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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