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Supervisor: Dr. Wolfgang J .R. Hoefer
Abstract Yumerical techniques for solving differential equations have been vigorously stud-
ied, and various techniques have been proposed and investigated for particular prob-
lems. Maxwell's equations are the system of partial differential equations which de-
scribe the behavior of electromagnetic fields. The rnethods for solving the equations
should be properlg cliosen depending on the purpose of the analysis and the available
computational resources.
In this thesis, we propose a time-domain electromagnetic field modeling technique
based on Haar wavelets. The multiresolution nature of the wavelets was used in
the formulation, and a time stepping algorithm that is similar to the conventional
finite-difference time-domain (FDTD) rnethod was obtained. The proposed technique
effectively models realistic structures by virtue of the multiresolution property: the
computational time is reduced approximately by half compared to the conventional
FDTD method.
In order to provide a comprehensive understanding of the proposed method. algo-
rithms for one, two and three space dimensions were formulated, validated in terms
of the accuracy, and actually applied to vanous realistic problems.
Various boundary conditions have been formulated and implemented, and in ad-
dition, the following applications are addressed: S-parameter extraction for two-
dimensional waveguide problems, combined with field singularity correction a t metal
edges and corners, resonant cavity analyses for validation purposes, and analyses of
microwave passive devices with open boundaries such as microstrip low-pas filters
and spiral inductors.
An algorithm that needs half the computational effort is equivalent to hardware
that is twice as fast. The purpose of this thesis is to make a contribution to the
improvement of cornputational speed in electromagnetic tirne domain solutions.
Contents
Table of Contents
List of Tables
List of Figures
Acknowledgments
Dedicat ion
iv
vii
viii
xiv
xvi
Notations xvii
1 Introduction 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background 1
1.2 Motivation - a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Time- Domain ~1uItiresolution Technique - 3 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 OriginalContributions 6
. . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Structure of the Thesis 9
2 Wavelets and Mult iresolution Analysis Il
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction 11
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Wavelet Theory 13
. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Haar Wavelets 13
. . . . . . . . . . . . . . . . . 2.2.2 Multiresolution Analysis (MFLA) 15 . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Orthogonal Wavelets 17
CONTENTS v
2.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 1-D Tirne-Domain Multiresolution Analysis 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1
3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 1-D Time Iterative Difference Equations . . . . . . . . . . . . 22
3.2.2 Perfect Electric Conductor Boundary Condition . . . . . . . . 31
3.3 Vumerical Dispersion Analysis . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.1 An Influence of Boundary Conditions to the Computational
Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 2-D Time-Domain Multiresolution Analysis 43
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction
4.2 Formulation and Implementation for TE Case . . . . . . . . . . . . . 4.2.1 2-D Basis Functions and Time Iterative Difference Equations . 4.2.2 Sampling of the Fields . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Perfect Electric Conductor (PEC) Boundary Condition . . . . 4.2.4 Perfect Magnetic Conductor (PMC) Boundary Condition . . . 4 . 2 3 Real Impedance Boundary Conditions for Transverse Elec tro-
. . . . . . . . . . . . . . . . . . . . . magnetic (TEM) Waves
1.2.6 Perfectly Matched Layer Absorbing Boundary Condition . . .
4.2.7 Conductor Edge and Corner Node Implementation with Field
. . . . . . . . . . . . . . . . . . . . . . Singularity Correction
4.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Numerical Dispersion Relation . . . . . . . . . . . . . . . . . .
4.4 Waveguide Analysis with 2-D Time-Dornain Multiresolution Analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technique
4 .41 Xnalysis of a Simple Rectangular Waveguide . . . . . . . . . . 4.4.2 Analysis of Waveguides with Inductive Irises and Singular Field
Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CONTENTS vi
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5 3-D Time-Domain Multiresolution Analysis
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 3-D Bai s Functions and Time Iterative Difference Equations . 5.2.2 Relation Between the Haar Basis Coefficients and the Actual
Field Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Perfect Electric Conductor (PEC) Boundary Conditions . . . 5.2.4 Absorbing Boundary Conditions . . . . . . . . . . . . . . . . .
5.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Analysis of Microstrip Planar Circuits . . . . . . . . . . . . . . . . .
5.4.1 Microstrip Low-Pass Filt er . . . . . . . . . . . . . . . . . . . . 2 Spiral Inductor . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Exact Formulation for Three-Dimensional Inhomogeneous Dielectric
Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 ;\lumerical Experiments . . . . . . . . . . . . . . . . . . . . . .
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Conclusions 126
6.1 Efficiency and Accuracy of the Method . . . . . . . . . . . . . . . . . 127
6.2 Overall Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.3 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Bibliography 130
Appendix
A A Finite-Difference Time-Domain Method
List of Tables
3.1 Resonant frequencies of 1-D strings with the two locations of PEC
conditions. (a): PEC located at a quarter ce11 size away from the
center of the basis function, (b): PEC located a t the center of the
basis function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.1 Cornparison of the number of degrees of freedom (WDF), the normal-
ized dominant resonant frequency and the cornputational time . . . . 103
5.2 Normalized dominant resonant frequencies of rectangular cavities . . 105
5.3 Higher-order resonant frequencies in a cavity analyzed with the pro-
posed technique with the third-order Lagrange interpolation of fields. 106
5 . Analysis conditions for the microstrip low-pass filter . . . . . . . . . . 108
5 . Analysis conditions for the spiral inductor . . . . . . . . . . . . . . . 112
List of Figures
2.1 Haar scaling function d and wavelet function @. . . . . . . . . . . . . 2.2 Yested subspaces of I.;. CVj is the orthogonal complement of Ci to
2.3 Orthonormal basis of Haar scaling (4 j ,k ) and wavelet (vjVk) functions.
The subscripts j , k denote the dilation and the translation, respectively.
3.1 The 1-D grid used for the TD-Haar-./IRA technique. n is the time
index. The electric and the magnetic fields are staggered in both space
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . andtirne.
3.2 Inner products in (3.12) and (3.13) for testing Maxwell's equation of
. . . . . . . . . . . . . . . . . . . . . . . . . . . . Faraday's law (3.1).
3.3 Inner products in (3.27) and (3.28) for testing Mauwell's equation of
. . . . . . . . . . . . . . . . . . . . . . . . . . . . Ampère's law (3.2).
3.4 Location of the sampling points with respect to the E and H field
nodes. The arrows and the circle represent E and H fields, respective15
and the black dots represent field sampling points. The upper and lower
side sampling points with respect to the original node are denoted with
superscripts " 1" and " un, respectively. . . . . . . . . . . . . . . . . . . 3.5 Schematic diagram of tangential electric fields near a one-dimensional
PEC boundary a t z = 2; = O. Long dashed lines (- - -) and short
dashed lines (- - -) show Haar scaling (Q) and wavelet (-$) functions,
respectively. Black dots (a) show the sampling points for the proposed
. . . . . . . . . . . . . . . . . . . . . . . . . . . multiresolution grid.
3.6 Two possible locations for the PEC boundary condition. Black dots
represent field sampling points. . . . . . . . . . . . . . . . . . . . . .
LIST OF FIGURES ix
Estimation of energy stored under the staircase approximation and
sinusoidal curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two-dimensional Yee grid for the TE case. Thick solid lines represent
the Yee grids, thin solid lines represent subcell boundaries. A "subcell"
is an elementary rectangular region that surrounds a field node on the
Yee grid. The hatched region is an example of the subcell. The wavelet
expansion coefficients (E;@. H$@, H!@ etc.) are defined at the field
nodes (O : Ey, t: Hz, +: Hz) on the standard Yee grids. . . . . . . . Two-dimensional Haar ba is functions for an Ey node. Hatched regions
represent + 1 and unhatched regions represent - 1. . . . . . . . . . . . Two-dimensional rectangular pulse basis functions for an E, node.
Hatched regions represent a magnitude of +2 and unhatched regions
represent zero magnitude. . . . . . . . . . . . . . . . . . . . . . . . . The sampling points in the two-dimensional Yee grid for the TE case.
Thick solid lines represent the Yee grids, thin solid lines represent sub-
ce11 boundaries. The field sampling points are represented by crosses
( x ) and located at the center of the subcells. The fields are sampled
at the center of the subcells, while the wavelet expansion coefficients
(E;? H:*, H:@ etc.) are defined at the field nodes (0 : Ev, T: Hz, +: HL) on the standard Yee grids. At each sampling point we sarnple
three field components E,? Hz and Hz. . . . . . . . . . . . . . . . . PEC boundaries in two-dimensional space. Thick solid lines (-1 r e g
resent Yee grids, dotted lines ( 0 - -) the position of PEC boundaries.
. . . . . . . . . . . . . . . . and small circles (O) the sampling points.
Schematic diagram of electric fields near PEC boundaries. Long dashed
lines (- - -) and short dashed lines (- - -) show Haar scaling (@) and
wavelet (-111) functions, respectively. Black dots (e) show the E, nodes
. . . . . . . . . . . . on Yee's grid. Circles ( O ) show sampling points.
4.7 The location of PEC and PMC boundaries in two-dimensional space.
4.8 Location of the impedance boundary implemented in this study. . . .
LIST OF FIGURES
4.9 Reflection a t the impedance boundary condition of a TEM transmis-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sionline
4.10 Spatial field distribution of a wave reflected by an impedance boundary.
4.11 Location of a sharp rnetal edge and a local polar coordinate system .
4.12 Implementation of a corner node. The corner node is located at the
"lu" sampling point of node ( i , k). Thin solid lines show the equivalent
. . . Yee grid. The hatched region shows a perfect electric conductor.
4.13 A 90 degree corner located at lu sampling point. . . . . . . . . . . . . 4.14 Implementation of a thin conductor and its edges. E, field samples
represented by large black dots are obtained by interpolation from the
peripheral known E, fields represented by open circles. The solid lines
represent the Yee grid lines. The expansion coefficients for the E,
components are located a t the corner of the Yee grids. . . . . . . . . . 4.15 Implementation of a thin conductor for the one dimensional case. An
Ey field sample represented by a large black dot is obtained by inter-
polation from the next neighbor E, field samples represented by open
circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Analytical dispersion relations for the 2-D FDTD and the 2-D TD-
Haar-MR4. Normalized angular frequency R = wAt is drawn as a
function of the normalized wavenumber x = khl. . . . . . . . . . . . 4.17 A top view of the rectangular cavity resonator for the numerical ex-
perirnents. a0 denotes the angle of propagation, and k the propagation
constant of the plane wave. Thin lines represent Yee's grid lines, PEC
. . . walls are located a t a quarter ce11 size away from the grid lines.
4.18 Experimentally obtained numerical dispersion relations for the 2-D
time-domain Haar-MRA technique. Normalized frequency error (0 - ~ s ) / ( ~ s ) is plotted as a function of normalized wavenumber x =
with stability factor of s = 0.9/& and s = 0.4/\/2. . . . . . . . . .
LIST OF FIGURES xi
1.19 Snapshots of the Ez field distribution for the TElo mode propagating
in the WR-28 waveguide section. The total E, field (a), the waveguide
configuration and the source location (b), wavelet decomposition of the
total field into the 2-D Haar b a i s functions (c ) : and the assignment of
the coefficients of the 2-D Haar basis functions on the xz-plane (d). . 82
4.20 Top view of the inductive irises in WR-28 waveguide. a = 7.112 mm.
. . . . . . . . . . . . . . . . . . . . . . . . . . . d = a / 2 and t = a/6. 83
4.21 Discretization of the inductive irises in WR-28 waveguide. . . . . . . 84
4.22 S-parameters for the thin iris in WR-28 waveguide. - : Ar = Az =
~ 1 4 8 , - - - : Ax = Ar = a/24. . e s : Ax = il- = a/12. . . . . . . . . . 86
4.23 S-parameters for the thick iris in WR-28 waveguide. - : (Ax =
a/48. Ar = a/51), - - - : (Ax = a/24, Az = a / 2 7 ) , . . ( A r =
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . a/12, Ar = a l l 5 ) . 87
5.1 Three-dimensional Haar basis functions for an Ez node. Hatched re-
gions represent a magnitude of +1 and unhatched regions represent
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 Eight E,-subcells (dashed lines) surround a standard FDTD node of Ex
on the Yee ce11 (solid lines). Circles ( O ) and black dots (a) represent
the magnetic and electric field components defined on the Yee cell.
respectively. The centers of the subcells represented by crosses ( x ) are
the field sampling points for the multiresolution grid. . . . . . . . . . 95
5 -3 Three-dimensional rect angular-pulse basis funct ions for an E, node.
Hatched regions represent a rnagnit ude of + J8 and un hatched regions
represent zero magnitude. Each function represents an individual subcell. 97
5.4 Location of a PEC boundary parallel to the xy-plane (thick solid line).
. . . . . . . . . . . . . . h unit Yee ce11 is marked by thin solid lines. 101
5.5 Three-dimensional rectangular cavities analyzed in this study. . . . . 104
3.6 Frequency spectrum of the higher-order modes in a cavity analyzed
. . . . . . . . . . . . . . . . . . . . . . . with the proposed technique. 106
LIST OF FIGURES xii
5.7 Field distribution at time step of 1200 for the TEija(i, j = 1,3 ,5: .)
higher-order mode analysis. . . . . . . . . . . . . . . . . . . . . . . . 5.8 Microstrip low-pass filter configuration [23]. The dimensions are in
millimeters, and the numbers in parentheses show the numbers of Yee
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cells.
5.9 Time signals of the low-pass filter computed with the proposed tech-
nique. The maximum time step is 2560. - : input port, - - - : output
port. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Computed S-parameters of the low-pass filter. - : proposed method,
- - - : conventional FDTD rnethod. . . . . . . . . . . . . . . . . . . . 5-11 Snapshots of the Ez field distribution at tirne 346.6 ps in the low-pass
filter immediately below the conductors. . . . . . . . . . . . . . . . . 5.12 Spiral inductor configuration. The dimensions are in rnillimeters. The
line widths and spacings are al1 2.0 mm. The height and the span of
the air bridges are 1 .O mm and 6.0 mm, respectively. . . . . . . . . . 5.13 Time signals of the spiral inductor computed with the proposed tech-
nique. The maximum time step is 231 10. - : input port. - - - : output
port. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Computed S-parameters of the spiral inductor. - : proposed method.
- - - : conventional FDTD method. . . . . . . . . . . . . . . . . . . . 5.15 Snapshots of the Ez field distribution at time 1.72 ns in the spiral
. . . . . . . . inductor immediately below the microstrip conductors.
5.16 The discretization of the inhomogeneous dielectric loaded rectangular
cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.17 EL field distribution in the centered-dielectric-slab loaded rectangular
cavity shown in Fig. 5.5 (b). . . . . . . . . . . . . . . . . . . . . . . . 5.18 The configuration of the slab loaded WR-90 rectangular waveguide.
a = 22.86 mm, b = 10.16 mm, L = 100 mm, t = 5.96 mm, ci = 1.0
and €2 = 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.19 Ez field distributions of the dominant mode propagation in the slab
loaded WR-90 waveguide. . . . . . . . . . . . . . . . . . . . . . . . .
LIST OF H G URES ... Xl l l
5.20 The configuration of the WR-90 waveguide loaded with a dielectric
post. a = 22.86 mm, b = 10.16 mm, L = 100 mm, €1 = 1.0 and €2 = 5.0.123
5.21 The discretization of the dielectric post and the distribution of the
dielectric constants. a : E Z , x : €3, 8 : e : es, a : O : €7 . . . . . 123
3.22 E, field distributions of the dominant mode propagating in the WR-90
waveguide containing a dielectric post. . . . . . . . . . . . . . . . . . 128
1 A three-dimensional Yee ce11 for the Cartesian coordinate system. Open
circles represent H fields. black dots represent E fields. hrrows indicate
the direction of the field components. . . . . . . . . . . . . . . . . . . 135
Acknowledgment s
My study on this thesis would not have been completed without the help and support
of several people as well as some good luck. Therefore, 1 would like to express rny
gratitude to al1 of thern and note how they helped me in accomplishing this work.
The evolution of my research activities was by no means straightforward or pre-
dictable. The subject of my rnaster thesis at Kobe University, Japan, was "An ex-
perimental study on the electron states in Bi-based high critical-temperature super-
conductor films by Raman scattering spectroscopy", and 1 enjoyed my first research
period at the university. After graduation. 1 joined the R&D center at Sumitomo
bletal Industries, Ltd., where 1 started numerical analysis of electromagnetic fields in
the context of a microwave passive device modeling. After some years, my Company
allowed me to study abroad, but for only two years.
Professor Wolfgang. J.R.Hoefer. rny supervisor, accepted me to st udy under his
supervision. and encouraged me to complete the work for the thesis in that short
time period. Before starting my Ph.D program, I had already published a few papers
on related subjects. However, it was still a challenge for nie to pursue the degree
in a limited period of time. With Prof. Hoefer's great foresight, 1 could clear the
many hurdles that were al1 first experiences to me. The International Microwave
Symposium held in Baltimore in 1998 was my first experience in presenting a paper
at an international conference. Although presenting a paper for the first time in the
second language is not an easy task, 1 was able even to enjoy my presentation, thanks
only to his advise and encouragement.
Doctor Sumio Kobayashi, my former superior at Sumitomo Metal, first introduced
me to numerical electromagnetic field analysis. He also recommended that 1 apply for
studying under Prof. Hoefer's supenision in spite of the difference in Our approaches;
the TLM and the FDTD. This discrepancy at last allowed me to abandon the fked
adherence to a particular method.
The members of the supervisory committee, Professors Kin-Fun Li and Jens
Bornemann did a thorough proof reading and provided many helpful comments. Pro-
fessor Bornemann also allowed me to choose a project on the Method of Moments in
his course; this project has become the basis for my understanding of the dcrivation
of the MRTD method through Galerkin's procedure. Professors Colin Bradley and
Peter Russer graciously agreed to join the examination committee.
During my study in Victoria, 1 enjoyed many discussions with the members of
the research group: Dr. Poman So, Dr. Eswarappa Channabasappa. Dr. Mario Righi.
Dr. Lucia Cascio, Dr. Giampaolo Tardioli. Dr. Enqiu Hu, Dr. Ismael Barba, Sarni
Saab. Wei Liu, and Charles Viennet.
My wife. Noriko. has been bringing up two girls and a boy with enormous patience
and humor. It is definitely hard work to take care of three children al1 by oneself. She
thus enabled me to concentrate on my work. Our good friends and neighbors Heidi
and Curt Waller with their lovely children Guinness and Soren significantly helped
us and showed us how to enjoy Our lives. Our children love our friends Masae and
Yevin Thompson. who often took care of them and often gave me advice on writing
in English.
Last but not least, my father, who passed away three p a r s ago, and my mother
have always been understanding and encouraging, both in regard to my study al1
through my life, and to my stay overseas for this long period.
Thank you very much indeed.
blasafurni Fujii,
Victoria, May 4, 1999
To Taiga, Haruka, Chika and Yoriko
Notations
List of Symbols
Symbol : Description
: inner product
: reflection coefficient
: spatial discretization intervals in the x-. y- and :-
directions. respectively
: time discret ization interval
: Cronecker's delta function.
{ 1 ,for i' = i dit = 0 ,for; # i
: Dirac's delta function in t
: permittivity of dielectric material
: permittivity of vacuum
: relative permittivity of dielectric material
: wavelength
: permeability of magnetic material
: permeability of vacuum
: relative permeability of magnetic material
: electric conductivity
1VO T4TIONS xviii
magnetic conductivity
scaling function with dilation j and translation k
normalized wavenumber x = kat wavelet func t ion with dilation j and translation k
normalized angular frequency R = dAt
angular frequency
basis transformation mat ~ L Y between the 3-D Haar b a i s
and the rect angular-pulse basis
basis transformation matrix between the 2-D Haar basis
and the rectangular-pulse basis
speed of light
electric flux density
electric field
trequency
magnetic field
current
source current
current density
rectangular pulse function in time
space indices in the x-. y- and 2- directions, respectivelÿ
wavenum ber
vector space of al1 square integable functions of real
argument
index in time t
real numbers
Courant stability factor
t ime
multiresolution subspace of the resolution level j
multiresolution detail subspace of the resolution level j ,
orthonormal compliment space of 4 against I/;.+i integer numbers
NOTATIONS XLX
: impedance
: impedance at a boundary
Notation : Description
w p f n l ~ k : expansion coefficient for the Fw field component (F =
E, H and w = x, y, z ) with respect to the basis functions
C, q , [ = qy 9 at time step n and spatial node ( 2 , j, k)
su bscrip ts
n, i, j, k : 6 ( = -112)
O (= +O)
h (= +1/2)
1 (= +1)
examples : p w O OOh
~ ~ w m Ohh
. . - - ; - E, uww ,J,, +,,, an expansion coefficient for E, field with
respect to the basis function d(x)@(y)+&(-) at time nAt
and spatial node ( 2 , jl k + 112) . - zv@@L" - - n 112 an expansion coefficient for voltages \/; with
respect to the ba i s function t$(z)qb(y)e(~) at time nAt
and spatial node ( 2 , j, k - 112) . - . - R- ,: ~ " + m ~ , ~ , ~ + ,2 an expansion coefficient for Hz field
with respect to the basis function #(x)yt(y)4(z) at time
(n - 1/2)At and spatial node (i, j + 1/2, k + 112)
Common Abbreviat ions
1-D : one-dimensional
2- D : two-dimensional
3-D : t hree-dimensional
TEM : transverse electromagnetic
TE : transverse electric
TM FDTD :
TLM :
FEM :
MRTD :
'v1R.A :
PEC :
PMC :
ABC :
WR-LX :
transverse magnetic
finit e-difference t ime-domain
transmission line matrix
finiteelement met hod
rnultiresoliition time-dornain
multiresolut ion analysis
perfect elec tric conductor
perfect magnetic conductor
absorbing boundary condition
name of rectangular waveguides
Chapter 1
Introduction
1.1 Background
The numerical analysis of systems of differential and integral equations has become
a major issue in science and engineering. There exist various methods or algorithms
to solve particular equations. However, we are still far from the ultimate goal of
developing fast and accurate algorithms for solving such equations.
Maxwell's equat ions
blauwell's equations. which are a system of partial differential equations that describe
the behavior of electromagnetic fields and waves, were established around 1870. More
than a century tias passed since then, and by virtue of the substantial progress of
cornputer hardware technology in the last couple of decades, a numerical analysis
technique called Finite-Difference Tirne-Domain (FDTD) method first proposed by
Kane S.Yee [35] in 1966 has stepped into the limelight in the field of microwave
engineering to solve Maxwell's equations.
It should be noted, however, that it took more than a decade for the method to
be widely acknowledged since Yee's pioneerîng paper on the space-grid time-domain
numerical analysis technique [35], Mlen Tdove and Morris Brodwin's developmental
work [30] and also their application of the method to computing the fields in biological
tissues [El].
Chapter 1. Introduction 2
Yee's FDTD algorithm
The finite-difference tirne-domain method, known by the acronym "FDTD", literally
solves partial differential equations in time-domain, without inverting matrices or
solving eigenvalues. The FDTD method has the following primary advantages [15,28]:
0 since the FDTD method is the direct differencing formulation of partial differ-
ential equations, it is robust in terms of the wide variety of materials. structures
and applications to be solved,
since it does not deal with matrix operation, the algorithm is simple and the
limitation of computer resources for matrix inversion is avoided,
visualization of the simulation results provides us with intuitive understanding
of the phenornena that occur in reality as time progresses.
the algorit hm fits any computer architecture from supercornputers to simple
persona1 machines, but is especially suitable for rnulti-processor parallel vector
cornputers because the computation is done only for the nearest neighbor nodes.
allowing the computational region to be partitioned.
.A significant amount of effort has been devoted to developing this method. and it
has become one of the most widely used numerical techniques for solving hlauwell's
equations. The capabilities of Yee's FDTD algorithm are almost unlimited:
it can handle hornogeneous or inhomogeneous, linear or nonlinear! dispersive or
non-dispersive, isotropic or anisotropic materials,
it can be formulated in any coordinate system: from orthogonal coordinate sys-
tems such as Cartesian. cylindrical and spherical, to nonorthogonal coordinate
systems of general curvilinear coordinates, and even in unstructured grids if
desired,
it can deal with unbounded computational regions by using absorbing boundary
conditions,
Chapter 1. Introduction 3
a it can mode1 perfect conductors as well as finitely conductive material by means
of surface impedance approximation or by rigorously representing the exact
properties of the material,
a output can be either single continuous wave responses or broad band frequency
responses that are obtained by Fourier transformation,
a it can solve eigenvalue problems of resonators by means of Fourier transforma-
tion,
a it can be applied to radiation of electromagnetic waves and antenna analysis
including the far field estimation from the near fields,
a it allows combinat ion with passive/act ive and linearlnonlinear lumped circuit
elements and transient analysis,
a it can be ernployed in the pico-second optoelectronic application including soli-
tons.
Disadvantages of the FDTD method
Whatever its advantages, the FDTD rnethod also has disadvantages. The perfor-
mance of the FDTD method is limited by available cornputer memory and compu-
tational time. At the time of writing of this thesis in 1998, engineering workstations
with clock rates of several hundred MHz and mernory storage of a few Gbytes are
available. When using such a machine, a computational task of ten million grids with
a hundred thousand time steps can be solved in approximately a day. This could
be a medium sized three-dimensional problem including, for example, a few layers
of dielectric substrates and a few distributed circuit components for microwave or
millimeter wave modules. This is still not satisfactory for solving either rnicrowave
or millimeter wave circuits, or even a simple module. It is also not fast enough for
optimizing a single distributed circuit element.
Cha~ter 1. Introduction 4
Emergence of wavelet analysis
Wavelet analysis has been vigorously studied in the field of mathematics [6, 18, 171
for the 1 s t decade. Although the prominent application of wavelet analysis was
first in signal or image compression, it has recently been applied to solve differential
equations by virtue of its orthogonal expansion capability.
In the field of microwave engineering. wavelet functions were first incorporated
into the method of moments as basis functions. It has been reported that wavelet
analysis can be successfully applied to frequency-domain eiectromagnetic analysis to
improve computational efficiency [25].
In the context of the time domain methods, the following wavelet-based techniques
have been recently proposed to improve computational efficiency: the pioneering work
by Krumpholz and Katehi, which is based on Battle-Lemarie wavelets and referred
to by the authors as the "multiresolution time-domain (MRTD) technique" [13, 141,
and another technique proposed by Werthen and Wolff, which is based on the method
of moments solved in time-domain and incorporates compactly-supported orthogonal
Daubechies wavelets (341.
These wavelet based time-domain techniques substantially reduce the computa-
tional effort because, in contrast to the conventional FDTD method that needs at
least ten unit cells per wavelength, the wavelet-based method requires only a few
unit cells per wavelength to attain the same level of accuracy; this is aimost the
Yyquist sampling rate. In addition, wavelet decomposition (or wavelet orthogonal
expansion) allows us to use thresholding techniques; by eliminating the coefficients
that are smaller than preassigned threshold value, we Save memory and reduce the
number of operations without degrading the computational accuracy. This thresh-
olding technique can result in an automatic adaptive grid technique, which is one of
the most desirable features of numerical analysis.
Chapter 1 . Introduction 5
1.2 Motivation
In spite of the advantages of the wavelet-based time-domain techniques mentioned
above, there is a serious disadvantage when using wavelets for electromagnetic field
analysis. Since the wavelet functions such as Battle-Lemarie wavelets and Daubechies
wavelets have a support larger than unity, or in other words, the width of the func-
tion is larger than one. the boundary conditions are not usually satisfied bv each
basis function individually, but only by the superposition of several ba i s functions.
Therefore, it is difficult to reslize boundary conditions when using wavelets.
I t is certainly possible to create, for example, perfect electric boundary conditions
by employing image theory as proposed in [l-L], but boundary conditions of this type
are not localized and are difficult to apply to complicated boundary structures. This
disadvantage prevents us from solving realistic problems such as waveguide compo-
nents and microstrip planar circuits with this approach. Yoreover. the image theory
based technique requires double the computation region. This increases the memory
requirements.
One possible solution is to use Haar wavelets. Although Haar wavelets are simple
rectangular shaped wavelet functions, t hey satisfy important propert ies of wavelet
basis functions. A Haar wavelet basis allows us to realize various boundary conditions
more easily, and allows us to solve realistic problems while maintaining the advantages
of a wavelet basis.
1.3 Time-Domain Multiresolut ion Technique
As briefly mentioned in the previous section, the following wavelet-based time-domain
techniques have been proposed so far: the Battle-Lemarie-wavelet multiresolution
time-domain (MRTD) technique [13, 141, the Daubechies-wavelet technique [34], and
a multigrid technique using Haar wavelets [7, 81. They have been applied to analyze
three-dimensional cavity resonator pro blems [14? 321, t hree-dimensional microstrip
planar circuits [34], two-dimensional field distribut ions in microst rip lines [3 1, 71, and
t hree-dimensional cavities wit h various inhomogeneous dielectric loads [22]. Those
Cha~ter 1. Introduction 6
applications are, however, far from addressing the requirements of realistic structures.
Moreover, t hese techniques have complex numerical dispersion properties (241.
The numerical dispersion relations are not linear functions of frequency. In general,
i t has been reported that the wavelet-based time-domain techniques are more advan-
tageous when coarser discretizat ion is employed. When the discretization is finer. the
accuracy is not always better than that obtained with coarser discretization.
1.4 Original Contributions
The main contribution of this thesis is to accelerate electromagnetic field analysis by
means of wavelet theory. It describes the derivation and the application of an FDTD-
like multiresolution technique based ou Haar wavelets. The proposed technique was
formulated in one-. two- and three-dimensional space and tirne using Haar scaling
and wavelet functions at one scaling level.
.A complete orthonormal basis in one-, tao- and three-dimensional real-spaces is
first created using Haar scaling and wavelet functions. The field components in the
E-H formulation of ;llaxwell's equations are then expanded in the orthonormal basis.
Application of Galerkin's procedure and the method of moments leads to FDTD-like
time-iterative difference equations that are individually applied to each basis function.
For reconstruction of the field distribution from the wavelet coefficients, a basis
transformation matrix was found to be useful; it is a real, orthogonal and symmetric
matrix; therefore, the inverse of the matrix is the same as the matrix itself. This
basis transformation mat rk is from the fast wavelet transformation, and has some
interesting properties to realize effective algorithms.
ID basic concepts
For the one-dimensional algorithm, derivation of the time-domain Haar waveiet mul-
tiresolution technique is precisely described. The numerical dispersion relation is also
derived for the technique. The resulting dispersion relation was the same as the FDTD
method; it disagrees with the results provided in the literature. This discrepancy is
discussed together with some experimental results.
Chapter 1. Introduction 7
The basic concepts for the perfect electric conductor boundary conditions are
addressed. It is first described using a linear interpolation approximation, and then
expanded using a higher-order interpolation. The location of the boundary conditions
is also discussed, and it is demonstrated that proper location of the boundary improves
the accuracy of the technique.
2D problems
In order to analyze rectangular waveguide structures, the two-dimensional transverse
electric (TE) case has been formulated, and various features have been implemented:
perfect electric and magnetic conductor (PEC and PMC) conditions, impedance
boundary conditions for TEM polarized waves, Berenger's perfectly matched layer
(PML) absorbing boundary conditions (ABC) [1], 90-degree and zero-degree corner
nodes with singular field correction by a quasi-static field approximation [20], as well
as S-parameter extraction.
For validation, the numerical dispersion relation has been verified bot h wi th nu-
merical experiments and with analytical formulas available in the literature. The
dispersion relation provides a clear insight into the accuracy of numerical techniques.
It is shown that, for the proposed technique, the irnprovement in accuracy over that
of the standard FDTD method is greatest when the discretization is coarsest. This
property agrees with observations made on techniques employing other wavelet fam-
ilies.
The proposed technique was first applied to a simple rectangular waveguide to
demonstrate the behavior of the Haar wavelet b a i s coefficients. It was then applied
to the analysis of waveguides with thin and thick inductive irises. The singular
field around the edges and the corners of the irises are corrected by the quasi-static
approximation of the fields. The effect of the singular field correction on the accuracy
and efficiency of the proposed technique will be demonstrated.
Chapter 1. Introduction 8
3D problems
The two-dimensional concept is then expanded into the three-dimensional full wave
formulation. Perfect electric conductor (P EC) boundaries are formulated using simple
forward- or backward-difference approximation, and then improved by using Lagrange
interpolation to analyze higher-order modes in a cavity. Mur's first order absorbing
boundary condition (ABC) has been irnplemented, as well.
Several rectangular cavities with inhomogeneous dielectric loading were analyzed
to validate the proposed technique. The results were then compared with analytical
results (when available) and with data obtained by conventional FDTD having the
same number of degrees of freedom; both methods were compared for situations in
which the same amount of computer memory was used.
The proposed technique was finally applied to analyze microstrip low-pass filters
and spiral inductors with open boundaries. These analyses demonstrate the capability
of this new technique for solving practical rnicrowave problems more efficiently than
the conventional methods.
Computer resources
Most of the numerical analyses in this thesis were performed with an engineering
workstation HP Cl60 - a 64 bit bus. a CPU clock rate of 160 MHz. a bus speed
of 120 MHz and a memory size of 160 'vlbytes. The required computer resources are
discussed and compared with those of the conventional FDTD method.
The multiresolution technique has the potential for reducing the computational
effort by thresholding small coefficients [32]; the unknown coefficients that are smaller
than a certain value can be omitted without affecting the computational accuracy.
However , t hresholding has not been implement ed in t his t hesis.
Although the accuracy and the memory requirement of this new procedure are
similar to those of a conventional FDTD method with the same number of degees
of freedom, the multiresolution technique based on Haar wavelets is approximat ely
twice as fast.
Chapter 1. Introduction 9
1.5 Structure of the Thesis
This thesis is organized into five chapters. After the introduction, Chapter 2 briefly
reviews the basic concepts of wavelet theory and the terminology that appears in the
following chapters. This description is intended to be concise rather than rigorous in
order to provide the reader with intuitive understanding.
In Chapter 3. the time-domain multiresoliition technique hased on Haar waveletrs
is formulated for the one-dimensional case. The derivation of the proposed tech-
nique is precisely described to present the outline of the formulation. We introduce
implernentation of the perfect electric conductor (PEC) boundary conditions in this
chapter. It is first implemented with a linear interpolation, and then extended by a
third-order interpolation. The numerical dispersion relation is also derived analyti-
cally from the results of the MRTD technique given by Krumpholz and Katehi [l-L].
Then the accuracy of the proposed technique is discussed with the location of the
boundary conditions with respect to the grid structures; it is shown that the proper
location of the boundary condition improves the accuracy.
In Chapter 4. the two-dimensional tirne-dornain multiresolution technique based
on Haar wavelets is formulated for the transverse electric (TE) case. Various bound-
ary conditions are also implemented in order to analyze waveguide structures. In
addition. the numerical dispersion relations are discussed to clarify the superiority of
the proposed technique over the conventional method in terms of accuracy. The two-
dimensional algorithm is then applied to waveguide problems. Inductive iris struc-
tures are analyzed, and the field singularity a t the corners of the perfect conductors
are corrected with a quasi-static field approximation technique.
Chapter 3 expands the idea into a full wave formulation in three-dimensional space.
resulting in eight basis functions for each field component. The algorithm is applied
to open microstrip planar structures. This is the most challenging analysis performed
in the thesis. We show that the proposed technique is applicable to realistic struc-
tures and has the same level of accuracy as the conventional FDTD technique while
requiring only about half the computational time. In these analyses, the interface
condition between different dielectric materials is treated in an approxîmate manner.
Chap ter 1. Introduction 10
An exact treatment for analyzing inhornogeneous dielectric materials is discussed in
the last section of this chapter; this exact formulation leads to a stable algorithm.
Chapter 6 draws an overall conclusion and discusses the advantages and disadvan-
tages of the time-domain multiresolution technique for electromagnetic field analysis.
New directions of research are also discussed.
Chapter 2
Wavelet s and Mult iresolut ion
Analysis
2.1 Introduction
Wavelet analysis has been vigorously studied for more than a decade. However.
the simplest wavelets, Haar wavelets, have been known since 1910 when they were
introduced by the German mathematician Alfred Haar [IO]. It is known chat any
continuous function can be approximated by a set of Haar wavelet b a i s functions.
The seminal works t hat stimulated the numerous subsequent studies in ivavelet t h e o ~
in both applied mathematics and engineering science are the lectures on the creation
of orthogonal bases of compactly supported wavelets in 1988 by Ingrid Daubechies [a], and the invention of multiresolution analysis (MRA) based on orthonormal wavelet
bases in 1989 by Stephane G.Mallat [18, 171.
Following these two contributions, wavelet analysis has been effectively applied to
various problems in signal processing such as signal or image compression, approxima-
tion and denoising. The most prominent application is the compression and storage
of finger prints standardized in the Federal Bureau of Investigation (FBI). Millions
of finger print images have been compressed and stored using a wavelet based image
compression technique.
Chapter 2. Wavelets and Multiresolution Analysis 12
Haar wavelets and electromagnetic field analysis
Although Haar wavelets are the sirnplest possible basis functions, they have the im-
portant properties of orthogonality and compact support that enable simple multires-
olution representation of signals. One of the prominent features of smooth wavelet
functions is t heir compactness (localizat ion) in bot h t ime and frequency domains, or
more generally, in both the real space and the reciprocal (Fourier) space. However,
because the Haar wavelets are step functions and the Fourier transform of a rect-
angular pulse function is a "sinc" function sin(x)/x, Haar wavelets are not compact
(localized) in the Fourier domain.
The non-localization in the Fourier domain is a disadvantage of the Haar wavelets.
and in the electromagnetic field analysis, it makes the resulting analysis technique
too simple, or in other words, an effect of incorporating a wavelet basis is not suf-
ficiently given for improving the accuracy of the analysis technique. This will be
discussed in detail in the context of the numerical dispersion of the proposed tech-
nique in Chapter 3. On the other hand. the simplicity of Haar wavelets allows us to
mode1 complicated boundary condit ions. Computational efficiency and realizability
of complicated boundary conditions are, in general. difficult to reconcile when using
wavelets for numerical analysis.
Two properties of wavelet bases, namely orthogonality and multireso1ution prop-
erty, play an important role in this thesis. The difference equations forming the
numerical algorit hm are derived from bIaxwellTs curl equations by using t hese two
properties. In this chapter, the basic concepts and terminology in wavelet theory.
which will appear in this thesis but might not be familiar to most engineers, are
briefly reviewed. More detailed and precise discussions can be found in the following
references: (27.26, 161 contain easy introductions, [6, 11, 12, 33) are general references
of wavelet theory, and some programs for the wavelet transforms are described in [21].
Cliapter 2. Wavelets and kfultiresolution Analysis 13
2.2 Wavelet Theory
2.2.1 Haar Wavelets
Haar wavelets
The simplest wavelet basis functions. Haar wavelets, are defined by
i f o r O < t < l
O elsewhere
1 f o r O < _ t < 1 / 2
-1 f o r 1 / 2 < t < l
O elsewhere
where 4 and w are called Haar scaling function and Haar wavelet function. respec-
tively, and are shown in Fig. 2.1.
Figure 2.1: Haar scaling function d and wavelet function W .
We will now review some concepts and terminology that are necessary for dis-
cussing the properties of wavelets.
Cha~ter 2. Wavelets and hfultiresolution .-lndysis 14
Vector space L2 (R)
We cal1 space a set of functions, and a space L2(R) denotes the vector space of al1
square integrable functions with independent variables of real numbers R. Square
integrable functions, in this case, are functions that have a finite inner product and
associated norm
where f* is the complex conjugate of f . The space L2(R) is an example of Hilbert
space.
Compact support
.A function f in the space L ~ ( R ) is said to have "compact support" if. for some
bounded interval [a, b] c R, the function f has nonzero value in [a. b ] , and / = O
almost everywhere outside of [a. b].
Therefore, Haar basis functions have compact support, because they have a certain
value in [O, 1) and have zero value outside [O, 1).
Dilat ion and t rans la t ion operations
The wavelet basis functions 9 ( t ) and ,w ( t ) become an orthogonal basis in a vector space
L2(R) through dilation and translation operations. They also have compact support.
which means that they have nonzero values in a certain interval of the independent
variable t.
Dilation and translation are the operations that change the scale and the location
of the functions defined by
where the superscript j denotes dilation, or changing the scale or resolution. and the
subscript k denotes translation, or shifting the function. Functions with a small value
Cbapter 2. Wavelets and Multiresolution Andysis 15
of j represent the detail of a function to be analyzed, and those with a large value
of j capture the coarse property of the function. The factor 2j I2 is a normalization
factor which ensures that the basis functions have unit norm.
It can be proven that the set of functions (2.5) and (2.6) forms an orthonormal
basis in L2(R). This will be shown in the next subsection.
2.2.2 Mult iresolution Analysis (MRA)
Definit ion of multiresolut ion analysis
A multiresolution analysis of L2(R) is defined as
"the nested sequence of closed subspaces { V ; } which approxirnote L2 (R)" .
where Z denotes integer numbers, and the subspace 'L; is generated with a scaling
function q5 by
C; = ~ ~ a n { 2 ~ / ~ & ( 2 J t - k) , k E 2) (3.7)
which means that every function in the subspace I.; can be expressed as a linear
combination of the functions 2 3 / * 9 ( 2 ~ t - k) with k integer.
'v1ultiresolution analysis is a recursive process. Any function in the space L2(R) c m be decomposed into coarser basis functions in space Ç;, and the detail of the
function is accumulated into the complement spaces C\.
The scaling function and the subspaces 1/;
For the set of subspaces 5 with j E Z generated with a scaling function #, the
following nested sequence is obtained:
which is also depicted in Fig. 2.2.
The subspaces V, have the following important p r ~ p e r t ies:
Chapter 2. Wavelets and Multiresolution Analysis 16
Figure 2.2: Sested subspaces of 1.;. I.V, is the orthogonal complement of Ç; to C i i l .
Scaling: f ( t ) E I.; * f ( W E b;+i Inclusion: I.; c I/;+I for each j
Density : u = L'(RI J E Z
Maximality: n 4 = {O) J E Z
Basis: { @ ( t - k), k E Z} is an orthonormal ba i s in Ci.
The wavelet function and the detail spaces Wj
The orthogonal complement space CVj of I.; in V,+l is generated with a wavelet func-
tion 11i by
Wj = ~ ~ a n { 2 ~ / ~ $ ( 2 ~ t - k), k E 2) (2.9)
which means that every function in the subspace Wj can be expressed as a linear
combination of the functions 2 ~ / * @ ( 2 ~ t - k) with k integer.
At every scale of j , I.;+l is represented by the direct sum of b; and CV, as
As shown in Fig. 2.2, the spaces yield the partial sum of the complement spaces
IVj
V,+, = W' @ Wj-i @ * @ bVo @ PVF1 (2.11)
C h a ~ ter 2. Wavelets and hfultiresolu tion ,ilnaZvsis 17
and, subsequently, the space L2(R) consists of the whole sum of the complement
spaces I.1; as
Note that the spaces 5 and Vk with j, k E Z are not orthogonal. but CV, and kt;; are orthogonal. and IV, and 5 are also orthogonal:
Those properties represent the theoretical background for the fact that any function
in the space L2(R) can be exactly represented by the superposition of scaling and
wavelet functions.
2.2.3 Orthogonal Wavelets
Orthonormal basis
Since the spaces CI;, with j E Z are al1 mutually orthogonal, the set of functions
{ ~ J / ~ I , ! J ( ~ J ~ - k) , j , k E Z} is an orthonormal basis in L2(R). Therefore. for any
function f ( t ) E L2 (R), the expansion
is an orthogonal expansion.
Practically, using a function fiV E Kv with an appropriate :V that approximates
a given function f E LZ(R) at a pre-required precision, we can obtain a wavelet
decomposition of f as an approximation
where the function f, is a function in the subspace 4, and the function gj a function
in the complement detail space Wj.
Cha~ter 2. Wavelets and Mul tiresolu tion Analysis 18
For many applications, most of the wavelet coefficients are so small that the signal
can be represented by truncated wavelet series with fewer terms than its Fourier series
requires. Moreover, by thresholding small wavelet coefficients, we can reduce the
number of coefficients in the expanded series.
The orthogonality of scaling and wavelet functions is summarized as follows: for
the integer numbers j, k, 1 , m, (i) the scaling functions are orthogonal for the same
resolution level 1 as
( (bj,k< 6 j .m ) = 6km : (2.15)
(ii) the scaling functions and the wavelet functions are always orthogonal if j 5 I as
(iii) the wavelet functions are always orthogonal as
where 6 denotes Dirack delta function. Those properties are used for calculating the
inner products when Galerkin's procedure is applied to rClauwell's equations with the
wavelet b a i s functions. The orthogonality of the functions helps significantly simplify
the resulting difference equations. This allows us to formulate an effective algorithm
for solving differential equations.
Orthonormal basis of Haar wavelets
Dilation and translation operations (2.5) and (2.6) create a set of orthonormal basis
functions in L2 (R). This, of course, holds for Haar wavelets. Figure 2.3 shows three
levels of orthonormal Haar scaling and wavelet functions. In this thesis, only one
lower-resolution level of the basis is incorporated to approxirnate original signais. It
is obvious from the figure that the wavelet functions are al1 orthogonal, and that the
scaling and the wavelet functions at the same resolution level are orthogonal. These
functions a t the resolution level of j = -1 will be the basis functions in this thesis.
The original signal a t the level of j = O will be decomposed into scaling and wavelet
funct ions wit h lower resolution level, which enables coarser discretization of grids,
result ing in the reduction of the computational effort required.
Cha~ter 2. Wavelets and Multiresolution Anaiysis 19
a l i n q functions
. . . p 0 . 0 tQo.i
etc.
velec functions
. . . p 0 . 0 +% 1 etc.
Figure 2.3: Orthonormal basis of Haar scaling (@j,k) and wavelet ( $ j , k ) functions.
The subscripts j, k denote the dilation and the translation, respectively.
Chapter 2. Wavelets and Multiresdution Analysis 20
2.3 Concluding Remarks
The introductory concepts and terminology in the field of wavelet theory were re-
viewed. Although Haar wavelets are the simplest basis, they exhibit the most promi-
nent characteristics of wavelets such as compact support and orthogonality. It is
known that the Haar wavelets are the only real-valued wavelets that are compactly
supported. symmetric and orthogonal $1. This simplicity cnables us to realizc corn-
plicated boundary conditions in the analysis of electromagnetic fields.
Chapter 3
1-D Time-Domain Mult iresolut ion
Analysis
Introduction
The t ime-domain Haar-wavelet- based rnultiresolution analysis (TD-Haar-MRA) tech-
nique has been formulated for the one-space-dimensional case in order to provide the
basic concepts involved in the technique.
The derivation of the time stepping algorithm is based on Galerkin's procedure
and the method of moments, requiring the calculation of inner products between the
scaling and the wavelet functions and their derivatives.
The perfect electric conductor boundary condition is irnplemented by enforcing the
superposition of the scaling and the wavelet functions of the tangential electric fields
to be zero at the boundary. This property allows arbitrary location of the boundary
conditions with respect to the grid lines. In this thesis, the PEC boundaries are
located at the position that is quarter ce11 size away €rom the center of the b a i s
function (quarter ce11 size shifted from the grid lines). Another possibility is that the
boundary is located a t the cent er of the basis funct ion (right on the grid lines). In t his
case, however, it is shown that the accuracy is the same as that of the FDTD method;
no improvement in the accuracy is obtained in the proposed technique compare to
the FDTD method.
Chapter 3. 1-D Time-Domain Mul tiresolution Andysis 22
The numerical dispersion relation is also derived analyticdly The results indicate
that the proposed technique has the same numerical dispersion relation as the conven-
tional FDTD method, which do not agree with the previously published results in [9].
Precise derivation is described in order to demonstrate that our results is correct. It
is also discussed from the view point of the accuracy in the experimental results.
In the discussions it is demonstrated that, although the dispersion relation is the
same as that of the FDTD method, the accuracy can be improved by choosing proper
location of the boundary.
3.2 Formulation
3.2.1 1-D Time Iterative Difference Equations
Starting with Maxwell's equations for a plane wave propagating in the :-direction.
the field components E,, Hy are expanded into two orthonormal basis functions, which
are the Haar scaling function m and wavelet function tb [6] multiplied by a pulse
function in time h(t) as
bk(4 h*(t) 9
@ k ( 4 hn(t) , (3.3)
where
and 1 for (n - 1/2)At t < (n+ 1/2)?lt
O elsewhere
Chapter 3. 1-D Tirne-Domain kfultiresolution .4ndysis 23
with the space and time discretization intervals Az and At. The functions 4 and v are defined as
1 for ls1 < 112
112 for ) S I = 112
O elsewhere
1/2 for s = -1!2
1 for -1/2<s<O
-1 for O < s < l / 2
-112 for s = 112
O for s = O and elsewhere
The b a i s functions (3.3) have the support (or the width of the function where it has
nonzero value) equal to the spatial discretization interval Ar. In order to obtain cor-
rect inner products when Galerkin's procedure is applied with those basis functions.
the definitions of the Haar scaling and wavelet functions (3.7) and (3.8) are slightly
modified frorn the definitions givcn by (2.1) and in Fig. 2.1.
Taking the electric and rnagnetic field nodes alternately along the z-auis as shown
in Fig. 3.1. then the expansions of the field variables are given by
Figure 3.1: The 1-D grid used for the TD-Haar-MM technique. n is the time index.
The electric and the magnetic fields are staggered in both space and time.
Chap ter 3. 1 -D Time-Domain iWd tiresolu tion Andysis 24
where the notations 'n with C = Q, denotes the expansion coefficients in terrns of
the Haar scaling and wavelet functions at time step n and position k.
Derivation of update equations for Maxwell's equation of Faraday's law
The field components (3.9) and (3.10) are substituted into Ma.xwell?s equation of
Faraday's law (3.1), and subsequently tested with the b a i s functions (3.3). Testing
(3.1) with the b a i s function 4 k + l / 2 ( ~ ) h n ( t ) , we obtain
where (f ( x ) ( g ( r ) ) denotes the inner product between the functions f (x) and g(x) as 36
(f (x) l g ( x ) ) = J f (x) . g(x)dx. Then testing the same equation (3.1) with the basis -Ca
function ut (2) hn+Il2 ( t ) , we obtain
The inner products between the b a i s functions are simply represented by virtue
of the orthogonal property. For the derivatives of the b a i s functions, however, thep
are no longer orthogonal. The inner products necessary for evaluating (3.12) and
(3.13) can be obtained from Fig. 3.2 as
Chapter 3. 1-D Time-Domain Multiresolution .Analysis 25
where b is Cronecker's delta function.
Thus? the inner products for the field components of (3.12) and (3.13) are given
Chapter 3. 1-D Time-Domain iMuItiresolution Andysis 26
4 ( f o r testing) 1-
p P k , I
k4-112 k 1 ~ * + I / s z Az
'6
Figure 3.2: Inner products in (3.12) and (3.13) for testing ZvIaxwell's equation of
Faraday's law (3.1).
Chapt er 3. I -D The-Domain Md tiresolu tion Andysis 27
Then the difference equations for Maxwell's equation (3 .1 ) are given by
Finally. the time update difference equation is given by
Derivation of update equations for Maxwell's equation of Ampère's l a w
Similarl. the field components (3.9) are substituted into Maxwell's equation of Ampère's
Law (3.2), and subsequently tested with the basis functions (3 .3 ) . Testing (3 .2 ) with
the b a i s Function Q~ (2) hnCLl2 ( t ) , we obtain
and testing the same equation (3.2) with the basis function d ~ ~ ( z ) h , + ~ / ~ ( t ) , we obtain
Cha~ter 3. I -D Time-Domain Multiresolu tion Analvsis 25
The inner products in (3.27) and (3.28) cao be evaluated as shown in Fig. 3.3:
The inner products for the field components of (3.27) and (3.28) are given by
Chapter 3. 1-D Tirne-Domain Mui tiresolu tion ,.lnalysis 29
A bk (for testing) Wk (for testing e 4 8
i
,k ' + 1 /
k g k'+1/2 r
- 1- ' -"x---' Yk. cl/',
4 +1,2 (for testing)
1.. - .r+r*+----+-J~.--.titi .... *-.. i
kao +
testing)
n'-1!2 n'
Figure 3.3: Inner products in (3.27) and (3.28) for testing hfaxwell's equation of
Ampère's law (3.2).
n'+1/2 L At
- 1-. (s
Chapter 3. 1-D Time-Domain Multiresolution ,4ndalysis 30
Chapter 3. I-D Time-Domain Multiresolu tion Analysis 31
Then the difference equations for kli~auwell's equation (3.2) are given by
Finally, the time update difference equations are given by
3.2.2 Perfect Electric Conductor Boundary Condition
PEC with linear interpolation
The implementation of PEC boundary conditions with a simple linear interpolation
technique is first described. As discussed in [7], the basis functions do not couple at
Chap ter 3. 1-D The-Domain Mul tiresolu tion Analysis 32
the inner computational nodes, but only at the boundary and the excitation nodes.
Therefore. the PEC condition is implemented by combining scaling and wavelet func-
tions at the boundaries such that the total tangential electric field at the boundaries
becomes zero. At the same time, the electric field in the subcell situated half a ce11
size away from the boundary must be found by interpolation so that the tangential
electric field varies smoothly in front of the b o u n d q .
We will first introduce a grid structure necessary for the implementation of PEC
boundaries. The nodes for E and H fields are located alternately along the spôtial
auis. then. as shown in Fig. 3.4. the basis functions d ( z ) and Q ( z ) represent field values
at the sampling points, which are located a quater ce11 size away from the original
grid nodes. Those sampled values are expressed with their expansion coefficients as
and
Figure 3.4: Location of the sampling points with respect to the E and H field nodes.
The arrows and the circle represent E and H fields, respectively, and the black dots
represent field sampling points. The upper and lower side sampling points with
respect to the original node are denoted with superscripts "1" and "u" , respectively.
Chapter 3. 1-D Time-Domain iLlultiresolution Analysis 33
Figure 3.5: Schematic diagram of tangent ial electric fields near a one-dimensional
PEC boundary a t 2 = zb = O. Long dashed lines (- - -) and short dashed lines (- - -) show Haar scaling (4) and wavelet ( - w ) functions, respectively. Black dots (e) show
the sampling points for the proposed multiresolution grid.
Figure 3.5 shows a one-dimensional PEC boundary placed a t : = 2; = O. In
contrast to the condition presented in [8], the boundary is located a t the position
that is shifted by 1 4 4 from the center of the basis functions. Since the tangential
electric field at the boundary is zero, the first equation is given by *
The tangential electric field a t z = r," is expanded into a Taylor series with respect
to z = 2: as
E(r,") = ~ ( r : ) + (zg - z:)E'(z~) +- . (3.52)
and the backward-difference approximation is used for the first derivative
then the second equation is given at z = zz by
Chapter 3. 1-D Time-Domain Multiresolution Analysis 34
Solving (3.51) and (3.54) in terms of the Haar basis coefficients E? and E$ leads to
the boundary condition at z = 0:
Similarly, by using a forward-difference approximation. the PEC condition for the
other side of the boundary at 2 = zh1 can be obtained as
PEC with higher-order interpolation
The PEC conditions (3.55) and (3.56) result in a slightly distorted field distr
which can be improved by using Lagrange interpolation instead of the forward- or
the backward-difference approximation. Since the tangential electric field near the
boundary is considered to be an odd function about the boundary, a third-order
interpolation polynomial is obtained by using only two reference points as known
field values. This is the same requirement as that required in the central-difference
approximation for the first derivative in the Taylor series of (3.52).
In the case of a PEC boundary at z = = 0, the electric field E," at 2 = -0 -" is interpolated from the field values at 2 = z: and 1. The third-order Lagrange
coefficient polynomials are given by
Thus, Et is given by Lagrange interpolation as
Chapter 3. 1-D Tirne-Domain Multiresolution Analysis 35
Solving (3.51) and (3.59) in terms of E: and E$ leads to an improved PEC boundary
condition at z = 0:
Similarly, the PEC boundary condition at r = :if can be obtained as
where the coefficient polynomials are given by
3.3 Numerical Dispersion Analysis
In this section, the numerical dispersion relation for the TD-Haar-MR-\ technique is
derived analytically. The numerical dispersion relation for the 3-D MRTD scheme
that is incorporating both the scaling and the wavelet functions in the y-direction
and only the scaling function in the z- and i-direction is given by (68) in [l4] as
with
Chap ter 3. 1-D Time-Domain fiil ul tiresolu tion .4nalysis 36
The operators are generally given by
2 j 1
Dr(?) = - x a ( i ) sin(i + 1/2)77 IY *=O
'zj 1
D,V(q) = -xb(i) sin(i + 1 / 2 ) q AY t=,
2j If1 D : ( ~ ) = -Cc(i) sin iv
' 2 / i=l
%j R dt(R) = - sin - .
At 2
w here
and the coefficients a(i) , b(i) and c(i) can be obtained by e
inner products
valuating the following
The number of terms 1 in the summation of (3.67) - (3.69) and (3.73) - (3.75) should
be taken so that they are large enough for the sums to converge.
Considering the one-dimensional case in which the field propagates in the y-
direction, eliminating D, and Dz, we obtain
For the Haar wavelet coefficients, comparing the definitions (3 .73)-(3.75) wit h (3 .29)-
(3 .32) . it is obvious that a ( 0 ) = b ( 0 ) = 1 and a ( i ) = b(i) = O for i = 1.2. . . I . rvhich
yields
- 2 j rl - -a(O) sin - AY 2 2 0 - - - sin - Ay 2
2 j - 0 - -b(O) sin - Ay 2
- 2 j q - - sin -
Ag 2
and the cross terms c ( i ) , i = O, f 1, I 2 . are obviously zero for the Haar wavelets.
then (3.76) leads to
2 ((2) sin :}' = { ($-) sin i} . Dispersion relation (3.80) is the same as the standard FDTD method (see Chap.5
of [28]). It is interesting to note that the dispersion relation can be improved by
choosing a proper location of boundary conditions. This will be discussed in the
following section.
Chapter 3. 1-D Time-Domain i2/lultiresolution Analysis 38
3.4 Discussions
3.4.1 An Influence of Boundary Conditions to the Compu-
tational Accuracy
Two different locations of the PEC b o u n d a ~ conditions have been investigated: one is
locat~d at a qiiarter ce11 size away from the center of the b a i s functions. as described
in the previous section, and the other is located at the center of the b a i s function.
The PEC boundaries were implemented with the linear interpolation approximation.
Resonant frequencies of one-dimensional strings are tested for each type of the PEC
b o u n d a ~ condition, and the results are compared. Fig. 3.6 schematically shows those
boundary locations.
For the PEC boundary located at a quarter ce11 size away from the center of the
basis function, the scaling and the wavelet coefficients at the boundary are deterrnined
by (3.55). Rewriting the equations for the case of uniform grids, we obtain
E? = ~ ( E ? + E : ) for location (a), A 4 4 shifted . (3.81)
E: = -L(EL+ 4 E?)
However, for the PEC boundary located at
Fig. 3.6 (b) a t r = O, we obtain
It is obvious that when the boundary is set
the center of the bais function. from
Location (b), at center . (3.82)
at the center of the b a i s function, the
scaling function coefficient E t is enforced to be zero, while in the case where the
boundary is located a t quarter ce11 size away from the boundary, the scaling and the
wavelet function coefficients have mutual interaction at the boundary.
Then the resonance of the one-dimensional string is tested with two locations of
the PEC boundary with the linear interpolation approximation. The length of the
string is chosen to be the half wavelength so that the resonant frequency is normalized
to un i t . The results are summarized in Table 3.1. When the discretization is coarse.
the PEC boundary of location (a), which is located a t a quarter ce11 size away from
Chap ter 3. 1-D Tirne-Domain Mul tiresol u tion Analysis 39
t t o t a l E-field
(a) PEC located at a quarter ce11 size away from the center
of the basis function.
(b) PEC located at the center of the basis function.
Figure 3.6: Two possible locations for the PEC boundary condition. Black dots
represent field sampling points.
Table 3.1: Resonant frequencies of 1-D strings with the two locations of PEC con-
ditions. (a): PEC located at a quarter ce11 size away from the center of the basis
function, (b): PEC located at the center of the basis function.
nurnber of Yee cells TD-Haar-bIRA FDTD
location (a) PEC location (b) PEC
the center of the basis function. gives much better results than the other location of
the PEC boundary. The accuracy of the PEC of location (a) with 4.5 cells is the same
as that of FDTD with 16 cells. It should be noted that in the case of the PEC located
at the center of the basis function (b), the resonant frequency is exactiy the same as
the FDTD method. This fact means that if the boundary is chosen at the center of
the basis funct ion, the proposed technique is not advantageous over the convent ional
FDTD method.
Next we discuss the reason why the accuracy is improved in the case of the PEC
of location (a). Figure 3.7 shows the staircase approximation of sinusoidal waveform
by means of both locations of PEC boundary conditions.
The normalized energy stored under the staircase approximation
is calculated and cornpared to the exact normalized energy stored under the sinusoidal
Chap ter 3. I -D Time-Domain .Multiresolu tion Andysis 41
4 . 5 unit c e l l s x=O
(a) PEC located at a quarter ceII size away from the center
of the b a i s function.
4 unit cells 1 x= l
(b) PEC located at the center of the b a i s function.
Figure 3.7: Estimation of energy stored under the staircase approximation and sinu-
soidal curve.
Chapter 3. 1-D Time-Domain Multiresolution Analysis 42
curve R
L' = 1 l ~ ~ 1 ~ s i n ~ r d z . (3.84)
From Fig. 3.7, the PEC of location (a) gives -1.9 x 10-~ error compared to the exact
value */21Eo)*, while the PEC of location (b) gives 4.3 x error. The PEC of
location (a) has 0.5 more ce11 than the PEC of location (b), but it is not the only
reason why the PEC of location (a) is more accurate. The staircase approximation
for the PEC of location (a) gives a better estimate of a sinusoidai curve than the
staircase for the PEC of location (b).
3.5 Conclusions
The time-domain Haar-wavelet-based multiresolution analysis (TD-Haar-MRA) tech-
nique for the one-dimensional case was derived. The dispersion relation obtained
analytically indicates t hat the proposed technique has the same dispersion as the
conventional FDTD method. The perfect electric conductor boundary condition was
first formulated with linear interpolation approximation, and then improved with
higher-order interpolation approximation. It was demonstrated that. although the
dispersion relation is the same as those of the FDTD method, the accuracy can be
improved by choosing the location of boundary at one quarter ce11 size away from the
center of the b u i s function.
Chapter 4
2-D Time-Domain Multiresolution
Analysis
4.1 Introduction
In this chapter, the time-domain Haar-wavelet-based multiresolution analysis (TD-
Haar-&IRA) technique for two-dimensional transverse-electric wave propagation is
formulated and demonstrated. It is formulated using a complete set of 2-D basis
functions of Haar wavelets, and applied to various waveguide structures.
In contrast to the conventional space-discrete time-domain analysis methods such
as the finite-difference time-domain (FDTD) method and the transmission line ma-
trix (TLM) method. special measures must be taken to realize boundary conditions
in the TD-MR4 technique. This effort, however, is compensated and rewarded by the
improved efficiency of the computational process. For the Haar-wavelet-based tech-
nique, although the memory requirement is the same as in the conventional FDTD
method, the computational time (CPU time) is reduced by approximately one-half.
In order to apply this technique to waveguide structures, various features are im-
plemented such as perfect electric or magnetic conductors (PEC or PMC), impedance
boundary conditions for TEM waves, Berenger's perfectly matched layer (PML) ab-
sorbing boundary conditions (ABC) (11, 90-degree and zero-degree corner nodes and
their singular field correction by a quasi-static field approximation (201 as well as S-
Chapter 4. 2-D Time-Domain Multiresolution Analysis 44
parameter extraction. Moreover, the numerical dispersion relation is compared with
numerical experirnents and with analytical formulas. The dispersion relation provides
a clear insight into the accuracy of the numerical technique.
The TD-Haar-ME4 technique is first applied to a simple rectangular waveguide
to demonstrate the behavior of the Haar wavelet basis coefficients. It is then ex-
tended to the analysis of waveguides with thin and thick inductive irises. The sin-
gular field around the edges and corners of the irises are corrected by a quasi-static
approximation of the fields. The effectiveness of the s inylar field correction in the
TD-Haar->IR;\ technique is also demonstrated.
4.2 Formulation and Implementation for TE Case
4.2.1 2-D Basis Functions and Time Iterative Difference Equa-
t ions
Consider Mauwell's curl equations for the TE polarization case
where the wave is propagating on the xr-plane and the electric field is in y-direction.
Yote that it could also be referred to as TM polarization when the waveguide is
uniform in y-direction and the wave is propagating along the same direction.
In the Cartesian coordinate systern, the field components E,, Hz, Hz are expanded
into four ort honormal basis functioos, which are the products of two-dimensional
combinations of the Haar scaling function # and wavelet function 11, [6] multiplied by
Cha~ter 4. 2-D Time-Domain Multiresolution Andysis 45
a pulse function in time h(t) as
w here
with ( W . m) = (x. 1 ) . ( 2 . k), and
1 for ( n - 1/2)At 5 t < ( n + 1/2)Lt
O elsewhere
with the space and time discretization intervals AI, At and At. The functions 4 and
iu are defined as
1 for I s ( < 1 / 2
1/2 for Isl = i / 2
O elsewhere
112 for s = -112
1 for - 1 / 2 < s < O
-1 for O < s < 1 / 2
-112 for s = 112
O for s = O and elsewhere
The basis functions (4.4) have the support (or the width of the function having
nonzero value) equal to the spatial discretization intervals l x and A;. The field
components are defined on Yee grids as çhown in Fig. 4.1, and the spatial basis
hinctions for an E, component are shown as an example in Fig. 4.2. The expansions
of the electric field E, and the source current density Jv are given by
Cbap t er 4. 2-D Time-Domain Ilf ul tiresolu tion Andysis 46
Figure 4.1: Two-dimensional Yee grid for the TE case. Thick solid lines represent the
Yee grids, thin solid lines represent subcell boundaries. A "subcell" is an elementary
rectangular region that surrounds a field node on the Yee grid. The hatched region is
an erample of the subcell. The wavelet expansion coefficients (E" HP, H f m etc.)
are defined at the field nodes (a : Ey, t: Hz, +: Hz) on the standard Yee grids.
Cha D t er 4. 2-D Time-Domain Multiresolu tion Ailnalysis 47
Figure 4.2: Two-dimensional Haar basis functions for an E, node. Hatched regions
represent +l and unhatched regions represent - 1.
Cha~ter 4. ID Tirne-Domain hdultiresohtion Analysis 18
with F = E, J, and the expansion of the magnetic fields Hz and Hz are given by
and
x <€ where the notations RItk with F = El J, H and C,c = 4. & denotes the expansion
coefficients in terms of the Haar scaling and wavelet functions at time step n and
position (i, k).
Then. the field components (4.10)-(4.12) are substituted into hIauwell's curl equa-
tions (4.1)-(4.3). and subsequently, tested with the basis functions (4.4). In order to
obtain the tirne-iterative update equation for n+1,3 H C ~ , ~ , ~ components, (4.1) is tested
with the basis function #t$h at position (i + 112, k) and time n as
Chap ter 4. 2-0 Time-Domain hl ultiresolution Analysis 49
where (f (2) lg(x)) denot es the inner product of the funct ions f (x) and g (x) given by OQ
(f (1) (g(x)) = J f ( x ) . g(x)dx. The left hand side inner product is then given by -OC
+ 5 GV, *4i+l/2(~) (2 )dk ( z ) hn] (t) hn (t) dx
and the right hand side inner product
where the following properties of the orthonormal basis functions were used:
Chapter 4. 2-0 Time-Domain Multiresolution And-ysis 50
Thus, (4.13) is given by the finite difference approximation as
It is efficient to define nodal voltages and nodal equivalent currents as follows, rather
than to use electric and magnetic fields as unknown variables because some division
operations can be eliminated in the updating equations:
In addition, source current is defined here as
Finally. the time iterative update equation is obtained as
=fV@ The equations for the other expansion coefficients n+ll,'121,2,k, n+L/l ,+ llÎ,k and n+,l~~~~,,,k can be obtained similarly and are found to have a structure identical to
(4.28).
Chapter 4. 2-D Time-Domain Multiresolution Anaiysis 21
With an analogous derivation, the update equations for other field cornponents
E, and Hz can be obtained similarly. Using the simplified notation
and so on. with <, C = 4, $, then the entire set of the update equations is given by
where the coefficients are defined by
The equations (4.36)-(4.38) are computed for each basis function, CC = p#, Q$, y4 or $@. The material constants pis? € i t and Oik are defined as common values in a
unit Yee ce11 at position i. k. Therefore, these update equations can be applied only
to homogeneous material regions. The formulation for the inhomogeneous materials
is discussed in Section 5.5.
4.2.2 Sampling of the Fields
The numerical calculation is conducted using only wavelet expansion coefficients, not
using the total field values. Therefore, when we need to sample the total fields,
we must reconstruct the total fields from the expansion coefficients that are obtained
through the calculation process. The relation between the total fields and the wavelet
expansion coefficients is well represented bv a matrix that leads to the so called Fast
CVavelet Transform [2], which is an efficient transformation between a vector (or
a discrete function) and its wavelet expansion coefficients. For reconstructing the
total field values in the proposed technique, we introduce four rectangular-pulse basis
functions, which are another set of b a i s functions. The basis transformation matrix
between the rectangular-pulse basis and the Haar-wavelet basis is equivalent to the
matrix incorporated in the Fast Wavelet Transform, and allows us to sample the total
fields efficient ly.
The four rectangular-pulse basis func tions are writ ten by
where
with (W. m) = (x. i), (y, j ) , (2, k). The functions 1 (s) and u ( s ) are defined by
JZ for - 1/2 < s 5 O
O elsew here
and JZ for O < s 5 112
O elsewhere
Chapter 4. 2-D Time-Domain kIultiresolution .halysis 53
In (4.46) and (4.47), the factor fi accounts for the orthonormal property of the func-
tions. These functions yield another 2-D orthonormal basis representing individual
subcells as shown in Fig. 4.3 for the case of an Ey node. To ensure the orthonormal
Figure 4.3: Two-dimensional rectangular pulse basis functions for an Ey node.
Hatched regions represent a magnitude of +2 and unhatched regions represent zero
magnitude.
property of the 2-D rectangular-pulse buis functions, the magnitudes of the functions
are chosen to be 2. Then the rectangular-pulse basis coefficients pz with 0, p = 1, u
can be related to the actual field values $z at sampling points in subcells op as
The location of the sampling points are depicted in Fig. 4.4.
Chapter 4. 2-0 Time-Domain hIultiresolution Anaiysis 54
Figure 4.4: The sampling points in the two-dimensional Yee grid for the TE case.
Thick solid lines represent the Yee grids, thin solid lines represent subcell boundaries.
The field sampling points are represented by crosses ( x ) and located at the center of
the subcells. The fields are sampled at the center of the subcells, while the wavelet
expansion coefficients ( E p , Hf4, H:@etc.) are defined at the field nodes (e : E,,
t: Hz. +: Hz) on the standard Yee grids. At each sampling point we sample three
field components E,, Hz and Hz.
Chapter 4. 2-D Tirne-Domain ~Multiresol ution Anaiysis 55
The rectangular-pulse basis coefficients with O, p = 1. u can be calculated
from the 2-D Haar b a i s coefficients pf, with c, C = 4, S, as
w here
1 +l -1 -1 +i J
which is a b a i s transformation rnatr~x between the 2-D Haar ba i s functions and the
rect angular-pulse basis funct ions.
Matrix A2D is orthogonal, i.e., A ~ ~ A ~ ~ = A ~ ~ A ~ ~ = 1. where denotes
the transposed matrix, and I the identity matrix. In addition. it is symnietric:
A:, = Therefore.
A;; = AzD . (4.51)
which allows a simple conversion between the expansion coefficients of the rectangular-
pulse basis functions and the Haar basis functions as
4.2.3 Perfect Electric Conductor (PEC) Boundary Condition
The perfect electric conductor (PEC) boundary conditions for a two-dimensional
uniform grid will be described. A simple linear interpolation technique is incorporated
here. The PEC boundary conditions incorporating higher-order interpolation in a
nonuniform grid will be described in Section 5.2.3.
Cha~te r 4. 2-0 Time-Domain Multiresolution halvsis 56
The spatial relation between Ey nodes and PEC boundaries is shown in Fig. 4.5.
To implement a PEC condition that is perpendicular to the x-axis, the 2-D Haar
Figure 4.5: PEC boundaries in two-dimensional space. Thick solid lines (-) represent
Yee grids, dotted lines ( - . + .) the position of PEC boundaries. and small circles (O)
the sampling points.
basis functions are divided into two pairs in such a way that the functions having the
same variation in z-directions form a pair as
The PEC boundary is located at a position that is shifted by 1x14 from the centers
of the basis functions as shown in Fig. 4.6.
Since the tangential electric field a t the boundary is zero, the first condition is
Chapter 4. 2-0 Sime-Domain Multiresolu tion Analysis 57
PEC PEC
Figure 4.6: Schematic diagram of electric fields near PEC boundaries. Long dashed
lines (- - -) and short dashed lines (- - -) show Haar scaling (4) and wavelet (-U)
functions, respectively. Black dots (O) show the Ey nodes on YeeTs grid. Circles (O)
show sampling points.
The second condition is that the electric field E,Y at x = xi is obtained by taking the
average between Ea(= 0) and E:, that is
In 2-D space, using the pair functions (4.53), those conditions are represented by
with k = 0, , K. Solving these equations in terms of the Haar basis coefficients E::
with 5T C = 4, $, then one can obtain the PEC conditions at x = O as
Y E ~ = i (YE~: + YEU,) (4.60)
YEO,~ = ok (4.61)
1 Y E ~ = - ( Y E ~ + Y E ? ~ )
4 (4.62)
y ~ , k U = - y ~ g ' (4.63)
(4.64)
Chapter 4. 2-0 Time-Domain Md tiresolution Andvsis 58
with k = 0, . . , K. Similarly, PEC conditions at x = x,, are given by
with k = 0, . , K.
PEC walls perpendicular to the --ais c m be obtained similarly, and the pair
4.2.4 Perfect Magnetic Conductor (PMC) Boundary Condi-
t ion
Perfect magnetic conductor (PMC) walls can be implemented in a way similar to the
definition of PEC walls. The only difference between the PMC and the PEC walls is
that the PMC boundary conditions are applied to the magnetic field. while the PEC
boundary conditions are applied to the electric field nodes. Therefore. the location
of the PEC and the PhIC boundaries are half a ce11 size apart as shown in Fig. 4.7.
The interpolation technique for obtaining a smooth field distribution in front of the
boundary is the same as for the PEC boundary condition.
4.2.5 Real Impedance Boundary Conditions for Transverse
Electromagnetic (TEM) Waves
Arbitrary real surface impedance conditions have been implemented for transverse
electromagnetic (SEM) waves. It can be achieved in various ways: one way is to
implement the boundary at the sampling points as in the case of PEC and PblC
conditions; another way is to implement it at the electric field nodes of Yee's grid.
The former incorporates interpolation techniques to obtain the field values in front of
Figure 4.7: The location of PEC and PMC boundaries in two-dimensional space.
Chapter 4. 2-D Time-Domain Multiresolution .4nalysis 60
the boundary, the latter requires "an opposite reflection coefficient''. which present
an interesting property associated with the wavelet functions.
It should be mentioned, however, that the accuracy in the latter case is exactly
the same as that of the conventional FDTD method, because the system of equations
associated with each basis function never couples; since the basis coefficients are
independent of each other. the numerical dispersion relation is not improved over the
FDTD formulation. Therefore. in this subsection, the inipedançe boundary conditiori
is implernented in terms of the opposite reflection property of a wave associated with
the Haar wavelet function.
Consider a TEM wave propagating in the +z-direction and an impedance bound-
ary located at E-nodes of Yee's grid at 2 = zma, as shown in Fig. 4.8. The field corn-
irnpedance boundary
1 TEM wave x Ey propagation
Figure 4.8: Location of the impedance boundary implemented in this study.
ponents propagating in +z-direction satisfy the impedance condition at the boundav
where Zb is the real impedance of the boundary. Since the magnetic field node
(-z$k+l) is not located on the boundary at the same time as the electric field node
Chapter 4. 2-0 Time-Domain ibfultiresolution AnaJysis 61
($?3k+l), it can be obtained by space extrapolation and time averaging of the magnetic
fields in front of the boundary as
Then the discrete form of the Faraday-hlauwell equation
leads to the time iterative update equation for the magnetic field
or by using simple V. I notation.
at r = where the coefficients are
For an impedance boundary at z = O and a wave propagating in -z-direction. the
update equation is similarly obtained as
with the same coefficients as given by (4.75) and (4.76).
Equations (4.74) and (4.77) can be applied to a wave associated with the Haar
scaling function. However, care must be taken when they are applied to a wave asso-
ciated with the Haar wavelet function, which has an asymmetric spatial distribution.
Chap ter 4. 2-0 Time-Domain M u l tiresolu tion ,.lnaiysis 62
-1 simple consideration leads to the following concept, which can be referred to as
"an opposite reflection coefficient". In the case of a PEC wall, the reflection coef-
ficient is l' = -1, which is applicable to the wave associated with the Haar scaling
function. However, for the wave associated with the Haar wavelet function, because
of the asymmetric spatial variation of the wavelet function, the reflection coefficient
for the wave must be the negative of r. that is, r" = -r = 1. When a waveguide
of characteristic impedance Zc is terminated with the real irnpedance Zb, the reflec-
tion coefficient is r = H. Then. the impedance for the wave associated with the
wavelet function 2: is given by
which is the inverse of the impedance of Zb when the space iç filled with air. For the
waveiet b a i s coefficient, this impedance is substituted in (4.74) and (4.77) for the
normal impedance Zb. For the two-dimensional case when the impedance boundary is
perpendicular to the :-direction, the normal reflection is applied to the b a i s functions
and a@, and the inverse reflection is applied to the basis functions qûw and UV.
For a boundary perpendicular to the x-auis, the impedance boundary equations
are, at x = x2,,,,
and at x = 0,
where the coefficients are given by
In this case, the normal reflection is applied to the basis functions Qd and #$, and
the opposite reflection is applied to the basis functions @# and $$.
To demonstrate the behavior of the impedance boundary, a simple TEM trans-
mission line of length 100 mm was analyzed with an impedance boundary having
different reflection coefficients. Figure 4.9 shows the time series data of the TEM
wave measured at a point 76 mm away from the impedance boundary. The incident
signals were al1 identical, and the signals reflected by the impedance boundaq with
several different impedance values were observed and plotted in the figure. The other
end at z = O mm of the transmission line was terminated with a matched irnpedance
boundary Zb = 1 (r = O). The Yee grid size was 2 mm in the direction of the
TEM transmission line. The normalized impedances of the boundary were set to
Figure 4.9: Reflection at the impedance boundary condition of a TEM transmission
line.
5 = 5 x iod6 (î = -1), Zb = 113 (r = -0.5), Zb = 1 (I' = O), Zb = 3 (î = 0.5) and
&, = 2 x 105 (î = 1). The errors of the reflection coefficients were in the range of
0.06% to 0.3%. Figure 4.10 shows the space distribution of the E, field reflected by the
impedance b o u n d q placed a t z = 100 mm. Those plots demonstrate smooth field
distribution of the reflected pulse for the various reflection coefficients; no distortion
C h a ~ ter 4. 2-D Tirne-Domain Multiresolu tion hdysis 64
or spurious mode was observed.
4.2.6 Perfectly Matched Layer Absorbing Boundary Condi-
t ion
Berenger's Perfectly Matched Layer Absorbing Boundary Conditions (PML-ABC) [l]
are implemented in the 2-D algorithm. In general? the ABC can be implemented in a
manner similar to the traditional FDTD method. The only difference is that the ABC
is implemented for each basis function (44. @lil, $4 and qo), and the wave associated
with each basis function is absorbed independently by the corresponding ABC.
Since the formulation of PbIL for the two-dimensional TE waveguide niode, which
is referred to as TM-to-y polarization in [l] in the context of two-dimensional free-
space scattering problems, is not described in detail in the literature, it will be de-
scribed below with the notation used in the previous sections.
The E, component is split into an x and a 2 component as E, = Eyz + EYI; then
(4.1)-(4.3) is rewritten as
where O: and (T; are magnetic losses that are chosen to satisfy the matching condition
in the PML medium given as 0 O* - - _ . é P
It has been pointed out that the decay of electromagnetic waves in PML media
is so rapid that the usual update formulation fails to follow the variation of the field,
and that another update scheme called exponential time stepping algorithm [28] is
required. Regarding (4.83)-(4.86) as first order ordinary differential equations in
terms of time t with a constant excitation source given by the rotation of the fields,
Chap ter 4. 2-D Time-Dom~n Mul tiresolu tion AnaJysis 65
(a) I' = 1 (b) 1: = 0.5
(d) I: = -0.5 (e ) I' = -1
Figure 4.10: Spatial field distribution of a wave reflected by an impedance boundary.
Chapter 4. 2-D Time-Domain Illuitiresolution ,-lnalysis
then ive obtain the solutions for the differentid equations [28], for Hz as an example,
The first term on the right hand side of (4.88) is the homogeneous solution and the
second term is the particular solution.
By using a simple notation in terms of voltages and currents (-1.29)-(4.34). and
considering that the equations are applied to each basis function. the updating equa-
tions in the PML medium are given by
with <, C = 4, ID, where the coefficients are
-G+ 1 /* .c~:~/, = ex, [-y At]
Chapter 4. 2-0 Time-Domain hfultiresolution Anaiysis 67
The loss factor can assume a linear or higher-order variation along the depth of
the PML region. In this study, it was chosen to Vary quadratically along the direction
into the PML region p as 2
(4.101)
where b is the depth of the P-IIL region and om, is chosen to bound the apparent
reflection coefficient, which is defined as the reflection coefficient â t the interface
between the regular and the P M L regions as
where c is the speed of light. The apparent reflection coefficient R is typically IO-''
to In the analysis. R was chosen to be 1 0 - ~ . The loss factor at a nodal point
is determined by taking the average over the ce11 as follows:
The magnetic loss factor is calculated using (4.87).
For waveguide analysis, the absorbing wall is placed between two PEC side walls.
When the absorbing wall is perpendicular to the ,--ais. loss factors dong the x-
direction are chosen to be O, = a; = O; then the coefficients (4.93). (4.94, (-4.99) and
(4.100) reduce to
y = q ' = At -- €
In addition, the side walls in the PML medium
by (4.60)-(4.68). The back wall a t opposite to
must satis& the PEC condition given
the interface is also a PEC wall.
Chap t er 4. 2-D Time-Domain Md tiresolu tion Andysjs 68
4.2.7 Conductor Edge and Corner Node Implementation with
Field Singularity Correction
In order to mode1 corners and edges of a conductor, special nodes are required to
enforce the field near the corners and the edges to have a smooth field distribution
around the discontinuity. In this subsection, the implernentation of special nodes
for 90 dpgree corners and zero degrw edges of perfect 4ectric condiict,ors will he
d iscussed . Furthermore, the singular fields around the corners and the edges are modeled
using a quasi-static approximation of the field distribution proposed for the FDTD
method by Mur (201 and for the TLM method by Cascio et al. [3]. It has been pointed
out that the fields in the vicinity of sharp metal edges do not change very much in time
compared to the variation of the fields in space. Therefore. the static approximation
using a-pnori knowledge of the field distribution around the edges provides a more
accurate representation t han the scaling and wavelet basis functions.
Field Singularity Correction
Following Mur's discussion of edge singularities in [20], the field distribution near
the edge is represented by means of a local polar coordinate system as shown in
Fig. 4.11 for the case of E-polarization. Maxwell's equation for E, and Hv in the
polar coordinates is given by
In the vicinity of the edge, Ey and H, can be approximated as
where the singularity factor un is given by
Cbapter 4. 2-D Tirne-Domain Multiresolution .4ndysis 69
Figure 4.11: Location of a sharp metal edge and a local polar coordinate system
Substituting the first term of (4.110) in (-1.109). one can obtain
where 1 1 = At or Ar. Using the central difference approximation for the time
derivative, we obtain the update equation with corner correction
If the correction factor u121-v1 is smaller than 1.0, the scheme maintains its stability.
This equation will be applied to correct the field singularity at the 90 degree corners
and zero degree edges discussed below .
90 Degree PEC Corner Node and Field Singularity Correction
For the implementation of a 90 degree corner node, E, fields at sampling points
that belong to the corner node are interpolated using the peripheral field values at
Chap ter 4. 2-D Tirne-Domain Mul t iresolu t ion .4nalysis 70
sampling points, as shown in Fig. 4.12, for the case of a corner node implernented at
the "lu" sarnpling point. The Ey fields around the corner node (black dots with a
,u i .......... 1
Xi-
Figure 4.12: Implementation of a corner node. The corner node is located at the "lu"
sampling point of node ( 2 , k). Thin solid lines show the equivalent Yee grid. The
hatched region shows a perfect electric conductor.
circle) are interpolated by using peripheral fields (double circles) as reference values.
By using second order Lagrange interpolation and considering that f;E& = 0, the
E, fields at the sampling points "Il" ,"ul" and "uu" are respectively given by
where the Lagrange polynomial coefficients are given by
These coefficients are the same in :-direction for uniform grid structures. Then.
Haar-basis coefficients at node (2, k) are obtained by using the linear transformation
(4.52).
For field singularity correction a t a 90 degree corner, ul = 7r/(2r - 2 ) = 213. The
correction factor is in this case
which is smaller than 1.0. Therefore. in the case of the FDTD method, the scheme
is stable. From (4.1 l-l), irnplementation of the correction for the 1 IL-side corner (see
Fig. 4.13), for example, is given by
Yee I b
Figure 4.13: A 90 degree corner located at lu sarnpling point.
or using C', I formulation,
Thin PEC, Zero Degree Edge and Field Singularity Correction
Implementation of a t hin perfect elect ric conductor imposes the PEC condition a t
each side of the conductor and edge nodes as illustrated in Fig. 4.14 for the one-
dimensional case. The PEC condition on one side can be that described already in
subsection 4.2.3. However. the condition on the other side is slightly different. Also.
the conductor edge node is implemented in such a way that the field has a smooth
variation near the edge.
The PEC condition on the other side is schematically shown in Fig. 4.15. This
PEC condition is achieved by interpolating the field value at r i with respect to the
reference field values sampled at 1- ,, z,", zC+, and so on. The simplest condition
is the linear interpolation between the neighbor field values sampled a t 2:-, and zk.
The two neighbor sampling points at z = t i and sf belong to the same Ev field
node at z = z t . In that sense, the PEC boundary is realized in a "self-consistent"
manner; in other words, only the basis functions assigned to the same field node are
involved in the boundary condition, thus it could be referred to as "a self-consistent
PEC condition". In this condition, as shown in Fig. 4.15, Et is first obtained by
evaluating the updated coefficients Ez and E: at z = zk as
and then EL is enforced as a function of EE by linear interpolation
Chapter 4. 2-D Time-Domain h/ldtiresolution .4nd.ysis
A
edge
I t i L
i l
i
thin PEC
edge
Figure 4.14: Implementation of a thin conductor and its edges. Ey field samples
represented by large black dots are obtained by interpolation from the peripheral
known E, fields represented by open circles. The solid lines represent the Yee grid
lines. The expansion coefficients for the E, components are located a t the corner of
the Yee grids.
Ch a D t er Li. 2-0 Tirne-Domain Mu1 tiresolu tion ..lndvsis
PEC
I ?
Figure 4.15: Implementation of a thin conductor for the one dimensional case. An
E, field sample represented by a large black dot is obtained by interpolation from the
next neighbor Ey field samples represented by open circles.
Then from the definition of the basis function coefficients, the self-consistent PEC
conditions are given by
In (4.128) and (4. U g ) , E i has been computed and stored in advance before calculating
(4.128) and (4.129). For the two-dimensional case, (4, w ) is replaced by (&D, @,b) or
( W Q , wb), and the same procedure can be applied for each pair of functions.
The self-consistent PEC condition can be improved by a third-order polynornial
interpolation. In this case, the first conditions, which yield knoivn Field values at
reference sampling points, are given by 1
After these values are cornputed and stored, the second condition fixes the interpo-
Chapter 4. 2-D Time-Domain $1 ultiresolu tion .4nalysis 75
lated field as
where the third-order Lagrange polynomial coefficients LI and L2 are given by
The implementation and the correction of zero degree edges can be achieved with
a procedure similar to the one described in the previous section for the 90 degree
corners. It must be modified such that the fields at five sampling points around an
edge are now interpolated and corrected. For the edge node at the "lu" sampling
point with a thin PEC perpendicular to the i-axis, referring to Figs.4.13 and 4.14.
additional fields at sampling points "ul" and "11" belonging to the node (i. k + 1) are
interpolated as Y lu Y lu 9% = h + % i f L2&'02 (4.133)
and
where the Lagrange polynomial coefficients are the same as those for the 90 degree
corner node given by (4.118). The above equations are calculated together with
(4.115)-(4.117), and the resulting field distribution is converted to Haar basis coeffi-
cients by the linear transformation (4.52).
For the field singularity correction at a zero degree edge, VI = 1/2; then. the
correction factor is 1 1
y21-v' = -2z 0.70710678 2
(4.137)
Chapter 4. 2-0 Tirne-Domain Multiresolution Andysis 76
which is also smaller than 1.0, and one can expect a stable scheme. The field around
the edge is corrected by
or. using I'I formulation,
together with (4.124) and (4.125) with the correction factor given by (4.137).
4.3 Validation
4.3.1 Numerical Dispersion Relation
Numerical techniques that involve discrete finite difference grids inherently have a
numerical dispersion error. The error is caused by the finite dimension of the grid
and cannot be ignored when the wavelength is comparable to the grid dimensions. .in
analytical dispersion relation is derived by assuming a time-harrnonic trial solution in
the finite difference form of Maxwell's equations [28](Chap.5). The dispersion error
is a function of the number of cells per wavelength, the angle of propagation and the
Courant stability factor that is defined by
where c is the speed of light, At the time step and 31 the spatial grid interval.
Analytical dispersion formula
The analytical dispersion relation for the conventional FDTD algonthm in the case
of two space dimensions is given by equation (5.5) on p.95 of (281 as
whereas the dispersion relation for the Haar wavelet based TD-MRA technique is
given in [9] as
These analytical dispersion relations are shown in Fig. 4.16 for two stability factors.
s = 0.9/& and O.+/&. and two directions of propagation. <r = 45" and 90" in the
case of square grids AI = A2 = Ai . The dispersion properties of the Haar-wavelet
based multiresolution technique are in general superior to those of the conventional
FDTD method. Since the dispersion relation for a plane wave is given by J = ck. the
ideal dispersion relation for certain values of At and 91 is given by
where R(= uAt) and y(= khl) denote the normalized angular frequency and the
normalized wavenumber, respectively; s is the stability factor (4.140).
Numerical experiments
We have tested the numerical dispersion relation experinientally by analyzing the
TEll mode resonance in
technique. The resonant
a and length b shown in
a rectangular cavity with the 2-D time-domain Haar-XIRA
Frequency of the TEli mode in a rectangular cavity of width
Fig. 4.17 is given by
and the wavenumber of the resonating field in the waveguide cavity is given by
The TEll mode field can be considered as the superposition of plane waves propa-
gating at the angle of 6
6 = arctan - a
(4.146)
with respect to the main coordinate ais.
Chap ter 4. 2-0 The-Domain iLlultiresolu tion Analysis 78
O 0.5 1 1.5 2 2.5 3 3.5 Normalized wavenumber
(a) 2-D FDTD
2.5 . 5 r r 1 r ' 45 deb.
, 90 deg.
O 0.5 1 1.5 2 2.5 3 3.5 Nomalized wavenumber
(b) 2-D TD-Haar-MM
Figure 4.16: Analytical dispersion relations for the 2-D FDTD and the 2-D TD-
Haar-MM. Normalized angular frequency R = wAt is drawn as a function of the
normalized wavenumber x = k ~ l .
Figure 4.17; -4 top view of the rectangular cavity resonator for the numerical exper-
iments. cro denotes the angle of propagation. and k the propagation constant of the
plane wave. Thin lines represent Yee's grid lines, PEC walls are located at a quarter
ce11 size away frorn the grid lines.
Therefore, by testing resonances for various aspect ratios of rectangular cavities.
one can experimentally obtain the numerical dispersion relation as a function of the
angle of propagation. The testing procedures are as follows: (i) fk propagation
constant k by means of (4.145), the angle of propagation O by means of (4.146), then
(ii) perform numerical analysis and obtain the resonant frequency f,. and finally, (iii)
plot y (= k ~ l ) versus R (= 27r f,At). The angles of propagation .LSo(a = 1, b = l) ,
63'(a = 1, b = 4, 78" (a = 1, b = 3) and 84" (a = 1, b = 10) were investigated and
compared to the analytical solution of the dispersion equations in Fig. 4.18. It is
shown in these figures that, in the limit of infinitesimal grids, the experimentally
ob t ained dispersion curves agree with the analytical dispersion relation of the FDTD
method, while for coarser grids, the TD-Haar-MRA technique is less dispersive than
the conventional FDTD. It should be mentioned here that the analytical dispersion
relation of (4.142) shown in Fig. 4.16 (b) disagrees with the experimentally obtained
dispersion relations shown in Fig. 4.18. The dispersion of TD-Haar-MRTD should
obviously follow that of FDTD in the b i t of infinitesimal grids (in the limit of the
Chapter 4. 2-0 Time-Domain hldtiresoIution Analysis 80
(FDTD) .
0.90 deg.
'. (FDTD) . .45 deg.
D
27,63 deg. . .
-. 12.78 deg. ' * 6,84 deg. .
4 cells/wavelength
O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Normalized wavenumber
(a) s = 0 . 9 / 4
-.- . b
.n -.- , w - - . -..
-- 12. 78 deg. (FDTD) '..(FDT$ 0.90 deg. '45 deg. 1
O 0.2 0.4 0.6 0.8 1 t.2 1.4 1.6 1.8 2 Normalized wavenumber
(b) s = 0.4/&
Figure 4.18: Experiment ally ob tained numerical dispersion relations for the 2-D time-
domain Haar-MRA technique. Normalized frequency enor (R - 1 s ) / (XS) is plot ted
as a function of normalized wavenumber x = LAI with stability factor of s = 0.9/&
and s = 0.4/&
Chapter 4. 2-0 Time-Domain i\.ldtiresolution -4nalysis 81
normalized wavenumber tending to zero). As the grids become coarser, the influence
of the PEC boundary conditions on the experirnent results becomes apparent. For
coarse grids? the dispersion relation changes abruptly because the opposite sides of
the PEC boundaries approach and mutually interact.
4.4 Waveguide Analysis wit h 2-D Time-Domain
Mult iresolution Analysis Technique
In this section. the analysis of rectangular waveguide structures with the 2D TD-
Haar-&IR-\ technique will be demonstrated. Simple WR-28 waveguide and sections
perturbed with several kinds of discontinuities such as thin and thick inductive irises
are modeled. The singular fields around the edges or corners of the discontinuities
are corrected with the quasi-static approximation technique.
4.4.1 Analysis of a Simple Rectangular Waveguide
.A simple rectangular waveguide is first analyzed to determine each field coefficient
associated with the wavelet basis functions &, &p. w @ , and ru@. The waveguide is
terminated with Berenger's PhIL absorbing boundaries. The total Ey field is recon-
structed from the Haar basis coefficients Em\ Ew, E*@ and E*? The distributions
of the total field and the Haar basis coefficients are shown in Fig. 4.19. Note that
the major part of the field is represented by E$@, and that the magnitude of EWv
is much smaller than E'? Therefore, most of the E*$ coefficients can be omit-
ted in the analysis without degrading the computational accuracy. This is called a
"thresholding technique", which can reduce the mernory requirement and increase
the computational efficiency in general. However, the thresholding technique has not
been implemented in this study and should be an important future research topic.
Chapter 4. 2-0 Time-Domain iMultiresolution Analysis 82
(a) Ey total field
(c) distribution of the expansion coefficients for
the four wavelet basis functions Eo*, EQW, Eu@ and Eww . Each of the four quadrants covers the
entire waveguide section depicted in (a).
(b) geometry of the waveguide
(d) assignment of the wavelet ex-
pansion coefficients for the plot in
( 4
Figure 4.19: Snapshots of the Ez field distribution for the TElo mode propagating
in the WR-28 waveguide section. The total Ez field (a), the waveguide configuration
and the source location (b), wavelet decomposition of the total field into the 2-D
Haar ba is functions (c), and the assignment of the coefficients of the 2-D Haar basis
functions on the xz-plane (d).
Chapter 4. 2-D Time-Domain ikf d tiresolution Analysis 83
4.4.2 Analysis of Waveguides with Inductive Irises and Sin-
gular Field Correction
Waveguides with thick and thin inductive irises (31 were then analyzed with the sin-
gular field correction. The geometry of these structures is depicted in Fig. 1.20. The
(a) thin iris
(b) thick iris
Figure 4.20: Top view of the inductive irises in WR-28 waveguide. a = 7.112 mm.
d = a / 2 and t = u / 6 .
irises are symmetrical and the aperture d = 4 2 , which is the worst situation in terms
of the accuracy of the numerical analysis; no perturbation technique works in this case
because the disturbance of the field by the iris is maximum. The length of the waveg-
uide is chosen to be 4a to eliminate the influence of the terminations. The waveguides
were terminated with Berenger's PhIL absorbing boundaries with a depth of 8 layers
and an apparent boundary reflection coefficient of W5. The discretizations of the
structures are shown in Fig. 4.21. Square uniform grids are used for the thin iris, while
non-square uniform @ds are used for the thick iris. The PEC walls are al1 shifted by
one quarter of the Yee ce11 size. PMC walls are used in the longitudinal plane of sym-
metry to reduce the computational dornain by half. Three different discretizations
Chapter 4. 2-0 Time-Domain iVfdtiresolution Analysis 84
(a) thin iris
, PMC
a/ 2
l PMC
a i2
' PEC
(b) thick iris
Figure 4.21: Discretization of the inductive irises in WR-28 waveguide.
Chapter 4. 2-0 Time-Domain 12fultiresolution ..lnalysis 85
were applied to each structure, and convergence of the resulting S-parameters was
tested. For the thin iris? we chose As = Az = a/(12n), n = 1,2 ,3 , while for the thick
iris, we used (Ax, 12) = ( ~ 1 1 2 , a/15), (a/21, a/27), (al.18, a /51 ) . The S-parameters
are shown in Fig. 1.22 for the thin iris and in Fig. 4.23 for the thick iris, each with and
wit hout singular field correction. These results show that singular field correction
ensures much faster convergence of the TD-Haar-&IRA results. Even in the case of a
zero degree edge, which has the strongest singular tields, the S-parameters converge
well. For the 90 degree corner, excellent convergence was obtained in al1 cases: the S-
parameters obtained with the singular field correction are almost the same as regular
resul ts.
4.5 Conclusions
The two-dimensional TD-Haar->IR1 technique for TE polarization has been de-
scribed. Various boundary conditions were formulated. and the numerical dispersion
relation was discussed. Although the analytical dispersion relation presented in [9]
shows behavior different frorn that of the conventional FDTD method, the dispersion
relation obtained by numerically analyzing the TEIi mode in rectangular cavities is
close to FDTD dispersion; the numerical dispersion of the TD-Haar-LIRA technique
is better than that of FDTD especially when the grid is coarse. It was also found
that the singular field correction based on the quasi-static field approximation was
also effective in the TD-Haar-MR4 technique. Better convergence of the S-parameter
values was obtained when the singular field was corrected.
Chapter 4. 2-0 Sime-Domain hfultiresolu tion Anaiysis 86
20 25 30 35 40 45 Frequency (GHz)
(a) wit h singular fietd correction
20 25 30 35 40 45 Frequency (GHz)
(b) wit hout singular field correction
Figure 4.22: S-parameters for the thin iris in WR-28 waveguide. - : Az = Az =
~ 1 4 3 , - - - : ix = Az = a l 2 4 : l x = A r = a/12.
-20 - '
-25 - y
-30 -
35 - I
20 25 30 35 40 45 Fmquency (GHz)
(a) with singular field correction
20 25 30 35 40 45 Frequency (GHz)
(b) wit hout singular field correction
Figure 4.23: S-parameters for the thick iris in WR-28 waveguide. - : (Ax =
a , 1% = a l ) - - - : (Ax = a/24, Az = a/27), . : (ilx = a/12. Az = a/ l5) .
Chapter 5
3-D Time-Domain Mult iresolut ion
Analysis
5.1 Introduction
This chapter describes the derivation and the application of a 3-D FDTD-like multires-
olution technique based on Haar wavelets. It is thus forniulated in three-dimensional
space and time using Haar scaling and wavelet functions at one scaling level. A corn-
plete set of orthonormal bases in three-dimensional real-space is first created using
Haar scaling and wavelet functions. The field components in the E-H formulation
of Maxwell's equations are then expanded in the orthonormal bases. Subsequently?
by applying Galerkin's procedure and the method of moments. we obtain FDTD-like
time-iterative difference equations that are individually applied to each basis function.
For structures with inhomogeneous dielectric materials, dielectric propert ies are
treated in an approximate manner where the relative permittivity has an anisotropic
property a t the interfaces of different dielectric rnaterials. An exact treatment for
analyzing inhomogeneous dielectric rnaterials will be discussed in the last section;
t his exact formulation leads to a stable algorit hm.
Perfect electric conductor (PEC) boundaries are first formulated using simple
forward- or backward-difference approximations. The PEC boundaries are then im-
proved by using Lagrange interpolation to analyze higher-order modes in a cavity.
Chapter 5. 3-0 Time-Domain Multiresolution ,And.ysis 89
Mur's first order absorbing boundary condition (ABC) is implemented in this chap-
ter as well. ABCs can be implemented just like in the conventional FDTD method.
In the case of the Haar scaling and wavelet basis functions, a basis transformation
m a t r k iç found to be useful for reconstructing field values from wavelet expansion
coefficients.
Several rectangular cavities with inhornogeneous dielectric loading are analyzed
to validate the proposed technique. The results are then compared with analytical
results (when available) and with data obtained by a conventional FDTD analysis
having the same nurnber of degrees of freedom; the comparison is made under the
condition that the same amount of computer memory is used in both methods. Fur-
thermore, the proposed technique is also applied to analyze microstrip low-pass filters
and spiral inductors with open boundaries to extract their S-parameters and field dis-
tributions. The results are compared to those obtained with the conventional FDTD
analysis. These analyses dernonstrate the suitability of t his new technique for solving
pract ical microwave pro blems.
The required CPU time is discussed and compared with that of the conventional
FDTD method. The rnultiresolution technique has the potential of reducing the
computational effort by thresholding small coefficients [32]; when coefficients of basis
lunctions are smaller than a certain value. they can be omitted without affecting the
computational accuracy. However, thresholding has not been iinplemented in this
thesis. Although the accuracy and niemory requirements of this new procedure are
similar to those of the conventional FDTD method when the number of degrees of
freedom is the same, the multiresolution technique based on Haar wavelets saves the
nurnbers of Roating point operations by half compared to the conventional FDTD
met hod.
5.2 Formulation
5.2.1 3-D Basis Functions and Time Iterative Difference Equa-
t ions
The field components in bIanvell's curl equations
are expanded in the following eight orthonormal basis functions. Those b a i s functions
are products of three-dimensional combinations of the Haar scaling function 4 and
wavelet function tu [6] multiplied by a rectangular pulse function in time h( t ) as
where
and 1 for (n - 1/2)At 5 t < (n + 1/2)At
hn(t) = { O othrrwise 7
with the space and time discretization intervals Ax, Ay, A r and At. Instead of the
definition of # and @ in [6], the following must be used to obtain appropriate inner
Chapter 5. 3-D Time-Domain Multiresolution -Inalysis 91
products when applying Galerkin's procedure
1 for lsl < 112
112 for lsl = 112 ,
O otherwise
1/2 for s = - 1 1 2
1 for - 1 / 2 < s < O
-1 for 0 < s < 1 / 2
-112 for s = 112
O for s = O and otherwise
The basis functions (5.3) have the support (or width over which the function has
nonzero value) equal to the spatial discretization intervals l x , Ay and 12. The
spatial basis functions for an Ez node are shown as an example in Fig. 5.1.
In Cartesian coordinates. the expansions of the electric field and the current den-
sity. Ez and J , for example. are given by
for F = E and J . and the expansion of the magnetic field, Hz for example. is given
Figure 5.1: Three-dimensional Haar basis functions for an Ez node. Hatched regions
represent a magnitude of + 1 and unhatched regions represent - 1.
Cliapter 5. 3-0 Tirne-Domain bIultiresolution Analysis 93
where the notations are consistent with those used in [Id] except that the field value
~Ify,: with F = E. J. H and cl q , = 4, lii denotes the expansion coefficients in terms
of the Haar scaling and wavelet functions a t time step n and position (1 . j. k ) . The
remaining field components can be expanded similarly.
Subsequently. each component is su bstituted in Maxwell's equations (5.1) and
(.5.2), and then. by following Galerkin's procedure and the method of moments. the
resulting expressions are tested with the basis functions (5.3). This leads to time iter-
ative difference equations in terms of the voltage across the E-node 3:;: z ~2: lx, x <T)€ the current flowing at the H-node E 3:;: lx, and the current source a t the
,{TOI€ * -q,,m€ - ( y / C r l € Y p ) } , O Olh O OOh O Oh1 - O Oh0
Y CT)C = q C 7 + u p J h 0 h h hOh hOh
. { y C q € - ? / cq t - O hOL O hOO ( ~ l ! , " h C - % % ' ) } y
and
% - C f 1 OOh
2 Ci)( - - <q€ 'ch00 ~ h $ + 'ch00 . {Jhho & i i ~
- (3#gIi - Y Cqt x-cqt
d h o a ) - J h 0 0 ) 7
YCO v C q € + y Ce x Ci)€ - r C~I€ Oh0 O Oh0 Oh0 M o h h d o h i
- ( - € Y -CM J,,hO) - doho 1 7
2 ~ 0 ~ C V C + LCOOh . Y ci)€ _ y O O ~ o O O ~ { d h ~ h J ~ o h
- (2;:: - x CqC J,,hh) - ?%} 7
where the left hand side subscripts O, h, h and 1 denote n, n + 112, n - 112 and n + 1. respectively, and the right hand side subscript, for example. ( h ~ h ) denotes (i + 112. j. k - 1/2), and so forth. The coefficients are given by
for x, 4 and z cyclic. The material constants p , ~ , f i j k and o i j k are defined as a common
value in a unit Yee cell. The equations (5.1 1)-(5.17) are the same as those appearing
in the traditional FDTD method. The only difference is that in the rnultiresolution
method. the equations are computed independently for each basis function (5.3).
5.2.2 Relation Between the Haar Basis Coefficients and the
Actual Field Values
In this multiresolution technique, the space is discretized into the conventional Yee
cells. However, to relate the expansion coefficients to the actual field vdues, the Yee
ce11 is divided into eight subcells in such a way that the original field node on the Yee
ce11 is surrounded by the eight subcells. We cal1 "subcell" an elementary cubic volume
that surrounds a point (node) at which a discrete field component is defined in 3-D
space. The example of an Ez node is shown in Fig. 5.2. The subcells are named Ill,
llu, Id, and so on, correspondhg to the lower (1) or upper (au) position with respect
Chapter 5. 3 -0 Time-Domain kfultiresolution .4nalysis 95
Figure 5.2: Eight E,-subcells (dashed lines) surround a standard FDTD riode of E,
on the Yee ce11 (solid lines). Circles ( O ) and black dots ( O ) represent the magnetic
and electric field components defined on the Yee cell, respectively. The centers of the
subcells represented by crosses ( x ) are the field sampling points for the multiresolution
Chapter 5. 3-D Time-Domain Multiresolution halysis 96
to the kée field node along the x-, y- and t-axes. The centers of the subcells are field
sampling points of the new multiresolution grid. As one can deduce from Fig. 5.2, each
subcell on the multiresolution g i d comprises three electric and t hree magiiet ic field
components. The nurnber of degrees of freedorn for the multiresolution technique is
eight times that of the traditional FDTD method having the sarne Yee grid size. This
means that for the same number of degrees of freedom, the multiresolution technique
allows a grid twice as coarse as that in the FDTD method.
The following eight rectangular-pulse functions are coasidered to be a set of 3-D
orthogonal basis funct ions t hat represent individual subcells:
where
with 1 (s) and U ( S ) defined by
Jci for - 1 / 2 < s s O
ot herwise
fi for O < s 1 1 / 2
O ot herwise
In (KU), the factor fi ensures the orthonormal property of the hinctions. The 3-D
rectangular-pulse basis functions are also shown in Fig. 5.3 in the case of an Ez node.
Chap ter 5. 3-0 Sime-Domain i\f ul tiresolu tion .-\nalysis 97
Figure 3.3: Three-dimensional rectangular-pulse basis functions for an Ez node.
Hatched regions represent a magnitude of +fi and unhatched regions represent zero
magnitude. Each function represents an individual subcell.
Chapter 5. 3-D Time-Domain ~Uultiresolu tion Analysis 98
So satisfi the orthonormal property of the 3-D rectangular-pulse b a i s functions, the
magnitudes of the functions are chosen to be fi. Then the rectangular-pulse basis x -opcl coefficients ~~~ for o , p , q = 1, u can be related to the actual field values Ji,, at
sampling points in subcells (opq) as
Thus the rectangular-pulse b a i s coefficients 3;: for o. p , q = 1. u can be calcu-
lated from the 3-D Haar basis coefficients 311°C for i, q, < = o. w as
r Il1 &i jk s llu S i j k r lu1 sijk x ull &ijk r luu &ijk
3;; r U U ~ &ijk 1: uuu $i jk
where
which is a basis transformation rnatrix between the 3-D Haar basis functions and the
rectangular-pulse basis functions.
M a t r k A has the orthogonality property A I A = A A ~ = 1 (or A-' = A ~ ) .
where A+ denotes the transpose matrix, and I the identity matr~u. Furthermore. it
is symrnetric: A+ = A. Therefore, it has the important property
which allows a simple conversion between the expansion coefficients of the rectangular-
pulse basis functions and the Haar basis functions as
Ur Ilu n Eij k w lu1 nEijk w ull nEijk
w luu n E i l k
p$ w uul nEijk w uuu
, nEtjk
. for u, =x .y . : .
5.2.3 Perfect Electric Conductor (PEC) Boundary Condi-
t ions
To implement the PEC condition that is perpendicular to the x-auis in three space
dimensions, the three-dimensional Haar b a i s functions are divided into four pairs in
such a way that the functions having the same variation in y- and z-directions form
a pair as follows:
1 [ i Y 7 #j(Y) bk(z)I
[$ i (x ) 4j (Y) l i k ( ~ ) i $i(x) 4 j ( ~ ) @k(z)I
[ $ i (x ) W ~ ( Y ) ~ ~ ( Z ) , rLi(x)~j(Y)bk(z)I
[ & i ( ~ ) uj(y) '&k(z)~ @ i ( x ) '@j (g) ~ k ( z ) ]
Then, the tangential electric fields E, and E, at the boundary are set to zero as
described for the one-dimensional case.
Chap ter 5. 3-D Tirne-Domain içlultiresolu tion dnalysis 100
At x = O (i = O), the 3-D PEC conditions are given by
f o r w = y , r .and q , < = o , G .
For thc other side of the boundary a t x = z~ (i = 11): with thc samc pairs. the
conditions are given by
for w = y , r 'and r),c = Q, w .
The boundary conditions for the other directions can be derived similarly. The equa-
tions (5.31) and (5.32) are coniputed for al1 the pairs of the b u i s functions. The
implementation of the Lagrange interpolation technique is also available in the three-
dimensional case.
To give a clear view of the implementation of the PEC condition. the implemen-
tation of the perfect electric planar conductor is now described. Referring to the
notations in Fig. 5.2, we assume, as shown in Fig. 5.4, that the PEC boundary is
parallel to the - p l a n e and cuts across the center of the four subcells 111. lul. d l and
uul at z = (k - l / l )h i . Then the field values in the upper subcells llu. luu. ulu
and uuu at 2 = ( k + l/-L)Az are determined by interpolation between the fields on
the conductor. which are zero, and the fields in the subcells Ill, h l , u11 and uul at
Chapter 5. 3-0 The-Domain &Iultiresolution ..lnaiysis 101
t = (k + 3/4)Az. Using simple averaging, we have
w 111 nEijk W E ~ ~ U n t j k w lu1 nEi jk
w ull nEijk+l
w ull nEijk w luu nEijk
, for w = x, y .
Hence, the 3-D Haar basis coefficients for Ez and Ey components can be calculated
- _ 1 - 3
using the basis transfomatilin matrix A as (5.29).
O w lu1 nEi jk+ 1
Figure 5.4: Location of a PEC boundary parallel to the xy-plane (thick solid line).
A unit Yee ce11 is marked by thin solid lines.
5.2.4 Absorbing Boundary Conditions
Mur's first order absorbing boundary condition (ABC) [19] has been implemented
in the three-dimensional context. The ABCs employed in the traditional FDTD
met hod can be implemented independently for each coefficient associated wit h the
three-dimensional Haar basis function. The outgoing wave associated with each basis
function is absorbed independently by each corresponding .ABC. Therefore. the im-
plementation of ABCs in the new multiresolution technique is sirnilar to that in the
traditional FDTD method.
5.3 Validation
The accuracy and the cornputational time of the new rnultiresolution technique were
first investigated by analyzing a rectangular cavity with normalized dimensions of
0.5a x 0.5a x 0.2 for a TEllo mode having a norrnalized dominant resonant fre-
quency of 1.0 (The normalized speed of light was assumed to be unity.). The number
of time steps was determined such that the computed resonant frequencies converged.
The excitation occurred a t the center of the cavity with a raised-cosine-modulated
sine-wave pulse, which had a normalized center frequency of approxirnately 1.0. The
tirne discretization interval was chosen to be 0.8 times the Courant limit for both
methods. By selecting a time discretization interval twice that of the traditional
FDTD method. the computational time was approximately half that of the tradi-
tional FDTD method for the same number of degrees of freedom. The results are
summarized in Table 5.1. Both the accuracy and the computational time of the pro-
posed technique lie between those of a conventional FDTD having the sarne number
of degrees of freedom and one having one-eighth of the number of degrees of freedom.
Four rectangular cavities loaded with inhomogeneous dielectric materials described
in [4] were then analyzed with the proposed technique. The dominant resonant fre-
quencies were compared with analytical values (when available) and those obtained
with the conventional FDTD rnethod. The geometries of the four cavities are shown
in Fig. 5.5 and the results are summarized in Table 5.2. The number of cells in the
Chapter 5. 3-D Time-Domain .Multiresolution Analysis 103
Table 5.1: Cornparison of the number of degrees of freedom (NDF), the normalized
dominant resonant frequency and the computational time -- -- -- - - - - -
No. of 30. of proposed convent ional
Yee cells time steps technique FDTD
NDF norm. CPU NDF norm. CPC'
freq. time' freq. time*
( sec (sec)
'CPU time on HP9000/C160 workstation
proposed technique \vas approximately one-eighth of the number of FDTD cells so
that the number of degrees of freedom was approximately the same for both meth-
ods. To discretize the geometry of the dielectric materiais accurately. nonuniform
grids were incorporated in the cases (b),(c) and (d). In the case of the homogeneous
dielectric cavity (a), the resonant frequencies obtained with both rnethods agreed
within kl% for the same number of degrees of freedom.
In t his new multiresolution technique, the inhomogeneous dielect ric interfaces have
an anisotropic property due to the approximate treatrnent of the interfaces. Suppose
the dielectric intexface is now located at the interface between Yee cells. Since each
unit Yee ce11 is divided into eight subcells, and each subcell includes three electnc and
three rnagnetic field sampling points collocated at the center of the subcell, the subcell
located at each side of the interface includes two tangential and one normal electric
field components. Then for the tangential electric fields, the dielectric constant is
the average of the dielectric constants on either side of the interface, whereas for the
normal electric field the dielectric constant is that specified on each side. Therefore,
the dielectric becomes anisotropic within a layer on each side of the interface.
Chapter 5. 3-0 Time-Domain Multiresolution Andysis 104
This artificial property causes instability problems when analyzing structures wit h
inhomogeneous dielectric materials, although for structures with homogeneous dielec-
tric materials the method is stable. The dielectric interface can be accurately rnodeled
by introducing the D-H formulation of Maxwell's equations which will be discussed
later. Also note that, in the analysis of open boundary structures such as microstrip
components. the approximate scheme is stable enough for time signals to converge.
Figure 5.5: Three-dimensional rectangular cavities analyzed in this study.
The analyses presented so far incorporate PEC boundaries modeled with simple
forward- or backward-difference approximations. PEC boundaries modeled with La-
grange interpolation improve the field distribution. The higher-order resonances in
a cavity with normalized dimensions 1/fi x 1/& x 1/& were analyzed subse-
quently using third-order Lagrange interpolation in the PEC formulation. The cavity
was discretized with 16x 16x 16 Yee cells and excited with a raised-cosine-modulated
sine-wave pulse, which had a normalized center frequency of 3.0. The calculation
C h a ~ t e r 5. 3-D Tirne-Domain IlIultiresolu tion Analysis 105
Table 5.2: Normalized dominant resonant frequencies of rectangular cavities
cavity proposed conventional % difference analytical technique FDTD
(Yee cells) (Eée cells)
(i) - (ii) (ii)
' nonuniform grids
Cha~ter 5. 3-D Tirne-Domain Multiresolution a4nalvsis 106
was done for 3000 time steps with At equal to 0.8 times the Courant limit, which
is At = 0.0198. The analyticd and computed normalized resonant frequencies are
compared in Table 5.3. The frequency spectrum and the field distribution at time
step 1200 are shown in Figs. 5.6 and 5.7, respectively.
Table 5.3: Higher-order resonant frequencies in a cavity analyzed with the proposed
technique with the third-order Lagrange interpolation of fields.
mode theoretical numerical % error
O 0.5 1 1.5 2 2.5 3 3.5 4 Notmalized frequency
Figure 5.6: Frequency spectrum of the higher-order modes in a cavity analyzed with
the proposed technique.
Chapter 5. 3-0 Time-Domain Multir~soIution rlnalysis 107
Figure 5.7: Field distribution at time step of 1200 for the TEijo(i. j = 1.3 .5 , . . .) higher-order mode analysis.
5.4 Analysis of Microstrip Planar Circuits
Two configurations, a low-pas filter and a spiral inductor, were analyzed with both
the proposed technique and the conventional FDTD method. Mur's first-order ABC (191
was implemented to extract their S-parameters. The results were compared to denion-
strate the capability of the new technique for analyzing realistic microwave compo-
nents. The cornputation was performed on a HP9000/C160 workstation.
5.4.1 Microstrip Low-Pass Filter
The proposed technique was applied to the analysis of the microstrip low-pass filter
shown in Fig. 5.8 1231. The Yee grid lines used in the analysis are shown in the figure
together with the geometrical dimensions. Xonuniform grids were incorporated only
in the proposed technique to accurately discretize the geornetry of the circuit.
The structure was also analyzed with the conventional FDTD method using the
spatial discretization described in [23]. The time discretization was chosen to be 0.98
times the Courant limit for both methods. The excitation pulse was a raised-cosine
Chapter 5. 3-0 Erne-Domain i~fultiresolution Andysis 108
pulse having a duratioo of 66.3 ps. The center frequency of the excitation pulse
was approximately 15 GHz. The analysis conditions for both methods are listed in
Table 5.4. The discretization was such that the number of degrees of freedom was
approximately the same for both methods. Since, in the proposed technique' the
minimum cell size was approximately twice that of the conventional FDTD method,
the time discretization interval could be approximately twice that of the conventional
FDTD method. The calculation time for the proposed niethod was therefore only
approximately half that of the conventional FDTD method.
Table 5.4: Analysis condit ions for the microstrip low-pass filter
proposed conventionai
technique FDTD
Yo. of Yee cells 4 9 x 3 9 ~ 8 1 0 0 x 8 0 ~ 16
(non-uniform) (uniform)
At 0.67694 ps 0.13325 ps
Yo. of time steps 2560 4000
computational time l l m 32.5s 2Om 45.5s
Figure 5.8: Microstrip low-pass filter configuration [23]. The dimensions are in mil-
limeters. and the numbers in parentheses show the numbers of Yee cells.
The resulting time signals and the S-parameters are shown in Fig. 5.9 and Fig. 5.10.
Chapter 5. 3-0 Time-Domah Multiresolution Analysis 109
respectively. The S-parameters indicate good agreement between both methods, ex-
cept for slight deviations in the high frequency range over 16 GHz and in the small
signal range below -30 dB. The time response was computed for 2560 time steps with
the new multiresolution technique. A long numerical analysis showed that the scheme
was stable until about 105 time steps.
The snapshots of the Ez field immediately below the microstrip conductors at tirne
346.6 ps are plotted in Fig. 5.11. In the proposed technique, a ripple was observed
on the excitation side of the filter, while in the conventional FDTD method, the
waveform was smooth. This ripple is caused by the reflection of the signal; since
the incident and the reflected waves both consist of the wavelet functions that have
an asymmetrical space distribution, the wavelet basis function associated wit h the
reflected signal has a field distribution that is inverse to that of the incident signal,
and this causes the ripple in the reflected signal. Therefore. the ripple did not appear
after passing the filter. If a smooth field distribution is desired. one can extract it
from the scaling function coefficients by using interpolation.
-0.3 I I I I I 1 O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Tirne (ns)
Figure 5.9: Time signals of the low-pass filter computed with the proposed technique.
The maximum time step is 2560. - : input port, - - - : output port.
Chapter 5. 3-D Tirne-Domain ibfultiresolution Analysis 110
Figure 5.10: Computed S-parameters of the low-pass filter. - : proposed method. -
- - : conventional FDTD method.
5.4.2 Spiral Inductor
-4 two-turn spiral inductor shown in Fig. 5.12 was investigated. The relative permit-
tivity and the dimension of the substrate were 9.6 and 50 mm x 50 mm. respectively.
The dimension of the inductor was 18 mm x 18 mm. The cutoff frequency of the
inductor was around 2.5 GHz. Uniform grids were incorporated to discretize the
structure for both methods except for the r-direction in the upper air region for the
multiresolution analysis. The discretization conditions and the calculation time are
listed in Table 5.5. The time discretization is also 0.98 times the Courant limit. The
excitation pulse is a raised-cosine pulse with a duration of 333 ps; the center frequency
of the excitation pulse is approximately 3 GHz. The time signals obtained with the proposed technique are shown in Fig. 5.13.
The time signals decay more slowly than those in the low-pass filter analyzed in the
previous subsection due to the long line length of the spiral inductor and the larger
permittivity of the substrate. .\ long cornputation showed that the proposed technique
was stable up to the time step of 10% The S-parameters of the inductor are shown in
Fig. 5.14 for both methods. It should be mentioned that, due to the large permittivity
Chapter 5. 3-D Tirne-Domain iLIultiresolution Analysis 111
(a) proposed technique
(b) conventional FDTD method
Figure 5.11: Snapshots of the Ez field distribution at time 346.6 ps in the low-pass
filter immediately below the conductors.
Chapter 5. 3-0 Tirne-Domain Multiresolution Andysis 112
Figure 5.12: Spiral inductor configuration. The dimensions are in millimeters. The
line widths and spacings are al1 2.0 mm. The height and the span of the air bridges
are 1.0 mm and 6.0 mm, respectively.
Table 5.5: Analysis conditions for the spiral inductor
proposed convent ional
technique FDTD --
Yee cells 6 2 x 4 2 ~ 1 3 1 0 0 x 6 8 ~ 2 6
( non-uniform) (uniform)
Ax, Ay 0.8 mm 0.5 mm
Az 0.5 mm, 0.4524 mm 0.25 mm
At 0.37274 ps 0.21535 ps
time steps 23110 40000
CPU time 3h 38m 2.9s 5h 4m 32.5s
Chapter 5. 3-0 Time-Domain Multiresolution ..lnaiysis 113
of the substrate, a large reflection from the Mur's first-order ABC was observed for
both results. Thus, in calculating the reference data a t the input port, computation
was terminated by the time in which the reflection from the ABC reached the input
port. This treatment made the energy of the reference signal smaller than that of the
signals from the inductor and resulted in the magnitude of the S-parameters being
larger than O dB. In order to eliminate this discrepancy. the S-paranieters in Fig. 5.14
were offset by about -0.3 dB. Since this error is due to the insufficient performance of
the implemented ABC, one can avoid offsetting the S-parameter by employing ABCs
of higher absorption such as Mur's second-order ABC [19], Berenger's PML [l] or
similar high-quality ABCs. The resulting S-parameters demonstrate good agreement
between both methods except for the small signal region below -10 dB.
Snapshots of the E, field immediately below the microstrip conductors a t tirne
1.72 ns are plotted in Fig. 5 - 1 5 In the proposed technique, a ripple waç observed on
the excitation side of the inductor similar to that in the low-pass filter analysis.
0 1 2 3 4 5 6 7 6 9 Time (ns)
Figure 5.13: Time signals of the spiral inductor computed with the proposed tech-
nique. The maximum time step is 23110. - : input port, - - - : output port.
-50 I I L
O 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (GHz)
Figure 5.14: Cornputed S-parameters of the spiral inductor. - : proposed method. - - - : conventional FDTD method.
5.5 Exact Formulation for Three-Dimensional In-
homogeneous Dielectric Structures
5.5.1 Formulation
The exact formulation for t hree-dimensional inhornogeneous dielect ric structures is
obtained by discretizing Maxwell's equations (5.1) and
and the rnaterial equation
where the conductivity of the materiai is assumed to be zero.
The electric flux density, D, for example, is expanded using the 3-D Haar b a i s
Chapter 5. 3-0 Time-Domain Multiresolution Analvsis 115
(a) proposed technique
(b) conventional FDTD method
Figure 5.15: Snapshots of the Ez field distribution at time 1.72 ns in the spiral
inductor immediately below the microstrip conductors.
Chap ter 5. 3-5 Time-Domain klul tiresolu tion Anal ysis 116
functions (5.3) as
where the notations are defined as in (5.9) and (5.10).
Then. a procedure similar to that used for (5.11) and (5.14) leads to the time
iterative difference equations with the weighted voltage across the E-node 31%
and
2 clic - p v c ) - YJC"} O ,,O Oh0
Chap ter 5. 3-0 Tirne-Domain Multiresolution Analysis 117
= O w'" OOh +
The material equation (5.35) cm be also discretized using 3-D Haar scaling and
wavelet bases (5.3) as discussed in [22] in the case of Battle-Lemarie scaling and
wavelet bases. However in the case of 3-D Haar scaling and wavelet bases. it is more
sirnply discretized by using the basis transformation matrix A. For isotropic dielectric
materials. the material equation (5.35) is written for a rectangular subcell (opq) for
o. p . q = 1. u in a unit Yee ce11 as
for cc = r. y. z. This can be rewritten in matrix form as
for w = x. y, 2. where
and
Since the matriv A is the basis transformation matrix between the rect angular-pulse
basis functions and the Haar basis functions, (5.44) can be transformed into the Haar
Cha D ter 5. 3-D Tirne-Domain Multiresolu tion Analysis 118
basis coefficients using relation (5.28) as
and
[Ci] ' j k Haar = A [ q ; I r e c A .
The transforrned rnatrix [$:] can be reduced to a simple rnatrix that is highly
where the elements of the mat ri^ are defined by the inner products
1 ai = - ai-^-', for i = 1 , 2 ; - . , 8 .
8
with the column vectors ai (i = 1, . '8) and E-' which are defined by
and
In the time stepping algorithm, (5.48) is computed after the flux density D is updated
by (5.40).
Chapter 5. 3-0 Time-Domain Md tiresolu tion Analysis 119
5.5.2 Numerical Experiments
Inhomogeneous dielectric loaded rectangular cavity
The TEl lo mode in the centered-dielectric-slab loaded rectangular cavity shown in
Fig. 3.5 (b) was analyzed with this formulation. The nurnber of Yee cells used in
the analpsis was 12.5 x 4.5 x 3.5, and non-uniform grids were incorporated. The
discretization of the structure is schematically shown in Fig. 3.16. At the dielectric
boundarv: the dielectric constant must be represented at the sarnpling point and
volume averaged in terms of the size of the Yee's grid. The dielectric constants at
Figure 5.16: The discretization of the inhomogeneous dielectric loaded rectangular
cavity.
each side of the boundary of different materials €1 and €2 will be determined as follows.
For the q-side of the boundary, it is defined as
and for the c2-side of the boundary as
For the structure analyzed here, el = 1.0, €2 = 3.75, e12 = 1.651783 and €21 =
3.050595. The time interval At was 0.8 times the Courant limit, which was At = 0.698
Chapter 5. 3-0 Time-Domain Multiresolution Analysis 120
ns . The computed resonant frequency rvas 0.05189, which is -0.61% in error compared
to the analytical resonant frequency 0.05221. This result is slightly less accurate than
that obtained with the FDTD method. A long computation showed that the exact
formulation was stable at 106 time steps, and no instability was observed. The Ez
field plot demonstrates smooth field distribution as shown in Fig. 5.17. Although this
scheme requires longer cornputational time than the formulation described in Section
5.2 which is for the approximate treatment of inhornogeneous dielectric materials.
it needs to be applied only at the dielectric interface to improve the computational
efficiency.
Ez lield
Figure 5.17: E, field distribution in the centered-dielectric-slab loaded rectangular
cavity shown in Fig. 5.5 (b).
Inhomogeneous dielectric loaded rectangular waveguide
The next example is an analysis of the dominant mode propagation in the slab loaded
rectangular waveguide (WR-90) shown in Fig. 5.18. The dielectric slab with c = 2.0
is 6 mm wide, and centered in the guide cross-section. The dielectric constants at the
boundaries are, according to the volume averaging definition of dielectric constant
Chapter 5. 3-0 The-Domain Multiresolution .4nalysis 121
Figure 5.18: The configuration of the slab loaded WR-90 rectangular waveguide.
a =?2.86 mm. b = 10.16 mm, L = 100 mm. t =5.96 mm, €1 = 1.0 and €2 = ? . O .
(5.56) and ( 5 . 3 ) , c = 1.75 in the dielectric slab. and É = 1.25 in the air region.
Uniform rectangular grids are incorporated in the analysis. The Yee cells are 11 in
width, 50 in length and 3 in height. The boundaries are al1 PEC walls.
Snapshots of the EL field distribution are shown in Fig. 5.19 Smooth field distri-
bution was obtained even at the dielectric boundary. Since EL is tangential to the
boundary. these results exhibit the correct behavior of the field.
Rectangular waveguide with a dielectric post
The last esample is a discontinuity consisting of a dielectric post in a WR-90 rectan-
gular waveguide. The dimensions and the location of the dielectric post are shown in
Fig. 5.20. The outer boundaries are al1 PEC conditions. The discretization of the di-
electric post and the distribution of the dielectric constants are depicted in Fig. 5.21.
The volume averaging definition of the dielectric constant gives five difFerent dielect ric
constants at the boundaries depending on the location of the nodes. r\ssume that the
dielectric constant outside the post is €1 and inside the post €2; then, by referring to
Fig. 5.21? the dielectric constants a t the boundaries are given by
Chapter 5. 3-D The-Domain Md tiresolu tion Analysis 122
(a) 50 time steps (b) 100
(c) 150 (d) 200
(e) 250
Figure 5.19: El field distributions of the dominant mode propagation in the slab
loaded WR-90 waveguide.
Chap ter 5. 3-D Time-Domain Md tiresolution Andysis 123
Figure 3.20: The configuration of the WR-90 waveguide loaded with a dielectric post.
a = 22.86 mm. b = 10.16 mm, L = 100 mm. = 1.0 and €2 = 5.0.
Figure 5.21: The discretization of the dielectric post and the distribution of the
dielectric constants. : €2, x : €3, 8 : €4, 9 : €5, a : €6, O : €7
Cha~te r 5. 3-D Tirne-Domain Multiresolution ..lnalysis 124
In this example. those volume averaged dielectric constants at the boundaries are
cl = 1.0. €2 = 5.0, €3 = J.O. €4 = 3.25. cs = 2.0, es = 1.75 and €7 = 1.25.
The propagation of the dominant TElo mode was then sirnulated in this waveguide.
The EL field distribution is plotted in Fig. 5.22. Smooth field distribution is obtained
cvcn around the dielectric post.
5.6 Conclusions
.-\ three-dimensional multiresolution analysis procedure similar to the FDTD method
has been derived by using a cornplete set of three-dimensional Haar scaling and
wavelet basis functions.
The resulting method has been tested and validated by analyzing several cav-
ity structures including inhomogeneously dielectric loaded rectangular cavities. The
method has also been applied to the analysis of rnicrowave passive structures such
as microstrip low-pass filters and spiral inductors. The resulting S-parameters are
in good agreement with those obtained with the conventional FDTD rnethod. How-
ever. the field distribution plots show small ripples in the fields computed with the
proposed method. The calculation time for the proposed method was approximately
half that of the equivalent conventional FDTD method.
Chapter 5. 3-0 Time-Domain Multiresolution Analysis 125
(a) 66 timesteps (b) 132
(c) 198
(e) 330
(d) 264
Figure 5.22: Ez field distributions of the dominant mode propagating in the WR-90
waveguide containing a dielectric post.
Chapter 6
Conclusions
The object of this thesis was to establish a general framework for time-domain elec-
tromagnetic modeling based on Haar-wavelet multiresolution analysis. Like other
numerical analysis techniques, the proposed technique has a number of advantages
and disadvantages.
As mentioned in the introduction. numerical techniques are still far from reach-
ing the ultimate goal of solving differential equations accurately and quickly. The
numerical technique employing wavelet theory also falls short of this expectation.
One major problem when applying wavelet theory in electromagnetic field rnodel-
ing is to properly mode1 complicated circuit structures found in practical applications.
To implement any boundary condition, the values of electric or magnetic fields at the
boundary must be enforced by ensuring that the superposition of the wavelet coeffi-
cients satisfy the boundary conditions.
For simple structures, b o u n d a ~ conditions can be established by means of image
theory, local combination of wavelet basis coefficients and so on, depending on the
wavelets used. The analysis of these simple structures with a wavelet based technique
can be accurate and efficient. However, for complicated structures, difficulties arise
when modeling the boundary conditions.
This thesis showed that it is possible to apply wavelet theory to realistic problems
involving relat ively cornplicated geometries. For more complicated applications such
as the analysis of human bodies and tissues, which include a number of dielectric
Chapter 6. Concl usions 127
materials with many interfaces, the wavelet approach becomes arduous. However, for
electric circuits and waveguide structures that are mainly composed of rectangular
shapes, the proposed technique is more readily applicable. This flexibility is achieved
by employing the simplest famiiy of wavelets, namely Haar wavelets.
6.1 Efficiency and Accuracy of the Method
When we discuss the superiotity of one numerical technique over others, it is impor-
tant to consider both cornputational efficiency and accuracy at the same time. Even
if a technique is highly effective, it might be inaccurate. Thus, the tradeoff between
computational efficiency and accuracy should be taken into consideration.
It was found that the accuracy of the Haar-wavelet based time-domain technique is
better than that of the conventional FDTD method when the discretization is coarse.
The computational burden is also reduced approximately by half compared to the
FDTD method when both models have the same number of degrees of freedom.
When modeling practical structures, the fine adjustment of boundary location is
not as simple as in the conventional techniques. However, this could be solved by an
automatic modeling interface program.
Moreover, perfect electric, perfect magnetic, impedance walls or other boundaries
must be pre-computed in the proposed technique. This represents an extra task in
progam coding; however. it is true that it saves computational effort in the actual
analysis. Similar situations can be found in frequency domain techniques such as the
Mode Matching technique, the Method of Moments, and so on. The computational
effort required by those techniques is often less than that required by the FDTD or the
Finite Element Method, but the former require extensive analytical preprocessing,
and those formulations are problem specific and less versatile.
One major advantage of wavelet decornposition is the potential for thresholding.
For image processing, it is an efficient technique because it reduces the amount of
stored information by omitting expansion coefficients that are smaller than a threshold
value. However, in electromagnetic field modeling, it rnight reduce the computational
efficiency when the field values must be checked at every time step. In this thesis,
Chap ter 6. Conclusions 128
a thresholding technique has not been implemented. Finding efficient algorithms for
t hresholding in electromagnetic field analysis should be the sub ject of future research.
6.2 Overall Conclusions
The solution of Maxwell's equations represents a continuing challenge even though
they were formulated more than a century ago. -1s computer hardware technology
develops, the problerns we wish to solve become more difficult as well. It is obvious
that problems will never be exhausted in the future; we will tackle ever larger. more
complicated problems, and problems involving several physical phenornena a t once.
Cutting computation time in half is equivalent to building hardware that runs
twice as fast. The proposed technique, "time-domain Haar-wavelet based multireso-
iution technique" reduces the computational burden by half compared to that of the
conventional FDTD rnethod, although the memory requirement is the same when the
number of degrees of freedom is identical.
We have demonstrated that the proposed technique is applicable to realistic prob-
lems such as two-dimensional waveguide analysis, three-dimensional planar circuits
with open boundaries, and resonant cavities loaded with inhomogeneous dielectric ma-
terials. The results were compared to those in the available references. We achieved
good agreement and demonstrated the feasibility of the Haar-wavelet based tirne-
domain technique.
6.3 Future Research
We conclude this thesis by discussing possible future research topics that have emerged
as a results of this work.
Thresholding is the most promising technique for reducing the memory require-
ment. When implementing the thresholding technique, care must be taken not to
increase computational overload. For thresholding, the field values must be evalu-
ated and checked against the threshold value at every time step; however, this slows
down the computation. Therefore, an optimum procedure must be found that reduces
Chao ter 6. Conclusions
rnemory requirements and maintains computational efficiency a t the same time.
It has been pointed out t hat O t her wavelet families such as Bat t le-Lemarie spline
wavelets and Daubechies' compactly supported wavelets significantly reduce the num-
ber of cells required per wavelength. However, a t the same time, boundary conditions
are more difficult to implement. For electromagnetic field analysis, formulations other
than the so called ~'multiresolution time-domain (MRTD) approach" might be neces-
sary;we need formulations that can handle boundaries and inhomogeneous materials
more systernatically.
hnother possible approach to modeling fine structures is to use multilevel wavelet
b a i s functions. Here, the wavelet function with higher resolution is used to mode1
discontinuities. The problem is that we have to deal with additiooal degrees of free-
dom, which increases memory requirement. Therefore. this must be only applied to
resolve highly singular fields. Xevertheless. the boundary conditions will be =ore
complicated.
Another measure that could alternate the modeling of boundaries would be to
combine wavelet analysis with other discrete methods such as FDTD. TL41 or FEM
(finite element method). The latter are more feasible when it cornes to modeling corn-
plex boundaries and could thus be used for this purpose, while the rest of the corn-
putational domain could be modeled with wavelets. To realize such hybrid schemes.
the connection between the various methods must be studied thoroughly.
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Appendix A
A Finite-Difference Time-Domain
Method
In t his appendix, the basic t heory and the discretization formulation of hIauwellTs
curl equations in a 3-D Cartesian coordinate system by means of the Finite-Difference
Time-Domain Method based on Yee's algorithm is summarized for reference.
To discretize 41âuwell's curl equations
in Cartesian coordinates, we write (Al) and (A.3) as follows
(A. 1)
(A.2)
(A. 3)
(A.4)
(A.5)
( A S )
(A* 7)
(A. 8)
The field components are defined as in Fig. A l showing a Yee cell, which contains
three electric and three magnetic components in a unit cell.
Figure A. l: A three-dimensional Yee ce11 for the
circles represent H fields, black dots represent E of the field components.
Cartesian coordinate system. Open
fields. Arrows indicate the direction
Update equations for currents
With the notation employed in this thesis, and by replacing the differentials by central
differences, (-4.3) can be discretized as
The subtraction in the numerator on the left hand side of (-4.9) must always be
taken between the upper and the lower field values in the direction defined by the
denominator with respect to the field node on the right hand side. In other words,
the temporal change in the magnetic field is represented by the spatial variation of
the four electric fields that surround the magnetic field. We cao thus simpliQ the
i l ~ ~ e n d i u A. '4 Finite-DiFerence Tirne-Domain Method 136
eauation as follows
where the prirned quantities are the lower side fields and the non-primed quantities
are the upper side fields. Multiplying (A.10) with AyAz leads to
('E - 'E' ) A; - (922 - P E I ) Ay
We introduce following quantities:
Although I is not defined as an actual current in terms of its direction of Bow, we refer
to this quantity I as "equivalent current" or "current!' for the sake of convenience.
Then we write (.\.Il) as
(A. 14)
By using voltage and current notations like this, we can avoid division operations
in the final differential equations as shown later. By solving (h.14) in terms of the
current 4 a t the new tirne step, the time stepping difference equation for jh.3) is
A further simplification of the notation is to write
and so on. We finally obtain the update equation for currents 7 as
Update equations for voltages
Similarly, for Maxwell's equation (A.6), the time variation of the electric field is
represented by the spatial difference of the four magnetic fields that surround the
electric field. Thus,
Changing the variables
(LI - 21') -
to voltages and currents. (A.20) becomes
(YI - Y I ' )
By solving (A.21) in terms of the voltage =V at the new time step, the time stepping
difference equation for (-1.6) is given by
where Y denotes the excitation source current
In simplified notation, the update equation for voltages V is given by
2At Ax + .- 2r + aht AyAr { i l h h O - i l / & O - ( f l h O h - YiIh0-h) - $00) +(-1.24)
Overall leap frog algorithm
The whole set of time-update difference equations for Maxwell's equations proposed
by K.S.Yee in 1966, which is called leap frog algorithm, is summarized as follows:
and
where the coefficients are
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