-
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
-
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
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[8] A. Buffa and S. H. Christiansen, “A Dual Finite Element
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3-D Dif-ferential Surface Admittance Operator for Lossy Dipole
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-
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
4.9 Schematic illustration of the scattering of an incident
electromagnetic field at a
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
4.13 Schematic illustration of the scattering of an incident
electromagnetic field at a
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
5.8 Schematic illustration of the scattering of an incident
electromagnetic field at a
junction of two blocks of material. . . . . . . . . . . . . . .
. . . . . . . . . . . . 62
5.9 Schematic illustration of the scattering of an incident
electromagnetic field at a
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.
-
FULL-WAVE SIMULATION METHODS 5
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.
-
FULL-WAVE SIMULATION METHODS 6
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.
-
FULL-WAVE SIMULATION METHODS 7
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)
-
FULL-WAVE SIMULATION METHODS 8
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
-
FULL-WAVE SIMULATION METHODS 9
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)
-
FULL-WAVE SIMULATION METHODS 10
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
-
FULL-WAVE SIMULATION METHODS 11
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.
-
FULL-WAVE SIMULATION METHODS 12
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.
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FULL-WAVE SIMULATION METHODS 13
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
-
FULL-WAVE SIMULATION METHODS 14
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
-
FULL-WAVE SIMULATION METHODS 15
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.
-
FULL-WAVE SIMULATION METHODS 16
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.
-
DOMAIN DECOMPOSITION METHOD FOR THE SCATTERING AT N OBJECTS
18
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.
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DOMAIN DECOMPOSITION METHOD FOR THE SCATTERING AT N OBJECTS
19
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
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DOMAIN DECOMPOSITION METHOD FOR THE SCATTERING AT N OBJECTS
20
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(