Improvements to a Fully Kinetic Hall Thruster
Simulation Code and Characterization of the ARCHsNE
Cylindrical Cusped Field Thruster
MASA CHU9Tl~iiI
)F TECHNOLOGY
by
UN 16 2014
Louis Boulanger
IBRARIES
Ing6nieur dipl6m6 de l'Ecole Polytechnique
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
Master of Science in Aeronautics and Astronautics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
@
Massachusetts Institute of Technology 2014. All rights reserved.
/a
Signature redacted
A uthor ........................
Department of A4ro)autics and Astronautics
May 22, 2014
Signature redacted
Certified by.....
Manuel Martinez-Sanchez
Professor
Thesis Supervisor
Signature redacted,
Accepted by ......
........... . -Paulo C. Lozano
Chairman, Department Committee on Graduate Theses
2
Improvements to a Fully Kinetic Hall Thruster Simulation
Code and Characterization of the Cylindrical Cusped Field
Thruster
by
Louis Boulanger
Submitted to the Department of Aeronautics and Astronautics
on May 22, 2014, in partial fulfillment of the
requirements for the degree of
Master of Science in Aeronautics and Astronautics
Abstract
This thesis presents an effort towards a better understanding of the operation
of miniaturized cylindrical Hall thrusters. This class of space propulsion devices
has come under attention since the 1990s as a possible candidate for the propulsion
of 100-1000 kg satellites. In first part, a fully kinetic simulation code developed
at the MIT Space Propulsion Laboratory (SPL) is described and applied to two
devices of interest: the Princeton Cylindrical Hall Thruster (CHT) and the MIT
DCFT (Diverging Cusped-Field Thruster). During this simulation effort, limitations
of PTpic were identified which prompted a major redesign, whose central idea is
a better parallelization of the workload. At the same time, possible candidates to
replace the leapfrog algorithm in the particle pusher have been studied. This work is
described in Chapter 3. Finally, chapter 4 presents the results of the testing of the
recently built Cylindrical Cusped-Field Thruster (CCFT) performed at the SPL.
Thesis Supervisor: Manuel Martinez-Sanchez
Title: Professor
3
4
Acknowledgments
My thanks go first to my advisor, Professor Martinez-Sanchez, for making these
two years possible, and for his guidance, patience and friendliness. This thesis would
not exist without him and all the friends and colleagues, members of the SPL or
visitors, who helped me at some point along the way. First, Anthony Pang, who
built the CCFT and made me feel welcome in the lab; Tom Coles, who provided
invaluable help in dealing with computer and programming issues; Steve Gildea, who
always answered my questions on "the code" in detail and very fast, even three time
zones apart; Professor Lozano, who fought with me to keep Astrovac working; Todd
Billings, who taught me Machine Shop 101 (several times); and Regina Sullivan and
Jaume Navarro. Special thanks to Jeff for helping with the experiments, and being
such a skilled plumber. Thanks to all my fellow SPLers for all the good moments and
beer Fridays we had together.
Grad School would have been much different without all my friends in the AeroAstro department and elsewhere. As we are about to leave for new horizons, I wish
them all the best. A special mention here for my two roommates and friends, R6mi
Lam and Alexandre Constantin.
Last but not least, I want to dedicate this thesis to my parents and my sister
Mathilde. They have supported me from kindergarten to MIT, and I look forward to
being reunited with them.
This project was funded by a grant of the Air Force Office of Scientific Research.
The Direction G6nerale de l'Armement also supported me financially through these
2 years.
5
6
Contents
1
2
Introduction
1.1
Space propulsion
. . . . . . . . . . . . . . . . .
17
1.2
Hall Effect Thrusters . . . . . . . . . . . . . . .
18
1.3
Miniaturized Hall thrusters . . . . . . . . . . . .
20
1.4
Hall thrusters numerical simulation . . . . . . .
. . . . . . . . . .
21
1.5
Thesis Overview . . . . . . . . . . . . . . . . . .
. . . . . . . . . .
21
23
The PTpic code: description and applications
2.1
Particle-in-Cell codes . . . . . . . . . . . . . . . . .
. . . . . . . . . .
23
2.1.1
Operations performed over one iteration
. . . . . . . . . .
24
2.1.2
Stability lim its
. .
. . . . . . . . . . . . . . . .
25
2.2
PTpic features . . . . . . . . . . . . . . . . . . . . .
25
2.3
Princeton Cylindrical Hall Thruster . . . . . . . . .
29
2.3.1
Simulation domain and Boundary conditions
29
2.3.2
Magnetic field . . . . . . . . . . . . . . . . .
30
2.3.3
R esults . . . . . . . . . . . . . . . . . . . . .
32
2.4
3
17
MIT Diverging Cusped-Field Thruster
. . . . . . .
41
2.4.1
Simulation domain and Boundary conditions
41
2.4.2
Magnetic field . . . . . . . . . . . . . . . . .
41
2.4.3
Results . . . . . . . . . . . . . . . . . . . . .
43
47
The PTpic code: improvements
3.1
Parallelization redesign . . . . . . . . . . . . . . . . . . . . . . . . .
7
47
3.2
4
3.1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.1.2
Implementation . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.1.3
Performance assessment
. . . . . . . . . . . . . . . . . . . . .
53
Implicit particle pusher . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.2.1
Stability limitations of the leapfrog algorithm
. . . . . . . . .
59
3.2.2
A framework for implicit PIC codes . . . . . . . . . . . . . . .
61
3.2.3
Semi-implicit field solver . . . . . . . . . . . . . . . . . . . . .
61
3.2.4
Particle Predictor-Corrector
63
Experimental Characterization of the Cylindrical Cusped-Field Thruster 67
4.1
Cylindrical Cusped-Field Thruster overview
4.2
4.3
. . . . . . . ...
. . . . .
67
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.2.1
Vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.2.2
Electrical setup . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.2.3
Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.2.4
Measurement devices . . . . . . . . . . . . . . . . . . . . . . .
71
R esults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
4.3.1
Anode overheating
. . . . . . . . . . . . . . . . . . . . . . . .
74
4.3.2
Regime transitions
. . . . . . . . . . . . . . . . . . . . . . . .
76
4.3.3
Voltage-Current characteristics
4.3.4
Faraday Cup measurements
4.3.5
Retarding Potential Analyzer Measurements
4.1.1
5
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
78
. . . . . . . . . . . . . . . . . . .
80
. . . . . . . . . .
84
Conclusion
87
5.1
Future work recommendations for PTpic . . . . . . . . . . . . . . . .
87
5.1.1
Electric field solver . . . . . . . . . . . . . . . . . . . . . . . .
87
5.1.2
Improved particle pushers
89
5.1.3
Refinement of the load metric
8
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
90
93
A Particle pusher integrators
A.1
Leapfrog method
A .2 Boris m ethod
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
9
10
List of Figures
19
. . . . . . . . . . . .
1-1
Hall Thruster schematic
2-1
The PTpic cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2-2
CHT simulation domain
. . . . . . . . . . . . . . . . . . . . . . . . .
30
2-3
CHT magnetic field, with superimposed streamlines. . . . . . . . . . .
31
2-4
Count of electron superparticles - CHT -ase A . . . . . . . . . . . . .
32
2-5
Anode current - CHT case A . . . . . . . . . . . . . . . . . . . . . . .
33
2-6
Electron density - CHT case A . . . . . . . . . . . . . . . . . . . . . .
34
2-7
Electron temperature - CHT case A . . . . . . . . . . . . . . . . . . .
34
2-8
Ion density - CHT case A
. . . . . . . .................
35
2-9
Anode current - CHT case B . . . . . . . . . . . . . . . . . . . . . . .
36
2-10 Anode current - CHT case C . . . . . . . . . . . . . . . . . . . . . . .
37
2-11 Electron density - CHT case C . . . . . . . . . . . . . . . . . . . . . .
38
2-12 Electron temperature - CHT case C . . . . . . . . . . . . . . . . . . .
39
2-13 Anode current - CHT case D . . . . . . . . . . . . . . . . . . . . . . .
40
. . . . . . . .................
42
2-15 DCFT magnetic field, with superimpose d streamlines . . . . . . . . . .
42
2-16 Anode current - DCFT simulation . . . . . . . . . . . . . . . . . . . .
44
2-17 Electron density in steady state . . . . . . . . . . . . . . . . . . . . .
44
. .................
45
2-14 DCFT simulation domain
2-18 Electron temperature in steady state
3-1
Principal partitioning generated by ParMETIS with 24 processes
.
50
3-2
Particles in a cell whose corners are owned by different processes
.
51
3-3
Average iteration time for the low density plasma, regular grid test
11
55
3-4
Average iteration time for the high density plasma, regular grid test
56
3-5
Average iteration time for the high density plasma, large grid test
56
3-6
Average time per iteration (high density, regular grid, 24 processes)
57
3-7
Computation time breakdown - low density . . . . . . . . . . . . . . .
58
3-8
Computation time breakdown - high density . . . . . . . . . . . . . .
58
3-9
Number of electron super-particles in various PPC configurations.
. .
65
4-1
Schematic DCFT diagram (from [9], p. 31 . . . . . . . . . . . . . . .
68
4-2
Simulated CCFT magnetic field (from [20], p. 36) . . . . . . . . . . .
70
4-3
Sketch of the electrical setup used for the CCFT experiments . . . . .
71
4-4
Stage system and thruster stand.........
72
4-5
The Faraday cup
4-6
Cutaway drawing of the RPA
4-7
Collected current as a function of the repelling voltage - Configuration
A,0
0 . . . . .
. .
......
...
..
..
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .
73
73
73
4-8
Graphite cap showing some fragments of molten steel . . . . . . . . .
74
4-9
Stainless steel rod showing evidence of melting (at the right tip) . . .
75
4-10 Red glow emitted by the anode assembly during operation
4-11 Normal plume
. . . . . .
75
. . . . . . . . . . . . . . . . . . . . . . . . . .
77
4-12 Bag plum e . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .
77
4-13 Jet plum e . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .
77
4-14 Anode current-voltage characteristics - I
. . . . . . . . . . . . .
79
4-15 Anode current-voltage characteristics - II
. . . . . . . . . . . . .
79
4-16 Plume current density - Case 1
. . . . .
. . . . . . . . . . . . .
81
4-17 Plume current density - Case 2
. . . . .
. . . . . . . . . . . . .
82
4-18 Plume current density - Case 3
. . . . .
. . . . . . . . . . . . .
82
4-19 Plume current density - Case 4
. . . . .
. . . . . . . . . . . . .
83
4-20 Plume current density - Case 5
. . . . .
. . . . . . . . . . . . .
83
. . . . . . . . . . . . .
85
4-21 Normalized
4-22 Normalized
at different angles - case A
- -
dV
at different angles - case B
12
85
4-23 Normalized
5-1
at different angles - case C
. . . . . . . . . . . . . . .
85
Mesh numbering scheme used by PTpic up to now . . . . . . . . . . .
88
d
13
14
List of Tables
2.1
Parameters for CHT simulation A . . . . . . . . . . . . . . . . . . . .
32
2.2
Performance parameters - CHT case A . . . . . . . . . . . . . . . . .
33
2.3
Parameters for CHT simulation B . . . . . . . . . . . . . . . . . . . .
35
2.4
Average performance parameters - CHT case B
. . . . . . . . . . . .
37
2.5
Parameters for CHT simulation C . . . . . . . . . . . . . . . . . . . .
37
2.6
Average performance parameters - CHT case C
. . . . . . . . . . . .
38
2.7
Parameters for CHT simulation D . . . . . . . . . . . . . . . . . . . .
39
2.8
Average performance parameters - CHT case D
. . . . . . . . . . . .
40
2.9
Parameters for the DCFT simulation . . . . . . . . . . . . . . . . . .
43
2.10 Performance parameters - DCFT simulation
. . . . . . . . . . . . . .
43
3.1
Common parameters for code speed experiments . . . . . . . . . . . .
54
3.2
Specific parameters for the low density plasma, regular grid test
. . .
54
3.3
Specific parameters for the high density plasma, regular grid test . . .
55
3.4
Parameters for the Particle Predictor-Corrector assessment . . . . . .
64
4.1
DCFT performance at the nominal operating point (from [201, p.30)
69
4.2
Configurations for the voltage-current characteristics
4.3
Comparison of measured anode current to the anode current expected
. . . . . . . . .
78
from full single ionization and full double ionization of the propellant.
80
4.4
Configurations for the Faraday cup scans . . . . . . . . . . . . . . . .
81
4.5
Integrated results for Faraday cup scans
. . . . . . . . . . . . . . . .
82
4.6
Configurations for the RPA measurements
. . . . . . . . . . . . . . .
84
15
16
Chapter 1
Introduction
1.1
Space propulsion
The constraints of cost and mass associated with spaceflight make the design of
any space propulsion device a challenging task. Aside fron considerations of cost and
reliability, two parameters are especially relevant to mission planners when considering the powerplant of a spacecraft: the thrust F (in N), and the specific impulse
I,, (usually expressed in s), which are connected by the following equation, with
g = 9.81m. s-
2
and T4 the total mass flow rate to the engine (including oxidizer if
applicable):
Isp = F
rmg
Thus, the specific impulse quantifies the "fuel efficiency" of a thruster: the higher
it is, the lower the fuel flow rate required to achieve a given level of thrust is. If
the thruster ejects only one type of particle at a single velocity c, and it operates in
vacuum (no pressure effects), there is a straightforward expression of F, and thus of
Isp:
17
F = Thc
C
(1.2)
-ISP = -
9
Thus, the specific impulse is strongly connected to the exhaust velocity achieved
by the engine.
The thusters developed up to this day can be broadly classified into two categories.
Chemical rockets are characterized by the ability to develop a very high thrust (up to
6.77 MN for the F-1 engine), at the expense of specific impulse. The best cryogenic
engines achieve a vacuum Ip of 450-460s (RL-10 or Space Shuttle Main Engine),
not far from the theoretical limit of about 500 s for the liquid hydrogen - liquid
oxygen mixture. On the other hand, electric thrusters, which use an electromagnetic
field to accelerate charged particles, can achieve specific impulses of several thousand
seconds. However, they usually generate less than a Newton of thrust, both because
of intrinsic limitations and because of the limited amount of electric power available
on spacecrafts.
1.2
Hall Effect Thrusters
Hall Effect Thrusters, also commonly referred to as "Hall thrusters", belong to the
electric thrusters family. The initial development effort, from the 1960s until the early
1990s, was largely carried out by Soviet engineers and scientists. The typical Hall
thruster, such as the Soviet/Russian SPT-100, is described in fig. (1-1). It features
an annular discharge channel, sealed at one end by the anode and open at the other
end.
Gaseous xenon is injected near the anode region; as it travels through the
discharge channel, it gets ionized by electrons coming from the external cathode, and
the resulting ions are ejected at a high velocity because they are repelled by the anode.
A radial magnetic field is set across the chamber, usually by electromagnets. Under
the combination of electric and magnetic field, the electrons, instead of going straight
to the anode, drift in the azimuthal direction, an effect known as the E x B drift
18
boron nitride
cathode
walls
neutralizer
anode /
gas distributor
inner
thruster
magnetic
coil
magnetic
circuit
outer magnetic
coil
Figure 1-1: Hall Thruster schematic
(see [4], p.50). This increases the electron residence time - and thus the probability
of hitting a neutral - by several orders of magnitude.
Compared to ion engines (see [13] for a detailed description), Hall thrusters have
several key advantages:
" Only one power supply - between the anode and the cathode - is required, while
ion engines need at least two.
" They can achieve a much higher thrust density because the plasma is quasineutral everywhere, and thus is not subject to the space charge limitation (see [13],
p. 85).
However, a few disadvantages must be noted. Most of them stem from the fact
that a single voltage difference is used for both ionization and acceleration, whereas
ion engines allow the operator to separately tune the ionization voltage and the acceleration voltage.
e Lower specific impulses, usually in the 1600s range.
19
* Lower efficiencies.
* Since the whole chamber contains high-energy ions, erosion of the lining material
(usually boron nitride) is a critical issue and ultimatley determines the lifetime
of the device.
An extensive flight experience has been accumulated by Soviet and Russian satellite operators since the 1970s, as well as Western operators since the late 1990s. Hall
thrusters are mostly used for station-keeping and orbit raising (in 2010, a Hall thruster
originally meant for station-keeping was used to raise the USA-214 satellite to geostationary orbit after the failure of its liquid-fuel apogee motor), but a few spacecrafts,
such as the European Space Agency's SMART-1 moon orbiter, have used them as
main propulsion.
1.3
Miniaturized Hall thrusters
Besides the "traditional" Hall thruster of fig. (1-1), characterized by an annular
discharge channel and a radial magnetic field, less conventional designs have been
investigated since the late 1990s. A primary driver for these efforts is the need to
miniaturize Hall thrusters in order to make them useful for a wider range of spacecrafts. When scaling down a Hall thruster, the designer typically wants to preserve
the following ratios:
" mean free path to characteristic length, in order to keep the collisionality level
constant. Since the device is smaller, this means that a higher plasma density
is required.
" electron gyroradius to characteristiclength, in order to keep the electrons magnetically confined. This requires a significantly higher magnetic field.
Consequently, the following characteristics are common among miniaturized Hall
thrusters:
20
"
A cylindrical discharge channel (no centerpiece), which offers a lower area-tovolume ratio than an annular one for small radii, and thus reduce losses to the
walls.
" An intense magnetic field with a complex topology, due to the absence of centerpiece which prevents the establishment of a radial magnetic field as in the
conventional design.
" While some designs, such as Princeton's Cylindrical Hall Thruster, retain electromagnets, permanent Samarium-Cobalt magnets tend to be preferred.
1.4
Hall thrusters numerical simulation
Significant efforts have been invested in the development of simulation codes for
Hall thrusters and other electric propulsion devices. Fully kinetic, or Particle-in-Cell
(PIC) codes, which model all species as a collection of superparticles, theoretically
have the greatest potential for high-fidelity simulation. The MIT SPL has developed
its own code, called PTpic (Plasma Thruster Particle-in-Cell), since the early 2000s.
1.5
Thesis Overview
This thesis comprises three main parts.
In the first one, we present the main
properties of PIC codes, describe the essential features of PTpic, and apply it to
two different devices, for which a large body of experimental data is available. The
second part deals with the improvements made to PTpic by the author, including a
major redesign which substantially enhances performance. The third and last part
is unrelated and deals with the testing of the Cylindrical Cusped-Field Thruster,
recently built at the SPL.
21
22
Chapter 2
The PTpic code: description and
applications
2.1
Particle-in-Cell codes
Due to the high cost of testing, and the inherent difficulty of instrumenting the
thruster without significantly disturbing its operation, numerical simulation is a useful
tool to predict the characteristics of a Hall thruster. However, due to the low density
and large mean free path prevalent in the device, a fluid treatment is not appropriate;
instead, part or all of the charged species must be simulated as a collection of particles,
which obviously consumes significant computational resources. If all species (neutrals,
electrons and ions) are treated as particles, as is the case in PTpic, the method is
said to be fully kinetic. If some species, usually electrons, are treated as a fluid, the
method is "hybrid".
The properties of a Hall thruster arise from the interaction of charged particles
with the electromagnetic field created partly by themselves, and partly imposed by
the boundary conditions. While it is possible to calculate directly the force exerted
by all other particles on the particle of interest, it is impractical because it involves
O(N 2 ) (N being the number of particles) calculation per step.
A more common
solution consists in projecting the charge created by the particles on a grid, solving
for the potential on this grid, and then interpolating to find the value of the field at
23
Figure 2-1: The PTpic cycle
the particles' positions. Thanks to this approach, the cost now grows linearly with
the number of particles. This method is called Particle In Cell (PIC) and has been
used by PTpic since its inception.
2.1.1
Operations performed over one iteration
During one iteration, an electrostatic PIC code such as PTpic performs the following operations:
o Calculation of the electric field; the magnetic field is assumed constant (electrostatic code).
0 "Particle pusher": moves the particles to their new position and handles collisions.
o Calculation of particle moments, including charge, from the new particle distribution.
24
2.1.2
Stability limits
This section applies to PIC electrostatic codes with the following characteristics:
" Particles are stepped forward with the leapfrog algorithm (see Annex A).
" The electric field used to move the particles at iteration n is determined by the
charge distribution at the previous iteration (n - 1):
Aon+l
--
pf
CO
Under these conditions, PIC codes are subject to the following instabilities:
" Plasma oscillations instability, when wpeAt > 2 (wpe being the plasma pulsation).
This instability can be demonstrated by deriving the modified plasma
dispersion equation when the time is discrete (with a timestep At) rather than
continuous (see [15]).
* Debye length instability, when a
is too small (AD being the Debye length). A
detailed analysis can be found in Birdsall ([3]).
* CFL-type instability, when VthA
> 1 (V
> 1 or vo
being the thermal veloc-
ity, and vo the beam velocity). It is usually not a problem for the simulations
we are running.
" Cyclotron instability, when wcAt > 2 (P.being the cyclotron pulsation. However, Parker ([21]) suggests that this instability is rather benign.
"
Nonlinear Instability ([16]), when x, =-
t
2
>> 1, where N, is the number
of superparticles per cell. In that case, the density of superparticles is too low,
so that the movement of a single superparticle induces a large variation of the
electric field. It is not an issue as of now.
2.2
PTpic features
Alteration of physics
The typical conditions prevailing in the channel of a Hall thruster (ne =i
o1-
10o 8 m-3, Te = 15eV) mean that the Debye length is of the order of 10-50 pm. For
25
a cylindrical Hall thruster, such as those we want to model with PTpic, the channel
size is usually 3-5cm length x 1.5-2cm radius. This means that, in order to resolve the
Debye length, a 4000x2000 mesh is required. The timestep, dictated by the cyclotron
frequency and the plasma frequency criterion (see above), has to be of the order of
10-11 s. Convergence time is usually estimated from the time taken by the slowest
species -i.e.
the neutrals- to cross the chamber.
Neutrals have a speed of about
200m/s, yielding a conservative estimate for the convergence time of 10-S. i.e. at
least 100 million iterations. With the computer facilities available at the SPL, such
a simulation would take several months to complete. In order to make the task more
tractable, PTpic uses an altered physics model developed by James Szabo ([28]),
which greatly alleviates the computational workload.
" The vacuum permittivity co is increased by -y'. The Debye length is thus multiplied by -y, allowing the use of a much coarser mesh. Additionally, it makes the
simulation less prone to the plasma oscillations instability, since it also divides
the plasma pulsation
pe by 7.
" The mass of heavy species (ions and neutrals) is reduced by a factor
multiplies their speed - and thus reduces the convergence time - by
f.
This
f.
" In order to compensate for the reduced neutral residence time, the electronneutral cross-sections are increased by V,/7.
The rationale backing this methodology is that the overall discharge properties
will not change if the Debye length remains small compared to the dimensions of the
thruster; nor will it change if the mass ratio of ions and neutrals to electrons remains
large enough.
It does however require a "post-processing" step in order to recover the "physical"
parameters from the simulated ones. For instance, the ion-beam current has to be
reduced by
f in order to recover a meaningful value. All these manipulations are
detailed in Szabo's PhD thesis ([28], p. 87).
26
Electric field solver
The electric potential governing equation is discretized by applying Gauss's theorem to a small control volume surronding each node ([12], p. 144).
This yields a
matrix equation linking the charge and the potential at each node:
AO = 4
(2.1)
where b and 4 are the vectors containing the normalized electric field and charge
at each node, respectively. In its current version, PTpic uses a direct solver which
computes a LU decomposition of A during initialization. Reusing this decomposition
throughout the run enables a significant saving of time compared to previous versions
of the code, which used an iterative solver ([12], p.166).
Particle pusher
The method used to advance the particles in time is the Boris algorithm, which
is the most common method for PIC codes (see Annex A). Its popularity stems
from the fact that it is second-order accurate and has interesting energy-conserving
properties, which some more accurate methods like the Fourth-Order Runge-Kutta
do not possess. However, it is bounded by the stability limits mentioned earlier.
Collisions management
The "particle pusher" step is also responsible for the handling of collisions. The
collisions currently included in PTpic are:
" Electron-Neutral elastic scattering.
" Ionization and double ionization collisions. Double ions can be generated from
single ions and neutrals.
" Electron-Neutral excitation in one lumped level.
" Ion-Neutral charge-exchange collisions.
27
* Ion-Neutral and Neutral-Ion scattering.
"
Electron and Ion wall recombination.
* Secondary Electron Emission.
All the particle-particle collisions are modeled with a Monte-Carlo Collisions
(MCC) methodology. This means that, for each collision, one species (the "'target")
is treated as a background of particles characterized only by its density ntarget and its
bulk speed Utarget. Then, for each superparticle of the other species, the probability p
of a collision over the timestep is calculated from the collision frequency v (see Szabo
[28], p. 192):
Q
P
Q
p-
1
*
-
IV - UtargetII * ntarget
(2.2)
e*
being the collision cross-section, evaluated at the incoming particle's velocity.
This probability is then compared to a random number r drawn with a uniform
probability distribution between 0 and 1; if r > p, then the collision is allowed to
proceed, provided that there are enough target superparticles to support the collision
event.
Particle-surface collisions are handled as follows:
" Neutrals undergo a diffuse reflection with an energy accomodation factor of 0.5.
* Ions are converted to neutrals and diffusely reflected, again with an energy
accomodation factor of 0.5.
* Electrons are destroyed ; if they hit a dielectric, their charge is transferred to
it and secondary electrons are emitted in accordance with the yield reported in
[13], p. 3 4 9 .
28
Moments calculation
Finally, certain particle moments have to be calculated at each iteration because
they are required by the field solver or the collision management functions.
The
electric charge is calculated with a first-order weighting, which has a lower "numerical
noise" than the nearest grid point (NGP) weighting. The particle moments which
do not directly contribute to the simulation but provide valuable information on the
operation of the thruster are only computed every few dozen thousand iterations.
Parallelization
PTpic started as a serial code (i.e. run on a single processor), but it was parallelized by Fox ([11])
in 2005.
MPI (Message-Passing Interface).
Inter-processes communication are handled with
The implementation of MPI used in this thesis
is OpenMPI. Chapter 3 deals with the efforts made in order to increase the speed
and scalability (i.e. the ability to run efficiently on a large number of processes) of
the parallelized code.
2.3
Princeton Cylindrical Hall Thruster
The first device studied here is the 2.6cm diameter Cylindrical Hall Thruster
(CHT), built at the Princeton Plasma Research Laboratory ([23]). This thruster was
selected because it is one of the best studied miniaturized Hall thrusters; thus a large
body of published data is available to evaluate the results of simulation.
2.3.1
Simulation domain and Boundary conditions
A 193x157 mesh with a uniform spacing of 0.25mm was used. A drawing of the
simulation domain is provided in fig.
2-2.
The light grey parts represent boron
nitride; the dark gray parts represent metal. The boundary conditions for the electric
potential are a combination of Dirichlet and Neumann type:
e The metal parts (dark grey) are set to a prescribed potential Om.
29
Free space: Q =
0
3.5
3
,-2.5
E
0
Ce2
1.5
1
F
Anode
0.5
,
Centerline:
09
, , , 1, ,1 , , , , 1 , , , ,(9 1,
0
0
0.5
1
1.5
2
2.5
3
=
0
,1
3.5
, l . ,
4
,
4.5
Z [cm]
Figure 2-2: CHT simulation domain
" The anode (red) is set to a prescribed potential.
" On the centerline (yellow), the electric field is tangent to the boundary.
" On the free space boundaries (blue), the potential is set to 0.
2.3.2
Magnetic field
The CHT has two electromagnetic coils which allow for an adjustable magnetic
field. All the simulations were performed with the magnetic field (fig.
2-3) corre-
sponding to a current of +1.4A in the back coil and -0.9A in the front coil. The
maximum magnetic field in the channel is 270 G.
This magnetic field was calculated with MAXWELL, a finite elements software.
30
4L
JBI M
0.7
0.422906
0.2555
0.154361
0.0932574
0.0563417
0.0340389
0.0205647
0.0124242
0.00750611
0.00453483
0.00273973
0.00165521
0.001
3.5
3
.- 2.5
E
S2
1.5
1
0.5
00
2
2.5
Z [cm]
Figure 2-3: CHT magnetic field, with superimposed streamlines.
31
Table 2.1: Parameters for CHT simulation A
Anode flow rate (seem)
Anode Voltage (V)
4
250
y
50
1000
Xenon
f
Propellant
Super-particle size (ions and electrons)
Super-particle size (neutrals)
Timestep (s)
Number of iterations
Thruster body potential OM (V)
108
108
10-11
3.106
20
1200000
1000000
CL
800000
£
600000
400000
E 200000
0
000E+00
5.00E-06
100E-05
1.50E-05
2.OOE-05
2.50E-05
3.00E-05
Time (s)
Figure 2-4: Count of electron superparticles - CHT case A
2.3.3
Results
Note: PTpic outputs the performance parameters (thrust, anode current...) at
every iteration. All graphs presented hereafter were time-averaged within groups of
1000 iterations.
Case A
A first round of simulations was performed with a conservative value for the artificial permittivity factor: -y = 50. The values of the other significant simulation
parameters are given in table 2.1.
After a large ionization peak, convergence is reached at about 8 x 106 s, i.e.
800000 iterations, as evidenced by figures 2-4 and 2-5. Performance parameters in
the steady state are presented in table 2.2, alongside the experimental values measured
at the PPPL ([26]).
The current utilization 71c is defined as the ratio of ion beam
current to anode current. The propellant utilization 7r is defined as the ratio of ion
32
2.5
C 1.5
0
0.5
0
0.DOE+00
5.OOE-06
1.50E-05
1.00E-05
2.OOE-05
2.50E-05
3.OOE-05
Time (s)
Figure 2-5: Anode current - CHT case A
Table 2.2: Performance parameters - CHT case A
Experimental
A
4.0
2.54
Thrust (mN)
0.57
0.181
Anode current (A)
0.288
0.139
Ion beam current (A)
-0.288
-0.128
(A)
current
beam
Electron
0.51
0.77
Current utilization nc
0.98
0.47
Propellant utilization r,
0.22
0.18
Efficiency j
beam current to the anode mass flow rate expressed in amperes, with 1 sccm Xe
corresponding to 0.0722399 A ([13], p. 464).
The simulated discharge is significantly weaker than the actual one, with a thrust
under-predicted by about 40% and an anode current about 3 times smaller than the
experimental value. However, the computed efficiency (0.18) is close to the experimental value of 0.22.
Let us now examine the structure of the plasma in its steady state. The electron
density (fig. 2-6) is rather homogeneous in the chamber, but features a narrow region
of higher density (peaking at 2.8 x 10
19
m- 3 ) on the center axis. The electron tem-
perature (fig. 2-7) is low, with most regions below 7 eV, while experimental values
for channel electrons put the temperature at 15 eV ([25]).
The ion density (fig 2-8) is very similar to the electron density in the channel
33
I
4
3.5
3
2.5
Ir2
1.5
0.5
0
0.5
1
1.5
2
2.5
3
3.5
.
Z [cm]
Figure 2-6: Electron density - CHT case A
416
3.5
15
14
3
13
12
11
2
7
1.5
4
6
5
1
10
3
2
1
Z [CM]
Figure 2-7: Electron temperature - CH T case A
34
4
3.5
3
-2.5
E
2
1.5
1
0.5
0
0.5
1
1.5
2
2.5
Z [CM]
3.5
4
4.5
Figure 2-8: Ion density - CHT case A
Table 2.3: Parameters for CHT simulation B
4 Super-particle size (ions and electrons)
Anode flow rate (sccm)
Super-particle size (neutrals)
250
Anode Voltage (V)
Timestep (s)
25
7
Number of iterations
1000
f
Thruster body potential Om (V)
Propellant Xenon
108
108
10-11
3.106
100
region. The plume, however, has a conical structure of its own, with a divergence
angle comprised between 30 and 45'. This rather large divergence angle echoes the
observations made at the MIT SPL on the DCFT ([18]) and the CCFT (see section
4.3.4).
Case B
In order to improve the fidelity of the simulation, another simulation was performed with -y = 25. Om was also raised to 100V in order to make the initiation of
the discharge easier by attracting a large amount of electrons to the thruster.
After an initial ionization surge, the discharge enters a highly oscillatory mode, as
evidenced by fig. 2-9, representing the anode current. On this graph, the "corrected
time" is used; it is equal to the simulation time multiplied by \/f.
35
The rationale
2
5
C'L5
a5
1,27EA23
1,32E-03
137E-03
1.42E-03
1.47E03
1.52E-03
1.57E-03
1.62L03
Corrected Time (s)
Figure 2-9: Anode current - CHT case B
for this manipulation is that low-frequency oscillations in Hall thrusters are usually
determined by the travel time of neutrals through the chamber. Since neutrals are
accelerated by a factor of v'f in our altered physics model (see 2.2), the period
of simulated oscillations must be multiplied by the same factor in order to make
comparison with experimental measurements possible. With this dilated time, the
period of the oscillations appear to be around 16.7 kHz, which agrees quite well with
the findings of the PPPL experiments, which detected oscillations at a frequency
slightly lower than 20 kHz ([24]).
These oscillations have the effect of increasing very significantly the average anode
current and thrust compared to the previous run, as evidenced by table 2.4. Thrust
is now about 25% higher than the experimental value, and the anode current is much
closer to 0.57A. An undesirable effect of the high thruster potential
#m
is that it
allows the thruster front surface to draw a large amount of electrons, thus leading to
an unphysical imbalance between the ion and the electron beam currents.
Case C
Pursuing our trend towards lower and more realistic values of -Y, we now attempt
a simulation with -y = 16. OM is set to the more realistic value of 20 V, as in run A.
Quite surprisingly, the discharge is steady (see fig. 2-10), without any trace of the
large amplitude oscillations observed in run B.
36
Table 2.4: Average performance parameters - CHT case B
Thrust (mN)
C
5.03
Anode current (A)
Ion beam current (A)
Electron beam current (A)
Current utilization rc
Propellant utilization rj
0.361
0.408
-0.15
1.13
1.39
Efficiency r/
0.35
Oscillations corrected period (ps)
Anode current oscillations relative amplitude R
60
5.1
Table 2.5: Parameters for CHT simulation C
Anode flow rate (secm)
Anode Voltage (V)
y
f
Propellant
4
250
16
1000
Xenon
Super-particle size (ions and electrons)
Super-particle size (neutrals)
Timestep (s)
Number of iterations
Thruster body potential Omu (V)
0.5
C
S0.3
C 0.2
0.1
3.20E-05
3.40E-05
3.60E-05
3.80E-05
4.0OE-05
4.20E-05
4AE-05
T4me (s)
Figure 2-10: Anode current - CHT case C
37
108
108
10-"
3.106
20
Table 2.6: Average performance parameters - CHT case C
C
Thrust (mN)
Anode current (A)
Ion beam current (A)
Electron beam current (A)
Current utilization qc
Propellant utilization r1
Efficiency q
5.25
0.407
0.283
-0.270
0.695
0.98
0.34
4
3.5
3
-2.5
E
1.5
1
0.5
00
Z[CI)
Figure 2-11: Electron density - CHT case C
In spite of this lack of oscillations, the performance parameters remain high, and
close to the value they had with y = 25. A noteworthy point is the good adequation
of the predicted (0.283A) and experimental (0.288A) beam current.
While the electron density map (fig. 2-11) shows the same general structure as in
run A, the electron temperature (fig. 2-12) is significantly higher. A large part of the
channel has an electron temperature above 10 eV, with peaks well above 20 eV.
Case D
In this last case, -y is set to 10 and Om = 20V.
This run shows a return of large discharge oscillations, although their amplitude
38
T e [eVI
24
3.5
22
20
18
16
14
12
10
8
6
4
2
0
3
-2.5
E
2
1.5
0.5
00
0.5
1
1.5
2
2.5
Z [CM]
3
O.U
+
14.;
Figure 2-12: Electron temperature - CHT case C
Table 2.7: Parameters for CHT simulation D
Anode flow rate (secm)
Anode Voltage (V)
7
f
Propellant
4
250
10
1000
Xenon
Super-particle size (ions and electrons)
Super-particle size (neutrals)
Timestep (s)
Number of iterations
Thruster body potential #m (V)
39
108
108
10-11
3.106
20
0.6
0.5
0.4
i 0.3
0.2
0.1
8.OOE-06 9.00E-06 1.00E-05 1.10E-05 1.20E-05 1.30E-05 1.4E-0S 1.5OE-05 1.60E-05
Time (s)
Figure 2-13: Anode current - CHT case D
Table 2.8: Average performance parameters - CHT case D
C
Thrust (mN)
Anode current (A)
Ion beam current (A)
Electron beam current (A)
Current utilization 77c
Propellant utilization T7H
Efficiency q
3.462
0.238
0.261
-0.268
1.097
0.904
0.256
is lower than in run B (fig. 2-13). While run C suggested that oscillations in run B
may be due to the high value of OM rather than the decrease in -y, this last simulation
invalidates this hypothesis.
The average anode current and thrust are lower than
for runs B and C (table 2.8).
This may be connected to the observed propensity
of cylindrical Hall thrusters to switch their operating mode. For instance, Raitses
([24]) reports that oscillations can be turned on and off in the CHT by changing the
cathode settings, which are currently not included in the PTpic model. Different
numerical treatments may end up simulating different modes; this should be further
investigated.
40
2.4
MIT Diverging Cusped-Field Thruster
This section deals with the application of PTpic to the Diverging Cusped Field
Thruster, the first cusped field thruster built at the SPL (see [9]).
This simulation
program was started by Gildea ([12], p.183), who extensively studied the y = 50 case.
At the time, he was using a feature of PTpic by which the thruster body potential Om
is allowed to change, based on the amount of charge collected ([12], p. 158). While
it is true that OM should be allowed to float in order to replicate the reality more
closely, implementation has been troublesome because of the difficulty of accounting
for the variations of capacitance caused by the presence of plasma. Thus we will use
a fixed Om, as is standard in the Hall thruster simulation community. Apart from
this change, the code used for these simulations is essentially the same as used by
Gildea; it does not include the modifications presented in the next chapter.
2.4.1
Simulation domain and Boundary conditions
A 213x113 mesh was used. It is based on the mesh used by Gildea in his thesis, but
focuses on the discharge channel in order to reduce the computational cost. Due to
the divergent shape of the thruster, this mesh is not mapped to a rectangular physical
space and thus the shape and size of its cells varies significantly; the spacing is smaller
near the anode and the cusps, where the highest plasma densities are expected. A
drawing of the simulation domain, with the electric potential boundary conditions, is
provided in fig. 2-14.
2.4.2
Magnetic field
The DCFT magnetic field (fig. 2-15) is created by permanent Samarium-Cobalt
magnets. It is much stronger than in the CHT, most of the channel being exposed
to a field greater than 2000G. The strong gradient from the center axis towards the
walls, as well as the cusped magnetic streamlines, are clearly visible.
This magnetic field was also calculated with MAXWELL.
41
r"I
E
0
&MEN
W 47
-
On
- 0
3
2
=0
1
01
2
1
Z [cm]
3
4
5
6
7
Figure 2-14: DCFT simulation domain
4
BI [T
|-I
3
0.9
0.8
0.7
*0.6
*0.5
*0.4
*0.3
2
0.1
0
1
3
4
Z [CM]
Figure 2-15: DCFT magnetic field, with superimposed streamlines.
42
Table 2.9: Parameters for the DCFT simulation
6 Super-particle size (ions and electrons)
Anode flow rate (sccm)
Super-particle size (neutrals)
300
Anode Voltage (V)
108
108
25
Timestep (s)
10-11
1000
Xenon
Number of iterations
Thruster body potential #m (V)
3.106
20
y
f
Propellant
Table 2.10: Performance parameters - DCFT simulation
Simulation
2.4.3
Thrust (mN)
5.4
Anode current (A)
Ion beam current (A)
Electron beam current (A)
Current utilization r7
Propellant utilization TH
Efficiency 17
0.309
0.296
-0.298
0.958
0.683
0.267
Results
The simulation presented in this section was performed with the parameters described in table (2.9). The value of -/ chosen represents a significant decrease compared
to the y = 50 used in [12].
Performance parameters in steady state are given in table 4.1. The anode current
prediction matches closely the experimental value of 0.3A (see [12], p.80). Unfortunately, direct experimental measurements are not available for the other performance
parameters.
However, comparison with the measurements made by Courtney ([9],
p.85) for a configuration at 300V anode voltage and 8.5 seem anode flow rate shows
that the simulation results are reasonable. The thrust and anode current are significantly higher than the values reported by Gildea ([12], p. 189), which were 2.72 mN
and 0.198A, respectively. The results of a 7 = 50, fixed OM run performed by the
author (not included in this thesis) strongly suggests that this increase is due to the
lower value of 7, rather than to the switch from a floating to a fixed body potential.
Finally, this discharge is clearly steady; the amplitude of fluctuations is very small
(fig. 2-16). Thus the oscillations seen by Gildea were likely due to the floating body
43
1.4
1.2
~0.8
0.2
0
0.00E+00
5,00E-06
1.OOE-05
1.50E-05
2.OOE-05
2.50E-05
Time (s)
Figure 2-16: Anode current - DCFT simulation
6 -
5
6528E+19
-8
5+31944E+1 8
4
-
00
1
2
3
[
4
5
6
7
Figure 2-17: Electron density in steady state
potential boundary condition.
Snapshots of the electron density and temperature in steady state are provided in
figs 2-17 and 2-18. The shaping of the electron density by the magnetic field cusps is
clearly visible. Interestingly, the high density regions (around 2 x 10 1 8 m-3 ) around the
cusps seem to be at a low temperature (about 5 eV). These density and temperature
maps globally agree with those obtained by Gildea ([121, p. 192 & 198). While the
performance parameters reported in table 4.1 are reasonable, the high electron density
and temperature seen in the anode region suggest that we may be locally close to
the onset of numerical instability. In fact, simulations attempted at 'y = 16 were
generally unsuccessful, contrary to the CHT case.
44
6T-0 fel/
5
25
5
17.5
15
]1245
4
E
U
2
10
7,5
5
-0
3~2.5
2
0
1
2
3
4
5
6
7
Z [CM]
Figure 2-18: Electron temperature in steady state
45
46
Chapter 3
The PTpic code: improvements
3.1
3.1.1
Parallelization redesign
Background
PTpic has proved itself a valuable research tool, but over the recent years its limitations have become more apparent. The Cusped-Field Thrusters studied since 2008
at the MIT SPL have proved especially challenging to model since their intense magnetic field and high plasma density make them prone to the numerical instabilities
mentioned in section 2.1.2, which in turn mandates the use of a very small timestep
(1 x 10-1 2 s), even with a high value of -y. For example, Steve Gildca's DCFT simulations were performed with -y = 50 (meaning that the vacuum permittivity is 2500
times higher in the simulation than in reality) and At =1 x 10-
12
s. Although these
simulations were able to predict the erosion rate along the dielectric surface with a
good accuracy, some other features, such as the location of the main ionization region,
were not resolved with the same success.
Tackling these issues and obtaining a more realistic simulation likely requires the
use of a vacuum permittivity as close as possible, and ideally identical, to its real
value. This, in turn, requires a much finer mesh in order to resolve the local Debye
length, as well as a much larger number of superparticles, so that the number of
superparticles per cell stays high enough to allow for the calculation of meaningful
47
statistics.
If run on 24 processes, as was standard during Gildea's work, such a
simulation would take several months to converge. Since PTpic is a parallel code, the
obvious solution consists in running it on more processes. Unfortunately, the version
of PTpic inherited by the author has a limited scalability; practically. using more
than 30 processes does not yield any reduction in computation time (also called "wall
time"), or even increases it. Thus, a prerequisite to the use of more realistic values
of 7 and
f
is the removal of this scalability limitation; and a general acceleration of
PTpic.
A key limiting factor to the speed and scalability of PTpic is the fact that each process manages particles spread all over the grid (see [11]). This non-local architecture
degrades performance for two reasons:
" Since each process needs, and contributes to, the particle moments everywhere
on the grid, the calculation of particle moments requires many time-consuming
calls to MPI collective communication functions per iteration. This effect is
especially significant when there are few particles, for instance at the beginning
of a run, because the statistics calculation takes a larger share of the overall
iteration time.
* PTpic is designed to ensure that the number of collision events per iteration in
a given cell is lower than the number of target superparticles present in this cell.
For instance, the number of Xe+ to Xe++ transitions in a given cell must be
smaller than the number of Xe+ superparticles in the cell. If this count check was
performed independently by each process, the collisionality would be artificially
decreased, because a process could exhaust its number of target superparticles
and thus reject a collision event while there are still target superparticles in the
cell belonging to other processes. Thus, each process has to send all its collision
candidates to a single process which approves or cancels them based on the total
count of target superparticles (see [11]). This introduces a serial component in
the code which degrades speed and scalability, especially when there is a large
number of particles.
48
Rather than this non-local architecture, a better program design consists in assigning a region of the domain to each process, so that all particles in the said domain
are managed by a single process (local architecture). Thus the calculation of particle
moments does not require inter-processor communication anymore, except for a few
boundary cells (and even in this case, the situation is much better than in the old
design, because the local process only needs to exchange information with its neighbours, rather than all processes), and each process can manage its collision events in
full autonomy.
This introduces a new issue though: load balancing. In Fox's design, it was automatically guaranteed because each new particle was randomly assigned to a process.
Now, we need to repartition the domain periodically so that each process manages
roughly the same number of particles. Fortunately, a team at the University of Minnesota has developed a freely available package, called ParMETIS, which is able to
quickly compute load-balanced partitionings. With ParMETIS available, it was de-
cided to go ahead with a major redesign of PTpic, switching from a non-local to
a local architecture. The resulting code will be referred to as PTpicFP (for "Fully
Parallel").
Implementation
3.1.2
Three different partitionings
It is essential to understand that PTpicFP alternatively uses three partitionings
of the grid. A partitioning is a division of the mesh so that each node is assigned to
one, and only one, process.
e The principal partitioning, which determines the domain over which each process is responsible for moving the particles and computing statistical moments.
An example with 24 processes on the CHT 193 x 157 mesh is provided in fig.
3-1.
e The field solver partitioning, used exclusively by the field solver. Each process
is responsible for computing the potential on a portion of the mesh.
49
4
3.5
3
72.5
2
1.5
1
0.5
0-m
'ZCM)
Figure 3-1: Principal partitioning generated by ParMETIS with 24 processes
e The ParMETIS partitioning, used exclusively by the ParMETIS functions.
The code would be slightly simpler if these three partitionings were identical.
However, this would be detrimental to performance because the relevant load metric
(a measure of the computation time spent on each cell) is different in each case. For
the principal partitioning, the relevant load is the number of superparticles present in
the cell. Ions and electrons have a higher weight than neutrals, since they are moved
at every iteration while neutrals are only moved every DT-N2E iterations. Thus, the
load associated to a cell
[k, j] is:
[h]load[k,j] = DT-N2E * (Nions[kij] + Neiectrons[kj]) + Nneutrais[k,j]
(3.1)
This means for instance that a process assigned to the plume of the thruster,
where the plasma is very diffuse, will manage several dozen times more cells than a
process responsible for a dense plasma region within the channel.
On the other hand, for the Field solver, and ParMETIS, the number of particles is
not relevant and we simply want to assign the same number of cells to each process.
50
D, proc. 4
C, proc.3
B, proc. 2
A, proc. 1
Figure 3-2: Particles in a cell whose corners are owned by different processes
Statistics exchange between processes
The local architecture requires the development of tools and structures in order to
handle the selective exchange of information between processes. The spatial specialization of each process drastically reduces the amount of statistics information that
needs to be exchanged between processes, but it does not completely eliminate it.
First, since the principal partitioning and the field solver partitioning are different,
a process may very well have to solve for the potential at a location which belongs
to its Field solver domain, but not to its Principal domain. To keep things simple,
it was decided to retain the logic of the conventional PTpic: all processes know the
value of the charge density everywhere.
All particles present in a given cell are managed by the process owning the lowerleft corner of this cell. However, they do contribute to the statistics at all four corners,
which means that some exchange of statistics may be required if some of the corners
are owned by other processes.
An example is given in fig.
3-2.
All particles in
the cell (red disks) are managed by process 1, because it owns the lower-left node
A. However, these particles contribute to statistics at nodes B, C and D, which are
owned by processes 2, 3 and 4 respectively.
The statistics calculation in PTpic is a two-stage process.
* Each time a particle is moved (in the "Particle Pusher" step, see fig. 2-1), its
contribution to the four neighboring nodes is added.
51
* Then, during the "Moments calculation" step, these raw statistics are divided
by the number of particles in order to obtain the moments.
The solution adopted is as follows:
* During the particle pusher step, process 1 computes the statistics contribution
of its particles on all nodes, including nodes owned by other processes.
* It then sends the raw statistics for nodes B, C and D to their owner process.
" Each process normalizes the statistics for the nodes it owns.
* Processes 2, 3 and 4 then send back the normalized moments for nodes B, C and
D to process 1. This is necessary because process 1 needs to know the correct
value of the moments at all four corners of the cell (for instance to interpolate the
neutrals density at the location of an electron in order to compute its collision
probability).
Obviously most of the cells have their four corners managed by the same process,
and thus do not require any external input.
Particle Exchange
After each move, each process must check if the particle is still in its area of
responsibility. If it has moved to another process's domain, one could send it to the
target process right away; however it is likely more efficient to pack all particles to
be moved in a single buffer and send them in one message. This requires creating a
separate array of particles to be displaced, as well as an array to keep count of their
number.
Non-blocking communications and communication-computation overlap
Care was taken to use non-blocking communications as much as possible. The
basic concept of non-blocking communications (for more details, see the MPI reference
[27] p.
79) consists in starting the transaction between two processes, leaving it
52
running in the background, doing some useful computation in the meantime and then
closing the communication channel once we make sure all data has been received. This
program design is called conmiunication-conputatiortoverlap and is a very desirable
feature in parallel computing.
To make it clearer, let's take the example of the statistics exchange described in
section 3.1.2. The first thing process 1 does is to initiate the sending of raw statistics
for nodes B, C and D to processes 2, 3 and 4 via a non-blocking send.
At the
same time, these processes open a non-blocking receive resource. Then, each process
normalizes the statistics for the nodes it owns which do not require external input
(i.e. the vast majority). Once this is done, processes 2, 3 and 4 wait for the message
sent by process 1. The key thing is that since they have been busy for some time with
the statistics calculation, this message has hopefully arrived so that they do not waste
any time waiting for it. Then, they can proceed with the calculation of statistics for
nodes B, C and D and send back the normalized moments to process 1.
Removal of unnecessary MPI function calls
The collision approval functions have been redesigned to remove the inter-processes
approval sequence, which is now unnecessary. A broader effort to reduce the number
of calls to MPI functions was implemented. Since the ratio of latency time to total
communication time is usually high for small messages, it is often beneficial to pack
as many messages as possible into a single buffer, then send it, rather than sending
them separately as was the case previously.
3.1.3
Performance assessment
The performance comparison between the FP and conventional versions of PTpic
was done for 8, 24 and 48 processes, on 2 different grids. The first one (regular grid)
is the 193x157 mesh used in the CHT runs (see previous chapter); the other one is a
301 x 151 mesh that was also used for a few CHT simulations, whose results are not
included in this thesis. Also, thanks to a feature of PTpic which allows the user to
53
Table 3.1: Common parameters for code speed experiments
Anode flow rate (sccm)
4.0
Anode voltage (V) 250
Timestep (s)
5 x 10-12
'Y
15
Table 3.2: Specific parameters for the low density plasma, regular grid test
Ion temperature (eV)
1.0
Electron temperature (eV)
10.0
Approximate ion and electron superparticles count 13000
"seed" a part of the domain with electrons and ions, various plasma states (density
and temperature) were investigated.
The parameters in table 3.1 were identical for all simulations.
The times reported hereafter are averaged over the first 1000 iterations.
Low density plasma, regular grid
In this experiment, a low density, low temperature (see 3.2) plasma is seeded in
the domain; the pusher component of the algorithm is thus lightly solicited.
This
configuration is representative of the beginning of a simulation. Since it can easily
take one million iterations before the thruster "ignites", reducing the iteration time
in this kind of situation is very important.
The average iteration time is given in fig. 3-3.
In this configuration, the bad scalability of the conventional PTpic clearly appears: the computation time is actually steeply increasing with the number of processes. With this low number of particles, adding more processes does not make the
particle pusher substantially faster, but it adds a considerable communication overhead due to the inefficient communication patterns. In contrast, the computation
time for PTpicFP decreases significantly between 8 and 24 processes and then stays
approximately constant, which is mostly due to the fact that the field solver currently
implemented does not scale well above about 30 processes for this kind of meshes (see
[12], p. 166).
54
0.12
0.1
0.08
3 FP
0.06
N Conventional
7~
0.04
0.02
0
8
48
24
Figure 3-3: Average iteration time for the low density plasma, regular grid test
Table 3.3: Specific parameters for the high density plasma, regular grid test
1.0
Ion temperature (eV)
20.0
Electron temperature (eV)
Approximate ion and electron superparticles count 1.3 x 106
High density plasma, regular grid
In this section, a much larger number of plasma particles is seeded. The electrons
energy is also increased in order to maximize the collisionality.
The average iteration time is given in fig. 3.3.
The difference in scalability between the two versions is not as striking as in the
previous case. Because of the amount of particles to manage, going from 24 to 48
processes helps, even for the conventional PTpic. However, the speedup achieved by
the Fully Parallel version is much larger: 34.2% versus 13.9 %.
High density plasma, large grid
Reducing -y will mean using much larger meshes. The number of plasma superparticles seeded and their temperature have the same value as in the previous section,
but the mesh size is now 301x151, a 50% increase. The average iteration time is given
in fig. 3-5.
The advantage enjoyed by the Fully Parallel code is significantly larger than in the
55
0.18
0.16
0.14
0.12
0.1
* FP
M0.06
I Conventional
0.06
0.04
0.02
24
48
Figure 3-4: Average iteration time for the high density plasma, regular grid test
0.3
0.25
0.2
M FP
0.15
* Conventional
0.1
0.05
24
48
Figure 3-5: Average iteration time for the high density plasma, large grid test
56
0.25
0.2
0.15
-Global
-Pusher
E
time
5
Pusher 9
Pusher 17
0.1
0,05
0
0
2000
4000
6000
8000
Iteration number
10000
12000
Figure 3-6: Average time per iteration (high density, regular grid, 24 processes)
high density - regular grid situation. On 48 processes, it is going more than twice as
fast as the conventional version, with a better speedup (35.6% against 24.5%). The
fact that the statistics calculation is completely serial in the conventional version is
obviously a big inconvenience on large meshes.
Occurence of load imbalance during a run of the Fully Parallel version
If we allow the Fully Parallel code to run for more than 1,000 iterations in the high
density - regular grid configuration, it appears than the time per iteration increases
rapidly. To investigate this undesirable phenomenon, we measure the time spent by
each process to step its particles forward (fig. 3-6).
Examining the results, we isolate three interesting processes (5, 9 and 17). While
their pusher times are initially close, they then drift rapidly from each other. Since
the code is only as fast as the slowest process, process 5 significantly slows down the
whole computation. What happened is simple: the partitioning was computed for
the initial, homogeneous plasma. Under the influence of the electric and magnetic
field, a fast redistribution takes place, which throws the work repartition between
processes off-balance.
The solution consists obviously in repartitioning the mesh
regularly. The whole repartitioning sequence, including particles redistribution and
statistics calculation, takes about 0.7-0.8s.
Thus, even with a repartitioning every
1000 iterations, the time penalty is less than 1%. After the initial redistribution, it
57
"Field solve r
" Pusher
- Particle Fxchange
a Statistics
Figure 3-7: Computation time breakdown - low density
* Field solver
* Pusher
a Particle Exchange
a Statistics
Figure 3-8: Computation time breakdown - high density
is likely that the need for such frequent repartitionings will disappear.
Breakdown of the computation time for the Fully Parallel code
Finally, timing the separate components of PTpicFP individually allows to establish the time profile of an iteration. We do it for the low (3-7) and high (3-8) density
configurations, on 24 processes and the regular mesh.
As expected, in the low-density case, the field solver is the most time-consuming
step, accounting for more than half of the global iteration time. For the high-density
case, the particle pusher consumes more than 80% of the iteration time. In both
cases, the time consumed by the particles exchange routine is negligible, which shows
58
that the principles that guided the design of the inter-process exchanges were sound.
Summary
PTpicFP shows promising performance. It outperforms the previous version substantially, scales up well in both low and high density plasma situations (unlike the
conventional PTpic, for which adding processes actually makes the code slower in a
low density context), and handles large meshes better. The overhead associated with
the periodic repartitionings and the exchange of particles between processes is very
small, even in the worst case.
3.2
3.2.1
Implicit particle pusher
Stability limitations of the leapfrog algorithm
A critical component of any particle-in-cell code is the algorithm used to advance
the particles (the "particle pusher"). All versions of PTpic up to now have used the
leapfrog algorithm:
X n+1
_ X" -+ Vn+1/2
(3.2)
Sn+1/2
_ Vn-1/2
+ F(xn)
.
At
where F is such that x = F(x).
In spite of its relatively low order of accuracy (2), it is still the most common
algorithm for PIC codes, due to its low computational cost (the force field F needs
to be evaluated only once per iteration, in contrast with higher order Runge-Kutta
methods for instance) and favorable energy conservation properties.
PTpic uses a
variant of the leapfrog known as the Boris algorithm, which further simplifies the
calculation when F is the Lorentz force. A detailed description is provided in Annex
A.
The traditional PIC method based on the leapfrog algorithm mandates the use
59
of a very small timestep, of the order of
to
10-12
10-11
s. This is several orders of
magnitude smaller than the timescale of any phenomenon of interest happening in
the thruster. With a more stable algorithm, it may be possible to use a much larger
timestep, without sacrificing the fidelity of the model.
In the field of ordinary differential equations (ODE), a common issue is the existence of so-called stiff differential equations, such as the following example (from
[22], p.727):
u' = 998u + 1998v
V = -999u - 1999V
(3.3)
u(0)
1
v(0)
0
The solution to this system is:
-e
u =2-
v=
eX +
10 00
X
(3.4)
-100x
As soon as one gets slightly away from x = 0, the e-100ox term becomes completely
negligible.
But if we are using an explicit Euler scheme (yn+i
are required to use a timestep h smaller than
2
=
= yn + hy' ), we
0.002, otherwise numerical
instability yields a result completely different from the exact solution.
Thus, with
the explicit Euler scheme, we are forced to use a timestep much smaller than the
characteristic variation time of the exact solution. This is exactly the same problem
we face with PIC codes: we have to resolve the cyclotronic motion of electrons and
plasma oscillations to avoid numerical instability, even if we have little interest in
them. In the field of ODEs, the solution to stiffness consists in using an implicit
method, where the derivative is evaluated at the new location Yn+1 instead of yn:
y7+1= y. + hy'
Thus it appears desirable to make PTpic implicit in order to
60
allow for a larger timestep (at constant -y) or a smaller, more realistic -y (at constant
timestep).
A framework for implicit PIC codes
3.2.2
There are two knobs on which one can act to make a leapfrog-based PIC method
implicit:
" The location Xn where the electric and the magnetic field are evaluated to
calculate the acceleration.
" The way we calculate E, since it is itself time-dependent.
A generic "implicit leapfrog" method thus reads:
Xn+1
Vn
1
2
-r
n
1
_ X
)
At
=
+- Vn+1/
2
.
q
-Enx)-
mn
(E"(xA)+
At
V
+1i/2)
+ V
2
-1/29
(3.5)
x
x
B
where R and E" are to be defined.
This formulation remains similar to the leapfrog Boris method and thus does not
encompass all possible implicit PIC schemes. These could be based on a higher-order
method, such as Runge-Kutta. However, we will limit our investigation to schemes
complying with 3.5.
In the straightforward, explicit leapfrog-based PIC (used in all versions of PTpic
up to now), we have
XT = xn,
and E" = E", where E" is calculated from the charge
density at t", pn:
V - En = $P
-760
3.2.3
(3.6)
Semi-implicit field solver
Cho recently reported very good results using a "semi-implicit field solver" ([6]).
The core of this method consists in using the particle densities as well as the current
61
densities in order to predict the charge distribution at t'+ 1 :
-En =
1+
1
I
(peAt)
F
2
pn --eoV
At . V
nj+ (1 -, vAt)ji
en
ie
+
eAt
-ji
me
x
B
B
N
(3.7)
where venj is the electron-neutral scattering collision frequency. One can recognize
on the right the ion and electron advection terms (including a collision attenuation
term for electrons) as well as the Hall current.
It does not however include any
pressure term.
This modification has been implemented in PTpic and tested on the CHT but
hasn't delivered a significant improvement in stability. Several reasons may explain
this:
" Cho used the 4 th order Runge-Kutta scheme to step the particles forward, while
the PTpic semi-implicit implementation retained the leapfrog algorithm, as ex-
plained in 3.5.
" The thruster studied by Cho was the SPT-100, a conventional Hall Thruster
with a lower magnetic field and plasma density than the CHT, and thus less
prone to numerical instabilities.
" Finally, instead of making ions and neutrals lighter as in PTpic, Cho increased
the electrons mass instead, which decreases the plasma and the cyclotron frequencies. But at the same time, he retained a small timestep of 10--s. This
might explain the increase in stability.
Another issue is that formula 3.7 is derived from a cold plasma model: it does not
include any pressure or temperature term. In the author's experience, the thermal
velocity of the electrons is not negligible with respect to their bulk velocity. Moreover,
given that a prominent numerical instability is linked to the ratio of the Debye length
to the grid spacing, it seems reasonable to expect that any scheme able to address this
instability should include a temperature term (since the Debye length itself depends
on electron temperature).
62
3.2.4
Particle Predictor-Corrector
The other implicit scheme investigated by the author, called the Particle PredictorCorrector (PPC) works at the particle level rather than with moments.
It can be
summarized as follows (0 E [0, 1]).
Vn+1/2
_
Vn-1/2
q
=-
(E'(x)+
Vn+1/2
Xn+l
-
x B(x'))
2
m
At
+ Vn-1/2
Xn
+ Vn+1/
Vp
-
1 + (1
At
2
0)pf
-
(3.8)
'Yo
V n+1/2
=/-
m
At
Xn+1
_
V n+1/2
-
q
_ V n-1/2
n
(En± (xn)
V+
+ Vn-1/2
2
x B(xn))
A
+ Vn+1/2
In this sequence:
e Particles are first advanced to a position x"2 1 with the classic, explicit leapfrog
("simulated push").
" The charge density pfl1 created by this new distribution is computed. Then
the composite charge density
Qpfl
+ (1 - Q)p" is calculated, as well as the
associated composite electric field En±.
" Particles are then "stepped back" to their initial position and advanced with
the standard leapfrog, but this time using the composite field E~ (final push).
The parameter 0 sets the "degree of implicitness" of the algorithm. A value of 0 for
instance corresponds to the classic, explicit leapfrog. Unlike the semi-implicit method,
we do not make any assumption about the nature of the plasma (cold or warm)
in order to obtain
p2,
and we fully account for the thermal motion of electrons.
Obviously, the computational cost is higher since we have to move the particles and
solve for the electric field twice at each iteration. However, this increase can be kept
reasonable by doing the simulated push for electrons only (since ions are much slower
63
and are not subject to numerical instabilities at the timesteps we use). Also, collisions
are not taken into account during the simulated push. Thus, the additional cost of
the PPC mostly consists in the extra call to the field sover; the extra electrons loop
executes fast enough to be of no concern.
The PPC was found to have a significant positive effect on stability. It was tested
on the CHT 193x157 mesh, with the parameters reported in table 3.4.
Table 3.4: Parameters for the Particle Predictor-Corrector assessment.
Anode flow rate (sccm)
Anode Voltage (V)
y
4
250
4
Super-particle size (ions and electrons)
f
Timestep (s)
W
1000
5 x 10-12
Fig. 3-9 represents the number of electrons superparticles in four cases: explicit
leapfrog, PPC 0 = 0.25, PPC 0 = 0.5, and PPC 0 = 1. The electrons count was
chosen because numerical instability increases the electron speed and temperature,
thus leading to a large, unphysical increase in ionization frequency and secondary
electron emission, which results in an exponential growth of the number of electrons.
Indeed we can see than the electron count diverges fast in the explicit leapfrog case,
while it remains bounded for the implicit cases. The extinction of the discharge after
the initial ionization surge was found to be caused by an accumulation of positive
charge on the dielectric surfaces, which gives rise to an unphysically high potential
in the chamber, expelling all ions and effectively ending the discharge. This problem
is likely unrelated to the PPC, but should be solved in order to allow for simulations
at low ' .
64
5000000
4500000
(U
t
4000000
, 3500000
: 3000000
-PC
0 2500000
U
-
(D 2000000
a)
o
E
Z
1.0
-EXPLICIT
4-,
PC 0.5
PC
025
1500000
1000000
500000
0
0.OOE+00
5.OOE-06
1.OOE-05
1.50E-05
2.OOE-05
2.50E-05
Time (s)
Figure 3-9: Number of electron super-particles in various PPC configurations.
65
66
Chapter 4
Experimental Characterization of
the Cylindrical Cusped-Field
Thruster
4.1
4.1.1
Cylindrical Cusped-Field Thruster overview
Background
The Cylindrical Cusped-Field Thruster (CCFT) builds on the experience gained
by the SPL with the Diverging Cusped-Field Thruster. Designed in 2008 by Daniel
Courtney ([9]), this device has been extensively tested at MIT, as well as in the Air
Force Research Laboratory facility at Edwards AFB, California.
The key features of the DCFT are the absence of centerpiece, the cusped magnetic
field created by permanent magnets rather than electromagnets, and obviously its
unusual divergent shape. The cusped magnetic field is a relatively recent concept in
Hall thruster engineering; it was first incorporated into the HEMP thruster ([14]).
Unlike the uniform radial magnetic field found in traditional Hall thrusters (see fig.
1-1), cusped-field thruster features regions of strong magnetic field gradient, called
cusps, between which electrons are axially confined, a process known as magnetic
bottling
([4],
p. 77). The strong magnetic field gradient from the center axis to the
67
Cathode
Ib
Axis
-1Ib
bb
E
s
Cusp
Magnets
N
Anode
Figure 4-1: Schematic DCFT diagram (from [9], p. 31
wall also keeps most electrons (and most ions, since they are electrostatically tied to
electrons) away from the walls, dramatically reducing erosion, except at the cusps.
The schematic structure of the magnetic field is given in fig. 4-1.
Testing showed good performance compared to other devices from the same category (see table 4.1), but several drawbacks were identified. First, the wide plume
divergence and the hollow conical plume structure (see
[9],
p.80) decrease the global
efficiency significantly. Also, the DCFT operates alternatively in a high anode current (HC) and a low anode current (LC) mode, and transitions are rather difficult to
predict ([9], p.62). The high current mode has a lower efficiency and features large
oscillations , first observed by Matlock ([18], p. 162), which may increase the erosion
of the boron nitride lining. Thus we want to keep the thruster operating in a single,
non-oscillatory mode.
The CCFT was built to address these deficiencies; a detailed account of the design
process can be found in [20]. It features a cylindrical discharge channel (37mm dia.
x 51.5 mm length) and an exit separatrix perpendicular to the center axis. These
features have been incorporated mainly in order to reduce the plume divergence angle.
Additionally, the cylindrical shape keeps the neutral density higher than in the DCFT,
68
Table 4.1: DCFT performance at the nominal operating point (from [20], p.30)
Anode Voltage (V)
550 V
Anode power (W)
Xenon mass flow rate (secm)
Anode efficiency
Anode potential (V)
242
8.5
44%
550
Specific impulse (s)
1640
thus increasing the ionization probability.
4.2
4.2.1
Experimental Setup
Vacuum chamber
All tests presented here were performed in SPL's largest - 1.6m diameter x 2.8m vacuum chamber, Astrovac. Pumping is provided by two CTI-Cryogenics cryopumps,
one OB-400 and one CT-10, both cooled down by a CTI-Cryogenics 9600 compressor.
Their combined pumping speed is rated at 7500 L/s for Argon.
Pressure during
high vacuum operation is manually recorded with an Instrutech IGM-401 Hornet
hot cathode vacuum gauge. Unless otherwise indicated, all pressures subsequently
mentioned were obtained with the gauge configured in Xenon mode. The background
pressure with the standard setup installed, no gas load and all valves closed is between
0.5 and 1 IpTorr.
Grade 5.0 (i.e. 99.999 % pure) Xenon is used for testing. Two OMEGA FMA6502-ST-XE flow controllers regulate the supply of gas to the anode and the cathode,
respectively.
The neutralizing cathode is a Busek BHT-1500, which can deliver an emission
current of up to 3A. It is a Barium Oxide impregnated cathode, and thus subject to
poisoning. To prevent this, a Restek 20600 high capacity oxygen trap and a Restek
22010 indicating oxygen trap are fitted to the cathode gas supply line; together, they
are rated to reduce the oxygen concentration to 0.1 ppm.
69
I ."
Figure 4-2: Simulated CCFT magnetic field (from 1201, p. 36)
4.2.2
Electrical setup
The basic electrical setup is presented in fig. 4.2.2. The main elements are the
anode (A), the cathode (C), the cathode heater (H) and the cathode keeper electrode
(K).
The anode and the keeper were powered by two computer-controlled Agilent
N5722A DC power supplies, rated for a maximum voltage of 600V and a maximum
current of 2.6A. The heater was supplied by an Agilent HPJA146OPS source, controlled manually.
4.2.3
Stage
Previous experiments conducted in Astrovac used a 2-axis stage sytem to move
the probes in the chamber, and a rotary stage to keep it pointed towards the thruster
(see [18], p.98)
.
Although this setup worked well, it was quite bulky and took a long
time to install. It was thus decided to build a much more compact and lightweight
70
Ac
HOC
Figure 4-3: Sketch of the electrical setup used for the CCFT experiments
stage. This stage provides azimuthal and radial mobility and is thus well suited to
azimuthal scans of the plume, which are the primary diagnostics for Hall thrusters.
The baseplate, the rail, the chariot and the instruments-bearing mast were machined
from 6061 Aluminum in order to minimize the outgassing. Both axes (azimuth and
radius) are powered by 6RPM DC motors; position feedback is provided by a rotary
and a linear encoder. All the electronics are commanded through an Arduino board.
Many components (motors, encoders, timing belt for the linear axis, ball bearings for
the chariot) are not certified for vacuum operation, and there were initially concerns
about whether the setup would outgass too much and contaminate the chamber.
These concerns were not founded since the background pressure (below 1 pfTorr)
was found to be small with respect to the pressure elevation caused by the injected
Xenon. This new stage has proved itself reliable and well suited for azimuthal-scan
based measurements. The only caveat is the vulnerability of the electronics to sparks.
4.2.4
Measurement devices
A Faraday Cup and a Retarding Potential Analyzer (RPA) were built for these
experiments. The Faraday cup (fig. 4-5) features a 9.91mm diameter stainless steel
collector plate, surrounded by a stainless steel guard ring.
71
Both are biased to a
Figure 4-4: Stage system and thruster stand
potential of about -27V during operation in order to repel electrons.
The RPA (see fig. 4-6) is made of an aluminum cylinder (outer diameter = 1")
onto which are screwed a backplate and a frontplate, both made of stainless steel.
This cylinder houses a stack of grids (made of a stainless steel mesh spot-welded on
a steel washer) separated by Macor washers (6). From front to back, the components
are: floating front grid (1), electron repelling grid (2), ion repelling grid (3), secondary
electron repelling grid (4) and collector plate (5). Insulation between the stack and
the hollow cylinder is provided by a layer of Kapton tape. During operation, the two
electron repelling grids are held to the same negative potential of about -27V, while the
ion repelling grid potential is progressively increased. Since only ions with an energy
higher than this potential can reach the collector, plotting the collected current as a
function of the repelling voltage gives access to the ion energy distribution. A typical
RPA profile, obtained during the tests reported in section 4.3.5, is provided in fig.
4-7.
72
Figure 4-5: The Faraday cup
Figure 4-6: Cutaway drawing of the RPA
*Data
Fit
points
2.5
2
1-5-
0,5[
0il
50
100
200
150
Ion repellng potential (V)
250
300
Figure 4-7: Collected current as a function of the repelling voltage - Configuration
A, 0
73
Figure 4-8: Graphite cap showing some fragments of molten steel
4.3
4.3.1
Results
Anode overheating
A salient feature noted during the operation of the CCFT is the very high temperature reached by the anode assembly. This assembly consists of a graphite cap
screwed on a metal threaded rod. During early testing, the temperature was high
enough to cause the graphite cap to glow red and the stainless steel threaded rod to
shear off. Pictures (figs. 4-8 and 4-9) clearly show evidence of melting at the location
where the steel sheared off. These observations are consistent with an anode temperature in excess of 800'C. All the stainless steel parts in the anode assembly were
subsequently replaced with molybdenum, which solved the rupture issue. However,
the red-orange glow (4-10) was observed during the whole test campaign.
Qualitative observation based on the intensity of the glow show that the temperature is strongly correlated to the anode voltage and, but much less to the anode
current or anode power level.
The most likely explanation for these unusually high temperatures is the magnetic
field, which funnels a large number of electrons in a narrow channel around the center
axis. It is worth emphasizing that it is the impact of these electrons which is the
cause,
74
Figure 4-9: Stainless steel rod showing evidence of melting (at the right tip)
Figure 4-10: Red glow emitted by the anode assembly during operation
75
and not ohmic heating, which is orders of magnitude too small to explain this effect.
4.3.2
Regime transitions
Over the course of the experiments, three distinct plume structures were observed.
" The first one, the most common, called the normal shape (fig. 4-11), is characterized by a globular, diffuse plume extending about 20 centimeters downstream
of the exit plane.
" The second one, dubbed the bag shape(fig. 4-12), is somewhat similar to the
normal plume, but features a distinct separation between a bright part centered
on the exit orifice, and a diffuse peripheral plasma around it.
* The third one has been called the jet regime (fig. 4-13). It is characterized by
a straight, solid plume coming out of the thruster, again surrounded by a much
dimmer plasma. This regime looks promising from the point of view of beam
collimation; unfortunately it is very elusive, and sometimes moving the probes
in the chamber is enough to cause a transition back to the normal or the bag
regime. Consequently, the only data available for this mode consists in a few
anode current-voltage points.
The factors governing the transition from one mode to another have not been conclusively identified. The thruster almost always starts in the normal mode; transition
to the "bag" mode, when it happens, usually takes place after several dozen minutes
of operation. One this transition has happened, the "bag" mode is rather stable and
will subsist even if the thruster is turned off for a few seconds. This suggests that the
transition may be due to a "warm-up" of the device. Temperature can change the
operation of the thruster by two main avenues: by affecting the velocity, and thus
the travel times of the xenon atoms; second, by modifying the magnetic field - the
Samarium-Cobalt S3212 magnets used in the CCFT are reported to have temperature coefficients of -0.03%/C for induction and -0.17%/'C for coercivity ([1]).
It
should be possible to fit some small magnetic field sensors between the magnets and
76
re 4-11: Normal plume
urp. 4-12: Bap n1ime
Figure 4-13: Jet plume
Table 4.2: Configurations for the voltage-current characteristics
Case
rha
Tc
P
Ik
A
B
C
D
E
F
G
H
I
J
K
(sccm Xe)
(sccm Xc)
(ptTorr, corrected for Xe)
(A)
4.0
6 .0
2.1
1.5
1.0
0.5
2.1
2.0
1.0
2.0
3.0
2.1
2.1
2.1
2.1
2.1
2.1
1.0
1.0
1.0
0.3
0.3
23.8-27.0
32.4
15.6 - 16.5
14.2 - 14.6
12.5 - 12.7
10.1 - 10.2
14.2 - 14.9
12.0 - 12.1
7.76
11.8 - 12.1
15
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.3
0.3
rha
Anode mass flow rate
rh,
Cathode mass flow rate
Background pressure
Cathode keeper current
Pb
Ik
the boron nitride lining in order to verify this hypothesis. Since three modes have
been observed, and the magnetic field has three cusps, the mode transitions may
correspond to a displacement of the ionization region from one cusp to another: the
well-focused jet beam would correspond to an ionization located at the cusp closest
to the anode, while the poorly collimated bag mode would come from the external
cusp.
4.3.3
Voltage-Current characteristics
The simplest measurement that can be made on a Hall thruster is the anode
voltage-current characteristic curve.
The configurations investigated are listed in
table 4.2.
The anode current appears to be rather weakly dependent on the anode voltage
(all other things being equal).
The correlation with the anode flow rate is much
stronger. This suggests that the CCFT achieves, at least at anode flow rates below 4
sccm, a near-complete ionization of the propellant, so that the only way to get more
current is to provide more gas. This hypothesis is compounded by the fact that the
78
12
1.1
1
0.9
0.8
C
-E
0.4
0.3
0.2
0.1
0
50
100
Anode voltage (V)
300
250
350
Figure 4-14: Anode current-voltage characteristics - I
0.6
0.5
0.4
-0- F
C
-~K
0.2
0.1
0
50
100
150 Anode voltage (V) 25o
300
350
400
Figure 4-15: Anode current-voltage characteristics - II
79
Table 4.3: Comparison of measured anode current to the anode current expected from
full single ionization and full double ionization of the propellant.
Case
Va
mha
la
Isingle Idouble
A
B
H
K
Va
Ia
Isingle
Idouble
(V)
150
100
300
300
(sccm Xe)
4.0
6.0
2.0
3.0
(A)
0.73
1.113
0.309
0.504
(A)
0.289
0.433
0.144
0.217
(A)
0.578
0.867
0.289
0.433
Anode voltage
Anode current
Current corresponding to the full single ionization of the anode flow.
Current corresponding to the full double ionization of the anode flow.
Note: the full single ionization of 1 sccm Xe corresponds to 0.0722399 A ([13], p.
464).
levels of anode current measured are close, or even exceed, what would result from
full double ionization of the propellant, as shown in table 4.3. However, the anode
current is likely boosted by the backflow of Xenon into the thruster caused by the
limitations of the pumping system. Testing in a facility with a higher pumping speed
would be appropriate.
4.3.4
Faraday Cup measurements
Several scans of the plume were performed with the Faraday cup described in
section 4.2.4, in order to ascertain the degree of collimation of the beam - obtaining a
solid plume was one of the main goals of the CCFT project ([20], p. 31). The probe
was moved on a circular arc from -90' to
+90' on each side of the center axis. An
interpolating current density function Ji(O) was then fitted to the data points and
this interpolant was integrated to compute the beam current Ib and its component
along the thrust axis Ib.
=I 27
0
80
Ji(6)sindO
(4.1)
Table 4.4: Configurations for the Faraday cup scans
Case
1
2
3
4
5
rha
(sccm Xe)
2.0
2.0
2.0
2.0
2.0
Va
la
Ik
Vk
(V)
250
325
250
250
250
(A)
0.282
0.319
0.265
0.298
0.258
(A)
0.5
0.5
0.5
0.5
19.7
(V)
19.3
19.3
23.7
19.7
0.5
Pb
(ptTorr, Xe corr.)
12.6
11.9
11.4-11.9
12.2
11.3-11.6
hc
(sccm Xe)
1.0
1.0
1.0
1.0
1
R
(cm)
26.0
26.0
12.9
12.9
12.9
fl~~VIN
1'
Vk
Cathode keeper voltage
R
Radius of the scan arc
1.4
1
0.8
0.6
C
0.4
0,2
0
-100
-80
-60
-40
-20
20
0
60
40
80
100
Angle (degree)
Figure 4-16: Plume current density - Case 1
Ibz =
2irR 2
(4.2)
J(0)cos0sin0d6
j/
The ratio I-z can be converted to a divergence angle
6
di,
via formula (4.3) which
quantifies the degree of collimation of the beam, a value of 0' corresponding to the
ideal case of a beam entirely directed along the main thrust axis.
cos(Odi,)
=
Ibz
(4.3)
IA
These results show that the CCFT never exhibits the hollow conical plume structure observed for the DCFT in high-current mode. However, the collimation of the
81
Mode
Normal
Normal
Normal
Bag
Bag
1.2
C
0.6
C
0)
0.4
0.2
0
-100
-80
-60
-40
-20
0
20
40
60
80
100
Angle (degree)
Figure 4-17: Plume current density - Case 2
4.5
3.5
E
3
2.5
C
2
V
C
1.5
0)
1
U
0.5
-100
-50
50
0
Angle
Figure 4-18: Plume current density - Case 3
Table 4.5: Integrated results for Faraday cup scans
0
Case
I
Iz
div
k
1
2
3
4
5
(A)
0.2637
0.2309
0.2194
0.1655
0.142805
(A)
0.1698
0.1382
0.1413
0.1010
0.0869
82
(0)
49.9
53.2
49.9
52.4
52.5
(-)
0.935
0.724
0.828
0.555
0.543
100
3.5
3
E
2.5
2
0.5
0,
100
50
0
-50
-100
Angle (degrees)
Figure 4-19: Plume current density - Case 4
3
2
E
C>
0,5
-100
-80
-60
-40
-20
0
20
40
60
Angle (degree)
Figure 4-20: Plume current density - Case 5
83
80
100
Table 4.6: Configurations for the RPA measurements
Case
A
B
C
mha
(sccm Xe)
2.0
2.0
2.0
Va
(V)
250
300
200
'a
(A)
0.311
0.281
0.422
Ik
(A)
0.5
0.5
0.5
Vk
(V)
17.7
25.5
23.2
he
(scem Xe)
1.0
1.0
1.0
P
Mode
(pTorr, Xe corr.)
17.9
13.6
17.2
Normal
Normal
Normal
beam is rather disappointing, with a characteristic angle on the order of 50'. We also
notice that the current utilization efficiency
(g)
drops sharply when the thruster goes
into the "bag" mode. However, contrary to what one may infer from visual inspection, the degree of collimation is similar in both cases. The poor current utilization
efficiency of the bag mode means that it should probably be avoided in the interest
of performance.
4.3.5
Retarding Potential Analyzer Measurements
RPA scans were taken for three different operating points, as reported in table
4.6, and at different off-axis angles.
An interpolating function was fitted to the raw data with the "smoothing spline"
option in the MATLAB Curve fitting toolbox; this interpolant was then differentiated
dI
and the resulting T
curve was normalized by its maximum value to enable easy
comparison across scans.
The normalized derivative curves show the expected aspect.
Ion energies are
clustered around a value equal or slightly lower than the anode potential in all cases.
The Full Width at Mid-Height (FWMH) is usually around 25V, and appears to be
larger for run A (one must keep in mind that the FWMH is not the ion temperature).
The literature reports that RPA scans sometimes show the existence of two distinct
ion populations: a high-energy population clustered around the anode voltage, and a
lower energy population created by charge-exchange collisions (see [13], p. 400). The
energy of the charge-exchange population decreases with the off-axis angle.
Given
the diffuse appearance of the plume in the normal mode (confirmed by Faraday cup
84
0
25
7>
p
0.8
0.6
0.4
0.2
-0.2
50
250
10
200
Ion repelling potential (V)
100
Figure 4-21: Normalized
dV
300
at different angles - case A
1.2
25"
80
1
0.8
JI
0.6
0.4
0
-02
50
0
100
250
300
150
200
Ion repelling potential (V)
350
400
y at different angles - case B
Figure 4-22: Normalized
1.2
25'
1
60"
0.8
0.6
0.4
0.2
0
-02
0
I
7-
50
100
200
150
Ion repelling potential (V)
Figure 4-23: Normalized
250
300
at different angles - case C
d
85
scans), one may expect to find a lot of low-energy charge-exchange ions, which are
created far from the anode and are expelled in a much more isotropic pattern than
the axially-directed high-energy ions. However, the RPA scans clearly prove that it
is not the case: ions with an energy close to the anode voltage dominate at all angles
and in all configurations. Thus, the causes for the CCFT plume divergence are yet
to be determined.
86
Chapter 5
Conclusion
5.1
Future work recommendations for PTpic
The improvements that can be made to PTpic broadly fall into two categories:
those related to performance (doing the same things faster), and those related to
the physical model being implemented (representing the physics of the thruster more
accurately).
The parallelization redesign work presented in this thesis is a major improvement
from the point of view of performance. However, in the author's opinion, there are
still several avenues to make PTpic faster. They are listed by order of importance.
5.1.1
Electric field solver
The experiments performed in section 3.1.3 prove that the electric field solver is
now the main obstacle to scalability for runs with a low number of plasma particles
(see fig. 3-3). This is problematic since a simulation usually runs in this low density
plasma regime for several hundred thousand iterations before the thruster "ignites".
Thus, removing this scalability limitation should be a priority of the future work on
PTpicFP. In order to understand how we may achieve this, some insight into the
internal operation of the field solver is required.
The purpose of the field solver is to solve a large linear system (5.1) at each
87
k 351
5
11
17
237
29
4
10
16
22
28
34
3
9
15
21
27
33
2
8
14
20
26
32
7
13
19
25
31
6
12
18
24
30
0
k
Figure 5-1: Mesh numbering scheme used by PTpic up to now
iteration.
A q
With the CHT 193x157 mesh for instance, there are N
(5.1)
= 21930 potential un-
knowns (this number is lower than the number of nodes since the anode and the
thruster body are held to a fixed potential), meaning that A is a 21930 x 21930 matrix. We take advantage of the fact that A is the same for all iterations by computing
a LU factorization A = LU during the initialization. Since L and U are triangular,
solving eq. 5.1 is much easier: it only requires a double backsubstitution.
general case, L and U have
-
In the
non-zero entries (the coefficients below and above the
diagonal, respectively). This number of non-zero entries in L and U is called the fillin. The number of operations requires to perform the backsubstitution is proportional
to the fill-in. Thus, minimizing it is critical in order to improve the performance of
the field solver.
Fortunately, A is a banded matrix, meaning that its coefficients are zero except in
a narrow diagonal band. This results from two things: the mesh nodes numbering,
and the finite difference scheme we use to discretize Poisson's equation.
In the current version of PTpic, the node numbering starts at the lower-left corner,
then goes column after column, from bottom to top (see 5-1).
88
Since we use a 9-point discretization of the laplacian ([121, p. 143), a given node
can only interact with its immediate neighbors.
Thus, the index number of any
node involved in the expression for the discretized laplacian at node n is within
[n - (N 1
+ 1); n + (Nj + 1)], N being the number of lines. For instance, in the
example given in fig. 5-1, we have Nj = 6. Then, the nodes involved in the discretized
laplacian at n = 21 have an index number between 14 and 28. Thus, A is banded,
with a bandwidth w = N + 1. This means that any coefficient which is more than 7V
away from the main diagonal is zero.
This banded structure drastically reduces the fill-in of the LU factors; more precisely it is now N
N/21930 = 148.
.
instead of N . For N, = 21930, this reduces the workload by
This is impressive but it is possible to do better.
With a clever
numbering of the mesh nodes, it is actually possible to bring the fill-in down to
Nlog 2 (Na).
One algorithm that can be used to compute this optimal numbering
scheme (or fill-in reducing ordering) is the nested-dissection algorithm.
Using a fill-in reducing ordering can thus reduce the operations count of the field
solver by
lo
. For Nu
=
21930, this is a factor of more than 10. This does not mean
that the field solver will be 10 times faster; in modern computers with a very high
clock frequency, the speed limiting part is often the loading of data into memory rather
than the actual calculation. However, it is certainly worth trying it. Implementation
would be easy since ParMETIS provides a function to calculate these fill-in reducing
orderings.
It is also important that PTpic users keep themselves informed of the state of the
art in numerical linear algebra. Right now we are using ScaLAPACK to handle the
LU factorization and the backsubstitutions, but it may very well be superseded by
another package in the near future.
5.1.2
Improved particle pushers
Further research on implicit particle pushers should also be a priority; being able
to relax the timestep and grid size stability limits would be a considerable advance.
Two interesting candidates have been identified and should be investigated in detail.
89
In both cases, the idea is again to predict the field at the next iteration En+1, then
advance the particles with a linear combination OE"+ 1 + (1 - O)E".
The first one is the Implicit Moments Method (IMM), developed at Los Alamos
National Laboratory by Brackbill and Forslund ([5]). Like Cho's semi-implicit method,
it uses particle moments to predict the electric field. The difference is that the IMM
takes the pressure into account, and that the predicted electric field and particle moments are iteratively refined until they fully agree (while the semi-implicit method
computes the electric field only once per iteration). Stable and accurate results at
large timesteps were reported by the authors.
The second possibility is the Direct Implicit Method (DIM) developed at Lawrence
Livermore National Laboratory by Langdon and Hewett ([16]). Unlike the IMM, it
works at the particle level and is thus conceptually close to the Particle PredictorCorrector method.
Finally, an alternative or complement to these methods is orbit-averaging, a technique which consists in filtering out the high-frequency electron oscillations by averaging their position over a group of iterations. More information can be found in
Denavit ([10]) and Cohen ([7] and [8]).
5.1.3
Refinement of the load metric
A somewhat tedious but useful task would consist in finding the optimal load
metric. Currently, the following formula is used to calculate the computational load
associated to the cell [k,j]:
load[k,j] =DTN2E * (Ni[k,j] + Ne[k,j]) + N,[k,j]
(5.2)
where DTN2E is the number of iterations that we allow to elapse before moving
the neutrals (values between 20 and 50 are customarily used), and Ni, N, and N,
are the number of ion, neutral and electron superparticles in the cell. The underlying
assumptions are that moving a superparticle of any type takes the same amount of
time, and that the cost of calculating the statistics for the cell is negligible. Both are
90
questionable. First, it is likely that moving a charged particle takes more time than
moving a neutral: the integration is more complicated because one must take the
electric and magnetic fields into account; also, the collision management functions
are different in each case, and are probably more time-consuming for the electrons
since they have to be individually checked for almost all the types of collision modeled
in PTpic. Second, the calculation of particle moments in a cell, even if it is empty,
does take some computation time. Thus, it may be appropriate to add a constant
cost to the load metric in eq. 5.2.
A good solution to this problem would consist in measuring the time spent on
each cell (including particle move and statistics calculation) directly, through timers
embedded into the code, rather than inferring it from a somewhat arbitrary formula
such as 5.2.
91
92
Appendix A
Particle pusher integrators
A.1
Leapfrog method
The leapfrog method is a numerical integrator for equations of the type:
(A.1)
x = F(x)
Its distinctive feature is that position and velocity are not calculated at the same
time, but with an offset equal to
Xn+1
:
_ X
-
Vn+1/
2
.At
(A.2)
V n+1/2
_
n-1/2
+ F(x")- At
Despite being only second-order accurate, it is widely used in PIC codes because
of the following properties:
2
1. Time-reversibility: if we start from (xn+1, Vn+1/ ) and apply the method with
timestep (-At), the result is (X.
V n- 1/ 2) exactly.
2. It is symplectic, i.e. it conserves exactly a modified particle Hamiltonian ([19],
[17]). Actually, this property may simply be a consequence of time-reversibility;
the literature I've seen is not conclusive on this point.
93
There is a formulation of the method which gives velocity and position at the
same time, it is known as the Synchronous Leapfrog Method. It may be useful for
theoretical studies, but is almost never used in actual codes.
Defining v" =nl/2+vn+1/2 and a' = F(x'), it reads:
1
~2
X
(A.3)
an+1
2a+
2
A.2
Boris method
The Boris method is an elegant reformulation of the leapfrog method when F is
the Lorentz force. In this case, the second leapfrog equation reads:
n-1/2
n+1/2
= mq(E +
'At
m
x B)
2
-
Let us introduce the auxiliary variables v
qE
-/
n+1/2I
7
-
rn 2
and v+
(A.4)
Vf+1/2
_
At
-2
Substituting into (A.4) yields (see [2])
V+
+-
=
(v+
At2m
A
We introduce t = 1B - At and s =
2t
v)xB
(A.5)
. Then it appears that, "from geometrical
considerations", v+ can be obtained by:
V
V-
+
V- X t
(A.6)
V- + V' X S
V+
Thus the sequence to obtain
e Calculate v-
= vn-1/2
+
q
rn
vn+1/
-t
2
from Vn-1/2 is:
2
e Then form v' = v- + v- x t
94
o Then calculate v+
SFinally, v'+1/2
= V+
v + v' x s
+
E - At
Hence the common description of Boris algorithm in the litterature: "Advance
velocity with half the electric field (step 1), then do the full magnetic rotation (steps
2 and 3), and finally add the other half of the electric field".
95
96
Bibliography
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2014.
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[6] Shinatora Cho, Kimiya Komurasaki, and Yoshihiro Arakawa. Kinetic particle
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