

To end this chapter on discretisation, we give two examples of basis functions from H−1/2(div; Γ)spanning a 2-D space (the boundary Γ of the 3-D region Ω) that will be employed over the course

of this master’s dissertation. A first set of div-conforming basis functions are the RWG functions,

named after their inventors Rao, Wilton and Glisson [25]. In a certain mesh, a proper RWG is

linked to each edge. The support is given by the two adjacent triangles. An illustration of an

RWG function is given in Fig. 2.5a. The arrows denote the direction of the RWG in each point,

the colours and the length of the arrows represent the magnitude. The analytical expression is

the following:

f(r) =

l2A+

(r − p+), ∀r ∈ T+,− l

2A− (r − p−), ∀r ∈ T−,0, ∀r /∈ {T+ ∪ T−},

(2.30)

with T+ the bottom and T− the upper triangle, l the length of the central edge, A± the areas

of the bottom and top triangle, respectively. f is zero everywhere outside these two supporting


triangles. The reader may notice that the divergence of f(r) is constant within the triangle.

The continuity conditions are satisfied with this definition, as the normal components along all

sides are continuous, viz. equal to those of the other constituent triangle along the common edge

and equal to zero along the other edges.

A second kind of div-conforming and quasi curl-conforming basis functions are the Buffa-

Christiansen (BC) functions [26], which are much more complicated, see Fig. 2.5b. They are

constructed as a linear combinations of RWG functions defined on the barycentrically refined

mesh. Analogous to the RWGs, a BC basis function is defined for each edge in the mesh.

As a final remark, we notice that often the same functions that are defined as expansion functions,

can be employed for testing as well. In this master’s dissertation, the curl-conforming rotated

RWG and BC functions are considered for this task.

(a) An RWG basis function. (b) A BC basis function.

Figure 2.5: Two examples of div-conforming basis functions [18].

2.3 Domain Decomposition Techniques

In the first section of this chapter, two famous full-wave simulation techniques were discussed.

Each of them had its strengths and weaknesses. The question of which method is superior,

cannot be answered unambiguously. The choice for one or the other solution method should

always be based on the situation (see also Fig. 2.1). This choice is a piece of cake when the

computational domain is bounded and heterogeneous (choose a volume method) or (un)bounded

and homogeneous (choose MoM). Nevertheless, when the domain consists of several clearly

discernible subdomains, one always seems to lose when choosing one of both. Either we capture

the correct behaviour in the heterogeneous regions but foist a huge number of unknowns and an

artificial truncation of the simulation domain on ourselves, or we limit the number of unknowns

and capture the correct radiation behaviour, but describe heterogeneous regions inadequately.

This seems a choice between the devil and the deep-blue sea.


Fortunately, a class of techniques has been developed the past decades to meet these demands.

These techniques are dubbed domain decomposition methods (DDM). Below, a roadmap de-

scribing the procedure of those methods is given [14]:

1. The computational domain is divided into clearly defined non-overlapping regions.

2. For each region, the suitable solution method is selected. The (incomplete) solutions for

the independent regions to the original incident electromagnetic field are premised. The

problem is expressed in terms of quantities at the boundaries of the regions.

For a heterogeneous bounded region, a volume mesh is used and the FE method is

the designated solution technique.

For a homogeneous bounded region or the background medium, a surface mesh is

employed and we utilise the MoM.

3. By applying certain transmission conditions at the interfaces between the different regions,

the unknowns of the independent regions become coupled. This is called the interface

problem.

4. The interface problem is solved for all the unknowns (i.e. magnetic and electric currents).

5. With this solution at hand, the fields inside the independent regions can be calculated.

DOMAIN DECOMPOSITION METHOD FOR THE SCATTERING AT N OBJECTS 17

Chapter 3

Domain Decomposition Method for

the scattering at N objects

In this chapter, the theoretical elaborations of the domain decomposition method are discussed.

Preceded by some definitions and conventions, the Huygens equivalence principle – which is at

the core of the theory – is explained. In Section 3.3, the matrix formalism that will be employed

in the following chapters, is deduced starting from the Stratton-Chu theorem [16]. First, the

simple case of a single dielectric scatterer is discussed. Then, we deal with the general case

involving N dielectric domains. It will become clear that most concepts, boundary conditions

and the total flow of elaborations can be extrapolated from the preliminary case of one scatterer.

Eventually, this general formulation will focus on scattering at two adjacent dielectric objects,

which is of great significance for this dissertation. This is what we call a junction. To conclude,

it is noted that the applications are not limited to dielectric objects. As an example of a simple

junction with general objects, a junction of a dielectric and a perfect electric conductor (PEC)

is treated.

One could go even further and have a look at a situation where some regions are filled with

heterogeneous materials. As explained in Chapter 2, this forms no limitation for domain decom-

position methods. However, these regions need to be treated with other computational methods

such as the finite element (FE) method or the finite-difference time-domain method (FDTD),

which fall outside the scope of this master’s dissertation.


3.1 Definitions and conventions

Figure 3.1: General 3-D volume with N+1 regions.

To start the theoretical elaborations of this master’s dissertation, we need to make clear what def-

initions, symbols and conventions will be employed. The electric and magnetic field are denoted

by e and h, respectively. We will consider a set of objects in free space, which we number starting

from 1 and represent with Ω1, Ω2, ... The outer region Ω0 is defined as R3 \ {Ω1 ∪ Ω2 ∪ ... ∪ ΩN}.The boundary between Ωp and Ωq is defined as Γpq. The total boundary surface of Ωp is then

given by:

Γp =⋃

q∈S(p)Γpq with S(p) the set of neighbours of Ωp. (3.1)

The normal on Γi is denoted as n̂i and is pointing into Ωi. This general configuration is clarified

in Fig. 3.1. The magnetic and electric current on the boundary surface are defined respectively

as:

mi = e× n̂i, (3.2)ji = n̂i × h. (3.3)

In order to relate the fields at both sides of a surface, a boundary condition has to be imposed.

In this dissertation, Robin transmission conditions (RTCs) are applied in a weak sense [14]. At

Γpq, the Robin transmission condition is stated as follows:

jp + etan,p = −jq + etan,q, (3.4)

with etan,p the tangential electric field at Γpq, given by n̂p × e× n̂p. It is clear that this RTC isnothing else than a linear combination of (rotated) magnetic and electric fields. The minus sign

in equation (3.4) is due to the substitution n̂q = −n̂p.

Finally, we suppose the unspecified sources of the incident fields to be located in Ω0 only. These

fields are denoted by einc and hinc, obeying the ejωt convention.


3.2 Huygens equivalence theorem

In order to use the relevant Green’s function G (see Section 3.3), a fully homogeneous space

needs to be considered. So far, this is not the case as the domain consists of several (N)

subdomains with different relative permittivity εr,i and permeability µr,i (with i ∈ {1, ..., N})and εr,0 = µr,0 = 1.

(a) The sources m1, j1, m2 and j2 gener-

ate the (unique) fields e1, h1 and e2, h2.

(b) The fields inside Ω1 are identical to

those in (a) when m2 and j2 are replaced

by equivalent Huygens sources on Γ12.

Figure 3.2: Illustration of Huygens equivalence theorem in a single volume Ω1 surrounded by free space

or any combination of other volumes.

The uniqueness principle dictates that a field is uniquely defined by the sources in the region,

and by the tangential part of e or h on the boundary of that region [27]. This is a necessary

but not a sufficient condition. In other words, different sources may as well result in identical

fields. This is where the equivalence theorem comes into play [15]. The concepts are depicted

in Fig. 3.2. If we cancel out m2 and j2 and introduce Huygens sources, i.e. equivalent currents

meq, jeq on Γ12 such that

meq = e× n̂1,jeq = n̂1 × h, (3.5)

the fields inside Ω1 stay identical as in the original problem, which can be proven using the

Lorentz reciprocity theorem [28]. Outside Ω1, the fields are zero. This can be deduced from the

tangential boundary conditions of the electric and magnetic field:

(e1 − e2)× n̂1 = meq,n̂1 × (h1 − h2) = jeq. (3.6)

From (3.5), it can be found that e2 and h2, the fields outside Ω1, are indeed zero. Therefore,

the equivalent Huygens sources are called non-radiating sources. The absence of fields outside

Ω1 in the equivalent formulation makes it possible to cancel and add objects at pleasure. A


logical choice is to replace everything with a homogeneous region characterised by εr,2, µr,2 and

to choose them equal to εr,1, µr,1, respectively.

This technique can be applied for each of the N objects and for the background medium Ω0. As

such, we haveN+1 problems characterised by a homogeneous space with εi,r, µi,r (i ∈ {0, ..., N}).For each problem, the Green’s function G can be constructed to use in the BIE formulation.

3.3 Scattering at a set of objects using RTCs

We continue our discussion by first considering a simpler case, namely one dielectric object

with permittivity ε and permeability µ, embedded in a background medium. Afterwards, the

general case with N + 1 regions, being N dielectric objects embedded in a background region is

discussed. For all cases, the Stratton-Chu representation theorem is at the centre:

(e±(r)× n̂n̂× h±(r)

)=

(K ± 12I −ηT

1ηT K ± 12I

)·(m(r)

j(r)

)+

(einc(r)× n̂n̂× hinc(r)

). (3.7)

This formulation can be deduced starting from Maxwell’s equations [18], and is an extension of

the representation formulas (2.21) we deduced in Chapter 2. Here, the ± stands for the fieldsjust at the inside or outside of the boundary of the dielectric under consideration. The symbol

η stands for the wave impedance and is given by√µ/ε. This right-hand term containing the

incident fields, einc and hinc, vanishes for all regions except for Ω0. Finally, T and K are theelectric and magnetic field integral operators, introduced to simplify our equations. They are

defined by:

T [x](r)=−jkn̂×∫

ΓG(|r − r′|)x(r′) dS′ + 1

jkn̂× PV

∫

Γ∇G(|r − r′|)∇′ · x(r′) dS′ (3.8)

and

K[x](r) = n̂× PV∫

Γ∇G(|r − r′|)× x(

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