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Nonintrusive microwave diagnostics of collisional plasmas in Hall thrusters and dielectric barrier discharges

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Nonintrusive Microwave Diagnostics of
Collisional Plasmas in Hall Thrusters and
Dielectric Barrier Discharges
DISSERTATION
Joshua Stults, Capt, USAF
AFIT/DS/ENY/11-15
DEPARTMENT OF THE AIR FORCE
AIR UNIVERSITY
AIR FORCE INSTITUTE OF TECHNOLOGY
Wright-Patterson Air Force Base, Ohio
APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED
The views expressed in this document are those of the author and do not reflect the official
policy or position of the United States Air Force, the United States Department of Defense
or the United States Government. This material is declared a work of the United States
Government, and is not subject to copyright protection in the United States.
AFIT/DS/ENY/11-15
NONINTRUSIVE MICROWAVE DIAGNOSTICS OF COLLISIONAL PLASMAS IN
HALL THRUSTERS AND DIELECTRIC BARRIER DISCHARGES
DISSERTATION
Presented to the Faculty
Graduate School of Engineering and Management
Air Force Institute of Technology
Air University
Air Education and Training Command
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
Joshua Stults, B.S.A.E., M.S.A.E.
Capt, USAF
September, 2011
APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED
UMI Number: 3474023
All rights reserved
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UMI 3474023
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AFIT/DS/ENY/11-15
NONINTRUSIVE MICROWAVE DIAGNOSTICS OF COLLISIONAL PLASMAS IN
HALL THRUSTERS AND DIELECTRIC BARRIER DISCHARGES
Joshua Stults, B.S.A.E., M.S.A.E.
Capt, USAF
Approved:
Richard E. Huffman Jr., LtCol, PhD
Chairman
Date
Dr. Mark F. Reeder
Member
Date
Dr. William F. Bailey
Member
Date
Accepted:
M. U. Thomas
Date
Dean, Graduate School of Engineering and Management
AFIT/DS/ENY/11-15
Abstract
This research presents a numerical framework for diagnosing electron properties in collisional plasmas. Microwave diagnostics achieved a significant level of development during
the middle part of the last century due to work in nuclear weapons and fusion plasma research. With the growing use of plasma-based devices in fields as diverse as space propulsion,
materials processing and fluid flow control, there is a need for improved, flexible diagnostic
techniques suitable for use under the practical constraints imposed by plasma fields generated
in a wide variety of aerospace devices.
Much of the current diagnostic methodology in the engineering literature is based on
analytical diagnostic, or forward, models. The Appleton-Hartree formula is an oft-used analytical relation for the refractive index of a cold, collisional plasma. Most of the assumptions
underlying the model are applicable to diagnostics for plasma fields such as those found in
Hall Thrusters and dielectric barrier discharge (DBD) plasma actuators. Among the assumptions is uniform material properties, this assumption is relaxed in the present research
by introducing a flexible, numerical model of diagnostic wave propagation that can capture
the effects of spatial gradients in the plasma state. The numerical approach is chosen for
its flexibility in handling future extensions such as multiple spatial dimensions to account
for scattering effects when the spatial extent of the plasma is small relative to the probing
beam’s width, and velocity dependent collision frequency for situations where the constant
collision frequency assumption is not justified.
The numerical wave propagation model (forward model) is incorporated into a general
tomographic reconstruction framework that enables the combination of multiple interferometry measurements. The combined measurements provide a quantitative picture of the
iv
spatial variation in the plasma properties. The benefit of combining multiple measurements
in a coherent way (solving the inverse problem for the material properties) is the reconstruction provides a stronger empirical constraint on the predictions of high-fidelity predictive
simulations than multiple un-reconstructed measurements in isolation. Use of the model for
reconstructions informs the choice of numerical discretization technique. The model must
be fast, low-storage and accurate to be useful for computing reconstructions.
An important part of experimental work is error analysis, or uncertainty quantification.
This becomes more difficult as sophistication of the measurement models increase. This
research presents an uncertainty quantification technique based on complex-step sensitivity
derivatives that is particularly well-suited for error propagation in sophisticated partial differential equation (PDE)-based measurement models, because it requires only trivial changes
to the PDE solver to implement.
v
Acknowledgements
Thanks to my wonderful wife for her long-suffering patience during my various academic
endeavors which generally involve prolonged periods of anti-social behavior.
Thanks to Dr John Schmisseur of Air Force Office of Scientific Research (AFOSR) for
providing partial funding support of this research.
Thanks to the Air Force Institute of Technology (AFIT) Department of Aeronautics
and Astronautics laboratory support staff for their help purchasing, setting-up and troubleshooting the equipment used in this research.
Thanks to the US Air Force for providing this training opportunity. Special thanks to
my fellow students (especially Dave, Dan and Steve) for your thoughtful criticisms and ready
help. The instruction of our shared struggle will bear fruit on other days and other fields.
Joshua Stults
vi
Table of Contents
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
I.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1
1.2
1.3
1.4
1.5
II.
DBD Plasma Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Hall Thrusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Problem Statement and Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Structure of the Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Plasma Diagnostic Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 Tomographic Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 Microwave Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Detailed Micro-discharge Measurement and Modeling . . . . . . . . . . . . . . . . . . . . . 31
2.3 Device Measurement and Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Applications of DBD Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Reconciling Models and Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
III. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1 Forward Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.1 Low-order Finite Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.2 Pseudospectral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 Tomographic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2.1 Axi-symmetric Material Distribution: Inverse Abel
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.2.2 Arbitrary Material Distribution: Total Variation
Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3 Complex-step Sensitivity Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.3.1 Interpolation in the Uncertain Parameter Space . . . . . . . . . . . . . . . . . . . . 86
3.4 Instrumentation and Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
IV. Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1 Forward Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 Hall Thruster Exhaust Plume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3 Dielectric Barrier Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
vii
Page
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
V.
Conclusions and Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.1 Numerical Forward Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.2 Hall Thruster Exhaust Plume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3 Surface Dielectric Barrier Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A.
Verification and Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.1 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.2 Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B.
Source Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
B.1 Forward Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
B.2 Inverse Abel Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
B.3 Total Variation Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
C.
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
C.1 Hall Thruster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
C.2 DBD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
viii
List of Figures
Figure
Page
1
DBD Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2
Hall Thruster Cross-section Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Interferometer Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4
Interferometry Setup of Yang for a DBD Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5
Basic Tomography Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6
Schematic of Cylindrical Medical Tomograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7
Measurement Rake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
8
DBD with Downstream Exposed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
9
Sliding Electrode Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
10
DBD Lumped-element Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
11
Equivalent DBD Circuit Including Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
12
Lumped-element Model of the DBD and Matching Circuits . . . . . . . . . . . . . . . . . 39
13
Appleton-Hartree Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
14
Estimates of Interferometer Operation (Appleton-Hartree) . . . . . . . . . . . . . . . . . 52
15
Noble Gas Cross-Sections for Momentum Transfer . . . . . . . . . . . . . . . . . . . . . . . . . 53
16
Atmospheric Gas Cross-Sections for Momentum Transfer . . . . . . . . . . . . . . . . . . . 54
17
One-dimensional Diagnostic Wave Propagation Model . . . . . . . . . . . . . . . . . . . . . 62
18
Eigenvalues of Low-order Finite Difference Spatial Operator . . . . . . . . . . . . . . . . 65
19
Error of Pseudospectral and Finite Difference Derivatives . . . . . . . . . . . . . . . . . . 66
20
Lanczos-σ Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
21
Spatial Operator Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
ix
Figure
Page
22
Effect of pre-recurrence truncation on eigenvalues (nx = 64,
m = 12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
23
Real-part of Eigenvalue with Maximum Imaginary Component . . . . . . . . . . . . . . 72
24
Spectral Convergence of Derivatives (y = cos(10x), x ∈ [−π, π]) . . . . . . . . . . . . . 74
25
General Tomographic Reconstruction Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 75
26
Constant Property Cylinder Theoretical Phase-Shift Derivatives . . . . . . . . . . . . 77
27
Inverse Abel Transform for Constant Property Cylinder, nx = 51 . . . . . . . . . . . . 78
28
High-resolution Inverse Abel Transforms of Constant Property
Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
29
Convergence of Inverse Abel Transforms for Constant Property
Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
30
Comparison of Total Variation for Functions with Equivalent
Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
31
Integral Sensitivity to Low-order DCT Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 85
32
Simple Microwave Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
33
Vertical DBD Microwave Interferometry Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
34
Horizontal DBD Gaussian optics antennas (GOA) Microwave
Interferometry Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
35
GOA Feed: Coaxial Cable to WR-28 Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . 90
36
Rhode & Schwarz ZVA 40 Vector Network Analyzer (VNA) . . . . . . . . . . . . . . . . 90
37
Tektronix AFG 3022B Arbitrary Function Generator . . . . . . . . . . . . . . . . . . . . . . 91
38
Crown XLS202 Audio Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
39
Oil Burner Ignition Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
40
DBD Driving Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
x
Figure
Page
41
DBD Circuit with Impedance Matching Network . . . . . . . . . . . . . . . . . . . . . . . . . . 93
42
Spectral Convergence of Derivatives (y = cos(10x), x ∈ [−π, π]) . . . . . . . . . . . . . 97
43
Time-stability of the Present Pseudospectral Discretization . . . . . . . . . . . . . . . . . 98
44
Time Complexity of the Present Pseudospectral Discretization . . . . . . . . . . . . . . 99
45
Convergence of Complex-step Sensitivity Derivative . . . . . . . . . . . . . . . . . . . . . . 101
46
Visible Emissions from a 200W Hall Thruster with Krypton
Propellant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
47
Two-parameter (σ, n) Calibration Model Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
48
Beam Intensity Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
49
Calibration Cylinder Index of Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
50
15GHz Microwave Transmission through 200W Hall Thruster
Plume, Krypton fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
51
15GHz Microwave Transmission through 200W Hall Thruster
Plume, Xenon fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
52
Average Phase Data for Krypton Propellant at 15GHz . . . . . . . . . . . . . . . . . . . . 108
53
Average Amplitude Data for Krypton Propellant at 15GHz . . . . . . . . . . . . . . . . 109
54
Average Phase Data for Xenon Propellant at 15GHz . . . . . . . . . . . . . . . . . . . . . 110
55
Average Amplitude Data for Xenon Propellant at 15GHz . . . . . . . . . . . . . . . . . 111
56
Relation of Significant Amplitude Response to Phase Variation
(Xe) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
57
Relation of Significant Amplitude Response to Phase Variation
(Kr) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
58
Imaginary Part of Appleton-Hartree Refractive Index . . . . . . . . . . . . . . . . . . . . . 113
59
Real Part of Appleton-Hartree Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . . 114
60
Xenon Phase Data, y = −205 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
xi
Figure
Page
61
Xenon Amplitude Data, y = −205 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
62
Xenon Phase Reconstruction, y = −205 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
63
Xenon Amplitude Reconstruction, y = −205 mm . . . . . . . . . . . . . . . . . . . . . . . . . 119
64
Forward Model Solution for Mean Center of Plume
Reconstruction,y = −205 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
65
Normalized Sensitivity Derivative for Collision Frequency . . . . . . . . . . . . . . . . . 122
66
Normalized Sensitivity Derivative for Electron Density . . . . . . . . . . . . . . . . . . . . 122
67
Uncertainty in Peak Center of Plume Electron Density,
y = −205 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
68
Uncertainty in Peak Center of Plume Collision Frequency,
y = −205 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
69
Visible Emissions from Wire and Flat-Tape Electrode DBDs . . . . . . . . . . . . . . . 127
70
Wire Exposed Electrode DBD Visible Emissions . . . . . . . . . . . . . . . . . . . . . . . . . 128
71
Phase Response from Wire-Electrode DBD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
72
Representative Phase Data Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
73
ne ≈ 1 × 1018 m−3 , νe0 = 1 × 1012 rad/s, f = 35GHz . . . . . . . . . . . . . . . . . . . . . . 130
74
ne ≈ 1 × 1018 m−3 , νe0 = 1 × 1012 rad/s, f = 160GHz . . . . . . . . . . . . . . . . . . . . . 131
75
Long-time Phase Response at 26.5GHz due to Dielectric Heating . . . . . . . . . . . 132
76
Real-part of refractive index (Appleton-Hartree) . . . . . . . . . . . . . . . . . . . . . . . . . 138
77
Estimate of Optimal Diagnostic Frequency for DBD Plasma
Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
xii
List of Tables
Table
Page
1
Numerical Forward-model Non-dimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . 49
2
Integral Sensitivity to Perturbations of DCT Coefficients . . . . . . . . . . . . . . . . . . . 86
3
Forward Model Fit to Center of Plume Reconstruction,
y = −205 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
xiii
Nomenclature
λ
wavelength
νt
total collision frequency for momentum transfer
νe0
electron-neutral collision frequency for momentum transfer
νei
electron-ith species collision frequency for momentum transfer
ω
diagnostic frequency
ω0
electron plasma frequency
σ
standard deviation or collision cross-section
B
magnetic field
c
speed of light
Ey
transverse electric field
Jy
transverse current density
me
electron mass
n
refractive index
ne
electron number density
q
electron charge
uy
transverse electron velocity
xiv
AFIT Air Force Institute of Technology
AFOSR Air Force Office of Scientific Research
AC alternating current
APP atmospheric pressure plasma
BDF backward difference formula
BC boundary condition
CAS computer algebra system
CFD computational fluid dynamics
COTS commercial off the shelf
CT computed tomography
DBD dielectric barrier discharge
DC direct current
DCT discrete cosine transform
DFT discrete Fourier transform
DoE design of experiments
EM expectation maximization
EHD electro-hydrodynamic
ESP electrostatic precipitator
FDTD finite difference time domain
xv
FEM finite element method
FFT fast Fourier transform
GGMRF generalized Gaussian Markov random field
GMRES generalized minimum residual
GOA Gaussian optics antennas
HDPE high-density polyethylene
IC initial condition
ICD iterative coordinate decent
IVP initial value problem
LTE local truncation error
MAP maximum a posteriori
MCMC Markov-chain Monte Carlo
MEM maximum entropy method
MHCD micro-hollow cathode discharge
MHD magneto-hydrodynamics
MMS method of manufactured solutions
MOL method of lines
MURI Multidisciplinary Research Initiative
MWI microwave interferometry
xvi
OAUGDP one atmosphere uniform glow discharge plasma
ODE ordinary differential equation
PDE partial differential equation
PIC particle-in-cell
PIV particle image velocimetry
RANS Reynolds-averaged Navier-Stokes
RCS radar cross section
SSOR symmetric successive over-relaxation
UAV unmanned air vehicle
UQ uncertainty quantification
VNA vector network analyzer
WKB Wentzel-Kramers-Brillouin
xvii
NONINTRUSIVE MICROWAVE DIAGNOSTICS OF COLLISIONAL PLASMAS IN
HALL THRUSTERS AND DIELECTRIC BARRIER DISCHARGES
I. Introduction
The motivation for this research is to develop improved techniques for diagnosing the
plasma properties in collisional plasmas such as those found in Hall thrusters or DBD plasma
actuators. The term diagnostic is borrowed from the medical community. In medicine,
diagnosis means that the symptoms or test results we observe are used to support inferences
about a disease that may remain unobserved by any direct means. In the context of a
plasma measurement, the state is our disease. The state (density, temperature, velocity, etc)
is often not directly observable for a variety of reasons that are often specific to the type of
plasma or plasma generating device. The symptoms we can observe are emitted, reflected
or transmitted electromagnetic waves. These observables, in combination with a diagnostic
model, are used to indirectly diagnose the plasma state.
Non-intrusive diagnostics can be passive (plasma emissions are collected), or active (an
externally generated wave is sent into the plasma). A well developed modality for active
plasma diagnostics is microwave (or millimeter-wave) interferometry [1]. This method relies
on a model of the plasma particle motion under the influence of the interrogating wave. For
the cold, collisional plasmas under consideration a Lorentz model is assumed for the particle
motion. These currents (moving charged particles), in turn, cause a phase and amplitude
shift of the interferometer beam which allows the electron number density, ne , and effective
collision frequency for total momentum transfer, νt , to be inferred. Since the electrons are
much lighter than the ions, only their current is included in the measurement model. At the
1
high frequencies used for the interrogating beam, the ions do not have time to accelerate to
significant speeds during each half-cycle of the diagnostic wave because of their relatively
large mass. For this reason ion currents are usually not included in the diagnostic, or forward,
model.
If multiple interferometry measurements are made along different chords through the
plasma under test, then these can be combined into a reconstruction of the full plasma field.
This reconstruction process is known as computed tomography, and is an ill-posed problem.
The ill-posedness requires that additional assumptions, or regularization, be introduced to
give the problem a unique solution with desirable properties.
The methods developed during this research are applied to diagnosing the properties of
a Hall Thruster exhaust plume as well as a DBD in air. The Hall Thruster plume provides a
large, low collisionality plasma which has been diagnosed using several techniques, so useful
comparisons can be made to the present methods. The Hall Thruster plume is roughly
axisymmetric, so simplified reconstruction techniques are available. Some of the challenges
that DBD plasmas have for diagnostics above those of the Hall Thruster plume are small
spatial extent, high collisionality, low ionization levels and a lack of spatial symmetry.
The plasma in a DBD is unsteady and highly collisional, so it provides an interesting
test case for microwave diagnostics. Producing a reconstruction of the electron density and
collision frequency in the highly collisional plasma of a DBD aerodynamic actuator would
provide insight into the operation of the device and add useful empirical constraints for
future modeling validation efforts.
The two plasma generating devices used in this research will first be briefly introduced.
Since the use of DBDs as flow control devices is relatively newer, their continuing development as actuators is covered. A discussion of some relevant plasma diagnostic techniques
follows that.
2
1.1
DBD Plasma Actuators
DBD plasma actuators are an emerging possibility for active flow-control in low-speed
aerodynamics. They rely on imparting small amounts of momentum into the boundary layer,
which can delay separation and increase lift at critical conditions such as stall in an airfoil or
off-design conditions in a turbine. They also offer the possibility of high-bandwidth control
(rapid response times) without the use of moving parts. This can increase reliability, options
for redundancy, and survivability while lowering the mass of actuator systems.
Plasma actuators operate on the same principle, a silent discharge or DBD, which Siemens
first used in 1857 to make ozone from oxygen or atmospheric pressure air [2]. The basic
plasma actuator consists of an electrode exposed to the working fluid (air) offset some distance from an electrode shielded by a dielectric. Exciting the electrode with a high voltage
causes a plasma to form over the dielectric. Figure 1 shows a two-dimensional schematic of
a basic linear DBD actuator.
While not directly applicable to flow-control, much of the work on developing an understanding of non-equilibrium air plasma generation and kinetics mechanisms grew from the
AFOSR Air Plasma Ramparts Multidisciplinary Research Initiative (MURI). This effort
was aimed at developing energy-efficient techniques for generating non-equilibrium plasmas
in air of appreciable volume and with electron densities on the order of 1013 cm−1 . Such
plasmas were intended to act as shield from electromagnetic waves for aircraft or critical
components. This application area has continued to receive research effort, but, as the basic
propagation equations used in the present study predict, negligible “shielding” effect is seen
in experiments. Obviously, the equations governing the propagation of diagnostic waves are
the same as those governing such “shielding” applications. Use of surface DBDs for flow
control has seen more success.
The plasma in the actuator is formed by the locally high electric field between the elec-
3
trodes accelerating electrons (present in small quantity in neutral air or produced through
emission at the cathode) across the electrode gap. These high energy electrons collide with
the neutral atoms, which results in ionization reactions. The ions thus created are accelerated by the electric field between the electrodes and transfer momentum to the neutral
gas through further collisions. Plasma actuators must have an asymmetric arrangement of
electrodes to achieve useful (net) momentum transfer. The ability of the plasma actua-
Flow Direction
Exposed Electrode
Dielectric
Buried Electrode
Figure 1. DBD Schematic
tors to transfer momentum to the neutral gas gives them utility as flow control devices in
aerodynamic applications at atmospheric pressure.
For plasma actuators to become an effective part of the aeronautics design toolbox the
physical processes underlying their performance needs to be well understood. An important
parameter in understanding the plasma generation process in these devices is the electron
density and collision frequency for momentum transfer. The small size of the devices and the
high potential of the electrodes mitigate against intrusive probe measurements of the plasma
parameters in the DBD. This leaves non-intrusive diagnostics for quantifying the plasma
parameters of interest in these devices. One of the primary non-intrusive means of measuring
plasma electron density and collision frequency is through microwave interferometry [1].
4
1.2
Hall Thrusters
Hall thrusters are electric space propulsion devices that rely on the Hall effect to ionize
their propellant (commonly Xenon or Krypton) and accelerate the ions thus created into
a high velocity exhaust jet, which creates low-levels of thrust very efficiently [3]. A crosssectional view of a Hall thruster channel is shown in Figure 2.
Figure 2. Hall Thruster Cross-section Schematic (taken from [3])
The radial magnetic field causes electrons from the cathode to experience a spiral motion
around the field lines. This prevents them from traveling directly down the thruster channel
to the anode, and allows them to participate in many ionization collisions with the propellant.
An important parameter for Hall thruster operation is the Hall parameter, a ratio of the
electron cyclotron frequency to the electron heavy particle collision frequency, given by
β=
eB
me ν
5
(1)
where e is the electron charge, B is the magnetic field, me is the electron mass and ν
is the collision frequency. In addition to the motion around the magnetic field lines, the
electrons experience an E × B drift azimuthally around the channel and scatter (diffuse) due
to collisions with heavy particles.
1.3
Problem Statement and Research Objectives
The problem addressed in this research is improving the analytical tools for generating
empirical constraints on the electron density and collision frequency in Hall thruster and
DBD actuator plasmas. The novel improvements developed in the present research include
a pseudospectral discretization that is particularly well-suited for diagnostic use, and is
broadly applicable to any type of linear wave propagation problem. Though the focus of
this research is on plasma diagnostics, the numerical methods developed here can be used
in acoustic or radar scattering, non-destructive ultrasound diagnostics, or modeling radio
wave propagation in the ionosphere. In addition, the Chebyshev pseudospectral PDE solver
used in the present research provides a novel application opportunity for the complex-step
uncertainty quantification method. This method is used to propagate uncertainty in the
measured data to the diagnosed physical properties.
Microwave interferometry is the diagnostic mode used used to demonstrate the analytical
techniques. Since the empirical constraints on electron state will hopefully be useful for
comparison with simulations of these devices, space and time resolution are desired. The
spatial variation in the field can be reconstructed from diagnostic measurements on the
boundaries through tomographic reconstruction methods.
The main hardware development tasks to support this research objective are to implement a microwave interferometer and the high-voltage driving circuitry for an atmospheric
pressure DBD. An existing Hall thruster that has been previously characterized using simpler diagnostic models will be used to facilitate comparison with results published in the
6
literature. The microwave interferometer is based on a commercial off the shelf (COTS)
vector network analyzer (VNA) with custom focusing optics. The driving circuit for the
DBD is based on an oil burner igniter and COTS audio amplifiers.
The main modeling and analysis tasks consist of implementing a numerical model of the
governing diagnostic wave propagation equations which allows measured phase and amplitude shifts to be related to plasma properties, and implementing a reconstruction algorithm
to take multiple interferometry measurements and reconstruct the spatially varying field of
electron state.
As with any experimental effort, error propagation is important so that the reported
results are credible and can be reliably compared to other diagnostics and simulation predictions. Since the measurements are heavily model dependent, special care must be taken in
performing this uncertainty quantification. A novel scheme for doing uncertainty quantification (UQ) for this plasma diagnostic technique based on complex-step sensitivity derivatives
is also implemented.
In summary, the main tasks for supporting the research objective of diagnosing electron
state in Hall Thruster exhaust plumes and sea-level pressure DBD in air are
• Build a microwave interferometer for characterization of the plasma in a DBD actuator
• Implement a one-dimensional wave propagation model
– Seamlessly incorporate the effect of material property gradients
– Extensible to multiple space dimensions
– Pseudo-spectral discretization: high accuracy, low memory, calculated with fast,
off-the-shelf transforms
• Implement novel uncertainty quantification technique, which is well suited to supporting use of PDE-based measurement models
7
• Reconstruct spatial field from multiple interferometry measurements
– Use an implicit regularization method (Abel Inversion) to compute reconstructions of a Hall Thruster plume including effects of electron scattering
– Use a explicit regularization method to compute reconstructions of the DBD
plasma
1.4
Methodology
An initial study using the exhaust plume of a Hall thruster will be performed. This
type of plasma has been extensively characterized by microwave and other diagnostics, and
simplifying assumptions can be made for the reconstruction [4–6]. The purpose of the pilot
study is to establish the measurement characteristics of the microwave interferometer. The
signal-to-noise ratio, temporal and spatial resolutions will be estimated. This information
then informs the use of the technique on the DBD plasma.
This axisymmetric plasma in the Hall Thruster plume can be used to validate various
tomographic reconstruction techniques. Since the plasma is axisymmetric the reconstructions
can be compared to the results of commonly used Abel inversion [7].
The preliminary testing will consist of a simple interferometer with one transmit and one
receive horn on either side of the plasma under test. The testing of the DBD uses the buried
electrode of the device as a reflecting surface for the diagnostic waves. Measurements can be
made at multiple incidence angles using two horns, and at normal incidence one horn can
be used for both the transmission and reflection measurement.
It is useful for discussion to divide the limitations of the present work into two main categories: those that depend on instrumentation hardware and those that depend on modeling
assumptions and numerics.
The hardware has a certain phase and amplitude resolution for making ratio-ed measure8
ments of transmitted and received diagnostic signals. There are also fundamental limitations
in the spatial resolution that is achievable by the focusing optics (in the direction transverse
to the direction of propagation). The spatial resolution in the direction along the wave propagation is governed by the frequency resolution of the instrument (i.e. through pseudo-time
domain techniques variations with frequency can be transformed to indicate variations along
the electrical path [8]). There is also a trade-off between high-resolution sampling in frequency with the time-resolution (since tuning to and sampling each frequency takes a finite
time).
The hardware-centric limitations just mentioned are common to any empirical work.
More important in diagnostic work (indirect measurements) are the limitations introduced
by modeling assumptions. These assumptions dictate what physical properties are inferred
from the diverse measurements. As an example for discussion, consider a wave propagating
through a plasma that has low ionization, and also has significant variations in the density of
the background neutral gas. The electron currents and collisions induced by the diagnostic
wave will naturally cause phase and amplitude shifts described by the diagnostic model (so
will the ions, though these are generally neglected this introduces yet another modeling
assumption which imposes its own limitations). The variations in neutral density will cause
phase shifts as well. Attributing the measurements to one or the other is a matter of critical
importance, and depends solely on the modeling choices made by the experimentalist. Ideally,
a band of frequencies is chosen to respond to one physical mechanism more strongly than
the other. Microwaves and millimeter waves respond strongly to electron motion, and less
strongly to variations in neutral density. Theory indicates the response from electron currents
has an optimum below commonly used optical (infrared) frequencies.
9
1.5
Structure of the Present Work
The present work is divided into five chapters and two supporting appendices as follows.
Chapter I introduces the emergence of dielectric barrier discharges as flow control devices,
and describes the present effort at developing non-intrusive diagnostics for the small plasmas
developed in these devices. Chapter II provides a review of the literature on plasma actuator flow control, as well as microwave-based diagnostic techniques for plasmas and other
materials (primarily biomedical applications). Chapter III develops the models of microwave
propagation used in the present diagnostic effort. Chapter IV presents quantitative results of
microwave measurements made of Hall Thruster exhaust plumes and atmospheric pressure
dielectric barrier discharges in air. Chapter V discusses the implications and limitations of
the present work, and provides suggestions for future research. Appendix A gives details on
the development of the correctness verification and uncertainty quantification for the diagnostic models. Appendix B provides listings of the source code for the diagnostic models
and reconstruction algorithms. Finally, Appendix C is an archive of plots for the raw data
collected during this research.
10
II. Literature Review
This literature review will cover plasma measurement techniques, specific modeling and
measurement efforts for Hall thruster and DBD plasmas, then briefly cover DBD plasma
actuator application areas since this is a relatively newer use for discharge plasmas. First
the most general diagnostic approaches will be covered, followed by detailed micro-discharge
behavior characterization efforts, then device-level modeling and measurement and the aerodynamic application areas will be covered, and finally the literature review will conclude with
the current state of reconciliation between simulation and empirical results of DBD behavior. Since a DBD at atmospheric pressure is composed of many micro-discharges occurring
over the spatial extent of the device and throughout the duration of the driving wave-form,
an understanding of these is useful to building up an understanding of the coarser physical
behavior of DBD actuator devices. Plasma actuators have been applied in a wide variety of
flow control applications, both internal and external flow, generally at low Reynolds number.
Since the actuator achieves its effect through momentum coupling in the near-wall, boundary layer portion of the flow, applications requiring re-attachment or delay of separation
are especially prevalent. Also, some direct attempts at modulating the growing turbulent
instabilities in transitional boundary layers have been attempted.
Applying plasma diagnostic techniques taken from nuclear weapons and fusion plasma
research communities to Hall thruster exhaust plumes has met with considerable success. The
plasma generated by the Hall thruster in a lab setting is similar enough to other laboratory
plasmas that the classical interferometry, reflectometry and emissions techniques work well.
It is more challenging to apply these techniques to the small surface discharge plasmas in
DBD actuators.
The measurements achieved to date of DBD actuators have been focused mainly on bulk
flow characterization (flow visualization, velocimetry, induced force, etc) rather than plasma
11
parameters per se. A notable exception to this is the early work of the research group at
the University of Tennessee. They attempted microwave interferometry measurements (at
10GHz) to quantify electron density and collision frequency of DBD arrays [9]. Related work
on microwave propagation in atmospheric pressure plasmas was also funded by the AFOSR
Air Plasma Ramparts MURI, though the plasma generating devices were not necessarily
DBDs in an actuator-representative configuration [10, 11]. Recent work on the chemical
kinetics and plasma properties of individual micro-discharges using spatially resolved emissions spectroscopy have also been performed [12]. While no reconstruction efforts have
been accomplished for DBD plasmas, relevant results on these measurement post-processing
techniques from the medical imaging and broader plasma diagnostics community will be
reviewed.
In addition to these quantitative plasma diagnostic efforts, other groups have performed
qualitative studies with high-speed digital photography of the discharge process. These studies indicate that even when the driving waveform of the actuator is a symmetric alternating
current, the discharge itself is quite different in the positive and negative phases, and that
the discharge is composed of many micro-discharges which occur at time-scales several orders
of magnitude faster than the driving wave-form (kHz) [13].
2.1
Plasma Diagnostic Techniques
As mentioned in the introduction, plasma diagnostics rely on measuring the changes in
transmitted and reflected waves on the boundaries of a plasma to infer the properties on its
interior. One of the main modes of making such measurements is interferometry. Figure 3
shows several different types of interferometers. Microwave interferometry is a non-intrusive
plasma diagnostic technique useful to estimate plasma characteristics from the amplitude
and phase shift of microwaves transmitted through the test plasma. Figures 3 illustrate
several of the different types of interferometer described in [7]. This type of measurement
12
In
Beamsplitters
Plasma
Plasma
In
In
Out
Beamsplitters
Beamsplitter
Out
Out
Out
(a) Mach-Zehnder
Out
(b) Fabry-Perot
(c) Michelson
Figure 3. Interferometer Types
relies on the difference in refractive index of the plasma compared to free-space propagation.
Various models of the measurement process, refractive index relation to plasma densities and
collision frequencies, and reconstruction methods can be applied to interferometry data.
Interferometry for diagnosing electron density in Hall thruster exhaust plumes has been
reported at Ku- and Ka-band frequencies (12-18 GHz and 26.5-40 GHz respectively) [6, 14,
15], and more recently at millimeter-wave (90 GHz) frequencies [5, 16]. The motivation for
applying non-intrusive techniques to this plasma are similar to those underlying the present
effort for DBD actuators. In the near field of the Hall thruster exit region, intrusive probes
can interfere with the operation of the thruster. The diagnostic models reported in the
literature for the Hall thruster plasma generally do not include the effect of elastic scattering
of the electrons on the wave propagation. The peak electron densities found by these methods
are on the order of 1010 to 1011 cm−3 . When diagnostic attenuation is reported it is generally
small and the effect is not included in the estimate of electron density [14]. In contrast
to the Hall thruster plasma, interferometry of DBD devices is less well developed, and the
collisional effects are not negligible.
Measuring the plasma characteristics in DBD actuators is critical to understanding and
optimizing their ability to transfer momentum to the working fluid. This process is governed
by the density of the charged species in the plasma, their collision frequency with the neutral
13
background gas and subsequent effects on the bulk gas state and velocity. Yang made
time-resolved measurements of the electron number density and collision frequency in a
fluorescent lamp and an atmospheric plasma using a 10GHz microwave interferometer ([9]).
His interferometry set-up was unique in that it used the buried electrode (sheet of copper)
of the DBD as a reflecting surface so that the interrogating beam passed through the test
plasma twice (shown in Figure 4).
In
Variable
Incidence
Out
Plasma
Buried Electrode
(reflecting surface)
Figure 4. Interferometry Setup of Yang for a DBD Sheet [9]
The complicating factor of using microwave interferometry on atmospheric pressure plasmas is the high electron-neutral collision frequency and the sometimes low electron number
density. This leads to difficulties in measuring small, low-density plasmas, such as those
that occur on DBD actuators [9, 11]. Laroussi discusses the more sophisticated techniques
required for using microwave interferometry diagnostics of atmospheric plasmas (including
collisional effects) as opposed to the low pressure variety [17]. He also showed the difficulty
in measuring low density atmospheric pressure plasmas such as those found in DBD actuators. Different measurement models can be attempted to ameliorate this difficulty of DBD
diagnostics. Shneider and Miles develop a Rayleigh scattering model rather than an interferometry model to perform microwave diagnostics of small plasma objects [18]. This model
applies when the plasma dimensions are much smaller than the microwave wavelength. The
14
plasma is treated as a dipole which is uniformly polarized by the incoming microwave.
In more traditional interferometry efforts, measurements of electron number density in
a fluorescent lamp were made by Howlader et al. with a network analyzer to measure the
phase shift and attenuation so that the Appleton-Hartree formula could be used to calculate
the electron number density and collision frequency directly without making assumptions
about the electron energy distribution function, electron temperature, or energy-dependent
collision cross-section for the collision process [19]. While the plasma being measured was low
pressure, the approach has bearing on successfully measuring high pressure plasmas. Akhtar
et al. presented a general formulation for measuring plasma density and effective collision
frequency for high and low collisional plasmas using microwave interferometry. They present
analysis showing that simultaneous measurement of the phase change and the amplitude
change is required to uniquely determine the plasma density for highly collisional plasmas
[20].
For cold, collisional plasmas of the DBD variety, early analytical efforts at quantifying
microwave propagation indicated that the plasma thickness needs to be greater than the
wavelength of the interrogating wave, and the electron number density needed to be on the
order of 1013 cm−3 to get measurable phase and amplitude differences [11]. Eckstrom et al.
demonstrated that for a cold, collisional plasma much greater microwave absorption could
be achieved if there were gradients in the electron density on the order of 2 wavelengths of
the interrogating wave [10, 21]. This result provides motivation for extending microwave interferometry (MWI) for DBD plasmas to higher frequencies. Better resolution and improved
signal to noise ratio are achievable. It also indicates that forward models which incorporate
the effect of non-constant material properties are valuable.
The need for numerical modeling of electromagnetic wave propagation in collisional plasmas was recognized as early as 1933 by Mary Taylor in pointing out directions for future
work following her extensive analytical analysis of radio wave propagation in the ionosphere
15
considering collisional losses [22, 23]. Numerical models of the linear wave phenomena of
electromagnetics that use low-order, centered finite differences are generally known as finite
difference time domain (FDTD) codes. More recent work on applying high-order compact
differencing schemes to classical wave equations and the full Maxwell’s equations in a finite
volume framework has been accomplished by Shang and Gaitonde [24–27]. A draw-back of
these compact schemes is the difficulty in achieving time-stability, and the complications of
achieving stable, non-reflecting boundary conditions. The ultimate extension of finite difference schemes to arbitrarily high-order is known as a spectral, or pseudospectral approach.
Limited work on establishing the stability of such schemes for the classical two-way wave
equation in one spatial dimension has been accomplished [28–30].
One of the important characteristics of collisional plasma diagnostics is that the frequency
of the interrogating wave need not be bound by the plasma frequency [11, 22, 23]. This gives
some additional flexibility in choosing the diagnostic frequency. Early measurements in this
area were based on 10GHz waves [9, 19, 31]. Recent work on low density (collision-less)
plasmas has shown the promise of better resolution that can be achieved by going to higher
interrogation frequencies [5]. These measurements at 90GHz were modeled using a FDTD
discretization of the three-dimensional Maxwell’s equations [32]. This is a significant advance
over the usual analytical diagnostic models based on the Appleton dispersion relation. The
model included effects of collisions as well as the magnetic field. It was shown that while
the differences in diagnosed properties could be on the order of 15%, the differences between
diagnostic models were still smaller than the experimental uncertainty.
Zhang et al. use an integral-differential wave equation to numerically evaluate the propagation of an electromagnetic wave through an atmospheric pressure plasma (APP) layer.
This numerical solution is then compared to Appleton’s equation derived from the WentzelKramers-Brillouin (WKB) solution of the integral-differential equation [33]. This type of
more complex numerical wave propagation model is an important component for integrating
16
interferometry measurements into a more detailed tomographic diagnostic technique (discussed below in section 2.1.1). Since the derivation of the Appleton-Hartree formula relies
on constant material properties, going to a numerical model allows to more easily and accurately incorporate the effects of gradients in the refractive index on the measurement wave
[23]. In a more complex forward model, Hu et al. used a FDTD discretization for the full
Maxwell’s equations to model the attenuation in a thin atmospheric plasma layer [34]. They
found that the shape of the electron density distribution in the thin plasma layer can have
a significant affect on the phase shift and attenuation of the wave.
Incorporating multiple frequencies into the interferometer measurement allows more accurate measurement of the relevant plasma characteristics. Dobson et al. used a polychromatic
microwave interferometer (three frequencies) to measure reflections between the transmitter
and receiver and analyze the effect on error estimation when taking these reflections into
account [35].
Extension of the basic microwave interferometry technique to infrared frequencies has
been attempted for a DBD plasma [36]. These reported results are problematic for a number
of reasons. The diagnostic model (forward model) used in that work did not include the
effects of electron-neutral collisions. In addition, as shown by Enloe et al. the actuator
can have a significant effect on the neutral gas density (on the order of 2% the background
density) [37]. The technique used to diagnose the density change relies on gradients in neutral
density to displace a laser beam (changes in index of refraction caused by density changes).
As predicted by the Appleton equation for the index of refraction due to electron motion
in a collisional plasma (discussed in section 3.1), the phase shift for a DBD sized plasma
approaches zero for these relatively high frequencies. Therefore, the measured result was
much more likely to be caused by gradients in the bulk gas density [38, 39] than the electron
density as attributed by the choice of incomplete forward model. Either a more complete
and accurate forward model that includes both effects should be used, or more appropriate
17
frequencies for each measurement (electron density, neutral density) should be chosen so that
the contribution to phase shift from each process can be properly attributed. This highlights
an important consideration that must be at the forefront of any diagnostic approach: the
physical inferences that can be drawn depend sensitively on the choice of forward model,
and there is little within the techniques themselves to tell when the choice of forward model
is insufficient. Auxiliary information from other diagnostics and simple physical reasoning
are invaluable for “sanity checking” the results.
There are several ways of separating the contributions to changes in refractive index from
charged particle currents, and those from variations in the neutral gas density. Leipold et. al.
present a method which uses infrared interferometry to simultaneously diagnose the electron
density and neutral density contributions by making assumptions about the time-scales over
which the changes for these two physical mechanisms take place [40]. The changes due
to electron currents are rapid compared to changes due to neutral density variation. This
fact can be exploited in the case of a transient discharge event to separate out the two
competing physical mechanisms. This approach was extended by Choi for diagnosing small
direct current (DC) plasma jets [41].
Another way of separating the physical mechanisms which cause phase shift is relying
on the diversity in the response for the mechanisms as a function of frequency. This would
be especially applicable if the transients in each mechanism do not have widely separated
characteristic times. Urabe et.al. combine several measurements at infrared and millimeterwave frequencies to achieve a diagnosis of electron density in a Helium / Nitrogen pulsed
DBD plasma [42].
Incorporate auxiliary measurements from intrusive devices is yet another approach to
validating the diagnostic modeling assumptions for non-intrusive techniques. Plasma density measurements were made by Deline et al. using a hybrid Langmuir probe microwave
interferometry technique. The Langmuir probe was traversed across the chord of the in18
terferometer measurement. This allows for correction of the interferometer measurements
to account for shot-to-shot variation and reduces the ill-posedness of any subsequent reconstruction of the plasma density field [43]. The applicability of these hybrid techniques may
be limited for DBD devices because of the unsteady evolution and small spatial extent of the
plasma. Novel combinations of several non-intrusive techniques in this hybrid spirit could
be quite beneficial though.
2.1.1
Tomographic Techniques.
A natural extension of microwave interferometric techniques on atmospheric plasmas is
microwave tomography. Tomography is the generation of three-dimensional spatial variation
of a quantity based on multiple chordal measurements, such as those arising in microwave
interferometry. This process of generating the internal structure of the plasma under test
from measurements on its boundary is known as reconstruction. In this type of diagnostic
technique, “[t]he objective is to extract all the possible information of some of the variables
based on knowledge of the measurements, model and information obtained prior to the
measurement” [44].
This section contains a review of methods for accomplishing the reconstruction. A
schematic of a modular tomographic method is shown in Figure 5. This modular organization
of the process high-lights the many options that are available for solving the tomographic
inverse problem. A variety of forward models, regularizations, and optimization techniques
can be combined into a reconstruction algorithm. This section will focus on the regularization and optimization aspects present in the literature, while section 3.1 will develop the
forward model specific to the present research effort.
Tomographic reconstruction is generally an iterative procedure. The method consists of
solving a forward model for the wave propagation through the unknown medium. This is
followed by some type of regularization which usually penalizes spurious detail or variation in
19
Initialize Reconstruction
Apply the forward model
A model of the measurement process
producing the data from the object
being reconstructed.
Calculate the objective function
Generally a function of the measured data,
data predicted by the forward model
and the reconstruction itself.
Measured data
Yes
No
Convergence criteria met?
Done
Perturb the reconstruction
Figure 5. Basic Tomography Process
20
the material property (tends to smooth edges in the reconstruction). Then an optimization of
the image based on an objective function, which generally includes a measure of agreement
with the measurements as well as the influence of the regularization, is performed. The
objective function can be a least-squares distance in the measurements, or a probability
based on a statistical model which includes prior information and structural assumptions (eg.
axisymmetry). This optimization can be accomplished in a pixel-wise fashion, or by using
gradient information for the entire image with respect to the objective function. The process
then repeats with solving the forward model based on this new distribution of properties.
Tomographic techniques can be divided into three main categories based on their choice
of objective function (regularization) [45]:
• Bayesian method: able to deal with noisy data and incorporate prior information
• maximum entropy method (MEM): less tolerant of noise, but able to incorporate prior
information consistently
• least squares: tolerant of noise, but unable to incorporate prior information consistently
Regularization methods range from the essentially ad hoc to the comprehensively Bayesian.
The ad hoc methods generally rely on a modified least squares objective function subject to
Tikhonov regularization.
fobj = kDobs − Df (u)k + σA(u)
(2)
where Dobs is the observed data, DF (u) is the prediction of the forward model, σ is a tunable
parameter and A(u) is a regularization functional. Total variation,
VT =
Z
|u0 (x)| dx
is a common choice of regularization functional [44, 46–49].
21
(3)
Bayes Methods.
As an alternative to the some-what ad hoc regularization approaches, a Bayesian approach
can be used to determine the maximum a posteriori (MAP) reconstruction. The posterior
probability of the image f given the data, P (f |d) is
P (f |d) = αP (f )P (d|f )
(4)
where P (f ) is the prior probability of the image, P (d|f ) is the likelihood of the data given
the image and α is a constant of normalization. An iterative optimization technique can be
used to find the most probable image.
The Bayesian formalism provides a coherent framework for probabilistic reasoning. Bayesian
approaches to reconstruction frame the problem in terms of Bayes Theorem,
p(A|B) = c · p(B|A)p(A)
(5)
where c is a normalization constant, p(A) is the prior probability, p(B|A) is the likelihood
and p(A|B) is the posterior probability. In this setting the prior would incorporate knowledge of the type of distribution being reconstructed and B would be the measurements.
The objective function in the Bayesian approach is then simply the conditional probability
p(A|B). The reconstruction which maximizes this function is known as the MAP estimate.
Hanson applies Bayes methods to incorporate prior information about the object being imaged to improve the quality of computed tomography (CT) reconstructions. The focus is on
evaluating the MAP method, with some discussion on extending to a more general Bayesian
approach [50].
The MEM is an approach that has been widely used in reconstruction efforts. Gull
and Daniell present an algorithm for noisy image reconstruction [51]. Gull and Newton
22
use the MEM for tomographic reconstruction and discuss the use of prior information [52].
Wu presented a revised version of the Gull-Daniell algorithm [53]. Mohammad-Djarari and
Demoment apply the MEM to the Fourier synthesis step in diffraction tomography and provide an extensive discussion of the limitations of methods that do not incorporate a priori
knowledge in a consistent way. They identify the implicit hypotheses (prior information)
incorporated out of computational convenience as the principle shortcoming of other reconstruction techniques.
Intuitively, the distribution that most honestly describes what we know, without
anything else, is the one that maximizes H [entropy] subject to the constraints
imposed by our information [54].
Even if a fully Bayesian method is not used for the reconstruction process, an analysis
of the method in the Bayesian framework is often useful to understand the virtues and
vices of a more expedient method [44, 55]. Since the structure of the plasma in a DBD has
not previously been measured the Bayes or MEM methods would not be of much utility
in initial measurements (no useful prior information to provide). A least squares objective
function seems most appropriate for initial testing. As high-fidelity physical and empirical
models of the plasma structure are built-up, a transition to MEM or Bayes methods would
be warranted to increase the fidelity and efficiency of measurements.
Local-search Methods.
The most basic reconstruction methods use pixel-at-a-time perturbations to the image
[56]. The constraints of the measurements are then satisfied approximately using an optimization technique such as simulated annealing [57].
Smith and Paxman et al. discuss algorithms and experimental results [58] as well as
code design [59] for image reconstruction. One of their important conclusions stated in the
23
abstract is “It appears that the fidelity of a reconstruction depends much more strongly
on the design of the data-taking system (the coded apertures) than on the reconstruction
algorithm” [58]. This matters to the current research effort because the choice of measurement chords can introduce unwanted artifacts in the reconstruction. Their coded apertures
are arrays of detectors rather than just the single detector / line-integral that would be
considered in a standard computed tomography. The example problem presented is of a
two-dimensional object with 64 × 64 pixels coded by two orthogonal one-dimensional 256
element apertures.
Their iterative back-projection algorithm is
1. The first iterate is fˆ1 = H T g, where H T is the back-projection operator (transposed
projection operator) and g is the coded data.
2. An image correction is formed by H T (g − ĝ 1 ), with ĝ 1 = H fˆ1
3. The correction is added to the image to find the next iterate
Formally:
h
i
fˆk+1 = fˆk + αH T g − H fˆk
(6)
where α is an acceleration parameter. The back-projection method is appropriate for the
present effort under the assumptions of one-dimensional wave propagation. A more general
optimization approach is required for two-dimensional propagation that includes the effects
of material property gradients (refraction).
They also present a simulated annealing based approach which uses an energy based on
the mean square difference between the true and the estimated coded images. “The smoothing is accomplished by adding a weighted term to the coded-image energy that depends on
the difference of the pixel value in question (in the reconstruction) from the average of its
four nearest-neighbor values.” [58]. The code design (design of coded apertures) method
24
they present is based on knowing the first and second order statistics of the object class
to be imaged. Using this knowledge allows to choose an optimum aperture design for that
object class [59].
The energy function to be minimized with simulated annealing is based on the difference
between the measurements predicted by the forward model and the actual measurements.
E=
s
PN
i=1
dtrue
− destimate
i
i
N
(7)
As optimization progresses the reconstruction gets closer and closer to satisfying the constraints (energy approaches zero).
Perturbations to the pixel values that lower the energy function are accepted, those that
raise the energy are accepted with a probability related to the cooling schedule, parametrized
by kT .
P (∆E) = e−∆E/kT
(8)
By sometimes accepting perturbations which increase the energy, this method avoids being
trapped by local minima in the cost function.
This is an outline of Geman & Geman’s algorithm for restoring degraded images [46]:
1. A local change is made to a pixel based on nearby pixels. The change is drawn from
a local conditional probability distribution.
2. The local conditional distributions are dependent on the simulated annealing temperature. At low temperature the distributions concentrate on states that increase the
posterior distribution, while at high temperature it is uniform.
3. The approach is model based, with a Bayesian hierarchical model captured in the prior
distribution.
25
The simulated annealing approach is more general and has the advantage that a backprojection operator does not have to be formed. The cost of this is that it tends to be more
costly computationally (in terms of forward model evaluations). Geman and Geman use
simulated annealing to find the MAP images using a general Bayesian image reconstruction
approach [46]. Smith et al. demonstrate a simulated annealing approach to image reconstruction which minimizes the difference between the measurements predicted for the current
image from the actual measurements. Simulated annealing is a local search-based method;
the optimization steps are taken at the individual pixel level [56].
Gradient Methods.
In contrast to local search such as simulated annealing, gradient based methods attempt
to use more information about the problem to speed up convergence. Saquib, Hanson and
Cunningham apply a conjugate gradient search method to find the MAP image reconstruction. The forward model is a diffusion equation solved with finite differences, and the gradient
is constructed using an adjoint differentiation method. Bent-line searching is used to enforce
a positivity constraint on the unknown coefficients. An edge-preserving generalized Gaussian Markov random field (GGMRF) is used as the image model [60]. The gradient data
required, as in all such methods, is the “gradient of data likelihood with respect to unknown
field” (see equation 5).
Outline of Hielscher, Klose and Hanson’s method for optical tomography [61]:
• Solve a diffusion equation using FDTD for the wave propagation (the forward problem)
• Compare the predicted values to the measured values
• Calculate the objective function based on the comparison of measured data and modeled data plus an arbitrary regularization
• Calculate the gradient using an adjoint differentiation method
26
• Update the data according to the gradient of the objective function
The methods of sensitivity analysis for computational fluid dynamics (CFD) and finite element method (FEM) are applicable to forming the gradient information needed by gradientbased inverse problem solutions. Kleb and Nielsen demonstrate the use of complex variables
for sensitivity analysis of unstructured CFD solutions [62]. This complex step approach, of
which Squire and Trapp provide a concise explanation [63], avoids the explicit formation of
the Jacobian as in the above adjoint approach. It also avoids the subtractive cancellation
inherent in the normal finite-difference-based approaches to sensitivity analysis. Sensitivity
derivatives, even if not obtained with the complex step method, can be used to improve the
efficiency of sampling (including Markov-chain Monte Carlo (MCMC)) approaches [64]. The
forward model for the present effort is based on a pseudo-spectral discretization of the wave
equation (developed in Section 3.1.2). This complex-step approach has been extended to
calculating the sensitivity derivatives for pseudo-spectral codes, and, with proper care taken
in the transforms of the complex solution, the favorable subtractive-cancellation avoidance
characteristics are retained [65].
Another approach, which combines the idea of local search and gradient decent, is based
on the Gauss-Seidel method from the iterative solution to linear systems. Bouman and
Sauer present a iterative coordinate decent (ICD) algorithm (known also as Gauss-Seidel)
for MAP estimation of tomographic reconstructions within the Bayesian formalism [66].
In a similar way, the expectation maximization (EM) algorithm iteratively calculates the
expected value of the unknown field based on current likelihood parameter estimates, and
then, given these, finds the parameters which maximize the likelihood [67]. Saquib, Bouman
and Sauer apply the EM algorithm to estimate shape parameters for the GGMRF used in
image reconstruction when the true image is unavailable [68]. Arridge and Schweiger use the
FEM to solve the forward problem and calculate the gradient of the (least squares) objective
function using an adjoint approach [69]. They also Arridge and Schweiger demonstrate
27
the performance of steepest descent and conjugate gradient based optimization for optical
tomographic reconstruction [70]. Their use of an adjoint method to construct the gradient
avoids the need to explicitly form the Jacobian, this being a computationally expensive
proposition.
The approaches based on simulated annealing, even though they require many forward
model evaluations, are motivated by a common problem in optimization, which is the curse of
dimensionality. In these reconstruction problems the dimensionality of the objective function
depends on the number of pixels in the reconstruction, this number grows as N 2 in 2D and
N 3 in 3D. As mentioned above, calculating the gradient with respect to this many variables
can be very expensive. One approach to tackle this problem is based on adapting techniques
for high-dimensional quadrature.
Just as methods from design sensitivity analysis (which tend to have low-dimensional design parameter spaces) can be adapted for use in inverse problem inference. Ideas developed
for UQ can be adapted as well. Stochastic collocation on sparse grids is a method developed
for UQ in CFD or FEM settings [71] which can be applied towards inference in inverse problems [72]. This approach avoids the need for expensive MCMC integration methods (many,
many evaluations of the forward model), while ameliorating the curse of dimensionality common to high-dimensional collocation approaches through the use of sparse, adaptive grids.
Rather than seeking a MAP (or other) estimate through an iterative search as in simulated
annealing or gradient search, the stochastic collocation approach seeks to approximate the
entire posterior distribution over the stochastic space. This approximation can then be used
to generate the desired functionals such as the MAP estimate for the reconstruction.
2.1.2
Microwave Tomography.
The reconstruction approaches in the literature for Hall thruster exhaust plumes rely
almost exclusively on methods based on inverse Abel transformation [5, 6, 14–16]. Various
28
different filtering, smoothing, differentiation and integration schemes may be used (a general,
robust numerical approach used in the present research is presented in section 3.2.1), but
the fundamental assumption allowing this approach is that the plume is axi-symmetric. The
DBD plasma has no such convenient geometrical symmetry to exploit in the reconstruction
process. While there are no published applications of tomography to DBD plasmas, work
from the medical imaging community on microwave tomography is germane to the present
effort. The basic operation of the medical tomograph shown schematically in Figure 6 is
Test Object
Transceivers
Figure 6. Schematic of Cylindrical Medical Tomograph
that each antenna operates as the transmitting antenna in a sequential fashion while the
other antennas receive the signals scattered from the test object. This approach could be
applied to the unsteady plasmas in DBD actuators, measurements would have to be made
with each antenna transmitting over several periods of operation so that a time-accurate
empirical model of the plasma could be built.
An alternative way of organizing the transmitters and receivers is in a rake. Figure 7
29
shows a measurement rake suitable for reconstruction. A single rake such as illustrated in
Measurement Chords
Material Distribution
Figure 7. Measurement Rake
Figure 7 can be used in an inverse Abel transform reconstruction where axisymmetry of the
object under test is assumed. This approach is detailed in Section 3.2.1. Reconstruction in
the absence of the axisymmetric assumption requires that additional measurement rakes at
various angles be taken.
Bulyshev et al. demonstrate solving the inverse problem in the full Maxwell’s equations
to perform three dimensional vector microwave tomographic reconstruction [73]. Rather
than solving the full Maxwell’s equations a promising one-dimensional wave model for APP
was presented in [33] and gives more accurate results than Appleton’s equation since arbitrary number density distributions are admissible, rather than the constant number density
assumed in the simpler model.
Hielsher et al. discuss optical tomography reconstruction techniques that perform a
30
gradient based optimization to arrive at the reconstructed image. Some of the discussion
has bearing on microwave tomography techniques because infrared is scattered as it passes
through the tissue (unlike x-ray, which tends to go in a straight line) [61].
The results from the medical imaging community on the microwave modality indicate the
importance of a physically realistic forward model. Going to more complex forward models
often means that the early results for computed tomographic reconstruction (eg. filtered
back-projection) do not apply, since the assumption of straight-line averages through the
material under test is not physically realistic. This highlights the possible need for twoand three-dimensional forward models of the measurement process (which generally requires
numerical solution as presented in section 3.1).
2.2
Detailed Micro-discharge Measurement and Modeling
Detailed modeling of the micro-discharge has proceeded from nanometer-scale devices in
simple gas compositions towards more realistic millimeter-scale devices in complex gas mixtures (air). The wide range of time-scales for the physics relevant to the discharge impose
significant constraints on the time-step and computational domain size of simulations. As
more computational capability becomes available, more realistic configurations (representative of aerodynamic actuators) can be simulated.
Improvements in modeling fidelity can be classed into three main categories: effects of gas
composition, physical size effects (which can include coupling to the neutral fluid inertia),
and driving wave-form frequency. Simple models including only electropositive gases do not
accurately mimic the empirically observed behavior of DBD actuators in air. The behavior of
nanometer-scale actuators, which are relatively easier to simulate, can not be scaled naively
to predict the behavior of millimeter scale devices. A related factor is the frequency of
the electrode-driving wave-form. It is relatively expensive to simulate the kHz frequencies
used in actuators because of the importance of physics on much faster time-scales. This
31
broad separation of time-scales leads to a stiff problem, which means that the governing
equations must be integrated for a prohibitively large number of time-steps (which drives up
the simulation wall-time).
Detailed modeling of the DBD discharge process including gas composition effects is a
fairly recent development [13]. PIC-DSMC simulations of the discharge cycle which included
only oxygen in the working fluid showed net momentum transfer in the appropriate direction
when average over the full discharge cycle, but showed a small upstream force on the order
of 1/20th the downstream force in one half of the cycle [74–76]. Also detailed kinetic calculations for the collision processes in a two-dimensional DBD plasma have been carried out
using simplifications of the Boltzmann equation for the electrons, loosely coupled with a fluid
model for the neutral gas [77]. These simulations show that the electron-neutral collisions
account for on the order of 1% of the momentum transfer. Even though the electrons directly
account for a small proportion of the momentum transfer in a DBD actuator, they play an
important role in ionization collisions, space- and surface-charge build-up, and subsequent
shaping of the electric field in the discharge gap.
Many modeling efforts of DBDs are for simple gas mixtures. This simple modeling can
give insight into some of the behavior of micro-discharges in DBDs such as the transition
from single discharges per driving cycle to multiple discharges per cycle [78]. Also variations
in discharge mode depending on gas composition. Massines et al. compared time-resolved
spectroscopy data with a model based on electron/ion continuity and momentum equations
coupled with Poisson’s equation [79]. The boundary conditions were set based on treating
the dielectric as an equivalent capacitor and the transport and ionization coefficients were
determined by a numerical solution of the Boltzmann equation. They showed that an atmospheric pressure discharge in Helium was glow-like, while the discharge in Nitrogen was
Townsend-like. The difference between a glow-like or Townsend-like discharge is governed by
the raio of relative time-scales of recombination times of charged species to the driving fre32
quency of the discharge. In a glow discharge the electrons and ions trapped in the electrode
gap between cycles play the predominant role in discharge initiation, while in a Townsend
discharge metastable species contribute to initiation by generating electrons at the cathode
through secondary emission. This time-scale dependence of the discharge initiation mechanism is apparent in studies of nano-second pulse discharges in atmospheric air. Ito et al.
made time-resolved measurements of the electric field in nano-second DBD discharges in air
that revealed the influence on the rapid rise-time of the applied potential on the build-up of
space-charge near the cathode, and the discharge initiation mechanism which was found to
be fundamentally different than the Townsend mechanism [80].
The force production mechanisms present in DBD actuators are discussed by Enloe et
al. [81]. Measurements of the time-averaged force produced by an asymmetricDBD in a
vacuum chamber at various ambient pressure levels were made. These empirical results
show that the force varies linearly with the ambient pressure and non-linearly with the
electric field strength and ion density. Time-averaged flow velocities and spatial profiles of
induced airflow over a DBD actuator have been made and related to the applied power /
voltage [82]. The effect of electrode gap and dielectric thickness on power dissipation and
induced velocity have been quantified.
The power dissipated into the neutral gas by a collisional plasma was investigated by
Fruchtman [83]. He showed the fraction of power deposited in the neutrals is independent
of the power level, but depends on the atomic cross-sections and the electron temperature.
The development addresses neutral-pumping through charge exchange collisions.
In the DBD many micro-discharges occur throughout the cycle of the driving wave-form as
well as distributed over the surface of the device. This means that the ionization is confined to
the micro-discharges, with the neutrals serving as a background sink for energy/momentum
[84]. The micro-discharge happens at time scales considerably faster than the driving voltage
cycle. Electrons reach equilibrium with the applied field on the order of picoseconds, and the
33
micro-discharges occur on the nanosecond time scale, but the chemistry of excited species
generated by the micro-discharges can reach micro to milliseconds [84]. Thus, the effect of
many micro-discharges averaged over a kHz driving cycle leads to effects on the bulk gas
properties and composition. Spectroscopic studies of micro-discharges and detailed kinetic
modeling show that the charge distribution on the dielectric surface is important to the
discharge development and that there are two areas of high chemical activity near the two
electrodes [85].
The empirical force measurements have shown a push-push behavior of DBD actuators.
This result is not found in models of the discharge in simple gases and in small geometries.
A four-fluid mixture (drift-diffusion approximation) model of a plasma actuator in air which
successfully captures the ’push-push’ behavior shows that including oxygen is important to
capturing the correct behavior, and that the size of the device matters as well [86, 87]. The
model results show the importance of the presence of negative ions for DBDs in air. This
model also provides enough insight into the micro-discharge process to identify inefficiencies
due to sinusoidal driving wave forms and suggest repetitively pulsed nanosecond discharges
superimposed on a DC bias as an optimum.
Other simulations using the drift diffusion approximation confirm the ’push-push’ behavior of the DBD in air [88]. These results found that the relative contribution of the
positive and negative ions depends on the applied voltage and frequency. The contribution
from negatively charged species tends to dominate at low frequency and high voltage. A
parametric study was performed for driving wave-forms between 1 and 10 kHz and 4 and 30
kV. The simulation shows the spatial distribution of the force in the positive and negative
half-cycles is different, so even with similar force magnitude the aerodynamic effects may not
be the same. This variation in the spatial distribution of the force during the two half-cycles
is indicated by the different nature of the discharge seen in the high-speed imaging to the
DBD actuators. Much more diffuse discharges are present as the electrons flow towards the
34
dielectric surface and build-up a locally self-limiting charge.
2.3
Device Measurement and Modeling
The insights from the detailed micro-discharge-level modeling informs our knowledge of
the bulk, device-level behavior of DBD actuators. As discussed above, the asymmetry in
the discharge characteristics is due to the presence of an exposed electrode and a virtual
electrode at the dielectric surface. When electrons flow towards the dielectric they create
high charge density regions, which do not get conducted away, as they would on the exposed
electrode. This space charge limits the formation of streamers or arcs in this phase of the
discharge, so the discharge is diffuse and spread over the dielectric surface which covers the
buried electrode. When the electrons are directed towards the exposed electrode they are
not prevented from forming arcs by any space charge build-up. This phase of the discharge
is thus dominated by arcs and streamers, and is much less diffuse.
As discussed above, the asymmetry of the discharge is driven by the asymmetry of the
electrode arrangement. The actuator is self limiting because charge builds-up on the downstream dielectric, which lowers the electric field between the electrodes, thus limiting the
power dissipated by the device. If the grounded electrode is not encapsulated, then plasma
will form on both sides of the dielectric, which is usually not desirable in flow-control applications [89]. The charge build-up can be advected downstream of the initial plasma actuator
location and remain on the aerodynamic surface (if it is a dielectric) for time-periods many
orders of magnitude longer than the driving waveform frequency [90]. These results are
consistent with other optical measurements of DBDs [91].
Opaits et al. showed that the surface charge build-up downstream of a standard DBD
plasma actuator is detrimental to the thrust produced by the actuator. The build-up shields
the plasma from experiencing the full voltage, a lower effective voltage leads to lower thrust.
They recommend exposing a down-stream portion of the covered electrode in order to prevent
35
Flow Direction
Buried Electrode
Exposed Electrode
Dielectric
Exposed Downstream
Portion
Figure 8. DBD with Downstream Exposed
Flow Direction
Buried Electrode
Exposed Electrode
Dielectric
Sliding Electrode
Figure 9. Sliding Electrode Configuration
charge build-up and increase thrust [90]. In a similar vein, Takashima et al. experimentally
studied a 3-electrode DBD configuration which has a grounded sliding electrode on the same
surface as the high voltage electrode [92]. This configuration is able to collect charge advected
downstream of the generation point and extend the plasma region. They found that the
presence of the sliding electrode was able to extend the plasma region significantly. Without
the sliding electrode the plasma region extended by 0.11 cm/kV, while it extended 0.16
cm/kV when the sliding electrode was present. The presence of an additional downstream
36
electrode is also shown to be important to shielding the high-potential electrodes from each
other in multi-actuator arrays [93].
Initial modeling efforts of DBD actuators focused on lumped element circuit models [94].
A lumped element circuit model consists of grouping discrete resistance, capacitance and inductance elements into an equivalent circuit for the DBD. The DBD is mainly a capacitive
device (the planar electrodes separated by a dielectric being reminiscent of the classic parallel
plate capacitor). Orlov studied a time-varying lumped-element model consisting of a single
dissipative element (the conduction current in the plasma) and three capacitive elements.
This simple model, shown in Figure 10, was able to achieve agreement with empirical results
for power dissipation that exhibited the expected power-law scaling [94, 95]. The power dissipated by the asymmetric DBD being proportional to the applied voltage to the 7/2 power.
Capacitor C3 is the capacitance between the exposed and buried electrodes. The surface
Figure 10. DBD Lumped-element Circuit Model
of the dielectric above the buried electrode provides a virtual electrode. The capacitance
between this virtual electrode and the buried electrode is C2, and C1 is the capacitance
which is shorted out by the plasma current through R1. The dissipation modeled by R1 is
caused by electron-neutral collisions (losses), the effective collision frequency describing this
loss is a primary parameter in the diagnostic model for the present study.
While the load presented by a DBD device is primarily capacitive, some investigators
have included an inductive element as well. The basic idea of a virtual electrode on the
37
DBD surface is retained, but an inductor is added. Figure 11 shows the equivalent circuit
used by Singh and Roy [96]. The physical reason for this inductance is electron oscillations
Figure 11. Equivalent DBD Circuit Including Inductance
between collisions (the high collision frequency in an atmospheric pressure discharge leads
to a small magnitude for this inductance).
The equivalent electrical parameters are governed by size, shape and material properties
of the DBD device [97]. A complicating factor in developing simple lumped element circuit
models is that the current is time varying (many micro-discharges), and the effective resistance is also non-linear and time varying as well since it depends on the plasma properties
throughout the discharge cycle.
The four main power flows in the actuator of engineering importance are: reactive losses
due to poor actuator/power-supply impedance matching, heating of the dielectric, power
required to ionize the atmospheric pressure air and finally power supplied to the neutral gas
through collisions [98]. Optimization of the DBD actuator requires putting the most power
into the last of these power sinks for the least cost in the others.
The power dissipated in the load (the plasma resistance in these lumped models), can
38
be optimized by impedance matching. Figure 12 shows an equivalent circuit model of the
DBD along with an impedance matching network as given in [96]. By adding an inductor
Figure 12. Lumped-element Model of the DBD and Matching Circuits
the reactance caused by the capacitor can be canceled, and the power reflected back to the
power supply can be minimized.
The simple lumped-element models presented above can be experimentally calibrated
[94] and then semi-empirically derived body force terms can be introduced into standard
fluid dynamics codes for modeling the aerodynamic effect of the DBD actuator [99]. This
approach has shown good results (after calibration) in Navier-Stokes calculations with a
body force vector based on solving for the electric field due to the applied AC voltage at the
electrodes and the electron density based on an empirical model [100]. Related empirical
work supported identification of some of the important physical parameters for efficiency
and power in DBD actuators. Dielectric material properties, electrode geometry, electrode
dimensions, driving frequency, and the gap between the exposed and buried electrode can
39
significantly affect the induced flow velocity above the electrode [98].
2.4
Applications of DBD Actuators
In the field of aerodynamic flow control DBD actuators have been demonstrated in a
variety of conditions. At low-speeds these uses depend on injecting momentum into the
boundary layer at critical times/locations to create substantial changes in the bulk flow. At
high-speeds flow-control is achieved by periodic heating effects causing compression waves
which induce vortical structures to entrain momentum down in to the boundary layer [101].
The most common method is a straightforward momentum injection in the direction of the
bulk flow to increase suction or reduce separation in airfoil-type applications. An alternative
approach is “finger” arrays which inject momentum in a cross-flow direction to generate
vortices. These vortices are then advected downstream and achieve separation control effects
in a similar manner to more traditional vortex generators. The velocities achieved by DBD
actuators reported in the literature have been in the range of 1 to 10 m/s at atmospheric
pressure [102–105]. This level of control power makes them suitable for actuation at low to
moderate Reynolds numbers [106, 107].
Internal flow applications of DBD actuators has focused on reducing losses on blades in
turbo-machinery. Much work has been done on applying DBD actuators to configurations
that mimic the environment of a turbine blade at an off-design, low Reynolds number condition. This is of significant interest because turbines are generally built for high on-design
efficiencies, but operation at off-design conditions that induce separation can lead to costly
pressure (and performance) loss. Lennart et al. demonstrated the use of a DBD to delay
separation on a flat plat used to simulate the suction surface of a low-pressure turbine blade
[108]. They used a phased-array of actuators fabricated on a printed circuit board to inject momentum and re-attach the flow. Corke et al. performed experiments on controlling
flow separation on a linear cascade of “PakB” low-pressure turbine blades at low Reynolds
40
number. Steady and unsteady actuation was found to work, and low duty cycle unsteady
actuation was just as effective as high duty cycle actuation [104]. Wall et al. measured
the effects of a pulsed-DC DBD actuator to re-attach the flow in a simulated low-pressure
turbine passage. They used particle image velocimetry (PIV) to measure velocity profiles of
the boundary layer for various pulse repetition rates and power settings of the actuator[109].
Controlling separation on airfoils intended for external flows (wings, wind turbine blades)
is another significant area of application for DBD actuators. While the flow conditions can
be substantially different in external flow applications, the basic modification of airfoil-like
flows by DBD actuators is similar to the above discussed internal-flow applications. A
variety of actuator arrangements have been tried ranging from leading edge actuators to
trailing edge actuators, vortex generators and actuator arrays. Corke et al. performed a
numerical and experimental analysis of using plasma actuators to re-attach the flow over
a bump in a channel. Several turbulence closures to the Reynolds-averaged Navier-Stokes
(RANS) equations as implemented in the commercial CFD tool Fluent were used for the
numerical effort, with the plasma actuator represented as a body force [99]. Orlov et al. [110]
demonstrate a lumped element circuit model of a plasma actuator coupled with Fluent’s
implementation of the SIMPLE algorithm and compare it to experimental results of the
same airfoil/leading-edge actuator configuration.
In the external-flow realm, more sophisticated arrays of actuators that govern bulk flow
around an entire body are possible. Roth et al. attempted to increase the velocity induced
by plasma actuators by incorporating a combination of para-electric (creating the body force
by an electric field on a polarized material) and peristaltic (creating the body force with a
traveling wave of actuation, like food moving through the intestine) actuator arrays. They
found that a single actuator can be modeled fairly closely by a Glauert wall jet, but other
effects cause the velocity profile to be significantly different for an array of actuators [102].
Nelson et al. [111] used DBD actuators to provide roll control at high angles of attack on
41
a blended wing-body plan-form unmanned air vehicle (UAV) with 47 degree leading-edge
sweep. The roll moment coefficients were found comparable to conventional moving control
surfaces.
The external flow application area has also seen a wider variety of device configurations
attempted. A counter-intuitive approach to DBD use was demonstrated by Visbal et al.
[112]. They compared use of co-flow or counter-flow asymmetric DBD actuators to suppress
stall on a NACA 0015 airfoil using pule-modulated actuation. The approach could control
laminar separation over a ramp and turbulent separation over a wall-mounted hump.
An emerging application area is the external aerodynamics of horizontal axis wind turbine
blades. Nelson et al. show computational and empirical results to demonstrate the feasibility
of using leading and trailing edge plasma actuators to control the blade loading of a wind
turbine for more efficient energy capture [113].
In addition to the work discussed above that is focused on very specific application areas,
modification and control of important canonical flows has been explored. Drag reduction
on a cylinder with a three-electrode DBD configuration which creates a sliding discharge
[114]. Jukes and Choi [115] also demonstrate the control of unsteady shedding on a circular
cylinder using DBD actuators.
At a more fundamental level than even the flow past a circular cylinder, some more limited
work has been done on turbulence modulation and drag reduction or pumping [116]. These
efforts tend to be focused on flat-plate flows in the context of classical viscous-flow theory.
Wilkinson attempted to achieve drag reduction on a flat plat by using the DBD plasma to
simulate an oscillating wall [105]. Fulgosi et al. [117] attempted to achieve drag reduction in
channel flow using high-voltage wires which discharge ions to impart momentum to the flow
and create large scale flow structures. These large scale structures are intended to modify
the behavior of the turbulent boundary layer and reduce the strength of the flow features
(streaks) that are believed to contribute to increased drag. Soldati et al. [118] discusses the
42
motivation and prospects for turbulence modulation by electrostatic precipitator (ESP)s.
ESPs are stream-wise wires suspended in the flow or in the walls of a channel bounded flow.
They are used to set-up electro-hydrodynamic (EHD) vortical structures to provide forcing
of the turbulent boundary layer in hopes of achieving drag reduction.
Though not concerned with flow control, the work funded under the AFOSR Air Plasma
Ramparts MURI improved the state of knowledge about the behavior and generation of
these non-equilibrium air plasmas [119, 120]. The electromagnetic wave “shielding” effect
has been shown to be negligible in experiments quantifying the change in radar cross section
(RCS) of a flat plate with a surface DBD in the Ka-band (26.5–40 GHz) [121]. This result
was predicted theoretically by Larousi using an analytical analysis of wave propagation in
collisional plasmas [11]. The flat-plate DBD set-up used for the RCS work is very similar in
nature to that used by Yang in the early low-frequency interferometry attempts which showed
measurable attenuation at 10GHz, but not measurable phase shift [9]. These several results
are consistent, as the Appleton-Hartree dispersion relation predicts higher attenuation at
lower frequencies for fixed material properties.
2.5
Reconciling Models and Measurements
Modeling of dielectric barrier discharge plasma actuators has proceeded from empirical
lumped-element circuit models [94, 110] to multi-fluid models including finite rate chemistry
and coupled electrostatics [86, 88] and particle-in-cell (PIC) calculations [122, 123].
Early modeling of atmospheric pressure dielectric barrier discharges in noble gas or including Nitrogen impurities showed net force in opposite directions during the two half-cycles.
This is contradicted by the observations of DBDs in air. The early, detailed modeling was
often at far shorter time and length scales than those at which DBD actuators are typically
operated. The other short-coming in much of the initial detailed discharge modeling is in
the gas composition. The fraction of oxygen content in the working fluid has been shown to
43
affect the discharge characteristics [37].
Modeling which includes a realistic O2 /N2 mixture shows that there is a threshold size,
voltage amplitude and frequency for the net force to be in the downstream direction in both
the positive and negative half-cycles [86]. This modeling result is confirmed by unsteady
force measurements made with a pendulum and Michelson interferometer by Enloe et al.
which shows the push-push behavior of the net momentum transfer over both half-cycles of
the discharge [124]. Time-resolved neutral density diagnostic results [37] indicate that fluid
inertial dynamics at kHz frequency time-scales are an important factor in understanding the
behavior of DBD actuators. Even though the more recent modeling includes a realistic gas
mixture, it still does not include fully coupled fluid motion (the neutral gas is assumed stationary / constant density). These modeling assumptions lead to a conclusion that dielectric
charging is the primary factor in determining force production, while the empirical results
tend to indicate a more complicated picture of the device behavior which probably depends
on coupling to fluid density oscillation at kHz time-scales.
The fluid inertial effects at kHz frequencies and length scales on the order of the actuator
size are shown to be a non-negligible contributor to the force production mechanism. The
changes in neutral density caused by the actuator are on the order of 1% of the background
density, and it is this fluctuating density change near the edge of the exposed electrode that
seems to drive the net momentum transfer [37]. While dielectric charging at medium timescales (on the order of the driving wave-form) plays an important role in the self-limiting
behavior of DBDs and may play a role in governing when the push-push behavior is achieved,
longer time-scale dielectric charging are important to understanding run-to-run variations in
atmospheric pressure DBDs [125].
Some of the empirical work in bulk-flow diagnostics have high-lighted important confounding affects to either minimize or control for in microwave diagnostics. There are startup variations due to surface chemistry and charging that slowly “bake-out” to approach a
44
steady-state behavior. These start-up and charging effects should be considered in experimental procedures (such as ensuring that the device is “wiped” of charge before each shot)
[125–127]. Also, depending on the dielectric, there can be significant surface heating (this
was seen in the present research when using silica glass dielectrics, see section 4.3). If these
medium to long time-scale variations are not accounted for in the experimental design or the
diagnostic model, then they can easily cause spurious results.
45
III. Methodology
The microwave diagnostic technique being applied in this research relies on a forward
model to relate phase and amplitude shifts of a diagnostic beam caused by plasma electron
currents to the state of those electrons (density and collision frequency). Each of these
measurements is made at the boundary of the test section containing the plasma, and is an
integration of the material properties along the wave’s path through the plasma. Multiple
measurements at different positions and angles are combined into a reconstruction of the
interior field. This process of combining multiple measurements on the boundary into a
reconstruction of the field is called computed tomography.
This chapter will introduce a simple analytical forward model as well as a numerical
PDE-based forward model. Subsequent sections will cover reconstruction techniques as well
as a novel uncertainty propagation and sensitivity derivative approximation technique particularly well suited for PDE-based forward models.
The reconstruction techniques depend on physical assumptions that can be made about
the plasma being measured. In the case where axisymmetry is a good assumption (as in a Hall
Thruster exhaust plume) then methods based on inverse Abel transforms are appropriate,
and the reconstruction has a direct solution. In more general cases an explicit regularization
must be incorporated so that the reconstruction algorithm is well-posed.
3.1
Forward Model
The basic measurement supporting any reconstruction is a chord-averaged measurement
that relies on a forward model to relate measured quantities to material properties. For cold,
collisional plasmas the Appleton-Hartree formula can be used in this manner [9, 17, 33]. This
section will develop a numerical model based on similar assumptions as those underlying the
development of Appleton-Hartree [128]. The flexibility of the numerical model allows certain
46
assumptions to be relaxed, most importantly the gradients in material properties can be
seamlessly treated, and non-linearities such as velocity dependent collision frequency could
be included.
A useful measurement model for microwave interferometry of atmospheric pressure plasmas is a one-dimensional wave equation including currents due to the electron motion.
2
∂ 2 Ey
∂uy
2 ∂ Ey
=0
−
c
+
4πnq
∂t2
∂x2
∂t
(9)
where Ey is the transverse electric field, uy is the electron velocity, n is the electron number
density, q is the electron charge and c is the speed of light. The current is assumed to be
due only to the motion of the electrons.
Jy = nquy
(10)
The momentum equation for the electrons is
∂uy
qE
=
− νe0 uy
∂t
m
(11)
where νe0 is the effective electron-neutral collision frequency (assumed independent of uy ).
This term contributes a drag force on the electrons and is a critical parameter in determining the amplitude shift of the interferometer wave transmitting through the plasma. The
present model assumes that the collision frequency is independent of the electron velocity,
this assumption does not always obtain [1]. This assumption can be checked experimentally
by changing the orientation of the polarization with respect to the bulk electron velocity.
At a parallel orientation the electric field of the microwave is in a direction that corresponds
to the field caused by the DBD itself, so there will be an additive bias in electron velocity.
When in a perpendicular orientation the velocity caused by the interrogating wave will be
47
perpendicular to the velocity induced by the DBD fields. If there is a significant dependence
of collision frequency on electron velocity over the chosen DBD operating range then this
experimental approach will allow that effect to be quantified. The numerical approach presented is flexible enough that a non-linear electron momentum equation incorporating this
velocity dependence can be used if the effect is indeed measurable.
Non-dimensionalizing equation 9 gives
∂ 2 Ey ∂ 2 Ey
−
+
∂x2
∂t2
nref qref xref
Eref
4πnq
∂uy
=0
∂t
(12)
where a judicious choice of the reference time, xref /c allows the elimination of the constant
in the wave equation and a single non-dimensional coefficient in front of the electron current
term. A convenient choice for Eref is nref qref xref which makes the coefficient one. A
convenient choice for xref in this application is c/fref , where fref is a reference frequency
(to be defined below in light of the momentum equation non-dimensionalization).
Applying the same reference quantities as above, and using fref as the reference quantity
for the collision frequency, gives the non-dimensionalized momentum equation as
∂uy
=
∂t
2
qref
nref
2
fref mref
A convenient choice for fref is
fref =
s
!
qE
− νe0 uy
m
2
qref
nref
mref
(13)
(14)
and if
qref ≡ e
(15)
mref ≡ m
(16)
48
then the non-dimensionalized momentum equation is simply
∂uy
= −E − νe0 uy
∂t
(17)
The choice of reference charge in Equation 15 simplifies the wave equation further to
∂ 2 Ey ∂ 2 Ey
∂uy
=0
−
+ 4πn
2
2
∂x
∂t
∂t
(18)
Table 1 summarizes the chosen non-dimensionalization. Dimensionally consistent values for
the reference charge, mass, number density and speed of light must be provided to properly
non-dimensionalize the governing equations. This non-dimensionalization also clearly highlights the two important parameters in this problem (the only ones remaining): electron
number density and electron-neutral collision frequency. While the numerical solution of the
Table 1. Numerical Forward-model Non-dimensionalization
Parameter
Scaling
r
Length
Time
Velocity
Electric field
Collision frequency
q
nref
mref
nref
mref
|qref | x
c
|qref | t
u
c
E
mref nref qref
r νe0
nref
|qref |
m
ref
Number Density
n/nref
model represented by equations 17 and 18 is used for the present research, it is useful to
examine the analytical result for insight into the diagnostic propagation. If plane-wave and
constant material properties are assumed then the analytical Appleton-Hartree formula gives
49
the complex refractive index as [1, 9, 23]
n2 = 1 −
X
1 − iZ
(19)
X = w02 /w2
(20)
Z = νe0 /w
(21)
where w0 is the previously identified plasma frequency. The variation in refractive index
according to 19 with X and Z is shown in Figure 13. Assuming a constant index of refraction along the measurement path, the phase shift caused by the plasma under test can be
calculated by
ω
d (Re(n) − 1)
c v
u r
u
u
1−
t
= dω 


∆φ =
ω02
2
ω +ν 2
2
+
2
ν 2 ω04
2
ω (ω 2 +ν 2 )2
+
1−
ω02
ω 2 +ν 2
2



− 1
 /c

(22)
where ω is the interrogating wave frequency, ω0 is the electron plasma frequency and d
is the length of the measurement chord. Figure 14 shows the estimated phase shift for a
plasma with parameters on the order of what is present in DBD actuators. The presently
accessible frequencies are high-lighted in Figure 14 as well. The current equipment (described
in more detail in Section 3.4) should be able to resolve this magnitude phase shift. The
instrumentation can achieve phase absolute phase accuracies of less than 2◦ [129], but relative
accuracies in interferometry applications on the order of 0.1◦ much higher as shown by the
noise characteristics of the phase difference measurements presented in section 4.3. This also
shows that extension to frequencies on the order of 100 GHz may prove fruitful for DBD
diagnostics.
50
(a) Real Part
(b) Imaginary Part
Figure 13. Appleton-Hartree Refractive Index
51
(a) Phase Shift
(b) Optimum Frequency
Figure 14. Estimates of Interferometer Operation (Appleton-Hartree)
52
There are two physical parameters in the diagnostic model presented above. The electron number density parameter, ne , is a physically meaningful quantity without any further
assumptions or modeling necessary. It is simply the count of electrons present in a volume. The effective collision frequency for momentum transfer, ν, requires some additional
assumptions to relate measured values to physical parameters such as electron temperature.
Estimates for electron collisions cross-sections for various molecules in air have been compiled [130–133]. Of interest in the present work are those for argon and krypton (see Figure
15), which are Hall thruster propellants, and oxygen and nitrogen (see Figure 16), which are
the main constituents of the neutral gas in atmospheric DBD plasma actuators.
(a) Xenon
(b) Krypton
Figure 15. Noble Gas Cross-Sections for Momentum Transfer (taken from [132])
A sum over the species participating in elastic electron scattering collisions is made to
arrive at the total scattering [1].
νtotal =
X
i
σi (Te )ni (ve − vi )
53
(23)
(a) Molecular Oxygen
(b) Molecular Nitrogen
Figure 16. Atmospheric Gas Cross-Sections for Momentum Transfer (taken from [132])
Where σi is the temperature (velocity) dependent collision cross-section, ve and vi are the
electron and collision partner velocities respectively, and ni is the number density of the
collision partner. It is convenient to separate the species into charged and neutral particles
to treat these collision process separately [1].
νtotal = νe0 + νei
(24)
For dielectric barrier discharge plasmas at atmospheric pressure, the ionization fraction is
relatively low, and neutral density is relatively high, so collisions with neutrals dominates
the momentum transfer.
νtotal ≈ νe0
(25)
This is not the case in a Hall Thruster plume where the ionization fraction is higher, and the
number density (collisionality) is relatively low, in this case elastic scattering from Coulomb
54
interactions dominates.
νtotal ≈ νei
(26)
Theoretical calculations of the electrical conductivity of non-equilibrium air have also been
accomplished using a two-fluid model (which allows separate electron and heavy particle
temperatures) for high temperature air [134].
Regardless of the model chosen for the collision processes, it is the collision frequency for
momentum transfer which can be diagnosed by non-intrusive wave propagation techniques.
As demonstrated by Heald and Wharton, the total cross-section and the cross-section for
momentum transfer are equal for electrons of moderate energy (≈ 1eV ) and hard-sphere
molecules [1]. This diagnosed quantity may be related to electron temperature (velocity) by
making the assumptions about the form of the right-hand side of equation 23 appropriate
to the plasma being diagnosed.
The numerical solution to the model requires characteristic decomposition in order to
specify non-reflecting boundary conditions [29]. To start, Equations 9 and 11 can be written
as a first order system.
 


d2
E   0 1 0  d t d x E  
0

−1 0 −4πne  
 
 
 2


 2

 d
 
 
 d
0 0
1 

  d t d x E  +  0 0 0  d x2 E  + νe0 u − E  = 0
 
 




d
d
u
−1 0 0
u
0
0 1
0
dt
dx


d2
d t2


(27)
Multiplying 27 through by the inverse of the temporal operator gives

d2
d t2



2
0  d tdd x
E   0 −1

 2
 

 d E  + −1 0 0 
dtdx  

 


d
u
0
0 0
dt



E  −4πne (νe0 u − E)
 

+
=0
d2
E
0
2



dx
 

d
u
νe0 u − E
dx
55
(28)
The Jacobian can be diagonalized




1
1
 0 −1 0 1 1 0 −1 0 0  2 2 0


 


−1 0 0 = 1 −1 0  0 1 0  1 − 1 0


 

 2
2


 


0 0 1
0
0 0
0 0 1
0 0 0


(29)
which shows there are two characteristic wave-speeds. As shown in equation 12, the wave
velocity has been non-dimensionalized so the characteristic speeds are ±1. Since the reference
velocity chosen for non-dimensionalization was the speed of light, this gives us the expected
result that electromagnetic waves propagate at the constant velocities of c and −c. The
characteristic variables are given by multiplying the primitive variables by the matrix of
left-eigenvectors.



1
2
1
2
R1 
  
R  =  1 − 1
 2  2
2
  
R3
0 0




d
d
d
0  d t E 
 d x E + d t E

 1

 d E =  d E − d E
0
 dx  2 dt

dx




1
u
2u
(30)
R1 is the right-going wave and R2 is the left-going wave. The electron velocity is not
described by a wave equation, so the third characteristic variable is the same as the third
primitive variable. The diagonalized system is given by multiplying 28 by the matrix of
left-eigenvectors.

  


R1  −1 0 0
R1  2πne E − 2πne νe0 u




  

∂   
 ∂ R  + 2πn E − 2πn ν u = 0
+
R
0
1
0
2
e
e e0 

 ∂x  2  
∂t 
  
  


R3
0 0 0
R3
νe0 u − E


(31)
To model an interferometry measurement the left boundary will be forced with a rightgoing wave and the left-going component will be calculated from the solution on the interior
56
of the domain. The right boundary will have a zero left-going wave, with the right-going
component calculated from the solution on the interior of the domain. In this way the interrogating wave can be sent into the numerical “test-section” from the left boundary and the
resulting phase/amplitude change can be measured on the right boundary (reflections can
also be measured by considering the left-going component on the left boundary). Specifying non-reflecting, characteristics-based boundary conditions in this way prevents spurious
reflections off the boundaries from contaminating the solution.
The forcing function for the left boundary is
p = sin (2 π f (x − t))
(32)
∂p
= 2 π f cos (2 π f t)
∂x
∂p
= −2 π f cos (2 π f t)
∂t
(33)
(34)
where f is the frequency of the interrogating wave and c is the wave speed (speed of light).
The characteristics on the left boundary due to the forcing function are




0
R1  

  

R  = −2 π f cos (2 π f t)
2
  

  

R3
ub
(35)
The characteristics due to information from the interior of the domain are



d
dx
R1 

  1
R  =  d
 2 2  d t

 
R3
E x=xint +
E
−
x=xint

E x=xint 


d
E
dx
x=xint 

d
dt
(36)
2uint
where the spatial derivatives are approximated using suitable one-sided differences (stencil
only containing interior points). To find the values on the left boundary we take the right57
going characteristic from Equation 35 and the left-going characteristic from Equation 36 (the
choice of u is arbitrary).



R1  
  
R  = 
 2 
  
R3
d
dx
E
+
x=xint
d
dt
E
x=xint 
−4 π f cos (2 π f t)
2u





(37)
The primitive variables on the boundary are then given by multiplying Equation 37 by the
matrix of eigenvectors.

d
 dt



E
1 1 0  

 1

 d E  = 1 −1 0 
dx  2 





0 0 1
u

d
E
 d x x=xint +

1
d
= 
E x=xint +

d
x
2
d
dx
E
x=xint
+
d
dt
E
x=xint 
−4 π f cos (2 π f t)
2u






E x=xint − 4 π f cos (2 π f t)


d
+
4
π
f
cos
(2
π
f
t)
E

dt
x=xint

2u
d
dt
(38)
(39)
On the right boundary no incoming (left-going) wave is specified so the characteristics are




0
R1 


  1

R  =  d E 
d
 2  2  d t x=xint − d x E x=xint 
 


R3
2u
58
(40)
and the primitive variables on the boundary can be recovered similarly

 

E
0
1 1 0 

 1
1



 d E  = 1 −1 0  d E d
−
E
 2  d t x=xint
dx  2 
dx
x=xint 


 


u
0 0 1
2u


d
d
 d t E x=xint − d x E x=xint 


1

d
d
= 
E
−
E
dx
dt
x=xint
x=xint 
2


2u

d
 dt

(41)
(42)
The method of lines (MOL) is more easily applied to integrate the solution on the interior
points by recasting the governing equations in slightly different form.

−1 0

0 0


0 1
3.1.1

2
−4πne   ddt2
1
0


d2
d x2

E
E 





  d E  = − ν u − E 
 e0

  dt 




d
d
u
−
E
dt
dt
(43)
Low-order Finite Difference.
Substituting second order backward differences for the time derivatives and second order
central differences for the spatial derivatives gives the discrete system as

3
− 2 k

 0


−1
1
h2
− h22
1
h2
0
−1
0
0
3
2k
0

∂E/∂ti,n



  Ei−1,n

2νe0 i k+3  
  Ei,n
2k


0
 Ei+1,n

ui,n
− 6πnk e i

 


2πn
(u
−4u
)
∂E/∂t
−4∂E/∂t
e
i
i,n−2
i,n−1
i,n−2
i,n−1

+
2k
k

 

 
ui,n−2 −4ui,n−1

=
−

 
2k
 


E
−4 E
− i,n−2 2 k i,n−1


(44)
where the first index (i) indicates the spatial discretization and the second index (n) indicates
the time-level. This implicit relation for the solution at the next time-step can be solved
59
with a point-Gauss-Seidel method according to

3
− 2 k

 0


−1
− h22
−1
3
2k

 
∂E/∂ti,n−2 −4∂E/∂ti,n−1
2πne i (ui,n−2 −4ui,n−1 )
+
−
2k
k
 ∂E/∂ti,n  

 
u
−4u
2 νe0 i k+3  
 
− i,n−2 2k i,n−1
  Ei,n  = 
2k

 
E
−4E
0
ui,n
− i,n−2 2k i,n−1
− 6πnk e i

Ei+1,n +Ei−1,n
h2

(45)
where the operator in Equation 45 can be inverted analytically and applied to the righthand-side at each point in an iterative fashion.
The boundary conditions given by Equation 39 and Equation 42 are applied to eliminate
the appropriate terms in Equation 45 at each boundary. Since the time derivative is specified
by the incoming and outgoing characteristics the unknowns at the boundary are only E and
u.
The i-min (left) boundary condition is given by

2
− h2


−1
− 6πnk e i


 Ei,n 


2νe0 i k+3
u
i,n
2k

4hk(∂E/∂x)−4Ei+1,n k+3h2 (∂E/∂ti,n )−4h2 (∂E/∂ti,n−1 )−16πh2 ne i ui,n−1 +h2 (∂E/∂ti,n−2 )+4πh2 ne i ui,n−2


2h2 k
= 
4ui,n−1 −ui,n−2
2k

(46)
where the time and space derivatives of E are given by Equation 39.
The i-max boundary condition is given by



6πne i
2
− h2 − k  Ei,n 



−1 2νe02ikk+3
ui,n


4hk(∂E/∂x)+4Ei−1,n k−3h2 (∂E/∂ti,n )+4h2 (∂E/∂ti,n−1 )+16πh2 ne i ui,n−1 −h2 (∂E/∂ti,n−2 )−4πh2 ne i ui,n−2
−

2h2 k
= 

4ui,n−1 −ui,n−2
2k
(47)
60




where the time and space derivatives of E are given by Equation 42.
The second order time integration with a backward difference formula is a multi-step
method, so it requires a first order initial step to start the solution.

1
− k

 0


−1
1
h2
− h22
1
h2
0
−1
0
0
1
k
0



∂E/∂ti,n
− 4πnk e i  

Ei−1,n

kνe0 +1  
  Ei,n
k


0
 Ei+1,n

ui,n

 


(∂E/∂t)
4πn
u
i,n−1
e
i
i,n−1

−
k
k

 −

 
ui,n−1

=

 
k
 


Ei,n−1

k

(48)
The boundary conditions are similarly simplified to depend on only the previous time-level
(initial condition). The i-min (left) and i-max (right) boundary conditions, respectively, are





2hk(∂E/∂x)−2Ei+1,n k+h2 (∂E/∂t)i,n −h2 (∂E/∂t)i,n−1 −4πh2 ne i ui,n−1


h2 k
4πne i
2
− h2 − k  Ei,n 

=

k νe0 +1
−1
ui,n
k
ui,n−1
k

(49)

 


2 (∂E/∂t)
2 (∂E/∂t)
2n u
2hk(∂E/∂x)+2E
k−h
+h
+4πh
4πne i
i−1,n
i,n
i,n−1
e i i,n−1
2

− h2 − k  Ei,n  −
h2 k
=

 


u
i,n−1
−1 k νe0k +1
ui,n
k
(50)
Figure 17 shows a solution to the one-dimensional model using a second order central
difference for the spatial derivatives and a first order forward Euler time integration. A
collision frequency of νe0 = 1 × 1012 s−1 is assumed [135], and a constant electron density of
n = 1 × 1015 m−3 is also used. The initial electron velocity is everywhere zero. In a short
time the electron velocity matches the electric field and an electron velocity perturbation
travels along with the electric field wave. This solution also shows the two characteristic
waves propagating out in opposite directions from the initial condition.
While the basic finite difference method can be used to solve the wave propagation
61
(a) Transverse electric field
(b) Transverse electron velocity
Figure 17. One-dimensional Diagnostic Wave Propagation Model
problem as illustrated in Figure 17, the Gauss-Seidel updates derived above are more useful
as a preconditioner (symmetric successive over-relaxation (SSOR)) to speed convergence of
the Krylov iterations used in the pseudo-spectral method presented below in section 3.1.2.
Using a grid transformation between physical and computational coordinates allows use
of arbitrarily spaced grid points, which can be adapted to the resolution needs of particular
problems much more efficiently than uniformly spaced grids. Applying the chain rule to
transform between computational (ξ) and physical (x) coordinates gives



d
dξ
x
d2
d ξ2
x



d
dx
ξ
d2
d x2
ξ
0
d
dξ
x
0
d
dx
ξ


2  


2  
d
dx




d
dξ

(51)
d
dx

(52)
E  E
= 2 
d2
d
E
E
d x2
d ξ2
d
dξ
E  E
= 2 
d2
d
E
E
d ξ2
d x2
This gives substitutions for transforming the model equations so that all derivatives are in
62
terms of computational coordinates.




d
dx
d2
d x2
ξ
0
d
dx
ξ
d
 dx ξ
 2
d
ξ
d x2
ξ
0
d
dx
ξ




 
2  = 
 

2  = 
d
dξ
x
d2
x
d ξ2
1
−
d
x
dξ
d2
d ξ2
x
3
( ddξ x)
0
d
dξ
x
−1
(53)

(54)

2 
0
1
2
d
( d ξ x)



Which gives the transformed forward model (equation 18) as
d2
d ξ2
E
2 −
d
x
dξ
d2
d ξ2
x
d
dξ
d
dξ
x
E
3
d2
− 2 E + 4 π ne
dt
d
u
dt
=0
(55)
Substituting second order differences for the first and second spatial derivatives gives the
semi-discrete, first-order method of lines system for the forward model as

d2
d
 t2


E  4 π ne (−ui,n νe0 − Ei,n ) +

 
 d E=
 dt  

 
d
u
dt
4 (Ei+1,n −2 Ei,n +Ei−1,n )
(xi+1,n −xi−1,n )2
−
dEdt i,n
−ui,n νe0 − Ei,n

4 (xi+1,n −2 xi,n +xi−1,n ) (Ei+1,n −Ei−1,n )

(xi+1,n −xi−1,n )3
(56)
Any of the standard time discretization methods can be applied to equation 56. Of particular
interest are time discretizations based on backward difference formulas (since we plan on
integrating stiff, pseudo-spectral discretizations of this model).
Though a roots grid, as defined above, is used throughout, the derivatives in physical
space are transformed to be suitable for arbitrary point distributions [136]. The relevant
63




substitutions in this case are
d
E=
dx
d
E
dξ
d
x
dξ
2
d
E=
d x2
(57)
d2
d ξ2
E
2 −
d
x
dξ
d2
d ξ2
x
d
dξ
x
d
dξ
3
E
(58)
This gives the governing equation in computational space as

d2


 d t2 E  
 (

 d E=
 dt  
 


d
u
dt
d2
E
d ξ2
2
d
x
dξ
)
−

( ddξ E )
+ 4 π ne (−E − νe0 u) − f1 
3
( ddξ x)


d

E
−
f
2

dt

−E − νe0 u − f3
d2
d ξ2
x
(59)
All of the grid metrics are calculated without truncation or smoothing. With a roots grid the
second term in equation 58 is identically zero. However, there is a non-zero contribution from
this term in the present implementation due to round-off error (this error on the order of the
machine precision is accepted so that the code may be used with arbitrary grid distributions
in the future).
The stability of the discretization can be assessed by calculating the eigenvalues of the
discrete spatial difference operator. If the x-locations of the grid points are given by
xi = cos
(2i − 1)π
2n
(60)
then the eigenvalues for the right-hand-side operator are as shown in Figure 18. The reason
for this choice of grid point spacing is explained in section 3.1.2. The spectral radius of the
MOL operator is much larger for the Chebyshev-roots grid than for a uniformly distributed
grid. This is an important consideration for the convergence of iterative methods, which are
governed by the spectral radius of the operator.
64
30
100
25
20
25
50
20
10
15
0
15
0
10
5
−10
−30
−7
−6
10
5
−50
0
−2.0−1.5−1.0−0.5 0.0 0.5 1.0
−20
20
−5
−4
−3
−2
−1
0
1
0
−2.0−1.5−1.0−0.5 0.0 0.5 1.0
−100
−400 −350 −300 −250 −200 −150 −100 −50
(a) Uniformly Spaced Grid
0
50
(b) Chebyshev Roots Grid
Figure 18. Eigenvalues of Low-order Finite Difference MOL Spatial Operator
SSOR based on this finite-difference discretization is a low-storage iterative method that
is useful as a preconditioner for higher-order methods [137]. SSOR has a tunable overrelaxation parameter ω. The spectral radius, or magnitude of the largest eigenvalue of the
SSOR operator governs the rate of convergence. The optimum value of ω depends on the
number of points in the grid, and increases with increasing n. The optimum will always be
in the interval [1, 2]. Values below 1 will converge too slowly, and values above 2 will fail to
converge [138].
3.1.2
Pseudospectral Method.
Using a pseudospectral method for the discretization of the wave propagation equations
is motivated by the fact that this forward model must be called many times inside the
“inner loop” of a reconstruction algorithm. This means that any reduction in run-time
or storage for the forward model evaluation will have significant pay-off in the context of
computing reconstructions. Pseudospectral methods promise fast, O(n log n), low-storage,
O(n), achievable optimal resolution (truncation error approximately equal to round-off error)
discretizations that are well suited for linear wave propagation problems.
Calculating the spatial derivatives with a Chebyshev-pseudospectral method, rather than
65
a standard finite difference gives improved accuracy with fewer points. Figure 19 shows
a comparison of the error in calculating the derivative of 10 wavelengths of cos(x) using
a second order central difference and a Chebyshev pseudo-spectral method. The pseudo-
(a) Error
(b) Error Norm
Figure 19. Error of Pseudospectral and Finite Difference Derivatives
spectral method converges to roughly the precision of the machine with only 64 points on
the interval. The second order finite difference method does not approach this level of
precision even with many more points.
The pseudo-spectral derivative can be calculated with full difference matrices which requires O(n2 ) operations, or more efficiently using discrete Fourier transform (DFT)s and a
short recurrence [139]. The set of orthogonal Chebyshev polynomials (of the first kind) can
be written in terms of cosine,
Tn (x) = cos(n arccos x)
(61)
So the coefficients of the solution in the Chebyshev basis can be found using a discrete cosine
transform (DCT) [140]
n−1
X
(j + 1/2)k
αk = 2
Yj cos π
n
j=0
66
(62)
and the coefficients of the solution’s derivative can be found by a simple recurrence [139]
βn = 0
(63)
βn−1 = 2(n − 1)αn
βk = βk+2 + 2kαk+1 , k = [n − 2, 1]
(64)
(65)
These derivative coefficients are then transformed back to the physical space with an inverse
DCT [140].
n−1
X
∂Y
j(k + 1/2)
= β0 + 2
βj cos π
∂x
n
j=0
(66)
Second derivatives are required for the solution to the diagnostic model. The recurrence
(equation 65) is applied twice to give coefficients for the second derivative, and these are
inverse transformed to give the second derivative values in the physical space. The spatial derivative operators can be written as a multiplication of the DCT, inverse DCT and
recurrence operators.
∂
≈ EBD
∂x
∂2
≈ EBBD
∂x2
(67)
(68)
where D is the DCT, E is the inverse DCT and B is the recurrence. Examining the eigenvalues of B and BB gives some insight into where numerical trouble may lie. Writing the
67
recurrence in matrix form gives
Bα = β

2(n − 1)



0


 2(n − 1)



..

.




 2(n − 1)

 [ n odd ↑





0
0

0
 


0
. . . 0
α
β
  n   n−1 

 

 βn−2 
2(n − 3)
. . . 0
α

n−1
 



 

..
... 



 αn−2  = βn−3 
.



 


.
.
 .   . 


2(n − 3) 
.
.







 




. . . 2
n even ↓ ]
α2
β1






0
0
2(n − 2)
0
..
.
0
2(n − 2)
...
(69)
B is a lower-triangular matrix, so it’s eigenvalues are just the diagonal entries [141]. The
largest eigenvalue is 2(n−1) and the smallest eigenvalue is 2. This means the ratio of smallest
to largest eigenvalues is linearly dependent on the number of grid points. However, for B 2 the
ratio of maximum to minimum eigenvalue grows as n2 − 2n + 1. This quadratic dependence
can quickly lead to a poorly conditioned matrix as n increases, and thus numerical difficulties
for large grids. The recurrence used here is susceptible to accumulation of round-off error
[139, 142].
The matter of grid transformations for unequally spaced grids is greatly simplified in this
approach if the collocation points are chosen at the zeros of the Chebyshev polynomials.
These locations are given by
xi = cos
(2i − 1)π
2n
(70)
Equation 70 gives a roots grid which clusters grid points near the boundaries of the domain, thus avoiding Runge phenomena which plagues high-order polynomial interpolations
on equally spaced grids.
68
The present approach requires Lanczos-σ factor smoothing [143] to stabilize.
σi =
sin(πi/n)
,
πi/n
i = 1 to n
(71)
Where the coefficients of the discrete cosine transform of the derivative (the βi ) are multiplied
by the σi before inverse transforming back to the physical space. Figure 20 shows the σ factor
(also shown is σ 2 , higher powers of σ can be used for more aggressive attenuation of the higher
modes).
1.0
σ
0.8
σ2
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
i/n
Figure 20. Lanczos-σ Factor
The 3nx − 2 largest magnitude eigenvalues for the semi-discrete operator in the present
MOL formulation (see equation 43) are calculated using an implicitly restarted Arnoldi
method [144]. The eigenvalues are shown in Figure 21 with and without Lanczos-σ smoothing. When no smoothing is applied to the Chebyshev coefficients there are several large69
(a) Without Lanczos-σ smoothing
(b) With Lanczos-σ smoothing
Figure 21. Spatial Operator Eigenvalues
magnitude eigenvalues with positive real part (Figure 21a). This leads to numerical instabilities when the MOL formulation is integrated in time. With smoothing (Figure 21b), all of
the eigenvalues remain on the left half of the complex plane, which allows stable time integration [145]. The insets in Figure 21 show the eigenvalues near the origin. A purely wave-like
solution with no dissipation would have eigenvalues along the imaginary axis, but the dissipation caused by the electron collisions moves the wave-like eigenvalues in the negative real
direction.
The effect of Lanczos-σ smoothing of the second derivative approximation suggest a
solution to the numerical instabilities. The recurrence can be modified so errors in the highorder modes of the first derivative (βi ) do not contaminate the low-order modes of the second
derivative (γi ).
γk = 0, k = [n, n − m]
(72)
γk = γk+2 + 2kβk+1 , k = [n − m − 1, 1]
(73)
70
Note that this is not smoothing the coefficients or truncation after the fact, but rather treating the source of difficulty, which is propagation of round-off error from poorly approximated
high-order modes of the first derivative into the low-order modes of the second derivative.
The importance of this distinction is illustrated by comparing the insets of Figure 21 with
Figure 22. Though both approaches result in time-stable schemes, smoothing the second
100
25
20
50
15
10
0
5
−50
−100
−300
0
−2.0
−1.5
−1.0
−0.5
0.0
0.5
1.0
raw
m-truncated
−250
−200
−150
−100
−50
0
50
Figure 22. Effect of pre-recurrence truncation on eigenvalues (nx = 64, m = 12)
derivative coefficients after the application of the recurrence completely changes the character of the oscillatory eigenvalues near the origin, while truncation prior to recurrence leaves
them intact. This requirement for pre-recurrence truncation is thus slightly different than
the low-pass filtering used with high-order compact schemes to achieve near time-stability
[25].
Figure 23 shows the effect of pre-recurrence truncation parameter (m) on the real part
of the problematic eigenvalues over a range of grid sizes (nx ). Increasing the truncation
parameter reduces the magnitude of the real-part until it crosses zero and the scheme becomes
71
stable. A linear fit is made to the stable truncation level curve in Figure 23 to predict how
18
stable truncation level
16
14
-40
-30
-10
-20
12
0
30
10
20
m
10
40
60
8
70
50
6
80
90
4
2
0
30
40
50
60
nx
70
80
90
Figure 23. Real-part of Eigenvalue with Maximum Imaginary Component
many modes will need to be truncated at different grid resolutions.
Rather than a Gauss-Seidel iterative method, a Krylov-subspace method is more appropriate for solving the implicit linear system resulting from a pseudo-spectral discretization.
This is because Krylov methods require only the action of the implicit operator on the unknown vector (iterate), rather than an explicit update formula as in point methods such as
Gauss-Seidel (developed in Equations 45 through 47) or SSOR (achieved by applying the
Gauss-Seidel sweep to the unknowns in both forward and backwards directions). However,
the effort in deriving the SSOR update formulas based on the finite difference approximation
can be leveraged by using them as a preconditioner for the Krylov methods [146, 147]. This
can substantially improve the convergence rate.
Equation 74 shows the pseudo-spectral governing system, and Equation 75 shows the
72
application of the implicit operator on the unknown (time-level n) iterate.




 (∂E/∂t)i,n 
6πne i
3
−
1
0
−

k
 2k



(∂ 2 E/∂x2 )i,n 


 0 0 −1 2νe0 i k+3  



2k



Ei,n


3


−1 0 2k
0
ui,n




=


(∂E/∂t)i,n−2 −4(∂E/∂t)i,n−1
2k
3(∂E/∂t)i,n
2k
−





−
2πne i (ui,n−2 −4ui,n−1 )
k
fmms 2 −
ui,n−2 −4ui,n−1
2k
fmms 3 −
Ei,n−2 −4Ei,n−1
2k
6πne i ui,n
k
2
2

+ (∂ E/∂x )i,n 

ui,n (2νe0 i k+3)

−
E
i,n

2k

3Ei,n
− (∂E/∂t)i,n
2k
(∂E/∂t)i,n−2 −4(∂E/∂t)i,n−1
2k


=


+
+
2πne i (ui,n−2 −4ui,n−1 )
k
fmms 2 −
ui,n−2 −4ui,n−1
2k
fmms 3 −
Ei,n−2 −4Ei,n−1
2k

(74)
+ fmms 1 





(75)
+ fmms 1 




The right-hand side vector is made up of solutions from the previous time-levels, and is
based on the chosen temporal discretization (in this case a second order backward difference
formula).
The convergence behavior for the first and second derivative calculated by the pseudospectral method is shown in Figure 24. Both the first and second derivative approximations show
the expected spectral (exponential) convergence. For ten periods of a sinusoidal function,
between 60 and 80 points (variation due to whether modes are truncated or not) are needed
to get to the round-off plateau using 32-bit precision arithmetic. Since the approximation
relies on fast Fourier transform (FFT)s, a grid of 64 points is chosen for subsequent calculations since this power-of-two grid-size is able to exploit a slightly faster factoring of the FFT
73
104
102
100
kerrork2
10−2
10−4
10−6
∂y
∂x
10−8
∂2y
∂x2
10−10
∂y
∂x
10−12
∂2y
∂x2
10−14
101
O(∆xn )
O(∆xn )
O(∆x2 )
O(∆x2 )
102
nx
103
Figure 24. Spectral Convergence of Derivatives (y = cos(10x), x ∈ [−π, π])
operator [140].
The two lines in Figure 24 for the pseudo-spectral approximation of the second derivative
show the effect of pre-recurrence truncation of the higher-order modes. The rate of convergence remains exponential, but is delayed slightly (a larger grid is required for the same
accuracy).
3.2
Tomographic Algorithms
This section will discuss two reconstruction techniques explored in this research. Figure
25 illustrates a high-level framework with which to organize reconstruction procedures in
general. The two main parts of any reconstruction algorithm are the forward model, which
relates measured data to material properties, and the objective function, which relies on a
regularization to give a measure of “goodness” of the reconstructed field. The regularization
74
Initialize Reconstruction
Apply the forward model
A model of the measurement process
producing the data from the object
being reconstructed.
Calculate the objective function
Generally a function of the measured data,
data predicted by the forward model
and the reconstruction itself.
Measured data
Yes
Convergence criteria met?
Done
No
Perturb the reconstruction
Figure 25. General Tomographic Reconstruction Algorithm
75
is required because there is no well-posed way in general to relate measurements on the
boundary of a domain to properties interior to the domain. There is always a significant
null space even though there may be many more measurements than parameters. An infinite
number of property distributions may satisfy the boundary measurements with equal mean
square deviation with nothing to choose between them. The measured data simply provide a
constraint on the possible reconstructions. There is no unique solution to the reconstruction
problem.
Figure 25 is meant to convey the modular nature of reconstruction algorithms. For a
given forward model, many different regularizations may be applied. For a given forward
model and regularization, many different perturbation heuristics may be used (e.g. gradient
decent, simulated annealing, etc.).
3.2.1
Axi-symmetric Material Distribution: Inverse Abel Transform.
The inverse Abel transform is given by [7]
−1
f (r) =
π
Z
r
∞
∂F
1
p
dy
∂y y 2 − r2
(76)
where F is our chord-averaged measurement, and f (r) is the radial distribution that we
would like to recover. The numerical procedures presented in this section will be checked
for correctness using the theoretical phase shift for a cylinder of constant refractive index,
which has the form
P (x) = width 2 − (x − x0 )2
(77)
∂P
= −2 (x − x0 )
∂x
(78)
The derivative is then
76
To calculate the inverse transform we first need an estimate of the derivative, ∂F/∂y, of
the data. Unfortunately, numerical differentiation tends to amplify noise. To avoid this we
convolve the derivative of a Gaussian distribution with the function we wish to differentiate.
The derivative of the Gaussian distribution is
(x−µ)2
(x − µ) e− 2 σ2
√
−
σ2 2 π σ2
(79)
This discrete convolution can be accomplished in O(n2 ) time with a naive implementation,
or in O(n log n) time using the FFT. In this case, the two functions to be convolved are
transformed, multiplied point-wise in the frequency space, and then the result is inverse transformed to recover the smoothed numerical derivative. Figure 26 shows the well-known Gibbs
phenomena that results from a spectral derivative approximation for functions containing
discontinuities (additive noise can be thought of as introducing discontinuities). Also shown
40
analytical
ik
σ = ∆x
σ = 2∆x
σ = 4∆x
30
20
∂y/∂x
10
0
−10
−20
−30
−40
−30
−20
−10
0
10
20
30
x
Figure 26. Constant Property Cylinder Theoretical Phase-Shift Derivatives
is the effect of changing the standard deviation of the Gaussian distribution for smoothing
77
the numerical derivative. It is convenient to choose the standard deviation of the smoothing
distribution in multiples of the spacing between data points (∆x). Choosing the smoothing distribution to have a standard deviation as small as the spacing between data points
(σ = ∆x) eliminates over-shoots around the discontinuities.
The true inverse Abel transform for a cylinder with uniform material properties is
Z
a
x − x0
√
dx
x2 − r 2
r
√
2 √ 2
a − r2 + a
x0 − log (2 r) x0 − a2 − r2
=−
log 2
π
2
p(r) =
π
(80)
This result is shown in Figure 27 along with transforms using the analytical derivative
and several levels of smoothing for the numerical derivative. The difference between the
true solution and the result using the analytical derivative shows the error of the integral
approximation code. The integral is approximated using a FFT as well. The integral is robust
12
true
analytical
10
8
∂
∂x
∂
∂x
using σ = ∆x
∂
∂x
using σ = 2∆x
∂
∂x
using σ = 4∆x
p
6
4
2
0
−2
0
5
10
15
r
20
25
30
Figure 27. Inverse Abel Transform for Constant Property Cylinder, nx = 51
with respect to noise, so no additional smoothing is required. The derivative approximation
78
requires multiplication in frequency space, in contrast to the integral which requires division.
∂
≈ G · (ik) · F
∂x
Z
d x ≈ G · (−i/k) · F
(81)
(82)
Where F is the forward FFT operator, G is the inverse FFT operator, i is the imaginary
unit and k is the wave-number.
The solution at r = 0 is extrapolated based on the axi-symmetric assumption which gives
zero slope at the origin. In the present calculations a fourth-order extrapolation is used.
p (x0) = −
3 p (x0 + 4 h) − 16 p (x0 + 3 h) + 36 p (x0 + 2 h) − 48 p (x0 + h)
25
(83)
In addition, at each radius the divisor in equation 76 is extrapolated towards zero from
above at the singular point (r = y). This prevents division by zero errors, and gives an
approximation that converges on the correct answer as the discrete resolution is increased.
For the present calculations the extrapolation is based on the numerical derivative calculated
using the convolution described above and σ = 0.25∆x.
The correctness of the inverse Abel transform code can be verified in much the same
manner as other simulation code [148]. However, because of the smoothing of the derivative
required due to measurement noise, and the singularity in the integral transform, high order
convergence rates should not be expected. This situation is similar to the slow convergence
rate of codes in the presences of solution discontinuities (shocks). Figure 28 shows the
solution with 501 points. The inset of the figure illustrates the low-order convergence at the
discontinuity (boundary of the cylinder). Figure 29 shows roughly first order convergence
for the numerical inverse Abel transform method across two orders of magnitude in the
resolution. This indicates that the present numerical implementation is correct: it converges
79
10
true
analytical
8
p
6
4
∂
∂x
∂
∂x
using σ = ∆x
∂
∂x
using σ = 2∆x
∂
∂x
using σ = 4∆x
3.0
2.5
2.0
2
1.5
1.0
0
0.5
0.0
14.0
−2
0
14.5
15.0
15.5
5
16.0
10
15
r
20
25
30
Figure 28. High-resolution Inverse Abel Transforms of Constant Property Cylinder
(albeit slowly) to the true solution.
There are two causes for the Abel inversion’s sensitivity to error: the derivative and
the integrable singularity. These two aspects of the transform are perturbed by two main
sources of error. Those are centering or positional error (recall the data is assumed to be
axi-symmetric) and additive error in the measurement itself (noise). The first source of error
is handled by centering the data, the second by filtering or smoothing.
Centering is accomplished by maximizing the real (symmetric) part of the FFT of the
side-on data [149]. The objective function for this optimization is
N/2−1
X 2
Re(e−2πmk ∆fn Lnh )
(84)
−N/2
Where Lnh is the Fourier transform of the data, and mk ∆ is a shift. Since the data is real,
the real part of its transform is the even part of the signal. The shift which maximizes
80
101
∂
∂x
using σ = ∆x
∂
∂x
using σ = 2∆x
∂
∂x
using σ = 4∆x
k error k2
100
10−1
10−2
101
102
nx
103
Figure 29. Convergence of Inverse Abel Transforms for Constant Property Cylinder
the magnitude of the even part is the one which makes the data most even (centered). A
qualitative graphical analysis of the data can provide a range of shifts to test. This error
can be minimized (but never completely eliminated) by careful test conduct.
The techniques presented above can be combined to fit a calibration model to interferometry measurements of a calibration cylinder (or any other object for which there exists a
theoretical phase-shift curve, i.e. known geometric properties). The purpose of a calibration
model is to provide estimates of parameters like the standard deviation of the Gaussian beam
for use in deconvolution and inversion of subsequent test measurements.
The steps in fitting the model are
1. Center the data
2. Find an error minimizing set of refractive index and smoothing σ (these are the parameters of our model).
81
(a) Define a function of the data and the σs which returns an error norm between the
theoretical phase shift curve and the smoothed, deconvolved data.
(b) Apply a minimization algorithm to this function. Choose an initial guess for the
deconvolution σ which is likely to be smaller than the true σ.
3.2.2
Arbitrary Material Distribution: Total Variation Regularization.
For the present gradient-based implementation, a total variation penalty term is chosen.
The objective function then is
f
f
f
f
Z ∂u =
(Fobs − F (u)) + dx
∂x
i
X
X ∂u ∂x −1 2
≈
(Fobs − F (u)) +
∂ξ ∂ξ ∆xi
i
i
X
X
∂u ∂x −1 ∂x 2
≈
(Fobs − F (u)) +
∂ξ ∂ξ ∆ξ ∂ξ i
i
X
X
∂u 2
≈
(Fobs − F (u)) +
∂ξ X
2
i
j
(85)
j
where u is the solution (reconstructed field), F (u) is the measurement functional, Fobs is
the actual measurement (e.g. phase shift) , x is the physical coordinate, and ξ is the computational coordinate. Eliminating the coordinate transformation in this way reduces the
computational cost (for the pseudo-spectral method presented in section 3.1.2 it eliminates
a transform). Figure 30 shows three functions that are normalized to have the same integral, though their total variations are quite different. The total variation regularization is
a minimum for smoothly varying functions. The sharp features in the two square shaped
distributions introduce Gibbs phenomena in the derivative estimate, so their total variation
is higher than the Gaussian distribution with equivalent area (integral). This illustrates
that a total variation regularization term penalizes sharp features in the reconstructed field
82
3.0
total variation = 4.51005
total variation = 13.9879
total variation = 42.5684
2.5
2.0
1.5
1.0
0.5
0.0
−1.0
−0.5
0.0
0.5
1.0
Figure 30. Comparison of Total Variation for Functions with Equivalent Integrals
(smooth reconstructions will have a smaller objective function).
3.3
Complex-step Sensitivity Derivatives
The complex-step method of estimating a derivative is easily derived by expanding the
function of interest in a Taylor series in the imaginary direction [63].
f (x0 + ih) = f (x0 ) + ihf 0 (x0 ) − h2 f 00 (x0 ) − ih3 f (3) (x0 ) + · · ·
Im [f (x0 + ih)] /h = f 0 (x0 ) − h2 f (3) (x0 ) + · · ·
(86)
(87)
Equation 87 gives a second order accurate approximation to the first derivative that, in
contrast to the standard finite-difference approximation, has no subtractive cancellation error. Care must be taken to preserve this desirable property when the method is applied to
approximating sensitivity derivatives of spectral or pseudo-spectral codes [65]. As long as
83
the real and imaginary components of the solution are transformed separately, subtractive
cancellation between the O(1) real part and the O() imaginary part can be avoided. This
method of derivative approximation is especially desirable in the case of a complicated deterministic solver since the only modification required to the code is that real variables be
redefined as complex variables [62].
When using the complex-step approach, a zeroth-order uncertainty quantification can
quickly be achieved by running the deterministic code at the expected value for each parameter. This gives the output and slope at that point in the parameter space. For this
case with two uncertain parameters and two outputs, the uncertainty quantification is easily
accomplished using the Jacobian of the forward model functionals



∂p
 ∂φ 

∂p
∂a
∂nx
 ∂φ
=
=
∂ne
∂a
∂a ∂φ
∂ν ∂ne

∂p
∂νe0
∂φ   ∂ne
∂νe0
∂a

∂p
∂νe0




∂a
∂νe0
1

∂a ∂φ 
− ∂ne ∂νe0 − ∂φ
∂νe0

∂p
∂a
− ∂n
e   ∂ne
∂φ
∂ne

∂p
∂νe0

(88)


The storage and work required for this approach doubles because an imaginary component
must be stored along with the real component of the solution.
The sensitivity derivatives can be examined to tell which parameter gives the most change
to which output quantity. When the derivative information is available, a solution at just a
single point in the parameter space can guide further grid adaption. In this case changing ne
has the greatest effect on phase shift, and νe0 has the greatest effect on the amplitude. This
is expected from the physical considerations. The collision term adds a drag force which
dissipates the energy of the diagnostic wave by transferring momentum to the neutrals,
leading directly to a loss in amplitude of the diagnostic wave.
The above analysis treated the two parameters as spatially uniform, however the moti-
84
vation for using numerical models for these diagnostics as opposed to analytical solutions is
treatment of material property gradients [23]. Since the forward model spatial dimension is
defined on a Chebyshev roots grid, the Chebyshev basis is a convenient one for parametrizing the spatial uncertainty. Figure 31b illustrates the complex-step perturbations to the low
order Chebyshev modes (h = 1 × 10−16 ) rather than individual point values in the physical
space.
The prototype functional for the reconstruction problem is an integral. Figure 31 shows
the variation in electron density caused by a perturbation to the first few lower-order Chebyshev coefficients. The coefficient is perturbed by making it a complex variable and adding a
small increment to its imaginary part. Inverse transforming this perturbed imaginary part
gives the spatial perturbations shown in Figure 31b. As might be expected from examining
2.0
40
ne perturbation
magnitude
60
20
0
−20
−40
0
5
10
15
20
mode
(a) Sensitivity Magnitude
×10−16
1.5
1.0
0.5
0.0
−0.5
−1.0
−1.5
−2.0
−30
mode 0
mode 1
mode 2
mode 3
mode 4
−20
−10
0
10
20
30
x
(b) Low-order Perturbations
Figure 31. Integral Sensitivity to Low-order DCT Coefficients
the mode shapes shown in Figure 31b, the integral value is not sensitive to changes in the odd
modes, but is sensitive to changes in the even ones. Table 2 shows the gradient of the integral
of a Gaussian electron density distribution with respect to variations in these parameters
(the sensitivity derivative). As can be seen from the table and Figure 31a, the magnitude
of the sensitivity rapidly falls off with increasing mode number. This indicates that this is
a good choice for a reduced order basis. As would be expected, the sensitivity magnitude
85
Table 2. Integral Sensitivity to Perturbations of DCT Coefficients
Mode
1
2
3
4
5
6
7
Gradient
DCT Coefficient
54.5959
31.2768
1.53343e-14
2.17199e-15
-36.3936
-27.1518
5.75083e-15
-5.06463e-15
-7.27653
18.2232
6.24001e-16
5.27677e-15
-3.11694
-10.5994
also correlates with the DCT coefficient magnitude. This allows for a way of choosing the
particular low-order modes to include in a gradient calculation so that the changes will have
the largest effect on the functional. Reducing the order in this manner allows O(100 to 101 )
parameters to be varied, (each of which requires a forward model solution), rather than
O(102 to 103 ) variations in the case of perturbing the distribution directly on the physical
grid.
Adding the first mode of ne to the stochastic space reveals an important characteristic
of this diagnostic problem. While average electron density does not cause amplitude shifts,
gradients in electron density can cause apparent amplitude shifts due to reflections. In order
to assess whether amplitude shifts are caused by collisional losses or reflections, measurements at both boundaries are needed (in VNA terminology, a “four port measurement” is
required [8]).
3.3.1
Interpolation in the Uncertain Parameter Space.
The uncertainty quantification method presented above relies on a local linearization.
The stochastic collocation technique can be used to discretize the uncertain parameter space
to provide a global method of uncertainty quantification which is valid for large parameter
variations [150]. In addition to the function values at the collocation points, the novel
approach described here uses the complex-step perturbations to provide slope information
86
at the points as well. This additional information increases the order of accuracy of the
stochastic collocation.
The osculating polynomial is the polynomial interpolant that matches the function derivatives at the interpolation points as well as its value. Using function and derivative values at
the points increases the accuracy of the interpolant up to n(m + 1) − 1 where n is the number
of points and m is the number of additional derivatives [151]. In other words, adding the
slope information roughly doubles the order of accuracy of the interpolant.

Xβ = Y



Y 
X 
 β =  
Y0
X0
(89)
(90)
While the convergence rate of the standard interpolation approach, and the approach using
slope information are both spectral, there is still value in using the slope information. In
a manner similar to Richardson’s extrapolation, we can estimate error by comparing the
interpolations of different order of accuracy (that with and without slope, equations 90
and 89 respectively). Just as the sensitivity information can inform the choice of important
parameters to expand initially, the comparison of different order approximations can indicate
which dimensions of the stochastic space need more resolution.
3.4
Instrumentation and Experimental Setup
This section describes the experimental equipment for the DBD measurements. The ex-
perimental equipment for the Hall thruster measurements are described in detail in Kenan’s
controlled-distribution thesis [6]. Both sets of measurements used commercially available
VNAs operating in the Ku and Ka bands. The VNA allows simultaneous phase and amplitude shift measurements of transmissions and reflections. In the case of the Hall thruster
87
Figure 32. Simple Microwave Interferometer
exhaust plume plasma the reflections off of the plasma are negligible. This is because the
gradients in material properties are gradual, i.e. there is no sharp “impedance mismatch” between the free-space propagating wave and the wave entering the plasma. The surface DBD
plasma, on the other-hand, has large electron density gradients (the electron density feature
is small with respect to the diagnostic wavelength) which causes reflections in a manner
analogous to the reflections from material discontinuities experienced in optical components.
The focusing optics for the horn antennas are based on the spherical high-density polyethylene (HDPE) lenses used in previous AFIT Hall thruster microwave diagnostics [6, 15]. Additional, integrated GOA for operation in the Ka-band were also used for bench-top testing
of the DBD devices. The integrated Ka-band solution removed a source of error caused by
relative motions of the horn and lens due to vibrations in the support frame, any steadystate misalignment of the two components and also reflections between the horns and the
extruded aluminum lens holders.
The DBD testing set-up using the HDPE lenses and standard microwave horns is shown
in Figure 33. The microwave transmission horn is mounted above the large reflecting (buried)
electrode. The horizontal bench-top setup using the twelve-inch-diameter, integrated GOAs
is shown in Figure 34. The GOAs are custom designed by Millitech, Inc. to achieve a beam
88
26.5 – 40 GHz Horn
26.5 – 40 GHz Horn
HDPE Lens
HDPE Lens
Vector Network Analyzer
Function Generator
Amplifier
Transformer
Transformer
Reflecting / Buried Electrode
Reflecting / Buried Electrode
(a) Side View
(b) Front View
Figure 33. Vertical DBD Microwave Interferometry Setup
Figure 34. Horizontal DBD GOA Microwave Interferometry Set-up
89
(a) Top
(b) Side
Figure 35. GOA Feed: Coaxial Cable to WR-28 Waveguide
waist diameter of 1.42 to 1.38 inches at 21 inches from the flat side (interior side) of the
focusing lens [152]. Figure 35 shows a close-up of the coaxial-to-waveguide adapter of the
GOA. The waveguide feed on the antenna is a standard WR-28 size.
The VNA used for the present DBD measurements is shown in Figure 36. The interfer-
Figure 36. Rhode & Schwarz ZVA 40 Vector Network Analyzer (VNA)
ometry measurements made with this VNA and focusing optics give spatial resolutions in the
transverse direction (beam waist) of less than 2 cm. The calibration model from section 3.2.1
is fit to the response measured from a foam cylinder of known diameter to give estimates of
the beam width at each frequency.
Driving the electrodes are a function generator which sends a signal to a two-channel
audio amplifier operated in bridged mode. The function generator is shown in Figure 37.
90
The amplifier is shown in Figure 38.
Figure 37. Tektronix AFG 3022B Arbitrary Function Generator
(a) Front
(b) Back
Figure 38. Crown XLS202 Audio Amplifier
The output of the amplifier (≈ 100V) is then used to drive an oil burner igniter, which
steps up the voltage (20kVrms , 16kVpp ). Figure 39 shows a cut-away view of the transformer
in the oil burner igniter. The transformer of the oil-burner igniter is mid-point grounded
(center-tapped) so it has two high-voltage terminals which can each drive the exposed electrode of a DBD. Figure 40 shows the equivalent circuit for the oil burner igniter and one-half
of the secondary attached to an equivalent circuit for a DBD (the secondary is center-tapped
91
High Voltage Terminals
Secondary
Winding
Primary Winding
Secondary
Winding
Figure 39. Oil Burner Ignition Transformer [153]
so there are two high-voltage terminals). The additional mutual inductance (coil 2) on the
transformer (Tr1) is a feedback coil that controls the transistor, which drives the resonant
circuit at approximately 25 kHz. The blocking diode (D1) in the circuit only passes one half
of the driving waveform. This allows the pulse-rate of the DBD to be controlled by changing
the frequency of the driving waveform (V1), which in the present set-up is the output of the
audio amplifier.
As discussed in chapter II, the most important addition to the impedance matching
network is the additional inductor which counteracts the primarily capacitive nature of
the DBD load. The present set-up does not include impedance matching, but impedance
matching could be incorporated in order to increase the volume of the plasma discharge.
Figure 41 shows a lumped element circuit for indicating schematically where an impedance
matching network could be added.
The DBD driving circuit used in this research is unique. Rather than using a relatively
much larger (and more expensive) high-voltage transformer optimized to operate the DBD
at 1 to 10 kHz frequencies that are reported in the literature, the tank circuit of the oil
92
Figure 40. DBD Driving Circuit
Figure 41. DBD Circuit with Impedance Matching Network
93
burner igniter oscillates at 25 kHz, which allows a much smaller transformer to be used.
Since the oil burner igniter is available from many large retail suppliers, the cost is roughly
one order of magnitude lower than custom, high-voltage transformers designed to operate at
approximately 5 kHz.
One of the limitations of the driving circuitry is that the frequency output of the amplifier
does not control the oscillation of the potential on the DBD electrodes. It controls the duty
cycle of the 25 kHz oscillations. Due to the blocking diode, the circuit will oscillate over only
half the period of the alternating current (AC) signal. A steady state (100% duty-cycle)
oscillation can be maintained for approximately 50 ms. This is a driving frequency of 20Hz.
Below this frequency the performance of the audio amplifier falls off significantly.
94
IV. Results and Analysis
This chapter presents the results and analysis of the novel forward model discretization,
and the application of this diagnostic model to two plasmas, a Hall thruster exhaust plume
and a surface DBD in atmospheric pressure air. The results presented for the numerical
model itself are the speed, accuracy and stability characteristics of the method. These
characteristics of the forward model are particularly important if it is to be used in the “inner
loop” of a reconstruction algorithm. The Hall thruster diagnostic results show that there
is some limited diagnostic value for using a two parameter model (including the effects of
collisional losses). The high-density region of the exhaust plumes have statistically significant
amplitude shifts which require the second parameter to treat. However, consistent with other
results in the literature, the correction in the electron number density estimate achieved by
introducing the second parameter is less than the uncertainty due to measurement noise.
The results for the DBD indicate that Ka-band diagnostics are too low-frequency to achieve
reliable phase and amplitude response from the small surface discharge plasma.
4.1
Forward Model
The novel forward model discretization developed in the present work improves on previ-
ous numerical methods for linear-wave diagnostic models in several ways. The stability of the
Chebyshev discretization of the second order wave equation with absorbing boundary conditions, including analysis of -pseudospectra, was accomplished by Driscoll and Trefethen
[30]. Further work, high-lighting the importance of boundary condition implementation details for this approach was accomplished by Jackiewicz and Renaut [29] and Bazán [28].
This previous work has in common that it begins with unforced, one-way wave equations as
boundary conditions. For a diagnostic model based on the classical wave equation, forcing
on the boundaries is required. Not only to send the diagnostic wave into the numerical “test
95
section”, but also to account for the currents due to the particle motions on the boundaries.
The present work shows characteristic-based boundary conditions which allow this boundary
forcing to be specified in an accurate and time-stable manner. In further contrast to the
previous work, the present work applies eigenvalue stability analysis to semi-discrete approximations which use standard transform libraries [140] and short recurrences for derivative
calculations, thus avoiding the need to store and apply the full difference matrix. This results in derivative approximations that have O(n log n) time complexity and require O(n)
storage. High speed and low storage are not without cost. The accumulation of round-off
error introduced by using the recurrence rather than transcendental function evaluations are
addressed by truncating several of the high-order modes in the spectral representation of
the second derivative. This results in a fast, low-storage, spectrally-accurate and time-stable
method.
The advantage of spectral convergence is seen in the relative grid resolution requirements
of the present method. Previous work on high-order compact difference schemes for plasma
diagnostic models [24–27] indicated that roughly 15 grid points per wavelength are needed
to achieve a low error solution, and that going below 10 points per wavelength increases
the point-wise error above O(10−4 ) [24]. The compact schemes offer an improvement in
performance over the standard FDTD approach which has also been used for microwave
diagnostic modeling, and uses explicit centered differences of second-order accuracy [32].
Figure 42 shows the convergence of the first and second derivative approximations given
by the present method along with a second order central difference for comparison (two
curves are shown for the second derivative, with/without m-truncation). The spectrally
convergent method developed here requires six to eight points per wavelength to reach the
“round-off plateau.” Optimal resolution, truncation error approximately equal to round-off
error, is achievable with half the number of points per wavelength required for compact
schemes to achieve “acceptable [not optimal] resolution.”
96
104
102
100
kerrork2
10−2
10−4
10−6
∂y
∂x
10−8
∂2y
∂x2
10−10
∂y
∂x
10−12
∂2y
∂x2
10−14
101
O(∆xn )
O(∆xn )
O(∆x2 )
O(∆x2 )
102
nx
103
Figure 42. Spectral Convergence of Derivatives (y = cos(10x), x ∈ [−π, π])
Another important feature affecting accuracy of the present method is time-stability. The
previously implemented compact schemes require low-pass filtering to maintain approximate
time-stability [25]. Figure 43 shows the transients in solution amplitude for an initially zero
field forced impulsively at t = 0 with an incoming wave on the boundary. The physical
damping in the electron momentum equation term is set to zero for the results in Figure 43,
so that it does not contribute towards the stability of the solution. As might be expected
from the examination of -pseudospectra [30], the start-up transients can be significant, but
the long-time stability is guaranteed by the position of the eigenvalues in the left half of the
complex plane (as shown in section 3.1.2).
The truncation used in the present implementation to achieve time-stability has a different motivation than the filtering used in other approaches. The recurrence used to calculate
the Chebyshev coefficients for the second derivative from the first derivative propagates errors from the poorly approximated higher order modes down into the low order modes. This
97
1.4
1.2
(max − min)/2
1.0
0.8
0.6
E, ne = νe0 = 0
0.4
∂E/∂t, ne = νe0 = 0
E, νe0 = 0, ne = 10−3
0.2
0.0
∂E/∂t, νe0 = 0, ne = 10−3
0
5
10
15
20
25
30
t (periods)
Figure 43. Time-stability of the Present Pseudospectral Discretization
contamination of the low-order modes is prevented by simply truncating m of the higher
order modes prior to application of the recurrence. This lowers the order of accuracy from
nx − 1 to nx − 1 − m for the second derivative, but spectral convergence is maintained (as
shown in Figure 24).
In addition to accuracy, the time required to calculate each forward model solution is
an important factor, since this forward model must be evaluated many times when used for
reconstruction. The time to calculate the spatial operator in the MOL formulation presented
in Section 3.1.2 using the present method is shown in Figure 44. The MOL operator includes
the boundary condition calculations in addition to the interior points, so both the first and
second derivatives are calculated. This requires three transforms, a forward transform of
the solution on the grid, an inverse transform to get first derivative values on the grid,
and an inverse transform to get second derivative values on the grid. The pacing item
for the time-complexity remains the DCT calculation. Performing multiple transforms just
98
102
101
linear
quadratic
observed
wall time
100
10−1
10−2
10−3
10−4
101
102
nx
103
104
Figure 44. Time Complexity of the Present Pseudospectral Discretization
adds a constant multiplier. A linear and a quadratic trend extrapolated from the smallest
grid time are shown for comparison. The implementation exhibits the expected O(n log n)
time complexity. The methods which store and apply the full difference matrix would scale
approximately with the quadratic trend illustrated in Figure 44. The time-scaling for fulldifference matrix methods becomes even more prohibitive when extension to multiple space
dimensions is considered. The scaling is then O(n4 ) for an n×n grid and O(n6 ) for a n×n×n
grid.
The numerical forward model approach developed here is fast, low-storage, time-stable
and spectrally accurate. The benefit of using a numerical approximation is that arbitrary
material property gradients can be treated, and additional complicating features can be
added. The approach presented here is extensible to two or three spatial dimensions, which
would be required to quantify diffraction / scattering effects. It is also possible to extend the
model to include velocity dependent collision frequency effects and currents from the heavy
99
ions if that is deemed necessary for a particular application.
Complex-Step Sensitivity Derivatives.
The sensitivity derivatives calculated by the complex-step method serve two purposes in
the present research. The uncertainty in model outputs given uncertain model inputs can
be quantified by using the sensitivity derivatives. Additionally, the same method can be
applied to the objective function of a reconstruction algorithm to support gradient-based
optimization of the reconstruction.
The great advantage of using the complex-step approach to calculating these derivatives
of solution functionals with respect to model parameters is that arbitrarily high accuracy can
be achieved in the absence of subtractive cancellation. This means that no optimal step-size
must be found as is the case with standard finite-difference-based sensitivity calculations.
As shown in Section 3.3, the complex-step approximation is expected to be second order
accurate. Figure 45 shows this expected convergence behavior. The functional being tracked
for convergence in Figure 45 is the norm of the error in the sensitivity ffield of the solution to
the numerical forward model. The same manufactured solution used to verify the convergence
of the forward model derivatives can be used to verify the parameter sensitivity derivatives.
For the manufactured solution chosen in Section A.1, only the electron velocity, u, is sensitive
to the value of collision frequency, ν, so the convergence of this derivative is shown in Figure
45.
Eventually the complex step becomes small enough that the error in the discretization of
the real-part dominates the error, so we see the familiar round-off plateau that occurs for the
pseudospectral derivative approximations. A second order central difference approximation
is shown in Figure 45 for comparison. The standard finite difference exhibits the same
convergence rate as the complex step method at large step sizes (h), but reaches the point
where the truncation error is approximately equal to the round-off error caused by subtractive
100
101
∂u
∂ν
10−1
complex step approximation
O(∆x2 ) finite difference
10−3
k error k2
10−5
10−7
10−9
10−11
10−13
10−15
10−17
10−16
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
h
Figure 45. Convergence of Complex-step Sensitivity Derivative
cancellation. This point is the optimum step-size, and is problem dependant. The error due
to subtractive cancellation continues to grow for step-sizes smaller than optimal, destroying
the accuracy of the slope approximation. The complex-step method removes the need to pick
an optimum step-size, and is robust in the sense that the approximation remains good even
if the step-size is chosen many orders of magnitude smaller than the size that first reaches
the round-off plateau.
4.2
Hall Thruster Exhaust Plume
This section presents microwave measurements of a 200W-class Hall thruster exhaust
plume using noble gas propellants. Figure 46 shows a close-up image of the visible emissions
from a Hall thruster. The image shows this 200W-class thruster is being run on Krypton
propellant. The main exhaust plume extends in the horizontal direction. The off-axis cathode
101
Figure 46. Visible Emissions of 200W Hall Thruster, Krypton Propellant (taken from [154])
is clearly visible at the top of the image.
The phase and amplitude differences measured on the boundary of the domain are turned
into a spatial field of plasma properties. This process of turning boundary data, or chordaveraged measurements, into fields is called reconstruction. As discussed in section 3.2.1,
for the roughly axi-symmetric plasma in a Hall thruster exhaust plume the inverse Abel
transform is a suitable reconstruction technique. Prior work on microwave interferometry
for Hall thruster exhaust plumes used a one-parameter (ne ) analytical model for the wave
dispersion in the plasma [5, 14, 15]. The present work uses a two-parameter numerical
model (ne , νe0 ) along with a robust, numerical inverse Abel transform approach to the
reconstruction. The uncertainty propagation method for the forward model is extended to
include uncertainty quantification of the reconstructed fields.
The electron density variation for two different noble gas propellants (Xenon and Kryp-
102
ton) was estimated in previous work using a one-parameter analytical model [6]. Other
diagnostics have also been applied to characterize the performance differences between these
two propellants [4, 155]. This work focuses on differences in collision frequency due to changes
in propellant which can be diagnosed by the two-parameter forward model for the microwave
diagnostic. Detailed descriptions of the Hall thruster itself are not publicly releasable, but
these can be found in Kenan’s controlled-distribution thesis [6].
The microwave interferometry data collected by myself and Capt Kenan in the AFIT
SPASS lab [6] provides a useful set of test data for the forward model presented above as
well as Abel inversion since the Hall thruster plume is quite close to axis-symmetric. The
data were taken in a vacuum tank with a VNA capable of time-gating. This is an important
capability because it allows multi-path reflections to be rejected, and the primary path (the
test section containing the plasma) to be analyzed in isolation.
The spatial resolution of the microwave measurements depends on the frequency of the
interrogating beam and the optics. To quantify this, an expanded polystyrene foam calibration cylinder was measured with the VNA. This allows the instrument (most importantly the
beam width / spatial resolution) to be characterized. The two-parameter calibration model
developed at the end of section 3.2.1 was fit to the phase shift data. Figure 47 shows the
results of the two parameter fit. The measurements across the Ku-band give the expected
result of better spatial resolution in the transverse direction (smaller beam waist) as the
diagnostic frequency increases. The standard deviation of the Gaussian interferometer beam
(Figure 48) is a function of the physical layout, lenses and frequency. The beam standard
deviation gets gradually smaller as the interferometer frequency is increased. The index of
refraction of the foam cylinder is roughly constant over the frequency range examined.
Figure 49 shows the various refractive indices at different frequencies and their uncertainty
based on the two-parameter calibration model fit to a measurement rake across the cylinder,
as well as an estimate of the index of refraction of the cylinder. The error bars in both
103
Figure 47. Two-parameter (σ, n) Calibration Model Fit
Figure 48. Beam Intensity Standard Deviation
104
Figure 49. Calibration Cylinder Index of Refraction
figures are two standard deviations and are based on resampling the residuals of the fit with
replacement (bootstrapping the residuals [156], see A.2).
Figure 50 and 51 show representative phase and amplitude shift data for this plasma
using the Ku-band microwave diagnostic. The x-direction is accross the exhaust plume, as
indicated in Figure 46.
The measurement in Figures 50 and 51 is a difference of ratioed
transmission measurements made with the VNA. Two base-line measurements are made
with no plasma present (before and after the measurements with plasma), and measurements are made with the Hall thruster ignited. The difference in phase and amplitude of the
transmissions in both directions (commonly designated S12 and S21) are taken using both
baseline vacuum measurements. This results in the four data-lines shown in the Figures. The
largest variation between the data sets is caused by the differences in the baseline measurements (the “reference leg” of the interferometer). Smaller differences can be seen between
the two transmission directions. If the plasma, instrumentation and surrounding equipment
105
0
0.010
0.005
−4
−6
−8
−10
−200
∆mag (u)
∆φ (deg)
−2
S12
S12
S21
S21
−150
−100
0.000
−0.005
S12
S12
S21
S21
−0.010
−0.015
−0.020
−0.025
−50
0
50
−0.030
−200
100
x (mm)
(a) Phase difference
−150
−100
−50
0
50
100
x (mm)
(b) Amplitude difference
0
0.010
−2
0.005
−4
−6
−8
−10
−200
∆mag (u)
∆φ (deg)
Figure 50. 15GHz Microwave Transmission through 200W Hall Thruster Plume, Krypton fuel
S12
S12
S21
S21
−150
−100
0.000
−0.005
−0.010
−50
0
50
100
x (mm)
(a) Phase difference
−0.015
−200
S12
S12
S21
S21
−150
−100
−50
0
50
100
x (mm)
(b) Amplitude difference
Figure 51. 15GHz Microwave Transmission through 200W Hall Thruster Plume, Xenon fuel
106
were perfectly symmetric these would be equivalent. The differences can be explained by a
combination of natural measurement noise and asymmetric plasma density gradients which
cause a “masking” effect. Masking is the term given for the response of one portion of
the material under test affecting or obscuring a subsequent portion along the measurement
chord. In standard circuit analysis it can be caused by changes or discontinuities in line
impedance [8]. For plasma diagnostics this effect is due to large gradients in index of refraction. The smallest component of variation in the data is illustrated by the error bars on
each data point. At each point in the traversal across the plume, 500 measurements at the
same frequency were taken the error bars shown are two standard deviations about the mean
of those measurement sets. The variation shown by these repeated measurements is due to
oscillations in the plasma itself, or the control equipment. The sampling rate of the VNA is
limited to resolving oscillation on the order of 3KHz. The high-frequency oscillations that
have been identified in these plasmas [157] were not resolved. This means they were aliased
to appear as power at low-frequencies, so temporal variation in the measurements is treated
as error for the present analysis.
Multiple sweeps across the plume in the transverse direction such as shown in Figures
51 and 50 are taken at several distances away from the thruster face in the plume-wise
direction (indicated as the y-direction in Figure 46). The average phase and amplitude
response for these combined sets of measurements are shown in Figures 52 through 55. This
plasma is at low enough pressure that the elastic scattering collisions of the electrons are
generally neglected in the diagnostic model [6]. Under this collisionless assumption, the
electron density can be recovered from the phase data alone, and the amplitude data makes
no contribution. However, there are statistically significant features in the amplitude data
for both propellants.
The phase response for the Krypton propellant gives the expected qualitative behavior.
The phase shift is high near the center of the plume, and close to the thruster face (bottom
107
of Figure 52, near x = 0). The density of the plasma gradually decreases moving away from
Figure 52. Average Phase Data for Krypton Propellant at 15GHz
the thruster face (increasing y). The region of significant phase shift also spreads out as the
plume expands.
The largest amplitude response for the Krypton plume occurs for the data nearest the
thruster face (small y, bottom of Figure 53). This amplitude response is off-centered from
the main phase-response of the exhaust plume. This asymmetric result was unexpected. The
only geometrical asymmetry in the device is the cathode (see Figure 46). The measurement
traverses were taken so that the cathode side is in the negative x-direction. So this asymmetry
can possibly be explained by an effect due to the cathode.
This thruster is designed for Xenon propellant, so the amplitude shift may be caused by
higher collisional losses at the cathode for Krypton propellant. Larger amplitude response
indicates that the collision frequency for momentum transfer is higher in this near-cathode
region of the exhaust plume. This can be caused by several physical mechanisms. The
collision frequency depends on the number density of the colliding pairs, and their cross108
Figure 53. Average Amplitude Data for Krypton Propellant at 15GHz
sections for collisions. The number density, as indicated by the phase response (neglecting
collisions), is slightly lower than the Xenon propellant so this does not explain the increased
collision frequency. The remaining physical mechanism is the cross-section. The cross-section
depends on the relative velocities (temperatures) of the colliding species. For electron neutral
collisions the cross-section is proportional to the velocity, while inversly proportional to the
cube of velocity for electron-ion collisions [1]. As pointed out in section 3.1, if species fraction
data is available, then the electron temperature could be infered from this diagnosed collision
frequency.
The phase response of the Xenon plume exhibits some of the same qualitative features
as the Krypton. Figure 54 shows the high density region in the center of the plume, near
the thruster face, with gradually decreasing density with increasing distance as the plume
expands. The response for the Xenon is higher than the Krypton plume, indicating higher
electron number density. One feature of the Xenon plume which is not exhibited by the
Krypton is the variations in phase response in the y-direction. Rather than a monotonic
109
Figure 54. Average Phase Data for Xenon Propellant at 15GHz
decrease in phase (number density) as y increases, there are alternating regions of higher
and lower response. This is oscillatory variation is an unexpected behavior in the phase
response data.
The amplitude response for the Xenon plume is qualitatively different than the Krypton plume. Figure 55 shows that the amplitude response for the Xenon does not have a
feature near the cathode as the Krypton plume does. There is an area of large amplitude
response in the high-density region of the plume corresponding to the maximum in the phase
response near the thruster face. There are also significant amplitude response “regions” in
the expanding plume away from the thruster face.
The standard error in the average amplitude response is used to examine if the regions of
large amplitude response away from the thruster face are significant, or due to measurement
noise. Figure 56 shows contours for amplitude response significantly different from zero
(greater than two standard deviations) superimposed on the phase response. These regions
of statistically significant amplitude response correspond to the high density region near the
110
Figure 55. Average Amplitude Data for Xenon Propellant at 15GHz
Figure 56. Relation of Significant Amplitude Response to Phase Variation (Xe)
111
thruster face, and the regions of low phase response in the oscillatory variations along the
y-direction.
The Krypton plume does not show the same strong association between significant amplitude response and abnormal variations in the phase down the length of the plume. If
Figure 57. Relation of Significant Amplitude Response to Phase Variation (Kr)
this were an installation effect, it is expected that both propellants would show a similar
behavior.
The differences between propellants in the high density region is understandable simply
due to the higher electron density indicated by the larger phase response of the Xenon plume
moving the imaginary part of the index of refraction increasingly negative (as shown in Figure
58). This has the effect of increasing the amplitude shift (collisional losses) for the Xenon
plume. Since the amplitude response is small compared to the measurement and process
noise, only the Xenon response is resolvable, while the expected smaller Krypton amplitude
response in this region is not.
Alternatively, if quasi-neutrality is assumed, then the phase response is an indication not
112
102
01
0
-0.
03
0
-0.
07
0
-0.
18
0
-0.
11
5
-0.
-0.
νe0 /ω
101
04
-0.
0
1
27
7
66
-0.
0
64
-1.
-4.0
32
100
-0
.0
18
10−1
10−1
100
101
102
ω02 /ω 2
Figure 58. Imaginary Part of Appleton-Hartree Refractive Index
only of the electron density, but indirectly of the ion density as well. Since the Xenon has a
higher over-all phase response magnitude, the density is greater. The significant amplitude
response in the high density regions of the Xenon plume and its absence in the Krypton plume
is explained by this higher density since the total collision frequency for momentum transfer
of the electrons is proportional to the number density of the collision partners (neutrals or
ions).
νt =
X
i
ni σi (Te ) (ve − vi )
(91)
where νt is the total collision frequency for momentum transfer, σi is the (electron temperature dependent) collision cross-section with the ith species, ve is the electron velocity and vi
is the velocity of the ith species. Higher density (ni ) is expected in the region closest to the
thruster, with lower densities further away from the thruster as the plume expands. While
this gives an explanation for the statistically significant amplitude shift seen close to the
113
thruster in the high density region, it does not explain the statistically significant amplitude
shifts in the low-density region of the plume that seem to correlate with “holes” in the phase
response of the Xenon plume.
These data indicate that the added complexity of a two parameter diagnostic model
(including both ne and νt ) is justified for some exhaust plumes at relatively low frequency
(in this case 15GHz). As indicated by Figure 58 going to higher diagnostic frequencies (such
as the 90GHz instrument in [5]) will lower the magnitude of the amplitude shift (likely
below the noise floor of the instrumentation), making the collisionless assumption more
well justified. Figure 59 shows the real part of the refractive index (which is the only part
considered by the simple one parameter model). At low electron densities (left of the plot)
103
01
1.0
1.000
Hall thruster, Ku diagnostic
102
64
1.1 525
1.
68
2.4
νe0 /ω
101
5.2
100
44
1.03
0.40 2
5
1
10−1
10−3
10−3
10−2
10−1
0.06
4
0.656
0.859
0.969
0.999
10−2
0.19
100
2
ω0 /ω 2
101
102
103
Figure 59. Real Part of Appleton-Hartree Refractive Index
and low collision frequencies (bottom of the plot), the refractive index is fairly insensitive
to changes in collision frequency (contours of constant refractive index roughly vertical).
114
This means that there will be little difference in the electron density diagnosed by the one
parameter model and the two-parameter model. This region of roughly vertical refractive
index contours is where the Hall thruster exhaust plume is expected to reside (which is
the reason a simple one parameter model is often used). At higher electron densities small
changes in collision frequency can have a significant effect on the electron density that is
predicted by the model (lines of constant refractive index with slope near horizontal). In
other words, given the measured phase shift and measurement chord, a required refractive
index is defined. In certain regions of the parameter space, by neglecting collisional effects,
the point on the refractive index contour moves strongly in the direction of increasing plasma
frequency (electron density), which means that the one-parameter model will over-predict the
electron density when collisional effects are inappropriately neglected. In the less sensitive
region of the parameter space, the point on the refractive index contour moves in the direction
of slightly reduced electron density. For the Xenon propellant exhaust plume of a 200W Hall
thruster measured at 15GHz, there is a statistically significant amplitude shift in the high
density region of the plume. This means that electron density in this peak density region
will be miss-predicted if only the phase shift data is considered. Assuming constant material
properties (appropriate for comparing chord-averaged measurements), the electric field is
given from the complex refractive index by [1]
E (x, t) = e−
x
iµωx
+χω
+i ω t
c
c
E0
(92)
where ω is the frequency of the diagnostic wave and µ is the real part of the refractive index
given by
q
p
√
(ω 2 + ν 2 ) ω04 − 2 ω 2 ω02 + ω 4 + ν 2 ω 2 + ω 4 + ν 2 ω 2 (−ω02 + ω 2 + ν 2 )
µ=
√
1
2 ω 2 + 2 ν 2 (ω 4 + ν 2 ω 2 ) 4
115
(93)
and χ is the imaginary part given by
q
p
√
(ω 2 + ν 2 ) ω04 − 2 ω 2 ω02 + ω 4 + ν 2 ω 2 + ω 4 + ν 2 ω 2 (ω02 − ω 2 − ν 2 )
χ=−
√
1
2 ω 2 + 2 ν 2 (ω 4 + ν 2 ω 2 ) 4
(94)
The collisionless case of ν = 0 gives the commonly used [7] real part as
µ=
r
1−
w0 2
w2
(95)
and the imaginary part as zero. For the peak chord averaged measurement of the Xenon
plume (−8.59◦ ,−0.0153) the electron density estimate using equation 95 is 9.85 × 1014 m−3 ,
while the estimate using the two parameter model is 9.96 × 1014 m−3 (an error of approximately 1%). So, while the Xenon exhaust plume has a statistically significant amplitude
shift in this region, it is in the regime where the electron density estimate is fairly insensitive
to the collisionality assumption.
The high density region of the Xenon plume which has a measurable phase and amplitude
response provides a useful data-set on which to apply the numerical forward model. Figures
60 and 61 show the raw and smoothed phase and amplitude data.
The smoothing is
accomplished as described in section 3.2.1, with a standard deviation chosen according to
the fit of the calibration model (see Figure 48) which includes a standard deviation parameter
for the Gaussian beam-width. This is using the measured beam width as a way to say what
variations are likely to be due to true spatial variations in the parameters and what are
likely to be due to uncontrolled variation in the experimental set-up. In other words, it
is unlikely for two measurements spaced 3mm apart to be different due to spatial gradients
when measured by a device that performs a spatial integration with Gaussian weights having
a standard deviation on the order of 15mm. If such variation does appear in the data it is
most likely to be noise and it is appropriate to average it out. In addition to averaging
116
Figure 60. Xenon Phase Data, y = −205 mm
Figure 61. Xenon Amplitude Data, y = −205 mm
117
out unwanted noise, the convolution provides a convenient way of extrapolating the data
smoothly to zero at the plume edge.
As shown in Figure 60, the phase response smoothly approaches zero without unphysical overshoots. The amplitude response, shown in Figure 61, is small enough in magnitude and has large enough noise that the smoothing procedure gives positive amplitude
changes near the plume edge. Amplification like this is unphysical in the context of this
two-parameter, one-dimensional model. Amplification has been reported in some microwave
diagnostic efforts, but this was likely due to diffraction effects focusing the diagnostic wave
on the instrument (which used bare horn antennas sans focusing optics) [26]. These sorts of
diffraction effects are second-order effects for this large (in relation to the diagnostic beamwidth) plasma, and require at least a two-dimensional model to resolve. In any case, the
best explanation for the small unphysical amplification given by the data smoothing is simple
measurement noise.
The chord-averaged data is transformed into a reconstructed field using the inverse Abel
transform approach described in section 3.2.1. Figures 60 and 61 show these inversions
along with the uncertainty in the reconstruction, which is calculated by a resampling method
described in Appendix section A.2. Due to the noise in the magnitude measurements, there
is a small region of unphysical amplification. This is truncated for use in the numerical
forward model (ν < 0 introduces unstable eigenvalues).
The reconstructed response per length must be integrated by a forward model to give
physically meaningful parameters. This is accomplished by fitting the scaling factors (reference electron density and collision frequency) for the reconstructed phase and amplitude
response that cause the forward model to give the measured phase and amplitude response.
Figure 64 shows the forward model solution for the mean reconstructed responses calculated above (the scaling on the electron density and collision frequency distributions are
exaggerated so that phase and amplitude shifts are visible on the plot) plotted against the
118
Figure 62. Xenon Phase Reconstruction, y = −205 mm
Figure 63. Xenon Amplitude Reconstruction, y = −205 mm
119
non-dimensionalized length. The length scaling comes from the non-dimensionalization de1.5
normalized ne , ν, E
1.0
0.5
0.0
−0.5
ne
ν
free space
with plasma
−1.0
−1.5
0
1
2
x
p
3
ne
q 2 /m
4
5
6
e /c
Figure 64. Forward Model Solution for Mean Center of Plume Reconstruction,y = −205 mm
veloped in Section 3.1 (scalings for all the parameters are listed in Table 1). The reference
length for this nondimensionalization is the electron plasma frequency divided by the speed
of light.
The solutions shown in Figure 64 (and all other solutions to the pseudospectral forward
model presented in this document) are interpolated onto a fine, uniformly-distributed mesh
for graphical display.
The actual parameters required for scaling the reconstructed field to achieve the measured
phase and amplitude at the plume center are shown in Table 3. The peak electron density is
on the same order (1010 cm−3 ) as the results from microwave-based diagnostics for a thruster
of similar propellant and power class (the raw phase data in the high-density region of the
plume is also roughly equivalent, so this is not surprising) [158].
There is a small amount of reflection off of the plasma, but the gradients are gradual
120
Table 3. Forward Model Fit to Center of Plume Reconstruction, y = −205 mm
Phase shift
Amplitude shift
peak ne
peak νt
−5.3◦
−8.1 × 10−3
1.2 × 1010 cm−3
9.9 × 109 s−1
enough that this reflection is negligible. This justifies making only the transmission measurement and attributing any amplitude change to collisional losses. This result indicating
negligible reflections is consistent with the one-dimensional results accomplished using a loworder FDTD modeling of microwave diagnostics in the near-field of a Hall thruster exhaust
plume [32].
This negligibility of reflections is a problem-specific feature. While reflections off the
plasma have not been noted as a significant problem for Hall thruster exhaust plume diagnostics, they have been the source of error in using microwave techniques for plasmas
characteristic of hypersonic magneto-hydrodynamics (MHD) flow-control applications [26],
and reflections are also expected in the target application of atmospheric pressure DBD
diagnostics. In these cases where reflections are expected to be significant (plasmas with
“sharp” electron density features) full “four port” measurements should be made (i.e. both
sets of reflections and transmissions).
Uncertainty in the forward model parameters is propagated to uncertainty in the solution using the complex step method described in section 3.3. Figures 65 and 66 show the
sensitivity derivatives of the last wave-length of the forward model solution with respect
to the uncertain parameters. Variations in the solution estimated from these complex-step
sensitivity derivatives caused by a ±1% change in the electron density and a ±100% change
in the collision frequency are shown as well. Such a large variation in collision frequency is
chosen to make the difference visible on the plot, which further underscores the conclusions
previously made about the relative insensitivity of the results in this case to the collisionless
assumption. The last wave-length of the solution is shown because this is a region of zero
121
1.5
normalized ∂E/∂ν, E
1.0
0.5
0.0
−0.5
−1.0
E
normalized sensitivity
−1.5
1.96
1.98
2.00
2.02
x
p
2.04
ne
q 2 /m
2.06
2.08
2.10
e /c
Figure 65. Normalized Sensitivity Derivative for Collision Frequency
1.5
normalized ∂E/∂ne , E
1.0
0.5
0.0
−0.5
−1.0
−1.5
1.96
E
normalized sensitivity
1.98
2.00
2.02
x
p
2.04
ne
q 2 /m
2.06
2.08
e /c
Figure 66. Normalized Sensitivity Derivative for Electron Density
122
2.10
electron density and collision frequency, so it can be used to estimate the phase and amplitude shift from the reference free-space beam since the wave speed is known. This approach
is taken for simple expediency over saving the solution on the boundary for a set amount of
time in order to calculate phase / amplitude data.
Examining the phase shift between the solution (E) and the sensitivity derivatives gives
an indication of which parameter has the most influence on the phase shift of the diagnostic
and which has the most influence on the amplitude shift. A parameter which has a high
sensitivity for a region where there is a zero-crossing of the wave will have a significant
impact on the phase, while a parameter that has a large sensitivity near a peak/trough of the
diagnostic wave will have a large impact on the amplitude. Figure 65 shows that the collision
frequency sensitivity is almost perfectly out of phase with the diagnostic wave. The large
values of sensitivity to this parameter correspond to peaks and troughs in the diagnostic
wave. This indicates that changes in collision frequency will have most of their impact
on the amplitude of the wave (and that increasing νt decreases the diagnostic amplitude).
Figure 66 shows that the electron density sensitivity is out of phase with the diagnostic by
approximately 1/4 wavelength (for the Ka-band diagnostics this wavelength is approximately
2 cm). The large electron density sensitivity regions correspond with diagnostic wave zerocrossings. This indicates that changes in electron density will have the most impact on the
phase of the wave.
It is important to note that the sensitivities displayed in Figures 66 and 65 are not
sensitivities to the local number density or collision frequency, they are sensitivities to the
scaling parameter for each of the reconstructed distributions. In other words, we’ve used
the Abel inversions as a way to reduce the dimensionality of the uncertainty in the spatial
field to only a two-dimensional stochastic space rather than a 2nx -dimensional one. This
dimensionality reduction means that the uncertainty quantification requires only two forward
model evaluations, one with the reference number density perturbed by an O() complex123
step, and another with a similarly perturbed collision frequency. Contrast this to the 1000
bootstrap samples that were used to quantify the uncertainty in the inverse Abel transform
reconstructions. Admittedly, the stochastic space in that case is of a much higher dimension.
In that case, the uncertain “parameters” are the data points themselves.
The analysis presented above is a local linearization of the uncertainty propagation.
Large changes in the parameters will move the solution to other parts of the parameter space
where these relationships between electron density/phase and collision frequency/amplitude
are not as clearly delineated. Even in this particular region of the parameter space changing
collision frequency has a small effect on the phase, and changing electron density has a
small effect on the amplitude. One of the important features of the forward model in this
region of parameter space is the large difference in sensitivity between the two parameters.
We see that the solution is much more sensitive to changes in the electron density than
change in the collision frequency. As pointed out in section 3.3.1, having access to the
slope information gives us a guide to which parameter we should add resolution, and expand
from a local linear approximation to a increasing-order stochastic collocation. We should
perform more forward model evaluations with changes in electron density if we want to better
understand the uncertainty. That is, it is prudent to retain the local linear approximation
for propagating uncertainty due to collision frequency, but sample the forward model at
more points of electron density. This lowers the dimensionality of the stochastic space by
one. For this two-dimensional stochastic space the pay-off for such selective resolution is not
especially high, but for multi-parameter problems, the curse of dimensionality means that
this information can have significant value.
This transformation from parameter (ne ,ν) to data (measured phase and amplitude)
space can then be inverted (see equation 88) to give a relation for expressing the uncertainty
in the fit electron density and collision frequency due to uncertainty in the measured phase
and amplitude data. Figures 67 and 68 show the uncertainty in the peak electron density and
124
collision frequency for the center of plume reconstruction presented above. The uncertainty
0.00008
0.00007
∂p/∂ne (cm3 )
0.00006
0.00005
0.00004
0.00003
0.00002
0.00001
0.00000
0
10000
20000
30000
−3
ne (cm )
40000
50000
60000
+1.199997×1010
Figure 67. Uncertainty in Peak Center of Plume Electron Density, y = −205 mm
in the electron density reconstruction is most tightly constrained by the phase measurement.
The collision frequency causes almost no significant phase shift, so the uncertainty in that
parameter is dominated by uncertainty in the amplitude measurement, which is small in magnitude and has significant measurement noise. Since the likelihood of the collision frequency
has significant probability density for unphysical values (ν < 0), the posterior distribution
obtained from the likelihood and a maximum entropy distribution [159] for a non-zero parameter with a mean of 1 × 1011 s−1 is displayed as well. The expected value of the posterior
distribution is 9.4 × 109 s−1 with 90% credible interval of [0.19 × 109 , 2.9 × 1010 ] s−1 . This
is a more physically realistic estimate of the peak collision frequency for the center of plume
reconstruction presented above since it eliminates the spurious negative estimates due to
measurement noise.
125
1.0
×10−10
likelihood
posterior
∂p/∂ν (s)
0.8
0.6
0.4
0.2
0.0
−6
−4
−2
0
−1
ν (s )
2
4
6
×1011
Figure 68. Uncertainty in Peak Center of Plume Collision Frequency, y = −205 mm
4.3
Dielectric Barrier Discharge
In flow-control applications of the DBD flat electrodes are generally used. This limits
perturbations to the flow when the actuator is not active, and it is more easily incorporated
into aerodynamic surfaces. The plasma from such an electrode has a relatively small spatial
extent and low emissions (indicating low density). Using the same driving circuitry, a much
larger plasma extent can be achieved by using an exposed wire electrode in place of the
flat tape. A side-by-side comparison of the visible emissions from the two different exposed
electrodes is shown in Figure 69. The DBDs pictured both used two layers of Kapton tape
dielectric with a nominal thickness of 0.002 in (2 mils).
With the presently available diagnostic equipment (limited to diagnostic frequencies below 40GHz), the plasma from the flat electrode is too small and low-density to generate a
measurable response. The wire electrode device generates a plasma that has a width on
126
Figure 69. Visible Emissions from Wire and Flat-Tape Electrode DBDs
the same order of the beam-width and the density of the plasma (as judged by the visible
emissions) is greater. Generating a plasma area that is large enough to cover most of the distribution of the diagnostic beam intensity is important to minimize diffraction effects. If the
area of the plasma is significantly smaller than the beam intensity distribution, then there
will be scattering (diffraction) of the diagnostic energy. This is a two-dimensional effect.
The one-dimensional diagnostic model used here would inappropriately attribute amplitude
changes due to diffraction to collisional losses. A two (or three) spatial dimensional model is
required to properly account for diffraction from small (relative to the beam width) plasmas.
Figure 70 shows the wire-exposed-electrode device is used in the present diagnostic effort
because of its larger signal. The visible emissions for the DBD are collected with long
shutter speeds on a COTS digital camera (Canon EOS Rebel Ti1). These time-averaged
emissions give a qualitative idea of the size of the plasma region generated by the device,
which is important to the assumption that significant diffraction is not occurring and causing
perceived amplitude loss which would be attributed to collisional losses by the present onedimensional forward model.
127
(a) Top View Unenergized
(b) Top View Visible Emissions
(c) End View Visible Emissions
Figure 70. Wire Exposed Electrode DBD Visible Emissions
The phase response caused by the wire-electrode DBD plasma is shown in Figure 71.
This measurement is made at 0 degrees obliquity, using the buried electrode as a reflecting surface over the Ka-band of frequencies (26.5 to 40 GHz). While the noise level of the
measurements is large, it is repeatable, so the mean phase response is significant. Figure
72 shows some representative time-traces of a multiple data collection events. The data is
collected by triggering the VNA to collect data over a range of frequencies, and the function
generator driving the DBD circuitry is time-delayed so that it does not turn on until this
initial sweep is complete (“plasma off” measurements). Timed to coincide with the initiation
of the function generator, the VNA then performs a second sweep of the same frequencies
(“plasma on” measurements). The difference between these two measurement sets is the
phase and amplitude change caused by the presence of the plasma. There is a statistically
significant mean phase shift response accross the band. Unfortunately there is not a significant variation in the response with frequency. Finding a measureable variation in the
response with frequency is desirable to provide more information for spatial reconstruction.
The small size of the plasma feature in relation to the diagnostic wavelength makes it
problematic to properly recover the amplitude shift caused by the DBD plasma feature. The
electron density feature in Figure 73 is Gaussian with a standard deviation of 1 mm and
128
Figure 71. Phase Response from Wire-Electrode DBD
Figure 72. Representative Phase Data Traces
129
an average electron density of 1 × 1018 m−3 . Previous work attempting to measure surface
DBD plasmas with diagnostics at 10GHz showed a significant amplitude shift, but not a
significant phase shift [9]. The ability to find a significant amplitude signal at the lower
frequency diagnostic is due to the larger imaginary part of the refractive index (see Figure
58). As Figure 73 shows, there is a significant reflection off the small electron density feature,
1.5
ne
free-space
with ne
normalized ne , E
1.0
0.5
0.0
−0.5
−1.0
−1.5
0
2
4
x
p
6
ne
q 2 /m
8
10
e /c
Figure 73. ne ≈ 1 × 1018 m−3 , νe0 = 1 × 1012 rad/s, f = 35GHz
which will confound measurement of the amplitude change caused in the portion of the signal
that traverses the plasma and returns off the buried electrode. This is because a significant
part of the diagnostic energy is reflected back to the recieving horn without interacting with
the plasma electrons on the interior of the DBD, and so does not experience amplitude
losses due to collisions. This result is similar to that found for microwave diagnostics of
MHD plasmas, where the amplitude measurement was confounded by diffraction effects due
to the plasma being smaller than the diagnostic beam size [26].
As discussed in section 3.1, an optimum in phase response is expected for the small DBD
130
plasma at roughly 160 GHz. Figure 74 shows the same Gaussian electron density feature as
that of Figure 73 being diagnosed by a 160GHz wave. Even though the amplitude shift is
1.5
ne
free-space
with ne
normalized ne , E
1.0
0.5
0.0
−0.5
−1.0
−1.5
0.0
0.5
1.0
x
p
1.5
ne
q 2 /m
2.0
2.5
e /c
Figure 74. ne ≈ 1 × 1018 m−3 , νe0 = 1 × 1012 rad/s, f = 160GHz
smaller than a diagnostic at 35 GHz, the phase shift is larger. This makes the diagnostic
easier to resolve as the transmitted and reflected waves are superimposed further out of
phase at the detector. Additionally, the forward model solution in Figure 74 shows that
the reflections off of the plasma surface are smaller at the higher frequency. This means
that more of the diagnostic intensity will traverse the plasma, causing electron motions and
subsequent collisions, which result in measureable amplitude shifts. Also, and the spatial
resolution (beam width) achievable at 160GHz is significantly smaller than that achievable
with a 40GHz system.
131
Silica Glass Dielectric.
A small study was accomplished using silica glass dielectric in place of the Kapton tape,
but retaining the same copper plate as the buried electrode and wire as the exposed electrode.
While no plasma diagnostics were performed for these devices, the phase measurements did
diagnose expansion and shrinkage of the dielectric caused by heating.
Figure 75 shows the long-time phase response of the microwave diagnostic as the plasma
is turned on and then turned off. The phase response shows the sort of exponential-in-
Figure 75. Long-time Phase Response at 26.5GHz due to Dielectric Heating
time variation expected for a material approaching a steady-state temperature as heating is
impulsively applied, and then cooling to a steady state as heating is impulsively removed.
The solid lines indicate an exponential fit to the data over each distinct time-interval: leadin, plasma-on and plasma-off. The discontinuities between the fits indicate the contribution
to the phase response from the presence of the plasma, which responds on a much faster
132
time-scale than thermal expansion of the dielectric. This contribution, on the order of 0.1◦ ,
is roughly equivalent to the response found for the Kapton tape devices (see Figures 71 and
72).
This long-time heating behavior was not apparent for the Kapton tape devices. From an
anecdotal perspective, the silica glass devices would get hot to the experimenter’s touch (and
would sometimes crack from highly localized heating) while the Kapton tape-based devices
would not heat perceptibly. This highlights a possible confounding factor for microwave
diagnostics of these surface discharge devices, which use components of the device itself as
part of the diagnostic instrument (the buried electrode providing a mirror in this case).
Changes in electrical path length caused by heating of the dielectric, or vibrations of the
device could easily be misdiagnosed as response due to plasma properties or their time
variation.
4.4
Summary
This research has shown the efficacy of a Chebyshev pseudospectral numerical diagnostic
model for collisional plasmas. An enabling technique for using this PDE-based forward model
is the complex-step method of sensitivity derivative calculations, which allows uncertainty
quantification of the model outputs. Two test plasmas were diagnosed using this numerical
model and data from a VNA-based microwave interferometer. Reconstruction techniques
applicable to the geometrical features of the test plasmas were used to infer the spatial
variation of material properties from the boundary measurements given by interferometry.
The present numerical approach is fast, low-storage, and spectrally accurate. It allows
material property gradients to be treated in its current one-dimensional state, and can be
extended to multiple spatial dimensions to account for diffraction and scattering effects
which may be significant for small plasma diagnostics. The numerical approach also allows
the possibility of including more complicated current models, such as heavy ion currents, if
133
these are deemed significant for the particular plasma under test.
The Hall Thruster exhaust plume results presented here are novel in their application of a
two-parameter diagnostic model which accounts for elastic scattering (momentum transfer)
of the electrons. This phenomena is usually neglected for microwave diagnostics of this type
of plasma reported in the literature. The collisionless assumption was found to be a good
one in a practical sense, because the error it introduces in estimates of electron density in
the peak density region of the plume in this case is only on the order of 1% for Xenon fuel,
and no significant amplitude shift was found in this region for the Krypton fuel.
The DBD remains a challenging plasma to diagnose non-intrusively. Previous efforts at
credible electron property diagnostics for this plasma have met with limited success, and this
effort was little different. The features of the DBD device had to be changed to use a wire as
the exposed electrode rather than the commonly used flat electrodes found in DBD plasma
actuators. This increased the size and intensity of the discharge so that a measurable phase
shift was found. However, the density gradients of this plasma have a small length-scale
with respect to the wave-length of a Ka-band diagnostic, and cause reflections which tend
to confound diagnosing material properties from the phase and amplitude shift of the return
signal. In contrast to the plasma of the Hall thruster exhaust plume, the collisional effects
in this atmospheric pressure surface DBD are expected to be significant, so both phase and
amplitude data are necessary for a credible reconstruction.
134
V. Conclusions and Recommendations
This research presents a flexible numerical framework for microwave diagnostic modeling
applicable to a wide variety of plasmas from Hall thruster exhaust plumes to atmospheric
pressure Dielectric Barrier Discharges in air. The strength of the numerical approach is that
it models a “numerical test section”, which includes in its one-dimensional form reflections
from material property gradients that go undiagnosed in simple analytical diagnostic models.
The numerical method is a novel and improved implementation of the Chebyshev pseudospectral method. The novelty lies in the approach to achieving time-stability. In contrast
to the methods found in the literature for high-order accuracy schemes for linear wavepropagation, the present method does not require ad-hoc filtering to achieve time-stability.
There is no theoretical justification for a filtering requirement in the discretization of a twoway linear wave equation since the domain of dependence extends in both directions of the
spatial coordinate, and there are no wave steepening behaviors requiring regularization as in
nonlinear wave propagation. The novel pre-recurrence truncation procedure developed in this
work treats the source of instabilities caused by round-off error propagation, and eliminates
any requirement for smoothing or filtering. The usefulness of the method is an improvement
over FDTD and compact differencing schemes in the lower memory requirements to achieve
not only useful, but optimal resolution (possible because of the spectral convergence of the
method). The usefulness is also an improvement over previous Chebyshev pseudospectral
methods especially for engineering applications, because it does not require storing and
applying the full difference matrix, which has an unfavorable time-complexity that scales
poorly with increasing grid size.
The following sections will summarize the results in the three main areas of research effort:
the numerical forward model, the Hall thruster exhaust plume and the surface dielectric
barrier discharge. Directions for future work in each specific area are also identified.
135
5.1
Numerical Forward Modeling
The forward model used here is based on one-dimensional linear wave propagation. Spe-
cific choices of what numerical treatment to use in solving the second-order wave equation
were driven by the requirement that this forward model would be used in the “inner-loop” of
a reconstruction algorithm which combines multiple measurements (in space or frequency)
into a spatial field of material properties. This means that the forward model must be accurate, fast, and low-storage. The pseudo-spectral technique achieves these requirements.
Because the scheme has spectral convergence, error proportional to ∆xnx , optimal resolution
(truncation error and round-off error approximately equal) is achievable. The grid points
per wave-length requirements of the scheme to achieve optimal resolution are lower than the
requirements to achieve “useful” resolution with other high-order difference schemes which
have been used for modeling microwave diagnostics. The time complexity of the derivative calculations is O(n log n), so scaling to large grids (multiple spatial dimensions) will
be tractable. The particular approach to calculating the derivatives in the transformedspace relies on a short recurrence, so only O(n) additional storage is required, whereas full
difference matrix approaches scale as O(n2 ) for storage and time complexity.
Since a modular approach to the reconstruction problem is taken, this forward model can
be used as part of a variety of reconstruction techniques. The two approaches demonstrated
in this research rely on inverse Abel transforms for axi-symmetric data, and a total variation
regularization for data with arbitrary spatial distribution.
5.2
Hall Thruster Exhaust Plume
The present diagnostic analysis found a statistically significant amplitude shift in mi-
crowave interferometry data at 15GHz for a thruster running on Xenon propellant. Amplitudes for microwave interferometry for these types of plasmas are not generally reported in
136
the literature. As instruments improve and get more sensitive, significant amplitude shifts
will become detectable at higher frequencies. The amplitude measurements, in conjunction
with measurements or assumptions about the species fractions of collision partners can give
indications of the electron temperature in the plasma. Since modern VNA record phase and
amplitude data automatically, it is worth looking for and reporting significant amplitude signals in future Hall Thruster diagnostic work. Including the additional model term necessary
to analyze the amplitude data is important for understanding additional plasma properties,
not necessarily for making better estimates of practical thruster performance. Including the
the collision frequency for total momentum transfer parameter (νt ) makes a correction over
the collisionless assumption of about 1% on the estimated electron number densities, and this
only in the high density regions near the thruster face for the Xenon propellant examined.
While the measurements for amplitude shift are statistically significant, and allow principled
use of a two-parameter diagnostic model, the error corrected in electron number density is
not practically significant in light of uncertainties between various diagnostic techniques.
5.3
Surface Dielectric Barrier Discharge
Figure 76 illustrates the variation in refractive index for an assumed material property
(electron density / collision frequency) as the diagnostic frequency is varied. The material
properties are chosen to give an index of refraction that results in the measured phase shift
found in the Ka-band interferometry. This constant material property line describes the
variation expected for the wire-electrode DBD device.
The area of the parameter space near the n = 1 contour results in low diagnostic response.
Increased spatial resolution coupled with increased phase response occurs as the frequency
of the diagnostic is increased.
The tape electrode device is expected to have lower electron density. Due to the curvature of the contour lines, there is an optimum frequency for which the phase response is
137
103
102
01
1.0
1.000
12–18 GHz (Ku)
26.5–40 GHz (Ka)
110–170 GHz (D)
64
1.1 525
1.
68
2.4
νe0 /ω
101
5.2
100
44
1.03
0.40 2
5
1
10−1
10−3
10−3
10−2
0.06
4
0.656
0.859
0.969
0.999
10−2
0.19
10−1
100
2
ω0 /ω 2
101
102
103
Figure 76. Real-part of refractive index (Appleton-Hartree)
the greatest. For rough estimates of the properties in a flat-tape DBD device plasma this
optimum occurs in the D-band (160GHz). As shown by the numerical forward model results,
this higher frequency should also experience lower reflections from the initial “face” of the
electron density features of the surface DBD, allowing a larger portion of the diagnostic
intensity through the plasma. An ancillary benefit of going to higher diagnostic frequencies
is the improved spatial resolution in the direction normal to the diagnostic propagation (i.e.
the beam can be focused onto a smaller spot).
With a smaller diagnostic beam-width, alternative modes of diagnostics become available.
The inter-electrode gap has been identified as an important design parameter for these surface
DBD devices. Inter-electrode gaps on the order of the beam-width (10s of millimeters) for
microwave frequencies do not sustain significant discharges. The optimum gap is found
to be on the order of 5 mm [160]. At the one to two millimeter beam-widths achievable at
millimeter wave frequencies it becomes possible to make transmission measurements through
138
0.05
ne = 1018 m−3 , ν = 1012 s−1
fopt = 159 GHz
∆φ/d, deg/mm
0.00
−0.05
−0.10
−0.15
−0.20
109
1010
1011
1012
1013
1014
f, Hz
Figure 77. Estimate of Optimal Diagnostic Frequency for DBD Plasma Actuator
the inter-electrode gap simultaneously with reflectivity measurements (the plasma density
gradient providing the primary reflecting surface rather than the buried electrode).
In conclusion, this research has shown that non-intrusive plasma diagnostics for aerospace
relevant plasmas provide an opportunity for a fruitful combination of knowledge from the
diverse fields of simulation and experiment. The numerical techniques developed in and for
the simulation communities [161] can be leveraged by the experimentalist to more accurately
model his measurements. This approach becomes even more attractive with the availability of open source scientific computing libraries that provide the supporting infrastructure
for these experimental measurement models. In the present research, the discrete cosine
transforms from the FFTW library [140] and the complex, L-stable backward difference formula time integrators from the VODE library [162] were critical enablers. The availability
of fast transforms libraries allowed model development work to focus on problem specific
pieces of modeling code, and the complex variable integrators allowed straight-forward use
139
of advanced uncertainty quantification techniques. Using the Maxima computer algebra system (CAS) allowed these problem-specific pieces of code to be generated reliably from the
governing equations. Another key enabler is the SciPy Python module, which makes calling
external subroutines in Fortran90 especially straight-forward (see Appendix B for selected
source listings), and includes the normal plotting and array handling features that modern
users have come to expect from scientific computing environments.
The research findings presented here indicate directions for future work in several areas.
The numerical models can be extended to multiple spatial dimensions to capture the effects
of finite beam size (diffraction/scattering). Quantifying error and uncertainty is a critical
task in any experimental endeavor, and the methods we use for this should improve apace
with the sophistication of our measurement models. The curse of dimensionality for large
stochastic spaces is always a consideration. Problem-specific dimensionality reductions, such
as the inverse Abel transform approach presented here, will continue to need development.
The Hall thruster diagnostics can better incorporate the effect of spatial variation in material
properties and use two-parameter models to analyze the effect of elastic scattering of electrons
in the exhaust plume. This information, when coupled with data on species fractions and
collision cross-sections can provide information on electron temperature. It will be beneficial
for surface DBD plasma diagnostics to go to the optimal diagnostic frequencies estimated
to be in the high end of the D-band (160GHz) to provide stronger empirical constraints on
the electron properties in the plasmas of these small, and increasingly useful, flow-control
devices.
140
Appendix A. Verification and Uncertainty Quantification
This appendix presents details on the numerical model verification methodology and the
uncertainty quantification methods.
A.1
Verification
The term verification is given slightly different definitions by different groups of practitioners [163]. In software engineering, the IEEE defines verification as
The process of evaluating the products of a software development phase to provide
assurance that they meet the requirements defined for them by the previous
phase.
while the definition now commonly accepted in the computational physics community is
The process of determining that a model implementation accurately represents
the developer’s conceptual description of the model and the solution to the model.
[164]
The numerical weather prediction community speaks of forecast verification, which someone
using the above quoted AIAA definition would probably consider a form of validation, and
a statistician might use the term validation in a way the computational physicist would
probably consider verification [165]. Arguing over the definitions of words [166] which, in
common use, are synonyms is contrary to progress under a pragmatic, “wrong, but useful”
conception of the modeling endeavor [167]. Rather, we should be clear on our meaning in
a specific context, and thus avoid talking past colleagues in related disciplines. Throughout
this work, the term verification is used consistent with currently accepted definitions in the
aerospace, defense and computational physics communities [164, 168–170].
141
In the present diagnostic effort, the forward model code seeks an approximate solution
to a discretized partial differential equation. This PDE is derived from Maxwell’s equations
augmented by conservation equations derived from the general hyperbolic conservation laws
through analytical simplification guided by problem-specific assumptions. The purpose of
formal verification procedures such as method of manufactured solutions (MMS) are to
demonstrate that the simulation code solves these chosen equations correctly. This is done
by showing ordered convergence of the simulated solution in a discretization parameter (such
as mesh size or time-step size).
Understanding the impact of truncation error on the quality of numerical solutions has
been a significant concern over the entire developmental history of such methods. Although
modern codes tend to use finite element, finite volume or pseudo-spectral methods as opposed
to finite differences, George Boole’s concern for establishing the credibility of numerical
results is generally applicable to all these methods. In his treatise, written in 1860, Boole
stated
...we shall very often need to use the method of Finite Differences for the purpose of shortening numerical calculation, and here the mere knowledge that the
series obtained are convergent will not suffice; we must also know the degree of
approximation.
To render our results trustworthy and useful we must find the limits of the error produced by taking a given number of terms of the expansion instead of
calculating the exact value of the function that gave rise thereto. [171]
In a related vein, Lewis Fry Richardson, under the heading Standards of Neglect in the 1927
paper which introduced the extrapolation method which now bears his name, stated in his
characteristically colorful language
An error of form which wold be negligible in a haystack would be disastrous in a
lens. Thus negligibility involves both mathematics and purpose. In this paper we
142
discuss mathematics, leaving the purposes to be discussed when they are known.
[172]
This appendix follows Richardson’s advice and confines discussion to the correctness of
mathematics, leaving the purposes and sufficiency of the proposed methods for comparison
with other diagnostic techniques in the context of intended applications.
While the development of methods for establishing the correctness and fitness of numerical approximations is certainly of historical interest, Roache describes why this effort
in code verification is more urgently important than ever before (and will only increase in
importance as simulation capabilities, and our reliance on them, grow).
In an age of spreading pseudoscience and anti-rationalism, it behooves those of us
who believe in the good of science and engineering to be above reproach whenever
possible. Public confidence is further eroded with every error we make. Although
many of society’s problems can be solved with a simple change of values, major issues such as radioactive waste disposal and environmental modeling require
technological solutions that necessarily involve computational physics. As Robert
Laughlin noted in this magazine, “there is a serious danger of this power [of simulations] being misused, either by accident or through deliberate deception.” Our
intellectual and moral traditions will be served well by conscientious attention
to verification of codes, verification of calculations, and validation, including the
attention given to building new codes or modifying existing codes with specific
features that enable these activities. [167]
There is then a compelling moral imperative underlying correctness checking, because of the
uses to which our results will be put.
The language Roache uses reflects a consensus in the computational physics community
that has given the name verification to the activity of demonstrating the impact of truncation error (usually involving grid convergence studies), and the name validation to the
activity of determining if a code has sufficient predictive capability for its intended use [164].
Boole’s idea of “trustworthy results” clearly underlies the efforts of various journals and professional societies [164, 168–170] to promote rigorous verification of computational results.
143
Richardson’s separation of the questions of correct math and fitness for purpose are reflected
in those policies as well. In addition, the extrapolation method developed by Richardson
has been generalized to support uniform reporting of verification results [173].
Two types of verification have been distinguished [148]: Code verification and calculation
verification. Code verification is done once for a particular code version, it demonstrates that
a specific implementation solves the chosen governing equations correctly. This process can
be performed on a series of grids of any size (as long as they are within the asymptotic range)
with an arbitrarily chosen solution (no need for physical realism). Calculation verification,
on the other hand, is an activity specific to a given scientific investigation, or decision
support activity. The solution in this case will be on physically meaningful grids with
physically meaningful initial condition (IC)s and boundary condition (BC)s (therefore no
a priori -known solution). Rather than monitoring the convergence of an error metric, the
convergence of solution functionals relevant to the scientific or engineering development
question at hand are tracked to ensure they demonstrate convergence (and ideally, grid/timestep independence).
The approach taken in this work to achieving verification is based on heavy use of a
CAS [174]. The pioneering work in using computer algebra for supporting the development
of computational physics codes was performed by Wirth in 1980 [175]. This was quickly
followed by other code generation efforts [176–179] demonstrations of the use of symbolic
math programs to support stability analysis [180] and correctness verification for symbolically
generated codes solving governing equations in general curvilinear body-fitted coordinate
systems [136].
The CAS handles much of the tedious and error prone manipulation required to implement a numerical PDE solver. It also makes creating the forcing terms necessary for testing
against manufactured solutions straight-forward for even very complex governing equations.
The MMS is a powerful tool for correctness checking and debugging. The parameters of the
144
manufactured solution allow the magnitude of the contribution of each term to the error
to be controlled. In this way, if a code fails to converge for a solution with all parameters
O(1) (note that this is the recommended approach, hugely different parameter values which
might obtain in a physically realistic solution can mask bugs). The parameter sizes can then
be varied in a systematic way to locate the source of the non-convergence (as convincingly
demonstrated by by Salari and Knupp with a blind test protocol [181]). This gives the code
developer a diagnostic capability for the code itself. The error analysis can be viewed as a
sort of group test [182]where the “dilution” of each term’s (member’s) contribution to the
total (group) error (response) is governed by the relative sizes of the chosen parameters.
Though we fit a parametric model (the error ansatz ) to determine rate of convergence, the
response really is expected to be a binary one as in the classic group test, the ordered convergence rate is maintained down to the round-off plateau or it is not. The dilution only
governs how high the resolution must rise (and the error must fall) for this behavior to be
confirmed. Terms with small parameters will require that convergence to very high levels is
used to ensure that an ordered error is not lurking below.
The correctness of the implementation is evaluated with the MMS [148]. To apply the
MMS, the governing equations are first put into a homogeneous form,
L(u) = 0
(96)
then a solution is chosen (umms ), and a resulting forcing function is calculated.
L(umms ) = fmms
(97)
When the MMS forcing is applied, the numerical implementation of the governing equations
should recover the chosen u in the limit of vanishing (∆t, ∆x). For this case (the governing
145
equations developed in section 3.1) the chosen solution is
u=
− cos(x − t)
νe0
E = cos(x − t)
which results in the forcing
fmms



=


0
0
sin(x−t)
νe0
(98)
(99)






(100)
For the convergence analysis both parameters (ne and νe0 ) are set equal to one.
The error ansatz for time convergence is
kumms − ui k = A (∆ti )p
(101)
kumms − ui k = B (∆xi )q
(102)
and for spatial convergence it is
where A (B) is the time (space) convergence coefficient, p (q) is the time (space) convergence
rate, and ∆ti (∆xi ) is a characteristic time (space) discretization size for the ith grid. The
use of Equation 101 and 102 to describe the convergence of the error norm requires that all
of the i grids be in the asymptotic range (i.e. the leading truncation error term is dominant).
The characteristic discretization time is simply the time-step. Since a Chebyshev-roots grid
is used for the spatial discretization, the spacing between grid points is continuously varying,
so the minimum and maximum ∆x in the grid are both used as characteristic sizes in the
convergence analysis. Figure 78 shows the convergence of the spatial derivative operator
(right-hand side of equation 59) with increasing grid resolution. The demonstrated spectral
146
10−1
10−2
10−3
10−4
min(∆x)
max(∆x)
k error k2
10−5
10−6
10−7
10−8
10−9
10−10
10−11
10−12
10−13
10−14
10−3
10−2
10−1
100
101
∆x
Figure 78. Exponential Convergence of Pseudo-spectral Discretization
convergence indicates a lack of any ordered error in the implementation. However, the effect
of the pre-recurrence truncation is exhibited in a stair-step character of the convergence plot
(since this truncation parameter changes in a discontinuous fashion with nx , see Figure 23).
The results in Figure 78 are sufficient to demonstrate code verification (the implementation is correct). However, problem specific features must be examined to demonstrate
solution verification (the solution for a particular problem is converged/reliable). As shown
in section 4.1, the present pseudospectral method achieves optimal resolution with roughly
six points per wavelength. That convergence study only took into account the resolution
requirements for the variations in electric field over a region of constant material properties.
The motivation for implementing a numerical model is to treat variations in material properties, and for some problems this can be the dominating factor for convergence. For the
surface DBD cases studied in this research that is the case. The wave itself is resolved far
sooner than the significantly smaller electron density feature. Figure 79 shows the Chebyshev approximation of a “sharp” Gaussian shaped electron density feature. The solution
is interpolated onto a uniform grid so that the unphysical variations in the under-resolved
147
1.0
nx = 32
nx = 64
0.8
nx = 128
nx = 256
ne
0.6
0.4
0.2
0.0
−0.2
−1.0
−0.5
0.0
0.5
1.0
x
Figure 79. Example of Resolution Requirement Driven by Material Property Gradients
electron density are clearly visible. These would not be apparent if the electron density at
the collocation points were simply plotted since the Chebyshev approximation interpolates
the data exactly (to within round-off). The unphysical negative electron densities caused by
under-resolution result in unstable transients in the solution. Rather than converging to a
steady periodic solution after the start-up transients are propagated out of the domain, the
solution exhibits chaotic-looking variation and sometimes divergence.
This difficulty is, in general, unavoidable. The non-dimensionalization can be based on
the wavelength of the diagnostic (as in the present implementation, see Table 1) or the
characteristic size of the electron density feature, but not both simultaneously. For cases
where the wavelength and the characteristic material property size are widely different, grid
resolution will continue to be a challenge. Thus the familiar difficulties of simulating “multiscale” physics challenges even the solution of linear wave propagation. This high-lights a
further possible difficulty, though not present in the two diagnostic problems treated in this
148
research, that of true material property discontinuities.
The basic idea of the MMS, demonstrating the design order of convergence to known
solutions, is broadly applicable to correctness checking of many numerical codes. This type
of correctness check was shown in section 3.2.1 for the numerical inverse Abel transform
method, and 4.1 for the complex-step sensitivity derivatives.
A.2
Uncertainty Quantification
Two primary uncertainty quantification methods are used in this research. The first
is a standard approach known in the statistical literature as “bootstrap” [156]. This is a
simulation approach to uncertainty quantification where samples are taken, either directly
from the distributions of the data, or from the residual distributions resulting from fitting
parametric models. The second approach is based on sensitivity derivatives calculated using
a complex-step method. While the resampling approach is tractable for models which are
inexpensive to evaluate, it becomes computationally burdensome for more costly models
such as those based on numerical PDE solvers. This is because estimates based on random
sampling only converge with the square root of the number of samples, so large numbers of
samples are usually required to achieve acceptable accuracy.
Since the inverse Abel transform is relatively fast (inexpensive to evaluate) the resampling
method is used to propagate uncertainty from the measured data to the reconstructed field.
This is done by resampling the residuals from the integration of the smoothed derivative
with replacement (bootstrap). Additionally, the measurement error at each spatial location
is assumed to be normally distributed, and the parameters of this distribution are estimated
from the repeated measurements at each measurement location. So samples from a normal distribution are added to the residuals from smoothing for each resample. Figure 80
illustrates this resampling approach to uncertainty estimation using a normally distributed
“smoothing error” with standard deviation of five, and a measurement error with standard
149
deviation of one. The resampling approach does not rely on parametric assumptions about
the distribution of the residuals, the normal distribution is used here simply for illustrative
purposes. This approach to partitioning the uncertainty is appropriate when the precision of
Figure 80. Resampling Approach to Error Estimation
the instrument is smaller than the variability caused by uncontrolled environmental factors
and intrinsic variability in the system. For instance, in the case of the Hall Thruster exhaust
plume measurement, the precision of the instrument is governed mainly by thermal noise.
The variability of the system has contributions from variation in the background pressure
and temperature of the vacuum tank, high-frequency oscillations [183] that are unresolved
by the sampling rate of the measurements, and drift of the current, potential or mass flow
around the set point.
To complete the error propagation, the reconstruction is calculated for each resampled
data set. This gives the approximate sampling distribution for the reconstructed field. Figure
81 shows the reconstructions for the resampled data from Figure 80. The dotted lines indicate
150
Figure 81. Reconstruction Uncertainty Propagation
the confidence intervals estimated from the bootstrap distribution. For this example onethousand bootstrap samples were generated, and the max/min of the reconstruction at each
location provides the limit for the interval. This illustrates the sensitivity of the reconstructed
field to measurement noise. The reconstruction is especially sensitive to noise at the origin
(r = 0).
The second method of uncertainty propagation relies on a basic transformation relation
from model inputs to model outputs. This mathematical machinery is similar to the grid
transformations in structured-grid PDE solvers. For the present case, the transformation
is between the measurement space (phase and amplitude) to the parameter space (electron
number density and collision frequency). The Jacobian of transformation is just a 2 × 2
151
matrix.



∂p
 ∂φ 

∂p
∂a
∂nx
 ∂φ
=
=
∂ne
∂a
∂a ∂φ
∂ν ∂ne

∂p
∂νe0
∂φ   ∂ne
∂νe0
∂a

∂p
∂νe0





∂p
∂ne

(103)
∂a
∂a
1
 ∂νe0 − ∂ne  



∂a ∂φ 
∂φ
∂φ
∂p
− ∂ne ∂νe0 −
∂νe0
∂ne
∂νe0
If we assume that this Jacobian is constant, then we have the local linearization which is used
in the present research for uncertainty propagation for the forward model. This linearizing
assumption means that only two forward model evaluations are needed. One evaluation for
each complex-step perturbation of the uncertain parameters.
152
Appendix B. Source Code
This appendix contains the source code for the forward model, and the various reconstruction procedures. The forward model is defined in a symbolic math language (Maxima),
which is used to generate compile-able code (Fortran90) for low-level subroutines. This code
is compiled into a shared library to be callable from a scripting language (Python). The
reconstruction routines rely on these low-level libraries and are implemented in the scripting
language.
B.1
Forward Model
The governing equations are defined symbolically in a CAS. This allows a variety of
discrete expressions (or function calls) to be substituted for the derivative operators, and
compile-able code for these expressions to be automatically generated. These automatically
generated expressions form the body of the “inner loops” of the simulation code, subroutine
boilerplate can be generated automatically [184], but is written by hand in the present
implementation.
Symbolic.
First, the dependencies between the dependent variables and the independent and curvilinear coordinates are defined. This allows the CAS’s knowledge of the chain rule to be
exploited to generate grid transformation expressions.
dep : [ E , u ] $
curv : [ x i ] $
ndep : [ x ] $
depends ( dep , f l a t t e n ( [ ndep , t ] ) ) $
depends ( ndep , curv ) $
153
t r a n s e q n s J : [ d i f f ( dep [ 1 ] , curv [ 1 ] ) , d i f f ( dep [ 1 ] , curv [ 1 ] , 2 ) ] $
tr an s un kwn J : [ d i f f ( dep [ 1 ] , ndep [ 1 ] ) , d i f f ( dep [ 1 ] , ndep [ 1 ] , 2 ) ] $
J : submatrix ( augcoefmatrix ( t r a n s e q n s J , trans unkwn J ) , 3 ) $
map(remove , dep , makelist ( dependency , i , 1 , length ( dep ) ) ) $
map(remove , ndep , makelist ( dependency , i , 1 , length ( ndep ) ) ) $
depends ( dep , f l a t t e n ( [ curv , t ] ) ) $
depends ( curv , ndep ) $
t r a n s e q n s K : [ d i f f ( dep [ 1 ] , ndep [ 1 ] ) , d i f f ( dep [ 1 ] , ndep [ 1 ] , 2 ) ] $
trans unkwn K : [ d i f f ( dep [ 1 ] , curv [ 1 ] ) , d i f f ( dep [ 1 ] , curv [ 1 ] , 2 ) ] $
K : submatrix ( augcoefmatrix ( t r a n s e q n s K , trans unkwn K ) , 3 ) $
t r a n s s u b s : f l a t t e n ( makelist (matrixmap(”=” , K, invert ( J ) ) [ i ] , i
,1 ,2) ) $
d e r i v s u b s : subst ( t r a n s s u b s , matrixmap(”=” , trans unkwn J ,
trans eqns K ) ) $
map(remove , dep , makelist ( dependency , i , 1 , length ( dep ) ) ) $
map(remove , curv , makelist ( dependency , i , 1 , length ( curv ) ) ) $
depends ( dep , f l a t t e n ( [ ndep , t ] ) ) $
depends ( ndep , curv ) $
Then the governing equations are defined, and the characteristic decomposition necessary
for defining the stable boundary conditions is calculated.
wave : d i f f (E , x , 2 ) − d i f f (E , t , 2 ) + 4∗% p i ∗ ne ∗ d i f f ( u , t ) = 0$
/∗ don ’ t t r a n s f o r m t o c o m p u t a t i o n a l c o o r d i n a t e s u n t i l t h e end
wave : s u b s t ( t r a n s s u b s , wave ) $ ∗/
momentum : d i f f ( u , t ) = − E − nu e0 ∗u$
/∗ momentum : s u b s t ( t r a n s s u b s , momentum) $ ∗/
154
/∗ n p i : f l o a t (% p i ) $ ∗/
onewave 1 : d i f f (E , t )+d i f f (E , x )=R [ 1 ] $
onewave 2 : d i f f (E , t )−d i f f (E , x )=R [ 2 ] $
unknowns : [ d i f f (E , t ) , d i f f (E , x ) , u ] $ /∗ p r i m i t i v e v a r i a b l e s ∗/
unknowns t : d i f f ( unknowns , t ) $
unknowns x : d i f f ( unknowns , x ) $
char unknowns : [R [ 1 ] , R [ 2 ] , R [ 3 ] ] $
/∗ t h e c r o s s d e r i v a t i v e term e q u a l s i t s e l f ∗/
t r i v i a l t : [ 0 , 1 , 0 , −d i f f ( d i f f (E , x ) , t ) ] $
t r i v i a l x : [ −1 , 0 , 0 , d i f f ( d i f f (E , x ) , t ) ] $
/∗ s p a t i a l o p e r a t o r ∗/
Aug x : augcoefmatrix ( [ wave , momentum ] , unknowns x ) $
Aug x : addrow( Aug x , t r i v i a l x ) $
A x : submatrix ( Aug x , 4 ) $
/∗ t e m p o r a l o p e r a t o r ∗/
Aug t : augcoefmatrix ( [ wave , momentum ] , unknowns t ) $
Aug t : addrow( Aug t , t r i v i a l t ) $
A t : submatrix ( Aug t , 4 ) $
/∗ f o r c i n g f u n c t i o n ∗/
F x : col ( Aug x , 4 ) − A t . transpose ( unknowns t ) $
F t : col ( Aug t , 4 ) − A x . transpose ( unknowns x ) $
155
check F : F x − F t$ /∗ s h o u l d be t h e same ∗/
F : F x $ /∗ e q u i v a l e n t l y ,
F : F t ∗/
/∗ e i g e n −s t r u c t u r e o f t h e J a c o b i a n ∗/
A : invert ( A t ) . A x$
AtinvF : invert ( A t ) . F$
[ A e i g v a l s , A e i g v e c t s ] : eigenvectors (A) $
Lam : diag matrix ( A e i g v a l s [ 1 ] [ 1 ] , A e i g v a l s [ 1 ] [ 2 ] ,
A eigvals [ 1 ] [ 3 ] ) $
T : addcol ( transpose ( A e i g v e c t s [ 1 ] [ 1 ] ) ) $
T : addcol (T, transpose ( A e i g v e c t s [ 2 ] [ 1 ] ) ) $
T : addcol (T, transpose ( A e i g v e c t s [ 3 ] [ 1 ] ) ) $
Tinv : invert (T) $
check A : fullratsimp (T. Lam . Tinv ) $
c h a r v a r s : Tinv . transpose ( unknowns ) $
TinvAtinvF : expand ( Tinv . AtinvF ) $
The forcing functions necessary for code verification using the MMS are generated from
the symbolically defined governing equations.
/∗ MMS f o r c i n g ∗/
s o l s : [ u=−c o s ( x−t ) / nu e0 , E=c o s ( x−t ) , p f=c o s ( x−t ) ] $ /∗ chosen
s o l u t i o n s ∗/
bc force terms : [ diff ( p f , x ,2) , diff ( diff ( p f , x) , t ) , diff ( p f , x) ,
diff ( p f , t ) ] $
156
rhs mms : subst ( s o l s , A 2inv . −F 2 ) $
rhs mms : ev ( rhs mms , nouns ) $
lhs mms : subst ( s o l s , unknowns 2 t ) $
lhs mms : ev ( lhs mms , nouns ) $
fmms : rhs mms − lhs mms $
fmms : ev ( fmms , nouns ) $
check : transpose ( lhs mms ) = rhs mms − fmms $
rhs bc1 mms : subst ( s o l s , b c 1 A i n v . −b c 1 F ) $
rhs bc1 mms : ev ( rhs bc1 mms , nouns ) $
rhs bc2 mms : subst ( s o l s , b c 2 A i n v . −b c 2 F ) $
rhs bc2 mms : ev ( rhs bc2 mms , nouns ) $
fmms bc1 : rhs bc1 mms − lhs mms $
fmms bc2 : rhs bc2 mms − lhs mms $
c h e c k b c 1 : transpose ( lhs mms ) = rhs bc1 mms − fmms $
c h e c k b c 2 : transpose ( lhs mms ) = rhs bc2 mms − fmms $
The low-order finite difference approximations are obtained by substituting the appropriate expression for the derivative terms. A Maxima script called explicit fde.mac is used
to calculate a difference expression for an arbitrary derivative using an arbitrary stencil of
points.
/∗
e x p l i c i t f i n i t e difference expressions
∗/
e x p l i c i t f d e ( s t e p s , d e r i v ) := block ( [ A, a f , b , c o e f l i s t , fde ,
f u n c v a l s , f c o e f s , f t a y l o r , M, n , p o i n t s , i , f , x , x0 , h ] ,
n : length ( s t e p s ) ,
157
f t a y l o r : taylor ( f ( x ) , x , x0 , n ) ,
p o i n t s : [ x=x0+s t e p s [ 1 ] ∗ h ] ,
for i : 2 thru n do
(
p o i n t s : append( p o i n t s , [ x=x0+s t e p s [ i ] ∗ h ] )
),
/∗ r i g h t −hand s i d e v e c t o r ( b a s e d on t h e d e r i v a t i v e r e q u e s t e d ) :
∗/
b : zeromatrix ( n , 1 ) ,
b [ d e r i v +1 ,1] : 1 ,
/∗ f u n c t i o n v a l u e s v e c t o r : ∗/
f u n c v a l s : matrix ( [ at ( f ( x ) , p o i n t s [ 1 ] ) ] ) ,
for i : 2 thru n do
(
f u n c v a l s : addcol ( f u n c v a l s , [ at ( f ( x ) , p o i n t s [ i ] ) ] )
),
/∗ c o e f f i c i e n t s l i s t ∗/
c o e f l i s t : [ at ( f ( x ) , x=x0 ) ] ,
for i : 2 thru n do
(
c o e f l i s t : append( c o e f l i s t , [ ’ at ( ’ d i f f ( f ( x ) , x , i −1) , x=x0 ) ]
)
),
f c o e f s : submatrix ( factor ( augcoefmatrix ( [ f t a y l o r ] , c o e f l i s t ) )
, n+1) ,
/∗ system m a t r i x : ∗/
158
A : addcol ( transpose ( subst ( p o i n t s [ 1 ] , f c o e f s ) ) ) ,
for i : 2 thru n do
(
A : addcol (A, transpose ( subst ( p o i n t s [ i ] , f c o e f s ) ) )
),
M : l u f a c t o r (A) ,
/∗ s o l v e f o r t h e c o e f f i c i e n t s : ∗/
a f : fullratsimp ( l u b a c k s u b (M, b ) ) ,
/∗ f i n i t e d i f f e r e n c e e x p r e s s i o n : ∗/
f d e : fullratsimp ( f u n c v a l s . a f ) ,
/∗ t r u n c a t i o n e r r o r , use t h e c o e f f i c i e n t s t o f i n d t h e l e a d i n g
truncation
e r r o r term ∗/
c o e f l i s t : append( c o e f l i s t , [ ’ at ( ’ d i f f ( f ( x ) , x , n ) , x=x0 ) ] ) ,
c o e f l i s t : append( c o e f l i s t , [ ’ at ( ’ d i f f ( f ( x ) , x , n+1) , x=x0 ) ] )
,
f t a y l o r : taylor ( f ( x ) , x , x0 , n+1) ,
f c o e f s : submatrix ( factor ( augcoefmatrix ( [ f t a y l o r ] , c o e f l i s t ) )
, n+3) ,
B : addcol ( transpose ( subst ( p o i n t s [ 1 ] , f c o e f s ) ) ) ,
for i : 2 thru n do
(
B : addcol (B, transpose ( subst ( p o i n t s [ i ] , f c o e f s ) ) )
),
l t e t : c o e f l i s t . (B. a f ) ,
return ( [ fde , l t e t , A, a f , B ] )
159
);
The pattern matching facilities of the CAS are then used to substitute these difference
expressions into the governing equations, based on the previously defined list of dependent
variables. First the pattern rules are defined.
/∗
t h e ’ e x p l i c i t f d e ’ s c r i p t r e t u r n s d i f f e r e n c e s i n f ( x0 ) i n s t e a d
of f [ i ]
∗/
matchdeclare ( [ fmatch , xmatch , nmatch ] , a l l ) ;
defrule ( f i n d e x s p a c e , f ( x0+nmatch∗h ) , f [ i+nmatch , n ] ) $
defrule ( f i n d e x s p a c e s t e a d y , f ( x0+nmatch∗h ) , f [ i+nmatch ] ) $
defrule ( f i n d e x t i m e , f ( x0+nmatch∗h ) , f [ i , n+nmatch ] ) $
dfdx : subst ( h=1, apply1 ( e x p l i c i t f d e ( [ − 1 , 0 , 1 ] , 1 ) [ 1 ] , f i n d e x s p a c e
)) $
d2fdx2 : subst ( h=1, apply1 ( e x p l i c i t f d e ( [ − 1 , 0 , 1 ] , 2 ) [ 1 ] ,
findex space ) ) $
/∗ our g r i d doesn ’ t move , so don ’ t use a time i n d e x ∗/
d f d x s t e a d y : subst ( h=1, apply1 ( e x p l i c i t f d e ( [ − 1 , 0 , 1 ] , 1 ) [ 1 ] ,
findex space steady ) ) $
d 2 f d x 2 s t e a d y : subst ( h=1, apply1 ( e x p l i c i t f d e ( [ − 1 , 0 , 1 ] , 2 ) [ 1 ] ,
findex space steady ) ) $
/∗ one−s i d e d d i f f e r e n c e s f o r t h e b o u n d a r i e s ∗/
d f d x l e f t : subst ( h=1, apply1 ( e x p l i c i t f d e ( [ 0 , 1 , 2 ] , 1 ) [ 1 ] ,
findex space ) ) $
160
d 2 f d x 2 l e f t : subst ( h=1, apply1 ( e x p l i c i t f d e ( [ 0 , 1 , 2 , 3 ] , 2 ) [ 1 ] ,
findex space ) ) $
d f d x r i g h t : subst ( h=1, apply1 ( e x p l i c i t f d e ( [ − 2 , − 1 , 0 ] , 1 ) [ 1 ] ,
findex space ) ) $
d 2 f d x 2 r i g h t : subst ( h=1, apply1 ( e x p l i c i t f d e ([ −3 , −2 , −1 ,0] ,2) [ 1 ] ,
findex space ) ) $
d f d x s t d y l f t : subst ( h=1, apply1 ( e x p l i c i t f d e ( [ 0 , 1 , 2 ] , 1 ) [ 1 ] ,
findex space steady ) ) $
d 2 f d x 2 s t d y l f t : subst ( h=1, apply1 ( e x p l i c i t f d e ( [ 0 , 1 , 2 , 3 ] , 2 ) [ 1 ] ,
findex space steady ) ) $
d f d x s t d y r g h t : subst ( h=1, apply1 ( e x p l i c i t f d e ( [ − 2 , − 1 , 0 ] , 1 ) [ 1 ] ,
findex space steady ) ) $
d 2 f d x 2 s t d y r g h t : subst ( h=1, apply1 ( e x p l i c i t f d e ([ −3 , −2 , −1 ,0] ,2)
[1] , findex space steady ) ) $
/∗ time d e r i v a t i v e s ∗/
d f d t 1 : subst ( h=dt , apply1 ( e x p l i c i t f d e ( [ − 1 , 0 ] , 1 ) [ 1 ] , f i n d e x t i m e
)) $
d f d t 2 : subst ( h=dt , apply1 ( e x p l i c i t f d e ( [ − 2 , − 1 , 0 ] , 1 ) [ 1 ] ,
findex time ) ) $
defrule ( Esubst , E , E [ i , n ] ) $
defrule ( xsubst , x , x [ i , n ] ) $
defrule ( usubst , u , u [ i , n ] ) $
defrule ( d1 , ’ d i f f ( fmatch , xi , 1 ) , subst ( f=fmatch , dfdx ) ) $
defrule ( d2 , ’ d i f f ( fmatch , xi , 2 ) , subst ( f=fmatch , d2fdx2 ) ) $
defrule ( d 1 s t e a d y , ’ d i f f ( x , xi , 1 ) , subst ( f=x , d f d x s t e a d y ) ) $
defrule ( d 2 s t e a d y , ’ d i f f ( x , xi , 2 ) , subst ( f=x , d 2 f d x 2 s t e a d y ) ) $
161
defrule ( d 1 l f t , ’ d i f f ( fmatch , xi , 1 ) , subst ( f=fmatch , d f d x l e f t ) ) $
defrule ( d 2 l f t , ’ d i f f ( fmatch , xi , 2 ) , subst ( f=fmatch , d 2 f d x 2 l e f t ) ) $
defrule ( d 1 r g h t , ’ d i f f ( fmatch , xi , 1 ) , subst ( f=fmatch , d f d x r i g h t ) ) $
defrule ( d 2 r g h t , ’ d i f f ( fmatch , xi , 2 ) , subst ( f=fmatch , d 2 f d x 2 r i g h t ) )
$
defrule ( d 1 s t d y l f t , ’ d i f f ( x , xi , 1 ) , subst ( f=x , d f d x s t d y l f t ) ) $
defrule ( d 2 s t d y l f t , ’ d i f f ( x , xi , 2 ) , subst ( f=x , d 2 f d x 2 s t d y l f t ) ) $
defrule ( d 1 s t d y r g h t , ’ d i f f ( x , xi , 1 ) , subst ( f=x , d f d x s t d y r g h t ) ) $
defrule ( d 2 s t d y r g h t , ’ d i f f ( x , xi , 2 ) , subst ( f=x , d 2 f d x 2 s t d y r g h t ) )
$
defrule ( t r h s , ’ d i f f ( fmatch , t , 1 ) , c o n c a t ( d , fmatch , dt ) [ i , n ] ) $
/∗ f i r s t o r d e r time−s t e p s ∗/
defrule ( t1 , ’ d i f f ( fmatch , t , 1 ) , subst ( f=fmatch , d f d t 1 ) ) $
defrule ( t 1 2 , ’ d i f f ( fmatch , t , 2 ) , subst ( f=c o n c a t ( d , fmatch , dt ) ,
dfdt 1 ) ) $
/∗ second o r d e r time−s t e p s ∗/
defrule ( t2 , ’ d i f f ( fmatch , t , 1 ) , subst ( f=fmatch , d f d t 2 ) ) $
defrule ( t 2 2 , ’ d i f f ( fmatch , t , 2 ) , subst ( f=c o n c a t ( d , fmatch , dt ) ,
dfdt 2 ) ) $
Then the patterns are applied to the governing equations.
f d e r h s : apply1 ( s t a b r h s , d 1 s t e a d y , d 2 s t e a d y , d1 , d2 , t r h s ,
Esubst , xsubst , u s u b s t ) $
f d e r h s b c 1 : apply1 ( s t a b r h s b c 1 ,
d 1 s t d y l f t , d 2 s t d y l f t , d 1 l f t , d 2 l f t , t r h s , Esubst , xsubst ,
usubst ) $
162
f d e r h s b c 2 : apply1 ( s t a b r h s b c 2 ,
d 1 s t d y r g h t , d 2 s t d y r g h t , d 1 r g h t , d 2 r g h t , t r h s , Esubst ,
xsubst , u s u b s t ) $
f d e l h s 1 : apply1 ( transpose ( unknowns 2 t ) , t1 , t 1 2 ) $
f d e e q n s 1 : f l a t t e n ( makelist (matrixmap(”=” , f d e l h s 1 , f d e r h s ) [ j ] ,
j ,1 ,3) ) $
gs updates 1 : linsolve ( fde eqns 1 , t i m e l e v e l n p t ) $
s o r u p d a t e s 1 : map(”=” ,map( lhs , g s u p d a t e s 1 ) ,(1 −w) ∗
t i m e l e v e l n p t + w∗map( rhs , g s u p d a t e s 1 ) ) $
f d e a u g 1 : augcoefmatrix ( f d e e q n s 1 , t i m e l e v e l n ) $
f d e l h s 2 : apply1 ( transpose ( unknowns 2 t ) , t2 , t 2 2 ) $
f d e e q n s 2 : f l a t t e n ( makelist (matrixmap(”=” , f d e l h s 2 , f d e r h s ) [ j ] ,
j ,1 ,3) ) $
gs updates 2 : linsolve ( fde eqns 2 , t i m e l e v e l n p t ) $
s o r u p d a t e s 2 : map(”=” ,map( lhs , g s u p d a t e s 2 ) ,(1 −w) ∗
t i m e l e v e l n p t + w∗map( rhs , g s u p d a t e s 2 ) ) $
f d e a u g 2 : augcoefmatrix ( f d e e q n s 2 , t i m e l e v e l n ) $
The pseudospectral code relies on outside subroutines for the derivative approximation,
so rules to replace the derivatives with temporary variables are defined and applied in this
case.
/∗ r e p l a c e d e r i v a t i v e s w i t h temp v a r i a b l e s , s i n c e t h e f i r s t and
second
d e r i v a t i v e w i l l b o t h be c a l c u l a t e d w i t h a s i n g l e f u n c t i o n c a l l ∗/
defrule ( d 1 p s p e c t , ’ d i f f ( fmatch , xi , 1 ) , c o n c a t ( d , fmatch , d x i ) ) $
defrule ( d 2 p s p e c t , ’ d i f f ( fmatch , xi , 2 ) , c o n c a t ( d2 , fmatch , d x i 2 ) ) $
163
defrule ( t r h s p s p e c t , ’ d i f f ( fmatch , t , 1 ) , c o n c a t ( d , fmatch , dt ) ) $
defrule ( b c 1 r h s p s p e c t , ’ d i f f ( ’ d i f f ( fmatch , t ) , x i ) , c o n c a t ( d2 ,
fmatch , dxi , dt ) ) $
defrule ( b c 2 r h s p s p e c t , ’ d i f f ( ’ d i f f ( fmatch , t ) , x i ) , c o n c a t ( d2 ,
fmatch , dxi , dt ) ) $
p s p e c t r h s : apply1 ( s t a b r h s , t r h s p s p e c t , d 1 p s p e c t , d 2 p s p e c t ) $
p s p e c t r h s b c 1 : apply1 ( subst ( d e r i v s u b s , apply1 ( s t a b r h s b c 1 ,
bc 1 rhs pspect ) ) , trhs pspect ) $
p s p e c t r h s b c 1 : subst ( f o r c e s u b s , apply1 ( p s p e c t r h s b c 1 ,
d1 pspect , d2 pspect ) ) $
p s p e c t r h s b c 2 : apply1 ( subst ( d e r i v s u b s , apply1 ( s t a b r h s b c 2 ,
bc 2 rhs pspect ) ) , trhs pspect ) $
p s p e c t r h s b c 2 : subst ( f o r c e s u b s , apply1 ( p s p e c t r h s b c 2 ,
d1 pspect , d2 pspect ) ) $
p s p e c t r h s b c 1 n o r m a l : apply1 ( subst ( d e r i v s u b s , apply1 (
stab rhs bc 1 normal , bc 1 rhs pspect ) ) , trhs pspect ) $
p s p e c t r h s b c 1 n o r m a l : subst ( f o r c e s u b s , apply1 (
pspect rhs bc 1 normal , d1 pspect , d2 pspect ) ) $
p s p e c t r h s b c 2 n o r m a l : apply1 ( subst ( d e r i v s u b s , apply1 (
stab rhs bc 2 normal , bc 2 rhs pspect ) ) , trhs pspect ) $
p s p e c t r h s b c 2 n o r m a l : subst ( f o r c e s u b s , apply1 (
pspect rhs bc 2 normal , d1 pspect , d2 pspect ) ) $
/∗ need t o f i n d a way t o do f o r t r a n 90 v e c t o r s l i c e s , maxima doesn
’ t l i k e : ∗/
p s p e c t p o i n t s u b s : [ dEdxi=dEdxi [ i , n ] , d2Edxi2=d2Edxi2 [ i , n ] ,
d2Edxidt=d2Edxidt [ i , n ] , dxdxi=dxdxi [ i , n ] , d2xdxi2=d2xdxi2 [ i , n ] , ne
164
=ne [ i , n ] , nu e0=nu e0 [ i , n ] , u=u [ i , n ] , dEdt=dEdt [ i , n ] , E=E [ i , n ] ] $
Numeric.
Convenience parameters for calling the FFTW [140] library from Fortran.
module c h e b y s h e v s p e c t r a l m e t h o d s
i m p l i c i t none
! t h e s e are t h e p a r a m e t e r s from ’ f f t w 3 . h ’ :
INTEGER FFTW R2HC
PARAMETER (FFTW R2HC=0)
INTEGER FFTW HC2R
PARAMETER (FFTW HC2R=1)
INTEGER FFTW DHT
PARAMETER (FFTW DHT=2)
INTEGER FFTW REDFT00
PARAMETER (FFTW REDFT00=3)
INTEGER FFTW REDFT01
PARAMETER (FFTW REDFT01=4)
INTEGER FFTW REDFT10
PARAMETER (FFTW REDFT10=5)
INTEGER FFTW REDFT11
PARAMETER (FFTW REDFT11=6)
INTEGER FFTW RODFT00
PARAMETER (FFTW RODFT00=7)
INTEGER FFTW RODFT01
PARAMETER (FFTW RODFT01=8)
165
INTEGER FFTW RODFT10
PARAMETER (FFTW RODFT10=9)
INTEGER FFTW RODFT11
PARAMETER (FFTW RODFT11=10)
INTEGER FFTW FORWARD
PARAMETER (FFTW FORWARD=−1)
INTEGER FFTWBACKWARD
PARAMETER (FFTWBACKWARD=+1)
INTEGER FFTW MEASURE
PARAMETER (FFTW MEASURE=0)
INTEGER FFTW DESTROY INPUT
PARAMETER (FFTW DESTROY INPUT=1)
INTEGER FFTW UNALIGNED
PARAMETER (FFTW UNALIGNED=2)
INTEGER FFTW CONSERVE MEMORY
PARAMETER (FFTW CONSERVE MEMORY=4)
INTEGER FFTW EXHAUSTIVE
PARAMETER (FFTW EXHAUSTIVE=8)
INTEGER FFTW PRESERVE INPUT
PARAMETER (FFTW PRESERVE INPUT=16)
INTEGER FFTW PATIENT
PARAMETER (FFTW PATIENT=32)
INTEGER FFTW ESTIMATE
PARAMETER (FFTW ESTIMATE=64)
INTEGER FFTW ESTIMATE PATIENT
PARAMETER (FFTW ESTIMATE PATIENT=128)
166
INTEGER FFTW BELIEVE PCOST
PARAMETER (FFTW BELIEVE PCOST=256)
INTEGER FFTW NO DFT R2HC
PARAMETER (FFTW NO DFT R2HC=512)
INTEGER FFTW NO NONTHREADED
PARAMETER (FFTW NO NONTHREADED=1024)
INTEGER FFTW NO BUFFERING
PARAMETER (FFTW NO BUFFERING=2048)
INTEGER FFTW NO INDIRECT OP
PARAMETER (FFTW NO INDIRECT OP=4096)
INTEGER FFTW ALLOW LARGE GENERIC
PARAMETER (FFTW ALLOW LARGE GENERIC=8192)
INTEGER FFTW NO RANK SPLITS
PARAMETER (FFTW NO RANK SPLITS=16384)
INTEGER FFTW NO VRANK SPLITS
PARAMETER (FFTW NO VRANK SPLITS=32768)
INTEGER FFTW NO VRECURSE
PARAMETER (FFTW NO VRECURSE=65536)
INTEGER FFTW NO SIMD
PARAMETER (FFTW NO SIMD=131072)
INTEGER FFTW NO SLOW
PARAMETER (FFTW NO SLOW=262144)
INTEGER FFTW NO FIXED RADIX LARGE N
PARAMETER (FFTW NO FIXED RADIX LARGE N=524288)
INTEGER FFTW ALLOW PRUNING
PARAMETER (FFTW ALLOW PRUNING=1048576)
167
INTEGER FFTW WISDOM ONLY
PARAMETER (FFTW WISDOM ONLY=2097152)
contains
This subroutine interpolates a function defined by a set of Chebyshev coefficients onto a
uniform grid. This is useful for making pretty plots of solutions.
subroutine c h e b y i n t e r p ( f , x , g , n ,m)
double precision , intent ( in ) , dimension ( n ) : : f
double precision , intent ( in ) , dimension (m) : : x
integer , intent ( in ) : : n , m
double precision , intent ( out ) , dimension (m) : : g
double precision , dimension ( n ) : : a , b
integer : : i , j
c a l l d ct ( a , f ( n : 1 : − 1 ) , n )
g = 0D0
do i =1,n
do j =1,m
i f ( i ==1)then
g ( j ) = g ( j ) + a ( i ) ∗ c o s ( d b l e ( i −1) ∗ a c o s ( x ( j ) ) ) /
d b l e (2∗ n )
else
g ( j ) = g ( j ) + a ( i ) ∗ c o s ( d b l e ( i −1) ∗ a c o s ( x ( j ) ) ) /
dble (n)
168
end i f
end do
end do
end subroutine c h e b y i n t e r p
This subroutine calculates the first and second derivatives of a function defined on an
arbitrary point distribution. This requires twice the work compared to a function defined at
the collocation points since transforms to get the grid metrics are required.
subroutine c h e b y o n e t r a n s f o r m b o t h d e r i v s s m o o t h ( dydx , d2ydx2 ,
y , x , m, n )
! use t h e d i s c r e t e c o s i n e t r a n s f o r m from f f t w t o c a l c u l a t e t h e
! f i r s t and second d e r i v a t i v e , works f o r a r b i t r a r y node
! distributions
integer , intent ( in ) : : n ,m ! number o f d a t a p o i n t s
double precision , intent ( in ) , dimension ( n ) : : y ! f u n c t i o n
values
double precision , intent ( in ) , dimension ( n ) : : x ! node v a l u e s
double precision , intent ( out ) , dimension ( n ) : : dydx ! 1 s t
derivative values
double precision , intent ( out ) , dimension ( n ) : : d2ydx2 ! 2nd
derivative values
integer : : n l o g i c a l , i ! t h e l o g i c a l s i z e o f t h e transform ,
size of
! t h e c o r r e s p o n d i n g r e a l symmetric DFT
! transforms for y
169
double precision , dimension ( n ) : : alpha ! f u n c t i o n
coefficients
double precision , dimension ( n ) : : beta ! d e r i v a t i v e
coefficients
double precision , dimension ( n ) : : gamma ! second d e r i v a t i v e
coefficients
! transforms for x
double precision , dimension ( n ) : : d e l t a ! f u n c t i o n
coefficients
double precision , dimension ( n ) : : e p s i l o n ! d e r i v a t i v e
coefficients
double precision , dimension ( n ) : : z e t a ! second d e r i v a t i v e
coefficients
! storage
double precision , dimension ( n ) : : dy , d2y , dx , d2x
! t h e l o g i c a l s i z e depends on t h e t y p e o f transform , c h e c k t h e
docs :
! h t t p : / /www. f f t w . org / doc /1d−Real 002deven −DFTs− 0028DCTs 0029
. html
n l o g i c a l = 2∗( n )
! f o r w a r d DCT:
c a l l d ct ( alpha , y , n )
c a l l d ct ( d e l t a , x , n )
! recurrence for the d e r i v a t i v e c o e f f i c i e n t s :
170
c a l l c h e b y s h e v d e r i v a t i v e r e c u r r e n c e n o s m o o t h ( alpha , beta , n )
c a l l c h e b y s h e v d e r i v a t i v e r e c u r r e n c e ( beta , gamma , n , m)
! recurrence for the transform c o e f f i c i e n t s :
call chebyshev derivative recurrence no smooth ( delta , epsilon ,
n)
c a l l c he b ys h ev d er i va t iv e r ec u rr e nc e n o s mo o th ( epsilon , zeta ,
n)
! i n v e r s e DCT:
c a l l i d c t ( dy , beta , n )
c a l l i d c t ( d2y , gamma , n )
c a l l i d c t ( dx , e p s i l o n , n )
c a l l i d c t ( d2x , z e t a , n )
! FFTW computes t h e un−n o r m a l i z e d t r a n s f o r m s , n o r m a l i z e by
logical size
dy = dy / d b l e ( n l o g i c a l )
d2y = d2y / d b l e ( n l o g i c a l )
dx = dx / d b l e ( n l o g i c a l )
d2x = d2x / d b l e ( n l o g i c a l )
dydx = dy / dx
d2ydx2 = d2y / dx ∗∗2 − d2x ∗ dy / dx ∗∗3
end subroutine c h e b y o n e t r a n s f o r m b o t h d e r i v s s m o o t h
subroutine c h e b y s h e v d e r i v a t i v e r e c u r r e n c e ( alpha , beta , n , m)
double precision , intent ( in ) , dimension ( n ) : : alpha
double precision , intent ( out ) , dimension ( n ) : : beta
integer , intent ( in ) : : n , m
171
integer : : i
! recurrence for the d e r i v a t i v e c o e f f i c i e n t s :
b e t a ( n−m: n ) = 0D0 ! t r u n c a t e t h e m h i g h e s t o r d e r c o e f f i c i e n t s
!
b e t a ( n−1) = 2D0 ∗ d b l e ( n−1) ∗ a l p h a ( n )
do i = n−m,2 , −1 ! n−1, 2 , −1
beta ( i −1) = beta ( i +1) + 2D0 ∗ d b l e ( i −1) ∗ alpha ( i )
end do
b e t a = − beta ! t h i s makes i t work , b e c a u s e t h e t r a n s f o r m s
expect
! t h e nodes i n r e v e r s e o r d e r ; t a k e c a r e o f i t a t
! t h i s l e v e l so t h e c a l l i n g f u n c t i o n s can use
nodes
! in ascending order
!
do i = 1 , n ! Lanczos sigma smoothing
b e t a ( i ) = s i n (3.141592653589793D0∗ d b l e ( i ) / d b l e ( n ) ) ∗ b e t a ( i
!
) &
!
!
/ 3.141592653589793D0 / ( d b l e ( i ) / d b l e ( n ) )
end do
end subroutine c h e b y s h e v d e r i v a t i v e r e c u r r e n c e
subroutine c h e b y s h e v d e r i v a t i v e r e c u r r e n c e n o s m o o t h ( alpha , beta
, n)
double precision , intent ( in ) , dimension ( n ) : : alpha
double precision , intent ( out ) , dimension ( n ) : : beta
172
integer , intent ( in ) : : n
integer : : i
! recurrence for the d e r i v a t i v e c o e f f i c i e n t s :
b e t a ( n ) = 0D0
b e t a ( n−1) = 2D0 ∗ d b l e ( n−1) ∗ alpha ( n )
do i = n−1, 2 , −1
beta ( i −1) = beta ( i +1) + 2D0 ∗ d b l e ( i −1) ∗ alpha ( i )
end do
b e t a = − beta ! t h i s makes i t work , b e c a u s e t h e t r a n s f o r m s
expect
! t h e nodes i n r e v e r s e o r d e r ; t a k e c a r e o f i t a t
! t h i s l e v e l so t h e c a l l i n g f u n c t i o n s can use
nodes
! in ascending order
end subroutine c h e b y s h e v d e r i v a t i v e r e c u r r e n c e n o s m o o t h
The spatial operator for the MOL formulation is defined by this Python class. Note that
the data type is complex so that the UQ method can be used. Also note that the real and
imaginary parts are transformed separately [65].
class cmplx rhs :
dtype=complex
def
init
( s e l f , n , m, x , ne , nu , d2pfdtdx , d2pfdt2 , d2pfdx2 ,
dpfdt , dpfdx , f , bcf , bclam ) :
s e l f . shape = (3∗ n , 3 ∗ n )
173
s e l f .m = m
self .x = x
s e l f . ne = ne
s e l f . nu = nu
s e l f . d2pfdtdx = d2pfdtdx
s e l f . d2pfdt2 = d2pfdt2
s e l f . d2pfdx2 = d2pfdx2
s e l f . dpfdt = dpfdt
s e l f . dpfdx = dpfdx
s e l f . f = f # f o r c i n g f o r mms v e r i f i c a t i o n
s e l f . bcf = bcf
s e l f . bclam = bclam
s e l f . dxdxi , s e l f . d2xdxi2 = (
oneD fort . chebyshev spectral methods . d c t d i f f 2 (
x) )
s e l f . dEdxi = sp . z e r o s ( n , dtype=complex , o r d e r= ’F ’ )
s e l f . d2Edxi2 = sp . z e r o s ( n , dtype=complex , o r d e r= ’F ’ )
s e l f . d2Edxidt = sp . z e r o s ( n , dtype=complex , o r d e r= ’F ’ )
def matvec ( s e l f , v ) :
u , E , dEdt = unpack unknown ( v )
# t r a n s f o r m r e a l and imaginary p a r t s s e p a r a t e l y
s e l f . dEdxi . r e a l , s e l f . d2Edxi2 . r e a l = o n e D f o r t .
c h e b y s h e v s p e c t r a l m e t h o d s . d c t d i f f 2 s m o o t h (E . r e a l ,
s e l f .m)
s e l f . dEdxi . imag , s e l f . d2Edxi2 . imag = o n e D f o r t .
c h e b y s h e v s p e c t r a l m e t h o d s . d c t d i f f 2 s m o o t h (E . imag ,
174
s e l f .m)
s e l f . d2Edxidt . r e a l = o n e D f o r t . c h e b y s h e v s p e c t r a l m e t h o d s .
d c t d i f f n o s m o o t h ( dEdt . r e a l )
s e l f . d2Edxidt . imag = o n e D f o r t . c h e b y s h e v s p e c t r a l m e t h o d s .
d c t d i f f n o s m o o t h ( dEdt . imag )
m v r e s u l t = o n e D f o r t . oned . c o m p l e x r h s (
s e l f . dEdxi , s e l f . d2Edxi2 , dEdt , s e l f . d2Edxidt , E , u , s e l f . nu
, s e l f . ne , s e l f . dxdxi , s e l f . d2xdxi2 , s e l f . d2pfdtdx , s e l f
. d2pfdt2 , s e l f . d2pfdx2 , s e l f . dpfdt , s e l f . dpfdx , s e l f . f )
return ( m v r e s u l t )
def r a t e s ( s e l f , t , v ) :
s e l f . d2pfdtdx , s e l f . d2pfdt2 , s e l f . d2pfdx2 , s e l f . dpfdt , s e l f .
dpfdx = s e l f . b c f ( s e l f . x , t , s e l f . bclam )
return ( s e l f . matvec ( v ) )
The stability of the method of lines formulation was explored by using this class.
class eig stab op :
dtype=f l o a t
def
init
( s e l f , n , x , ne , nu , d2pfdtdx , d2pfdt2 , d2pfdx2 , dpfdt
, dpfdx ) :
s e l f . shape = (3∗ n , 3 ∗ n )
self .x = x
s e l f . ne = ne
s e l f . nu = nu
s e l f . d2pfdtdx = d2pfdtdx
s e l f . d2pfdt2 = d2pfdt2
175
s e l f . d2pfdx2 = d2pfdx2
s e l f . dpfdt = dpfdt
s e l f . dpfdx = dpfdx
s e l f . dxdxi , s e l f . d2xdxi2 = (
oneD fort . chebyshev spectral methods . d c t d i f f 2 (
x) )
def matvec ( s e l f , v ) :
u , E , dEdt = unpack unknown ( v )
dEdxi , d2Edxi2 = o n e D f o r t . c h e b y s h e v s p e c t r a l m e t h o d s .
d c t d i f f 2 (E)
d2Edxidt = o n e D f o r t . c h e b y s h e v s p e c t r a l m e t h o d s .
d c t d i f f n o s m o o t h ( dEdt )
m v r e s u l t = o n e D f o r t . oned . e i g s t a b (
dEdxi , d2Edxi2 , dEdt , d2Edxidt , E , u , s e l f . nu , s e l f . ne , s e l f .
dxdxi , s e l f . d2xdxi2 , s e l f . d2pfdtdx , s e l f . d2pfdt2 , s e l f .
d2pfdx2 , s e l f . dpfdt , s e l f . dpfdx )
return ( m v r e s u l t )
class eig stab smooth :
dtype=f l o a t
def
init
( s e l f , n , m, x , ne , nu , d2pfdtdx , d2pfdt2 , d2pfdx2 ,
dpfdt , dpfdx ) :
s e l f . shape = (3∗ n , 3 ∗ n )
s e l f .m = m
self .x = x
s e l f . ne = ne
176
s e l f . nu = nu
s e l f . d2pfdtdx = d2pfdtdx
s e l f . d2pfdt2 = d2pfdt2
s e l f . d2pfdx2 = d2pfdx2
s e l f . dpfdt = dpfdt
s e l f . dpfdx = dpfdx
s e l f . dxdxi , s e l f . d2xdxi2 = (
oneD fort . chebyshev spectral methods . d c t d i f f 2 (
x) )
def matvec ( s e l f , v ) :
u , E , dEdt = unpack unknown ( v )
dEdxi , d2Edxi2 = o n e D f o r t . c h e b y s h e v s p e c t r a l m e t h o d s .
d c t d i f f 2 s m o o t h (E , s e l f .m)
d2Edxidt = o n e D f o r t . c h e b y s h e v s p e c t r a l m e t h o d s .
d c t d i f f n o s m o o t h ( dEdt )
m v r e s u l t = o n e D f o r t . oned . e i g s t a b (
dEdxi , d2Edxi2 , dEdt , d2Edxidt , E , u , s e l f . nu , s e l f . ne , s e l f .
dxdxi , s e l f . d2xdxi2 , s e l f . d2pfdtdx , s e l f . d2pfdt2 , s e l f .
d2pfdx2 , s e l f . dpfdt , s e l f . dpfdx )
return ( m v r e s u l t )
The classes defined above rely on these Fortran subroutines.
module oneD
use c h e b y s h e v s p e c t r a l m e t h o d s
! One−d i m e n s i o n a l f o r w a r d model f o r l i n e a r wave p r o p a g a t i o n i n a
! c o l l i s i o n a l plasma
177
i m p l i c i t none
contains
subroutine e i g s t a b ( mol op , dEdxi , d2Edxi2 , dEdt , d2Edxidt , E , u , nu , ne ,
dxdxi , d2xdxi2 , d2pfdtdx , d2pfdt2 , d2pfdx2 , dpfdt , dpfdx , nx )
double precision , intent ( in ) , dimension ( nx ) : : dEdxi , d2Edxi2 ,
dEdt , d2Edxidt , E , u , nu , ne , dxdxi , d2xdxi2
double precision , intent ( in ) , dimension ( 2 ) : : d2pfdtdx , d2pfdt2 ,
d2pfdx2 , dpfdt , dpfdx
integer , intent ( in ) : : nx
double precision , intent ( out ) , dimension (3∗ nx ) : : mol op
!
d o u b l e p r e c i s i o n , dimension ( nx ) : : dEdx , d2Edx2
integer : : i
double precision : : np i = 3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3D0
!
c a l l d c t d e r i v ( dEdx , d2Edx2 , E, x , nx )
! interior points
mol op ( 1 : nx ) = 4D0∗ n pi ∗ ne∗(−E−nu∗u )−d2xdxi2 ∗ dEdxi / dxdxi ∗∗3+
d2Edxi2 / dxdxi ∗∗2
mol op ( nx +1:2∗ nx ) = dEdt
mol op (2∗ nx +1:3∗ nx ) = −E−nu∗u
! boundaries
mol op ( 1 ) = −4D0∗ ne ( 1 ) ∗E( 1 ) ∗ npi −4D0∗ ne ( 1 ) ∗nu ( 1 ) ∗u ( 1 ) ∗ n pi+
d2Edxidt ( 1 ) /&
&dxdxi ( 1 )+d2pfdx2 ( 1 )−d2pfdtdx ( 1 )
178
mol op ( nx+1) = dEdxi ( 1 ) / dxdxi ( 1 )−dpfdx ( 1 )+d p f d t ( 1 )
mol op (2∗ nx+1) = −E( 1 )−nu ( 1 ) ∗u ( 1 )
mol op ( nx ) = −4D0∗ n pi ∗ ne ( nx ) ∗E( nx )−d2Edxidt ( nx ) / dxdxi ( nx )−4D0∗
n p i ∗ ne(&
&nx ) ∗nu ( nx ) ∗u ( nx )+d2pfdx2 ( 2 )+d2pfdtdx ( 2 )
mol op (2∗ nx ) = −dEdxi ( nx ) / dxdxi ( nx )+dpfdx ( 2 )+d p f d t ( 2 )
mol op (3∗ nx ) = −E( nx )−nu ( nx ) ∗u ( nx )
! psubsts
! mol op ( 1 ) = d 2 E d x i d t ( 1 ) / d x d x i ( 1 )
! mol op ( nx+1) = dEdxi ( 1 ) / d x d x i ( 1 )−d p f d x ( 1 )+d p f d t ( 1 )
! mol op (2∗ nx+1) = −E( 1 )−nu ( 1 ) ∗u ( 1 )
! mol op ( nx ) = −d 2 E d x i d t ( nx ) / d x d x i ( nx )
! mol op (2∗ nx ) = −dEdxi ( nx ) / d x d x i ( nx )+d p f d x ( 2 )+d p f d t ( 2 )
! mol op (3∗ nx ) = −E( nx )−nu ( nx ) ∗u ( nx )
end subroutine e i g s t a b
subroutine c o m p l e x r h s ( mol op , dEdxi , d2Edxi2 , dEdt , d2Edxidt , E , u , nu ,
ne , dxdxi , d2xdxi2 , d2pfdtdx , d2pfdt2 , d2pfdx2 , dpfdt , dpfdx , f , nx )
double complex , intent ( in ) , dimension ( nx ) : : dEdxi , d2Edxi2 ,
dEdt , d2Edxidt , E , u , nu , ne , dxdxi , d2xdxi2
double complex , intent ( in ) , dimension ( 2 ) : : d2pfdtdx , d2pfdt2 ,
d2pfdx2 , dpfdt , dpfdx
double complex , intent ( in ) , dimension ( 3 , nx ) : : f
integer , intent ( in ) : : nx
179
double complex , intent ( out ) , dimension (3∗ nx ) : : mol op
!
d o u b l e p r e c i s i o n , dimension ( nx ) : : dEdx , d2Edx2
integer : : i
double precision : : np i = 3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3D0
mol op ( 1 : nx ) = 4D0∗ ne ∗ n p i ∗(−E−nu∗u )−d2xdxi2 ∗ dEdxi / dxdxi ∗∗3+
d2Edxi2 / dxdxi∗&
&∗2− f ( 1 , : )
mol op ( nx +1:2∗ nx ) = dEdt−f ( 2 , : )
mol op (2∗ nx +1:3∗ nx ) = −E−nu∗u−f ( 3 , : )
mol op ( 1 ) = −4D0∗ ne ( 1 ) ∗E( 1 ) ∗ npi −4D0∗ ne ( 1 ) ∗nu ( 1 ) ∗u ( 1 ) ∗ n pi+
d2Edxidt ( 1 ) /&
&dxdxi ( 1 )−f ( 1 , 1 )+d2pfdx2 ( 1 )−d2pfdtdx ( 1 )
mol op ( nx+1) = −f ( 2 , 1 )+dEdxi ( 1 ) / dxdxi ( 1 )−dpfdx ( 1 )+d p f d t ( 1 )
mol op (2∗ nx+1) = −f ( 3 , 1 )−E( 1 )−nu ( 1 ) ∗u ( 1 )
mol op ( nx ) = −4D0∗ n pi ∗ ne ( nx ) ∗E( nx )−d2Edxidt ( nx ) / dxdxi ( nx )−4D0∗
n p i ∗ ne(&
&nx ) ∗nu ( nx ) ∗u ( nx )+d2pfdx2 ( 2 )+d2pfdtdx ( 2 )−f ( 1 , nx )
mol op (2∗ nx ) = −dEdxi ( nx ) / dxdxi ( nx )−f ( 2 , nx )+dpfdx ( 2 )+d p f d t ( 2 )
mol op (3∗ nx ) = −E( nx )−nu ( nx ) ∗u ( nx )−f ( 3 , nx )
end subroutine c o m p l e x r h s
These subroutines for doing the forward and inverse DCT are just thin wrappers around
the special case for real even FFTs provided by FFTW.
subroutine d ct ( out , in , n )
180
integer , intent ( in ) : : n
double precision , intent ( in ) , dimension ( n ) : : in
double precision , intent ( out ) , dimension ( n ) : : out
integer : : plan1
! f o r w a r d DCT:
c a l l d f f t w p l a n r 2 r 1 d ( plan1 , n , in , out , FFTW REDFT10,
FFTW ESTIMATE)
c a l l d f f t w e x e c u t e r 2 r ( plan1 , in , out )
c a l l d f f t w d e s t r o y p l a n ( plan1 )
end subroutine d ct
subroutine i d c t ( out , in , n )
integer , intent ( in ) : : n
double precision , intent ( in ) , dimension ( n ) : : in
double precision , intent ( out ) , dimension ( n ) : : out
integer : : plan2
! i n v e r s e DCT:
c a l l d f f t w p l a n r 2 r 1 d ( plan2 , n , in , out , FFTW REDFT01,
FFTW ESTIMATE)
c a l l d f f t w e x e c u t e r 2 r ( plan2 , in , out )
c a l l d f f t w d e s t r o y p l a n ( plan2 )
end subroutine i d c t
181
This subroutine actually calculates the first and second derivatives and the grid transformation metrics.
subroutine c h e b y o n e t r a n s f o r m b o t h d e r i v s s m o o t h ( dydx , d2ydx2 ,
y , x , m, n )
! use t h e d i s c r e t e c o s i n e t r a n s f o r m from f f t w t o c a l c u l a t e t h e
! f i r s t and second d e r i v a t i v e , works f o r a r b i t r a r y node
! distributions
integer , intent ( in ) : : n ,m ! number o f d a t a p o i n t s
double precision , intent ( in ) , dimension ( n ) : : y ! f u n c t i o n
values
double precision , intent ( in ) , dimension ( n ) : : x ! node v a l u e s
double precision , intent ( out ) , dimension ( n ) : : dydx ! 1 s t
derivative values
double precision , intent ( out ) , dimension ( n ) : : d2ydx2 ! 2nd
derivative values
integer : : n l o g i c a l , i ! t h e l o g i c a l s i z e o f t h e transform ,
size of
! t h e c o r r e s p o n d i n g r e a l symmetric DFT
! transforms for y
double precision , dimension ( n ) : : alpha ! f u n c t i o n
coefficients
double precision , dimension ( n ) : : beta ! d e r i v a t i v e
coefficients
double precision , dimension ( n ) : : gamma ! second d e r i v a t i v e
coefficients
182
! transforms for x
double precision , dimension ( n ) : : d e l t a ! f u n c t i o n
coefficients
double precision , dimension ( n ) : : e p s i l o n ! d e r i v a t i v e
coefficients
double precision , dimension ( n ) : : z e t a ! second d e r i v a t i v e
coefficients
! storage
double precision , dimension ( n ) : : dy , d2y , dx , d2x
! t h e l o g i c a l s i z e depends on t h e t y p e o f transform , c h e c k t h e
docs :
! h t t p : / /www. f f t w . org / doc /1d−Real 002deven −DFTs− 0028DCTs 0029
. html
n l o g i c a l = 2∗( n )
! f o r w a r d DCT:
c a l l d ct ( alpha , y , n )
c a l l d ct ( d e l t a , x , n )
! recurrence for the d e r i v a t i v e c o e f f i c i e n t s :
c a l l c h e b y s h e v d e r i v a t i v e r e c u r r e n c e n o s m o o t h ( alpha , beta , n )
c a l l c h e b y s h e v d e r i v a t i v e r e c u r r e n c e ( beta , gamma , n , m)
! recurrence for the transform c o e f f i c i e n t s :
call chebyshev derivative recurrence no smooth ( delta , epsilon ,
n)
183
c a l l c he b ys h ev d er i va t iv e r ec u rr e nc e n o s mo o th ( epsilon , zeta ,
n)
! i n v e r s e DCT:
c a l l i d c t ( dy , beta , n )
c a l l i d c t ( d2y , gamma , n )
c a l l i d c t ( dx , e p s i l o n , n )
c a l l i d c t ( d2x , z e t a , n )
! FFTW computes t h e un−n o r m a l i z e d t r a n s f o r m s , n o r m a l i z e by
logical size
dy = dy / d b l e ( n l o g i c a l )
d2y = d2y / d b l e ( n l o g i c a l )
dx = dx / d b l e ( n l o g i c a l )
d2x = d2x / d b l e ( n l o g i c a l )
dydx = dy / dx
d2ydx2 = d2y / dx ∗∗2 − d2x ∗ dy / dx ∗∗3
end subroutine c h e b y o n e t r a n s f o r m b o t h d e r i v s s m o o t h
B.2
Inverse Abel Transform
This Python function calculates the inverse Abel transform of a function given its derivative using the method described in section 3.2.1.
def a b e l f f t i n t ( dfdx , x ) :
nx = l e n ( x )
i n t e g r a l = sp . z e r o s ( nx , dtype=f l o a t )
dx = x [1] − x [ 0 ]
for i in xrange ( nx−1) :
184
x2 = sp . append (# add x s t o g e t b e t t e r e x t r a p o l a t i o n s
x [ i : nx ] , dx∗ sp . l i n s p a c e ( 1 , i , i ) + x [ nx −1])
d i v i s o r = sp . s q r t ( x2 ∗∗2 − x [ i ] ∗ ∗ 2 )
L = x2 [ nx −1] − x2 [ 0 ]
dddx = i f f t ( f f t ( sp . append ( d i v i s o r , sp . z e r o s ( nx ) ) )
∗ f f t ( p e r i o d i c g a u s s i a n d e r i v ( sp . l i n s p a c e ( 0 , 2 ∗
L , 2 ∗ ( nx ) ) , 0 . 2 5 ∗ dx ) ) ) [ 0 : nx−i ] . r e a l
# e x t r a p o l a t e towards the s i n g u l a r i t y :
d i v i s o r [ 0 ] = d i v i s o r [ 1 ] − dx∗dddx [ 1 ]
i n t e g r a n d = dfdx [ i : nx ] / d i v i s o r [ 0 : nx−i ]
L = x [ nx −1] − x [ i ]
a n t i d e r i v = f f t i n t ( i n t e g r a n d , x [ i : nx ] , L)
na = a n t i d e r i v . shape [ 0 ]
i n t e g r a l [ i ] = − ( a n t i d e r i v [ na −1] − a n t i d e r i v [ 0 ] ) / sp . p i
# s o l v e f o r i n t e g r a l a t r=0 u s i n g 3 rd o r d e r z e r o s l o p e
estimate
#i n t e g r a l [ 0 ] = (2∗ i n t e g r a l [3] −9∗ i n t e g r a l [2]+18∗ i n t e g r a l [ 1 ] )
/11.0
# 4 th order
i n t e g r a l [ 0 ] = −(3∗ i n t e g r a l [4] −16∗ i n t e g r a l [ 3 ] + 3 6 ∗ i n t e g r a l
[2] −48∗ i n t e g r a l [ 1 ]
) /25.0
return ( i n t e g r a l )
The derivative can be calculated in a variety of ways from the function’s point data. The
convolution smoothing method described in section 3.2.1 is used in the present work. This
185
requires one line of Python, e.g.
cyl smooth deriv dx = i f f t ( f f t ( cyl ) ∗ f f t ( periodic gaussian deriv (
x , 1 . 0 ∗ dx ) ) ) . r e a l
B.3
Total Variation Regularization
This simple Python function calculates the total variation of a function defined on a
one-dimensional grid using the same Chebyshev pseudo-spectral derivative routines as the
forward model, and the trapezoidal integration rule.
def t o t a l v a r i a t i o n ( ne , x ) :
# c h e b y o n e t r a n s f o r m b o t h d e r i v s ( dydx , d2ydx2 , y , x , n )
dnedx , d2nedx2 = o n e D f o r t . c h e b y s h e v s p e c t r a l m e t h o d s .
c h e b y o n e t r a n s f o r m b o t h d e r i v s ( ne , x )
tv = sp . t r a p z ( abs ( dnedx ) , x )
186
Appendix C. Data
This appendix is an archive of the Hall thruster and DBD phase and amplitude data.
Hall Thruster
0
0.010
−2
0.005
−4
−6
−8
−10
−200
∆mag (u)
∆φ (deg)
C.1
S12
S12
S21
S21
−150
−100
0.000
S12
S12
S21
S21
−0.005
−0.010
−50
0
50
−0.015
−200
100
x (mm)
−150
−100
−50
0
50
100
x (mm)
Figure 82. Xenon fuel, -225 mm
0
0.010
0.005
−4
−6
−8
−10
−200
∆mag (u)
∆φ (deg)
−2
S12
S12
S21
S21
−150
−100
0.000
−0.005
S12
S12
S21
S21
−0.010
−0.015
−0.020
−0.025
−50
0
50
100
x (mm)
−0.030
−200
−150
Figure 83. Krypton fuel, -225 mm
187
−100
−50
x (mm)
0
50
100
2
0.006
0.004
0.002
∆mag (u)
∆φ (deg)
0
−2
−4
−6
−8
−10
−200
S12
S12
S21
S21
−150
−100
0.000
−0.002
S12
S12
S21
S21
−0.004
−0.006
−0.008
−0.010
−50
0
50
−0.012
−200
100
x (mm)
−150
−100
−50
0
50
100
x (mm)
0
0.020
−1
0.015
−2
0.010
−3
−4
−5
−6
−7
−8
−200
∆mag (u)
∆φ (deg)
Figure 84. Xenon fuel, -215 mm
S12
S12
S21
S21
−150
−100
S12
S12
S21
S21
0.005
0.000
−0.005
−0.010
−0.015
−50
0
50
−0.020
−200
100
x (mm)
−150
−100
−50
0
50
100
x (mm)
0
0.010
−1
0.005
−2
−3
−4
−5
−6
−7
−200
∆mag (u)
∆φ (deg)
Figure 85. Krypton fuel, -215 mm
S12
S12
S21
S21
−150
−100
S12
S12
S21
S21
0.000
−0.005
−0.010
−50
0
50
100
x (mm)
−0.015
−200
−150
Figure 86. Xenon fuel, -205 mm
188
−100
−50
x (mm)
0
50
100
0.004
0
0.002
−1
∆mag (u)
∆φ (deg)
1
−2
−3
−4
−5
−6
−7
−8
−200
S12
S12
S21
S21
−150
−100
S12
S12
S21
S21
0.000
−0.002
−0.004
−0.006
−0.008
−0.010
−50
0
50
−0.012
−200
100
x (mm)
−150
−100
−50
0
50
100
x (mm)
0
0.006
−1
0.004
−2
0.002
−3
−4
−5
−6
−7
−200
∆mag (u)
∆φ (deg)
Figure 87. Krypton fuel, -205 mm
S12
S12
S21
S21
−150
−100
S12
S12
S21
S21
0.000
−0.002
−0.004
−0.006
−50
0
50
−0.008
−200
100
x (mm)
−150
−100
−50
0
50
100
−50
0
50
100
x (mm)
1
0.020
0
0.015
−1
0.010
−2
−3
−4
−5
−6
−7
−200
∆mag (u)
∆φ (deg)
Figure 88. Xenon fuel, -195 mm
S12
S12
S21
S21
−150
−100
0.005
0.000
S12
S12
S21
S21
−0.005
−0.010
−0.015
−50
0
50
100
x (mm)
−0.020
−200
−150
Figure 89. Krypton fuel, -195 mm
189
−100
x (mm)
2
0.010
S12
S12
S21
S21
∆φ (deg)
0
−1
−2
−3
0.005
∆mag (u)
1
−4
−5
S12
S12
S21
S21
−0.005
−0.010
−0.015
−6
−7
−200
0.000
−150
−100
−50
0
50
−0.020
−200
100
x (mm)
−150
−100
−50
0
50
100
x (mm)
0
0.010
−1
0.005
−2
−3
−4
−5
−6
−7
−200
∆mag (u)
∆φ (deg)
Figure 90. Xenon fuel, -185 mm
S12
S12
S21
S21
−150
−100
0.000
S12
S12
S21
S21
−0.005
−0.010
−50
0
50
−0.015
−200
100
x (mm)
−150
−100
−50
0
50
100
−50
0
50
100
x (mm)
Figure 91. Krypton fuel, -185 mm
1
0.010
0.005
−1
−2
−3
−4
−5
−6
−200
∆mag (u)
∆φ (deg)
0
S12
S12
S21
S21
−150
−100
0.000
S12
S12
S21
S21
−0.005
−0.010
−50
0
50
100
x (mm)
−0.015
−200
−150
Figure 92. Xenon fuel, -175 mm
190
−100
x (mm)
0.015
−1
0.010
−2
0.005
−3
−4
−5
−6
−7
−200
∆mag (u)
∆φ (deg)
0
S12
S12
S21
S21
−150
−100
S12
S12
S21
S21
0.000
−0.005
−0.010
−0.015
−50
0
50
−0.020
−200
100
x (mm)
−150
−100
−50
0
50
100
−50
0
50
100
−50
0
50
100
x (mm)
Figure 93. Krypton fuel, -175 mm
2
0.010
S12
S12
S21
S21
−2
0.005
∆mag (u)
∆φ (deg)
0
−4
−6
−8
−200
0.000
S12
S12
S21
S21
−0.005
−0.010
−150
−100
−50
0
50
−0.015
−200
100
x (mm)
−150
−100
x (mm)
Figure 94. Xenon fuel, -165 mm
1
0.005
0.000
−1
−2
−3
−4
−5
−6
−200
∆mag (u)
∆φ (deg)
0
S12
S12
S21
S21
−150
−100
−0.005
−0.010
−0.015
−50
0
50
100
x (mm)
−0.020
−200
S12
S12
S21
S21
−150
Figure 95. Krypton fuel, -165 mm
191
−100
x (mm)
0.002
−1
0.000
∆mag (u)
∆φ (deg)
0
−2
−3
−4
−5
−6
−200
S12
S12
S21
S21
−150
−100
S12
S12
S21
S21
−0.002
−0.004
−0.006
−0.008
−0.010
−50
0
50
−0.012
−200
100
x (mm)
−150
−100
−50
0
50
100
x (mm)
0
0.015
−1
0.010
∆mag (u)
∆φ (deg)
Figure 96. Xenon fuel, -155 mm
−2
−3
−4
−5
−6
−200
S12
S12
S21
S21
−150
−100
S12
S12
S21
S21
0.005
0.000
−0.005
−0.010
−0.015
−50
0
50
−0.020
−200
100
x (mm)
−150
−100
−50
0
50
100
0
50
100
x (mm)
0.0
0.004
−0.5
0.002
−1.0
∆mag (u)
∆φ (deg)
Figure 97. Krypton fuel, -155 mm
−1.5
−2.0
−2.5
−3.0
−3.5
−4.0
−4.5
−200
S12
S12
S21
S21
−150
−100
0.000
−0.002
S12
S12
S21
S21
−0.004
−0.006
−0.008
−0.010
−50
0
50
100
x (mm)
−0.012
−200
−150
Figure 98. Xenon fuel, -145 mm
192
−100
−50
x (mm)
1
−1
−2
−3
0.000
−0.005
−0.010
−4
−5
−200
S12
S12
S21
S21
0.005
∆mag (u)
∆φ (deg)
0
0.010
S12
S12
S21
S21
−150
−100
−50
0
50
−0.015
−200
100
x (mm)
−150
−100
−50
0
50
100
x (mm)
Figure 99. Krypton fuel, -145 mm
2
0.010
S12
S12
S21
S21
1
0.005
−1
−2
−3
−4
−5
−6
−200
∆mag (u)
∆φ (deg)
0
S12
S12
S21
S21
−150
−100
0.000
−0.005
−0.010
−50
0
50
−0.015
−200
100
x (mm)
−150
−100
−50
0
50
100
x (mm)
Figure 100. Xenon fuel, -135 mm
−0.5
0.004
0.002
−1.5
∆mag (u)
∆φ (deg)
−1.0
−2.0
−2.5
−3.0
−3.5
−4.0
−4.5
−5.0
−200
S12
S12
S21
S21
−150
−100
0.000
−0.002
S12
S12
S21
S21
−0.004
−0.006
−0.008
−50
0
50
100
x (mm)
−0.010
−200
−150
Figure 101. Krypton fuel, -135 mm
193
−100
−50
x (mm)
0
50
100
0.015
−1
0.010
∆mag (u)
∆φ (deg)
0
−2
−3
−4
−5
−6
−200
S12
S12
S21
S21
−150
−100
S12
S12
S21
S21
0.005
0.000
−0.005
−0.010
−50
0
50
−0.015
−200
100
x (mm)
−150
−100
−50
0
50
100
x (mm)
Figure 102. Xenon fuel, -125 mm
0
0.006
S12
S12
S21
S21
−2
−3
−4
−5
−200
S12
S12
S21
S21
0.004
∆mag (u)
∆φ (deg)
−1
0.002
0.000
−0.002
−0.004
−0.006
−0.008
−150
−100
−50
0
50
−0.010
−200
100
x (mm)
−150
−100
−50
0
50
100
x (mm)
Figure 103. Krypton fuel, -125 mm
−0.5
0.002
∆mag (u)
∆φ (deg)
−1.5
−2.0
−2.5
−3.0
−3.5
−200
S12
S12
S21
S21
0.000
−1.0
S12
S12
S21
S21
−150
−100
−0.002
−0.004
−0.006
−0.008
−0.010
−0.012
−50
0
50
100
x (mm)
−0.014
−200
−150
Figure 104. Xenon fuel, -115 mm
194
−100
−50
x (mm)
0
50
100
−1.0
0.006
−2.0
0.002
0.004
∆mag (u)
∆φ (deg)
−1.5
−2.5
−3.0
−3.5
−4.0
−4.5
−5.0
−5.5
−200
S12
S12
S21
S21
−150
−100
0.000
−0.002
S12
S12
S21
S21
−0.004
−0.006
−0.008
−0.010
−50
0
50
−0.012
−200
100
x (mm)
−150
−100
−50
0
50
100
−50
0
50
100
−50
0
50
100
x (mm)
0.0
0.004
−0.5
0.002
−1.0
∆mag (u)
∆φ (deg)
Figure 105. Krypton fuel, -115 mm
−1.5
−2.0
−2.5
−3.0
−3.5
−4.0
−4.5
−200
S12
S12
S21
S21
−150
−100
0.000
S12
S12
S21
S21
−0.002
−0.004
−0.006
−50
0
50
−0.008
−200
100
x (mm)
−150
−100
x (mm)
Figure 106. Xenon fuel, -105 mm
−1.0
0.015
0.010
−2.0
−2.5
−3.0
−3.5
−4.0
−4.5
−200
∆mag (u)
∆φ (deg)
−1.5
S12
S12
S21
S21
−150
−100
0.005
S12
S12
S21
S21
0.000
−0.005
−50
0
50
100
x (mm)
−0.010
−200
−150
Figure 107. Krypton fuel, -105 mm
195
−100
x (mm)
2
0.004
S12
S12
S21
S21
0
−1
−2
−3
−4
−200
S12
S12
S21
S21
0.002
∆mag (u)
∆φ (deg)
1
0.000
−0.002
−0.004
−0.006
−0.008
−0.010
−150
−100
−50
0
50
−0.012
−200
100
x (mm)
−150
−100
−50
0
50
100
x (mm)
Figure 108. Xenon fuel, -95 mm
0
0.008
∆mag (u)
∆φ (deg)
−2
−3
−4
−5
−6
−200
S12
S12
S21
S21
0.006
−1
S12
S12
S21
S21
−150
−100
0.004
0.002
0.000
−0.002
−0.004
−0.006
−50
0
50
−0.008
−200
100
x (mm)
−150
−100
−50
0
50
100
x (mm)
Figure 109. Krypton fuel, -95 mm
6
0.005
S12
S12
S21
S21
2
0
0.000
∆mag (u)
∆φ (deg)
4
−2
S12
S12
S21
S21
−0.010
−0.015
−4
−6
−200
−0.005
−150
−100
−50
0
50
100
x (mm)
−0.020
−200
−150
Figure 110. Xenon fuel, -85 mm
196
−100
−50
x (mm)
0
50
100
0.010
−1.5
0.005
∆mag (u)
∆φ (deg)
−1.0
−2.0
−2.5
−3.0
−3.5
−4.0
−200
S12
S12
S21
S21
−150
−100
0.000
−0.005
−0.010
−50
0
50
100
x (mm)
−0.015
−200
S12
S12
S21
S21
−150
Figure 111. Krypton fuel, -85 mm
197
−100
−50
x (mm)
0
50
100
C.2
DBD
The phase and amplitude data shown below is for the wire-electrode device described
in Section 4.3. Each data series consists of two 128 point frequency sweeps of the VNA
(segmented sweeps). The first sweep happens before the plasma turns on and the second
sweep occurs while the plasma is on. Each sweep takes 17.28 ms. The driving wave-form has
a frequency of 29Hz, which gives means the plasma is “on” for the full time of the frequency
sweep. The device is aligned so that the diagnostic beam is directed at the middle of the
plasma.
0.8
0.0004
0.6
Mag (u)
Φ ( ◦)
0.4
0.2
0.0
−0.2
−0.4
0.0002
0.0000
−0.0002
−0.0004
0
50
100
150
200
250
0
50
collection order
100
150
200
250
200
250
collection order
Figure 112. Set 1 trace 0
1.0
0.0004
0.8
Mag (u)
Φ ( ◦)
0.6
0.4
0.2
0.0
0.0002
0.0000
−0.0002
−0.2
−0.4
0
50
100
150
200
250
−0.0004
0
collection order
50
100
150
collection order
Figure 113. Set 1 trace 1
198
0.5
0.0004
0.4
Mag (u)
Φ ( ◦)
0.3
0.2
0.1
0.0
−0.1
0.0000
−0.0002
−0.2
−0.3
0.0002
0
50
100
150
200
−0.0004
250
0
50
collection order
100
150
200
250
200
250
200
250
collection order
Figure 114. Set 1 trace 2
1.0
0.0004
0.8
Mag (u)
Φ ( ◦)
0.6
0.4
0.2
0.0
−0.2
0.0002
0.0000
−0.0002
−0.0004
−0.4
0
50
100
150
200
250
0
50
collection order
100
150
collection order
Figure 115. Set 1 trace 3
1.0
0.0004
0.8
Mag (u)
Φ ( ◦)
0.6
0.4
0.2
0.0
0.0002
0.0000
−0.0002
−0.2
−0.4
−0.0004
0
50
100
150
200
250
0
collection order
50
100
150
collection order
Figure 116. Set 1 trace 4
199
0.3
0.0006
0.2
0.0004
Mag (u)
Φ ( ◦)
0.1
0.0
−0.1
−0.2
−0.3
0.0000
−0.0002
−0.4
−0.5
0.0002
−0.0004
0
50
100
150
200
250
0
50
collection order
100
150
200
250
200
250
200
250
collection order
Figure 117. Set 1 trace 5
1.0
0.0006
0.0004
Mag (u)
Φ ( ◦)
0.5
0.0
−0.5
0.0002
0.0000
−0.0002
0
50
100
150
200
−0.0004
250
0
50
collection order
100
150
collection order
1.5
0.0004
1.0
0.0002
Mag (u)
Φ ( ◦)
Figure 118. Set 1 trace 6
0.5
0.0
0.0000
−0.0002
−0.0004
0
50
100
150
200
250
0
collection order
50
100
150
collection order
Figure 119. Set 1 trace 7
200
Set 2 has the same instrumentation and driving wave-form settings, but a different device
0.6
0.0006
0.4
0.0004
Mag (u)
Φ ( ◦)
and different day. No significant difference is found in the response.
0.2
0.0
0.0000
−0.0002
−0.2
−0.4
0.0002
−0.0004
0
50
100
150
200
250
0
50
collection order
100
150
200
250
200
250
collection order
Figure 120. Set 2 trace 0
0.8
0.0006
0.0004
0.4
Mag (u)
Φ ( ◦)
0.6
0.2
0.0
0.0002
0.0000
−0.2
−0.0002
−0.4
−0.0004
−0.6
0
50
100
150
200
250
0
collection order
50
100
150
collection order
Figure 121. Set 2 trace 1
201
0.0004
Mag (u)
Φ ( ◦)
0.5
0.0
−0.5
0.0002
0.0000
−0.0002
0
50
100
150
200
−0.0004
250
0
50
collection order
100
150
200
250
200
250
200
250
collection order
0.6
0.0006
0.4
0.0004
Mag (u)
Φ ( ◦)
Figure 122. Set 2 trace 2
0.2
0.0
0.0002
0.0000
−0.0002
−0.0004
−0.2
0
50
100
150
200
−0.0006
250
0
50
collection order
100
150
collection order
Figure 123. Set 2 trace 3
0.6
0.0004
Mag (u)
Φ ( ◦)
0.4
0.2
0.0
−0.2
−0.4
−0.6
0.0002
0.0000
−0.0002
−0.0004
0
50
100
150
200
250
0
collection order
50
100
150
collection order
Figure 124. Set 2 trace 4
202
0.0006
0.6
0.0004
0.4
Mag (u)
Φ ( ◦)
0.2
0.0
−0.2
−0.4
−0.6
−0.8
0.0002
0.0000
−0.0002
−0.0004
0
50
100
150
200
250
0
50
collection order
100
150
200
250
200
250
200
250
collection order
0.6
0.0006
0.4
0.0004
Mag (u)
Φ ( ◦)
Figure 125. Set 2 trace 5
0.2
0.0
−0.2
0.0002
0.0000
−0.0002
−0.0004
−0.0006
−0.4
0
50
100
150
200
−0.0008
250
0
50
collection order
100
150
collection order
Figure 126. Set 2 trace 6
1.0
0.0004
Mag (u)
Φ ( ◦)
0.5
0.0
0.0002
0.0000
−0.0002
−0.5
−0.0004
0
50
100
150
200
250
0
collection order
50
100
150
collection order
Figure 127. Set 2 trace 7
203
0.0006
0.8
0.0004
0.6
0.0002
Mag (u)
Φ ( ◦)
1.0
0.4
0.2
0.0
−0.0002
−0.0004
−0.2
−0.4
0.0000
0
50
100
150
200
−0.0006
250
0
50
collection order
100
150
200
250
200
250
200
250
collection order
Figure 128. Set 2 trace 8
1.0
0.5
Mag (u)
Φ ( ◦)
0.0005
0.0
−0.5
0
50
100
150
200
0.0000
−0.0005
−0.0010
250
0
50
collection order
100
150
collection order
Figure 129. Set 2 trace 9
0.0006
0.3
0.0004
Mag (u)
Φ ( ◦)
0.2
0.1
0.0
−0.1
−0.2
0.0000
−0.0002
−0.3
−0.4
0.0002
0
50
100
150
200
250
−0.0004
0
collection order
50
100
150
collection order
Figure 130. Set 2 trace 10
204
0.0008
0.0006
Mag (u)
Φ ( ◦)
1.0
0.5
0.0
0.0004
0.0002
0.0000
−0.0002
−0.0004
0
50
100
150
200
250
0
50
collection order
100
150
200
250
200
250
200
250
collection order
Figure 131. Set 2 trace 11
1.5
0.0006
0.0004
Mag (u)
Φ ( ◦)
1.0
0.5
0.0002
0.0000
−0.0002
−0.0004
0.0
−0.0006
0
50
100
150
200
250
0
50
collection order
100
150
collection order
Figure 132. Set 2 trace 12
1.0
0.0006
0.0004
Mag (u)
Φ ( ◦)
0.5
0.0
0.0002
0.0000
−0.0002
−0.5
−0.0004
0
50
100
150
200
250
0
collection order
50
100
150
collection order
Figure 133. Set 2 trace 13
205
0.0006
0.6
0.0004
Mag (u)
Φ ( ◦)
0.4
0.2
0.0
0.0002
0.0000
−0.0002
−0.0004
−0.2
−0.0006
0
50
100
150
200
250
0
50
collection order
100
150
200
250
200
250
collection order
0.6
0.0006
0.4
0.0004
Mag (u)
Φ ( ◦)
Figure 134. Set 2 trace 14
0.2
0.0
0.0002
0.0000
−0.2
−0.0002
−0.4
−0.0004
0
50
100
150
200
250
0
collection order
50
100
150
collection order
Figure 135. Set 2 trace 15
206
The settings for the function generator are different on the third set. A driving waveform with a 880Hz frequency (which gave the largest visible area of plasma for the wireelectrode device) was used to see if any time variation in the response could be resolved.
The responses were not significantly different from the previously shown measurement sets
which had plasma “on” for the entire frequency sweep.
0.0006
0.6
0.0004
Mag (u)
Φ ( ◦)
0.4
0.2
0.0
0.0000
−0.0002
−0.2
−0.4
0.0002
0
50
100
150
200
−0.0004
250
0
50
collection order
100
150
200
250
200
250
collection order
Figure 136. Set 3 trace 0
0.0006
1.0
Mag (u)
Φ ( ◦)
0.0004
0.5
0.0
−0.5
0.0002
0.0000
−0.0002
−0.0004
0
50
100
150
200
250
0
collection order
50
100
150
collection order
Figure 137. Set 3 trace 1
207
0.0006
0.6
0.0004
Mag (u)
Φ ( ◦)
0.8
0.4
0.2
0.0
−0.2
0.0002
0.0000
−0.0002
−0.4
0
50
100
150
200
−0.0004
250
0
50
collection order
100
150
200
250
200
250
200
250
collection order
Figure 138. Set 3 trace 2
0.0006
0.2
Mag (u)
Φ ( ◦)
0.0004
0.0
−0.2
−0.4
0.0002
0.0000
−0.0002
−0.6
0
50
100
150
200
250
0
50
collection order
100
150
collection order
Figure 139. Set 3 trace 3
0.0006
0.4
0.0004
Mag (u)
Φ ( ◦)
0.2
0.0
−0.2
−0.4
0.0002
0.0000
−0.0002
−0.0004
0
50
100
150
200
250
0
collection order
50
100
150
collection order
Figure 140. Set 3 trace 4
208
1.0
0.0004
Mag (u)
Φ ( ◦)
0.5
0.0
−0.5
0.0002
0.0000
−0.0002
−0.0004
0
50
100
150
200
250
0
50
collection order
100
150
200
250
200
250
200
250
collection order
Figure 141. Set 3 trace 5
1.0
0.0006
0.0004
Mag (u)
Φ ( ◦)
0.5
0.0
0.0002
0.0000
−0.0002
−0.0004
−0.5
−0.0006
0
50
100
150
200
250
0
50
collection order
100
150
collection order
1.0
0.0006
0.8
0.0004
0.6
0.0002
Mag (u)
Φ ( ◦)
Figure 142. Set 3 trace 6
0.4
0.2
0.0
−0.2
−0.4
0.0000
−0.0002
−0.0004
−0.0006
0
50
100
150
200
250
−0.0008
0
collection order
50
100
150
collection order
Figure 143. Set 3 trace 7
209
0.6
0.0006
0.4
Mag (u)
0.0004
Φ ( ◦)
0.2
0.0
−0.2
0.0002
0.0000
−0.0002
−0.4
−0.0004
0
50
100
150
200
250
0
50
collection order
100
150
200
250
200
250
200
250
collection order
Figure 144. Set 3 trace 8
0.0006
0.0004
Mag (u)
Φ ( ◦)
1.0
0.5
0.0
0.0002
0.0000
−0.0002
−0.0004
−0.0006
−0.0008
0
50
100
150
200
250
0
50
collection order
100
150
collection order
Figure 145. Set 3 trace 9
0.4
0.0004
0.2
Mag (u)
Φ ( ◦)
0.0
−0.2
−0.4
0.0000
−0.0002
−0.6
−0.8
0.0002
−0.0004
0
50
100
150
200
250
0
collection order
50
100
150
collection order
Figure 146. Set 3 trace 10
210
0.0006
0.0004
Mag (u)
Φ ( ◦)
1.0
0.5
0.0
−0.5
−1.0
0.0002
0.0000
−0.0002
−0.0004
0
50
100
150
200
250
0
50
collection order
100
150
200
250
200
250
200
250
collection order
Figure 147. Set 3 trace 11
0.0006
0.0004
Mag (u)
Φ ( ◦)
0.5
0.0
−0.5
−1.0
0.0002
0.0000
−0.0002
−0.0004
0
50
100
150
200
−0.0006
250
0
50
collection order
100
150
collection order
Figure 148. Set 3 trace 12
1.2
0.0006
1.0
0.0004
Mag (u)
Φ ( ◦)
0.8
0.6
0.4
0.2
0.0002
0.0000
−0.0002
0.0
−0.0004
−0.2
−0.0006
0
50
100
150
200
250
0
collection order
50
100
150
collection order
Figure 149. Set 3 trace 13
211
0.0006
0.4
0.0004
Mag (u)
Φ ( ◦)
0.2
0.0
−0.2
0.0000
−0.0002
−0.4
−0.6
0.0002
0
50
100
150
200
−0.0004
250
0
50
collection order
100
150
200
250
200
250
collection order
Figure 150. Set 3 trace 14
0.0006
0.4
0.3
0.0004
Mag (u)
Φ ( ◦)
0.2
0.1
0.0
−0.1
−0.2
0.0002
0.0000
−0.0002
−0.0004
−0.3
0
50
100
150
200
250
0
collection order
50
100
150
collection order
Figure 151. Set 3 trace 15
212
The phase and amplitude data shown below is for the wire-electrode device with silica
glass dielectric described in Section 4.3. These measurements are taken at a single frequency
over long time periods. There is a 10 s lead-in in each trace before the plasma turns on.
This is followed by 20 s of plasma on, and finally 30s with the plasma off.
213
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Vita
Joshua A. Stults was born in Ulm, Germany in 1981. After completing high school
at Virgil I. “Guss” Grissom High School in 1999, he entered the US Air Force Academy
and earned a Bachelor of Science in Aeronautical Engineering and was commissioned a 2nd
Lieutenant in the US Air Force in 2003. His first assignment as an officer was a monthlong language immersion with the University of Hawaii at Manoa to improve his ChineseMandarin speaking and reading skills. He then attended AFIT and earned a Masters of
Science in Aeronautical Engineering in 2005. Prior to returning to AFIT in 2008 to pursue
a Ph.D., Capt Stults lead a team of Live Fire Test and Evaluation professionals, and was a
lead test engineer for flight-testing a variety of developmental weapon systems.
His primary professional interests is in using high-fidelity simulations and test results for
decsion support. Finding rigorous ways of informing test design and analysis with simulation
output is needed for product develpments that are increasingly complex and expensive. Using
the outputs of both test and simulation efforts to support sound decisions will be a continuing
challenge for US Air Force product developments that he is excited about tackling.
An error of form which would be negligible in a haystack would be disastrous in a lens.
Thus negligibility involves both mathematics and purpose.
–Lewis Fry Richardson, Standards of Neglect
238
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Nonintrusive Microwave Diagnostics of Collisional Plasmas
in Hall Thrusters and Dielectric Barrier Discharges
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Joshua A Stults, Capt, USAF
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14. ABSTRACT
This research presents a numerical framework for diagnosing electron properties in collisional plasmas. Microwave
diagnostics achieved a significant level of development during the middle part of the last century due to work in
nuclear weapons and fusion plasma research. With the growing use of plasma-based devices in fields as diverse as
space propulsion, materials processing and fluid flow control, there is a need for improved, flexible diagnostic
techniques suitable for use under the practical constraints imposed by plasma fields generated in a wide variety of
aerospace devices.
Much of the current diagnostic methodology in the engineering literature is based on analytical diagnostic, or
forward, models. The Appleton-Hartree formula is an oft-used analytical relation for the refractive index of a cold,
collisional plasma. Most of the assumptions underlying the model are applicable to diagnostics for plasma fields
such as those found in Hall Thrusters and DBD plasma actuators. Among the assumptions is uniform material
properties, this assumption is relaxed in the present research by introducing a flexible, numerical model of
diagnostic wave propagation that can capture the effects of spatial gradients in the plasma state. The numerical
approach is chosen for its flexibility in handling future extensions such as multiple spatial dimensions to account for
scattering effects when the spatial extent of the plasma is small relative to the probing beam's width, and velocity
dependent collision frequency for situations where the constant collision frequency assumption is not justified.
The numerical wave propagation model (forward model) is incorporated into a general tomographic reconstruction
framework that enables the combination of multiple interferometry measurements. The benefit of combining
multiple measurements in a coherent way (solving the inverse problem for the material properties) is the
reconstruction provides a stronger empirical constraint on the predictions of high-fidelity predictive simulations than
multiple un-reconstructed measurements in isolation.
15. SUBJECT TERMS
plasma diagnostics, linear wave propagation, Chebyshev pseudospectral, inverse problem
16. SECURITY CLASSIFICATION OF:
a.
REPORT
b.
ABSTRACT
c. THIS
PAGE
U
U
U
17. LIMITATION
OF ABSTRACT
UU
18. NUMBER
OF PAGES
256
19a. NAME OF RESPONSIBLE PERSON
Dr. Mark Reeder
19b. TELEPHONE NUMBER (Include Area Code)
(937)255-3636, x6154
Standard Form 298 (Rev. 8–98)
Prescribed by ANSI Std. Z39.18
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