RL  2Z
Multiresolution methods for simulation and
design of antennas
by
Lars S. Andersen
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Electrical Engineering)
in The University of Michigan
1999
Doctoral Committee:
Professor John L. Volakis, Chair
Professor Linda P.B. Katehi
Professor Noboru Kikuchi
Emeritus Professor Thomas B.A. Senior
RL992 = RL992
o Lars S. Andersen 1999
All Rights Reserved
ACKNOWLEDGEMENTS
I sincerely want to thank John L. Volakis for encouraging me to come to the
University of Michigan and for serving as my PhD advisor for the past four years.
We most certainly have our differences but I feel that we both understand those and
that we have built a relationship of tremendous mutual benefit. I thank the Danish
Research Academy for the generous financial support that made my stay in Michigan
possible. In this context, I also want to thank Thomas B. Thrige's Fond for providing
valuable travel support and Olav Breinbjerg for being my Danish contact person.
I want to thank all the past and current students and postdocs in the Rad Lab in
general and in my research group in particular with whom I have interacted in the past
four years. Several technical discussions have been fruitful for me professionally and
numerous other conversations have benefitted me at a more personal level. I especially
want to acknowledge Stephane R. Legault, Mark D. Casciato, Tayfun Ozdemir and
Thomas F. Eibert, four superb gentlemen whose friendship and assistance have been
no less than invaluable to me. I also want to thank the entire Rad Lab and EECS
support staff for providing valuable assistance.
Finally, I want to thank al the friends and family members who have provided
me with significant support throughout the past four years.
11
TABLE OF CONTENTS
ACKNOWLEDGEMENTS.......................... ii
LIST OF TABLES................................ v
LIST OF FIGURES............................... vi
LIST OF APPENDICES............................ x
CHAPTERS
1 INTRODUCTION............................. 1
1.1 M otivation............................ 1
1.2 Fundamental concepts...................... 5
1.3 Proposed approach.................................. 7
1.4 Organization.................................... 8
2 BACKGROUND.............................. 11
2.1 Vector wave equations................................. 11
2.2 Review of TVFEs.............................. 13
2.3 Sum m ary.............................. 20
3 TWODIMENSIONAL TVFES.......................... 21
3.1 Hierarchical mixedorder TVFEs for triangular elements... 21
3.2 Application to closeddomain problems................ 28
3.2.1 Formulation for closeddomain problems........... 28
3.2.2 Numerical results for closeddomain problems.. 31
3.3 Application to opendomain problems................... 33
3.3.1 Formulation for opendomain problems............... 33
3.3.2 Numerical results for opendomain problems.....38
3.4 Summary....................................... 45
4 THREEDIMENSIONAL TVFES.......................... 46
4.1 Hierarchical mixedorder TVFEs for tetrahedral elements.. 46
4.2 Application to closeddomain problems.................... 49
iii
III
4.2.1 Formulation for closeddomain problems......... 49
4.2.2 Numerical results for closeddomain problems.... 50
4.3 Application to opendomain problems............... 54
4.3.1 Formulation for opendomain problems........ 55
4.3.2 Numerical results for opendomain problems..... 61
4.4 Summary..................................... 6(5
5 ANALYSIS OF CONDITION NUMBERS.................. 66
5.1 Background.......................... 6...... 6
5.2 Formulation for closeddomain problems.................
5.3 Homogeneous application of TVFEs................ 69
5.3.1 Twodimensional case.................. 69
5.3.2 Threedimensional case........................ 71
5.4 Inhomogeneous application of TVFEs................. 73
5.5 Summary..................................... 77
6 ADAPTIVE TVFE REFINEMENT.................. 79
6.1 Background........................... 79
6.2 Adaptive refinement strategies...................... 82
6.3 Numerical results........................ 84
6.3.1 Square patch antenna........................ 84
6.3.2 Printed bowtie antenna.......................93
6.3.3 Discussion......................... 99
6.4 Summary............................. 104
7 ANALYSIS OF TAPERED SLOT ANTENNAS............... 105
7.1 Background.......................................... 105
7.2 Tapered slot antenna analysis.................. 108
7.3 Summary................................... 1:6
8 SUMMARY, CONCLUSIONS AND FUTURE WORK........... 127
8.1 Summary and conclusions......................... 127
8.2 Future work.......................................... 130
APPENDICES............................. 132
BIBLIOGRAPHY..................... 140
iv
LIST OF TABLES
Table
3.1 Comparison of relevant parameters for the FEM results in Fig. 3.8.. 13
3.2 Comparison of relevant parameters for the FEM results in Fig. 3.10.. 43
3.3 Comparison of relevant parameters for the FEM results in Fig. 3.12.. 13
4.1 Definition of Case 13................................... 53
4.2 Computational effort for Case 16 for the antenna in Fig. 4.104.11.. 64
5.1 Condition numbers for the global matrices resulting from FEM analysis
of the waveguide in Fig. 5.1......................... 70
5.2 Condition numbers for the global matrices resulting from FEM analysis
of the cavity in Fig. 5.6........................... 73
5.3 Condition numbers for the global matrices resulting from FEM analysis
of the cavity in Fig. 5.11............................ 77
6.1 Definition of Case 19................................... 85
6.2 Definition of Case 1012.......................... 96
7.1 Definition of Case 14............................ 117
v
LIST OF FIGURES
Figure
2.1 Illustration of a triangular element........................... 14
2.2 Illustration of a tetrahedral element.......................... 14
3.1 Geometry of a triangular element and illustration of the vectors n x
(r  rn), n {1,2, 3}, describing the directions of the vector basis
functions at the point P............................. 22
3.2 Illustration of the proposed vector basis function (CVj  (jVCi for a
triangular element................................... 29
3.3 Illustration of the proposed vector basis function ((i  (j)(iV(Cj 
(iV(i) for a triangular element................................ 29
3.4 Illustration of the proposed vector basis function (k((7V(j  C('V) for
a triangular element............................ 29
3.5 Error of the dominant transverse propagation constant for a rectangular waveguide as a function of the number of unknowns. TE modes
(left) and TM modes (right)........................ 32
3.6 Geometry of a scatterer illuminated by a TE polarized incident electromagnetic field and giving rise to a TE polarized scattered electromagnetic field.................................. 34
3.7 Square cylinder with a groove illuminated by a TE polarized plane wave. 40
3.8 Bistatic RCS of the cylinder in Fig. 3.7................ 40
3.9 Square cylinder with a groove loaded by a dielectric slab illuminated
by a TE polarized plane wave.................................. 42
3.10 Bistatic RCS of the cylinder in Fig. 3.9.................. 412
3.11 Grating structure on top of a grounded dielectric illuminated by a TE
polarized plane wave........................... 44
3.12 Bistatic RCS of the cylinder in Fig. 3.11................. 44
4.1 Homogeneous and isotropic rectangular cavity.............. 51
4.2 Convergence rate for expansion of the field within the homogeneous
and isotropic rectangular cavity in Fig. 4.1 using mixedorder TVFEs
of order 0.5 and 1.5...........................51
4.3 Inhomogeneous and isotropic rectangular cavity............. 52
vi
4.4 Eigenvalue error (Eformulation) for the inhomogeneous and isotropic
rectangular cavity in Fig. 4.3.............................. 54
4.5 Eigenvalue error (Hformulation) for the inhomogeneous and isotropic
rectangular cavity in Fig. 4.3.................................... 54
4.6 Magnetic field on the back PEC wall for the TM111 mode of the inhomogeneous and isotropic rectangular cavity in Fig. 4.3 with mixedorder
TVFE of order 0.5 applied (Case 1)............................... 55
4.7 Magnetic field on the back PEC wall for the TM111 mode of the inhomogeneous and isotropic rectangular cavity in Fig. 4.3 with mixedorder
TVFEs of order 0.5 and 1.5 applied (Case 3)............. 55
4.8 Side view of a cavitybacked patch antenna recessed in an infinite PEC
ground plane................................................. 56
4.9 Top view of a cavitybacked patch antenna recessed in an infinite PEC
ground plane for the case of a triangular patch and a circular cylindrical
cavity............................................. 56
4.10 Side view of a square metallic patch antenna backed by a dielectricfilled rectangular cavity recessed in an infinite metallic ground plane. 62
4.11 Top view of a square metallic patch antenna backed by a dielectricfilled rectangular cavity recessed in an infinite metallic ground plane. 62
4.12 Real part of the input impedance of the antenna in Fig. 4.104.11 for
C ase 16.................................. 65
4.13 Imaginary part of the input impedance of the antenna in Fig. 4.104.11
for Case 16..................................65
5.1 Illustration of a rectangular waveguide............................69
5.2 Condition numbers for the individual element matrices for the waveguide illustrated in Fig. 5.1; TE/TMformulation with unnormalized
vector basis functions............................ 71
5.3 Condition numbers for the individual element matrices for the waveguide illustrated in Fig. 5.1; TE/TMformulation with normalized vector basis functions....................................... 71
5.4 Ratios of condition numbers the for individual element matrices for the
waveguide illustrated in Fig. 5.1; TE/TMformulation with unnormalized vector basis functions...................................... 72
5.5 Ratios of condition numbers for the individual element matrices for the
waveguide illustrated in Fig. 5.1; TE/TMformulation with normalized
vector basis functions................................... 72
5.6 Illustration of a rectangular cavity........................... 73
5.7 Condition numbers for the individual element matrices for the cavity
illustrated in Fig. 5.6; E/Hformulation with unnormalized vector basis
functions........................................ 74
5.8 Condition numbers for the individual element matrices for the cavity
illustrated in Fig. 5.6; E/Hformulation with normalized vector basis
functions................................................. 74
vii
5.9 Ratios of condition numbers for the individual element matrices for
the cavity illustrated in Fig. 5.6; E/Hformulation with unnormalized
vector basis functions...........................
5.10 Ratios of condition numbers for the individual element matrices for the
cavity illustrated in Fig. 5.6; E/Hformulation with normalized vector
basis functions...............................
5.11 Illustration of an inhomogeneous rectangular cavity...........
75
76
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
Real and imaginary part of input impedances
Real and imaginary
Real and imaginary
Real and imaginary
Real and imaginary
Real and imaginary
Real and imaginary
Real and imaginary
Real and imaginary
part
part
part
part
part
part
part
part
of input
of input
of input
of input
of input
of input
of input
of input
impedances
impedances
impedances
impedances
impedances
impedances
impedances
impedances
for Case 1
for Case 2
for Case 3
for Case 4
for Case 5
for Case 6
for Case 7
for Case 8
for Case 9
(EIl
(EIl
(EI1
(EI2
(El2
(EI2
(EI3
(EI3
(EI3, Tetra). 87,Prism). 87,Brick). 87, Tetra). 88,Prism). 88,Brick). 88, Tetra). 89, Prism). 89, Brick). 89..... 90..... 90..... 90.... 91.... 91.... 91.... 92..... 92..... 92
Regions of' refinement
Regions of refinement
Regions of' refinement
Regions of' refinement
Regions of' refinement
Regions of refinement
Regions of' refinement
Regions of refinement
Regions of' refinement
for Case 1
for Case 2
for Case 3
for Case 4
for Case 5
for Case 6
for Case 7
for Case 8
for Case 9
(EI1,
(EI1,
(EII1,
(E I2,
(E I2,
(EI2,
(EI3,
(E13,
(EI3,
Tetra) at 4.15 GHz..
Prism) at 4.30 GHz..
Brick) at 4.30 GHz..
Tetra) at 4.10 GHz..
Prism) at 4.30 GHz..
Brick) at 4.30 GHz..
Tetra) at 4.10 GHz..
Prism) at 4.30 GHz..
Brick) at 4.30 GHz..
6.19 Side view of a metallic printed bowtie antenna backed by an air and
absorberfilled rectangular cavity recessed in an infinite metallic ground
plane...................................
6.20 Top view of a metallic printed bowtie antenna backed by an air and
absorberfilled rectangular cavity recessed in an infinite metallic ground
plane....................................
6.21 Real and imaginary part of the input impedance for the metallic bowtie
patch antenna in Fig. 6.196.20......................
6.22 Real and imaginary part of the input impedance for the metallic bowtie
patch antenna in Fig. 6.196.20.....................
6.23 Regions of refinement for the metallic bowtie patch antenna in Fig. 6.19 6.20 for Case 10 (left), Case 11 (middle) and Case 12 (right)...
6.24 Final number of iterations as function of frequency for different iterative solutions...............................
6.25 Final number of iterations as a function of frequency for adaptive solutions for Case 10 (EIl, Prism) with different starting guesses for the
iterative solver..............................
94
94
98
98
99
101
102
viii
6.26 Final number of iterations as a function of frequency for adaptive solutions for Case 11 (EI2, Prism) with different starting guesses for the
iterative solver.............................. 102
6.27 Final number of iterations as a function of frequency for adaptive solutions for Case 12 (EI3, Prism) with different starting guesses for the
iterative solver.................................... 102
6.28 Relative residual as a function of the iteration number for adaptive
solutions for Case 10 (El1, Prism) with different starting guesses for
the iterative solver.............................. 103
6.29 Relative residual as a function of the iteration number for adaptive
solutions for Case 11 (EI2, Prism) with different starting guesses for
the iterative solver....................................... 103
6.30 Relative residual as a function of the iteration number for adaptive
solutions for Case 12 (EI3, Prism) with different starting guesses for
the iterative solver............................. 103
7.1 Illustration of a metallic (dark gray) ETSA (left), LTSA (middle) and
CWSA (right) backed by a dielectric (light gray)................ 106
7.2 Geometrical parameters for a LTSA (not drawn to scale)........... 109
7.3 Eplane patterns for the LTSA in Fig. 7.2...................11
7.4 Hplane patterns for the LTSA in Fig. 7.2................. 111
7.5 Co and crosspolarized Eplane patterns for the LTSA in Fig. 7.2... 114
7.6 Co and crosspolarized Hplane patterns for the LTSA in Fig. 7.2... 114
7.7 Co and crosspolarized Dplane patterns for the LTSA in Fig. 7.2.. 114
7.8 Geometrical parameters for a LTSA (not, drawn to scale)......... 115
7.9 Real and imaginary part of the input impedance of the LTSA in Fig. 7.8
for different refinement schemes..................... 118
7.10 Number of iterations to reach a relative tolerance of l03 with a QMR
solver for the LTSA in Fig. 7.8 for different refinement schemes.... 119
7.11 Regions of refinement for the LTSA in Fig. 7.8 for Case 1 with 3%
refinem ent............................... 121
7.12 Regions of refinement for the LTSA in Fig. 7.8 for Case 2 with 3%
refinement......................................... 121
7.13 Regions of refinement for the LTSA in Fig. 7.8 for Case 3 with 3%
refinement.................................. 1'22
7.14 Regions of refinement for the LTSA in Fig. 7.8 for Case 4 with 3%
refinem ent............................... 122
7.15 Eplane patterns for the LTSA in Fig. 7.8..................... 124
7.16 Hplane patterns for the LTSA in Fig. 7.8................ 124
7.17 Eplane patterns for the LTSA in Fig. 7.8......................125
7.18 Hplane patterns for the LTSA in Fig. 7.8.................. 125
A.1 Illustration of the variation of ul and v1 over a triangle.......... 134
ix
LIST OF APPENDICES
Appendix
A Explicit expressions for Wi, W2 and W3................ 133
B Expressions for vector basis functions.................. ]136
x
CHAPTER 1
INTRODUCTION
In this chapter, a brief motivation for the dissertation is offered, some fundamental
concepts are presented, a highlevel description of the proposed approach is given and
the organization of the dissertation is outlined.
1.1 Motivation
Most practical electromagnetic problems dealing with microwave circuit characterization, scattering analysis or wireless communication contain regions where the
field exhibits rapid variations and others where it varies slower. Rapid spatial field
variations are often encountered in dense materials, at material boundaries and in
the vicinity of corners, edges and small geometrical details while the opposite typically is the case away from such regions. For microwave circuits, examples are
stripline, microstrip, slotline or coplanar line structures used extensively for constructing complex microwave components such as inductors, capacitors, resonators,
filters and transformers of use in various system circuitry. For scattering analysis,
I
examples are different airborne or ground vehicles for which accurate radar signature predictions are necessary for target recognition and observability minimization
in commercial and military frameworks. For wireless communication, examples are
numerous narrowband and broadband metallic printed antennas (rectangular or circular patches, bowties, spirals, logperiodics and tapered slots) possibly backed by
dielectric or magnetic substrates of interest for antenna miniaturization and multifrequency communication purposes.
Brute force application of traditional sub or entiredomain method of moments
(MoM) is of limited applicability for largescale threedimensional electromagnetic
problems involving inhomogeneous media. Whether solving for volumetric polarization currents or repeatedly invoking equivalence principles to solve for equivalent and
induced surface currents at material boundaries, the MoM leads to a set of linear
equations described by a fully populated system matrix. As the number of unknowns
increases, the central processing unit (CPU) time and memory requirements for cornputing and storing matrix elements become unrealistic. Moreover, for electromagnetic
problems characterized by regions with different degrees of field variation, uniform
discretization at a rate appropriate in the regions where the field varies the most
typically leads to excess discretization elsewhere contributing even further to an Unmanageable computational problem. Acceleration methods like the fast multipole
method [71] or the adaptive integral method (AIM) [12] can be invoked for avoiding
the storage of a full matrix and for speeding up matrixvector products within the
iterative solver but these approaches add significant complexity to the formulation
and implementation. The problems associated with uniform discretization through2
out the computational domain can partially be remedied by changing the sampling
rate within the computational domain. However, such nonuniform meshing does not
address the problems pertaining to a full system matrix and is unattractive in practice
since robust software packages for nonuniform meshing are rare and expensive.
Brute force application of lowest order scalar or vector finite element (FE) approaches only partially alleviates the problems related to traditional sub or entiredomain MoM. The finite element method (FEM) as well as hybrid finite element
/ boundary integral (FE/BI) methods have been demonstrated to be attractive for
modeling complex materials and fine geometrical details [91] and FE approaches for
solving partial differential equations inherently overcome the MoM problem of a full
system matrix by leading to sets of linear equations described by sparsely populated
system matrices. However, the problems related to uniform versus nonuniform discretization are as serious for FE approaches as for MoM approaches.
Robust and efficient modeling of electromagnetic problems characterized by regions with different degrees of field variation calls for a numerical method that is
capable of accurately and efficiently modeling complex materials and geometric details, leads to a set of linear equations described by a (partly) sparse system matrix
and in addition offers the possibility of locally increasing resolution. To this end,
various MoM solutions utilizing wavelets have been proposed. Wavelets are classes
of functions based on the elegant mathematical theory of a multiresolution analysis
(MRA) [21, 90, 99] and therefore have several attractive properties such as containment, upward and downward completeness, scaling, translation and, possibly, orthogonality or semiorthogonality. They have been applied extensively in mathematics,
3
electromagnetics and signal processing for the past decade [21, 73, 90].
Within the context of the MoM in electromagnetics, wavelets have been applied as
basis and weighting functions to directly produce a set of linear equations described
by a system matrix that can be sparsified via thresholding [84]. Alternatively, one can
use traditional subdomain MoM to generate a set of linear equations described by
a full system matrix and subsequently utilize a wavelet based transformation matrix
and thresholding for sparsification [102]. Similar to the fast computation of a discrete
Fourier transform via the fast Fourier transform [90], wavelet based MoM matrix
elements can be computed in an extremely efficient manner using Mallat's fast wavelet
algorithm [56, 73]. In addition, matrix elements small enough for thresholding can
under certain circumstances be predicted a priori [73], lowering the computational
and storage requirements even further. Moreover, wavelet based MoM_ matrices are
inherently wellconditioned [73]. Despite all these attractive features, wavelet based
MoM analysis is not free of serious drawbacks. The definition and implementation
of wavelets in the vicinity of discontinuities and material boundaries is tedious and
unattractive [73]. Also, wavelets on arbitrarily curved threedimensional surfaces are
notoriously difficult to implement. Furthermore, the choice between a host of different
wvavelets (Haar, Daubechies, Shannon, Meyer, BattleLemarie and LittlewoodPaley
to mention just a few) is far from obvious when considering such factors as tradeoff
in resolution between domains, threshold limit, infinite versus compact support and
orthogonality versus semiorthogonality.
In summary, disjoint regions with different degrees of field variation are a nearly
inevitable part of any practical electromagnetic application. Accurate simulation of
4
electrornagnetic problems comprised of such regions is therefore of significant practical
importance but traditional numerical methods often have difficulties doing so. It is the
goal of this dissertation to develop a FE based multiresolution method for accurate
and efficient analysis of electromagnetic problems of that particular nature.
1.2 Fundamental concepts
In order to understand the approach proposed in this dissertation, certain fundamental concepts must be familiar. These are presented in the following.
For the initial application of the FEM in mechanical and civil engineering [108], the
FE expansion of an unknown quantity was based on groups of scalar basis functions
whose unknown expansion coefficients are associated with nodes of a finite element
mesh. These groups known as node based finite elements are suitable for modeling scalar quantities but typically not so for simulating vector electromagnetic fields.
When assigning vector field values to element nodes, values may need to be specified
at locations where the true field is undefined (corners, edges), spurious modes can
be generated and the enforcement of the boundary conditions occurring in electromagnetics can be a challenging task. These drawbacks prompted the introduction of
groups of vector basis functions whose unknown expansion coefficients are associated
with edges, faces and cells of a finite element mesh. These groups are known as tangential vector finite elements (TVFEs) [15] and have been shown to be free of the
shortcomings of node based finite elements [96].
A TVFE is referred to as polynomialcomplete to a given order n if all possible
5
polynomial variations up to and including order n are captured within the element
and on the element boundary. Nedelec pointed out [61, 62] that it is not necessarily advantageous to employ polynomialcomplete TVFEs. It was proven that a
polynomialcomplete expansion of a vector field A can be decomposed into a part
representing the range space of the curl operator (V x A 7 0, A Z VO) and a part
representing the null space of the curl operator (V x A = 0, A = V7). For representation of electromagnetic fields in a source free region, it suffices to employ a
TVFE that is complete only in the range space of the curl operator. Since such a
TVFE captures polynomial variations of order n interior to the element and polynomial variations of order n  1 for the tangential field along edges, it is referred to
as a mixedorder TVFE. In line with the syntax adopted by Webb [97], it will be
referred to as complete to order n  0.5 in the remainder of this dissertation. F'or
extensive discussions of mixedorder TVFEs versus polynomnialcomplete TVFEs, see
[15, 61, 62, 67, 68, 85].
A TVFE is referred to as interpolatory if the vector basis functions forming the
TVFE interpolate to (tangential) field values within or on the boundary of the element. For an interpolatory TVFE, the meaning of the expansion coefficient corresponding to a given vector basis function is typically easy to interpret physically.
A class of TVFEs is referred to as hierarchical if the vector basis functions forming
the TVFE of order n  0.5 are a subset of the vector basis functions forming the
TVFE of order n + 0.5. This desirable property allows use of different order TVFEs
in different regions of the computational domain for efficient discretization of the
unknown quantity  an approach we will refer to as selective field expansion. This
6
selective choice of TVFEs over the computational domain can lead to a memory and
CPU time reduction as well as improved accuracy.
1.3 Proposed approach
The FEM as well as hybrid FE/BI methods have been demonstrated to be attractive for modeling complex materials and fine geometrical details [91]. They implicitly
allow a local increase of resolution, either by locally increasing the mesh density
(hrefinement), the polynomial order of the expansion (prefinement) or both (liprefinement). In the context of FE or FE/BI methods, a discretization scheme employing hierarchical mixedorder TVFEs permits robust combination of expansions of
different orders within a computational domain. In a straightforward manner, it allows use of lowest order expansions where the field varies slowly and use of (different)
higher order expansions where the field varies rapidly for an appropriate discretization
of the unknown electromagnetic field. Although not formulated within the stringent
framework of a MRA, such a discretization scheme is truly a multiresolution discretization scheme in the sense that it allows different levels of field modeling within
a computational domain. The concept of hierarchality is widely known but the applications hereof in electromagnetics seem few and far between. Nevertheless, FE or
FE/BI discretization exploiting hierarchality appears to be a nearly ideal candidate
for simulating and designing a large class of complex electromagnetic systems.
In this dissertation, hierarchical mixedorder TVFEs for triangular (twodimensional
problems) and tetrahedral (threedimensional problems) elements are developed and
7
tested and the effectiveness of selective field expansion is investigated for solution of
electromagnetic scattering problems as well as for analysis of various narrowband and
broadband antennas.
1.4 Organization
Chapter 2 introduces background material. Vector wave equations used throughout the dissertation are presented and TVFEs for triangular and tetrahedral elements
used for discretizing partial differential equations are reviewed.
Chapter 3 deals with twodimensional TVFEs. Hierarchical mixedorder TVFEs
of order 0.5, 1.5 and 2.5 for triangular elements are proposed and tested for solution
of closed as well as opendomain problems. For solution of certain classes of electromagnetic problems, field expansion using hierarchical mixedorder TVFEs of order
0.5 and 1.5 selectively is demonstrated to be a very promising approach in terms of
accuracy, memory and CPU time requirements as compared to a more traditional
approach.
Chapter 4 extends the work in chapter 3 to threedimensional TVFEs. Hierarchical
mixedorder TVFEs of order 0.5, 1.5 and 2.5 for tetrahedral elements are proposed and
tested for solution of closed as well as opendomain problems. Again, selective field
expansion using hierarchical mixedorder TVFEs of order 0.5 and 1.5 is demonstrated
to be a very promising approach for accurate and efficient solution of certain classes
of electromagnetic problems.
Chapter 5 discusses matrix condition numbers. The condition numbers resulting
8
fromn FEM analysis using the hierarchical mixedorder TVFEs of order 1.5 for triangular and tetrahedral elements proposed in chapter 3 and chapter 4 are contrasted to
those of existing interpolatory and hierarchical mixedorder TVFEs of order 1.5 for
triangular and tetrahedral elements. The proposed hierarchical mixedorder TVFEs
of order 1.5 are proven to be better conditioned than existing hierarchical mixedorder TVFEs of order 1.5 and thus the analysis fosters no concerns for potential
future convergence problems due to excessive matrix condition numbers. In addition, an approach for improving the condition numbers of FEM matrices resulting
from selective field expansion is suggested and tested. The improvement comes at
the expense of a more complicated formulation and computer code but does not alter
accuracy.
Chapter 6 focuses on adaptive TVFE refinement. A review of existing error estimators and indicators is given and the effectiveness of the proposed hierarchical
mixedorder TVFEs of order 0.5 and 1.5 for tetrahedra is investigated when some
of the reviewed error indicators are applied in the context of a very simple adaptive refinement strategy. The results are extremely promising for both narrowband
and broadband antennas provided the refinement is carried out on a subdomain by
subdomain basis as opposed to an element by element basis.
Chapter 7 integrates the work in the previous chapters and presents an advanced
application of the approach proposed in this dissertation. Specifically, adaptive TVFE
refinement with hierarchical mixedorder TVFEs of order 0.5 and 1.5 for tetrahedra
is used to analyze tapered slot antennas (TSAs) with large impedance and pattern
bandwidths. The adaptive inclusion of a very small percentage of higher order TVFEs
9
is found to have a dramatic effect on the accuracy of the computed input impedances
and far field patterns, thus justifying the approach proposed in this dissertation for
large and complex problems.
Chapter 8 closes the dissertation. Brief summaries and the most important conclusions for the individual chapters are given and several future tasks to be completed
are suggested.
10
CHAPTER 2
BACKGROUND
In this chapter, vector wave equations used throughout the dissertation are presented and tangential vector finite elements (TVFEs) for triangular and tetrahedral
elements used for discretizing partial differential equations are reviewed.
2.1 Vector wave equations
Consider a possibly inhomogeneous and anisotropic region V characterized by the
permittivity tensor E and the permeability tensor /I. The electric and magnetic field
intensities E and H, respectively, then fulfil Maxwell's equations [40] 1
V'xE=jw/7.HM (2.1)
V x H= j E+J (2.'2)
where J and M, respectively, denote the impressed electric and magnetic volume
current densities. Elimination of H in (2.1) or E in (2.2) then leads to the vector
1Throughout this dissertation, a time factor eijt is assumed and suppressed.
11
wave equations
V x (f1 V x E) 2. E E= V x (I M)jJ (2.3)
V x (E1 V x H) w2~ H = V x ('1. J)jwM (2.4)
for E and H, respectively. By applying Galerkin's method with the vector weighting
function T and invoking elementary vector identities, we derive their weak forms
T(( ) (it V xE)  w2 T ( E)) dV =j T n x(71 Vx E) cdS
JV
T.Vx ( t  M) dV
 w T.JdV (2.5)
JV((V x T). ( 1 V x H) w2 T~ (7 H)) dV = Jf Tn x (1 * x H) (lS
+ / T V x (1 J) dV
j' T MdV (2.6)
where n, denotes the outward directed unit normal vector to the boundary Fv of V.
Boundary conditions at FV are incorporated in (2.5)(2.6) through the first term on
each righthand side. These terms vanish at a perfectly electrically conducting (PEC)
or perfectly magnetically conducting (PMC) surface.
The tensors E and,u are constant if V is homogeneous while they are functions of
the spatial variables if V is inhomogeneous. Letting 6o and,o denote the permittivity
and permeability of free space, respectively, we introduce the relative permittivity and
permeability tensors ~ and dr by
c = c o (2.7)
12
Yu = oir UO. (2.8)
For an isotropic material, E, and Plr are given by
r = er I (2.9)
Ir =/ I (2.]0)
Twhere I denotes the 3 x 3 identity tensor while Er and,vr are the scalar relative
permittivity and permeability, respectively. Specialized weak vector wave equations
can be obtained by substituting (2.7)(2.8) or (2.7)(2.10) into the general weak vector
wave equations (2.5)(2.6) but for brevity such weak vector wave equations are not
given here.
2.2 Review of TVFEs
Consider a triangular element with nodes 1, 2 and 3 and a tetrahedral element
with nodes 1, 2, 3 and 4, as illustrated in Fig. 2.12.2. The area of the triangle is
denoted by A and the volume of the tetrahedron is denoted by V. Simplex coordinates
at a point P of the triangle or the tetrahedron are defined in the usual manner [91],
i.e.
Ai
for the triangle and
i {1, 2, 3, 4} (2.12)
1;
13
for the tetrahedron where Ai denotes the area of the triangle formed by P and the
nodes of the edge opposite to node Z and VK denotes the volume of the tetrahedron
formed by P and the nodes of the triangular face opposite to node i.
3
P (l1,~2, 3)
1
2
Figure 2.1: Illustration of a triangular element.
4
3
1 P((1, ~2, 3,)4)
2
Figure 2.2: Illustration of a tetrahedral element.
In his seminal papers, Nedelec determined the number of vector basis functions
required for a mixedorder TVFE of order n  0.5, n E N [61, 62]. The starting point
is the number of vector basis functions required for a polynomialcomplete expansion
of order n. By excluding the degrees of freedom representing the null space of the
curl operator, one arrives at
=n(n + 2) (2.13)
and
~, =n(n + 2)(n + 3) (2.)
for triangular and tetrahedral elements, respectively. These equations hold for interpolatory as well as hierarchical mixedorder TVFEs. The vector basis functions can
14
be separated into three types, namely edgebased, facebased and cellbased vector
basis functions. General verbal descriptions of each of these types of vector basis
functions are given in the following while pictorial illustrations of specific vector basis
functions appear in section 3.1.
An edgebased vector basis function and its corresponding edgebased expansion
coefficient can be associated with an edge in a, triangular or a tetrahedral mesh.
Edgebased vector basis functions affect tangential as well as normal field components
at edges and field continuity requirements across edges force expansion coefficients
in adjacent elements corresponding to an interior edge to be related. Edgebased
expansion coefficients are therefore referred to as global unknowns. They provide
tangential field continuity across edges but allow normal field variation across edges.
A facebased vector basis function and its corresponding facebased expansion
coefficient can be associated with a face in a triangular or a tetrahedral mesh. Facebased vector basis functions affect normal field components at edges and tangential as
well as normal field components at faces. For triangular elements, the corresponding
expansion coefficients are not related across element boundaries. Facebased expansion coefficients for triangular elements are therefore referred to as local unknowns.
They allow normal field variation across edges. For tetrahedral elements, field continuity requirements across faces force expansion coefficients in adjacent elements
corresponding to an interior face to be related. Facebased expansion coefficients for
tetrahedral elements are therefore referred to as global unknowns. They provide tangential field continuity across faces but allow normal field variation across edges and
faces.
15
A cellbased vector basis function and its corresponding cellbased expansion coefficient can be associated with a tetrahedron in a tetrahedral mesh. Cellbased vector
basis functions affect only normal field components at edges and faces and the corresponding expansion coefficients are not related across element boundaries. Cellbased
expansion coefficients are therefore referred to as local unknowns. Thev allow normal
field variation across edges and faces.
Given a number N of vector basis functions W, j 1.., N, for anelement e,
the expansion of the unknown electric or magnetic field intensity EU or He within the
element is
N
E = EVW (2.15)
j=l
or
A'
H6 =  H7Ww (2.16)
j=l
where E and H6 are unknown expansion coefficients. Their physical interpretation
is determined by the vector basis functions WJ.
In the following, vector basis functions for different mixedorder TVFEs are reviewed. The vector basis functions presented in this dissertation are not normalized.
To normalize them, each edgebased vector basis function must be multiplied by the
length of the edge it is associated with. Furthermore, the indices i, j and k are imnplicitly assumed to belong to the set {1, 2, 3} for vector basis functions for triangular
elements and the set {1, 2, 3, 4} for vector basis functions for tetrahedral elements.
For n  1 leading to Nti = 3 and Net 6, a, mixedorder TVFE of order 0.5 is
obtained for triangular and tetrahedral elements. Such a TVFE provides a constant
16
tangential / linear normal (CT/LN) field along edges and a linear field at faces and
inside an element.
Whitney [98] initially presented a mixedorder TVFE of order 0.5. It is characterized by the edgebased vector basis functions
vj  CV, i < j. (2.17)
The vector basis function iV(j  (jV( provides a constant tangential component
along the edge between node i and node j, zero tangential component along all other
edges and a linearly varying normal component along all edges. A detailed explanation
of the characteristics of this vector basis function is given in [13].
For n = 2 leading to Nri = 8 and Net = 20, a mixedorder TVFE of order 1.5
is obtained for triangular and tetrahedral elements. Such a TVFE provides a linear
tangential / quadratic normal (LT/QN) field along edges and a quadratic field at
faces and inside an element.
Peterson [66] (triangular elements) and Savage and Peterson [77] (tetrahedral eLements) proposed an interpolatory mixedorder TVFE of order 1.5. It is characterized
by the edgebased vector basis functions
0VCj, i < (2.1 8)
C(jVi, < (2.19)
and the facebased vector basis functions
C(CVVCj  jVi), i < j < k (2.20)
17
((kV(1  (CVk), i < j < k. (2.21)
Graglia et al. 137] proposed an interpolatory mixedorder TVFE of order 1.5. It
is characterized by the edgebased vector basis functions
(3(C  1)(CVCj  jV), i < j (2.22)
(3(j  1)((iV(j  C), i < j (2.23)
and the facebased vector basis functions (2.20)(2.21).
Webb and Forghani [97] proposed a hierarchical mixedorder TVFE of order 1.5.
It is characterized by the edgebased vector basis functions
C<VCj  jV~, i < j (2.24)
C(V(9 + (V(, i < j (2.2_5)
and the facebased vector basis functions (2.20)(2.21).
For n = 3 leading to N~r = 15 and N3*t = 45, a mixedorder TVFE of order 2.5 is
obtained for triangular and tetrahedral elements. Such a TVFE provides a quadratic
tangential / cubic normal (QT/CuN) field along edges and a cubic field at faces and
inside an element.
Peterson and Wilton [68] (triangular elements) and Savage and Peterson [77]
(tetrahedral elements) proposed an interpolatory mixedorder TVFE of order 2.5.
A correction of the vector basis functions in [68, 77] was later given by Peterson
[65]. This corrected set of vector basis functions is the one presented here. It is
18
characterized by the edgebased vector basis functions
i(2C 1)V(j, i + j (2.26)
Cj(Vi Vj), i <j, (2.27)
the facebased vector basis functions
(k(2 k  1)((Cv,  jVC), i < j < k (2.28)
j(2(,  1)((kV(  (CVC(), i < j < k (2.29)
V(CjC), i < < k (2.30)
k((cVj  vci), <j, i j k i (2.31)
and (for tetrahedral elements only) the cellbased vector basis functions
C(j((CV(i  iV,), i,j,k > 1, i # j I k < i, j < k. (2.32)
The vector basis functions for the above mixedorder TVFEs of order 0.5, 1.5
and 2.5 were given explicitly since we shall use them in this dissertation. However,
several other TVFEs for triangular and tetrahedral elements have been presented in
the literature. We mention the work of Carrie and Webb [14], Cendes [15], GarciaCastillo and SalazarPalma [27], Lee et al. [53, 54], Mur and de Hoop [60], Wang [94],
Webb [95], Wu and Lee [100] and Yioultsis and Tsiboukis [103, 104, 105].
We note that a vast number of TVFEs for several other element shapes have also
been presented in the literature. A comprehensive review of these is beyond the scope
19
of this dissertation. However, we stress that triangular and tetrahedral elements offer
a, geometrical modeling flexibility that does not place serious restrictions on the classes
of problems to which they can be applied. For this reason, it is not unreasonable to
devote our attention to triangular and tetrahedral elements only. TVFEs for curved
triangular and tetrahedral elements would provide even greater modeling flexibility
but such TVFEs can be constructed from those proposed in this dissertation for
straight triangular and tetrahedral elements via a straightforward mapping, see for
instance [37].
We also note that Mur [59] throughout the past decade has strongly criticized
the use of TVFEs in general. However, his viewpoints are not accepted within the
computational electromagnetics community.
2.3 Summary
In this chapter, vector wave equations used throughout the dissertation were presented and TVFEs for triangular and tetrahedral elements used for discretizing partial
differential equations were reviewed.
20
CHAPTER 3
TWODIMENSIONAL TVFES
In this chapter, hierarchical mixedorder TVFEs of order 0.5, 1.5 and 2.5 for
triangular elements are proposed and tested for solution of closed as well as opendomain problems. The work in this chapter is published in [7, 8].
3.1 Hierarchical mixedorder TVFEs for triangular elements
The proposed class of hierarchical mixedorder TVFEs is based on an expansion
introduced by Popovic and Kolundzija [50, 69] for the surface current on a PEC
generalized quadrilateral. For this expansion, it is demonstrated in [50, 69] that the
surface current can be expanded using approximately ten unknowns per square wavelength as opposed to approximately one hundred unknowns per square wavelength
for traditional subdomain pulse basis functions. This suggests that the expansion
introduced by Popovic and Kolundzija is very efficient. Below, corresponding vec
21
tor basis functions applicable for finite element (FE) expansion are constructed and
hierarchical mixedorder TVFEs of order 0.5, 1.5 and 2.5 are presented.
As a degenerate case of the generalized quadrilateral considered in [50, 69], we
consider a triangular element of area A with nodes 1, 2 and 3 described by position
vectors ri, r2 and r3 with respect to the origin 0 of a rectangular coordinate system,
see Fig. 3.1. The edges from node 1 to node 2, node 2 to node 3 and node 3 to node
1 are referred to as edge #1, #2 and #3, respectively. Simplex coordinates (1, (2
and '3 at a point P described uniquely by a position vector r are defined in the usual
manner, see section 2.2. We let n denote a unit normal vector to the surface of the
triangle.
( #2 (
r3 r2
\ nx(rr,) x(rr3) /
w rh
Figure 3.1: Geometry of a triangular element and illustration of the vectors n x (r 
rn), n C {1,2, 3}, describing the directions of tile vector basis functions
at the point P.
Popovic and Kolundzija expands the surface current J, over the triangle as [69]
3 3
Js = AJsn= nV, (3.1L)
n=l n=l
where
r  rn
V = (3.2)
22
22
is a vector whose direction is from node n to P and Tn is a polynomial function of
position that provides the amplitude variation of the vector current component Js, =
nVn,. The polynomial On contains a number of unknown expansion coefficients.
Its specific form is irrelevant at this point and will be given later. As in the RaoWiltonGlisson expansion [70], Js has no normal component along the two edges
sharing, node n and Js has both a normal and a tangential component along the
edge opposite to node n. Thus, the quantity
F, = n x Jn = Only x V, = n x rW) (3..3)
2A  (3.3)
with
W,  x (r r) (3.4)
2A
has no tangential component along the two edges sharing node n and has both a
tangential and a normal component along the edge opposite to node n. This suggests
that the vector basis functions multiplying the expansion coefficients in the expansion
(3.3) of F,z can be employed as vector basis functions for the edge opposite to node
n when applying the finite element method (FEM). Considering all three edges, the
FEM expansion of an unknown vector quantity F becomes
3 3 3
F = x J = n x Ja, S F, =,nWn (3.5)
n=l n=l n=l
where expressions for tI (depending upon the order of the expansion) and Wn (independent of the order of the expansion) are to be presented.
Introducing normalized coordinates over the triangle and using relations from [69],
23
it is shown in Appendix A that
Wl = ~2V3  ((3V.2 (3.6)
W2 = C3V(l  (1V(3 (3.7)
W3 =1V2  C2V1i. (3.8)
The polynomial.n is a function of position that in terms of a number of unknown
expansion coefficients provides the amplitude variation of the vector component W,.
It can be defined using normalized coordinates u, and vn over the triangle. Specifically, we choose u, = 0 at node n, un = 1 along the edge opposite to node n and
Un = ~1 along the two edges sharing node n. A detailed description of the variation
of un and vn is given in Appendix A. From [69], we have
= 2 b +Z a (u2 1) v — (3.9)
j=l i=3
where n, and n, are integer constants determining the order of the approximation
and bJn and ai< are the expansion coefficients. Also, l = (2 + (31 ulVl  (2 (3
U2  43 + 4C 1 U2V2 == 43  41, U3 = 4 + 42 and u3z,3 = 4  42, as shown in Appendix
A.
The expansion (3.5) for F along with the expressions (3.6)(3.8) for Wi, W2 arnd
W3 and the expression (3.9) for 'n describes the proposed vector basis functions.
However, a certain simplification provides a more familiar form. By regrouping tihe
terms in (3.9) for fn, the expansion (3.5) for F can be cast into
3 \ntv n u
F = E E c nu vnu (3.10)
k=l m=l
24
where N"'vu = n(inu  1) denotes the total number of vector basis functions per edge
for the given values of nu and nr. Also, c j~ are expansion coefficients corresponding
to edge #k while Wk, V are vector basis functions associated with edge #k. Wk"m is
given as a function of (1, <2 and <3 times a direction vector (C1V(2  C2V(1 for k = I,
2V(3  C3VC2 for k = 2 and C3VC1  C1V(3 for k = 3). Except for normalization
constants, the vector basis functions W":u are directly used for forming hierarchical
mixedorder TVFEs.
From (3.10), we recover for (n,r n) = (1,2) the three vector basis functions
introduced by Whitney [98], see section 2.2. For larger values of nv and nu, (3.10)
includes additional vector basis functions all of which maintain the same fundamental
direction vectors ((iV(2  (2V(1 for edge #1, C2V(3  (3V(2 for edge #2 and C3V(i 
(IV 3 for edge #3). Thus, the proposed higher order vector basis functions differ from
the lowest order vector basis functions only in magnitude and hence in a given point of
the triangle, the field is represented as a linear combination of vector basis functions
having only three fundamental directions. This is one of the major differences between
the proposed and traditional hierarchical TVFEs [14, 97]. For the latter, the higher
order vector basis functions differ from the lowest order vector basis functions in
both magnitude and direction. The field in a given point of the triangle is again
represented as a linear combination of vector basis functions but in this case the
number of fundamental vector directions used for representing the field grows with
the order of the TVFE.
An important property of (3.10) is that the vector basis functions W,u^ for k =
1, 2,3 and n = 1. *., Nn u are a subset of the vector basis functions W(n,+l)(n+u1)
25
for k =1,2,3 and m = 1,, N,(+l)(ni+l). This shows that TVFEs based on the
above presented vector basis functions are hierarchical.
Based on the vector basis functions in (3.10) for different values of n, and n,
along with knowledge of Nedelec spaces [61, 62] and existing interpolatory mixedorder TVFEs, hierarchierarchical mixedorder TVFEs of order 0.5, 1.5 and 2.5 will now be
proposed and compared to existing interpolatory mixedorder TVFEs. The method
has the potential of providing hierarchical mixedorder TVFEs of even higher orders
if so desired. Explicit expressions for vector basis functions are given in Appendix B.
For the case (nPnu) = (1,2), we obtain from (3.10) a set of three vector basis
functions forming a mixedorder TVFE of order 0.5 identical to that introduced by
Whitney, see section 2.2. This result is expected since the lowest order expansion
adopted by Popovic and Kolundzija is identical to the RaoWiltonGlisson expansion
[70] whose vector basis functions reduce to the vector basis functions introduced by
Whitney when converted using the procedure applied above.
For the case (n,,nu) = (2,3), we obtain from (3.10) a set of twelve vector basis
functions. To make a comparison to the interpolatory mixedorder TVFEI of order
1.5 presented by Peterson [66], see section 2.2, the twelve vector basis functions are
reduced to eight vector basis functions forming a hierarchical mixedorder TVFE of
order 1.5, see Appendix B.
Peterson's interpolatory mixedorder TVFE of order 1.5 [66] has the desirable
property of being complete to second order in the range space of the curl operator.
This property ensures a complete second order expansion of a field with nonzero
curl and guarantees eigenvalue solutions free of spurious nonzero eigenvalues. Since
26
Peterson's interpolatory mixedorder TVFE of order 1.5 and the proposed hierarchical mixedorder TVFE of order 1.5 span the same space (the existence of a linear
transformation from Peterson's eight vector basis functions to the proposed eight
vector basis functions is demonstrated in Appendix B), the proposed hierarchical
mixedorder TVFE of order 1.5 has the same desirable property. However, the two
mixedorder TVFEs are not identical as they have different properties and may not
be equally efficient numerically. For both mixedorder TVFEs, six edgebased vector
basis functions provide a linearly varying tangential component along edges while
the remaining two facebased vector basis functions (identical for the two different
mixedorder TVFEs  added to provide a complete linear representation of the curl
of the field that is expanded) provide a quadratic variation of the normal component
along edges. However, the linear variation of the tangential component along edges is
obtained in two different ways. For Peterson's interpolatory mixedorder TVFE, the
two unknowns per edge represent the magnitude of the field at edge endpoints. F'or
the proposed hierarchical mixedorder TVFE, the two unknowns per edge represent
the average field value along the edge and the deviation from this average value at
edge endpoints.
A generalization to even higher order hierarchical mixedorder TVFEs is possible.
For the special case of (n1, nu) = (3,4), (3.10) gives vector basis functions that based
on knowledge of Nedelec spaces [61, 62] and the interpolatory mixedorder TVFE of
order 2.5 presented by Peterson and Wilton [65, 68], see section 2.2, can be used to
form a hierarchical mixedorder TVFE of order 2.5, see Appendix B. The similarity
between the hierarchical mixedorder TVFEs of order 1.5 and 2.5 is apparent and one
27
gets an impression of the needed generalizations to obtain even higher order TVF Es.
However, use of TVFEs beyond order 2.5 does not seem to be of practical interest.
The proposed vector basis function (V(j ~jV( providing a constant tangential component along an edge and a linearly varying normal component along all
edges is pictorially illustrated in Fig. 3.2. The proposed vector basis function (i 
j)( ( jV( i) providing a linearly varying tangential component along an edge
and a quadratically varying normal component along all edges is pictorially illustrated
in Fig. 3.3. The proposed vector basis function (k((iCVj  jV(i) providing zero tangential component along all edges and a quadratically varying normal component
along two edges is pictorially illustrated in Fig. 3.4.
3.2 Application to closeddomain problems
In this section, various TVFE options are employed for solution of closeddomain
problems. The pertinent formulation is given and numerical results are presented.
3.2.1 Formulation for closeddomain problems
Consider a source free homogeneous and isotropic waveguide with PEC walls. The
waveguide region is denoted by V and its cross section is denoted by S with Yv and
Fs being the boundaries of V and S, respectively. The waveguide is filled with a
material characterized by a constant relative permittivity E, related to the complex
permittivity tensor ~ via (2.7) and (2.9) and a constant relative permeability Fi,
related to the complex permeability tensor 7 via (2.8) and (2.10).
28
Figure 3.2: Illustration of the proposed vector basis function VCVj  QCVjC for a
triangular element.:L:::::::::::::l:l::::::::::::::::: 9iit:!i:: ii:i i;ii: i;ii i i:t
Figure 3.3: Illustration of the proposed vector basis function ((i  j)((:i7q — (3Vi)
for a triangular element.
40 4*04.4 40 4 
Figure 3.4: Illustration of the proposed vector basis function Ck((iV(j  (jV() for a
triangular element.
The weak vector wave equations (2.5)(2.6) hold for the electric and magnetic
field intensities E and H. However, a simplification is provided by reducing these to
29
weak vector wave equations for the transverse field components Et (transverse electric
(TE) modes) and Ht (transverse magnetic (TM) modes) over S only. Assuming the
waveguide is in the zdirection, we can let E = Etejz, H = Htejsz, V = Vt + zt
and T = Tt in the general weak vector wave equations (2.5)(2.6) and substitute
(2.7)(2.10) for e and ju to arrive at the TE and TM weak vector wave equations
s((Vt x Tt) (Vt x Et)  72 Tt Et) dS 0 (3.11)
s
((Vt x T,) (Vt x H,)  72 Tt Ht) dS= 0 (3.12)
with 7y = /2 being the transverse propagation constant and k == w:6rLrCo(
being the wave number. We now discretize S into Ns triangular elements via
Ns
S  SC (3.13)
e=l
and enforce the TE and TM weak vector wave equations (3.11)(3.12) in each element
S6. Next, we expand the transverse electric and magnetic fields EC and He in Se via
(2.15)(2.16). Choosing the vector weighting functions Tt = We, i = 1,, N, then
leads to
N N
(V, x W) * (V, x Wp) dSE We * WS dSE`' (3.14)
J1 =
N N
(Vt x> W) _ (Vt x W )dSH = 7 / w dSHW C. (3.15)
These element equations can be formulated in the matrix form
[A']{x } =  [B] { } (3.16)
30
with the element matrices [Ae] and [Be] having the matrix elements
A j = f (Vt x W?). (V, x W;)dS (3.17)
Se
jB = WX w w dS (3.18)
Se
for both TE and TM modes. Assembly of element equations then leads to a global
matrix equation system of the form
[A]{x} = y2[B]{x} (3.19)
for both TE and TM modes. We note that the assembly process includes the enforcement of boundary conditions along Fs. For TE modes, this leads to a condensation
of the equation system making the equation system for TE modes smaller than that
for TM modes.
3.2.2 Numerical results for closeddomain problems
To validate the proposed hierarchical mixedorder TVFE of order 1.5 and to make
a comparison with the interpolatory mixedorder TVFE of order 1.5 presented by
Peterson [66], see section 2.2, we consider a homogeneous rectangular waveguide of
side lengths a and a/2. The dominant transverse propagation constants for this
waveguide are 7) =< w/a and T1M = /a for TE and TM modes, respectively
[40]. The percentage of error in determining 7TE and 7TM as a function of the
number of unknowns is given in Fig. 3.5 (loglog plot) for the mixedorder TVFE of
order 0.5 (denoted 'Basis 1'), Peterson's interpolatory mixedorder TVFE of order
31
10~ 10~
e Basis 1  Basis 1
o Basis 2  Basis 2
Bai3  Basis 3 Basis 3
102 10
10) 2
1 0)
210
1) 02)10
10 5 10 3
101 10 10 10 1
Number of unknowns [] Number of unknowns []
Figure 3.5: Error of the dominant transverse propagation constant for a rectangular
waveguide as a function of the number of unknowns. TE modes (left) and
TM modes (right).
1.5 (denoted 'Basis 2') and the proposed hierarchical mixedorder TVFE of order 1.5
(denoted 'Basis 3').
In all cases, we observe the expected behavior that as the number of unknowns
increases, the error decreases. Furthermore, higher order TVFEs are seen to be superior to the lowest order TVFE for a given number of unknowns. Matrices based on
the proposed hierarchical mixedorder TVFE of order 1.5 were observed to be better
conditioned than matrices based on Peterson's interpolatory mixedorder TVFE of
order 1.5, something we discuss in more detail in chapter 5. More importantly, Basis 2 and Basis 3 give indistinguishable results. This is not surprising since the two
mixedorder TVFEs are related through a linear transformation, see Appendix 1B,
and consequently span the same space within each triangle. Thus, the proposed hierarchical mixedorder TVFE gives results that are identical to those obtained by using
a traditional interpolatory mixedorder TVFE and in addition has the advantages of
being hierarchical and (for this particular application) leading to better conditioned
matrices. Similar observations were made for higher order modes.
32
3.3 Application to opendomain problems
In this section, various TVFE options are employed for solution of opendomain
problems. The pertinent formulation is given and numerical results are presented.
3.3.1 Formulation for opendomain problems
Consider a twodimensional scattering problem where the scatterer and all fields
are independent of a spatial coordinate, say z. The scatterer can be composed of PElC
and isotropic dielectric and / or magnetic materials and is situated in free space. The
relative permittivity and permeability is denoted by Er and [,r respectively. The scatterer is illuminated by a TE polarized ' (the derivation for TM polarization closely
parallels the one for TE polarization) incident electromagnetic field (EHZ) (subscript 't' denotes 'transverse to z', subscript 'z' denotes 'zdirected' and superscript
'i' denotes 'incident') and the scatterer then gives rise to a TE polarized scattered
electromagnetic field (Es, Hs) (superscript 's' denotes 'scattered'). The configuration
is illustrated in Fig. 3.6. The total electric field (transverse to z) is then
Et = El + Es (3.20)
and the total magnetic field (zdirected) is
Hz = Hz + Hs. (3.21)
Similar to the derivation in section 3.2.1, a weak vector wave equation for the
total electric field Et can be derived from the general weak vector wave equation (2.5)
'In this section, polarization is with respect to the zaxis.
33
Hi
E;S EE l. /H:
H Et
Scatterer
Figure 3.6: Geometry of a scatterer illuminated by a TE polarized incident electromagnetic field and giving rise to a TE polarized scattered electromagnetic
field.
and (2.7)(2.10) for E and 7 by carrying out the substitutions E ~ Et, T + Tt,
V ~ Vt + ^Z = Vt, V 4 S = rFv  Fs, dV ~ dS and dS ~ dL and letting J  0
and M = 0. We arrive at
((Vt x T, (V x t) V )  k 2 Tt (r,Et)) dS
fi
T= Tt, V x ( E) dL (32)
with ko = VC0/oo( being the free space wave number. The computational domain S
is of infinite extent at this point but will become finite through the introduction of
a mesh truncation scheme, see below. Introducing the separation (3.20) of Et in the
weak vector wave equation (3.22) gives
 Tt' (Vt x (Vt X E')  k ErE) dS (3.23)
where the righthand side vanishes in free space since the incident field is Maxwellian
and therefore fulfils the vector wave equation in free space. We now discretize S into
Ns triangular elements via
Ns
S Z (3.24)
e=l
34
and enforce (3.23) in each element,S. Next, we expand the scattered electric field E6t
in 5' via (2.15) and choose the vector weighting functions Tt = W, i = 1,., N,
to arrive at
N 1
/ ((Vt x W). (Vt x We)  k W (rWJ)) d E+ w x (n V x Es) (dL
= W  (Vt x (Vt x E) r2 E dS. (3.25)
JSe Pr
Above, the boundary integral over Fse was not considered. This terms requires
special attention as it is through this term boundary conditions, including the radiation condition, are incorporated. Two different mesh truncation schemes are considered, an artificial absorber (AA) and a boundary integral (BI).
With an AA mesh truncation scheme, the mesh is terminated by a layer of a
fictitious homogeneous and isotropic material backed by a PEC surface and placed a
certain distance from the scatterer. It is important to note that the AA influences
only the scattered field, not the incident field. In this case, evaluation of the boundary
integral in (3.25) along all pertinent boundaries is trivial. In conclusion, each element
leads to a set of element equations that can be formulated in matrix form as
[A'] {EC} = {gC} (3.26)
where [Ae] is a known N x N matrix, {Ee} is an unknown N x 1 vector and {g&}
is a known N x 1 excitation vector. Subsequent assembly of element equations then
leads to a sparse global matrix equation system of the form
[A] {E} = {g} (3.27)
35
where [A] and {g} are a known matrix and vector, respectively, and {E} is a vector
containing the unknown expansion coefficients EJ for the electric field as introduced
by (2.15).
With a BI mesh truncation scheme, the mesh is terminated by a contour where an
exact integral equation incorporating the radiation condition is enforced. We denote
this contour by ro. Consistent with the discretization (3.24) of S into triangular
elements, we discretize Fo into No piecewise straight segments via
No
Fo =. (3.28)
v=1
For an element Se that does not bound FO, the discretization of (3.25) is again straightforward and we arrive at a set of element equations of the form
[A6] {Ef} = {g6} (3.29)
where [Ae] is a known N x N matrix, {Ee} is an unknown N x 1 vector and {ge} is
a known N x 1 excitation vector. For an element Se that bounds Fo, we are faced
with the incorporation of the integral
W n x (Vt x E) dL = —jwto f W. *. x Hz dL (3.30)
with the index 7y (describing a segment of Fo) being a known function of the index e
(describing a triangular element bounding Fo). To discretize (3.30), we introduce an
expansion of Hs over Fo via
No
H  HZ Z X HZ WZ Z (3.3 1)
Y=1
where Hz are unknown expansion coefficients and Wj are scalar pulse basis functions. In this case, we arrive at a set of element equations that can be formulated in
36
matrix form as
[Ao] {E} + {Bo} H0 {go} (3..32)
where [A]j is a known N x N matrix, {Eoe} is an unknown N x 1 vector, {Boe} is a
known N x 1 vector, Hoe is the unknown expansion coefficient Hzs, for the yth segment
of Fo corresponding to the eth triangular element of S and {gO} is a known N x 1
excitation vector. The expansion (3.31) of Hs introduces No additional unknowns
and hence No additional equations must be constructed. These are provided via the
exact integral equation [3]
H(p) = 4 j H (ko p  p'l (E(p') + E (p)) x dn' dS'
4 ( Jro
+. / V x (H (kop  ) x (H(p) + H(p')) d S' (3.33)
3 o
relating Es and Hs on Fo. In this equation, Co = VU//o is the intrinsic impedance
of free space, H(2) is the Hankel function of zeroth order and second kind and p is
the usual polar vector in a circular cylindrical coordinate system. Discretization of
Es and Hs leads via testing to No equations that can be formulated in matrix form
as
{Ho} = [Ao] {Eo} + [Bo] {Ho} + {go} (3.31)
where {Ho} is an unknown No x 1 vector, [Ao] is a known No x No matrix, {Eo} is an
unknown No x 1 vector, [Bo] is a known No x No matrix and {go} is a known No x 1
excitation vector. Upon assembly of (3.29), (3.32) and (3.34), we arrive at a partly
sparse / partly full global matrix equation system of the form
[A] {E H} {g} (3.35,)
37
where [A] and {g} are a known matrix and vector, respectively, and {E H}' is
a, vector containing the unknown expansion coefficients E. for the electric field as
introduced by (2.15) and the unknown expansion coefficients H"s3 for the magnetic
field as introduced by (3.31).
3.3.2 Numerical results for opendomain problems
Finite element / artificial absorber (FE/AA) and finite element / boundary integral (FE/BI) computer codes were developed to evaluate the scattering of a TE or TM
polarized plane wave by an arbitrary infinite cylinder composed of PEC and isotropic
dielectric and / or magnetic materials. These are based on the FEM in conjunction
with an AA or BI mesh truncation scheme as detailed above. The AA termination
scheme is approximate but attractive for reasons of simplicity. We use a fictitious
material of relative permittivity and permeability 1  j2.7 and thickness (.25Ao (A0
denotes the free space wavelength) backed by a PEC surface and placed a distance
0.5A0 from the scatterer. The BI termination scheme is exact until discretized and
coupled with a FE system and hence it is attractive for rigorous truncation of 1F'E
meshes. For our test, the integral contour is situated a slight distance away from the
scatterer so that piecewise constant (lowest order) expansions can be employed for
discretizing and testing the BI. The resulting matrix equation system is solved using
a quasi minimal residual (QMR) solver [72] with a relative tolerance of 10.
In the following, we compare the scattering by various cylinders using different
TVFE options and different uniform discretizations to demonstrate the merits of
the proposed hierarchical mixedorder TVFEs of order 0.5 and 1.5 when the field is
selectively expanded over the computational domain. The reference results in this
chapter are obtained using the method of moments (MoM) code RAM2D developed
by Northrop. They are all found with a very fine discretization and thus can be
considered accurate.
To test the AA termination scheme, we consider a square PEC cylinder of side
length A0 situated in free space. Centered on the upper side of the cylinder is a
rectangular groove of length Ao/2 and height Ao/4. The groove is filled with a material
characterized by the relative permittivity Or 2  j2 and the relative permeability
/ar = 2 j2. The cylinder is illuminated by a TE polarized homogeneous plane
wave whose propagation vector forms a 45~ angle with all sides of the cylinder, as
illustrated in Fig. 3.7. In Fig. 3.8, we give a comparison of results for the twodimensional radar cross section (RCS) or echo width J21) normalized to Ao0 as a
function of the observation angle 4 2. The MoM result is denoted 'MoM'. For a mesh
where the generic edge length is 0.15Ao, the FE/AA result using the mixedorder
TVFE of order 0.5 is denoted 'FEM  1 TVFE  Coarse mesh' and the FE/AA result
using selective field expansion (with the groove and a layer surrounding the scatterer
as the region in which the proposed hierarchical mixedorder TVFE of order 1.5 is
employed) is denoted 'FEM  2 TVFEs  Coarse mesh'. For a, mesh where the generic
edge length is O.1o, the FE/AA result using the mixedorder TVFE of order 0.5 is
denoted TFEM  1 TVFE  Denser mesh'.
The 'FEM  1 TVFE  Coarse mesh' result is seen to compare reasonably well with
2 =_ 450 corresponds to backscatter and < = 225~ corresponds to forward scattering, see Fig. 3.7.
39
H1,o xo xo
4 2 4 P
4 4
  
PEC r =2j2
tr =2j2
to
Figure 3.7: Square cylinder with a groove illuminated by a TE polarized plane wave.
20 1,,,
MoM
FEM  1 TVFE  Coarse mesh.. — FEM  1 TVFE  Denser mesh
 FEM  2 TVFEs Coarse mesh
4
45 65 85 105 125 145 165 185 205 225
Observation angle ~ [degrees]
Figure 3.8: Bistatic RCS of the cylinder in Fig. 3.7.
the MoM result. However, discrepancies can be seen and this is not surprising since
the mesh is relatively coarse. For the denser mesh, the 'FEM  1 TVFE  Denser
mesh' result shows a slight improvement. However, by keeping the original mesh
and employing the proposed hierarchical mixedorder TVFE of order 1.5 close to the
scatterer where the field can be expected to vary rapidly and accurate modeling is
therefore necessary. the 'FEM  2 TVFEs  Coarse mesh' result shows a significant
40
improvement. It matches the MoM result exactly except in regions surrounding nulls
and it was obtained using less computational resources (less unknowns, less nonzero
matrix entries and less matrix solution time) than the 'FEM  1 TVFE  Denser mesh'
result. In conclusion, we observe selective field expansion to be superior to the more
traditional approach of using a denser mesh and the mixedorder TVFE of order 0.5
throughout the computational domain.
We now consider a slightly different cylinder geometry by introducing a slab of
length Ao and height Ao/4 on top of the cylinder. As depicted in Fig. 3.9, the groove
is filled with free space and the slab has the relative permittivity r = 2  j2 and the
relative permeability,r = 1. For the same illumination as before, results similar to
those in Fig. 3.8 are given in Fig. 3.10 and they reinforce the conclusions from the
previous case: The 'FEM  1 TVFE  Coarse mesh' result compares reasonably well
with the MoM result and the 'FEM  2 TVFEs  Coarse mesh' result is, though found
using less computational resources than the 'FEM  1 TVFE  Denser mesh' result,
significantly more accurate than the 'FEM  1 TVFE  Denser mesh' result.
Explicit parameter values quantifying the computational savings for the results in
Fig. 3.8 and Fig. 3.10 are given in Tab. 3.13.2, respectively. In both cases, improved
accuracy is obtained for less nonzero matrix entries (i.e., less memory) and less
solution time.
To test the BI termination scheme, we consider a rectangular PEC cylinder of
width 3Ao and height 0.25Ao covered by a dielectric material of width 3Ao and height
Ao whose relative permittivity is or = 2  j0.5. On top of the dielectric is a grating
structure of height 0.25Ao consisting of three PEC strips of lengths 0.75A0, 0.5A0
41
H1
^ i t h
4 l r r P.. 4
EC
2 ^.._.  —. . —
PEC r =2j2
Pr=1
To
Figure 3.9: Square cylinder with a groove loaded by a dielectric slab illuminated by
a TE polarized plane wave.
20 l l l l
MoM
FEM  1 TVFE  Coarse mesh
10   FEM  1 TVFE  Denser mesh
 FEM  2 TVFEs  Coarse mesh:a 0
10 r
20 ,i
t;
30
45 65 85 105 125 145 165 185 205 225
Observation angle ( [degrees]
Figure 3.10: Bistatic RCS of the cylinder in Fig. 3.9.
and 0.75A0, respectively, separated by dielectric inserts of length 0.5Ao having the
relative permittivity r, = 10. The structure is illustrated in Fig. 3.11. A structure
of this type (but of different size and different material composition) is of practical
interest for guiding electromagnetic waves and below we demonstrate how a selective
field expansion can lead to accurate modeling of the fields in and near the grating
structure and hereby accurate prediction of the scattered field. The structure is
42
Nonzero Matrix RMS
FE/AA approach Unknowns matrix entries solution time error
1 TVFE  Coarse mesh 811 3955 9 seconds 3.8657 d]
1 TVFE  Denser mesh 3977 19664 111 seconds 2.1059 d]
2 TVFEs  Coarse mesh 1070 7292 21 seconds 1.5614 d]
Table 3.1: Comparison of relevant parameters for the FEM results in Fig. 3.8.
Nonzero Matrix RMS
FE/AA approach Unknowns matrix entries solution time error
B
B

1 TVFE  Coarse mesh 900 4398 11 seconds 2.1952 d]B
1 TVFE  Denser mesh 4469 22118 133 seconds 1.6213 d]3
2 TVFEs  Coarse mesh [ 1280 9466 ] 34 seconds  0.6114 dB
Table 3.2: Comparison of relevant parameters for the FEM results in Fig. 3.10.
Nonzero Matrix RMS
FE/BI approach Unknowns matrix entries solution time error
1 TVFE  Coarse mesh 1230 19662 172 seconds 5.3980 dB3
1 TVFE  Denser mesh 2421 43961 301 seconds 0.9807 d13
2 TVFEs  Coarse mesh 1716 25884 534 seconds 0.7705 d13
Table 3.3: Comparison of relevant parameters for the FEM results in Fig. 3.12.
situated in free space and illuminated as the previous two cylinders. Results similar
to those in Fig. 3.8 and Tab. 3.1 are given in Fig. 3.12 and Tab. 3.3. The results again
reinforce the conclusions reported above, except that the matrix solution time for the
'FEM  2 TVFEs  Coarse mesh' result is larger than that for the 'FEM  1 TVFE 
Denser mesh' result. We note that the preprocessing, matrix element evaluation and
assembly time is significantly larger for the 'FEM  1 TVFE  Denser mesh' result
than the 'FEM  2 TVFEs  Coarse mesh' result due to the larger BI system. This
must be kept in mind when interpreting Tab. 3.3.
We note that higher order TVFEs (either Peterson's interpolatory mixedorder
TVFE of order 1.5 or the proposed hierarchical mixedorder TVFE of order 1.5) could
43
6, ILo
4ko
4
Hi
3ko
4
2o 2o o
2 2 2
4,
ko
PI —
PEC plrasas PEC IB1~p BB P C
I 7
I i Ira rI \
S I
S r sit ilii 5i~U~i
I:::~ II":: 'L::
ei;?~i!E I.~: ~ "ii d:
*: IPs
P
7 
k0
4 I / Irk; \ I
I /
~r=10 ~r=2j0.5
Figure 3.11: Grating structure on top of a grounded dielectric illuminated by a TE
polarized plane wave.:10 .
0
I 0
10
20...
45 75 105 135 165 195 225
Observation angle o [degrees]
Figure 3.12: Bistatic RCS of the cylinder in Fig. 3.11.
alternatively be applied throughout the computational domain. This approach was
tested and the two mixedorder TVFEs gave similar and accurate results. However,
the approach could not measure up with the selective one in terms of computational
resources.
44
3.4 Summary
In this chapter, hierarchical mixedorder TVFEs of order 0.5, 1.5 and 2.5 for
triangular elements were proposed and tested for solution of closed as well as opendomain problems. For solution of certain classes of electromagnetic problems, field
expansion using hierarchical mixedorder TVFEs of order 0.5 and 1.5 selectively was
found to be a very promising approach in terms of accuracy, memory and central
processing unit (CIPU) time requirements as compared to a more traditional approach.
45
CHAPTER 4
THREEDIMENSIONAL TVFES
In this chapter, hierarchical mixedorder TVFEs of order 0.5, 1.5 and 2.5 for
tetrahedral elements are proposed and tested for solution of closed as well as opendomain problems. The work in this chapter is published in [5, 9] and expected to be
published in [25].
4.1 Hierarchical mixedorder TVFEs for tetrahedral elements
Hierarchical mixedorder TVFEs for tetrahedral elements have been proposed up
to and including order 1.5 by Webb and Forghani [97], see section 2.2. These were
written up by inspection. The purpose of this section is to propose a set of hierarchical
mixedorder TVFEs for tetrahedral elements beyond order 1.5. Specifically, hierarchical mixedorder TVFEs are presented up to and including order 2. 5 where the
mixedorder TVFE of order 1.5 differs from the one presented by Webb and Forghaii
46
[97]. We derive the hierarchical mixedorder TVFEs from the interpolatory mixedorder TVFEs for tetrahedral elements proposed by Savage and Peterson [65, 77], see
section 2.2, and the hierarchical mixedorder TVFEs for triangular elements proposed
in section 3.1 in a fashion that makes tht he proposed set of hierarchical mixedorder
TVFEs for tetrahedral elements the direct threedimensional equivalent of the set of
hierarchical mixedorder TVFEs for triangular elements proposed in section 3.1. ]ierarchical mixedorder TVFEs for higher orders than 2.5 can be derived by modifying
the TVFEs proposed by Graglia et al. [37].
Consider a tetrahedral element with nodes 1, 2, 3 and 4 for which simplex coordinates are defined in the usual manner, see section 2.2. Below, vector basis functions
are formulated in terms of these coordinates.
A mixedorder TVFE of order 0.5 providing CT/LN variation along edges and
linear variation at faces and inside the element is characterized by NIt = 6 linearly
independent vector basis functions. The threedimensional equivalent of the twodimensional CT/LN vector basis functions presented in section 3.1 is identical to the
vector basis functions presented by Whitney [98], see section 2.2. The 6 e(lgebased
vector basis functions are given by (2.17).
A mixedorder TVFE of order 1.5 providing LT/QN variation along edges and
quadratic variation at faces and inside the element is characterized by N2ett = 90
linearly independent vector basis functions. Savage and Peterson [77] proposed the
12 edgebased vector basis functions (2.18)(2.19) and the 8 facebased vector basis
functions (2.20)(2.21). The 20 linearly independent vector basis functions (2.18)(2.21) do not compare to the vector basis functions (2.17) in a hierarchical fashion.
47
We propose to replace the 12 edgebased vector basis functions (2.18)(2.19) by (2.17)
and
( Cj)((iV  (jVi), i < j. (.1)
The 20 linearly independent vector basis functions (2.17), (4.1) and (2.20)(2.21)
form a mixedorder TVFE of order 1.5 that compares hierarchically to the proposed
mixedorder TVFE of order 0.5.
A mixedorder TVFE of order 2.5 providing QT/CuN variation along edges and
cubic variation at faces and inside the element is characterized by Net = 45 linearly
independent vector basis functions. Savage and Peterson [65, 77] proposed the 18
edgebased vector basis functions (2.26)(2.27), the 24 facebased vector basis functions (2.28)(2.31) and the 3 cellbased vector basis functions (2.32). The 45 linearly
independent vector basis functions (2.26)(2.32) do not compare to the vector basis
functions (2.17) in a hierarchical fashion. We propose to replace the 18 edgebased
vector basis functions (2.26)(2.27) by (2.17), (4.1) and
(  j)2((.V(o  QV), < j. (4.2)
Further, we propose to replace the 8 facebased vector basis functions (2.28)(2.29) by
(2.20)(2.21). The 45 linearly independent vector basis functions (2.17), (4.1)(4.2),
(2.20)(2.21) and (2.30)(2.32) form a mixedorder TVFE of order 2.5 that compares
hierarchically to the proposed mixedorder TVFEs of order 0.5 and 1.5.
48
4.2 Application to closeddomain problems
In this section, various TVFE options are employed for solution of closeddomain
problems. The pertinent formulation is given and numerical results are presented.
4.2.1 Formulation for closeddomain problems
Consider a possibly inhomogeneous and anisotropic cavity V characterized by the
permittivity tensor E and the permeability tensor 7J. The cavity V is Eassumed to be
free of sources and the boundary IV of V is assumed to be a PEC surface. The electric
and magnetic field intensities E and H, respectively, then fiulfil the weak vector wave
equations (2.5)(2.6) with the surface integrals over Fv eliminated. Introducing now
the relative permittivity and permeability tensors gr and 'r by (2.7)(2.8), we obtain
((V x T) ( .1 V x E) k2 T ( * E)) dV = 0 (4.3)
j((V x T) (j1' V x H) k2 T ( r H)) dV 0 (4.4)
with k = wv/ o/o being the wave number. We now discretize V into Nv tetrahedral
elements via
ANv
e=l
and enforce (4.3)(4.4) in each element V'. Next, we expand the electric and magnetic
fields Ee and He in Ve via (2.15)(2.16). Choosing the vector weighting functions
T = We, i = ],., N, then leads to
N N
(V X W) ( 1 V x WJ) dlE = k2 We *j ( d (4.6)
r z Jz::
49
N N
/ (V XW,).( 1 *V XHWE) Jdll =k2 Wt ( r W) dl/H;. (4.7)
These element equations can be formulated in the matrix form
[Ae]xz} = k2[Be]{x} (4.8)
for both electric field formulation (Eformulation) and magnetic field formulation ('Hformulation). Assembly of element equations then leads to a global matrix equation
system of the form
[A]{x} = k2[B]{x (4.9)
for both E and Hformulation. We note that the assembly process includes the
enforcement of boundary conditions along Fv. For Eformulation, this leads to a
condensation of the equation system making the equation system for Eformulation
smaller than that; for Hformulation.
4.2.2 Numerical results for closeddomain problems
Homogeneous and isotropic rectangular cavity
Consider a homogeneous and isotropic rectangular cavity of normalized dimensions
1 x 0.75 x 0.5, see Fig. 4.1. The exact eigenvalues for this geometry are wellknown
[40]. A FEM solution (Eformulation) for the eigenvalues of the cavity is carried
out for various tetrahedral meshes of different average edge length with the proposed
mixedorder TVFEs of order 0.5 and 1.5 used for field expansion.
The convergence rate for the two cases is illustrated in Fig. 4.2 where the average
error of the first eight eigenvalues is plotted in percent as a function of the average edge
50
0.5 0.
i,...........
0.75
Figure 4.1: Homogeneous and isotropic rectangular cavity.
length in the mesh (loglog plot). The approximate distribution around a straight line
suggests that the average error decreases as xn for a decreasing average edge length.
For the mixedorder TVFE of order 0.5, the exponent is 2.37 which is slightly larger
than the expected value 2 [77]. This is due to the very low average error 0.56% for
the average edge length 0.175. Similarly, for the mixedorder TVFE of order 1.5, the
exponent is 4.66 which is again larger than the expected value 4 [77] and the exponent
3.86 found in [77] for a different mixedorder TVFE of order 1.5. This demonstrates
that the mixedorder TVFE of order 1.5 proposed in this dissertation has slightly
better convergence properties than the one in [77] for this particular geometry and
for the employed meshes.
30
v v TVFE of order 0.5
10 TVFE of order 0.5  curve fit
o o TVFE of order 1.5
   TVFE of order 1.5  curve fit
0 ' 1  1 0 /
< 0.3 o
o /,
0.1
0.0 o
0.1 0.2 0.3 0.4 0.5 0.6
Average edge length
Figure 4.2: Convergence rate for expansion of the field within the homogeneous and
isotropic rectangular cavity in Fig. 4.1 using mixedorder TVFEs of order
0.5 and 1.5.
51
Inhomogeneous and isotropic rectangular cavity
Consider a rectangular cavity of normalized dimensions 1 x 0.75 x 0.5. It is composed of a rectangular cavity of normalized dimensions 0.25 x 0.75 x 0.5 filled with a
homogeneous and isotropic dielectric of relative permittivity Er = 10 and a rectangular cavity of normalized dimensions 0.75 x 0.75 x 0.5 filled with a homogeneous and
isotropic dielectric of relative permittivity r, = 1, see Fig. 4.3. The true eigenvalues
can be obtained via accurate numerical solution of an exact analytical transcendental
equation, see for instance [40].
0.25=
0.5 Q'...............
05 "'" 10:
0.75
Figure 4.3: Inhomogeneous and isotropic rectangular cavity.
We compare the error in computing the eigenvalues for the first few modes for
three different cases. In Case 1, we apply the mixedorder TVFE of order 0.5 for a
fine mesh consisting of 778 tetrahedral elements. In Case 2, we apply the mixedorder
TVFE of order 1.5 for a coarse mesh consisting of 130 tetrahedral elements. In Case 3,
we apply the mixedorder TVFEs of order 0.5 and 1.5 selectively for the same coarse
mesh consisting of 130 tetrahedral elements. The dielectric material is modeled with
the mixedorder TVFE of order 1.5. In free space, the region away from the dielectric
is modeled with the mixedorder TVFE of order 0.5. This makes the region close
to the dielectric a transition region where incomplete mixedorder TVFEs (complete
to order 0.5 but not to order 1.5) are applied. The three cases are summarized in
52
Tab. 4.1 where also the number of unknowns and number of nonzero matrix entries
are given for E and Hformulation.
Eformulation Hformulation
TVFE Matrix Matrix
Case order(s) Elements Unknowns entries Unknowns entries
I
1 0.5 778 695 9161 1151 16051
2 1.5 130 604 17876 1084  38080
i i i.....
I
3 0.5/1.5 130 354 9120 668 20060
Table 4.1: Definition of Case 13.
The eigenvalue error of the first six modes for the three different cases is given in
percent in Fig. 4.44.5 for E and Hformulation, respectively.
Comparing Case 2 and Case 3, we see that the average error is approximately the
same. This is the case for both E and Hformulation. Thus, we do not compromise
accuracy by modeling only part of the cavity with the mixedorder TVFE of order 1.5
and the remainder of the cavity with the mixedorder TVFE of order 0.5 as compared
to using the mixedorder TVFE of order 1.5 throughout the cavity. However, the
storage and CPU time requirements drop significantly making selective field expansicn
an attractive option.
Comparing Case 1 and Case 3, we see that Case 1 is best for Eformulation while
Case 3 is best for Hformulation. This demonstrates that the choice of TVFE(s)
is not necessarily unambiguous. This unambiguity can be further demonstrated by
considering the eigenmode corresponding to a given eigenvalue. As noted above, Case
3 is generally better than Case 1 for Hformulation. However, for the TM111 mode,
Case 1 gives a more accurate eigenvalue than Case 3. The tangential magnetic field on
the back PEC wall for the TM111 mode is plotted for Case 1 and Case 3 in Fig. 4.64.7,
53
1 1 Case 1
/... Case 2
2 Case 3
TE110 TM111 TE101 TE111 TM121 TE120
Figure 4.4: Eigenvalue error (Eformulation) for the inhomogeneous and isotropic
rectangular cavity in Fig. 4.3.
3 Case 1
L, ~~.Case 2
i 2 Case 3
0a
A r1'~ L: H.. 41 Il
TE110 TM111 TE101 TE111 TM121 TE120
Figure 4.5: Eigenvalue error (Hformulation) for the inhomogeneous and isotropic
rectangular cavity in Fig. 4.3.
respectively. Clearly, transitions are much smoother for Case 3 than for Case 1 and
significantly more accurate fields at edges can be observed for Case 3 as compared
to Case 1 (the tangential magnetic field for Case 1 possesses normal components at
edges that do not exist for Case 3). Thus, for a mode where Case 1 gives a more
accurate eigenvalue than Case 3, Case 3 gives a more accurate eigenmode than Case
1. Further discussion of TVFE ambiguity is given in [75].
4.3 Application to opendomain problems
In this section, various TVFE options are employed for solution of opendomain
problems. The pertinent formulation is given and numerical results are presented.
54
Figure 4.6: Magnetic field on the back PEC wall for the TM11l mode of the inhomogeneous and isotropic rectangular cavity in Fig. 4.3 with mixedorder
TVFE of order 0.5 applied (Case 1).
R., ~;s:. i:  S.
Figure 4.7: Magnetic field on the back PEC wall for the TM,11 mode of the inhomogeneous and isotropic rectangular cavity in Fig. 4.3 with mixedorder
TVFEs of order 0.5 and 1.5 applied (Case 3).
4.3.1 Formulation for opendomain problems
Consider a PEC patch antenna backed by a PEC cavity and recessed in an infinite PEC ground plane. The cavitybacked patch antenna is situated in free space
characterized by the permittivity eo and the permeability do as illustrated in Fig. 4.8
(side view) and Fig. 4.9 (top view) for the case of a triangular patch backed by a finite
circular cylindrical cavity. The volume of the possibly inhomogeneous and anisotropic
cavity is denoted by V and characterized by the permittivity tensor E and the permeability tensor 1u. The region V can include internal PEC surfaces. Internal resistive
and impedance surfaces can easily be incorporated [92] but we restrict ourselves to
internal PEC surfaces in this dissertation. The boundary of the cavity V defined as
55
O toin
PEC
A
Figure 4.8: Side view of a cavitybacked patch antenna recessed in an infinite PEC
ground plane.
Figure 4.9: Top view of a cavitybacked patch antenna recessed in an infinite PEC
ground plane for the case of a triangular patch and a circular cylindrical
cavity.
the top surface (the PEC patch or the nonPEC part between the PEC ground plane
and the PEC patch), the PEC side surfaces, the PEC bottom surface and any internal
PEC surfaces is denoted by Fv. The nonPEC part of Fv is denoted by S while the
PEC part of Fv is denoted by Fv S.
Assuming the antenna feed is described by an electric volume current density
J within 1, the electric and magnetic field intensities E and H, respectively, fulfil Maxwell's equations (2.1)(2.2) for M = 0 and E fulfils the weak form (2.5) of
the vector wave equation (2.3) for M = 0. Combination of the weak vector wave
equation (2.5) with Maxwell's equation (2.1) for M = 0 and introduction of the rel
56
ative permnittivity and permeability tensors ar and r via (2.7)(2.8) then leads to E
fulfilling
((Vx T). (1 V xE)  T (r E)) dV = jWo I T (i x H)dS
ji ofT TJdf (4.10)
Jv
where ko = wvtoo1 is the free space wave number, where n denotes the unit normal
vector to S directed out of V and where a surface integral over the PEC part Fv\5,S
of Fv has been eliminated since it vanishes.
We now proceed to discretize (4.10) using the hybrid FE/BI method. We initially
neglect the surface integral in (4.10) and discretize
j ((VxT r V x T) ( r. V xE T E) T)dV ( j E) JdV (4.11)
to obtain a FE system of linear equations. Upon discretization of the surface integral
jio / T (n x H) dS (4.1)
in (4.10) using a BI (an expression for H in terms of E) and subsequent correction
of the FE system of equations according to (4.10), the final FE/BI system of linear
equations is obtained.
We consider (4.11). Let us discretize V into Nv tetrahedral elements via
Nv/
V VCe (4.13)
eeeee eeeee e
and enforce (4.11) in each element V6". Next, we expand the electric field E" in Ve via
57
(2.15) and choose the vector weighting functions T = We, i = 1,, N, to arrive at
N 3
Z / ((V x W) * ( V X Wj)
j=1
 k We (. W)) dllE =  jio ' We J dV. (4.14)
e
The N element equations (4.14) for the eth element can be formulated in the matrix
form
[A6]{E6} = {g} (4.15)
where [A6] is the element matrix with entries
A7 j/ ((V x we) (,1 V x We)  ko2 W. ( r W;)) (4.16)
33v<3
{ge} is the excitation vector with entries
g = jwo i We J dV (4.17)
and {Ee} is a vector containing the unknown expansion coefficients EJ. Assembly of
the element equations (4.15) and elimination of unknowns along Fv\S then leads to
a global matrix equation system that can be formulated
Al, AIS E g I
(4.18)
ASI AsS
As, As Es gS
where [A,], [AIS], [AsI] and [Ass] are known matrices, {g1} and {gS} are known
vectors and {EI} and {ES} are vectors containing the unknown expansion coefficients
associated with the interior of V (superscript I) and the boundary S of V (superscript
S). This is the desired FE system of linear equations resulting from discretization of
(4.11).
58
We now consider the surface integral (4.12) which is in terms of the magnetic
field H. As mentioned earlier, we seek an expression in terms of the electric field E.
This is obtained by using the surface equivalence principle [45] to relate H to E just
outside S via
H = 2jwso Go(r, r'). (E' x h') dS' (4.19)
where Go is the dyadic free space Green's function given by
Go(r, r') = (I + VV ) Go(r, r') (4.20)
0
with I being the identity tensor and Go being the scalar free space Green's function
given by
 jko rr'I
Go(r, r) 47ir r' (4.21)
Substitution of (4.19) for H into the surface integral (4.12) yields
jWbo jT (h x H) dS = 2 k 2 T (n x j Go(r, r') (E' x n') dS' ) dS (4.22)
which is in terms of E and can be directly discretized. Before doing so, it is advantageous to substitute Go from (4.20) and use elementary vector identities to obtain
jwuo J T. (n x H) dS =2 ko j j Go(r, r') (E' x nh') (T x n) dS' dS
2 is Go(r, r') V'. (E' x n') V (T x n) dS' dS. (4.23)
The singulatities of the integrands on the righthand side of (4.23) are of a lower
order than the singulatity of the integrand on the righthand side of (4.22.) and hence
the tangential electric field in (4.23) can be expanded using an expansion of a lower
59
order than that in (4.22). Consistent with the discretization of V into Nvi elements
via (4.13), we now discretize S into Ns triangular faces via
Ns
S S. (4.24)
Next, we expand the tangential electric field Etg linearly in Se' via
Egn S = E EVJ (4.25)
j=l
where Etj are unknown expansion coefficients and
V' = W '  (W ~' n) n) (4.26)
are known vector basis functions, respectively, for the e'th segment. With the field
representation (4.25) over S, the field uniqueness requirements at S mandate that all
higher order edge and facebased vector basis functions associated with edges and
faces of S are eliminated. This implies that mixedorder TVFEs for elements with a
triangular face bounding S can be complete to order 0.5 or incomplete to order 1.5
but not complete to order 1.5. Further discussion is given in section 8.2. Choosing
the vector weighting functions T = Ve, i = 1,.., M, we obtain the equations
Ns M
j W/oJ T (n l H) dS = j / / Go(r, r') (2 ko VJ V.
2 V' (VJ' x n') V. (V x n)) dS' dS Ej. (4.27)
Upon assembly of these equations, we end up with
0 0 E'
jWto /f T (h x H) dS' (4.28)
0s Bss Es
60
which is the desired discretization of (4.12).
The FE/BI discretization of (4.10) now follows from the FE discretization (4.18)
of (4.11) representing the FE part of (4.10) by combining it with the BI discretization
(4.28) of (4.12) representing the BI part of (4.10). We obtain the final FE/BI system
All AIS El g: A {. (4.29)
AS ASS  Bss Es gS
We note that the adaptive integral method (AIM) [11, 12] can be invoked for
avoiding the storage of a full BI matrix and for speeding up matrixvector products
within an iterative solver. For conservative choices of AIM parameters (dense AIM
grid and large AIM nearzone threshold), these computational improvements come
with virtually no compromise in accuracy. The AIM was conservatively taken advantage of for some of the computations in this dissertation but since the AIM is not
essential to this dissertation the formulation is not repeated here.
4.3.2 Numerical results for opendomain problems
Consider a square metallic patch antenna backed by a rectangular cavity recessed
in an infinite metallic ground plane, as illustrated in Fig. 4.10 (side view) and Fig. 4.11
(top view). The cavitybacked patch antenna is situated in free space characterized
by the permittivity Co and the permeability,/o. The cavity is of dimensions 1.85 cm
x 1.85 cm x 0.15 cm and filled with a dielectric material of permittivity 10 Eo and
conductivity 0.0003 S/cm. The patch is of side length 0.925 cm and centered in the
cavity aperture. It is fed by a vertical coaxial line whose outer conductor is attached
61
Co P0
0.4625 cm 0.925 cm 0.4625 cm
0.15 cm
10~% 0 0.0003 S/cm
Figure 4.10: Side view of a square metallic patch antenna backed by a dielectricfilled
rectangular cavity recessed in an infinite metallic ground plane.
Figure 4.11: Top view of a square metallic patch antenna backed by a dielectricfilled
rectangular cavity recessed in an infinite metallic ground plane.
to the ground plane and whose inner conductor is attached to the patch at the mid
point of an edge, as illustrated in Fig. 4.104.11. The coaxial feed is modeled as a
vertical probe of constant current.
An almost identical antenna was considered by Schuster and Luebbers [81]. In [81],
the cavity walls and the ground plane was removed and a similar patch on a similar
but finite grounded dielectric substrate was analyzed using the finite difference time
domain (FDTD) method. In spite of these geometrical differences, the two antennas
are expected to have the same input impedance and, consequently, the same resonant
frequency since the dominant fields are confined to a volume under and in the vicinity
of the patch. The resonant frequency was found in [81] to be 4.43 GHz. The resistance
62
at resonance was found to be 400 Q while the reactance was in the range of 230 Q
to 170 Q close to resonance. We note that the results in [81] were found with an
extremely fine discretization and hence can be considered accurate.
The patch antenna, is analyzed using the FE/BI method in conjunction with an
iterative QMR solver. We discretize the cavity into tetrahedral elements and consequently discretize the surface forming the boundary between the cavity and free
space into triangular faces. Two different TVFE options are applied. The first TVFE
option is to use the mixedorder TVFE of order 0.5 throughout the mesh. For a
mesh of average edge length 0.260 cm (Case 1), the input impedance is determined
as a function of frequency and the resonant frequency of the patch is predicted. The
coarse discretization of Case 1 means that this resonant frequency is most likely not
accurate. For meshes of average edge lengths of 0.188 cm (Case 2), 0.153 cm (Case
3) and 0.133 cm (Case 4), more accurate resonant frequencies but also higher conmputational costs can be expected. The second TVFE option is to use the hierarchical
mixedorder TVFE of order 1.5 close to the radiating edges and the mixedorder
TVFE of order 0.5 elsewhere. For the meshes of average edge length 0.260 cm (Case
5) and 0.188 cm (Case 6), the input impedance is again determined and the resonant
frequency is again predicted. The effectiveness of this approach (Case 56) in terms
of accuracy, CPU time and memory requirements is compared to the previous one
(Case 14). The six cases are summarized in Tab. 4.2.
Real and imaginary parts of the input impedance as a function of frequency are
given in Fig. 4.124.13 for Case 16 and corresponding resonant frequencies are provided in Tab. 4.2. For Case 14, a larger and larger resonant frequency is observed
63
Average Time per
edge Resonant frequency
TVFE length frequency BI Matrix point
Case order(s) [cm] [GHz] Unknowns unknowns entries [sec]
1 0.5 0.260 3.974 345 120 17119 7.52
2 0.5 0.188 4.147 817 288 89695, 44.78
3 0.5 0.153 4.258 1489 528 291359 222.92
4 0.5 0.133 4.302 2361 840 725791
771.59
5 0.5/1.5 0.260 4.323 827 120 30675 17.33
6 0.5/1.5 ~ 0.188 4.437 1467 288 107963 77.28
6

Table 4.2: Computational effort for Case 16 for the antenna in Fig. 4.104.11.
as the mesh becomes denser and denser. However, even for Case 4, the error as
compared to the result obtained by Schuster and Luebbers is quite large (2.98 X%)
for resonant frequency computation. Use of selective field expansion (Case 56) leads
to a, significant accuracy improvement. Case 5 (error 2.42 %) gives a more accurate
result than Case 14 and Case 6 (error 0.16 %) matches the result by Schuster and
Luebbers almost exactly. The computational cost (number of unknowns, number of
BI unknowns, number of nonzero matrix entries (memory usage) and CPU time per
frequency point) to obtain these results are also given in Tab. 4.2. It is evident that
the second TVFE option corresponding to Case 56 is significantly more attractive
than the first TVFE option corresponding to Case 14. Case 5 gives a nmore accurate
result than Case 4 but uses only 4.22 % of the memory and 2.15 % of the CPU time
that Case 4 does. The accuracy of Case 6 is vastly superior to that of Case 4 and yet
Case 6 uses only 14.88 V% of the memory and 10.02 %o of the CPU time that Case 4
does. We note that the savings in Case 56 are reached in part because coarlse meshes
with higher order TVFEs lead to significantly smaller BI portions of' the resulting
matrix equation systems than fine meshes with lowest order TVFEs.
64
3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7
Frequency [GHz]
Figure 4.12: Real part of the input impedance of the antenna in Fig. 4.104.11 for
Case 16.
300 C ase 16.Case 5
the results are not included in this dissertation.
200 Case Case 2 promising approach for accurate and efficient
20solution7 38 39 4 41 42 43 44 4'5 4'.6 4.7
Frequency [GHz]
Figure 4.13: Imaginary part of the input impedance of the antenna in Fig. 4.104.11
for Case 16.
the results are not included in this dissertation.
4.4 Summary
In this chapter, hierarchical mixedorder TVFEs of order 0.5, 1.5 and 2.5 for
solution of certain classes of electromagnetic problems.
65
CHAPTER 5
ANALYSIS OF CONDITION NUMBERS
In this chapter, the condition numbers resulting from FEM analysis using the
hierarchical mixedorder TVFEs of order 1.5 for triangular and tetrahedral elements
proposed in chapter 3 and chapter 4 are contrasted to those of existing interpolatory and hierarchical mixedorder TVFEs of order 1.5 for triangular and tetrahedral
elements. In addition, an approach for improving the condition numbers of FEM
matrices resulting from selective field expansion is suggested and tested. The work
in this chapter is published in [6].
5.1 Background
Linear equation systems associated with the FEM and hybrid versions hereof are
often solved using iterative solvers which are known to perform relatively poorly
(require many iterations) when the system matrices are badly conditioned. This is
the case for (generalized) eigenvalue problems as well as excitation problems. This
drawback makes it desirable to construct system matrices having small condition
66
numbers. It is generally postulated that higher order TVFEs lead to larger condition
numbers than the lowest order TVXFE and that hierarchical higher order TVFEs lead
to larger condition numbers than interpolatory higher order TVFEs [75]. These are
two postulates that make selective field expansion using hierarchical higher order
TVFEs significantly less attractive.
The condition numbers of element matrices based on the mixedorder TVFEs of
order 1.5 developed by Graglia et al. [37] (interpolatory) and Webb and Forghani [97]
(hierarchical) have been studied before [75]. However, no detailed study has been presented that examines the interrelationships between the condition numbers of element
as well as global matrices based on various interpolatory and hierarchical mixedorder
TVFEs. It is the aim of this chapter to carry out such a comparison. Specifically,
we compare the interpolatory mixedorder TVFEs of order 1.5 developed by Peterson
[66] (twodimensional problems), Savage and Peterson [77] (threedimensional problems), Graglia et al. [37] and Webb and Forghani [97], see section 2.2, with the
hierarchical mixedorder TVFEs of order 1.5 proposed in this dissertation. The study
includes results for two as well as threedimensional problems, TE field formulation
(TEformulation) as well as TM field formulation (TMformulation) (twodimensional
problems), E as well as Hformulation (threedimensional problems), element as well
as global matrices and unnormalized as well as normalized vector basis functions.
Based on the study, an approach for improving the condition numbers of system
matrices resulting from selective field expansion is suggested and tested.
67
5.2 Formulation for closeddomain problems
Given a mixedorder TVFE, different applications lead to structurally different
global matrix systems and hence it is impossible to uniquely define a global matrix
whose condition nurber characterizes the mixedorder TVFE for all applications. If
we consider a waveguide or a cavity with metallic walls, FEM analysis based on a
given mixedorder TVFE leads to element matrix equation systems of the form (for
specifics, see chapter 3 and chapter 4)
[A6]{x6} = A[B6]{x} (5.1)
where each entry of [Ae] is the integration of the dot product of the curl of two vector
basis functions over the element and each entry of [B6] is the integration of the dot
product of two vector basis functions over the element. Assembly of element equations
then leads to a global matrix equation system of the form
[A]{x}= X[B]{x}. (5.2)
The matrices [A6] and [A] are singular while [B6] and [B] are nonsingular. In this
chapter, we perform FEM analysis of waveguides and cavities with metallic walls
based on different mixedorder TVFEs and use the condition numbers of [Be] and [B]
as indicators of the matrix condition numbers that the different mixedorder TVFEs
lead to. We note that the condition number of a matrix can be defined in a variety
of ways [36]. The condition number used in this dissertation is the absolute value of
the ratio of the maximum to the minimum eigenvalue.
68
5.3 Homogeneous application of TVFEs
In this section, condition numbers for FEM matrices are computed when mixedorder TVFEs of order 0.5 and 1.5 are used throughout computational domains.
5.3.1 Twodimensional case
Consider a rectangular waveguide of normalized dimensions 1 x 0.5, as illustrated
in Fig. 5.1. For TE and TMformulation, the eigenvalues are determined using the
FEM (64 triangular elements) with the mixedorder TVFEs by Whitney, Peterson,
Graglia et a/., Webb and Forghani and that proposed in this dissertation used for
field expansion (unnormalized as well as normalized vector basis functions).
0.5,
Figure 5.1: Illustration of a rectangular waveguide.
In Tab. 5.1, the condition number of the global matrix [B] is given for TE and
TMformulation with unnormalized as well as normalized vector basis functions. The
abbreviations 'Wh' 'Pe' 'Gr' 'An' and 'We' denote the mixedorder TVFEs developed byr Whitney, Peterson, Graglia et a/., that proposed in this dissertation and that
developed by Webb and Forghani, respectively. The higher order TVFEs are seen to
lead to much larger condition numbers than the lowest order TVFE. Also, unnormalized vector basis functions are seen to lead to much larger condition numbers
than normalized vector basis functions for all TVFEs. For interpolatory TVFEs,
69
the TVNFE by Graglia et al. leads to better conditioned matrices than the TVFE
by Peterson while for hierarchical TVFEs, the TVFE proposed in this dissertation
leads to better conditioned matrices than the TNFE by Webb and Forghani. Tlhe
hierarchical TVFE proposed in this dissertation leads to better conditioned matrices
than the interpolatory TVFE by Peterson, especially when the vector basis functions
are normalized. The interpolatory TVFE by Graglia et al. leads to slightly better
conditioned matrices than the hierarchical TVFE proposed in this dissertation.
TEformulation TMformulation
TVFE Unnormalized Normalized Unnormalized Norma]lized
Wh 3 3 5 6
Pe 428 90 479 149
Gr 144 37 144 61
An 402 43 451 72
We 827 71 926 101
Table 5.1: Condition numbers for the global matrices resulting from FEM analysis of
the waveguide in Fig. 5.1.
To possibly correlate the global matrix condition numbers to those of the individual element matrices, condition numbers for the element matrix [Be] for each of the 64
elements used to discretize the waveguide are displayed in Fig. 5.25.3 for unnormalized andl normalized vector basis functions, respectively. For higher order TVFEs, the
results are presented in a slightly different form in Fig. 5.45.5 where for each element
the condition numbers of [Be] obtained with the mixedorder TVFEs by Peterson,
Webb and Forghani and those proposed in this dissertation are given relative to the
one obtained with the mixedorder TVFE by Graglia et al. Fig. 5.25.5 show that all
conclusions for global matrices also hold for the individual element matrices.
70
TE/TM Unnormalized
TVFE of order 0.5  Whitney
— * TVFE of order 1.5  Peterson..........>.......... TVFE of order 1.5  Graglia
 TVFE of order 1.5  Andersen
 TVFE of order 1.5  Webb
c
==..ro
h
1g
IO
0 10 20 30 40 50 60
Element number
Figure 5.2:
Condition numbers for the individual element matrices for the waveguide
illustrated in Fig. 5.1; TE/TMformulation with unnormalized vector basis functions.
TE/TM Normalized
TVFE of order 0.5  Whitney
* TVFE of order 1.5  Peterson....TVFE of order 1.5  Graglia... v..... TVFE of order 1.5  Andersen
4  TVFE of order 1.5  Webb
c.2.a
a
a
10 20 30 40 50 60
Element number
Figure 5.3:
Condition numbers for the individual element matrices for the waveguide
illustrated in Fig. 5.1; TE/TMformulation with normalized vector basis
functions.
5.3.2 Threedimensional case
Consider a rectangular cavity of normalized dimensions I x 0.75 x 0.5, as illustrated
in Fig. 5.6. For E and Hformulation, the eigenvalues are determined using the FEM
(130 tetrahedral elements) with the mixedorder TVFEs by Whitney, Savage and
71
TE/TM Unnormalized
E 10
0.5
0 10 20 30 40 50 60
Element number
Figure 5.4: Ratios of condition numbers the for individual element matrices for the
waveguide illustrated in Fig. 5.1; TE/TMformulation with unnormalized
vector basis functions.
TE/TM Normalized
4
* C(Peterson)/C(Graglia)........... . C(Andersen)/C(Graglia)
3 —.5  C(Webb)/C(Graglia)
C 3 "
2.5 
2.5
0.5
0 10 20 30 40 50 60
Element number
Figure 5.5: Ratios of condition numbers for the individual element matrices for the
waveguide illustrated in Fig. 5.1; TE/TMformulation with normalized
vector basis functions.
Peterson, Graglia et al., Webb and Forghani and that proposed in this dissertation
used for field expansion (unnormalized as well as normalized vector basis functions).
Condition numbers similar to those in Tab. 5.1 and Fig. 5.25.5 are given in
Tab. 5.2 and Fig. 5.75.10 for E and Hformulation with unnormalized as well as
normalized vector basis functions. The conclusions for the twodimensional case are
72
05....................
0.75
0.75
Figure 5.6: Illustration of a rectangular cavity.
Eformulation Hformulation
TVFE Unnormalized Normalized Unnormalized Normalized
Wh 8 12 23 31
Pe 2173 287 5939 958
Gr 684 99 1387 239
An 1834 195 5341 656. We 4238 472 11564 141.4
Table 5.2: Condition numbers for the global matrices resulting from FEM analysis of
the cavity in Fig. 5.6.
also seen to be valid in the threedimensional case.
5.4 Inhomogeneous application of TVFEs
In this section, condition numbers for FEM matrices are computed when mixedorder TVFEs of order 0.5 and 1.5 are combined selectively within a computational
domain.
Consider a rectangular cavity of normalized dimensions 1 x 0.75 x 0.5. It is
composed of a rectangular cavity of dimensions 0.25 x 0.75 x 0.5 filled with a dielectric
of relative permittivity er = 10 and a rectangular cavity of dimensions 0.75 x 0.75 x 0.5
filled with a dielectric of relative permittivity sr = 1, as illustrated in Fig. 5.11.
The field and its derivatives within the cavity are expected to be largest in the
dielectric and in the immediate vicinity hereof. An effective field expansion would
73
D
E 1.5
= 1
0.
0 20 40 60 80 100 120
Element number
Figure 5.7: Condition numbers for the individual element matrices for the cavity illustrated in Fig. 5.6; E/Hformulation with unnormalized vector basis
functions.
E/H Normalized
3000 I
 TVFE of order 0.5  Whitney
 TVFE of order 1.5  Peterson
 TVFE of order 1.5  Graglia
2500 j..........:.  TVFE of order 1.5  Andersen
 TVFE of order 1.5  Webb
2000
~1500 
C:
500
0 20 40 60 80 100 120
Element number
Figure 5.8: Condition numbers for the individual element matrices for the cavity illustrated in Fig. 5.6; E/Hformulation with normalized vector basis functions.
therefore use a higher order TVFE within the dielectric and in the part of the airfilled
region closest to the dielectric and the lowest order TVFE elsewhere. The field continuity requirements between the lowest and higher order regions make hierarchical
higher order TVFEs attractive and the detailed investigation of matrix conditioning
in the previous section suggests using the hierarchical mixedorder TVFE of order ].5
74
E/H Unnormalized
  C(Peterson)/C(Graglia)............................ C (A ndersen)/C (G raglia)
  C(Webb)/C(Graglia)
12 j'
4 * l l l l
0
0 20 40 60 80 100 120
Element number
Figure 5.9: Ratios of condition numbers for the individual element matrices for the
cavity illustrated in Fig. 5.6; E/Hformulation with unnormnalized vector
basis functions.
E/H Nonrmalized
20c
— * — C(Peterson)/C(Graglia)
18 .................... C(Andersen)/C(Graglia)16  C(Webb)/C(Graglia)
16 
2 ' '" " " " '.
2 
0 20 40 60 80 100 120
Element number
Figure 5.10: Ratios of condition numbers for the individual element matrices for the
cavity illustrated in Fig. 5.6; E/Hformulation with normalized vector
basis functions.
proposed in this dissertation rather than the one developed by Webb and Forghani.
However, the field continuity requirements between the lowest and higher order regions only mandate use of a hierarchical higher order TVFE at the boundary to the
lowest order region and hence away from this boundary an interpolatory higher or
75
,r =10
0.5 1
0.75
Figure 5.11: Illustration of an inhomogeneous rectangular cavity.
der TVFE could be used. When the field within an empty cavity is expanded using
the mixedorder TVFEs of order 1.5 proposed in this dissertation as well as those
developed by Savage and Peterson and Graglia et al. throughout the cavity, the interpolatory TVFE by Savage and Peterson leads to larger condition numbers than
the hierarchical TVFE proposed in this dissertation while the interpolatory TVFE by
Graglia et al. leads to smaller condition numbers than the hierarchical TVFE proposed in this dissertation. This suggests that combination of the hierarchical TVFE
proposed in this dissertation with the interpolatory TVFE by Savage and Peterson
within the higher order region will lead to larger condition numbers than those when
the hierarchical TVFE proposed in this dissertation is used throughout the higher
order region whereas combination of the hierarchical TVFE proposed in this dissertation with the interpolatory TVFE by Graglia et al. within the higher order region will
lead to smaller condition numbers than those when the hierarchical TVFE proposed
in this dissertation is used throughout the higher order region. Below, we examine
the three different approaches for modeling the inhomogeneous cavity. They span the
same space within each element and give identical eigenvalues. However, they are
not equally efficient numerically as the different TVFEs provide matrices of different
condition numbers.
Condition numbers are given in Tab. 5.3 for E and Hformulation. The abbreviations 'Wh/An', 'Wrh/An/Pe' and 'Wh/An/Gr' denote the three different approaches.
For both E and 11 —formulation, the expected relative size of the different condition
numbers is observed. Hence, an approach for improving the condition numbers of
FEM matrices resulting from selective field expansion has been suggested and tested.
The improvement comes at the expense of a more complicated formulation and computer code but does not alter accuracy. In essence, the suggested approach uses
hierarchical TVFEs to transition between regions with lowest and higher order interpolatory TVFEs. Such use of transition TVFEs is directly equivalent to the use of
scalar transition elements for efficient transition between coarse and fine meshes or
between regions with lowest and higher order expansions [38, 87, 108]. This is a wellknown concept that has been applied successfully in mechanical and civil engineering
for decades.
TVFEs Eformulation Hformulation
Wh/An 1188 654
Wh/An/Pe 1747 996
Wh/An/Gr 534 282
Table 5.3: Condition numbers for the global matrices resulting from FEM analysis of
the cavity in Fig. 5.11.
5.5 Summary
In this chapter, the condition numbers resulting from FEM analysis using the
hierarchical mixedorder TVFEs of order 1.5 for triangular and tetrahedral elements
77
proposed in chapter 3 and chapter 4 were contrasted to those of existing interpolatory and hierarchical mixedorder TVFEs of order 1.5 for triangular and tetrahedral
elements. The proposed hierarchical mixedorder TVFEs of order 1.5 proved better
conditioned than existing hierarchical mixedorder TVFEs of order 1.5 and thus the
analysis fostered no concerns for potential future convergence problems due to excessive matrix condition numbers. In addition, an approach for improving the condition
numbers of FEM matrices resulting from selective field expansion was suggested and
tested. The improvement comes at the expense of a more complicated formulation
and computer code but does not alter accuracy.
78
P17
CHAPTER 6
ADAPTIVE TVFE REFINEMENT
In this chapter, a review of existing error estimators and indicators is given and
the effectiveness of the proposed hierarchical mixedorder TVFEs of order 0.5 and 1.5
for tetrahedra is investigated when some of the reviewed error indicators are applied
in the context of a very simple adaptive refinement strategy. The work in this chapter
is expected to be published in [4].
6.1 Background
Numerous methods for a posteriori error estimation or indication have been studied extensively in mathematics and engineering for decades and a vast amount of
literature exists on the subject. For reviews of the various methods, see for instance
[2, 16, 23, 46, 63, 74]. Following the approximate solution of a partial differential
equation using a lowest order FE or hybrid FE/BI approach, each method seeks to
identify local regions with large error for subsequent adaptive refinement of the mesh
(hrefinement), basis functions (prefinement) or mesh and basis functions simulta79
neously (hprefinement) leading to an improved FE or FE/BI solution. This process
is repeated until a desired accuracy is deemed to be reached. The identification of
local regions can be performed by estimating the actual local error (error estimation)
or by determining a local quantity that indicates whether the local error is small or
large without estimating the actual local error (error indication). The local error
estimation or indication can be carried out on an element by element basis or clusters
of elements can be grouped together in subdomains whereby the local error estimation or indication can be carried out on a subdomain by subdomain basis. Methods
for which the error estimator or indicator can be computed directly from the initial
solution are referred to as explicit methods whereas methods for which computation
of the error estimator or indicator requires the solution of a local boundary value
problem are referred to as implicit methods.
The various methods for a posteriori error estimation or indication can be grouped
in several different ways and hence a unique classification of these is not possible. The
one presented here closely follows that of [74]. Implicit residual methods are based
on the solution of a local Dirichlet or Neumann boundary value problem constructed
from the lowest order FE solution [1, 76]. Explicit residual methods are based on
local error estimation or indication by computation of a residual directly from the
lowest order FE solution [10, 32, 34, 39, 47, 48, 93]. Among these, explicit complete
residual methods take into account both local interior and boundary effects, explicit
incomplete residual methods take into account only local interior effects and explicit
interface residual methods take into account only local boundary effects. Recovery /
gradient / average / smoothing methods are based on local comparison of the gradi80
ent of the original lowest order FE solution to a smoothened version of this gradient
[109]. Dual / complementary / variational / mixed / hybrid methods are based on
local comparison of solutions of two dual / complementary problems [16, 32, 33]. Perturbation methods are based on the estimation of local errors from differences between
FE solutions of different orders [42, 57]. Interpolation and extrapolation methods are
based on interpolation and extrapolation theory to compute approximations to higher
order derivatives [31, 86]. For comparisons of the different methods, see for instance
[16, 20, 24, 29, 33, 35, 41]. We note that the above classification is not complete. In
addition to the main classes of methods presented above, other methods have been
presented [74].
Hierarchical mixedorder TVFEs for tetrahedral elements have been proposed up
to and including order 1.5 by Webb and Forghani [97], see section 2.2, and up to and
including order 2.5 in this dissertation. The hierarchical mixedorder TVFE of order
1.5 proposed by Webb and Forghani was tested [76] for adaptive refinement using an
implicit residual method. The purpose of this chapter is to investigate the merits of
various explicit residual methods for a posteriori error indication. This is done via
adaptive prefinement of hybrid FE/BI solutions using the hierarchical mixedorder
TVFEs of order 0.5 and 1.5 for tetrahedral elements proposed in this dissertation.
We restrict ourselves to an adaptive prefinement approach in this dissertation and
refer to [22, 74, 82, 89] or additional references in [74] for adaptive h or hprefinement
approaches.
81
6.2 Adaptive refinement strategies
Consider a general threedimensional electromagnetic problem. We perform a
lowest order FE or FE/BI analysis (mixedorder TVFEs of order 0.5 applied for field
expansion) with the computational domain discretized into N6 tetrahedral elements
denoted by Te, = 1,., Ne, each having 4 faces denoted by F/, i = 1,.,4.
The center of Te and Fi is denoted by C(Te) and C(Fi), respectively., and the unit
normal vector to Ft directed out of the element Te is denoted by fn. The lowest
order FE or FE/1I solution leads to approximations of the electric field intensity E
and the electric flux density D within and on the boundary of each element Te. On
the face Fie we let D"in denote the value of D evaluated in Te and let Diout denote
the value of D evaluated in the element bounding Te. Based on these, we present
three different methods for indicating the error in a given region of the computational
domain. For each method, we give the error indicator corresponding to an element.
The corresponding error indicator for a subdomain is simply the maximum of the
error indicators for the elements comprising the subdomain.
The magnitude of the electric flux density generally does not correlate with the
error associated with the electric flux density. Nevertheless, regions with high flux
densities often give the dominant contributions to the physical response of a given
electromagnetic eigenvalue, radiation or scattering problem. This justifies accurate
modeling of such regions and hereby use of the simple error indicator
EII(e)= [D(C(T6))[. (6.1)
Although we are strictly not computing a residual, we will refer to this method as an
82
explicit incomplete residual method.
The trivial threedimensional extension of the twodirnensional error indicator
applied by Wang and Webb [93] for adaptive refinement in surface Mo)M problems is
the error indicator
EI2(e) = max [n (D"  Do )C(Fe)} (6.2)
This method is an explicit interface residual method.
A slightly different and computationally more expensive error indicator initially
proposed by Golias and Tsiboukis [34, 35] is the error indicator
E3(e) = max { >, ~ (D'in  Dut)2d. (6.3)
This method is also an explicit interface residual method.
In this dissertation, a conceptually very simple adaptive refinement strategy is
adopted. Following the lowest order FE or FE/BI analysis, we determine the degree
of error in each element via an error indicator and compute a refined solution where
a certain prespecified percentage of the elements having the highest degree of error
are modeled with mixedorder TVFEs of order 1.5 and the remaining elements are
again modeled with mixedorder TVFEs of order 0.5. A more advanced refinement
strategy would estimate the optimal percentage of refinement for the improved solution, use TVFEs of more orders for refinement and incorporate a feedback loop
leading to multiple error indications and refined solutions. However, given the lack of
previous applications of adaptive refinement for practical electromagnetic problems,
the simple adaptive refinement strategy described above was deemed sufficient in this
dissertation.
83
6.3 Numerical results
In this section, the merits of the error indicators EIji, ElI2 and EI3 presented in
the previous section are investigated for determining input impedances of metallic
patch antennas backed by materialfilled cavities recessed in infinite metallic ground
planes. We apply a standard hybrid FE/BI formulation with mixedorder TVFEs of
different orders used for field expansion, as detailed in section 4.3.1.
6.3.1 Square patch antenna
In section 4.3.2, we considered a square metallic patch antenna backed by a rectangular cavity recessed in an infinite metallic ground plane. In this section, we consider
the same antenna and we therefore refer to the beginning of section 4.3.2 for a presentation of the antenna.
We discretize the BI surface and the patch into 8 x 8 = 64 squares each of which
are broken into two triangles. This surface mesh is extruded into the cavity to form
a prism layer and each prism is broken into three tetrahedra. Four different TVFE
options are applied: First, the mixedorder TVFE of order 0.5 is applied throughout
the cavity. Second, the mixedorder TVFE of order 1.5 is applied throughout the
cavity. Third, the mixedorder TVFE of order 0.5 is applied in conjunction with the
mixedorder TVFE of order 1.5 in the vicinity of the radiating edges of the patch
(38% of the TVFEs are of order 1.5). Fourth, the mixedorder TVFE of order 0.5
is applied in conjunction with the mixedorder TVFE of order 1.5 in regions found
adaptively (40% of the TVFEs are of order 1.5). The refinement is carried out for
84
each of the error indicators EIl, EI2 and EI3 given by (6.1)(6.3) on a tetrahedron
by tetrahedron (384 elements), prism by prism (128 subdomains) as well as brick by
brick (64 subdomains) basis. The nine different cases are defined in Tab. 6(.1.
Error Base of adaptive Percentage of
Case indicator refinement refinement
1 EI1 Tetra 40
2 EI1 Prism 40
3 EIL Brick 40
4 E12 Tetra 40
5 EI2 Prism 40
6 EI2 Brick 40
7 EI3 Tetra 40
8 EI3 Prism 40
9 EI3 Brick 40
Table 6.1: Definition of Case 19.
For the mixedorder TVFEs of order 0.5 and 1.5, the particular mesh is too coarse
to yield the correct resonant frequency of 4.43 GHz as obtained by Schuster and
Luebbers [81] and confirmed in chapter 4 for finer meshes. Nevertheless, the mesh
is very useful for evaluating the merits of the various error indicators. The dynamic
range of the differences in resonant frequency between solutions where mixedorder
TVFEs of order 0.5 and 1.5 are applied throughout the computational domain are
the largest for coarse meshes and hence coarse meshes are ideal for investigating how
well shifts in resonant frequency are predicted by adaptively refined solutions based
on various error indicators. This approach can of course only be justified for problems
where proper convergence has been ensured by very accurately predicting the correct
resonant frequency by using a more accurate approach (finer meshes, higher order
TVFEs). As demonstrated in section 4.3.2, such convergence was observed for this
85
particular antenna.
Real and imaginary parts of the input impedance as a function of frequency are
given in Fig. 6.16.9 for each of the four TVFE options for Case 19. The results
for the first three TVFE options are the same for Case 19. The resonant frequency
predicted with the mixedorder TVFE of order 1.5 applied throughout the cavity
is larger than that when the mixedorder TVFE of order 0.5 is applied throughout
the cavity. Application of the mixedorder TVFE of order 1.5 in the vicinity of the
radiating edges only and the mixedorder TVFE of order 0.5 elsewhere is seen to
predict this shift very well. Most importantly, we observe that adaptive refinement
on an element by element basis (Case 1, 4, 7) is seen to predict an inaccurate upward
shift while adaptive refinement on a subdomain by subdomain basis (Case 23, 5 6, 89) is seen to predict the upward shift very well. For EI2 and E13, prism by
prism refinement is slightly more accurate than brick by brick refinement but more
complicated antennas must be studied before decisive conclusions can be reached
regarding the relative merits of these two subdomain refinement schemes.
Corresponding regions of refinement at the resonant frequency are displayed in
Fig. 6.106.18. These figures show a top view of the mesh where the cavity and
patch boundaries are marked with thick black lines. A white / light gray / average
gray / dark gray triangle indicates that 0 / 1 / 2 / 3 of the three elements in the
prism beneath the triangle are being refined with a mixedorder TVFE of order 1.5.
Adaptive refinement on an element by element basis is seen to lead to very sporad(lic
regions where the mixedorder TVFE of order 1.5 is applied. Such combination of
lowest and higher order TVFEs is inefficient and consequently leads to less accurate
86
400, 1 I I I I
300
8
0
200u
Nbc
200
01
3~0
200
 0.5
    1.5
  0.5+1.5(edges)
0.5+1.5(adaptive)
95 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.
Frequency [GHz]
4 — 0.5
* — 1.5
 0.5+1.5(edges)
0.5+1.5(adaptive);requency [GHz]
Figure 6.1: Real and imaginary part of input impedances for Case 1 (El,, Tetra).
4001 1 1 1 1 1 I I I 
300
8
0
N100
200
'0' 100
0
 100
200
e 0.5+1.5(edges)
0.5+1.5(adaptive)
05 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4~
Frequency [GHz]
4 0.5
1.5
ae 0.5+1.5(edges)
0.5+1.5(adaptive).5 3.6 317 3.8 3.9 4 4.1 4.2 4.3 4.4 4.
Frequency [GHz]
5
Figure 6.2: RearI and imaginary part of input impedances for Case 2 (El,, Prism).
40 0 r 1 1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
300
0
i200
N100
4 0.5
 l — 1.5
e 0.5+1.5(edges)
a 0.5+1.5(adaptive)
    .     .5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4
Frequency [GHz]
208........
3.1 0 3.6.... 373.3..1.. 3 4 4
FrqecEGz
Fiue.: el n iainr pr o nptimeane frCae3 E1,Bic)
08
400, N I I I I I I
300
S
E
200
N1
100
0 — 0.5
K  1.5
 '0.5+1.5(edges)
0.5+1.5(adaptive)
_J
r     ~ *  
5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4
Frequency [GHz]
Figure 6.4: Real and imaginary part of input impedances for Case 4 (El2, Tetra).
400,,,, 1 I I I
30C
E
C
20
rr
100
 0.5:
1.5
 0.5+1.5(edges)
0.5+1.5(adaptive).. X,, ^.;.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5
Frequency [GHz]
Figure 6.5: Real and imaginary part of input impedances for Case 5 (EI2, Prism).
3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5
Frequency [GHz]
Figure 6.6: Real and imaginary part of input impedances for Case 6 (EI2, Brick).
88
4001,, I I I I I I
300
0
E
0
I200
N0
100
 0.5
 * — 1.5
E 0.5+1.5(edges)\ 
0.5+1.5(adaptive) \ \ \\
rA.
a   . 1
3
5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4
Frequency [GHz]
4.5
Figure 6.7: Real and imaginary part of input impedances for Case 7 (EI3, Tetra).
400,.......
300
E
200
100
1001
  0.5 A
1.5
E 0.5+1.5(edges)
 0.5+1.5(adaptive). \
IL.I
5 3.6 3.7 3.8 3.9 4 4.1 4.2
Frequency [GHz]
4.3 4.4 4
5
Figure 6.8: Real and imaginary part of input impedances for Case 8 (EI3, Prism).
Frequency [GHz]
Frequency [GHz]
Figure 6.9: Real and imaginary part of input impedances for Case 9 (El, Brick).
89
Figure 6.10: Regions of refinement for Case 1 (El1, Tetra) at 4.15 GHz.
Figure 6.11: Regions
of refinement for Case 2 (EI1, Prism) at 4.30 GHz.
Figure 6.12: Regions of refinement for Case 3 (Ell, Brick) at 4.30 GHz.
90
Figure 6.13: Regions of refinement for Case 4 (EI2, Tetra) at 4.10 GHz.
Figure 6.14: Regions of refinement for Case 5 (EI2, Prism) at 4.30 GHz.
Figure 6.15: Regions of refinement for Case 6 (EI2, Brick) at 4.30 GHz.
91
~_ ~_ ~_ ~_ ~_ ~
Figure 6.16: Regionm
s of refinement for Case 7 (EI3, Tetra) at 4.10 GHz.
Figure 6.17: Regions of refinement for Case 8 (EI3, Prism) at 4.30 GHz.
Figure 6.18: Regions of refinement for Case 9 (EI3, Brick) at 4.30 GHz.
92
results, as observed. Adaptive refinement on a subdomain by subdomain basis
inherently circumvents this problem and leads to very accurate results, as observed.
We note that all adaptive refinement schemes were tested for less than 40% refinement. 30% refinement was found to lead to very accurate results with EI but
slightly less accurate results with EI2 and EI3. Generally, 20% refinement was not
enough to significantly improve the lowest order solution. For brevity, these results
are not included.
6.3.2 Printed bowtie antenna
Consider a metallic printed bowtie antenna backed by a rectangular cavity recessed
in an infinite metallic ground plane, as illustrated in Fig. 6.19 (side view) and Fig. 6.20
(top view). The cavitybacked patch antenna is situated in free space characterized
by the permittivity so and the permeability,uo. The cavity dimensions are 48 cm x 64
cm x 12 cm. The interior metallic cavity walls are covered with an AA of permittivity
(1 j2.7):o, permeability (1 j2.7)[to and thickness 4 cm. This absorber layer would
not be part of an actual antenna but is merely a wellestablished computational
tool [64] that serves to approximately simulate a metallic printed bowtie antenna
situated in free space. Such an antenna is expected to exhibit broadband behavior in
contrast to the narrowband square patch antenna considered above. Consequently,
the printed bowtie antenna can provide a second and independent evaluation of the
error indicators presented in section 6.2. The specific printed bowtie antenna consists
of two isosceles triangular patches characterized by the opening angle 67.38~ and the
93
~o ho
4cm 8cm 24cm 8cm 4cm
0o 1o 8 cm
I(j2.7) (Ij2 4cm
Figure 6.19: Side view of a metallic printed bowtie antenna backed by an air and
absorberfilled rectangular cavity recessed in an infinite metallic ground
plane.
         1 1
8 cm 24 cm 8 cm i i:A
18 cm
I: I 18 cm
absorberfilled rectangular cavity recessed in an infinite metallic ground
plane.
maximum width 24 cm. The bowtie patches are centered in the cavity aperture and
fed by a probe of constant current connecting the two triangular patches.
An antenna very similar to the above was discussed by Collin [18] and is expected
to cover the UHF channels 14 to 83 spanning the frequency range [450 MHz, 900 MHz]
when used with a 300 Q feed line. That is, the real and imaginary parts of the input
impedance are expected to hover around 300 Q and 0 Q. respectively, in this frequency
range.!!:::1 Is
32~~~~'g 71';i~~>~ol'~IlgC. ysmla ote bv vs icse b oln 1]adisepce
94
The distance from the antenna, to the absorber is 8 cm which translates into
Ao/8.33 at 450 MHz and Ao/4.17 at 900 MHz. Similarly, the thickness of the absorber
is 4 cm which translates into Ao/16.67 at 450 MHz and Ao/8.33 at 900' MHz. These
distances should be sufficient to simulate an antenna in free space. That the absorber
actually works is confirmed by the fact that the behavior of the antenna is significantly
altered if the absorber is removed and the antenna is backed by a purely metallic
cavity. The main purpose of the absorber is to prevent excitation of modes norral
to and bounded by the metallic antenna and the bottom of the metallic cavity. If
such modes are excited, the antenna ceases to operate as an antenna situated in free
space but starts to function in the presence of the cavity which is not intended here.
The absorbers covering the side walls are not essential for this purpose and therefore
can be expected to have less influence on the behavior of the antenna. Indeed., it
was confirmed that results strikingly similar to those presented in the following are
obtained when the absorbers covering the side walls are removed. The extent of the
feed region is 4 cm (Ao/16.67 at 450 MHz and Ao/8.33 at 900 MHz) and hence it
is reasonable to assume a constant current over the probe in the frequency range in
which the antenna is operated. For larger feed regions, a phase variation of the probe
current will have to be accounted for. Doing so is trivial within the context of the
FE/BI method but it was deemed unnecessary in this case.
For analysis and evaluation of the proposed error indicators, we discretize the BI
surface and the patch into a coarse (56 antenna triangles and 330 BI triangles) and
a fine (160 antenna triangles and 1030 BI triangles) mesh. These surface meshes
are extruded into the cavity to form three prism layers. Each prism is broken into
95
three tetrahedra and four different TVFE options are applied: First, the mixedorder
TVFE of order 0.5 is applied throughout the cavity for the coarse mesh. Second, the
mixedorder TVFE of order 0.5 is applied throughout the cavity for the fine mesh.
Third, the mixedorder TVFE of order 1.5 is applied throughout the cavity for the
coarse mesh. Fourth, the mixedorder TVFE of order 0.5 is applied in conjunction
with the mixedorder TVFE of order 1.5 in regions found adaptively (20% of the
TVFEs are of order 1.5) for the coarse mesh. Supported by the findings for the
square metallic patch antenna, the refinement is carried out for each of the error
indicators EIl, EI2 and El3 given by (6.1)(6.3) on a subdomain by subdomain
basis only with a subdomain being three adjacent prisms extruded from a surface
triangle (386 subdomains for the coarse mesh). That is, we opt not to examine
adaptive refinement on an element by element basis since it did not prove efficient in
the previous analysis and we further limit ourselves to only one type of subdomain
by subdomain refinement. The three different cases are defined in Tab. 6.2.
Error Base of adaptive Percentage of
Case indicator refinement refinement
10 E,1 Prisms (3 adjacent) 20
11 EI2 Prisms (3 adjacent) 20
12 EI3 Prisms (3 adjacent) 20
Table 6.2: Definition of Case 1012.
Real and imaginary parts of the input impedance as a function of frequency are
given in Fig. 6.21 for the first three TVFE options (no adaptivity). Instead of the
strongly resonant behavior characterizing cavitybacked patch antennas, we observe
the expected slightly oscillatory behavior of the real and imaginary parts. Mixed
96
order TVFEs of order 0.5 for the fine mesh and mixedorder TVFEs of order 1.5 for
the coarse mesh give similar results that are better (real parts closer to 300 Q and
imaginary parts closer to 0 Q) than those with mixedorder T''VFEs of order 0.5 for the
coarse mesh. Although mixedorder TVFEs of order 0.5 for the fine mesh and mixedorder TVFEs of order 1.5 for the coarse mesh give similar results, the latter approach
is more attractive than the former in terms of memory and CPU time requirements.
To demonstrate the merits of adaptive refinement, Fig. 6.22 shows the real and
imaginary parts of' the input impedance as a function of frequency for the coarse
mesh with mixedorder TVFEs of order 1.5 throughout the cavity and with 80%
mixedorder TVFEs of order 0.5 and 20% mixedorder TVFEs of order 1.5 found
via adaptive refinement using EIl, EI2 and EI3 on a subdomain by subdomain
basis (Case 1012). The results are almost indistinguishable expressing that we can
accurately predict the behavior of the antenna using only 20% mixedorder TVFEs of
order 1.5. This presents a significant memory and CPU time improvement at virtually
no cost.
To illustrate the regions of refinement for Case 1012, we consider a crosssection
of the mesh parallel to the antenna with the boundaries of the metallic cavity and
antenna, marked with thick black lines. As mentioned previously, the tetrahedral
volume mesh is grown from a triangular surface mesh in a cut in the plane of the
bowtie patches by extruding it into three prism layers and breaking each prism into
three tetrahedra. Since we consider subdomain by subdomain refinement only, 0 or
9 tetrahedra can be refined corresponding to a given triangle in the crosssection. In
the following, a white / dark gray triangle indicates that 0 / 9 tetrahedra are being
97
400
300
0
N 0 0.5 (coarse)
r 100   0.5 (fine)
0.4 0.5 0.6 0.7 0.8 0.9 1
Frequency [GHz]
300
0 0.5 (coarse)
200.............  0.5 (fine)........
E    E 1.5 (coarse)
0
100
N
E 0
10).
0.4 0.5 0.6 0.7 0.8 0.9 1
Frequency [GHz]
Figure 6.21: Real and imaginary part of the input impedance for the metallic bowtie
patch antenna in Fig. 6.196.20.
300
 100   20% E12
E_ 20%  E13.
100
0.4 0.5 0.6 0.7 0.8 0.9 1
Frequency [GHz]
3001!i 0;  100%.................:............ 2 0 %  E ll................
100 E3!. ......
0.4 0.5 0.6 0.7 0.8 0.9 1
Frequency [GHz]
Figure 6.22: Real and imaginary part of the input impedance for the metallic bowtie
patch antenna in Fig. 6.196.20.
refined. The regions of refinement at 0.7 GHz (close to the center of the frequency
band of operation) are shown in Fig. 6.23 for Case 1012. Although they are seen to
differ slightly for the three different error indicators, they all show the general trend of
predicting the feed area and, to a lesser extent, the corners of the triangular patches
as the regions where mixedorder TVFEs of order 1.5 are applied.
We note that all approaches predict almost the same far field patterns a.s well as a
98
Figure 6.23: Regions of refinement for the metallic bowtie patch antenna in Fig. 6.19 6.20 for Case 10 (left), Case 11 (middle) and Case 12 (right).
linearly polarized far field (parallel to the feed) in the direction normal to the bowtie
patches. For brevity, these expected results are not included.
6.3.3 Discussion
The error indicator EIl is an explicit incomplete residual indicator taking into
account only local interior effects while the error indicators EI2 and EI3 are explicit
interface residual indicators taking into account only local boundary effects. Explicit
complete residual indicators take into account both local interior and boundary effects
and thus are expected to offer superior performance. An obvious construction of
an explicit complete residual indicator is to simply combine EIl with E12 or E'3.
That is, a certain fraction of the elements or subdomains are refined based on El11
and the remaining elements or subdomains are refined based on EI2 or EI3. This
approach was tested and found accurate for analysis of square metallic patch antennas.
However, more complicated antennas must be studied before decisive conclusions can
be reached regarding the efficiency of this approach.
99
An apparent drawback of adaptive refinement is that it requires multiple (in this
case two) FE/BI solutions for each frequency. This seems computationally expensive
since it involves the consecutive solution of different equation systems using an iterative solver for which zero is the traditional starting guess. A possible remedy might be
use of the previous FE/BI solution at a given frequency as the starting guess for the
iterative solver all adaptive FE/BI solutions. As stated by Golias and Tsiboukis
[33], this should greatly accelerate the convergence of the adaptive FE/BI solutions.
We note that each adaptive FE/BI solution requires a starting guess for numerous
unknown expansion coefficients that have not previously been solved for. For these
unknowns, we shall continue to use zero as the starting guess.
To investigate the effectiveness of this approach for the metallic printed bowtie
antenna considered above, Fig. 6.24 shows the final number of iterations to reachl a
103 relative residual with a QMR solver as a function of frequency. Convergence
curves are given for the approaches where TVFEs of order 0.5 are used for a coarse
mesh, TVFEs of order 0.5 are used for a fine mesh and TVFEs of order 1.5 are
used for a coarse mesh. These three cases correspond to 3704, 11825 and 19745
unknowns, respectively. The number of iterations is seen to be slightly under 500
for all frequencies when mixedorder TVFEs of order 0.5 are applied for the coarse
mesh. As expected, this number increases when a finer mesh or mixedorder TVFEs
of order 1.5 are applied. It increases more in the latter case but relative to the number
of unknowns this case has the best convergence properties.
Fig. 6.256.27 shows the same information for Case 1012 with zero as well as
the solution with mixedorder TVFEs of order 0.5 used as the starting guess for the
100
9rnnI
N"  0.5 (coarse)., + 0.5 (fine)
2 1.5 (coarse)
 15000 ,
500
0.4 0.5 0.6 0.7 0.8 0.9 1
Frequency [GHz]
Figure 6.24: Final number of iterations as function of frequency for different iterative
solutions.
iterative solver. The number of unknowns is around 6400 for all frequencies and
error indicators. The number of iterations is lower than when mixedorder TVFEs of
order 1.5 are applied throughout the coarse mesh. Also, the number of iterations is
consistently seen to drop when the solution with mixedorder TVFEs of order 0.5 is
used as the starting guess for the iterative solver instead of zero. The savings come
at no cost since we need to carry out the solution with mixedorder TVFEs of order
0.5 in order to determine the regions of refinement. However, even if we preselect
the higher order regions and thus have no need to find a solution with mixedorder
TVFEs of order 0.5, it is still worthwhile to do so. Subsequent use of such a solution
as the starting guess for the iterative solver can reduce the iteration count by several
hundred iterations for systems with approximately 6400 unknowns for the price of
less than 500 iterations for systems with 3704 unknowns.
Fig. 6.286.30 shows the relative residual as a function of the iteration number at
the frequency 0.7 GHz (close to the center of the frequency band of operation) for
Case 1012. The results reinforce the conclusions from above.
101
en
C:
0._
o
ca
z
E=
2500
Zero
Order 0.5 solution
2000
150d
1000
500
i1
6.4 0.5 0.6 0.7 0.8 0.9 1
Frequency [GHz]
Figure 6.25: Final number of iterations as a function of frequency for adaptive solutions for Case 10 (EIl, Prism) with different starting guesses for the
iterative solver.
9.(nn II
2000
o
'Z 1500
150C
a)
 100C
z
50C
 Zero
Order 0.5 solution::
0.4 0.5 0.6 0.7
Frequency [GHz]
0.8 0.9
Figure 6.26: Final number of iterations as a function of frequency for adaptive solutions for Case 11 (EI2, Prism) with different starting guesses for the
iterative solver.,n(.....
2000
I 1500<
o
E 1000
z
 Zero
 Order 0.5 solution
* 0 J:F 4: '  '.::
500
i I
b.4 0.5 0.6 0.7 0.8
Frequency [GHz]
0.9
Figure 6.27: Final number of iterations as a function of frequency for adaptive solutions for Case 12 (EI3, Prism) with different starting guesses for the
iterative solver.
102
10~.o
E3
a)
a1)
nrr
500 1000
Iteration number
1500
Figure 6.28: Relative residual as a function of the iteration number for adaptive solutions for Case 10 (El1, Prism) with different starting guesses for the
iterative solver.
10~
1 0.=====................................
Zero........... Order 0.5 solution
10 ~..........:...
co
103
104 ll
0 500 1000 1500
Iteration number
Figure 6.29: Relative residual as a function of the iteration number for adaptive solutions for Case 11 (EI2, Prism) with different starting guesses for the
iterative solver.
1 0::::::::::.::::::::::
Zero...    Order 0.5 solution:
1 0
10.....................
104
0 500 1000 1500
Iteration number
Figure 6.30: Relative residual as a function of the iteration number for adaptive solutions for Case 12 (EI3, Prism) with different starting guesses for the
iterative solver.
103
We note that the above comments for reducing the iteration count do not apply
to resonant antennas like the square metallic patch antenna considered previously.
This is due to the fact that the field values throughout the computational domain
experience a strong frequency dependence.
6.4 Summary
In this chapter, a review of existing error estimators and indicators was given and
the effectiveness of the proposed hierarchical mixedorder TVFEs of order 0.5 and
1.5 for tetrahedra, was investigated when some of the reviewed error indicators were
applied in the context of a very simple adaptive refinement strategy. The results
were extremely promising for both narrowband and broadband antennas provided
the refinement was carried out on a subdomain by subdomain basis as opposed to
an element by element basis.
104
CHAPTER 7
ANALYSIS OF TAPERED SLOT ANTENNAS
In this chapter, the hierarchical mixedorder TVFEs of order 0.5 and 1.5 for
tetrahedra proposed( in this dissertation are used in conjunction with a simple adaptive
refinement strategy to analyze the impedance and pattern characteristics of tapered
slot antennas (TSAs) using the hybrid FE/BI method. The work in this chapter is
expected to be published in [4].
7.1 Background
Metallic TSAs such as exponentially tapered slot antennas (ETSAs, also called
Vivaldi antennas), linearly tapered slot antennas (LTSAs) and constant width slot
antennas (CWSAs), see Fig. 7.1, are of practical interest as reflector antenna or
lens feeds, as elements of broadband antenna arrays or (for larger TSAs) directly as
transmit or receive antennas [106]. Whether isolated or backed by thin dielectrics,
TSAs have a number of appealing properties. They are broadband with typical
impedance and pattern bandwidths of 5:1 [83] although values as large as 40:1 have
105
been reported [52], they can provide an almost symmetric main beam despite their
entirely planar nature [44] with typical 10 dB beamwidths of 3040~ [106i], they have a
moderate directivity of approximately ten times the antenna, length normalized to the
free space wavelength [107] and they have reasonable crosspolarization characteristics
with a typical isolation of 20 dB in the principal planes [44]. Their beamwidth and
directivity are somewhat controllable when the length of the antenna is varied [106]
and a narrow pattern beamwidth versus broad pattern bandwidth tradeoff exists
[107]. In addition, they have the practical advantages that they are inexpensive and
(feed region excluded) easy to fabricate [106], that their transmitting or receiving
portion is well separated from the necessary circuitry providing ample space for the
latter [106] and that the transverse spacing between TSA array elements can be made
very small [106].
Figure 7.1: Illustration of a metallic (dark gray) ETSA (left), LTSA (middle) and
CWSA (right) backed by a dielectric (light gray).
TSAs are traveling wave antennas (TWAs) of the surface wave type, i.e. a traveling wave propagates along the antenna structure with a phase velocity smaller than
the speed of light. A review of TWAs was given by Zucker [110]. During the past two
decades, the characteristics of TSAs have been measured as well as simulated using
106
approximate methods and more rigorous numerical techniques such as the MoM, the
FDTD method and the transmission line method. TSAs in air have been characterized in [43, 44, 19] while TSAs backed by thin homogeneous dielectrics to better
confine the field to the slot and thereby provide narrower beamwidths have been
treated in [30, 43, 44, 49, 106, 107]. TSAs backed by inhomogeneous dielectrics (homogeneous dielectrics with holes) to lower the effective dielectric constant and provide
a directivity improvement have been measured [58] and simulated using the FD'I'D
method [17]. The bandwidth bottleneck of the ETSA is the transition between the
feed and slot region [52]. The antipodal ETSA was introduced in [28] and comprehensively studied in [26] to overcome this difficulty while still providing beamwidths
and directivity comparable to the ETSA. The antipodal ETSA, however, gives rise
to a high degree of crosspolarization and experiences a significant polarization tilt
for high frequencies [52]. The balanced antipodal ETSA introduced in [52] removes
these limitations while maintaining the attractive features of the antipodal ETSA.
Parametric studies of various TSAs are given in [26, 51, 83, 88].
TSAs are often used as elements of large phased arrays [19, 78, 79, 80, 83, 88].
When used as array elements, TSAs are usually much smaller than standalone TSAs
and in many cases so small that they cease to work as broadband TWAs [107]. Such
small TSAs are useless as standalone antennas but can be useful within an array
framework [80].
Accurate determination of the phase velocity of the traveling wave within the
tapered slot is important for accurate prediction of the beamwidth in the Eplane
and, especially, the Hplane of a TSA [44]. This suggests that TSAs can be more
107
accurately characterized by improving the field modeling within the slot. It is the
aim of this chapter to investigate the validity of this hypothesis for TSAs situated in
free space. To this end, the TSAs are placed within a metallic cavity whose bottom
and side walls are covered with an AA that, as discussed in section 6.3, would not
be part of an actual antenna but is merely a wellestablished computational tool
to simulate an antenna situated in free space. After a validation against results and
trends reported in the literature, the hierarchical mixedorder TVFEs of order 0.5 and
1.5 for tetrahedra proposed in this dissertation are used in conjunction with a simple
adaptive refinement strategy to analyze the impedance and pattern characteristics of
a LTSA using the hybrid FE/BI method.
7.2 Tapered slot antenna analysis
We consider a LTSA uniquely characterized by the height 15.46 cm, the width
2.68 cm + 3.98 cm + 2.68 cm = 9.34 cm, the opening angle 14~ and the narrowest
slot width 0.18 cm. The antenna operates at 10 GHz and is fed by a probe of constant
current situated at the narrow end of the tapered slot, as illustrated in Fig. 7.2. The
probe excites a traveling wave in the tapered slot resulting in endfire radiation. Apart
from some minor geometrical differences that do "not change the physics that governs
the radiation mechanism of the antenna" [43], this LTSA was measured and simulated
using the MoM in [43] 1.
e use the F/B method to simulate the LTSA situated in free space by placing it
1The LT'SA in [43] is actually a slightly larger antenna operating at 9 GHz. It is the same size
in wavelengths as the one considered here.
108
2.68 cm 3.98 cm 2.68 cm
15.46 cm
0.18 cm
Figure 7.2: Geometrical parameters for a LTSA (not drawn to scale).
in a rectangular cavity with an open top surface and metallic bottom and side surfaces.
The interior metallic cavity walls are covered by an AA of relative permittivity and
permeability 1  j2.7 and thickness 0.15 cm. Thicker absorbers were found to give
similar results. The top of the antenna (the wide end of the tapered slot) is aligned
with the open boundary of the cavity. The vertical distance from the bottom of the
cavity to the bottom of the antenna is 1.65 cm while the horizontal distance from the
side wall of the cavity to the antenna is 1.65 cm parallel to the antenna and 2.45 cm
perpendicular to the antenna. The perpendicular dimension is larger since it is the
main purpose of the absorber to prevent excitation of modes normal to and bouncded
by the metallic antenna and the metallic cavity walls parallel to the antenna. If such
modes are excited, the antenna ceases to operate as an antenna situated in free space
but starts to function in the presence of the cavity which is not intended here. Such
operation can, however, be useful for arrays where suppression of scan blindness is of
importance [101].
The measured and MoM Eplane (parallel to the antenna) and Hplane (perpendicular to the antenna) far field patterns (polar cuts) reported in [43] were scanned
109
and digitized and are given in Fig. 7.37.4 along with FE/BI1 results using the mixedorder TVFEs of order 0.5 for a coarse mesh (129492 elements, 264513 faces and 159460
edges). Since the geometrical symmetry of the LTSA ideally translates into symmetric E and Hplane patterns, only half of the E and Hplane patterns are actually
given in [43]. The measured and MoM patterns from [43] were therefore mirrored for
comparison with the full E and Hplane patterns found via the FE/BI method as
given in Fig. 7.37.4. However, had the full patterns been measured and computed
in [43], minor discrepancies due to measurement and mesh asymmetries would have
been present. Also, since the measured and MoM data in [43] are normalized to 0 dB
at endfire (0 = 0), the same normalization is applied to the FE/BI data.
Overall, the FE/BI data agree well with the measured and MoM data in both
the E and the Hplane. The FE/BI main beam in the Eplane is slightly narrower
than the measured and MoM main beams while the FE/1BI main beam in the Hplane is almost identical to the measured main beam. This is quantified by the
approximate Eplane 10 dB beamwidths 22~ (FE/BI), 23~ (MoM) and 26~ (measured)
and the approximate Hplane 10 dB beamwidths 370 (FE/BI), 35~ (MoM) and 36~
(measured). In the Eplane, the measured data have a shoulder at 30  40~ and a
null at 50~ while the MoM data have nulls at 30~ and 50~. The FE/BI data fall in
between with a null at 40~. The MoM and measured data stay below 15 dB beyond
50~ whereas the FE/BI data stay below 12 dB for these angles. In the Hplane, the
FE/BI data follow the MoM and measured data very well down to 20 dB (50~) at
which point the measured and MoM data level off while the FE/BI data continue to
decrease.
110
Eplane
0
0 [degrees]
Figure 7.3: Eplane patterns for the LTSA in Fig. 7.2.
Hplane

Li
0
0 [degrees]
Figure 7.4: Hplane patterns for the LTSA in Fig. 7.2.
111
Considering the fact that the above FE/BI results were found using mixedorder
TVFEs of order 0.5 with a coarse discretization, better overall agreement cannot be
expected. The results show acceptable agreement with independent measured and
MoM data, and display trends (such as an almost symmetric main beam with the
Hplane being somewhat broader than the Eplane) that agree with the literature
[43, 49, 106]. However, several other trends that we can confirm with the above
antenna but cannot confirm from the above plots have been reported for LTSAs.
Before proceeding to demonstrate the merits of hierarchical mixedorder TVFEs for
LTSA modeling, we shall therefore discuss some of these trends to provide further
validation.
LTSAs have very low crosspolarization in the principal E and Hplanes and
larger crosspolarization in the diagonal Dplane [49]. The actual values of the crosspolarization in the Dplane will inevitably depend on how co and crosspolarization
is defined as this definition is not unique [55] 2. Following Ludwig's Definition III [55],
it is demonstrated in [49] that for a LTSA backed by a dielectric, the crosspolarized
field in the Dplane is significantly smaller than the copolarized field close to the
endfire direction 0 = 0 but rises to a level comparable to and in some cases higher
than the copolarized field offaxis. To investigate the polarization characteristics of
the above antenna, we plot the co and crosspolarized patterns as defined by Ludwig's
Definition III [55] in the E, H and Dplanes, see Fig. 7.57.7. They are seen to be
in full agreement with the trends reported in [49]. The lack of symmetry for the
2Ludwig's Definitions II and III [55] suitable for antenna patterns lead to the same definitions
for co and crosspolarization in the E and Hplanes but different definitions in the Dplane.
112
crosspolarized fields in the E and Hplanes is not unexpected and is due to the fact
that the crosspolarized field values in these planes are very small and thus extremely
sensitive to any asymmetries introduced by the FE/BI analysis process (asymmetrical
meshes, integrations and so on). We note that similar patterns were not given in [43].
LTSAs are inherently broadband antennas with typical bandwidths up to 5:1 [83].
Using only one mesh for accurate simulation over such a wide range of frequencies
is inefficient. It is necessary to keep the distance to the absorber large enough in
terms of wavelengths at low frequencies (large wavelengths) which in combination
with the requirement of a discretization rate small enough in terms of wavelengths
at high frequencies (small wavelengths) results in an unmanageable computational
domain. The solution is to remesh the geometry so the analysis at each frequency
is carried out with a mesh that provides approximately the same electrical distance
to the cavity wall and approximately the same electrical discretization rate. Since
it is not the purpose of this chapter to investigate LTSA bandwidth limitations but
rather to demonstrate the effectiveness of modeling LTSAs using a multiresolution
FE/BI approach, this remeshing approach was not adopted for the LTSA in Fig. 7.2.
However, the antenna in Fig. 7.2 was simulated at 8 GHz, 9 GHz, 11 GHz and 12 G Hz
with the mesh used at 10 GHz to show essentially similar patterns at all frequencies.
Although this does not demonstrate the extreme bandwidths reported in the literature
[52], sufficient bandwidth for validation purposes has been observed.
To demonstrate the merits of hierarchical mixedorder TVFEs for multiresolution
FE/BI modeling of LTSAs, extensive numerical simulations are carried out. We therefore opt to consider a slightly smaller antenna and cavity than above. Specifically, we
113
30 . :  '
50
60 opol
1, _Crosspol
70
80 60 40 20 0 20 40 60 80
e [degrees]
Figure 7.5: Co and crosspolarized Eplane patterns for the LTSA in Fig. 7.2.
Hplane
10
20
30
40
50
60 
60 Copol l
Crosspol
70
80 60 40 20 0 20 40 60 80
e [degrees]
Figure 7.6: Co and crosspolarized Hplane patterns for the LTSA in Fig. 7.2.
Dplane
mo"  .
30,;
40..
60  Copolt
Crosspol
70
80 60 40 20 0 20 40 60 80
0 [degrees]
Figure 7.7: Co and crosspolarized Dplane patterns for the LTSA in Fig. 7.2.
114
consider a LTSA uniquely characterized by the height 9.00 cm, the width 1.50 cm +
2.55 cm + 1.50 cm = 5.55 cm, the opening angle 15.20 and the narrowest slot width
0.15 cm, as illustrated in Fig. 7.8. As above, the antenna operates at 10 GEHz and is
fed by a probe of constant current situated at the narrow end of the tapered slot, see
Fig. 7.8, for excitation of a traveling wave in the tapered slot. For FE/BI analysis of
this antenna situated in free space, we place it in a rectangular cavity with an open
top surface and metallic bottom and side surfaces. The interior metallic cavity wails
are covered by an AA of relative permittivity and permeability 1j2.7 and thickness
0.15 cm and the top of the antenna (the wide end of the tapered slot) is aligned with
the open boundary of the cavity. The vertical distance from the bottom of the cavity
to the bottom of the antenna as well as the parallel and perpendicular horizontal
distances from the side walls of the cavity to the antenna are all 1.65 cm.
1.50cm 2.55 cm 1.50cm
9.00 cm
0. 15cm
Figure 7.8: Geometrical parameters for a LTSA (not drawn to scale).
For analysis of the LTSA, we employ a coarse and a dense tetrahedral volume
mesh grown from a coarse and a dense triangular surface mesh in a cut in the plane
of the antenna. Each surface mesh is extruded into a number of prism layers (ten for
the coarse mesh and fourteen for the dense mesh) and each prism is broken into three
115
tetrahedra. The coarse mesh has 44370 elements, 91269 faces and 55621 edges while
the dense mesh has 148638 elements, 303013 faces and 182109 edges. Four different
TVFE options are applied for FE/BI analysis: First, the mixedorder TVFE of order
0.5 is applied throughout the coarse mesh. Second, the mixedorder TVFE of order
0.5 is applied throughout the dense mesh. Third, the mixedorder T VFE of order
0.5 is applied within the coarse mesh in conjunction with the mixedorder TVFE of
order 1.5 in the four prism layers closest to the metallic antenna (40% of the elements
within the mesh) where the dominant fields are expected and accurate field modeling
is therefore necessary. Fourth, the mixedorder TVFE of order 0.5 is applied within
the coarse mesh in conjunction with the mixedorder TVFE of order 1.5 in regions
found adaptively (Case 14). The refinement is carried out by confining the higher
order TVFEs to the two (Case 12) or four (Case 34) prism layers closest to the
metallic antenna and performing the refinement with the error indicator El1 (shown
in chapter 6 to be at least as good as EI2 or EI3) on an element by element (Case
1,3) or subdomain by subdomain (Case 2,4) basis with 0%, 1%,..., 9% of the 44370
elements in the entire mesh being refined, as defined in Tab. 7.1. With the refinement
confined to the two or four prism layers closest to the metallic antenna, a subdomain
is defined as the two or four adjacent prisms extruded from a given surface triangle,
respectively.
Since the input impedance of an antenna is an extremely sensitive parameter, it is
expected to be influenced more than any other through improved field modeling. We
therefore expect the different TVFE options to result in different input impedances.
With the mixedorder TVFE of order 0.5 applied throughout the coarse mesh, the real
116
Error Base of adaptive Percentage of Constraint on
Case indicator refinement refinement refinement
1 E I Tetra 09 2 middle layers
2 EI, Prisms (2 adjacent) 09 2 middle layers
3 Eh Tetra 09 4 middle layers
4 El1 Prisms (4 adjacent) 09 4 middle layers
Table 7.1: Definition of Case 14.
part of the input impedance is 77 Q while the corresponding imaginary part is 78 Q.
The latter value is comparable in magnitude to the real part which is physically
unrealistic for a broadband antenna like the LTSA considered here. With the mixedorder TVFE of order 0.5 applied throughout the dense mesh, the real part of the input
impedance increases to 162 Q whereas the corresponding imaginary part of 81 Q is
almost unchanged. However, if the mixedorder TVFE of order 0.5 is applied within
the coarse mesh in conjunction with the mixedorder TVFE of order 1.5 in the four
prism layers closest to the metallic antenna (40% of the elements), the real part of the
input impedance increases to 234 Q while the corresponding imaginary part of 36 Q
is almost an order of magnitude smaller. The real and imaginary parts of the input
impedance for Case 14 are given in Fig. 7.9 as a function of the percentage of higher
order TVFEs within the computational domain. The addition of just a few percent
of higher order TVFEs is seen to have a dramatic influence on the input impedance.
It converges very quickly to a real part of around 220  260 Q and an imaginary part
that is significantly smaller. This is very close to the result when 40'% of the carvity
is modeled with the mixedorder TVFE of order 1.5, i.e. a very small percentage of
higher order TVFEs is needed provided these are placed properly. This is true for all
the Cases 14, i.e. the specific refinement scheme is unimportant for this antenna.
117
E 200
15 //:
rr  Case 2
50 :  : Case 3: Case 4
C L
0 1 2 3 4 5 6 7 8 9
Percentage of higher order TVFEs [%]
50
E 0
0 1 2 3 4 5 6 7 8 9
Percentage of higher order TVFEs [%]
Figure 7.a: Real and imaginary part of the input impedance of the LTSA in Fig. v.8
for different refinement schemes.
When mixedorder TVFEs of order 0.5 are applied throughout the coarse mesh,
the system with 17823 unknowns converges in 2143 iterations (103 relative tolerance
with a QMR solver). Improvement of accuracy via a denser mesh or (adaptive)
refinement with mixedorder TVFEs of order 1.5 leads to increased memory and C'U
time requirements. When mixedorder TVFEs of order 0.5 are applied throughout
the dense mesh, the system with 164222 unknowns converges in 8858 iterations (103
relative tolerance with a QMR solver). When mixedorder TVFEs of order 0.5 are
applied within the coarse mesh in conjunction with mixedorder TVFEs of order 1].5
in the four prism layers closest to the metallic antenna (40% of the elements), the
system with 130967 unknowns converges in 7801 iterations (103 relative tolerance
with a QMR solver). When mixedorder TVFEs of order 0.5 are applied within the
coarse mesh in conjunction with mixedorder TVFEs of order 1.5 in regions found
adaptively (Case 14), an initial solution with mixedorder TVFEs of order 0.5 must
be computed (47823 unknowns and 2143 iterations for a 10i3 relative toler ance with
118
a QMR solver), regions of refinement must be determined and a refined solution must
be computed. The number of iterations to reach this refined solution (103 relative
tolerance with a QMR solver) with the previous solution used as the starting guess
is given in Fig. 7.10 for Case 14 as a function of the percentage of higher order
TVFEs within the computational domain. As for the input impedance, the number
of iterations required for convergence reaches a plateau (60007000 iterations) after a
few percent of higher order TVFEs have been added. That the number of iterations
goes up as higher order TVFEs are added is fully expected and in agreement with
previous observations in the literature and in this dissertation. In fact, the numbers
of iterations when mixedorder TVFEs of order 1.5 are applied are by no means
excessive since we are dealing with problems of 49155 (Case 1, 1% refinement) to
130967 (40% refinement) unknowns. Note that all approaches applying hierarchical
mixedorder TVFEs of order 0.5 and 1.5 selectively lead to less unknowns and less
iterations than when mixedorder TVFEs of order 0.5 are applied throughout a denser
mesh for accuracy improvement.
8000
4 0 0 0 .....................................
5000
CO 3 0 0 0 .......................
o 4000
1 3000 .......................
1 I000:: ': ' i. Case 2 H
u:O Case 3
0 1 2 3 4 5 6 7 8 9
Percentage of higher order TVFEs [%]
Figure 7.10: Number of iterations to reach a relative tolerance of 10t with a QMR
solver for the LTSA in Fig. 7.8 for different refinement schemes.
119
To illustrate the regions of refinement for Case 14, we restrict ourselves to 3%o
refinement. We consider a crosssection of the mesh parallel to the antenna with
the boundaries of the metallic cavity and antenna marked with thick black lines. As
mentioned previously, the coarse tetrahedral volume mesh is grown from a coarse
triangular surface mesh in a cut in the plane of the antenna by extruding it into
ten prism layers and breaking each prism into three tetrahedra. With the refinement
constrained to at most four prism layers around the antenna, anywhere between 0 and
12 tetrahedra can be refined corresponding to a given triangle in the crosssection.
In the following, a white / light gray / average gray / dark gray triangle indicates
that 0 i/ 14 / 58 / 912 tetrahedra are being refined. The regions of refinement
for Case 14 (3% refinement) are given in Fig. 7.117.14. For Case 1 (Fig. 7.11), 06
elements can be refined and hence white, light gray or average gray is used. For
Case 2 (Fig. 7.12), 0 or 6 elements can be refined and hence white or average gray
is used. For Case 3 (Fig. 7.13), 012 elements can be refined and hence white, light
gray, average gray or dark gray is used. For Case 4 (Fig. 7.14), 0 or 12 elements can
be refined and hence white or dark gray is used. The figures show that the elements
within and immediately bounding the tapered slot as well as elements close to the
bottom metallic antenna edges in the vicinity of the feed are being refined. This is
fully in agreement with the facts that the antenna works as a TWA and that certain
fringing effects can be expected due to the sharp metallic edges.
Electromagnetic near field parameters are generally more sensitive than far field
parameters. A wellknown example is electromagnetic scattering problems where
fairly crude approximations to induced currents can often yield very accurate scat120
Figure 7.11: Regions of refinement for the LTSA in Fig. 7.8 for Case 1 with 3%
refinement.
Figure 7.12: Regions of refinement for
refinement.
the LTSA in Fig. 7.8 for Case 2 with 3%
121
Figure 7.13: Regions of refinement for the LTSA in Fig. 7.8 for Case 3 with 3%
refinement.
Figure 7.14: Regions of refinement for
refinement.
the LTSA in Fig. 7.8 for Case 4 with 3%
122
tered fields since integration of induced currents via radiation integrals is a very
forgiving process. This also holds for certain antenna problems where crude approximations to equivalent near field currents can be integrated to provide very accurate
far field patterns. To investigate the far field characteristics of the above LTSA as
predicted with the different TVFE options, E and Hplane patterns with mixedorder TVFEs of order 0.5 applied throughout the coarse mesh, mixedorder TVFEs
of order 0.5 applied throughout the dense mesh and mixedorder TVFEs of order 0.5
applied within the coarse mesh in conjunction with mixedorder TVFEs of order 1.5
in the four prism layers closest to the antenna are given in Fig. 7.157.16 with the last
pattern normalized to 0 dB at endfire. The need for accurate field modeling within
the tapered slot is obvious from these figures as the pattern found with mixedorder
TVFEs of order 0.5 applied throughout the coarse mesh differs significantly from the
two others. The reason is that the poor field modeling within the tapered slot accumulates and leads to aperture fields so inaccurate that the integration of equivalent
aperture currents cannot provide the correct far field patterns. More accurate field
modeling via a denser mesh or addition of higher order TVFEs provides more accurate patterns. They are similar although the levels at endfire (0 = 0) and close to
grazing are different.
To investigate the merits of adaptive refinement and determine whether 40%
higher order TVFEs are really needed, the E and Hplane patterns with mixedorder
TVFEs of order 0.5 applied within the coarse mesh in conjunction with mixedorder
TVFEs of order 1.5 in the four prism layers closest to the antenna are repeated in
Fig. 7.177.18 where also E and Hplane patterns for Case 14 with 3% refinement
123
Eplane
80 60 40 20 0 20 40 60 80
0 [degrees]
Figure 7.15: Eplane patterns for the LTSA in Fig. 7.8.
Hplane
80 60 40 20 0 20 40 60 80
0 [degrees]
Figure 7.16: Hplane patterns for the LTSA in Fig. 7.8.
124
Eplane
80 60 40 20 0
e [degrees]
20 40 60 80
Figure 7.17: Eplane patterns for the LTSA in Fig. 7.8.
Hplane
80 60 40 20 0 20 40 60 80
0 [degrees]
Figure 7.18: Hplane patterns for the LTSA in Fig. 7.8.
125
are given (same normalization as above). We observe that Case 14 give almost identical patterns. Discrepancies around the Eplane shoulders at 40  60~ can be viewed
but overall they are of similar value and shape, i.e. the specific refinement scheme
is unimportant for this particular application. More importantly, the patterns for
Case 14 with 3% refinement are seen to be very similar to that corresponding to 40%
higher order TVFIEs, expressing again a need for very few higher order TVFEs for
accurate field modeling provided these are placed properly.
7.3 Summary
In this chapter, the hierarchical mixedorder TVFEs of order 0.5 and 1.5 for tetrahedra proposed in this dissertation were used in conjunction with a simple adaptive
refinement strategy to analyze the impedance and pattern characteristics of TSAs
using the hybrid FE/BI method. The adaptive inclusion of a very small percentage of higher orderor TVFEs was found to have a dramatic effect on the accuracy of
the computed input impedances and far field patterns, thus justifying the approach
proposed in this dissertation for large and complex problems.
126
CHAPTER 8
SUMMARY, CONCLUSIONS AND FUTURE
WORK
In this chapter, brief summaries and the most important conclusions for the individual chapters are given and several future tasks to be completed are suggested.
8.1 Summary and conclusions
In chapter 1, the work presented in this dissertation was introduced. After a brief
motivation, some fundamental concepts were presented, a highlevel description of the
proposed approach was given and the organization of the dissertation was outlined.
In chapter 2, background material was given. Vector wave equations used throughout the dissertation were presented and tangential vector finite elements (TVFEs) for
triangular and tetrahedral elements used for discretizing partial differential equations
were reviewed.
In chapter 3, hierarchical mixedorder TVFEs of order 0.5, 1.5 and 2.5 for tri
127
angular elements were proposed. An efficient set of vector basis functions for the
expansion of the surface current on a perfectly electrically conducting (PEC) generalized quadrilateral was converted to vector basis functions applicable for finite element
(FE) analysis and hierarchical mixedorder TVFEs of order 0.5, 1.5 and 2.5 were proposed for triangular elements. The proposed hierarchical mixedorder TVFEs of order
0.5 and 1.5 were tested for solution of closed as well as opendomain problems. For
solution of certain classes of electromagnetic problems, field expansion using hierarchical mixedorder TVFEs of order 0.5 and 1.5 selectively was found to be a very
promising approach in terms of accuracy, memory and central processing unit (CPU)
time requirements as compared to a more traditional approach.
In chapter 4, hierarchical mixedorder TVFEs of order 0.5, 1.5 and 2.5 for tetrahedral elements were proposed. They were constructed as the direct threedimensional
equivalents of the hierarchical mixedorder TVFEs of order 0.5, 1.5 and 2.5 for triangular elements proposed in chapter 3. The proposed hierarchical mixedorder TVIEs
of order 0.5 and 1.5 were tested for solution of closed as well as opendomain problems. Again, selective field expansion was found to be a very promising approach for
accurate and efficient solution of certain classes of electromagnetic problems.
In chapter 5, the condition numbers resulting from finite element method (FEM)
analysis using the hierarchical mixedorder TVFEs of order 1.5 for triangular and
tetrahedral elements proposed in chapter 3 and chapter 4 were contrasted to those of
existing interpolatory and hierarchical mixedorder TVFEs of order 1.5 for triangular
and tetrahedral elements. The proposed hierarchical mixedorder TVFEs of order 1.5
proved better conditioned than existing hierarchical mixedorder TVFEs of order 1.5
128
and thus the analysis fostered no concerns for potential future convergence problems
due to excessive matrix condition numbers. In addition, an approach for improving
the condition numbers of FEM matrices resulting from selective field expansion was
suggested and tested. The improvement comes at the expense of a more complicated
formulation and computer code but does not alter accuracy.
In chapter 6, a review of existing error estimators and indicators was given and
the effectiveness of the proposed hierarchical mixedorder TVFEs of order 0.5 and
1.5 for tetrahedra was investigated when some of the reviewed error indicators were
applied in the context of a, very simple adaptive refinement strategy. The results
were extremely promising for both narrowband and broadband antennas provided
the refinement was carried out on a subdomain by subdomain basis as opposed to
an element by element basis.
In chapter 7, the hierarchical mixedorder TVFEs of order 0.5 and 1.5 for tetrahedra proposed in this dissertation were used in conjunction with a simple adaptive
refinement strategy to analyze the impedance and pattern characteristics of tapered
slot antennas (TSAs) using the hybrid finite element / boundary integral (FE/BI)
method. The adaptive inclusion of a very small percentage of higher order TVFEs was
found to have a dramatic effect on the accuracy of the computed input impedances
and far field patterns, thus justifying the approach proposed in this dissertation for
large and complex problems.
129
8.2 Future work
The hybrid F'E/BI formulation for threedimensional opendomain problems presented in this dissertation is based on a lowest order boundary integral (BI) formulation even when higher order TVFEs are (partly or fully) used within the interior
of the computational domain. This implies that even when higher order TVFEs are
(partly or fully) used within the cavity, higher order vector basis functions associated
with edges and faces on the BI surface are eliminated to maintain field uniqueness.
Since the BI surface is partly in the near field of the metallic antenna, a (partly or
fully) higher order BI formulation is expected to provide superior accuracy. Such
a formulation would require a (partly or fully) higher order testing scheme for the
pertinent integral equation. This formulation could be carried out and the results
could be compared to those based on the lowest order BI formulation.
Motivated by the superiority of hierarchical mixedorder TVFEs of order 1.5 over
those of order 0.5, the merits of even higher order hierarchical mixedorder TVFEs
could be investigated. Those of order 2.5 for triangular and tetrahedral elements
proposed in this dissertation could be implemented and their effectiveness could be
assessed. If deemed necessary, even higher order TVFEs could be suggested, implemented and evaluated following the principles outlined in this dissertation.
Although methods for adaptive refinement have been studied in mathematics and
engineering for decades, use of such methods for solution of practical engineering
problems is only beginning to emerge. This dissertation includes a study of adaptive
TVFE refinement for the hierarchical mixedorder TVFEs proposed in this disserta
130
tion using three different error indicators in the context of a very simple adaptive
refinement strategy. Alternative hierarchical mixedorder TVFEs, other error indicators, various error estimators and more complex adaptive refinement strategies could
be studied to provide an understanding of the interrelations between the different
approaches and their applicability for solution of practical electromagnetic problems.
This dissertation is limited solely to polynomial order refinement (prefinement)
techniques with a uniform mesh density. The effectiveness of mesh density refinement (Ihrefinement) techniques with a uniform polynomial order could be studied for
similar problems and contrasted to that offered by prefinement techniques with a uniform mesh density. Subsequently, hprefinement techniques could be developed. Dual
hprefinement techniques are theoretically always superior to isolated hrefinement
or prefinement techniques but often difficult to implement practically [29]. Useful
progress to this end has been reported in [22, 89].
As justified previously, this dissertation focuses on triangular and tetrahedral elements due to their geometrical modeling flexibility. However, hierarchical mixedorder
TVFEs for alternative element shapes could be developed and contrasted to those proposed in this dissertation. Hierarchical mixedorder TVFEs for curved triangular and
tetrahedral elements could be constructed from those proposed in this dissertation for
straight triangular and tetrahedral elements via a straightforward mapping, see for
instance [37], and similar hierarchical mixedorder TVFEs for other elements (rectangles, bricks, prisms, pyramids) could be constructed by direct analogy. A comparative
study of all these hierarchical mixedorder TVFEs could be carried out for solution
of practical electromagnetic circuit, scattering or radiation problems.
131
APPENDICES
132
Appendix A
Explicit expressions for Wi, W2 and W3
To derive an expression for W1, we introduce two coordinates (u1, v1) over the
triangle. These are degenerates of similar coordinates for a generalized quadrilateral
[69]. ul takes its minimum value ul,min 0 at node 1 and its maximum value
Ul,max = 1 along edge #2 while v1 takes its minimum value vl,i — 1 along edge
#3 and its maximum value Vrmax = 1 along edge #1. u1 is constant and v, is linear
along straight lines parallel to edge #2 while u1 is linear and v1 is constant along
straight lines starting a node 1 and ending at edge #2, as illustrated in Fig. A.I.
Using these coordinates, the position vector r defining P can be expressed as [69]
r r= + ul ru1 + uivl ruil (A.1)
where
r,  [(r3 ri) + (r2 ri)] (A.2)
rul = (r2r3). (A.3)
Further, u1 and v1 can be shown to be related to the simplex coordinates (1, (2 and
133
Figure A.1: Illustration of the variation of u1 and v1 over a triangle.
43 via
U1 = (2 + ~3 (A.4)
(2  C3
V1  (A.5)
C2 + (3
From (3.4) for n = 1, trivial algebra then leads to
W1 = C2V(3  C3VC2. (A.6)
To derive expressions for W2 and W3, we can similarly introduce coordinates
(u2, v2) and (U3, V3) where U2,3 = 0 at node 2, 3, U2,3 = 1 at the edge opposite to node
2, 3 and V2,3 = ~1 along the two edges shared by node 2,3. The algebra is similar
and we arrive at
U2 = C3 + C1 (A.7)
V2 3  A.8)
3 + (A.S)
13 + 4I
W2 = C3VCi  C1VC3 (A.9)
134
U3 (=,i + (2
V3 _ _ _ _ 2
w3 I (
(A.1O )
(A.1I1)
(A.[12)
135
Appendix B
Expressions for vector basis functions
Peterson's interpolatory mixedorder TVFE of order 1.5
Peterson's interpolatory mixedorder TVFE of order 1.5 is characterized by the
eight vector basis functions
W: = (2V(3 (B.1)
WI = C3VC2 (B.2)
W1 = C3V(C (B.3)
W = CVC3 (B.4)
W = lVC2 (B.5)
W1 = C2VCl (B.6)
W = C3(C1VC2  (2VCi) (B.7)
W = Cl (2VC33  C3V2). (B.8)
136
Proposed hierarchical mixedorder TVFE of order 1.5
The proposed hierarchical mixedorder TVFE of or der 1.5 is characterized by the
eight vector basis functions
W1 = 2VC3  C3V(2
w2 = C3V(l  (iV(3
(B.9)
(B.10)
w2 
3  1V2  (2V(l
(B. l 1)
W4 = ((2  3)(C2V(3 ,3V(2)
W2 = ((3  l1)(3V(l  CiV(3)
(B.12)
(B. 13)
W2  (  2)(iV(2  2V(l)
(B.1L4)
W7'  (3 ((I V (2  (2 V (1)
(B.15)
W' = (l((2V(3  (3V(2)
(B.] 6)
Transformation between mixedorder TVFEs of order 1.5
The two mixedorder TVFEs of order 1.5 presented above are related through
a linear transformation. Let [W1] be a column vector containing Peterson's eight
vector basis functions WJ, j = 1,.., 8, and [W2] be a column vector containing the
137
proposed eight vector basis functions W?, i 1,., 8. In this case, [W2] is related
to [WI] via
jVLIj II
[W2] =[Aj][Wl]
(B.17)
where [A i] is the sparse 8 x 8
transformation matrix
[A.j] =
1
0
0
1
0
0
0
0
1
0
0
1
0
0
0
0
0
1
0
0
1
0
0
0
0
1
0
0
1
0
0
0
0
0
1
0
0
1
0
0
C
C
C
0
1
0
0
O
) 0
0
1 0
) 2
0
0
0
1
1
2
0
1
I
(B. L8)
1
1
1
0
Proposed hierarchical mixedorder TVFE of order 2.5
The proposed hierarchical mixedorder TVFE of order 2.5 is characterized by the
fifteen vector basis functions
W3I = C2V(3  C3VC2
w3 = (3VC,  Cl V3
W3 = GCV~2  2V(C
(B.19)
(B.20)
(B.2 )
w3= (2 
C3)(82V(3  (3V(2)
138
(B.22)
W3  ((3 _ (1)((3V(l _
5 
O V(3)(B2)
(B. 2, J)
 ((I ((2)(V(2 
w7 ((2 ((2V( 3
9NV12  (2 (( V (2 
(B.24)
(3V(2)
(B.25)
(B. 2)(~
 (2V(l)
(B. 27)
(B.28)
(B.29)
2V6)
(B.3 0)
w13 ( 1((2V7(3  (3V(2)
w3 (2((V I (I V(3)
(B.3[)
(B.32)
(B.3 33)
139
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140
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