ELIMINATION OF SPURIOUS MODES IN THE

VECTORIAL FINITE ELEMENT ANALYSIS OF

ISOTROPIC AND ANISOTROPIC WAVEGUIDES BY

THE METHOD OF CONSTRAINTS

Xiang Gao

A thesis submitted in conformity with requirements

for the degree of Master of Applied Science

Department of Electrical and Corn puter Engineering

University of Toronto

@Copyright by Xiang Gao 1996

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Abstract An improved vectorial finite-element met hod for the analysis of dielectric waveg-

uide problems solving for magnetic-field components is forrnulated by combining the

vectorial finite-element method with the rnethod of constraints. In the method of con-

st raints, the proper boundary condit ions for arbitrary boundaries are implemented

and the non-divergence relation V - B = O is enforced. As a result. the spurious.

nonphysical solutions which have been necessarily included in the solutions of earlier

vectorial finite-elernent methods are eliminated in the whole region of the propagation

diagram. This method is generalized for solving homogeneous/inhomogeneous and

isotropic/anisotropic waveguide problems with arbitrary cross sections. To verify the

validity of the present method. numerical results for some typical waveguide problems

are presented and compared with exact and earlier vectorial finite-element solutions.

Acknowledgement s

First. I would like to express my deepest gratitute to my supervisor. Professor .-1.

Konrad. for his invaluable technical guidance and financial support throughout the

whole process of accomplishing t his t hesis.

1 would also like to thank the people in the Power Group for providing an envi-

ronrnent conductive to research.

Finally. 1 wish to thank my husband and parents for their overwhelming love.

understanding, encouragement and support that is tremendously important for me

to achive this thesis and more.

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $1 - 1.3 Thesis Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

. 1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Vectorial Finite Element Method 9

2.1 41axivel11s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Vector Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 . :3 Functional Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Mathematical Representation of Traveling CVaves . . . . . . . . . . . 13

2.5 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . 13

'2.6 Interface Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Boundary Conditions 16

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Summary 10

3 Spurious Modes 21

. . . . . . . . . . . . . . . . . . . 3.1 The Appearance of Spurious Modes 21

3.2 Explanation for the Appearance of Spurious Modes . . . . . . . . . . 29

3.3 Methods of Identifying Spurious Modes . . . . . . . . . . . . . . . . . 3%

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 The Method of Constraints 41

4.1 Description of the Method of Constraints . . . . . . . . . . . . . . . . 41

4.2 Implementation of Boundary Conditions . . . . . . . . . . . . . . . . 13

4.2.1 Implementation of Boundary Conditions for Perfect Electric

Conductor Walls . . . . . . . . . . . . . . . . . . . . . . . . . 4 3

4.2.2 Implementation of Boundary Conditions for Perfect Magnetic

Conductor Walls . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Implementation of Non-Divergence Constraint . . . . . . . . . . . . . 54

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Description of the Computer Program 65

6 Results and Applications 68

6.1 Homogeneous Isotropic Waveguide Problem . . . . . . . . . . . . . . 68

6.2 Inhomogeneous Isotropie Waveguide Problem . . . . . . . . . . . . . SS

6.3 Circular Waveguide Problem . . . . . . . . . . . . . . . . . . . . . . . 92

6.4 Magnetically Anisotropic Waveguide Problem . . . . . . . . . . . . . 96

6.5 Coaxial Waveguide Problem . . . . . . . . . . . . . . . . . . . . . . . 103

7 Conclusions 105

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.2 Advantages and Disadvantages of the Method . . . . . . . . . . . . . 106

7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10s

A Directional Derivatives of Interpolation Polynomials in Finite Ele-

ment Problems Using Triangular Elements 109

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

. . . . . . . . . . . . . . . . . . . . . . . . . . A 2 Derivat ion of Formulas 1 10

. . . . . . . . . . . . . . . A.3 Derivation of Matrices G2 and Gg from GI 112

. . . . . . . . . . . . . . . . . . . . . . . . . A.4 The Computation of Gi 113

List of Figures

An arbitrary boundary . . . . . . . . . . . . . . . . . . . . . . . . . .

Element subdivision for an empty rectangular waveguide involving

eight t hird-order triangular elernents. . . . . . . . . . . . . . . . . . .

k - 3 diagram for the empty rectangular waveguide problem. . . . . .

,i?/k- k diagram for the empty rectangular waveguide problem. . . .

Cross section for a rectangular waveguide hâlf-filled with a dieiectric

material of relative permittivity 2.45. . . . . . . . . . . . . . _ . . . .

Mesh for the waveguide shown in Figure 3.3. . . . . . . . . . . . . . .

k - 9 diagram for the rectangular waveguide half-filled with a dielectric

material of relative permittivity 2.35. . . . . . . . . . . . . . . . - . .

k - /.? diagram for the rectangular waveguide half-filled wi t h a dielect ric

material of relative permit tivity 2.45. . . . . . . . . . . . . . . . . . .

H field vector for the 90th mode in the homogeneous waveguide with

mesh division (Ne, N p ) = (8,51) . . . . . . . . . . . . . . . . . . . . .

H field vector for the 92nd mode in the homogeneous waveguide with

mesh division (ive, N p ) = (8,Yl) . . . . . . . . . . . . . . . . . . . . .

3.10 H field vector for the 93rd mode in the homogeneous waveguide with

mesh division ( N e , IV,) = (8, Y 1) . . . . . . . . . . . . . . . . . . . . .

3.1 1 Contour lines for Hz and H,, for the 90th mode of the homogeneous

waveguide with mesh division (Ne, Y,) = ( Y . 31) . . . . . . . . . . . .

3.12 Contour lines for Hz and H, for the 92nd mode of the homogeneous

waveguide with mesh division (Xe, :V,) = (8,s 1 ) . . . . . . . . . . . .

3.13 Contour lines for H, and H, for the 93rd mode of the homogeneous

waveguide wit h mesh division ( Ne. N p ) = (S. Y 1) . . . . . . . . . . . .

3.14 H field vector for the 76th mode in the inhomogeneous waveguide wi t h

mesh division (Ne: Np) = (Y. 81) . . . . . . . . . . . . . . . . . . . . .

3.1.5 H field vector for the 77th mode in the inhornogeneous waveguide wit h

mesh division ( Ne. N p ) = ( Y , Y 1 ) . . . . . . . . . . . . . . . . . . . . .

3.16 H field vector for the 78th mode in the inhomogeneous waveguide wit h

mesh division (WC, Y,) = (8. Y 1 ) . . . . . . . . . . . . . . . . . . . . .

3.17 H field vector for the 79th mode in the inhomogeneous waveguide wit h

mesh division (Ne, :Vp) = (8.81) . . . . . . . . . . . . . . . . . . . . .

3-18 H field vector for the 80th mode in the inhornogeneous waveguide with

mesh division (Y,. Y,) = (8.81) . . . . . . . . . . . . . . . . . . . . .

3.19 H field vector for the Y 1st mode in the inhomogeneous waveguide wit h

mesh division (:L;. N p ) = (3 .81) . . . . . . . . . . . . . . . . . . . . .

3.20 H field vector for the S2nd mode in the inhomogeneous waveguide with

rnesh division ( N e , A$) = ( Y , Y 1) . . . . . . . . . . . . . . . . . . . . .

3.21 H field vector for the S3rd mode in the inhomogeneous waveguide with

mesh division ( N e , N p ) = ( S . YI) . . . . . . . . . . . . . . . . -

3-22 H field vector for the S4th mode in the inhomogeneous waveguide wit h

mesh division ( Ne, IV,) = (8,51) . . . . . . . . . . . . . . . . . . . . .

3.23 H field vector for the 85th mode in the inhomogeneous waveguide with

mesh division ( N e , N p ) = (5,81) . . . . . . . . . . . . . . . . . . . . .

vii

3.24 H field vector for the 86th mode in the inhomogeneous waveguide with

mesh division (Ney N p ) = (8 . Y i ) . . . . . . . . . . . . . . . . . . . . . 38

3.25 H field vector for the SFth mode in the inhomogeneous waveguide wi th

meshdivision(Ne.~Vp)=(Y. 81) . . . . . . . . . . . . . . . . . . . . . 38

Point on a bound- parallel to the y-axis . . . . . . . . . . . . . . .

Point on a boundary parallel to the x-axis . . . . . . . . . . . . . . .

Point on a sloping boundary . . . . . . . . . . . . . . . . . . . . . . .

&True3 corner point . . . . . . . . . . . . . . . . . . . . . . . . . . . .

"False" corner point due to the finite element approximation of smooth

surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . Point on a boundary parallel to the y-auis

. . . . . . . . . . . . . . . Point on a boundary pardlel to the x-axis

Point on a sloping boundary . . . . . . . . . . . . . . . . . . . . . . .

"True- corner point . . . . . . . . . . . . . . . . . . . . . . . . . . . .

*False7 corner point due to the fini te element approximation of' smoot h

surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.11 The cross-section of a hollow square waveguide with two second-order

triangular elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.1 The hollow square waveguide . . . . . . . . . . . . . . . . . . . . . . 68

6.2 The cross section and the mesh of the waveguide shown in Figure 6.1 69

6.3 k . ,û diagram for the hoilow square waveguide before the constraints

are enforced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.4 ,O/ k. k diagram for the hollow square waveguide before the constraints

are enforced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.5 k . .8 diagram for the hollow square waveguide after the constraints

-6 are enforced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i 2

... Vll l

$/k - k diagram for the hollow square waveguide after the constraints

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . are enforced.

The cross section and the mesh of the same waveguide. as in Figure 6.2.

a t the different position. . . . . . . . . . . . . . . . . . . . . . . . . .

The cross section and the mesh of the right half of the waveguide shown

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . in Figure 6.2.

The cross section and the mesh of a quarter of the waveguide shown

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . in Figure 6.2.

hnother mesh of the same waveguide a s in Figure 6.2 . . . . . . . . .

k - J diagram for the hollow square waveguide with mesh shown in

. . . . . . . . . . . . Figure 6.10 before the constraints are enforced.

k - 9 diagram for the hollow square waveguide with mesh shown in

. . . . . . . . . . . . . Figure 6.10 alter the constraints are enforced.

. . . . The cross section of an inhomogeneous rectangular waveguide

k - 3 diagrarn for the inhomogenous rectangular waveguide before the

. . . . . . . . . . . . . . . . . . . . . . . . . constraints are enforced.

k - ,$' diagram for the inhomogeneous rectangular waveguide after the

. . . . . . . . . . . . . . . . . . . . . . . . . constraints are enforced.

The cross section of a circular waveguide and the mesh for one-fourth

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . of the structure.

The line diagram showing the positions of analytical cutoff frequencies

of some low-order modes compared with the dominant TEii modes. .

. . . . The same line diagrarn of the computed results as Figure 6.17.

k-P diagram for the circular waveguide after the constraints are enforced. 94

k - P diagram for the circular waveguide before the constraints are

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . enforced.. 95

. . . . . . . . . . . . . . . 6.21 Cross Section of the Anisotropic waveguide 96

6.22 k - . 8 diagram for the magnetically anisotropic waveguide before the

. . . . . . . . . . . . . . . . . . . . . . . . . . constraints are enforced 100

6.23 k - diagram for the magnetically anisotropic waveguide after the

. . . . . . . . . . . . . . . . . . . . . . . . . . constraints are enforced LOI

. . . . . . . . . . . . . . . . . . . . . 6.24 Coaxial waveguide cross section 103

. . . . . . . . . . . . . . . 6.25 Cutoff characteristics of coaxial waveguide 104

. . . . . . . . . . . . .A 1 A triangular elernent with the interpolation points 110

List of Tables

2.1 Relations between (j' t) and material tensors against v. . . . . . . . L 1

3.1 The geometric means of divH of al1 interpolation points for 12 modes in

the inhomogeneous waveguide shown in Figure 3.1. 3 = 1 .O. ( .V,. Y,) =

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( Y 1 . Y ) 40

6 Reiative errors of k of the first 14 modes for i3 = O before (ILI ) and

after (k2) enforcing the constraints. . . . . . . . . . . . . . . . . . . .

6.2 Comparison of the results. a t cutoff. computed for the same waveguide

with the same mesh division in two different positions. . . . . . . . .

6.3 Comparison of the results from computing H m d E when 3 = O and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 = 1

6.4 Results returned by calculating one-half of the waveguide shown in

Figure 6.8 at cutoff. . . . . . . . . . . . . . . . . . . . . . . .

6.5 Results returned by calculating one-fourth of the waveguide shown in

Figure 6.9 at cutoff. . . . . . . . . . . . . . . . . . . . . . . . . .

6.6 The comparison of the results returned by calculating the whole waveg-

uide, one-half of the waveguide and one-fourth of the waveguide. . . .

6.7 Cornpuison of analytical results and computed solution of 7 modes for

5 ,B values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3 The comparison of k for mode no.1 to no.10 in Figure 6.23 with those

for no. 58 to no. 67 in Figure 6.22. . . . . . . . . . . . . . . . . . . . 09

6.9 The wavenumber /ci , k2 and k3 returned by our program for (mode no.

1. no2 and 00.3: respectively, in Figure 6.23) and their percent errors. 102

-4.1 Table of indices r(i) for iV = 1.. . . .5 . . . . . . . . . . . . . . . . . . 114

xii

Chapter 1

Introduction

1.1 Background

In applications at microwave and millimeter mave frequencies. an accurate knowledge

of the propagating modes in general waveguide structures is essent ial .These waveg-

uides may have irregular cross-sect ions which contain inhomogeneous. anisot ropic

dielectric or magnetic media. The development of methods to solve the associated

electromagnetic field problems has attracted the attention of many researchers. 01

the many numerical techniques possible. the finite element method (FEM) is prob-

ably the most versatile and widely used, since it was first applied to homogeneous

waveguide problem by Silvester in 1969 [II.

The numerical analysis of complicated waveguide structures mentioned above re-

quires at least two dependent variables and is therefore vector. rather than scalar.

in nature. The earliest vectorial finite element method was formulated in terms of

the two axial components of field, E, and HL [2]. Subsequently. attention shifted to

formulations involving the transverse field components. One met hod used t hree corn-

ponents of electric field and three components of magnetic field [3]. a computationally

expensive approach. The remainder employed just the rnagnetic field (or. by duality.

just the electric field) [4].

Unfortunately, al1 of these vectorial methods give rise to spurious modes. A survey

of t his early work appears in [SI. Spurious modes are numerical solut ions of the vector

wave equation that have no correspondence to physical reality: t hese solut ions are

simply wrong answers, and should not be confused with experimental spurious modes

t hat are unwanted-but-obviously-p hysically existing modes. . b o t her name somet imes

used for numerical spurious modes is 'vector parasites'. The issue of spurious modes

has been a puzzle in the elect romagnet ic communi t y

-4 review of the literature about the reasons for the existence of spurious modes

and methods to eliminate them is presented in the next section.

1.2 Literature Review

For many years, the identification and suppression of spurious modes have been a

major considerat ion in waveguide analysis.

The curl-curl vector wave equation was first solved using the FELI by Konrad

[6], who was also the first to describe the problem of spurious modes arising frorn

this equation. Konrad proposed, in his PhD thesis, that spurious modes are caused

by the nonsolenoidal nature of vector finite element approximation procedures [ I I . Beginning with Konrad, a series of papers expounded this idea and presented various

techniques to enforce the solenoidal nature of the flux. They did eliminate part or al1

of the spurious modes.

Attempts have been made t o solve the coupled vector and scalar potential equa-

tions of elect romagnetics [5, Y. 9,101. Such methods lead to very large and complicated

matrix equations. The penalty functioo method [L 1. 12. 13. 14. 15. 16. 171 adds a

penalty term multiplied by a penalty parameter to t h e conventional functional so that

explicitly the Euler equations are vector wave equations plus natural conditions of

the vanishing of divB. The eigenvalues of spurious modes increase with the penalty

parameter while the errors associated with the physical modes are quite low. particu-

Lady for the lower order modes. Thus the spurious modes are pulled out of the range

of interest. The biggest difficulty wi th the penalty function rnethod is that a good

value for the penalty parameter is not known in advance. If the parameter is set too

low. there will be spurious modes mixed with the true modes. If the penalty pararn-

eter is set too hi&. undue emphasis is placed on the solenoidality of the solution at

the expense of other rneasures of accuracy. In view of this dificulty. Lynch et al.

introduced the modified penalty method [17]. Rather than leaving the penalty pa-

rameter arbitrary and problem-dependent. the aut hors prescribed an expanded weak

form which in effect reduces the algebraic curl-curl operator to a Laplacian on ho-

mogeneous subregion. In [1S]. it is shown that the Helmholtz algebra has monotonic

dispersion curves which mimic their analytic counterpart such that proper specifi-

cation of physical boundary conditions ensures suppression of spurious .'divergence"

modes w hereas the curl-curl algebra has double-valued dispersion surface such t hat

the synthesis of physical and spurious modes is unavoidable in problems driven by

physically correct boundary conditions.

The method of constraints [19. 20, 21- 22. 23: 241 is another approach to render

finite element vector field solutions rigorously non-divergent. it is based on the idea

of a posteriori imposition of a set of constraints on the finite element matrix equation.

It may also be referred to as the method of reduction because a beneficial consequence

of such a procedure is the reduction of global finite element matrix size. This methocl

was tried by Konrad [20, 211, however. his attempts to generalize this approach was

not successful. In one of his papers [24], he stated that

.' 'lumerical experiments with multi-element 2D models indicate that

zero divergence constraint can only be applied a t non-common nodes on

an element-by-element ba i s . The met hod of constraints for enforcing the

zero divergence condition in vectorial finite element schemes is problem-

atic when applied to rnulti-element models. The reduction in matrix size

alone would justify the development of such a technique for general rnulti-

element grids but it will require the implementation of a global approach

to the method of constraints."

In this present research a global approach to the method of constraints is developed.

In recent years, a body of literature has existed developing the edge elernents

[25. 26. 27: 28. 29. 30. 3 1 : :32, 33. :34], which solve for the field values at t,he edge

of elements instead of at the node of elements. Edge elernents have an advantage

over node-based elernent S. Xode- based fini te elements over-const rain the cont inui ty

of the normal component of the field vector at materiai interfaces. Therefore. material

parameters are restricted; eit her perrnittivity or permeabili ty should be constant in

al1 regions. However. in edge elernents? the tàngential components of the field vector

are continuous while the normal components rnay be discontinuous. Thus. there is no

restriction on the discontinuity of either permittivity or perrneability of the medium.

Another advantage of edge elements is that they can eliminate spurioiis modes of

non-zero eigenvalues. In [35], Cendes et al. considered that solutions of the vector

wave equation rnust either be static (the eigenvalue k = O, static field forms the nul1

space of the curl operator), or be associated with a solenoidal flux. They stated that:

"The early thinking about the cause of spurious modes in finite-element

methods is wrong. The true cause of spurious modes is the incorrect ap-

proximation of the null space of the curl operator."

In another paper by Cendes et al. [30]: the authors state that:

"Provided the finite-element trial functions are able to approximate

the null space of the curl operator correctly. the eigenvalue k = O will be

computed exactly and it is only necessary to ignore these zero solutions.-'

Their main idea is that. because of the poor approximation of the nul1 space of t h e

curl operator in n o d ~ b a s e d finite elements. the zero eigenvalues (static solutions) are

approximated by Large numbers which f o m the spurious modes. However. the static

solutions are not physical solutions. Only those associated with a solenoidal flus are

physicai modes. Therefore, it is not wrong to attribute the spurious mode problem

to a deficiency in imposing the solenoidal nature of the field in the approximation

process. Furtlierrnore. it will be shown in this thesis that the static solutions can

also be eliminated using the method of constraints. The drawback of edge elements

is that more unknowns are produced for a given number of elements in cornparison

with node-based grids. Therefore. the size of the matris to be solved is large.

In this present research, based on [XI, a global approach to the method of con-

straints is developed. Only those modes associated with solenoidal flux and corre-

sponding to the physical modes are left.

1.3 Thesis Objective

The main objective of this thesis is to eiiminate the spurious modes appearing in the

vectorial finite element analysis of waveguides by using the method of constraints.

which is a valid, economical and easiiy implementable method.

-4s mentioned in the li terature review. several explanat ions for the appearance of

spiirious modes and various suggested methods to remove the nonphysical solutions

have been proposed during the past fifteen years. However. much room is still left

in the sense of solving the problem a t the least cost. This research esarnined the

possible reasons for explaining the nonphysical solutions and the methods to identify

the spurious modes. Difficulties arise frcm the spurious standing wave solution which

does not satisfy the divergence free condition. In addition. spurious modes are excited

whenever the proper bouildary conditions are not satisfied.

The method of constraints is a brute force approach to impose the non-divergence

constraint and proper boundary conditions on the node-based finite element matris

equation. It is also referred to as the method of reduction since the size of matrices

in the eigenvalue equation to be solved are reduced to roughly two-thirds. However.

the reduced matrices are more dense than the original matrices.

Based on [31], in which the method of constraints was demonstrated to be valid

for single finite element models and it is also evident that a global approach to this

method must be developed, the rnethod of constraints h a . been implemented for

multi-element models to analyze waveguides wit h arbi trary boundaries in t his re-

search. The applications considered have been homogeneous/ inhomogeneous isotropie

waveguides and magnetically anisotropic waveguide.

1.4 Thesis Outline

This dissertation contains sis chapters which are out lined below:

Chapter 2:

The vectorial finite element method is reviewed. A vector wave equation called

t h e curl-curl equation is derived from Maxwell's equations. The boundary and in-

terface conditions associated with physical problems are treated. The mat hemat ical

representat ions of traveling waves are given. Variat ionaily stat ionary funct ionais are

derived for specific wave types in planar two-dimensional geometries. .A finite element

formulation is generated by minirnizing the discretized functional.

Chapter 3:

The appearance of spurious modes is shoivn. Reasons for the existence of the

spurious modes are presented. Several methods to identify the spurious modes are

illustrated with an exarnple problem.

Chapter 4:

The rnethod of constraints is implemented . The boundary conditions are derived

for various specific shapes of both perfect electric conductor boundaries and perfect

magnetic conductor boundaries. The expressions of non-divergence constraint are

given for both magnetically isotropie and anisotropic waveguides wi t h a constant

structure in the z-direction. h finite element formulation of the non-divergence con-

dition is derived. Both boundary conditions and the non-divergence constraint are

enforced on the finite element matrix equat ion.

Chapter 5:

A computer program, which implements the vectorial finite elernent method and

uses the method of constraints to remove the spurious modes. is presented.

Chapter 6:

.A homogeneous waveguide is solved for demonstrative purposes. More com-

plicated problems such as an inhomogeneous isotropic waveguide. a magnetically

anisotropic waveguide. circular and coaxial waveguides are also solved in order to

illustrate potential app tications. Analysis of the results is carried oiit.

Chapter 7:

The contributions of this thesis are stated. The limitations of t he method are also

mentioned. The recommendations for future work are given.

Chapter 2

Vectorial Finite Element Met hod

2.1 Maxwell's Equat ions

The behaviour of electromagnetic fields is governed by Maxwell's equations. Maxwell's

equat ions for t ime- harmonic fields are expressed as

V x E = - jwB

V x H = j u D + J

v - B = O

V - D = p

where E and H are the electric and magnetic field intensities. D and B are the electric

and magnetic flux densities, J is current density and includes impressed ciment Ji

and induced conduction current J,, p is the electric charge densi ty.

The constitutive relationships for anisotropic media are

where ji, 2, & are the tensor permeability. tensor permitt ivity and tensor conduct ivi ty

of the media, respectively.

2.2 Vector Wave Equation

Maxwell's equations can be cast into a

treatment . The vector wave equat ion

tion. The derivation is kept as general

anisotropic media.

form which lends itself easily to variational

derived is referred to as the curlcurl equa-

as possible and it includes the treatment of

Substituting equations (2.5) through ('2.7) into equations (2.1) and (2.2) respec-

tively and then taking curl of both sides give

The substitution for V x H in (2.5) and V x E in (2.9) from equations ( 2 . 2 ) and

(2.1 ), respect ive15 leads to

For nonconductive media, where 5 is identically zero, equations (2.10) and ('7.11)

reduce to

Table 2.1: Relations between (P. 4) and material tensors against v.

For source-free problems, where Ji is zero, the above equations can be written as

Equations (2.14) and (2.15) are special cases of the following general equation de-

pending on the interpretation of f i , e, v.

where v denotes either E or H, and 6, are the material tensors as shown in Table

2.3 Funct ional Formulation

An energy functional associated with the operator of equation (2.16) is given by

F ( v ) = < v x @ - I V X V ? V > - w ~ <&?v> (2.17)

Recall the definition of inner product

where the asterisk denotes complex conjugate, hence the functional in (2.17) can be

rewritten as

Consider the following vector identity

Integrating both sides and then applying the divergence theorem to the left-hand side

yields

Thus (2.19) is converted into

F(v) = J[(v x v') 6-' (V x V ) - W ~ V = &]dR

J

Let v represent E, then

The cross product En x H is the complex

/[E* x (B-lV x E)] nds (2.23)

-jw (E' x H) dS f (2.24)

Poynt ing vector represent ing the densi ty

of power flux. The surface integral in (2.22) therefore represents the net power flow

across the boundary surface. If the boundary is a perfect conductor, then no energy

is transferred and there would be no surface integral term included in (2.22). The

functional may be written as

F ( v ) = J[(v x v*) $-'(v x v) - w2v'pv]dR

2.4 Mat hemat ical Representat ion of Traveling Waves

It is very important to realize that the character of a wave that can be suppcrted in

any given medium depends upon the form of the material property tensors descri bing

the medium.

Assuming that in the z-direction the wave varies as e x p ( - j 3 z ) . where 3 is the

propagation constant. then only traveling waves of the form

may exist in media characterized by perrneability and permittivity tensors of the

following form

respectively. These tensors are characteristic of transversely magnetized ferromag-

netic materials and plasma. The magnetization may be in any direction. but restcicted

to the x-y plane. The waves described by (2.39) and (2.40) are linearly polarized with

respect to the transverse plane. Ooly this type of propagating wave will be considered

in the following.

2.5 Finite Element Formulation

By substituting a vector Function v(x? y. z ) of the form (2.26) or (2.27) and tensor p

and @ of the form (2.25) and (2.29) into the functional ( 2 2 5 ) . a particular functional

for traveling waves is derived. By substituting the polynomial approximations of

the field into the functional. a matrix expression is obtained which is the discretized

equivalent of the original functional. Finally. by rninimizing the discret ized funct ional.

a matrix equation is generated as

[St] [vt] - ~ ~ [ T t l [vt] = 0

where the column matrix [v] is given by

[vt] =

Equation (2.30) is an eigenvalue equation. The eigenvalues of the equation are d'

and the eigenvectors are the nodal values of either the magnetic or the electric field

intensity vectors depending on the choice of j. 4 and the boundary conditions. For

every value of the propagation constant 9' an eigenvalue - eigenvector spectrum set

can be obtained. The size of the coefficient matrices [St] and [Tt]: for an assembly of

triangles with a total of nt interpolation nodes. will be 3n, x 3nt. It will be seen that

the size of the coefficient matrices is reduced by more than one-third when boundary

conditions and non-divergence constraint are imposed on the finite element equation

(2.30).

2.6 Interface Condit ions

A t interfaces between two elements, if the electric field E is solved. E should satisfy

the condit ion t hat its tangential component is cont inuous

where n is a surface normal unit vector. Additionall- the electric flux density D

should satisfy the condition that its normal component is continuous

If the magnetic field H is solved, the corresponding interface conditions are

n x H 1 = n x H 2 (2.;35)

n -p lHi = ne p2H~ (2.36)

Therefore. in general, the interface continuity requires that

n x (vl - v2) = O

n . (tj1v1 - &vZ) = O

which means, in node-based h i t e elements, eit her permittivity O

( 2 3 7 )

(2.:35)

r pemeability should

be constant and both normal and tangential components of the desired field should

be continuous.

2.7 Boundary Conditions

Let v represent H. Consider the boundary shown in Figure 2.1. Part L is the inside

of the computed structure and part 2 is the boundary wall.

Figure 2.1: An arbitrary boundary

For perfect electric conductor boundaries. according to the general boundary con-

dition between two materials. we have

where B,, Ht and k represent the normal component of magnetic flux density. the

tangential component of magnetic field intensity and linear density of surface current.

Additionally, in the perfect electric conductor, we have

If part 1 is filled with magnetically isotropic materials.

rvhere is perneability tensor without any nonzero non-diagonal elemnets.

Therefore. we can obtain

H*, = O

which means H is tangential to the boundary.

If the ~vaveguide is filled wi t h rnagnet icaily anisot ro pic materials. t hen

where is a permeability tensor with nonzero non-diagonal elements. According to

Section 2.4. has the form

Hence.

Therefore, from BnI = O, we can not clerive Hn1 = O. as in magnetically isotropic

waveguide. For erample, if the boundary is parallel to y-axis or BnI = B I , = O. we

have

HnI(= HI,) is not necessarily zero. So for magnetically anisotropic waveguide prob-

lem. the boundary condition is more complicated.

From (2.48), we can obtain

Biy(x. y. z ) = p r y H l ~ j x . ~ . -) + pyyHtg(x= y. =) + jpryHiz( .~ . y. =) (?..X )

Assuming that. at a boundary point p in Figure 2.1. the dope of the outward normal

vector is a. Tlius,

B n i = Bi, + Bi,,

= Bl,cosa + Bl,sina = O

Substituting (2.50) and (2.51) into (3.52). we have

From (3.27). we can derive

Substituting (2.54) to (2.56) into (2.53) and simplifying the equation. we have

where Hl,(x, y), Hl,(x, y) and H L , ( x . y) are the desired variables in the finite element

formulation. So (2.57) can be further sirnplified as

(p,,cosa + pyrsina)HL, + (p,cosa + ,u,sincr)Hl, + (p,,coscr + ,u,z.sina)Hiz = O

(2258)

This is the boundary condition on elect ric wall for magnet ically anisot ropic \va\-eguide.

In this research, the symmetry plane of the waveguide is treated as perfect mag-

netic wall. For the magnetic wall. we have

Bit = O

For magnet ically isot ropic waveguide,

where c l is a pemeability tensor without any nonzero non-diagonal elements.

Thus

T herefore.

which means Hl is normal to the boundary.

For magnet ically anisot ropic waveguide,

where ci is a tensor with nonzero non-diagonal elements. According to a similar

derivat ion, we can obtain the boundary condition on magnetic wall for magnetically

anisotropic waveguide as

where a is the dope of the outward normal vector of the boundary.

2.8 Summary

This chapter provides the necessary background introduction for the three cornpo-

nent vectorial finite elernent formulation of an arbitrary shaped waveguide filled wit h

iossless dielectrics. This finite element formulation is the bais of this research. In

chapter 3 , the spurious modes yielded From solving t his formulation will be s hown

and exarnined. In chapter 4. the proposed rnethod to eliminate these spurious modes

will be implemented based on this finite element formulation.

Chapter 3

Spurious Modes

3.1 The Appearance of Spurious Modes

.i t hree-component magnet ic field vector fini te element program was wri t ten by Iion-

rad [4]. This program assembles the matrices [Ç,] and [Tt] in eqiiation (2.30) for a

collection of triangles in the x-y plane and implements the triangular finite element

met hod for linearly polarized traveling waves.

Consider the hollow rectanguiar waveguide shown in Figure 3.1. The region has

been subdivided into eight third-order triangular elements. The eigenvalttes k ob-

tained from the program are plotted in Figure 3.2 as a function of the propagation

constant 0. In Figure 3.3. the results are replotted as d / k versiis k. .ifter comparing

with analytical solutions, we find that

(1) modes 1 to 51 (part A ) have no correspondence to physical modes.

(3) modes -57 and 61 (part B) are also not included in physical modes.

which means that modes 1 to 51. 57 and 61 are spurious modes. We can see that

spurious modes c m be divided into two parts. In part A, spurious modes are below al1

Figure 3.1 : Element subdivision for an empty rectangular waveguide involving eight

third-order triangular elements.

the physical modes. On the k - /3 diagram, t hey are the lines starting from the origin

and below the Air Line (straight iine at 45 degrees wit h respect to the fl axis). On the

p/k - k diagram, they are the curves above the Air Line (straight line of 3 / k = 1).

For the homogeneous problem, this part of the spurious modes are separate from the

physical modes. So it is easy to distinguish them from physical modes. In part B.

spurious modes are mixed with physical modes. It is difficult to distinguish them

from true modes.

Now consider an inhomogeneous waveguide problem. Figure 3.1 shows a rect-

angular waveguide half-filled with a dielectric material of relative permittivity 2.15.

Figure 3.5 shows the mesh of ( N p , Ne ) = ( 4 9 4 , which means that there are 49 in-

terpolation points and 8 elements. The corresponding k - ,d and P / k - k diagrams

appear in Figure 3.6 and Figure 3.7, respectively. Spurious modes appear in this

problem as they had before in the homogeneous waveguide problem. By using the

methods introduced in Section 3.3, the spurious modes can be distinguished from

true modes. In Figure 3.6 and Figure 3.7, spurious modes are indicated using 'x',

Figure 3.3: k - ,d diagram for the empty rectangular waveguide problem.

Figure 3.3: B/k - k diagram for the empty rectangular waveguide problem.

Figure 3.4: Cross section for a rectangular waveguide half-filled wi th a dielrctric

material of relative permittivity 2.43.

Figure 3.5: Mesh for the waveguide shown in Figure 3.3.

Figure 3.6: k - B diagram for the rectangular waveguide half-filled with a dielectric

material of relative permittivity 3.15.

Figure 3.7: k - P diagram for the rectangular waveguide half-filled with a dielectric

material of relative permittivity 2.15.

while physical modes by '0'. Like in the case of the homogeneous waveguide. spurious

modes can also be divided into two parts. Part A is from modes 1 to 53 and part B

is modes 60 and 62. In the inhomogeneous case. even some spurious modes in part -4

are mixed with physical modes.The k - L3 diagram and $/k - k diagram show that

curves No. 36 through 55 intersect with curves No. 56 and above. These intersections

result in a change in the sequence in which the eigenvalues appear for different values

of 0. This phenornenon can only occur in inhomogeneous waveguides since the curves

intersect only in the region between the air and the dielectric lines.

From the above two examples. we have seen how spurious modes appear in the

computational results of the vectorial finite element program. In the case of the

homogeneous waveguide. one kind of spurious modes are mixed with physicai modes.

In the inhornogeneous case. both kinds of spurious modes are mixed wit h true modes.

As one increases the mesh to improve the accuracy of the solution. the number of

spurious solutions also increases.

3.2 Explanation for the Appearance of Spurious

Modes

Different explanations for the appearance of spurious modes motivate the ernergence

of different methods to eliminate the nonphysicai solutions.

One reason is believed to be that boundary conditions are not satisfied. In section

2.C1 the boundary conditions for both electric wali and magnetic wall of isotropic and

anisotropic waveguide are derived. These boundary conditions are principal bound-

ary conditions. which are not satisfied automatically. Not enforcing t hese boundary

conditions or fading to enforce them properly are considered to cause some of the

spurious modes, which can be shown from the vector diagram of the field.

Different views about another reason, ais0 the most important reason. result in

the appearance of mainly two different categories of methods to solve the problern.

Consider the curlcurl equation derived in section 2.2:

Take the divergence of both sides of equation (3.1 ). Since the divergence of the ciirl

of any vector is identically zero, it follows that

Hence, either w = O, which is static solution, or V - B = 0, where B is the magnetic

flux density, B = GH.

The reseàrchers represented by Ibnrad considered that other spurious modes are

caused by solving equation (3.1) alone, wit hout explicitly enforcing the solenoidal

nature of flux and atternpted to find methods for imposing the non-divergence of flux

in the finite-element met hod. Ot her researchers considered t hat the zero-divergence

of the flux is built into equation (3.1) for al1 but static solutions. So they look to

static solutions of equation (3.1) for the source of spurious modes. Associated with

these static solutions is a potential function. o. satisfying the equation

Static magnetic fields form the nul1 space of the curl operator. since V x H = -V x

V+ = O. Substituting equation (3.3) into equation ( 3 4 shows that magnetic fields

derived frorn equation (3.3) are eigenvectors of the curlcurl equation. with eigenvalue

zero:

In node-based finite elements. the scalar potential. 4. can b e approrimated by

where r ] ; ( + . y ) , i = L, - * - . !V represent the set of ?j linearly independent basis functions.

and ai is the value of scalar potential d at the i-th interpolation point. Equation ( 3 . 3 )

states t hat O exists such that its negative gradient equals H. For t his to be possible

in this case, d must have continuous x. y and z derivatives. called CL continuity.

However, reference [37] shows that CL piecewise polynomials. in generai. do not exist

over an arbitrary mesh. On arbitrary meshes, the approximation of the nuil space is

very poor, and the zero e igendues are approximated by large numbers. which form

the spurious modes. Quoting from [30], "Provided the finite-element triai functions

are able to approximate the nullspace of the curl operator correctly. the eigenvalue

k = O will be computed exactly and it is only necessary to ignore these zero solutions."

Consequent ly, the edge element met hod was developed in w hich the approximation

for H is such that its tangential components are continuous. but its normal compo-

nents are discontinuous. In edge elements, CL continuity of o can be satisfied. Some

researchers holding the second view consider the first view wrong [33]. In my opinion.

it is not wrong. Static solutions, whether calculated exactly or not. are just mathe-

matical solutions, not physical solutions, so the enforcement of dzvB will also remove

al1 these static solutions and only the physical modes remain.

In this thesis, two reasons are believed to be responsi ble for spurious modes. One

is that proper boundary conditions are not satisfied. The other is that dicB = O is not

satisfied. When the permeability tensor has no nonzero non-diagonal elements. or

say in the case of isotropic waveguide problem, divB = O can be written as dicH = 0.

3.3 Met hods of Ident ifying Spurious Modes

As shown in Section 3.1, spurious solutions are found to spread al1 over the eigen-

value spectrum, sorne of them appearing below any true mode and some between the

physical modes. Identifying spurious modes from eigenvalue solutions is the basis of

further investigating and solving the spurious mode problem. In this section. two

ways to distinguish between spurious modes and physical modes are given.

One way is to plot and observe the eigenvectors of modes. In the case of a

nonphysical mode, the fields vary in an unreasonable. sometimes random fashion

over the waveguide cross section. Both the contour of one field component (i.e. H,

or Hy field contour) and the vector diagram can provide a good test. Consider the

homogeneous waveguide shown in Section 3 . 1 . When the mesh division is ( ?Y,. Ne) =

(81,s) and propagation constant /3 = 1.0, vector diagrams of the 90th. 92nd and 93rd

eigenmodes are plotted in Figure 3.8, 3.9 and 3.10. respectively. The corresponding

H, and H, field contours for these three modes are plotted in Figure 3.1 1. 3.12 and

3.13, respectively. Obviously, for the 90th and 93rd modes. the Hz and Hg spatial

variations and t heir vector combinations are reasonable and consistent. but not for

the 92nd mode. Therefore, the 90th and 93rd modes are physical and the 92nd mode

is spurious.

Now consider the inhomogeneous ivaveguide shown in Figure 3.4 as another ex-

ample. When the mesh division is (X,,, Xe) = (81,s) and the propagation constant

@ = 1.0, vector diagrarns of 12 modes are plotted in Figures 3.14 through 3.25. These

modes begin with the 76th mode which is the dominant mode. From the vector di-

agrams, obviously, the conclusion is that the 76th, 8-jth, S6th and 87th modes are

physical, while the 77th through 84th modes are spurious.

Figure 3.S: H field vector for the 90th mode in the homogeneous waveguide with

mesh division (Y,, N p ) = (8'81)

Figure 3.9: H field vector for the 92nd mode in the homogeneous waveguide with

mesh division (ive' N p ) = (Y. YI)

Figure 3.10: H field vector for the 93rd mode in the homogeneous waveguide with

mesh division ( N e , IV,,) = (8,81)

Figure 3.11: Contour lines for Hz and Hy for the 90th mode of the homogeneous

waveguide with mesh division (Ney N p ) = (8.81)

Figure 3.12: Contour Lines for H, and H, for the 9Znd mode of the hornogeneous - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - -

waveguide wit h mes h division (1%. Y,) = (S. 8 1 )

Figure 3.13: Contour lines for Hz and H, for the 93rd mode of the homogeneous

waveguide with mesh division ( N e , IV,) = (8, Y 1 )

Figure 3.13: H field vector for the 76th mode in the inhomogeneous waveguide with

mesh division (Ne! Y,) = (Y. 81)

Figure 3.1.5: H field vector for the 77th mode in the inhomogeneous waveguide with

Figure 3.16: H field vector for the 78th mode in the inhomogeneous waveguide with

mesh division ( N e , N p ) = (S,81)

Figure 3.17: H field vector for the 79th mode in the inhomogeneous waveguide with

mesh division (!K. N p ) = (Y. Y 1)

Figure 4.18: H field vector for the 80th mode in the inhomogeneous waveguide ivith

mesh division (!V,: :V,) = (Y. 81)

Figure 3.19: H field vector for the Slst mode in the inhomogeneous waveguide wi th

mesh division ( N e , N p ) = (8, Y 1)

Spurious modes can also be checked out by calculating divergence of the solution

with mesh refinements. di.H for a spurious mode is quite high compared with that

of a physical eigenmode. The reason for this phenomenon has been explained in

Section 3.3. This criterion has also been successfully used to filter out most of the

spurious solutions. Consider the same inhomogeneous waveguide with mesh division

of ( Np, Ne ) = (Y 1. Y) . divH of every interpolation point for modes are calculated.

Geometric means of divH of al1 points are aiso computed for modes. When the

propagation constant 0 = L.O. the geometric means of divH for the same L2 modes

as in the first method are shown in Table :3.1. From Table 3.1. we can see that diuH

of the 76th. 85th. 86th and 87th modes is much smaller cornpared with that of the

-c

r 4 th to 84th modes. Therefore. the same conclusion can be reached as by the first

method.

Rather t h m calculating the spectrum and eigenvectors and tlien filter out. a

posteriori. any spurious modes. they should be removed. a priori.

3.4 Summary

In this chapter, the appearance of spurious modes is shown. Explanations for t h e

emergence of spurious solutions are given. Methods of identifying nonphpical so-

lutions are examined. Based on these detailed examinations of the spurious mode

probIern, a met hod referred to as the "method of constraints' will be applied to solve

the problem in chapter 1.

1 Mode Xo. 1 Eigenvalue k 1 divH 1

Table 3.1: The geometric means of divH of a11 interpolation points for L2 modes in

the inhomogeneous waveguide shown in Figure 3.4. 0 = [.Ol ( N p . :V,) = (31.8)

Chapter 4

The Method of Constraints

4.1 Description of the Method of Constraints

To describe the Method of Constraints clearly. the following three questions are an-

swered.

1. What is the Method of Constraints?

The method of constraints is an approach based on the idea of a posteriori impo-

si tion of a set of constraints on the finite element matrir equation. derived in Chapter

2. in order to eiiminate the spurious modes appearing in the vectorial finite element

solution of waveguides.

3. What are the constraints to be enforced?

In Section 3.2, it is stated that the spurious modes appear because of two reasons:

(1) Boundary conditions are not satisfied.

(2) Non-divergence condition dzvB = O is not satisfied.

Accordingly, to remove spurious solutions, the two corresponding constraints to be

imposed are:

(1) Proper boundary conditions which are expressed in a form which is easy to be

implemented.

(3) divB = 0: where H can substitute for B if permeability tensor tias no

non-diagonal elements.

3. Why is this method chosen to be the object of study:'

Compared with many methods which are used to solve the spurious mode problem.

this method has some advantages. One is that this method is a very direct method

which is easily understand and implemented. Another is that this method is an

economical rnethod. In fact it is aiso referred to as the method of reduction because

a beneficial consequence of the procedure is the reduction of global finite element

matrix size. The matrices in the eigenvalue equation to be solved are reduced by

roughly one third. Meanwhile, the symmetry of the matrices remains. Therefore.

this method is a good choice for solving the spurious mode problem.

The Method of Constraints is not new. Iionrad put forward thjs method more

than ten years ago [3Y] and Hayata et ai. also used a similar method [2]. Konrad

showed that this technique worked well on single and two element rnodels but was

problematic when applied to multi-element models. Thus. there is much room left

for research into the implemention of this method. In this research. the method is

im plemented for general multi-element grids and arbitrary boundaries. which was not

shown in [El.

In the following sections of this chapter, it will be shown in detail how to implement

t hese constraints. The magnetic field intensity H is taken as the desired variable. .A

parallel treatment is possible in terms of the electric field intensity E.

4.2 Implementation of Boundary Condit ions

In this research. the boundary of the ivaveguide is a perfect electric conductor wall.

Symmetry planes. which can be exploited to reduce the matrix size. act like a perfect

magnetic conductor wall in electromagnet ics. Therefore. the boundaries here are

made up of bot h perfect electric conductor boundaries and perfect magnet ic conductor

boundaries. For both cases. the boundary conditions are given in Section 2.7. The

boundary conditions for both will be implemented. separately. in the following two

subsections.

4.2.1 Implementation of Boundary Conditions for Perfect

Electric Conductor WalIs

.As shown in Section 2.7. the boundary condit,ion for the perfect electric cond~tctor

wall of the isotropic waveguide is

which means the magnetic field is tangential to the boundary. or put anotlier w-.

the normal component of H at boundaries must be zero. The boundary condition

may be expressed in t e m s of Hz, Ky and Hz at the interpolation points on t he

boundaries, and can be easily enforced on the finite element matrices. To facilitate

the implementation. the interpolation points are classified into 5 cases.

Casel: Point on a boundary parallel to the y-axis

V

Figure 4.1: Point on a boundary parailel to the y-axis

Since the normal component of H at the boundary must be zero.

Case?: Point on a boundary parailel to the x-axis

Y

Figure 4.2: Point on a boundary parallel to the x-axis

In this case,

Case3 Point on a sloping boundary

Hn -- normal component Ht - tangential component ol - sloping angle of boundary

Hxn a O

Since the normal

Because

Then

Figure 4.3: Point on a sloping boundary

component of H at the boundary must be zero.

sinû cosa

From Figure 4.3. it can be seen that when the slope of the boundary is greater than

O, Hz and Ky have the same signs. otherwise they have opposite signs. Therefore.

where k(= tana) is the siope of the boundary.

O

Figure 4.4: "True' corner point

Case4: "True" corner point

For this case.

for obvious reason.

Cases: jFalseY corner point due to the finite element approximation of smooth

surfaces

Figure 4.5: "FaIse" corner point due to the finite element approximation of smooth

surfaces

In Figure 1.5, line 1 is the tangent of the boundary curve at the interpolation point

p. y is the sloping angle of line 1. Lines l1 and l2 are two lines which approximate

the curve surface. a and /3 are the sloping angles of line I l and 1 2 , respectively. n. ni

and n2 are the outer normal vectors of line 1. l 1 and 12, respectively. Here we assume

that line I I and line l2 are symmetric with respect to line 1 so that the angle between

nl and n and that between n and n2 are equal. Mie have

As in Case 3. to make the normal component of H at point p zero. we have

szn y Hy = -

cosy Hz

Substituting (4.10) into the above equation. we have

a+a COS -5-

H, = - a+d sin

HZ

According to trigonometry. we can rewrite the above equation as

cosa+cosd where k = - sina+3,nLI.

Equations (a.-), (1.3), ( l . ï ) , (4.9) and (4.13) are boundary conditions in terms of

Hz and Ky for 5 kinds of interpolation points. respectively. These conditions can be

easily imposed on the finite element matrices of the matrix eigenvalue equation. For

H, = O or H,, = O, we just need to remove the row and column corresponding to Hz or

H, in the finite element matrices. For H, = k H z , we multiply the H,-corresponding

row and column by factor k, add this changed H,-corresponding row and column to

the Kr-corresponding row and column, and t hen rernove the H,-corresponding row

and colunin.

The boundary condition for a perfect eiectric wall of an anisotropic waveguide is.

as shown in Section 2.7,

where a is the dope of the outward normal vector of the houndary.

or

Hz = k l H z + k2Hy

w here

The implementation of this boundary condition is similar CO that of (4.8). First we

rnultiply the Hz-corresponding row and column by factor kI and add them to the

Hz-corresponding row and column. Xext . we rnultiply the Hz-corresponding row and

column by factor k2 and add them to the H,-corresponding row and column. And

then remove the Hz-corresponding row and column.

We can see that the order of finite element matrices will be reduced hy 1 whenever

a boundary condition equation is enforced. In the post-processing of t he results. the

H,s and H,s corresponding to the removed rows and columns of the finite element

matrices will be calculated.

4.2.2 Implementation of Boundary Conditions for Perfect

Magnetic Conductor Walls

It is shown in Section 2.7 that the boundary condition for a perfect magnetic con-

ductor wall of an isotropie waveguide can be expressed as

which means that the magnetic field is normal to the boundaries, or put another way.

the tangential components of H at boundaries must be zero. In the same way a s

what is done to perîect electric conductor boundaries? magnetic boundary points are

divided into five cases and the boundary condition is therefore expressed in terms of

Hz. Hy and Hz at the boundary points.

Case 1: Point on a boundary parallel to the y-axis

Figure 4.6: Point on a boundary parallel to the y-axis

Case 2: Point on a boundary parallel to the x-axis

49

Figure 1.7: Point on a boundary parallel to the x-axis

H, = Hz = O

Case 3: Point on a sloping boundary

Y 4 Hn - nomai component

Ht - tangential component

a - sloping angle of boundmy

Figure 4.8: Point on a sloping boundary

To ensure thot the tongential component of H at the boundary is zero.

I&tI = IHytl

Since 1 f f d I = I c o s Q H ~ ~ ~ IHytl = IsznaHyl ,

IcosaHzl = IsinaHyl

sina COSQ'

From Figure 4.8, we can see that when the dope of the boundary is greater than O.

Hz and Ky have opposite signs. otherwise. they have the same signs. .lccordingly.

H, = kH,

where k = -tana. the negative of the slope of the boundary. .\dditionally.

Hz = O

Case 3: "Truey corner point

Figure 1.9: "True7 corner point

Case5: "False' corner point due to the finite elernent approximation of srnooth

surfaces

Hxt n l t \ Y Y!

Figure 4.10: 'False" corner point due to the finite elemeot approximation of smoot h

surfaces

With the same assumption as in Case 5 of Subsection 4.2.1 and a similar derivation.

tve obtain cosû + cos3

H f = sina + szno H Y

cos of cos^ .,d where k = linoi+sinLI - H: = O (4.l31)

Equations (4.20), (4.21), (3.25) and (1.27). (4.28): (4.29) and (C31) are boundary

conditions in terrns of Hz, H, and HI of 5 kinds of interpolation points. respectively.

For a magnetic wall of an anisotropic waveguide. the boundary condition is

where a is the slope of the outward normal vector of the boundary. The above

condition can also be expressed as

Hz = k l H z + k2Hy

where

The method of implernenting the boundary cooditions on magnetic wall is the

sarne as that on electric wall. Compared to the boundary cooditions for an electric

wall. one more condition is added for every magnetic boundary point of isotropie

waveguide. i.e. Hz = 0.

Before enforcing the boundary conditions. the fini te element ma t r i x equation

(2.30) is expressed. in terms of H. as

After enforcing the boundary conditions. the above equation becomes

where [s~'] and [T~ ' ] differ in size frorn [S;] and [Tt]- respectively.

4.3 Implementat ion of Non-Divergence Constraint

As shown in Section 3.3, it is considered that the nontrivial solenoidal nature of the

flux is responsible for the rest of the spurious solutions after boundary conditions are

imposed. Therefore. i t is believed t hat . by enforcing the non-divergence coost raint

and accordingly further reducing the finite element matrix size. al1 the spurious modes

wiIl be elirninated.

The non-divergence constraint for the magnetic field is given by

If tensor ii has no non-diagonal elements. i.e. for isotropic waveguide. (-1.38) can be

rewritten as

V - H = O

In the Cartesian coordinate system. (4.39) can be expressed as

-4s shown in Section 2.4. traveling waves can be represented as

The k terms in the above a i s e from the fact that the z vector component may either

lead or lag the other two components by a time phase of $ radians. Therefore. we

have

By substituting (4.42). (4.43) and (4.44) into (1.40) and removing the e x p ( j u t ) in

every term. we obtain

(4.43) can be further simplified as

where H,(z- y), H,(I. y) and H , ( x . y) are the desired variables in the finite element

equation and appear as Hz' H, and Hz in the following expressions.

By using the Galerkin procedure on (4.46) and adding the contribution of al1

different elements for the region 0. we can obtain a compact rnatrix equation form a s

where the (i.j)-th element of [Dr] and [Dy] are given by the x- and y- derivatives.

respectively. of the j-th interpolation polynomial evaluated at the i-t h interpolation

node as foliows

da @ i . j ) = - . . 1 i Y

k = x, y. l . J = 1:-- ..v dl;

in which [Il is the identity matrix, and

The detailed derivation of matrices [D,] and [Dy] will be shown in Appendix .A.

For anisotropic waveguide. pemeability tensor p has non-diagonal elements and is

expressed as

Substituting (4.42), (4.43) and (4.14) into above equation and simplifing it. we can

obtain

By using the same Galerkin procedure, we can obtain the same matrix equations

(4.47) and (4.18) from the above equation.

Since the boundary conditions have been enforced in the finite element matrix

equation and the desired variables are [H'] in (1.37) instead of [Hl. the same boundary

conditions must also be enforced on the non-divergence constraint equations (4.47)

and (4.48). The method of implementing boundary conditions on the finite elernent

mat rix equation is somewhat diflerent from t hat of implement ing boundary conditions

on the non-divergence const raint equat ion. In t h e former case. as s tated in su bsect ions

4.2.1 and 4 -22 , both row and column treatments are involved to keep finite element

matrices [Si] and [TI] symmetric. In the latter case. only column treatment is involved.

For Hz = O or H, = O. just the column corresponding to Hz or H, is removed. For

H, = kHz or N, = kH,. we rnultiply the H,- or Hz- corresponding columns of [D,,;]

by the factor k. add the changed H,- or Hz- corresponding columns to the Hz- or

H,- corresponding columns. and t hen remove the H,- or Hz- corresponding colurnns.

After enforcing the boundary conditions. the non-divergence rnatrix equation (4.4s)

can be expressed as

[ ~ : , z l [ ~ ' l = 0

where [H'] is the same as in the finite element matrix equation (4.37). -4ssuming

that there are n miables in [H'], the size of [D:,,] is m x n, where rn is equal to the

number of interpolation points N.

LVe assume t hat

where r is less than or equal to m. Thus, there are r independent variables and

n-r dependent variables in matrix equatioo (4.60). which means r variables can be

expressed in terms of the other n-r variables. Therefore. in the finite element matrix

equation (4.37). we can Say that there are r dependent variables and n-r independent

variables, which are ex~ressed as [H;] and [HJ' respectively.

We need to find a set of r dependent variables and the corresponding n-r indepen-

dent variables. To obtain this, two steps are taken:

Step i: Choose r rows from [D:~=] to form a new matrix. whose rank is still r. The

new rnatrix is still expressed by [D:,;]. the size of which is oow r x n. instead of

m x n.

Step 2: Choose r columns from [D:~,] to form rnatrix [D:,,~]. whose rank is r and size

is r x r. The remaining columns of [D:,,] form matrix [D:,,,]. whose size is r x ( n - r ) .

Change the positions of columns in (D:,_] to obtain anot her matrix [[D:,,,] [~:,,,]h Meanwhile. change the position of the corresponding variables in [H'] to form another

vecior { i:i 1. where [H:] itmcis for independent variable vector and [H;] stands

for dependent variable vector. Therefore. we have

Thus we can further obtain

where the dependent variables are expressed in terms of the independent variables in

matrix form.

The process of finding a set of r dependent variables and the corresponding n-r

independent variables can be further illustrated with a simple numerical example.

Consider a hollow square waveguide with two second-order triangular elements as

shown in Figure 1.1 1. The matrix form of the non-divergence constraint. at cutoff.

Figure 1.11: The cross-section of a hollow square waveguide with two second-order

t riangular elernents.

for this problem is

where

Enforcing the following boundary conditions,

H,, = O . H I , = 0 .

H2= = O. = O.

H3= = O. H3y = O.

H4, = O . H4, = O .

Hs, = o.

Hsy = 0,

HSy = O ?

Hg, = o.

the matrix forrn of the non-divergence coostraint becornes

[DL,] [H'] = O

w here

Since rank[D&] = 5 . there are 5 dependent variables and 1 independent variables in

[H']. In order to find a set of 5 dependent variables. tivo steps are taken:

Step 1: Choose .5 rows from [DA,] to form a new [D:,;]. whose rank is still 3. We get

Step 2 Choose 5 colurnns from the above [D:.~],,, to form rnatrix whose

rank is still 5. The remaining columns of [DL,,] form [D~,,~]. W e obtain

Then change the position of columns in [D:.,],., to obtain [[D~.~,][D~,~,]]. In the

meantirne. change the position of the corresponding variables in [ H ' ] to form 1 i:; 1. Therefore. the matrix form of non-divergence constraint becomes

which is the relationship between independent and dependent variables.

By su bstituting each dependent variable in the finite element matris equat ion

(4.37) by the expression of independent variables according to equation (1.6%) and

making row and column treatrnents on the finite element matrices [ S I ] and [T;] in the

same way as that of impiementing boundary conditions, we can obtain the final finite

element matrix equation as

where [H"] = [H:] . The order of [s:] and [T;'] is reduced by r based on that of ($1

and [T;]! and the number of eigenvalueç is also reduced by r compared to tha t of

eigenvalue equation (4.37). That rneans r more spurious modes are removed alter the

non-divergence constraint is enforced.

4.4 Summary

In this chapter? boundary conditions for both perfect electric conductor walls and

perfect magnetic conductor walls are enforced. The non-divergence constraint is also

imposed on the finite element inatrix equation. As a result. the size of finite element

matrices is significantly reduced and the number of the matrix order which is reduced

is equal to that of spurious modes which are eliminated. In the next chapter. a brief

description of the computer program will be given. In Chapter6. some applications

will be presented to show the validity of the proposed method.

Ckapter 5

Description of tke Computer

Program

A cornputer program which implements the triangular finite element met hod for lin-

early polarized traveling waves has been provided by Konrad [4]. Based on this

program. the implementation of boundary conditions and non-divergence constraint

bas been added to eliminate the spurious modes appearing in the original program.

The program has been written in fortran under UNIX environment and tested on Sun

Workstation.

The body of the program could be broken into six major parts:

1. Read-ln - Al1 the physical and geometrical information about a list of the triangle

vertices and coordinates, a list of the triangles and a list of boundaries. Using this

information, subroutine READIN generates additional points corresponding to the

desired polynomial approximation over each triangle. The relative locations of these

points are mapped approximately by the subroutine MhP.

2. Matrix Assembly - Based on the information supplied by READIN. subroutine

ASSEMB assembles the global matrices [St] and [Tt] for a collection of triangles in

the x-y plane.

3 . Forcing the Boundary Conditions - In subroutine BOUNPT. the nodes that fall

on the electric walls are identified. In subroutine SYMMEPT. the nodes that fa11 on

the magnetic walls are identified. Subroutine CONSTR.4INTl sets boundary condi-

tions in terms of H z , H, and H, of each and every boundary point and enforces them

to the global matrices [St] and [Tt] so that [St] and [Tt] are reduced to ($1 and [Si].

This part is generalized to al1 kinds of shapes of boundaries including straight Iine

and circular boundaries.

4. Forcing the Non-Divergence Constraint - In subrout ine CONSTRAINTT non-

divergence constraint D - B = O is discretized by approximation of vector fields rvith

high-order polynornial triangular finite elements and a compact matrix form is ob-

tained. The same boundary conditions are enforced to this non-divergence matrix

equation. Through this equation. dependent variables are expressed in terms of in-

dependent variabies. Finally, global matrices [$] and [$] are reduced to [$'] ancl

[s;'] mhen dependent variables are substituted by independent variables in t he

fini te element equation.

5 . Solving Eigenvalue Equation - Subroutine EIGVAL and EIGVEC is cal

solve the rnatrix eigenvaiue equation

[$ ' ] [~ f ] - x-~[T~"][v;) = O , W / L ~ T Y + = &,/PX

global

led to

EIGV.4L uses the Choleski method to decompose the positive definite symmetric ma-

trix [TJ i n t ~ lower and upper triangular factors and to câst the matrix eigenvalue

equation (5.1) into standard form. Householder's method and a modified Sturm se-

qiience procedure are employed to compute the eigenvalues. The eigenvectors are

computed in the subroutine EIGVEC by Wielandt iteration and are properly trans-

formed to give the eigenvectors of equat ion ( 5 . 1 ).

6. Solution - After eigenvalue equation is solved. the eigenvalues stand for the

propagation factor of modes. The aurnber of eigenvalues n" stands for the first n"

modes propagat ing in the calculated waveguide. The eigenvecton can be interpreted

either as the approximate nodal values of the magnetic field H or as those of the

electric field El depending on what the material propertjr tensor j and q in equation

(2.30) represents. Because dependent variables are removed. only the approximate

nodal values of the field which are independent variables are obtained from eigenmlue

equation. The nodal values of the field which are dependent variables are calculated

in the postprocessing according to the relations between dependent and independent

variables.

This program is to calculate waveguide with same structure in the z-direction and

arbitrary cross-section. In the next chapter- this program is used to calculate soine

typical waveguide problems to show t he validity of the proposed method.

Results and Applications

6.1 Homogeneous Isotropic Waveguide Problem

Choose the hollow square waveguide, shown in Figure 6.1, as a test problem to check

the validi ty of the proposed method and the correctness of the implementat ion.

Figure 6.1: The hollow square waveguide

As 8 triangular elements and 3rd-order interpolation polynomiaIs are used. the

mesh division of the cross-section of the waveguide is shown in Figure 6.2. Before

Figure 6.2: The cross section and the mesh of the waveguide shown in Figure 6.1

enforcing any constraints, there are 49 interpolation nodes, and the coefficient matrix

size is 147 x 141. Thus 147 eigenmodes are returned by the program. The correspond-

ing k - B and ,O/k - k diagrams appear in Figure 6.3 and 6.4. respectively. When

comparing the computed results with the analytical solution of the ernpty square

waveguide problem [39], we find chat spurious modes spread al1 over the physical

modes. Some are below al1 the ~hysical modes, such as modes 1 through .5 1. which

are below the air line. Some mix with physical modes, such as modes .56. 63 and 66.

These spurious modes are indicated by l x f in the k - 0 and P/k - k diagrams. while

p hysical modes are represented by 'of .

If H is the desired variable, the boundary is a perfect electric wall. There are

33 boundary conditions. After the boundary conditions are enforced, the coefficient

matrix size is reduced to 114 x 114. The rank of the non-divergence rnatrix. which is

also applied boundary conditions, is 44. Therefore, after the non-divergence constraint

is enforced, the coefficient matrix size is fioally reduced to 70 x 70. Thus only 70

eigenmodes are returned, while 77 spurious modes are eliminated. The corresponding

k - ,O and Plk - k diagrams are shown in Figure 6.5 and 6.6.

Figure 6.3: k - 13 diag~am for the hollow square waveguide before the constraints are

enforced.

Figure 6.1: P / k - k diagram for the hollow square waveguide before the constraints

are enforced.

Figure 6.5: k - /3 diagram for the hollow square waveguide after the constraints are

enforced.

Figure 6.6: P / k - k diagram for the hollow square waveguide after the constraints

are enforced.

Comparing the results before and after enforcing constraints. rve can see that

1) At cutoff frequency, al1 the zero eigenmodes except one are eliminated. The mag-

netic field vectors for this remaining zero eigenvalue mode have vanishing H, and Hg

values and constant Hz values at al1 interpolation nodes.

2) At propagation frequencies, al1 the spurious modes are eliminated including al1

those below or among the physical modes.

.4fter enforcing constraints, spurious modes are eliminated. Is the accuracy of

physical modes decreased? Let's compare the reiative errors of the wave-number k

of the first 14 physical modes before and after the boundary conditions and non-

divergence constraint are enforced. The relative error e is given by

where k and & are computed and analytical solut,ions, respectively. The percent errors

For iji = O are shown in Table 6.1. where kl and k2 represent wavenurnber before and

after constraints are enforced? respectively. From Table 6.1. me can see that the

difference of accuracy of the results before and after the constraints are enforced is

within 1.5%. Therefore. when we use the method of constraints to eliminated the

spurious modes. the effect on the physical modes is slight.

In the above calculation, the boundaries of the structure are al1 parallel to the

x-mis or y-axis. To test the correctness of boundary condition implementation on

sloping boundaries, we calculate the same waveguide in a difFerent position shown

in Figure 6.7. The same mesh division is adopted as the waveguide in Figure 6.2.

The results at cutoff computed for the same waveguide with the sarne mesh divisions

on two different positions are compared in Table 6.2. We can see that almost exact

same results are returned for the same waveguide at different positions, which means

that the implementation of boundary conditions on both the x-axis, y-axis parallel

Physical Mode 1 ,i3 = O 1

rtc % %

No. Type Analytical kl error k2 error

Solut ions in kl in &

Table 6.1: Relative errors of k of the first 14 modes for 3 = O before (kl) and after

(k2) enforcing the constraints.

Figure 6.7: The cross section and the mesh of the same waveguide. as in Figure 6.2.

at the different position.

boundaries and the sloping boundaries are correct. The correctness of boundarp

condition implementation on circular boundaries will be tested in a later section to

calculate a circular waveguide.

We also need to check the correctness of boundary condition implementation on

perfect magnetic wall. One way to check it is by calculating electric field intensitp

E, instead of magnetic field intensity H. since on the elctric wall E need to satisfy

n x E = O, which is same as the boundary condition H needs to satisfy on magnetic

wall. To calculate E, we only need to interchange p and c in the input data for

calculating H. The waveguide shown in Figure 6.2 is computed. The results from

calculating both H and E are listed in Table 6.3 to compare them.

Rom Table 6.3, when ,û = 0, the only remaining spurious mode in calculating

H also disappears when calculating E. The reason for this is that when enforcing

Mode

30.

k

Position 1

:3.14177

3.141 7'7

4.44668

4.45525

6 .LM93

6.25493

7.07015

1.07S51

1.03162

7.16853

9.03103

9 -23365

9.64013

9.64150

10.2194

k

Position 2

Table 6.2: Cornparison of the results, at cutofF, computed for the same waveguide

with the same rnesh division in two different positions.

H is calculated

Mode No

zero

:3.14 177

:3.14177

4.44668

4.45535

6.25493

6.25493

7.07015

'ï.O'ï85 1

7.08163

7.16853

9.03102

9.22365

9.64013

9.64150

10.2194

- - - -

E is calculated

Mode 1\50

H is calculated

Mode NO

E is calculated

Mode Xo

Table 6.3: Cornparison of the results from computing H and E mhen ,3 = O and ,$ = 1

boundary condition of E, on al1 boundary nodes. El is enforced to zero. Thus it is

impossible for E, on every interpolation node to be same nonzero values. which is the

character of that only remaining spurious modes. When ,O > O. ail the spurious modes

are removed as well. The results returned from calculating H and E are very close.

especially for the first several modes. For the latter modes. the difference between

two sets of results becomes larger because the accuracy of calculation is decreased for

the latter modes. Therefore. it is proved that boundary condition irnplementation for

boundary nodes. on magnetic wall. of case 1 (parallel to the y-axis). case 2 (parallel

to the x-axis) and case 1 (the corner point) is correct. We calculate the waveguide

in Figure 6.7 as well. .Us0 very close results are returned by calculating H and E.

which means the implementation of boundary condition for magnetic wall boundary

nodes of case 3 (on the sloping boundaries) is correct as well.

Another case which involves magnetic wall is symmetry plane. We can calcu-

late just one-half or one-fourth of the waveguide by applying appropriate boundary

conditions on symmetry plane. Now we calculate the right half of the waveguide in

Figure 6.2, shown in Figure 6.8. The symmetry plane p is x = O plane. which can be

electric wall or magnetic wall. The mesh division is (:V,. .V,) = (2s. 4). wliich is same

as the same part of the waveguide in Figure 6.2. If the symmetry plane is electric

wall. the results at cutoff are shown in the first column of Table 6.4. If the symmetry

plane is magnetic wall, the results at cutoff are shown in the second column. The

third column shows the total modes in two cases. which is the modes propagating in

the whole waveguide. Then we calculate one-fourth of the same waveguide shown in

Figure 6.9. The symmetry plane is x = O, y > O plane ( p plane) and y = 0 . x > O

plane (q plane). The mesh division is ( N p , L V ~ ) = (16,2). which is same as the same

part of the whole waveguide in Figure 6.2 and one-half of the waveguide in Figure 6.8.

There are four cases to be calculated depending on the different combination of elec-

Figure 6.8: The cross section and the mesh of the right half of the waveguide shown

in Figure 6.2.

Figure 6.9: The cross section and the mesh of a quarter of the waveguide shown in

Figure 6.2.

tric wall and magnetic wall on two symmetry planes. The corresponding results for

every case are shown in the first four columns of Table 6.5. The Iast colurnn shows

the total modes in four cases, which is the modes propagating in the whole waveg-

uide. In Table 6.6, the results returned by calculating the whole structure. half of

the structure and one-fourth of the structure are listed together. The relative errors

and CPC' tirne are compared. From Table 6.6, we can see that the difference arnong

relative percent errors in 3 calculation methods are slight when the mesh divisions for

the same part of the structure are the same. However. the required CPC time for cal-

culating the whole structure is much longer t han that of calculating half or one-fourt h

of the structure. Therefore, making use of symmetry conditions and calculat ing part

of the whole structure can simplify the computation. shorten calculation tirne and

Save memory.

1 P plane is electric wall 1 P plane is magnetic wall 1 Total modes 1

Table 6.3: Results returned by calculating one-half of the ivaveguide shown in Fig-

ure 6.8 at cutoff.

Mode

1

3 -

3

3

5

6

- 1

(3

9

k

zero

3.13115

6.23493

6.25493

7.08 10:3

1.13506

9.17360

9.23319

9.63959

Mode

1

3 - :3

4

5

6

- 1

Mode

1

'3

13

4

5

6

7

8

9

10

11

I I

1:3

14

1.5

16

k

3.1317'7

3.34684

3.45935

7.08359

1.146'10

9.63323

10.2332

k

zero

3.1111.5

3.14177

4.44684

4.359:35

6.25493

6.2S493

1.08103

1.08359

1.13506

1.14670

9.1260

9.2'2419

9.62333

9.6:3989

10.1333

* e.w. : electric wall:

* m.w. : magnetic wall;

Table 6.5: Results returned by calculating one-fourth of the waveguide shown in

Figure 6.9 at cutoff.

p is e.w.

q is

Mode

1

2

:3

4

5

e.w.

k

zero

6.25493

6.23493

~ . g w i r 9.22115

9.63989

p is e.w.

q is

Mode

1

'3 - :3

4

p is m.w.

m.w.

k

3.141 15

1.08252

1.14253

9.60940

q is

Mode

1

2

3

4

e-W.

k

:3.1417.5

7.08252

1.14253

9.60940

p is msv.

q is

Mode

1

2

Total modes

m m .

k

4.33341

3.36542

I

3

Mode

1

-3 - t3

4

3

6

1

3

9

10

11

LI

13

14

k

zero

3.14175

:3.14ll.5

4.44347

4.46542

6.1849:3

6.2S49:3

7.08253

1.08252

1.14253

7.14253

s.gc)~'ï"1

9.22115

9.60940

9.60940

10.1256 10.1256

15

16

1 1 T h e whole waveguide 1 Half of the waveguide 1 One-fourth of the waveguide (

CPU time

( sec

I

a

3

78

for one run

31

for two runs

zero

3.1417'7

3.14177

3 6

far four runs

Table 6.6: The cornparison of the results returned by calculating the whole waveguide.

0.00573

0.00573

one-half of the waveguide and one-fourth of the waveguide.

zero

3.14175

3.1317'7

0.00509

0.00513

zero

3 . 13175

3.141 75

0.00509

0.00509

Now the same hollow square waveguide with a different mesh division is computed.

shown in Figure 6.10. The rnesh division is (Np. N e ) = (49.2). where 2 triangular

elements and 6th-order interpolation polynomial are used. The number of interpo-

Lation points is 19, which is sarne as that in Figure 6.2. Before enforcing boundary

conditions and non-divergence constraint. the corresponding k - 3 diagram is shown

in Figure 6.11. where spurious modes are represented by Y . Comparing k - LI di-

agrams in Figure 6.3 and 6.11, ive find that the position of spurious modes in the

spectrum of whole eigenmodes are diiferent when mesh division is different alt hough

the nurnber and position of interpolation points are same. In Figure 6.1 1. al1 spuri-

ous modes are below physical modes: while in Figure 6.3, some spurious modes mix

with physical modes and some below physical modes. However. al1 spurious modes

should be eliminated after enforcing boundary conditions and non-divergence con-

straint wherever they locate before the method of constraints is used. After imposing

constraints. the k - ,d diagram appears in Figure 6.12. which is similar to that in

Figure 6-5. Therefore. by using the method of constraints. al1 spurious modes. except

one at cutoff. can be eliminated. This is independent of the mesh subdivision.

Figure 6.10: Another mesh of the same waveguide as in Figure 6.2

Figure 6.11: k - 9 diagram for the hollow square waveguide with mesh shown in

Figure 6.10 before the constraints are enforced.

Figure 6.12: k - /3 diagram for the hollow square waveguide with mesh shown in

Figure 6.10 after the constraints are enforced.

6.2 Inhomogeneous Isotropic Waveguide Problem

In Section 3.1, we have shown how spurious modes rnix with physical modes in in-

homogeneous waveguide problem. In this section, we will calculate the inhomogeous

waveguide, whose cross section is shown in Figure 6.13, to further justify the validity

of our method.

Figure 6.13: The cross section of an inhomogeneous rectangular waveguide

Before enforcing any const raints, the results returned by Iionrad's program. the

whole structure is used and mesh disvision is ( N , , N . ) = (63?24), are plotted in

Figure 6.14 as k versus P . When compared with analytical results based on formulas

in Young [40], spurious modes are found and represented by 'x'. while physical modes

are represented by 'O?.

In my research, the symmetry in this mode1 is exploited and only half of the

problem is computed. The mesh division used is ( N p , Ne ) = (35,121. The computed

results are plotted in Figure 6.15. Comparing Figure 6.14 and 6.15, we can see that

al1 the modes represented by ' X ' disappear and none of the modes represented by 'O'

Figure 6.14: k - 3 diagrarn for the inhomogenous rectangular waveguide before the

cons traints are enforced.

are eliminated, which means that our method successfully eliminated al1 the spurious

modes. Meanwhile, ou. results are shown in Table 6.7, along with analytical results in

order to see the accuracy of physical modes. For each propagation constant J. seven

modes are displayed. We can see that our results coincide with analytical solutions

quite well. The maximum percent error is less than 5%.

Figure 6.15: k - 13 diagram for the inhomogeneous rectangular waveguide after the

constraints are enforced.

Mode

1

3 - 3

4

a

6

7

1

2

3

3

5

6

7

% error k-analytic

1.7666

2.3053

2.6779

2.9548

3.2987

3.5524

4.1380

2.0000

2.4637

2.8137

3.0699

3.6363

3.6528

4.2187

2.5135

2.8832

3.1535

3.3886

3.9376

4.3941

4.4513

Mode

1

2

3

4

5

6 - I

l

3 L

:3

4

5

6 C

I

k-FEM

1.7681

3.3076

2.7115

2.9747

3.3097

3.7226

4.2273

1.0063

2.4738

2.5457

3.1041

3.6571

3.7776

4.3442

3.5176

2.8955

3.2152

3.4194

4.0309

4.4256

4.5631

k-analytic % error

Table 6.7: Cornparison of analytical results and computed solution of 7 modes for 3

,L? values.

6.3 Circular Waveguide Problem

In order to test the applicability of the proposed method to the waveguide with curved

boundaries, a hollow circular waveguide is analyzed. The computation is carried out

with the mesh shown in Figure

with 37 second order triangular

6.16. where a quarter of the waveguide is considered

elements.

v

Figure 6.16: The cross section of a circular waveguide and the mesh for one-fourth of

the structure.

By using the boundary conditions of electric mal1 on circular boundaries and the

boundary conditions of magnetic wall on two symmetry planes. and enforcing non-

divergence constraints on every interpolation nodes, the k - 3 diagram for the first

6 modes is shown in Figure 6.19. In (411, a line diagram showing positions of the

cutoff frequencies of some of the lower-order modes as compared with the dominant

T Ell mode is given for a circular guide in Figure 6.17. Figure 6.18 shows the same

line diagram of Our results for cutoff. We can see that our results coincide with the

analytical results very well.

Figure 6.17: The line diagram shorving the positions of anaiyt ical cutoff frequencies

of some low-order modes compared wi th the dominant T EiI modes.

Figure 6.18: The same iine diagram of the computed results as Figure 6.17.

If boundary conditions and non-divergence constraint are not enforced and the

whole structure is computed, the corresponding k - 3 diagram is shown in Figure

6.20. The curves in Figure 6.19 are very close t o those represented by .O' in Figure

6.20, which are considered as physical modes. The remaining curves represented by

' x ' in Figure 6.20 are spurious modes. -4s other problems computed. our method

elirninates al1 spurious modes and keeps al1 physical modes. By now. the correctness

of al1 kinds of boundary conditions have been tested.

Figure 6.19: k - 3 diagram for the circular waveguide alter the constraints are en-

forced.

6.4 Magnetically Anisotropic Waveguide Problem

The first t hree exarnple waveguides are al1 magneticail- isotropic waveguides. -4s

shown in Chapter 2 and Chapter 4. the boundary conditions and non-divergence con-

straint for anisot ropic waveguide are relatively more difficult t han t hose for isot ropic

w avegui des.

New: consider a rectangular waveguide with a 2: 1 widtb to height ratio completely

filled with a ferrite material characterized by a relative permittivity of 2.0 and a

relative perrneability tensor Ur given by

Figure 6.21 shows the cross section and the mesh of the waveguide. 1. 2 . :3 and 1 in

Figure 6.2 1 represent 4 boundaries.

Figure 6.21: Cross Section of the Xnisotropic waveguide

Rewrite the boundary condit ion on elect ric wall for magnet icdly aniso t ropic waveg-

96

uide (2.55).

where p,, p,,, pry. pyyo Pr and p,; are eiements of the permeability tensor p. a is

the slope of outward normal vector of the boundary. Substituting the elements of @.

we get

:l.OcosaH, + L.OsinaHy + O.YcoscrH, = O (6.4 )

For boundary L and 3 in Figure 6.21. a = 90'. So. cosa = O. sina = 1 and the

boundary condition becomes

which means. on boundary 1 and 4. H is tangential to the boundary. as same as

that for isotropie waveguide. For boiindary 1 and 1 in Figure 6.". a = O0 and 180'.

respectively. Thus. cosa = kl. sina = O and the corresponding boundarlr condition

New' consider the non-divergence constraint . Rewrite the non-divergence con-

straint for rnagnetically anisotropic waveguide (4.58).

For the specific f i in this problem, the above equation becomes

which is the non-divergence constraint to be implemented in the program.

Xow. let us compare the results before and after enforcing boundary conditions

and non-divergence constraint. the k - $ diagrams of which are shown in Figure 6.2'1

and 6.23. respectively. In Figure 6.22, curve no. 1 to no. 57 are from origin and

intercross with curve no. 58 and above. In Figure 6.23. a11 the curves through origin

dissappear and curves no. 58 and above remain. In Table 6.8. we compare the

wavenumber k of modes no. 1 to no.10 in Figure 6.23 with those of no. 5S to no. 67

in Fi y r e 6.22 for ;3 = 1. We can see that the values of two columns are very close.

Therefore. we can corne to the conclusion that the modes in Figure 6.23 are physical.

Analytical solution cm be obtained for the dominant mode and also for some of

the high order modes. .A characteristic of these modes is Ky = O. In Figure 6.23.

these modes are numbered 1. 3 and 7. respectively. The analytical solution for these

modes are derived in [a] as follow

A cornparison of the analytical solution with computed results has been made. The

percent error in the computed wavenumbers axe given in Table 6.9 for the dominant

mode (n=l ) and two high-order modes (n=2 and 3) . We can see that our results

coincide with analytical solutions very well.

1 In Figure 6.22 1 In Figure 6.23 1

Table 6.8: The cornparison of k for mode no.1 to 110.10 in Figure 6.23 with those for

no. 58 to no. 67 in Figure 6.22.

Figure 6.22: L - P diagram for the magnetically anisotropic waveguide before the

constraints are enforced.

Figure 6.23: k - ,8 diagram for the magnetically anisotropic waveguide after the

constraints are enforced.

analytical solution Ki = &[b2 + (n5)2]

Table 6.9: The wavenumber kl, k2 and k3 retumed by our program for (mode no. 1.

n o 2 and no.3. respectively, in Figure 6.23) and their percent errors.

6.5 Coaxial Waveguide Problem

Pis the last example investigated in this thesis, we solve a coaxial waveguide problem

shown in Figure 6.24. Making use of the symmetry condition. we calculate jiist one-

Figure 6.24: Coaxial waveguide cross section

fourth of the structure. The radius of the outside circle. a. is kept unchanged. CVe

alter the radius of the central conductor. b. The first six modes for several radius

ratios c = a / b are obtained. Figure 6.25 shows the cutoff characteristics of the

coaxial waveguide. The results have been compared with analytical solutions and

good agreement was found. especially for the small value of radius ratio c. Except

one zero-eigendue-mode, as indicated before, no other spurious modes appear. If

calculating the structure at other values of propagation constant. we will obtain all-

physicai-modes result S.

/

* Analytical

O Computed

- - - - -

Figure 6.25: Cutoff characterist ics of coaxial waveguide

Chapter 7

Conclusions

Based on a vectorial finite element formulation of bounded electromagnetic field proh-

lem. the method of constraints is combined tvith it in order to eliminate the spurious

modes appearing in the original solutions. In this vectorial finite elernent formulation.

the generalized curlcurl equation was adopted as the governing differential equation

and the high-order polynomial triangular finite elements were chosen to solve t h e elec-

tromagnetic boundary value problem. The medium may be isotropic or anisotropic

but must be linear and loss-free. Lots of spurious modes appear in the solution of

this electrornagnetic boundary value problem.

In this research? the appearance of these spurious modes is attributed to two

reasons: (1) Proper boundary conditions are not satisfied: (2) The non-divergence

constraint is not included. Then, the generd boundary conditions for bot11 isotropic

and anisotropic waveguides are derived and enforced on the global finite element

matrices. After that, the general non-divergence constraint for both isotropic and

anisotropic waveguides are derived and imposed on the finite element matrices which

have already been reduced by enforcing the boundary conditions. Therefore. t he order

of finite element matrices are furt her reduced to less than two-t hirds of the original

matrices.

The three component magnetic field K is solved, by the computer program de-

veloped, for some typical waveguide problems. The solutions are the same as what

is expected from the proposed method. When in the propagation region. where the

propagating constant B is greater than zero, no spurious modes appear. When at

cutoff, where ,û is equal to zero, one zero-eigenvaiue mode exists' which also appears

in the solution of scalar finite element methods.

7.2 Advantages and Disadvantages of the Method

Compared to the penalty function method and the edge element method. the other

main two met hods developed to solve the spurious-mode problem. t his met hod sig-

nificantly reduces the number of degrees of freedom by roughly one third, or s a - just

two field components are solved. For the penalty function method, the size of the

finite element matrices remains the same as that of the original matrices. The edge

element method also suffers from the disad~antage that it requires large and dense

matrices for computation and effectively needs more computer storage and CPU time

than the ordinory FEM. Meanwhile, in the method of constraints, the reduced finite

element matrices remain symmetric, which also minirnizes the cost of computatioo.

However, the sparsity pattern of the finite element matrices is destroyed. In 14-1' it

was stated that

" To minimize the cost of computation, a solver should, ideally, be formulated using

just two field cornponents and have sparse symmetric matrices. Table 1 (cornparison

of al1 formulations for modal waveguide analysis) shows chat none of the formula-

tions meet this requirement ; the reduction from t hree field components to two always

results in the loss of either sparsity or symmetry in the matrix equation."

Another feature of the method presented in this thesis is that it works for both

homogeneous and inhomogeneous dielectric loaded waveguides and is applicable to

bot h isotropie and anisotropic waveguides. Therefore, this approach can be used to

solve a wide range of waveguide problems.

As shown in one of the example problems the solutions are independent of the

position of the waveguide in the Cartesian coordinate system because of the derivation

and implementation of the general boundary conditions.

As for disadvantages. it must be pointed out t hat it rvould be difficult to apply t be

met hod of constraints to problems with both dielectric and magnetic materials. Since

both normal and tangential components of the vector field must be continuous along

al1 inter-element boundaries, the material parameters are restricted: eit her permitt iv-

ity or permeability should be constant in al1 regions. In fact. this is a disadvantage

of the nodal finite element method.

Also, because of the over-restriction of the normal continuity of the field. this

method is not suited to problerns with sharp conducting edges. The singular field at

the edge can be modeled by introducing special singular trial functions [43]. However.

if the normal continuity of the field is relaxed, good accuracy is generally possible

even without singular trial functions [44.

7.3 Future Work

It should be pointed out that the approach in this thesis is not restricted to two

dimensions and could equally be applied to three-dimensional resonant cavity prob-

lems. For three-dimensional problems. the boundary conditions and non-divergence

constraint need to be derived and implemented in a t hree-dimensional vectorial finite

element program.

Addit ionally, the extension to waveguide analysis where losses may be significant

can be considered. From lossless media to lossy media. the main work is to derive a

new funct ional from which vectorial finite element formulation can be ob t ained.

Appendix A

Directional Derivat ives of

Interpolation Polynomials in Finite

Element Problems Using

Triangular Element s

- - - - - - - -

A.1 Introduction

In Section 1.3, we need to express the non-divergence constraint (-1.46) and (4.58) in

a compact matrix equation form as (1.47). Therefore, the directional derivatives of

interpolation polynomials in two-dimension finite element problem using triangular

elements needs to be derived, which was shown in [45].

Consider the interpolation polynomial

( A . 1)

over the triangular element shown in Figure A.1. The directionai derivative 2 in any

specified direction n will be determined at the interpolation points of the element.

Figure A. 1: A triangular element wit h the interpolation points.

A.2 Derivation of Formulas

where X = cosa and p = cos@ are the direction cosines of the unit vector n.

Concisely. (A. 1 ) becomes

where n = $( N + 1 ) ( N + 2) is the total number of interpolation points.

Substituting (A.3) into (A.2) gives

Since

with

It follows that

If we substitute (A.11) and (-4.12) into (A.5) and let

We now obtain

( A . L3)

(A . 14)

which can be written in matrix form as

8' = (dlG1 + d2G2 + d3G3)Q

w here

S . . . . - S . .

. . - . .. S . .

the (i.j)th elernent of matrices Gi, G2, G3 are. respectively.

(3) - Ba, Sil - - li ah

A.3 Derivation of Matrices G2 and G:] fkom G1

According to the following proposition

proof is omitted, we have

For each value of Pi, the corresponding r may be established in Table A.1 .

A.4 The Computation of Gi

To determine G1. for each triplet (i.j,k). i + j + k = !V, we muçt evaluate

According to (-4.26). we have

Therefore. the last :V + 1 columns of G1 are null.

According to ( A . 9 7 ) .

An explicit formula for ~ ; ( g ) is needed only in the case p < i. Therefore. in the 1-th

column of Gi ( l < 1 < n - iV - l),

(1) the first 1-1 entries are zeros:

aa, k (2) if % stands for +-' then the diagonal elernent is always followed by at l e s t

j + k zeros.

Table -4.1: Table of indices r(i) for .Ri = 1, . . .*, 5

In (A.24). by definition.

Hence.

Similarly.

For i = 1.

Bibliography

[l] PSilvester, "Finite Elernent Solution of Homogeneous Waveguide Problem.' .-Ilta

Frequenza. 38, 1969, pp. 313-317.

[" P.Daly. "Hybrid-Mode Analysis of Microstrip by Finite-Element Methods."

IEEE Transaction on Microwave Theonj and Techniques. MTT- 19. .lanuary

1971, pp. 19-25.

[3] J. A. M. Svedin. "A Yumerically Efficient Finite-Element Formulation for the

General Waveguide Problem Without Spurious Modes.' lEEE Transaction on

Micro~wove Theory a n d Techniques. MTT- 19. Xovern ber 1989. pp. 1708- 1 71 5.

[4] A. Iionrad. "Triangular Finite Elements for Vector Fields in Elect romagnet ics ."

P hD Thesis. Department of Electrical Engineering, McGill University. 1974.

[5] P-Silvester and G. Pelosi, Finite Element for Wave Problems' Yew York. IEEE

Press. 1994.

[6] -4. lionrad. "Vector Variational Formulation of Electromagnet ic Fields in

Anisotropic Media," IEEEE Transaction on iblicrowave Theory and Techniques.

MTT-24, 19'76, pp. 553-559.

[FI M. V. K. Chari, A. Konrad. M. A. Palmo and J. Dy ANgeIo. Three-Dimensional

Vector Potential Analysis for Machine Field Problems.' IEEE Transaction on

Magnetics, MAG-18. No. 2. March 1982. pp. 436-446.

[SI M. V. K. Chari. M. A. Palmo and .J. D' AXgelo, - Three-Component. Tm-

Dimensional Analysis of the Eddy Current Diffusion Problems." lEEE Transnc-

Lion on iklagnetics. MAG-20, No. 5 . September 1984, p1989.

(91 C. J . Carpenter, "Cornparison of Alternative Formulations of Three-Dimensional

Magnetic-Field and Eddy-Current Pro blems at Power Frequencies." IEE Pro-

ceedings. Vol. 121. No. I l . November 1977. pp. 1026-1034.

[IO] T. Morisue. "Magnetic Vector Potential and Electric Scalar Potential in Three-

Dimensional Eddy Current Pro blem,' IEEE Transaction on .\.fagnetics. .LI AG-

18. 30. 2: March 1982, P531

[ I l ] J. L. Coulomb. "Finite Element Three Dimensional Magnetic Field Computa-

tion." IEEE Transaction on Magnetics. .L[i\G-17. Yo. 6. Yovember 1981. pp.

334 l-iE46.

[I 21 K . Bathe. Finite Element Procedure in Engineering .4nalysis. Englewood Cliffs.

3 J: Prentice-Hall, 1982.

[13] M. Hara, T. Wada, T. Fukasawa and F. Kikuchi, " .A Three-Dimensional Analysis

of RF Electromagnetic Fields by the Finite Element Method." lEEE h n s a c t i o n

on kfugnetics, MAG-19, 1983, pp. 2417-2420.

[14] B. M. A. Rahman and J.B. Davies, "Penalty Function Improvement of Waveg-

uide Solution by Finite Elements," IEEE Transaction on !Microwave Theoty and

Techniques, MTT-32, 1984, pp. 922-928.

[15] M. Koshiba, ILHayata and M.Suzuki, "Vectorial Finite-Element Method With-

out Spurious Solutions for Dielectric Waveguide Problems." EIectronic Lett ers.

Vol. 20, May 1984, pp. 109-410.

[16] M. Koshiba, K. Hayata and M.Suzuki. "Irnproved Finite-Element Formulation in

Terms of the Magnetic-Field Vector for Dielect tic Waveguide," iEEE Transaction

on ~Wicrowave Theory and Techniques. MTT-33. March 1985. pp. 227-233.

[ l T ] Iieith D. Paulsen and Daniel R. Lynch. *Elimination of Vector Parasites in

Finite Element Maxwell Solutions." IEEE Transaction on .Kcrowaue Th eory

and Techniques. Vol. 39. No. 3. March 1991. pp. 395-304.

[lY] Daniel R. Lynch and Iieith D. Paulsen. "Origin of Vector Parasites in Numerical

kf axwell Solution," IEEE Transaction on Microwaue Theory and Techniques.

MTT-39, March 1991. pp. 353-394

[19] J. P. Webb. "Efficient Generation of Divergence-Free Fields For the Finite Ele-

ment Analysis of 3D Cavi ty Resonances.' lEEE Transaction on .Ifagnetics. Vol.

24. No. 1. January 1988. pp. 162-165.

[-O] A. Iionrad, "A Direct Three-Dimensional Finite Element Met hod for the Solut ion

of Electrornagnetic Fields in Cavities." IEEE Transaction on .\~agnetics. Vol. 21.

No. 6, November 1985. pp. 2276-2279.

[SI] A. lionrad, "A Method for Rendering 3D Finite Elernent Vector Field Solutions

Non-Divergent," IEEE Transaction on Magnetics. Vol. 25' 1989, pp. 2822-2834.

[22] K. Hayata, M. Koshiba. MEguchi and M.Suzuki, "vectorial Finite-Element

Mathod Wi thout Any S purious Solutions for Dielectric Waveguiding Problems

using Transverse Magnetic-Field Component." IEEE Transaction on fificrowave

Theoty and Techniques, MTT-34, 1986, pp. 1120-1 124.

[23] -4. J. Kobelansky and J . P. Webb, "Eliminating Spurious Modes in Finite-

Element Waveguide Problems by Using Divergence-Free Fields,' Electronic Let-

ters, Vol. 22. I986? pp. 569-30.

[34] A. lionrad. "The zero Divergence Constraint in 3D Vector Finite Element Meth-

o ~ s , " The International Journal for Cornpvtation and .kfathernatics in Elect7ical

.And Electronic Engineering,' Vol. 9. 1990. Supplement A. pp. 7-16.

[25] M. Hano, "Finite-Element Analysis of Dielectric-Loaded Waveguides." lEEE

Transaction on Microwaue Theory and Techniques. MTT-32. 1984. pp. 1275-

1219.

[26] M. Hano.Yector Finite-Element Solution of Anisotropic Waveguides Using

Novel Triangular Elements." Electronics and Communications in .Japnn. Part

2, Vol. 'il7 No. (S. 1988. pp. 71-80.

[27] A. Bossavit and 1. Mayergoyz. Edge-Elements for Scattering Problems." [EEE

Transaction on !Lfagnetzcs. Vol. 25. 'JO. 1. July 1989. pp. "16-22221.

[%] -4. Bossavit, "Solving Maxwell Equations in a Closed Cavity. and the question

of 5purious Modes',*? IEEE Transaction on Magnetics. Vol. 26. Yo. 2. March

1990, pp.702-705.

(291 C. W. Crowley, P. P. Silvester and H. Hurwitz, "Covariant Projection Elements

for 3D Vector Field Problems," I E E Transaction on 1Vfagnetics. M AG-24, 1988.

pp. 397-400.

[30] S. H. Wong and 2. J. Cendes. "Combined Finite Element-Modal Solution of

Three-Dimensional Eddy Current Problems," lEEE Transaction on Magnet ics.

MAG-24, 1988, pp. 2685-2687.

[31] Jin-Fa Lee, "Analysis of Passive Microwave Devices by Using Three- Dimensional

Tangential Vector Finite Elements? International Journal of !Vumerical .Mod-

elling: Electronic networks. Devices and Fields. Vol. 3 . December 1990. pp. 235-

246.

[32] Jin-Fa Lee, Din-Iiow Sun and 2. J. Cendes. "Full-Wave Analysis of Dieiectric

Waveguides Using Tangential Vector Finite Elements.' IEEEE Transaction on

iMicrowave Theory and Techniques. Vol. 39, No. S. August 1991. pp. 1-62-1271.

(331 T.Nakata. N.Takahashi. K. Fujiwara and Y. Shiraki. " Cornparison of Different

Finite Elements for :3D Eddy Current Anal ysis." IEEE Transaction on .Magn e t -

ics, Vol. 26, Xo. '7. Warch 1990. pp. 434-437.

[34] Akihisa Iiameari. "Calcuiation of Transient 3D Eddy Current Using Edge Ele-

ments," IEEE Transaction on Magnetics. Vol. 26. Yo. 2. klarch 1990. pp. 466-

369.

[35] D. Sun, J. Manges. X. Yuan and 2. Cendes. "Spurious Modes in Finite-Element

Method," IEEE -4ntennas and Propagation :Magazine. Vol. 37. So. 5 . Octo ber

1995. pp. 12-24.

[36] S. Ramo. J. R. Whinnery and T. Van Duzer. Fields and CVaves in Communica-

tion Electronics, Xew York: John Wiley & Sons, 1967.

[37] Z. J. Cendes and S. H. Wong, "Cl Quadratic Interpolation Over Arbitrary Point

Sets," IEEE Cornputer Graphics and .Applications. November 1987. pp. 8- 15.

[38] -4. Konrad, "On the Reduction of the Number of Spurious Modes in the Vectorial

Finite-Element Solution of Three-Dimensional Cavi ties and Waveguides." IEEE

Transaction on lMierowave Theory and Techniques, MTT-34, No. 2, February

1986, pp. 224-227.

[39] bfatthew N. O. Sadiku, Elements of Electrornagnetics. Saunders College Pub-

Iishing, 1989.

[40] L. Young, Advances in iI.licrowaves, Vol. 1, Academic Press. London. 1966.

[41] Marcuvitz. Wavwguide :randbook, McGraiv-Hill. 195 1.

(421 B. M. Dillon and J. P. Webb, " A Cornparison of Formulation for t h e Vector Fi-

nite Element Analysis of Waveguides." IEEE Transaction on Microwave Theory

and Techniques. Vol. 32, No. 2? February 1994, pp. 308-316.

[43] .J. P. Webb. ;' Finite-Element Analysis of Dispersion in Waveguides Wit h Sharp

Metal Edges," IEEE Transaction on Micro wave Theory and Techniques. Vol. 36.

December 1988, pp. 18 lg-LS24.

[44] R. Minowitz and J. P. Webb. " Covariant-Projection Quadrilateral Elements For

the .4nalysis of Waveguides Wi t h Sharp Edges." lEEE Transaction on .Vitro wu ve

Theory and Techniques, Vol. 39. March 19910 pp.501-505.

[43] E.Iiicak. Directional Derioatioes O/ Interpolation Polynomials in Two-

Dimensional Finite Element Problems L~'sing triangular Elements. April 1973.

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