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

код для вставкиСкачатьNonintrusive 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 INFORMATION TO ALL USERS The quality of this reproduction is dependent on the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3474023 Copyright 2011 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 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. 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Cited on page 150. [184] Cook, G. O., “ALPAL: A tool for the development of large-scale simulation codes,” Tech. Rep. UCID-21482, Lawrence Livermore National Lab, August 1988. Cited on page 153. 237 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 Form Approved OMB No. 0704–0188 REPORT DOCUMENTATION PAGE The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704–0188), 1215 Jeﬀerson Davis Highway, Suite 1204, Arlington, VA 22202–4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD–MM–YYYY) 2. REPORT TYPE 15-09-2011 PhD Dissertation 3. DATES COVERED (From — To) Aug 2008 – Sept 2011 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Nonintrusive Microwave Diagnostics of Collisional Plasmas in Hall Thrusters and Dielectric Barrier Discharges 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER Joshua A Stults, Capt, USAF 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Air Force Institute of Technology Graduate School of Engineering and Management (AFIT/ENY) 2950 Hobson Way WPAFB OH 45433-7765 9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) Air Force Office of Scientific Research Aeronautics/Chemical and Materials Sciences Directorate Dr John Schmisseur, John.Schmisseur@afosr.af.mil (703) 696-6962 Air Force Office of Scientific 875 N. Randolph, Ste.325, Rm. 3112, Arlington Virginia, 22203 8. PERFORMING ORGANIZATION REPORT NUMBER AFIT/DS/ENY/11-15 10. SPONSOR/MONITOR’S ACRONYM(S) AFRL/AFOSR/RSA 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION / AVAILABILITY STATEMENT APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED 13. SUPPLEMENTARY NOTES This material is declared a work of the US government, and is not subject to copyright protection in the United States. 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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