Download Solution methods for Electric Field Integral Equations

Solution methods for Electric Field
Integral Equations
Master Thesis
Iris Koster, 1171585
Delft, 16th December 2012
Supervisors:
Dr. Koen W.A. van Dongen
Dr. Conor Brennan
Laboratory of Acoustical Wave Field Imaging
Faculty of Applied Sciences
Delft University of Technology
RF Modeling and Simulation Group
School of Electronics and Engineering
Dublin City University
1
Abstract
This thesis is part of the Master program Applied Physics. A majority of the work has been done
at the RF Modeling and Simulation Group from Dublin City University with the aim to initiate
collaboration with the Laboratory of Acoustic Wave Field Imaging from Delft University of Technology.
In this thesis Volume Electric Field Integral Equations (VEFIE) have been used to model electromagnetic scattering problems. The obtained results from VEFIE for a plane wave scattering
on cylindrical and spherical objects are compared with the analytical Mie series solutions. This
validated the conclusion that VEFIE is properly implemented and does describe electromagnetic
scattering problems with high accuracy.
In addition, several methods to improve the way of solving the VEFIE were introduced and
studied. Unfortunately attempts to optimize the solution method by filtering out irrelevant frequencies in the Fourier domain, using low frequencies solutions for the prediction of high frequencies
or by using pre-conditioning did not result in something useful directly. However, the use of FFTs
by computing convolutions instead of doing full matrix multiplication is confirmed to reduce the
amount of computations. Also the Reduced Forward Operator (RFO) is again proved to be an useful operator and reduces the number of iterations significant by decoupling the interaction between
locations in the spatial contrast domain at which there is non-zero contrast and those positions at
which there is zero contrast. However, the same reduction in 3D can be obtained by only leaving
out the outer boundary of the region during the iteration. Hence the reduction of RFO on 3D
scattering problems is less than expected. Still, it is remarkable that literature does not show a
lot of examples where RFO is actually used for solving VEFIE.
Contents
1 Introduction
3
1.1
Scattering Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2 Theory
2.1
2.2
2.3
5
Electromagnetic background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1.1
Maxwells equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1.2
Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1.3
Electric waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Solving scattering problems by the VEFIE . . . . . . . . . . . . . . . . . . . . . . .
7
2.2.1
Equivalent principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2.2
VEFIE
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Mie series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3 Numerical implementation
3.1
3.2
3.3
12
Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.1.1
Dirac delta function and weak approximation . . . . . . . . . . . . . . . . .
13
3.1.2
Discrete VEFIE
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Iteration schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.2.1
The Conjugate Gradient iterative method . . . . . . . . . . . . . . . . . . .
18
3.2.2
The Stabilized Bi-Conjugate Gradient iterative method . . . . . . . . . . .
19
Computer performance improving methods . . . . . . . . . . . . . . . . . . . . . .
20
3.3.1
Rapid matrix-vector multiplication using FFT . . . . . . . . . . . . . . . .
21
3.3.2
Reduced forward operator . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1
4 Results
4.1
24
VEFIE vs Mie series; result validation . . . . . . . . . . . . . . . . . . . . . . . . .
25
4.1.1
2D solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4.1.2
3D solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.2
Optimization 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
4.3
Optimization 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
5 Conclusions and recommendations
5.1
5.2
34
Validation by comparing with Mie series . . . . . . . . . . . . . . . . . . . . . . . .
34
5.1.1
2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
5.1.2
3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Optimisation of the solving methods . . . . . . . . . . . . . . . . . . . . . . . . . .
35
5.2.1
2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
5.2.2
3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
A Mathematical formulations
37
A.1 Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
A.2 Fourier transform; convolution sum . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
B Matlab Implementations
40
B.1 Legendre, Bessel and Hankel terms . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
40
Chapter 1
Introduction
This thesis is concerned with computer simulations for modeling the scattering and diffraction
of electromagnetic waves by an object. The project was performed in cooperation with the RF
Modeling and Simulation Group at the School of Electronics and Engineering from Dublin City
University and the Laboratory of Acoustical Wave Field Imaging at the Faculty of Applied Sciences
from Delft University of Technology.
1.1
Scattering Problems
Figure 1.1: Schematic representation of a forward scattering problem.
An electric scattering problem, graphically shown in figure 1.1, refers to the situation where electromagnetic waves are scattered and diffracted by an object (contrast), due to differences in electric
permeability between the object and the background medium. These problems arise in a wide scope
of applications. In figure 1.2 an overview of some electromagnetic wave applications is shown. In
general two types of scattering problems are considered; the inverse and the forward problem.
The inverse problem refers to translating the measured total field, that arises as a result from
the scattering and diffraction of a known incident electromagnetic field by an unknown object, into
the electromagnetic properties of the object. There is for example an electrical contrast between
malignant tumors and normal tissue. The identification of such a difference in electromagnetic
permittivity allows detection of breast cancer [19].
The forward problem works the other way around. Knowledge about the electromagnetic
properties of tissue in the volume of interest and knowledge about the electromagnetic field to
which it is exposed provides the opportunity to predict the scattered and diffracted fields. Think
about the prediction of a Wi-Fi signal strength in a building or how electromagnetic waves from a
mobile phone penetrates into the human brain.
3
Figure 1.2: Electromagnetic spectrum; an indication of sources and applications for different frequency ranges.
1.2
This thesis
In this thesis, Volume Electric Field Integral Equations (VEFIE) are introduced to model electromagnetic fields. These equations form a mathematical model which may be use to describe electric
scattering problems. Research has been done to optimize the numerical methods for solving these
equations. The accuracy of the results is monitored by comparing them with analytical solutions.
There are also other methods that may be applied to solve scattering problems. Ray tracing
methods for example solve the problem very efficient by calculating and combining various possible travel paths of a wave. However, this method only includes the most dominant pathways and
therefore does not achieve the accuracy that can be obtained with VEFIE. Accurate analytical
solutions may be obtained with Mie series. Unfortunately these series are limited to spherical
and homogeneous contrast shapes. VEFIE does not have this limitation and therefore has a much
wider range of applications compared to the Mie series.
The benefits of VEFIE in accuracy and range of applications are accompanied with a large
amount of calculations. Improvements in numerical methods to perform these calculations are
desirable.
The research performed in the Laboratory of Acoustical Wave Field Imaging in Delft is focused
on acoustic waves. The RF Modeling and Simulation Group in Dublin has the focus on electromagnetic waves. Since both scattering problems are described by integral equations of the first
kind, knowledge is exchangeable and collaboration between the two research groups has lead to
new ideas on modeling.
4
Chapter 2
Theory
2.1
Electromagnetic background
In this section a number of concepts from electromagnetic wave field theory are summarized. The
equations that form the foundation of electromagnetic theories (Maxwell’s equations) are given.
The most important constitutive relations are specified and some basics about electromagnetic
waves are discussed. These relations and basics are used in section 2.2 to formulate and solve
scattering problems using electric field integral equations.
2.1.1
Maxwells equations
Electromagnetism is concerned with the interaction between charged particles. Stationary charges
cause an electric field, moving charges (currents) both an electric and a magnetic field. The way
in which charges and currents interact with the electromagnetic fields is described by Maxwell’s
equations, which read [1] [6]
∂B
,
∂t
∂D
Jf +
,
∂t
ρf ,
∇×E =
∇×H
=
∇·D
=
−
∇·B =
0,
(2.1)
(2.2)
(2.3)
(2.4)
with
E
B
H
D
Jf
ρf
=
=
=
=
=
=
Electric field [N/C = V /m],
Magnetic field [T = W b/m2 ],
Auxiliary field [A/m],
Displacement field [C/m2 ],
volume density of electric current [A/m2 ],
volume density of electric charge [C/m3 ].
The force acting on a particle of electric charge q with instantaneous velocity v, due to an
5
external electric and magnetic field, is given by the Coulomb and Lorentz force, and equals [6]
F = q(E + (v × B)).
2.1.2
(2.5)
Constitutive relations
The displacement and magnetic fields D and B are related to the electric and auxiliary fields E
and H via the constitutive relations, whose precise form depends on the material in which the fields
exist. For homogeneous, isotropic, dielectric and magnetic materials these relations read
D = εE,
B = µH.
(2.6)
Where ε and µ are respectively the electric permittivity and magnetic permeability of the medium.
Permittivity is the physical quantity for the ability of a material to polarize due to an electric
field. Permeability is a material property that indicates the ability of a material to support the
formation of a magnetic field within itself. Their numerical values in vacuum equal [13]
ε0 = 8.85 · 10−12 C 2 /N m,
µ0 = 4π · 10−7 N/A2 .
(2.7)
The relative permittivity εr and relative permeability µr are defined as the material properties
relative to vacuum, i.e.
εr =
ε
,
ε0
µr =
µ
.
µ0
(2.8)
From the two quantities ε and µ we can define two other physical constants, namely the speed
of light c and the characteristic impedance η, who equal [8]
r
1
c= √ ,
µε
η=
µ
.
ε
(2.9)
Their numerical values in vacuum read
c0 = 3.00 · 108 m/s,
2.1.3
η0 = 376.7 Ω.
(2.10)
Electric waves
For a linear, isotropic, non-dispersive medium without sources, the Maxwell’s equations can be
combined into the wave equation for the electric field
∇2 E − εµ
∂2E
= 0.
∂t2
(2.11)
In the above equation (2.11), the wave velocity can be recognized as the speed of light, see equation
(2.9).
6
A well known straigtforward time-independent solution for the one-dimensional wave equation
is a plane wave;
E(r) = C1 eık·r + C2 e−ık·r ,
(2.12)
with imaginairy number ı and where the wave number k is defined via the angular frequency ω as
kkk = k =
ω
√
= ω εµ.
c
(2.13)
This time independent solution corresponds to the time-harmonic solution
T (t) = D1 eıωt + D2 e−ıωt .
2.2
(2.14)
Solving scattering problems by the VEFIE
The volume electric field integral equation (VEFIE) is an integral equation for the electric field in
scattering problems. It is derived from the Maxwell equations with the use of the volume equivalent
principle. A principle that converts the original scattering problem into an equivalent problem for
which a formal solution may be found directly.
2.2.1
Equivalent principle
Maxwell’s equations (2.1)-(2.4) for electromagnetic fields in the presence of an inhomogeneity can
be transformed into the frequency domain, with angular frequency ω, to produce [14]
∇ × Ê(r)
= −iωµ0 Ĥ(r) − K̂(r),
(2.15)
∇ × Ĥ(r)
= iωε0 Ê(r) + Ĵ(r),
(2.16)
∇ · (ε0 Ê(r))
= ρe (r),
(2.17)
∇ · (µ0 Ĥ(r))
= ρm (r),
(2.18)
= iωµ0 (µr (r) − 1)Ĥ(r),
(2.19)
with
K̂(r)
= iωε0 (εr (r) − 1)Ê(r),
1
ρe (r) = ε0 εr (r)Ê(r) · ∇
,
εr (r)
1
ρm (r) = µ0 µr (r)Ĥ(r) · ∇
,
µr (r)
Ĵ(r)
(2.20)
(2.21)
(2.22)
where we use the carret symbol ˆ to denote the frequency dependency of a quantity. The equations
(2.19)-(2.22) satisfy the Maxwell’s equations for induced sources K̂(r), Ĵ(r), ρe (r) and ρm (r).
Therefore K̂(r), Ĵ(r), ρe (r) and ρm (r) can be thought of being sources radiating in free space.
We refer to the procedure of introducing for the spatially varying dielectric or magnetic material
7
properties the induced sources K̂(r), Ĵ(r), ρe (r) and ρm (r) as the equivalence principle. Since
the equivalent sources are radiating in free space, it is much more straightforward to find the
electromagnetic fields caused by these sources than the original burden of solving directly in the
inhomogeneous environment.
In the case of a homogeneous dielectric contrast it is sufficient to locate the mathematical
equivalent sources on the original location of the contrast surface [14]. This is refered to the
surface equivalent principle, illustrated in figure 2.1. If the equivalent sources of an inhomogeneous
contrast take up a volume it is refferd to as the volume equivalent principle. This principle is used
to formulate the VEFIE in section 2.2.2.
Figure 2.1: Scattering problem of a homogeneous object in vacuuum. (a) Original problem, involving a homogeneous body. (b) Exterior problem equivalent to the original problem. (c) Interior
problem equivalent to the original problem.[14]
Consider the situation in which there is a contrast with respect to the homogeneous background
medium (ε = ε0 , k = k0 ). The contrast is assumed to be spatially invariant in the z-direction and
the electromagnetic waves are TM-polarized and oriented in z-direction. As can be observed from
equation (2.20), the equivalent electric currents Jˆz (r) and the total field Ê z (r) from the original
scattering problem are than related according [14]
Jˆz (r) = iωε0 [εr (r) − 1]Ê z (r).
(2.23)
In a 3D situation, the source-field relationships for electric and magnetic current sources Ĵ(r),
K̂(r) in homogeneous background medium are in general described in Maxwell’s equations (2.15),
(2.16) and can be rewritten in a wave equation [25] [27]
∇2 Ê(r) + ω 2 µ0 ε0 Ê(r) = iωµ0 Ĵ(r) + ∇ × K̂(r).
2.2.2
(2.24)
VEFIE
In order to formulate the integral equation for the unknown electric field in a scattering problem
we do not consider the scattering problem with the dielectric contrast anymore, but the equivalent
situation, explained in previous section 2.2.1. In other words, we now have to deal with sources
located in the original contrast volume, instead of the contrast itself.
Consider a scattering problem in absence of any contrast. The field generated by the transducers
will propagate without any disturbance in the homogeneous background medium. In general this
field is referred to as the incident field Êinc (r). However, in reality, contrasts are present and the
actual wave field observed is referred to as the total wave field Êtot (r). The difference between the
8
total and incident wave field is referred to as the scattered wave field Êscat (r), hence
Êtot (r) = Êinc (r) + Êscat (r).
(2.25)
In the equivalent situation, Êinc (r) is the electric field excited by the impressed sources Ĵ(r) in
the background medium. For a 2D situation where we have an inhomogeneous contrast in the
xy-plane and an incident field Ê inc
z (r) which is TM-polarized in the z-direction, the incident field
can be described using electric currents in the z-direction Jˆz (r) on the surface S in the xy-plane,
hence
Ê inc
z (r)
ZZ
1 (2)
Jˆz (r)
+ iωµ0
Jˆz (r0 ) H0 (k0 R)dS(r0 ),
=
iωε0 [εr (r) − 1]
4i
S
(2.26)
(2)
where R = |r − r0 | and H0 is the Hankelfunction. In this equivalent situation the sources Jˆz (r)
are chosen as the primary unknowns rather than the total field Ez (r).
Equation (2.26) is only valid on surface S 0 . To get an equation useful everywhere in space we
use the equivalent principle again. Substitution of Jz (r) by using equation (2.23) gives [14]
Ê inc
z (r) = Ê z (r) +
ik02
4
ZZ
(2)
H0 (k0 R)χ(r0 )Ê z (r0 )dS(r0 ),
(2.27)
S
where χ(r) = k 2 (r)/k02 − 1 is the contrast function. The integral equations for the equivalent
sources are now rewritten in an integral equation with total field Ê z (r) as the only unknown. This
is the VEFIE for the particular two-dimensional TM-polarized situation of a scattering problem.
In general, for a three-dimensional situation, it is the equation for the source-field relationship
for the unknown equivalent sources, equation (2.24), in combination with the Maxwell’s equations
(2.15), (2.16) that gives the equation for the electric field [25] [27]
Êinc (r) = Ê(r) − k02 + ∇∇
Z
G(|r − r0 |)χ(r0 )E(r)0 dr0 ,
(2.28)
V
where G(|r − r0 |) is the scalar Greens’s function for the background medium wich equeals
G(r) =
2.3
e−ik|r|
.
4π|r|
(2.29)
Mie series
The simplest version of a scattering problem and at the same time one of great practical interest
is that of a plane wave falling upon a sphere (or an infinite cylinder). Mie series are well known
analytical solutions of the Maxwell equations to this kind of problems. In this thesis Mie series
are used as a reference to check whether solutions gained in this study are correct. The Mie Series
formulation is quite straightforward and easy to use, however, restrictions on the contrast shape
and the requirement that the contrast has to be homogeneous limit the applicability. In this section
only the Mie series for the 3D situation are explained, which describe a plane wave falling upon a
sphere. The 2D Mie series, which describe a plane wave falling upon an infinite cylinder, are used
in this research as well. Since the 2D formulation is quite similar to the 3D version, we will not
discuss this in more detail.
9
Figure 2.2: Configuration of the coordinates as used in the Mie Series formulation: Cartesian
coordinates (x,y,z) and spherical coordinates (r,θ,φ,) with spherical unit vectors (i1 ),(i2 ),(i3 ).
In the Mie series, solutions to spherical (or cylindrical) scattering problems are expanded into
vector spherical (or cylindrical) wave functions. Again a distinction is made between the total,
incident and scattered field according to equation (2.25). The scattered field is devided in two
parts; one for the interior of the contrast, Êt , the other part is valid at all external points with
the necessary regularity at infinity, Êr . By enforcing the boundary condition on the spherical (or
cylindrical) surface of the contrast, the expansion coefficients of the scattered field are computed.
The Mie series theory is described in more detail by Stratton [18]. As a reference, it is sufficient
to check either the interior or the exterior field. Therefore only a formulation for points outside
the contrast sphere (or cylinder) is given in this thesis. The total external field can be written as
Êtot (r) = Êinc (r) + Êr (r).
(2.30)
Consider a 3D homogeneous dielectric spherical contrast of radius a, propagation constant
kd and magnetic permeability µd , placed in an infinite, homogeneous medium with propagation
constant k0 and magnetic permeability µ0 . Stratton [18] formulates the time independent expansion of an incident plane wave, whose electric vector is linearly polarized in the x-direction and
propagated in z-direction, according
Êinc = E0
∞
X
n=1
in
2n + 1
(m1 − in1e1n ),
n(n + 1) o1n
(2.31)
where E0 represents the amplitude of the incident field. The terms m1o1n and n1e1n are given by
m1o1n =
n1e1n
=
1
∂P 1
jn (k0 R)Pn1 (cosθ)cosφi2 − jn (k0 R) n sinφi3 ,
sinθ
∂θ
n(n + 1)
1
∂P 1
jn (k0 R)Pn1 (cosθ)cosφi1 +
[k0 Rjn (k0 R)]0 n cosφi2
k0 R
k0 R
∂θ
1
−
[k0 Rjn (k0 R)]0 Pn1 (cos)sinφi3 ,
k0 Rsinθ
10
(2.32)
(2.33)
where R is the distance from the point of interest with respect to the origin of the spherical
dielectric contrast. The angles θ and φ according the spherical coordinates with unit vectors i1 ,
i2 and i3 are illustrated in figure 2.2. Pn and jn are respectively the Legendre Polynomial and
Spherical Bessel function. Which are solutions to the spherical Laplace equation.
The time independent scattered field for points outside the spherical contrast is given by
Êr = E0
∞
X
n=1
ın
2n + 1 r 3
(a m
− ibrn n3e1n ).
n(n + 1) n o1n
(2.34)
(1)
The functions m3o1n and n3e1n are obtained by replacing jn (k0 R) by hn in (2.32) and (2.33).
Satisfying the boundary conditions leads to equations for the expansion coefficients. The
solutions for these coefficients of the external field are
arn = −
brn = −
µd jn (kd a)[k0 ajn (k0 a)]0 − µ0 jn (k0 a)[kd ajn (kd a)]0
(1)
(1)
µd jn (kd a)[k0 ahn (k0 a)]0 − µ0 hn (k0 a)[kd ajn (kd a)]0
,
µd jn (k0 a)[kd ajn (kd a)]0 − µ0 ( kkd0 )2 jn (kd a)[k0 ajn (k0 a)]0
(1)
(1)
µd hn (k0 a)[kd ajn (kd a)]0 − µ0 ( kkd0 )2 jn (kd a)[k0 ahn (k0 a)]0
(2.35)
.
(2.36)
The equations described in this section cannot be implemented in MATLAB directly, since
MATLAB is using slightly different notation. In appendix B.1 is shown which formulation is used
to apply the equations in MATLAB.
11
Chapter 3
Numerical implementation
This thesis is concerned with the introduction of VEFIE for scattering problems and optimalization of the numerical methods for solving them. There are several aspects that must be taken into
account [2]; the accuracy that is reached, the amount of memory it costs and the time in which
the calculations are done. The way in which the analytical formulation is implemented numerically
influences these aspects, and therefore influence the performance of the computer simulations.
In this chapter the numerical implementation is discussed. First, the discretization of the VEFIE is described. Secondly, the iterative schemes that are used to solve the VEFIE are introduced.
Finally, methods to optimize the performance of the computer simulations are discussed.
3.1
Discretization
Discretization refers to the process of transferring continuous models and equations into discrete
counterparts, which is the first step toward making them suitable for implementation on digital
computers. In this case we are working in a spatial domain, therefore the spatial region of the
domain is divided into discreet squared cells, as shown in Figure 3.1. The size of the cell is denoted
by ∆x×∆y in 2D and by ∆x×∆y×∆z in 3D.
Figure 3.1: Cross section of a dielectric cylinder, discretized into square cells. [14]
12
3.1.1
Dirac delta function and weak approximation
There are several ways to transfer continuous equations and field quantities into discrete counterparts associated with cells from the discretized space (Figure 3.1). In this theses the Dirac Delta
function and the weak approximation are used to do so.
Using the Dirac Delta function for discretization means that the Dirac Delta function describes the
distribution in which the continuous model f (r) contributes to the discrete value fn for a certain
cell (cell ”n”). As a result only the value of the continues model located in the middle of the cell
contributes to the discrete value,
fn = f (rn ),
(3.1)
where rn is the vector to the center of cell n. In this thesis, the Dirac Delta function is used
to discretize the permeability εn = ε(rn ), wavenumber kn = k(rn ) and the incident plane wave
Eninc = E inc (rn ).
An uniform cylindrical and spherical function is used to descretize p
respectively the Hankel and
Greens functions. The Hankel and Greens functions scale with 1/ |r| and 1/|r| respectivaly.
These functions are singular as they will go to infinity for |r| = 0. Using a uniform distribution in
which the continuous model contributes to the discrete value will overcome this problem, since the
area or volume of a cell goes faster to zero than the Hankel or Greens function is going to infinity
respectively. This is called the weak approximation.
The rectangular average of the Hankel and Greens function is not known [16]. Therefore, the
area or volume over which the average is taken is a cylinder or sphere respectively where the radius
ac and as equals the radius of the equivalent cylinder and sphere respectively, hence
ac =
as =
q
q
3
∆x∆y
π
3∆x∆y∆z
4π
(cylindrical configuration),
(3.2)
(spherical configuration).
The weak approximation of the Hankelfunction reads [14]
2
R 2π R ac 2 0 0 0
3
H0 (kR) = 4πa
H0 r dr dφ
3
0
0
c
(
4
j
2πac
3
2
J
(kr)H
(ka
)
−
for |r| = 0,
3
2
0
c
1
4πac
k
k
=
3
2
J
(ka
)H
(kr)
for
|r| ≥ ac ,
c
0
2a2 k 1
(3.3)
c
where (r,φ) represents the conventional cylindrical coordinates, J0 and J1 are respectively the zeroand first-order Bessel functions and H1 is the first-order Hankel function.
The weak approximation of the Greens function reads [?]
R as R π R 2π
3
[G] (r) = 4πa
G(|r + r0 |)r02 sinθ0 dφ0 dθ0 dr0
3
0
0 0
s
(
−ikr
6e
for |r| ≥ as ,
s ) − kas cos(kas )]
8k3 πa3s r [sin(ka
=
6
−ikas
−1
for |r| = 0,
8k2 πa3 (1 + ikas )e
(3.4)
s
where (r,θ,φ) represent the spherical coordinates illustrated in figure 2.2.
These weak approximations are used to create the discrete values
Gn = [G](rn ),
Gmn = [G](Rmn ),
where Rmn = |rm − rn |.
13
H02 (kRmn ) = [H02 ](kRmn ),
(3.5)
3.1.2
Discrete VEFIE
In this section the VEFIE is discretized with the use of the Dirac Delta function and the weak
approximation, from section 3.1.1. Starting with the 2D-VEFIE, equation (2.27), followed by the
3D-VEFIE, equation (2.28).
Two-dimensional VEFIE
Figure 3.2: A schematic summary of the transformation from the original scattering problem to a
matrix equation.
Applying previous equations (3.1) and (3.5) on the 2D-VEFIE equation (2.27), gives the discrete
form of the VEFIE in a 2D TM-polarized situation
Eninc = En + ∆x∆y
N
ik02 X 2
H (k0 Rmn )χm Em ,
4 m=0 0
(3.6)
2
where χm = km
/k02 − 1 is the discreet contrast of cell m. Notice that En is the discrete unknown
value for the amplitude of the total field in cell n and Eninc the discrete known value for the
amplitude of the incident field in cell n.
The equation can be rewritten into a matrix equation which equeals
 inc  
E1
Z1,1
E2inc   Z2,1

 
 ..  =  ..
 .   .
Z1,2
Z2,2
..
.
···
···
..
.
 
Z1,N
E1

 
 E2 
Z2,N 
 
..   ..  ⇒ Ēinc  = 
.  . 
ZN,1
ZN,2
···
ZN,N
inc
EN
EN
14
 
Z̄
 Ē .
(3.7)
where the indices 1 to N denotes the cell number and where
(
ik02
2
∆x∆y
4 H0 (k0 Rnm )χn
Znm =
ik02
1 + ∆x∆y 4 H02 (k0 Rnm ) χn
m 6= n
m = n.
(3.8)
The unit values on m = n are equivalent to the identity matrix. The Contrast χn only depends on
the column n, so could be written as a multiplication with a diagonal matrix. The other terms form
together the Dense Matrix, dependent on Rnm . This gives the opportunity to write the discrete
2D-VEFIE according matrix equation as
Einc = Z̄E = Ī + Ḡχ E,
(3.9)
where Ī is the identity matrix, Ḡ the dense matrix only dependent on Rnm and χ the diagonal
contrast matrix.
In Figure 3.2, the process of rewriting the problem into a matrix equation is summarized.
The matrix equation in which the discreet 2D-VEFIE is written, has a pattern in the structure.
The structure of Ī and χ is clear; both are diagonal and Ī even only contains ones on the diagonal.
Furthermore, χnm = 0 for cells without contrast. This is recognizable in the equations (3.6) and
(2.27), since only the cells for which χnm 6= 0 (or χ(r0 ) 6= 0) will contribute to the summation (or
integral) on the right hand sides of these equations. Therefore, it is only necessary to take the cells
with non-zero contrast into account, in order to find a solution for En everywhere. As a result we
do not have to discretize the whole space, a discretization of the scatterer as illustrated in figure
3.1 is sufficient.
But, in order to find a pattern in the Dense Matrix Ḡ, we do discretize the whole rectangular
space. Ḡ only depends on Rmn , which is equal to |rm − rn |. Therefore, the way in which values
repeat is similar to the values of |rm − rn |, which will give a nice pattern when the space is
rectangular and equidistant discretized. How the structure of Ḡ arises is illustrated in figure 3.3
in more detail.
Ḡ has a so called Blocked Symmetric Toeplitz (BST) structure, shown in figure 3.4. It are
Toeplitz matrix structures, symmetric over there diagonals, structured as diagonal symmetric
Toeplitz matrixes again. This structure allows the use of FFT, explained in section 3.3.1.
Figure 3.3: The pattern that arises in the structure of the Dense matrix Ḡ due to rectangular
discretizing of the space.
15
Figure 3.4: The pattern in the structure of the two dimensional Dense matrix Ḡ: Diagonal symmetric Toeplitz matrixes are structured as a diagonal symmetric Toeplitz matrix again.
Three-dimensional VEFIE
The way in which the 3D discreet VEFIE is formulated is similar to the formulation of the discreet
2D-VEFIE. One big difference; where in the 2D case the irrotational part of the wave is canceled
out, the configuration of the 3D case leaves us with this irrotational part. As a result, it is not
that easy anymore to construct a matrix equation with such a clear pattern in the structure like
the pattern in the 2D case.
Discretizing the 3D VEFIE equation (2.28) using (3.1) and (3.5) gives
2
Einc
n = En − k 0
N
X
∆V gmn χm Em − ∇∇ ·
m=0
N
X
∆V gmn χm Em ,
(3.10)
m=0
where ∆V = ∆x · ∆y · ∆z.
Consider the vectors En and Einc
n , in the first two terms on the right hand side of equation
(3.10), as three independent amplitudes for each direction. In that case, these terms are similar
to the terms in the discreet 2D-VEFIE equation (3.6). But, instead of the Hankelfunction, now
the Greens function is used. The Greens function depends also only on Rmn , but now three
dimensional vectors are used. Therefore they could be written as Ī + Ḡ χ E again, where Ḡ is a
three dimensional dense matrix that includes the Greens function and constant k02 ∆V .
The pattern in the structure of the 3D Dense matrix is similar to the pattern in the structure
of the 2D Dense matrix, but then structured as a diagonal symmetric Toeplitz matrix again. The
structure is illustrated in figure 3.5.
The last term on the right hand side of equation (3.10) is called the irrotational part of the field
Eirr . Using this notation gives
Einc = L [E] = E − k02 A − Eirr ,
where
An = ∆V
N
X
gmn χm Em ,
(3.11)
(3.12)
m=0
and where
Eirr = ∇∇A.
16
(3.13)
Figure 3.5: Illustration of the pattern in the structure of the three-dimentional dense matrix Ḡ.
Diagonal symmetric Toeplitz matrixes, structured as a diagonal symmetric Toepliz matrixes, are
structured as diagonal symmetric Toepliz matrix again.
To implement it in a numerical way, a central finite differencing scheme is used to approximate the
gradient divergence of A [25],
irr
Ex;j,k,l
=
irr
Ey;j,k,l
=
irr
Ez;j,k,l
=
1
(Ax;j−1,k,l − 2Ax;j,k,l + Ax;j+1,k,l )
∆x2
1
+
(Ay;j−1,k−1,l − Ay;j−1,k+1,l − Ay;j+1,k−1,l + Ay;j+1,k+1,l )
4∆x∆y
1
+
(Az;j−1,k,l−1 − Az;j−1,k,l+1 − Az;j+1,k,l−1 + Az;j+1,k,l+1 ),
4∆x∆z
1
(Ay;j,k−1,l − 2Ay;j,k,l + Ax;j,k+1,l )
∆y 2
1
+
(Ax;j−1,k−1,l − Ax;j−1,k+1,l − Ax;j+1,k−1,l + Ax;j+1,k+1,l )
4∆x∆y
1
+
(Az;j,k−1,l−1 − Az;j,k−1,l+1 − Az;j,k+1,l−1 + Az;j,k+1,l+1 ),
4∆y∆z
1
(Az;j,k,l−1 − 2Az;j,k,l + Az;j,k,l+1 )
∆z 2
1
+
(Ax;j−1,k,l−1 − Ax;j−1,k,l+1 − Ax;j+1,k,l−1 + Ax;j+1,k,l+1 )
4∆x∆z
1
+
(Ay;j,k−1,l−1 − Ay;j,k−1,l+1 − Ay;j,k+1,l−1 + Ay;j,k+1,l+1 ).
4∆y∆z
(3.14)
In order to calculate Eirr , knowledge about the spatial configuration of the cell is needed.
Therefore, the subscript (j, k, l) is used, which refers to the discrete Cartesian coordinate component values (nx , ny , nz ) of the earlier used subscript n. This leads to a difficulty at the borders of
the spatial domain, since information about the neighbours is not available in each direction. To
overcome this problem, we ensure that the contrast lies entirely in region D00 , illustrated in figure
3.6. Cells in this region do have neighbours in each direction. Since the equivalent principle has
the effect that only cells with non-zero contrast contribute to the total field in the entire region, it
is then possible to find A for the entire space D = D00 + D0 . So when the contrast lies entirely in
the region D00 , limitations in spatial domain due to the numerical implementation of Eirr will not
influence the solution.
17
Figure 3.6: Spatial region devided into two parts. D00 is the region with cells that contains neighbours in each direction; Basis With Neighbours (BWN). D = D00 + D0 represents the intire region
of interest.
3.2
Iteration schemes
For big scattering problems it is not possible to solve the VEFIE directly for the unknown total or
actual wave field E. Therefore, an iterative solution method is used. This procedure starts with a
guess for the solution of the unknown total field, from which the associated incident field may be
calculated. As long as this calculated incident field does not match the original incident field of
the problem sufficiently, the approximated total field is adjusted.
There are several iterative schemes, where each scheme has its own method to find the update direction. Some methods are capable of finding a solution for problems which have a poor
convergence, while others are limited to problems that converge easily. This last class of iteration
methods are generally simple to implement and require a low amount of calculations compared
to the more advanced iterative methods that are capable of solving ill-conditioned problems. An
example of a simple and easily implementable method is the Neumann iteration method. Unfortunately, this method will not work for the intended application. Hence, advantage will be taken
of the more advanced schemes described in the next section.
3.2.1
The Conjugate Gradient iterative method
The most common class of solvers are based on the generation of Krylov subspaces. Traditionally these methods have been favored due to their robust convergence properties. The conjugate
gradient (CG) method is an example of the simplest and most popular Krylov technique. It is an
effective method for solving symmetric positive definite systems. In this study the CG method is
used to solve the linear equation as a result of the 2D discrete VEFIE. [11]
There are several different ways in which the CG algorithm can be understood. For instance,
the algorithm can be viewed as a procedure to minimize an error functional or as a process for
constructing an orthogonal set of expansion functions for the solution. Actually, these two ideas
are linked together since each functional is associated with a specific orthogonal expansion. The
CG algorithm reduces to the process of generating the orthogonal vectors and finding the proper
coefficients to construct the desired solution. It builds the expansion from the most dominant part,
so a reasonable accuracy will be reached without the need of completing the whole expansion. [15]
Consider a nonsingular matrix equation
Ax = b,
18
(3.15)
Table 3.1: Conjugate Gradient Algorithm solving Ax = b [14]
Initial steps:
Guess x0
r0 = Ax0 − b
p1 = −A† r0
Iterate (n = 1, 2, . . . , Nmax )
n ,rn−1 i
αn = − hAp
=
kAp k2
n
kA† rn−1 k2
kApn k2
xn = xn−1 + αn pn
rn = Axn − b = rn−1 + αn Apn
βn =
kA† rn k2
kA† rn−1 k2
pn+1 = −A† rn + βn pn
Terminate when a norm of rn falls
below some predetermined value, or
when the maximum allowable number
of iterations Nmax is reached.
where A is a known square matrix, b a known vector and x the vector containing the unknowns.
The iterative CG scheme seek an estimate of the solution in the form
xn = xn−1 + αn pn ,
(3.16)
where xn−1 is a previous estimate of the solution, pn contains an update ”direction” and αn is a
scalar coefficient that determines how far the algorithm moves in the direction pn .
The CG algorithm to be presented, minimizes the error functional krn k2 with residual vector
rn = Axn − b. The step length αn from equation (3.16) that minimizes the error is given by [14]
αn =
−hApn , rn−1 i
.
kApn k2
(3.17)
Besides this step, an improvement step is made in a plane spanned by two update directions. As
step directions, conjugate update directions are used. [15]
The conventional CG algorithm discussed in many computer science literature is restricted to
the special case of a Hermitian positive-definite matrix A. To extend the algorithm to arbitrary
linear systems, the matrix equation is pre-multiplied by A† . This results in the equation A† Ax =
A† b
Finally, the CG algorithm for general matrix equations is given in Table 3.1. [14]
3.2.2
The Stabilized Bi-Conjugate Gradient iterative method
To solve some 3D scattering problems the Stabilized Bi-Conjugate Gradient (BiCGSTAB) iterative
method is used. The BiCGSTAB method is a variant of the BiCG method, developed by H.A. van
19
Table 3.2: BiCGSTAB Algorithm solving L[x] = b [25]
Initial steps:
Guess x0
r0 = b − L[x0 ]
ρ0 = α = ω0 = 1
v0 = p0 = 0
Iterate (n = 1, 2, . . . , Nmax )
ρn =< r̂0 , rn−1 >
ρn
α
β = ρn−1
ωn−1
pn = rn−1 + β(pn−1 − ωn−1 vn−1 )
vn = L[pn ]
α=
ρn
<r̂0 ,vn >
s = rn−1 − αvn
t = L[s]
ωn =
<t,s>
<t,t>
xn = xn−1 + αpn + ωn s
rn = s − ωn t
Terminate when relative residual error er =
krn k/kbk falls below some predetermined value,
or when the maximum allowable number
of iterations Nmax is reached.
der Vorst [24].
The irrotational part of the 3D-VEFIE makes it hard to rewrite the problem into a matrix equation
for which the adjoint matrix can be constructed easily. Since the CG iterative method involves
the adjoint matrix an alternative iterative method is desirable. The BiCGSTAB iterative method
doesn’t require the adjoint matrix and therefore seems to be a good alternative.
The BiCGSTAB has one major drawback; for ill-conditioned problems convergence is slow or
even not obtained. In that case, a higher (l) order BiCGSTAB can be used (BiCGSTAB(l)). This
can avoid break-down, but constructing updates becomes more demanding and more vectors need
to be stored. The discreet VEFIE is, due to Rmn dependency, strongly diagonal which results in
a low condition number. This allows us to use BiCGSTAB(2). [17]
The operations that are executed to solve L[x] = b according BiCGSTAB(2) can be found in
Table 3.2.
3.3
Computer performance improving methods
As seen in section 2.2.2, solving a scattering problem with VEFIE is accurate and applicable to all
kind of inhomogeneous contrast shapes. However, it has one big drawback; solving with VEFIE
20
is computationally intensive. This makes the duration of finding a solution unreasonably long and
the memory necessary to fulfill the calculations unreasonably big.
In this section methods are discussed, which are used during this study, to improve the computer performance in the process of solving a scattering problem with VEFIE. First it is discussed
how FFT can be used to compute matrix-vector multiplication efficiently. After that, a Reduced
Forward Operator is introduced, which is used to improve convergence of the iteration scheme.
3.3.1
Rapid matrix-vector multiplication using FFT
A cyclic matrix multiplied with a vector can be written as a convolution sum of two vectors. In
the Fourier Domain convolutions are just normal multiplications.

1
4

3
2
2
1
4
3
3
2
1
4
  
 
     
A
1
A
4
1 
A 







  
 


     
3
 B  = 4 ∗ B  = F −1 F 4 . ∗ F B 
 C  ,
2  C  3  C 

3





 





D
2
D
2
1
D
(3.18)
where F and F −1 implies the Fourier transform and inverse Fourier transform respectively. And
where ∗ and .* are respectively the discreet convolution sum and the element-by-element multiplication. Appendix A.2 illustrates this in more detail.
Replacing a convolution sum by a normal multiplication in Fourier domain will save memory
and computational effort. The computational savings gained from this method depends on the
algorithm used to compute the discrete Fourier transform and its inverse. Therefore the efficient
algorithm Fast Fourier Transform (FFT) is used to do so.
This research is not dealing with cyclic matrixes, but with diagonal symmetric Toepliz matrixes or
blocked diagonal symmetric Toepliz matrixes, which can be extended into cyclic matrixes, see figure
3.7. To ensure that the dimensions of the vector which is multiplied by has the same dimensions,
zero’s are added. Known as zero padding.
Figure 3.7: Diagonal symetric toeplitz matrix extended into a cyclic matrix.
Mathematical implementation
Unfortunately, it is not only matrix equation (3.9) we need to calculate. Using the conjugate gradient algorithm from section 3.2.1 implies that also the transposed conjugate matrix multiplication
need to be done and that the equation sometimes need to be pre-multiplied. Simple math shows
21
that this could be implemented according
Ḡv = F −1 {F(G0 ) · F(v0 )}≡F F T (Ḡ, v),
Ī + Ḡχ v
P Ī + Ḡχ v
†
Ī + Ḡχ v
†
P Ī + Ḡχ v
(3.19)
= v + F F T (Ḡ, χv),
(3.21)
= Pv + PF F T (Ḡ, χv),
(3.22)
†
†
= v + χ F F T (Ḡ , v),
(3.23)
= P† v + χ† F F T (Ḡ† , P† v),
(3.24)
where v symbolized a random vector. In practice, v often takes the form of the electric field E. The
extended and zero padded G and v are denoted by G0 and v0 respectively . F F T (Ā, v) represents
the algorithm that replaces the matrix-vector multiplication Av by a normal multiplication in
the Fourier domain, using FFT. P stands for any kind of vector where the equation could be
pre-multiplied with.
Savings
There are two savings gained from this method; savings in storage and savings in computational
effort. Instead of storing the whole matrix, only one array needs to be stored of the extended
matrix. This will reduce storage from O(N 2 ) to O(2N ). The computational effort will reduce
from O(N 2 ) calculations to O(4N log2 4N ) calculations.
3.3.2
Reduced forward operator
The reduced forward operator is an operator developed by K.W.A. van Dongen and C. Brennan.
It reduces the iterations for solving electromagnetic scattering problems using a volume integral
equation in conjunction with a conjugate gradient fast Fourier transform scheme. The reduction
is obtained by decoupling the interaction between locations in the spatial computational domain
at which there is non-zero contrast and those positions at which there is zero contrast.[4]
In section 3.1.2it is explained why the whole space is discretized while only the non-zero contrast
contribute to the solution. The unknown field values at the location with zero contrast do not
affect the values of the unknowns at any other location in space. Unfortunately including these
not-contributing cells does have a negative effect on the convergence of the CG scheme. The
adverse influence of these unknowns on the solution scheme is removed by pre-multiplying both
sides of equation (3.9) with a diagonal matrix Ĩ whose diagonal elements reflect the presence or
absence of contrast. Hence,
Ĩmm =
1
0
∀(m ∈ N0 ),
∀(m ∈ N00 ),
(3.25)
where N0 represent the indices of the basis functions whose domains possess non-zero contrast and
N00 the complementary set; basis functions whose domain posses no contrast.
The matrix in the pre-multiplied equation
ĨEinc = ĨZ · E
(3.26)
has a lower condition number than the matrix in the original equation
Einc = Z̄ · E
22
(3.27)
due to a more compact spread of eigenvalues. This will lead to a more rapid convergence when
used in a CG scheme.
Note that the pre-multiplied matrix equation is only valid for cells whose domains possess
non-zero contrast. In order to obtain the solution everywhere in space, the original equation is
calculated once, after the iterative process is completed. Using the Reduced Forward Operator
does not compromise the ability to rapid BDST-matrix multiplication with the use of FFT. It
merely introduces a trivial extra multiplication by a diagonal matrix at each iteration. [4]
23
Chapter 4
Results
Figure 4.1: An example of a 2D forward scattering problem; the prediction of the strength of a
WiFi signal through a building, obtained with the use of VEFIE. (a) Configuration of the region
of interest. (b) View of the distribution of the real part of the field.
In section 2.2.2 it is shown how VEFIE describe electromagnetic scattering problems, while in
chapter 3 it is shown how these equations can be solved numerically. An example of a solution of a
scattering problem, obtained during this studies with VEFIE, is shown in Figure 4.1. This example
shows the 2D forward scattering problem, in which the configuration and the electric-permeability
of different materials in space, define the way in which WiFi signal propagates through space. As
explained in the introduction Chapter 1, the scope of EM applications is not limited to this kind of
forward problems, but is ranging from the prediction of all kind of electromagnetic signal strengths
to the detection of tumor cell’s.
In this section the results of simulations, performed with the VEFIE as applied to two- and
three- dimensional forward scattering problems, are presented. Rather than looking to all kind of
results for different applications, comparable to Figure 4.1, it is focused on a limited number of
results that all contribute to the optimalisation of the modeling process. Starting with validating
the solutions obtained with VEFIE by comparing them with the analytical Mie series solutions.
Followed by several cases regarding the optimisation of the numerical methods for solving VEFIE.
Like the effect of using FFT, the Reduced Forward operator and different iteration schemes.
24
Figure 4.2: Positions of the centers of the discreet domains, with the outline of the cylindrical
contrast shape.
Figure 4.3: (a) Distribution of the total reflected field through the region of interest. The spatial
discretisation is 10 points per wavelenghts for the contrast which has a relative permittivity of
εr = 4 (b) Relative error (in dB) of the VEFIE solution for the reflected field, with the Mie series
used as a reference.
4.1
VEFIE vs Mie series; result validation
In section 2.3 an analytical solution for a particular class of scattering problems was formulated;
the Mie series. This analytical solution is limited to problems with homogeneous cylindrical or
spherical contrast shapes and therefore does not have a wide scope of applications. However, this
solution is very useful to validate the results found with VEFIE.
In this section, 2D and 3D solutions of the VEFIE, for cylindrical and spherical contrasts
respectively, will be compared with the exact solutions obtained via Mie series.
4.1.1
2D solutions
Different configurations are used to validate the VEFIE solutions for 2D scattering problems.
Since they all produce similar results, only the configuration shown in Figure 4.2 is presented in
this section. As incident wave, a 300MHz plane wave is used. For the background medium the
corresponding wavelength equals λ0 = 0.9993m, where as the dielectric contrast has a relative
permittivity εr = 4 resulting in a corresponding wavelength of λd = 0.4997m. The size of the
region of interest is 1.6λ0 × 1.6λ0 , with a discretization size of λd /10, which has led to 32 × 32
unknowns.
To validate VEFIE truly the matrix division operator from Matlab used. This operator
produces the solution using Gaussian elimination, which is a computational intensive approach and
not a realistic way to solve VEFIE for larger realistic problems, but very accurate for validation.
25
Figure 4.4: Scattered electric field, normalized with respect to the amplitude of the applied incident
plane wave. Computed with the use of VEFIE and Mie series respectively, on positions of the
discrete domains in a straight line through the centre of the region of interest.
In Figure 4.3(a) the absolute value of the total field is shown, while in figure 4.3(b) the error (in dB)
of the scattered field with respect to the Mie series is shown. The incident field is not taken into
account since this part of the field does not exhibit any error. To understand the error presented in
figure 4.3 in more detail, solutions on a line through the centre of the region of interest are plotted
in figure 4.4. In this graph, both results show two kinds of trends; a low frequency oscilation with
a high frequency oscilation super imposed on it. The error is only localized in the extrema of this
high frequency trend, as the peaks of the waves do not match each other properly.
4.1.2
3D solutions
For the 3D situation, results obtained with the VEFIE method for various configurations have
been compared with the exact solutions obtained via Mie series. Due to the similarity between
these results, only one of the configurations that is used to validate the VEFIE solutions for 3D
scattering problems is presented in this section.
As incident wave, a 800MHz plane wave propagating in x-direction is used. For the background
medium the corresponding wavelength equals λ0 = 0.3747m, where as the dielectric contrast has
a relative permitivity εr = 4 resulting in a corresponding wavelength of λd = 0.1874m. The size of
the region of interest is λ0 ×λ0 ×λ0 , with a discretization size of λd /30, which has led to 60×60×60
unknowns.
Besides the fact that in 3D the number of grid points scales cubic as compared to quadratic for
2D, it is also the case that in 3D all three vector components need to be accounted for. This creates
much more unknowns as compared to the situation in 2D and therefore exclude the opportunity to
solve the VEFIE via the matrix division operator present in Matlab. Hence, a BiCGstab iterative
method is used to solve the scattering problem with VEFIE. To validate the method, the relative
iteration error is set to -100dB. Only the region with zero contrast is taken into account, as this is
sufficient for validation the whole space.
In Figure 4.5 the absolute value of the total scattered field obtained with VEFIE is shown, where
26
Figure 4.5: Solutions for the reflected field obtained with VEFIE and Mie series. The contrast
has a relative permittivety of of ε = 4 resulting in a descretisation of 30 points per wavelength.
(a),(c) Distribution of the absolute value of the scattered field in respectively the yz-plane and the
xz-plane of the region of interest. (b),(d) Relative error (in dB) of the VEFIE solutions of the
absolute scattered field in respectively the yz-plane and the xz-plane of the region of interest, with
the Mie series used as a reference.
(a) and (b) show respectively the absolute electric field and the error, in the yz-plane, while (c)
and (d) show respectively the absolute electric field and the error, in the xz-plane.
The largest error is observed in the y and z-component of the electric scattered field in the
yz-plane. These errors are shown in Figure 4.6, with the y-component in (a) and the z-component
in (b). There are two critical areas recognizable where the error is relatively large; on the edge of
the spatial compulation domain and near the edge of the contrast.
4.2
Optimization 2D
Several methods to optimize the numerical solution method for VEFIE for scattering problems are
applied and studied in this thesis. A few 2D results of these methods are shown in this section.
Starting with the effect of the use of FFT. Followed by the effect of different discretizations and
the effect of the reduced Forward Operator.
FFT
27
Figure 4.6: (a), (b) Relative error (in dB) of the VEFIE solutions of respectively the y and zcomponent of the scattered field in the yz-plane of the region of interest, with the Mie series used
as a reference.
Table 4.1: Overview of the relative permitivities of materials used in configuration Figure 4.1(a)
εr
free space
1-0ı
concrete 6-0.06ı
wood 2 - 0.4 ı
glass
4-0ı
In section 3.3.1 it is shown how FFT could rapid a matrix-vector multiplication and reduce the
amount of data needed to store. In Figure 4.7 it is shown how much time is needed to do the full
matrix-vector multiplication in Matlab and how much time is required when FFTs are used. A
blocked diagonal symmetric Toeplitz matrix that arises from discretization of the VEFIE is used.
For each size, an average is taken over 10 multiplications with different random vectors. Applying
FFT does not affect the solution, which remains exactly the same.
In the graph the number of unknowns is going to 60. Full-matrix multiplication becomes
difficult when the number of unknowns gets larger. Eventually the scattering problems that need
to be solved contain billions of unknowns, which make the benefit of using FFT even larger.
In the generation of the remaining 2D results, FFTs are used to compute the spatial convolutions.
Discretization size
The computational effort, and therefore the optimization of the simulation, is strongly dependent
on the number of discretization points. A rule of thumb for discretization of a wave is that 10
points per wavelength are needed to generate an accurate representation of the wave. However, if a
reduction in the number of points per wavelength still yields acceptable results, this would reduce
the computing intensity and data storage. To investigate whether really ten discretization points
per wavelength are needed, solutions for different discretization’s are shown in this section.
In Figure 4.1(a) the used configuration is shown. The size of the spatial domain is 11.3m×7.5m.
A 300MHz point source is used as the incident source. Details of the different materials are
presented in Table 4.1. The different discretization sizes can be found in Table 4.2. A CG-iterative
scheme with a stopping criteria based on a relative error of 0.001 is used to obtain the solutions.
In Figure 4.8(a) the electric field on a line through the spatial domain is shown, with respectively a
discretization size of λ/2.5, λ/5, λ/7.5 and λ/10. The electric fields are normalized with respect to
the incident field. Because the sizes of the discretization domains differ, not all lines are localized on
28
Figure 4.7: Full matrix multiplication time vs. the time needed with FFT
Table 4.2: Overview of the different discretization characteristics and there computation times.
λ/10
λ/7.5
λ/5
λ/2.5
# unknowns 276 × 148 = 50784 208 × 138 = 28704 140 × 92 = 12880 72 × 48 = 3456
computation time (sec)
2.44 · 103
8.46 · 102
2.65 · 102
3.42 · 101
# iterations
2829
2511
2077
1419
the same place. The lines are located on y=95mm, y=84mm, y=80mm and y=78mm respectively.
To be able to judge about the different results, a smooth version is presented in figure 4.8(b).
This is computed by taking the average of adjacent results. To overcome the difference in y-position
the results were interpolated in that direction as well. Since interpolation did not affect the results,
this is not included in this theses.
Reduced Forward Operator
In section 3.3.2 it is shown how the Reduced Forward Operator reduces the number of iterations
for solving scattering problems. This effect is viewed under different conditions for 2D contrast
shapes and discretisation densities. In this section the results for a contrast shape consisting of
three solid rectangular shapes, placed on the diagonal of the spatial domain are presented.
The positions of the centers of the voxels with contrast are shown in Figure 4.9. As the
rectangular contrast shapes are solid, the distance between two centers do not represent empty
space but a difference in discretization density. In part (a) of the figure, a discretization size of
λ/2.5 is used and in part (b) λ/5. Therefore, the discretization density differs by a factor 4. In
Figure 4.10 the convergences for four different discretization sizes are shown. The discretization
sizes used are λ/2.5, λ/5, λ/7.5 and λ/10, resulting in a relative discretization density of 1, 4, 9
and 16, respectively
It can be seen that the use of RFO reduces the number of iterations, needed to gain a certain
minimum in the error, approximately by half. The amount of reduction is strongly dependent on
the shape and size of the contrast. However, in all cases, it is clear that RFO reduces the amount
of iterations.
29
Figure 4.8: The solutions for the scattered total field for different discretisation sizes (λ/2.5, λ/5,
λ/7.5, and λ/10) obtained with VEFIE, on positions of the discrete domains in a straight line
through the region of interest. (a) Original version, (b) Smooth version, obtained by taking the
average of adjacent results.
4.3
Optimization 3D
In this section a few 3D results from methods that are applied and studied to optimize the process
of solving VEFIE for scattering problems, are presented. Starting with the effect of FFT, followed
by the effect of the Reduced Forward Operator in combination with different iteration schemes.
FFT
In figure 4.11 it is shown how much time is needed to do full matrix-vector multiplication and
how much time to do this with the use of FFT. A blocked diagonal symmetric Toeplitz matrix
that arises from discretization of 3D VEFIE is used. Again applying FFT does not affect the
accuracy of the solution; this remains exactly the same. Be aware that the advantage of reducing
the multiplication time will occur every iteration.
In 3D the amount of unknowns can rapidly go to an order of billions or trillions. In this
section, the results are limited to 28 unknowns because otherwise full matrix multiplication is not
realistic and takes too much time.
In the generation the remaining 3D results, FFTs are used to compute the spatial convolution.
30
Figure 4.9: The positions of the centers of the voxels wiht non-zero contrast; relative grid densities
used with RFO for a 2D configuration. The number of grid points is increased by a factor when
going from (a) to (b).
Figure 4.10: Effect of RFO on the convergences for four different discretization densities.
ReducedCG vs BiCG and CG iterations scheme.
The configuration of the region of interest used in this section has a dimension of 1m × 1m × m,
with 9% occupied by the spherical contrast shape with radius of 0.8 meters. For the background
medium λ0 = 1m and k0 = 6.28. In the contrast kd = 8.89, so ε = 2. Discretisation size is λd /10,
and therefore contains 44 × 44 × 44 unknowns. Two different iteration schemes are used; CG and
BiCGstab.
A 3D scattering problem is much more complex to solve than a 2D scattering problem due to
the presence of the irrational part of the field (see equation 3.10). Because this irrational part is
calculated with central differences approach, there is a risk at the edges of the region. Therefore,
sometimes discretization domains without neighbours in all directions are not taken into account
during the iterative process. Using the term BWN (basis with neighbours) refers to the process of
only taken the basis with neighbours in all directions into account during the iterative process. The
term Reduced refers to the use of the Reduced Forward Operator. Like BWN also RFO limit the
region that is taken into account during the iteration. Where BWN exclude the outer boundary
of the region, RFO exclude the region with zero contrast, explained in section 3.3.2.
In this thesis RFO is only applied in combination with CG-iteration since it did not converge
in combination with BiCGstab iteration scheme.
31
Figure 4.11: The time it takes to create greens, to do Full matrix multiplication and to do the
same multiplication with FFTs.
Figure 4.12: Convergence for solving a 3D scattering problem with VEFIE, (a) using BiCGstab
iterative method and (b) using CG iterative method
The convergence of iterative solutions for VEFIE are presented in figure 4.12. In part (a) BiCGstab iteration scheme is used to gain a solution, in part (b) CG iteration scheme is used. In both
cases it can be seen that the iterative proces will converge faster when only BWN are taken into
account.
In Figure 4.13 the convergence of different iteration methods are shown; CG, BiCGstab and
Reduced-CG. In part (a) full space iterative convergences are presented, in part (b) convergence
of iteration are presented in which only BWN are taken into account. The graph shows that BWN
converges as rapidly as the Reduced version, regarding to the number of iterations that are needed.
If also is taken into account the time that it takes, shown in Figure 4.14, BWN converges even
faster.
32
Figure 4.13: Convergence for solving a 3D scattering problem with VEFIE (a) full space, (b) only
basis with neighbours
Figure 4.14: Computing time voor different iteration schemes
33
Chapter 5
Conclusions and recommendations
In this thesis Volume Electric Field Integral Equations (VEFIE) have been used to model electromagnetic wave field scattering problems. The method is validated by comparing the obtained
results from VEFIE for a plane wave scattering on cylindrical and spherical objects with the analytical Mie series solution. In addition, several approaches to improve the solution method have
been introduced and studied. In this section the main findings are summarized and recommendations are given. A majority of the work has been done at the RF Modeling and Simulation Group
from Dublin City University with the aim to initiate collaboration with the Laboratory of Acoustic
Wave Field Imaging from Delft University of Technology.
5.1
5.1.1
Validation by comparing with Mie series
2D
The error found in the solutions of the 2D VEFIE are dominated by the errors due to discretisation.
This meets the expectations since the Mie series form an exact solution to the cylindrical scattering
problem and the VEFIE a discretised one. This does not mean that VEFIE is not accurate, but
that discretisation should be handled carefully. Moreover, the relative error found the solution is
in the order of -30dB for a spatial discretisation of ten points per wavelength.
5.1.2
3D
The largest errors in the solutions of the 3D VEFIE were located in the y and z-components
of the electric field, on the edges of the spatial computational domains and near the boundaries
of the contrast in the yz-plane. The errors near the edge of spatial computational domain may
be explained via the observation that the irrational part of the field is computed with a finite
differences approach, which depends on the adjacent discrete grid points and therefore cannot be
computed correctly on the edges of the spatial computational domain.
The largest reflections are found near the boundaries of the contrast where the boundary of
the discretised object does not coincide with the actual boundary. This explains the relative larger
error is found here as well. Especially in consideration that Mie series give the exact solution to the
spherical scattering problem and VEFIE a discretised one. That these errors are typically seen in
the y and z-component could be caused by the fact that the incident field has only a x-component.
The reflection caused by this x-component is expected to be relatively small in y and z-direction,
and therefore small absolute errors in these directions will yield large relative errors.
34
In the evaluation of the errors for the 3D configuration it must be mentioned that an iterative
solution method is used. The tolerance of the iterative method is satisfied with respect to the
average error, which allows relative large errors when they are localized in a small area. This
means that the error does not need to be caused by an error in VEFIE but could be caused by the
iterative numerical solving method.
For the 3D configurations studied it may be concluded that the VEFIE is properly implemented
and does describe electromagnetic scattering problems with high accuracy. Hence, for a spatial
discretisation of 30 point per wavelength the maximum relative error found in the solution is in
the order of -10 dB in the direct neighbourhood of the boundary of the contrast falling rapidly
down to below -30 dB at a distance of only a few gridpoints away from the contrast. Attention
should be given to the discretisation of the region and to the iterative method used for solving the
VEFIE.
5.2
5.2.1
Optimisation of the solving methods
2D
When solving 2D scattering problems with VEFIE it is important for computational efficiency to
use FFTs to compute convolutions instead of doing full matrix multiplication. Using FFTs will not
affect the accuracy of the solution and will reduce the computation time in theory approximately
2
with a factor (N/ log(N )) for each dimension, with N the number of unknowns in one direction.
Consider that the number of unknowns can easily go to thousands and that multiplications must be
performed during every iteration, which makes the advantage of using FFT practically necessary.
While determining the discretisation size it is important to keep the aim of the modeling in
mind, as the size may affect the accuracy of the result significantly. A spatial discretisation size
of ten points per wavelength seems to give a good representation of the field. However, when
a general gradient of the solution is sufficient, it may be concidered to lower the discretisation
density. Verweij and Huijssen developed a filtered convolution method [??], which can be used to
reduce the number of discretisation points. Instead of discretise=ing with λ/10 they claim that
this can be reduced to λ/2 with retention of the accuracy. When this method could be applied
on solving VEFIE it could improve the process of solving VEFIE. Therefore, it seems worthy for
further examination.
What already is proven to be a good method for optimizing the process of solving VEFIE
for 2D scattering problem is the Reduced Forward Operator. This decoupling of the interaction
between location in the spatial computational domain at which there is non-zero contrast and
those positions at which there is zero contrast, reduces the number of iterations remarkable. The
advantage of this operator is strongly dependent on the size and shape of the contrast. But,
implementing this operator is so simple that the benefits do not need to be considered against the
effort it costs to realize the implementation. Remarkable is that literature does not show a lot of
examples where RFO is actually used for solving 2D VEFIE, a lost opportunity.
There are more aspects of solving 2D scattering problems with VEFIE that are studied in
this thesis for optimisation possibilities. For example, it has been attempted to take advantage of
filtering out irrelevant frequencies in the spatial Fourier domain, the convergence is tried to speed
up using pre-conditioning and it has been investigated whether the modeling results obtained for
a low frequency signal may be used to predict the outcome for a high frequency simulation.
5.2.2
3D
Also in 3D it is important to apply FFT instead of doing full matrix multiplication. Using FFT
will not affect the accuracy of the solution and will reduce the time and the amount of storage
needed.
During the 3D optimization stage of the research, it is found that BiCGstab did not converge
35
in combination with the use of the Reduced Forward Operator. An explanation could be that
BiCGstab only works for weak contrasts. However, this is in contradiction with the fact that RFO
actually improves the condition of the linear operator. Hence, the problem is probably related to
the observation that with RFO a boundary is generated between the non-zero and zero contrast,
while the irrational part of the 3D VEFIE is not capable in handling these boundaries. A recommendation is to try whether this problem also arises when adding an imaginary edge on the outer
part of the contrast shape for which the values will be updated during the iterative process.
RFO did as expected show a reduction in the number of iterations in combination with the CGiterative method. However, the amount of reduction is a bit disappointing. Even with a contrast
shape that only takes up 9% of the volume, the amount of reduction was equal to the reduction
that can be obtained by only leaving out the interaction of the outer boundary of the region. Since
3D scattering problems involves many unknowns they need badly optimisation methods for solving
them. RFO is already proven to be able to reduce the number of iterations. Therefore it is worthy
to further investigate the possibilities to apply RFO on 3D VEFIE properly.
36
Appendix A
Mathematical formulations
A.1
Vector Analysis
The relationship used to transform spherical coordinates into Cartesian coordinates read
x =
r sin(θ) cos(φ),
y
=
r sin(θ) sin(φ),
z
=
r cos(θ).
(A.1)
The above equation (A.1) can be rewritten in matrix form as [1]

 
Ar
cos(θ) cos(φ) − sin(φ)
cos(θ) sin(φ) cos(φ)   Aθ  .
− sin(θ)
0
Aφ
 
Ax
sin(θ) cos(φ)
Ay  =  sin(θ) sin(φ)
cos(θ)
Az
(A.2)
The nabla operator (∇) is used to execute the following operations: The gradient;
∇A =
∂A
∂A
∂A
x̂ +
ŷ +
ẑ,
∂x
∂y
∂z
(A.3)
∂vx
∂vy
∂vz
+
+
,
∂x
∂y
∂z
(A.4)
the divergence;
∇·v =
and the curl;
∇×v =
∂vz
∂vy
−
∂y
∂z
x̂ +
∂vx
∂vz
−
∂z
∂x
37
ŷ +
∂vy
∂vx
−
∂x
∂y
ẑ.
(A.5)
A.2
Fourier transform; convolution sum
The Fourier transform
Periodic functions can be written as the sum of sinus and cosines functions (or eıω ), according the
study of Fourier series. The Fourier transform is an extension of the Fourier series that results
when the period of the represented function is allowed to approach infinity, in other words; it
allows non- periodic functions to be transformed as well. For these non- periodic representations
an integral is used, instead of a sum of simple waves.
The Fourier representation of a continues function f (x) can be written as [22]
Z
1
F (ω) = √
2π
f (x)e−ıωx dx,
(A.6)
where F (ω) is refered to as Fourier transform of f (x), notated as F(f (x)) with ω the angular
frequency of the waves that together represent the original function.
The inverse Fourier transform describes how a function in the frequency domain can be
transformed to the original domain and is defined as [22]
1
f (x) = √
2π
Z
F (ω)eıωx dω
(A.7)
Where f (x) is the original function, obtained by taking the inverse fourier transform of F (ω),
notated as F −1 (F (ω)).
Functional relationships; the convolution sum
Table A.1: Functional relationships in conventional and frequency domain [22]
Conventional domain
Fourier domain
f (x − a)
e−ıaω F (ω)
dn f (x)
dxn
(ıω)n F (ω)
f (x) ∗ g(x)
F (ω)G(ω)
In the frequency domain the functional relationships differ from the relationships in the conventional domain. Some relationships are more complicated in the Fourier domain than in the conventional domain, for other relationships it is the other way around. In table A.1 a few common
relationships are given. For example for the convolution sum, notated as ∗, which in discrete form
is defined as;
xn ∗ hn =
N
−1
X
hm xn−m
m=0
38
mod N
(A.8)
Instead of calculating the convolution in conventional domain, it is possible to transform the
function to frequency domain and perform a simple multiplication;
g(x) ∗ h(x) = F −1 [F(g(x)) · F(h(x))]
(A.9)
Computational consequences
Figure A.1: Illustration of the effect in Fourier domain of sampling and windowing in the original
domain.
Doing Fourier calculations on the computer got some consequences. Computer calculations are
involved with discretizing and windowing, which has impact on the Fourier transform. In figure
A.1 this is illustrated.
Assume that we are interested in the Fourier transform of an original continuous function.
Using a computer will limit the original function to a discrete version, which leads to a repetition
of the function in the Fourier domain. The lower the sample frequency, the smaller the period in
the frequency domain. When the period becomes smaller than two times the maximum frequency
in the Fourier domain, overlapping occurs. This is called aliasing and makes it more difficult to
reconstruct the original function.
Using a computer will also limit the infinite original function to a certain domain, known
as windowing. In the Fourier domain this will lead to an approximation of the original Fourier
transform. The bigger the window, the lower the error.
39
Appendix B
Matlab Implementations
To implement the theory in MATLAB it is important to verify that the formulations used in the
theory corresponds to the formulations used by MATLAB. It is also important to make sure that
restrictions and limitations for certain equations or details of the process used in the theory are
implemented properly. This appendix highlight the most important ones.
B.1
Legendre, Bessel and Hankel terms
Using Legendre functions in MATLAB gives a column of Legendre values [10]
Legendre(n, x) = Pn0 (x); Pn1 (x); . . . ; Pnn (x) ,
where Pnm is defined as
Pnm (x) = (−1)m (1 − x2 )m/2
dm
Pn (x),
dxm
(B.1)
(B.2)
and Pn (x) is used for the Legendre polynomial of degree n,
Pn (x) =
n
1
d
2
n
(x
−
1)
.
2n n! dxn
(B.3)
Therefore Pn1 used in (2.32) and (2.33) is in MATLAB written as the 2nd value of Legendre(n, x)
The derivative term in (2.32) and (2.33) is written in MATLAB as [3]


∂Pn1
=

∂θ
cos(θ)Pn1
1−cos2 (θ) √
cos(θ)Pn1 + 1−cos2 (θ)Pn3
1−cos(θ)2
for n=1
(B.4)
for n >1.
The Bessel function used in MATLAB differs with a constant and a shift in ’n’ from the
Spherical Bessel used in the theory
r
jn (x) =
π
besselj(n + 0.5, x),
2x
40
(B.5)
where besselj is the command used in MATLAB. For the Bessel derivative term we use
0
[xjn (x)] = 0.5jn (x) + 0.5
πx
(besselj(n − 0.5, x) − besselj(n + 1.5, x))
2
(B.6)
The Hankel function rewritten in a MATLAB compatible language reads
r
hn (x) =
π
(besselj(n + 0.5, x) − i · bessely(n + 0.5, x),
2x
(B.7)
with derivative
0
[xhn (x)]
=
πx
[(besselj(n − 0.5, x) − i · bessely(n − 0.5, x))
2
−(besselj(n + 1.5, x) − i · bessely(n + 1.5, x))].
0.5hn (x) + 0.5
41
(B.8)
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44