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Data-driven Closure for Fluid Models of Hall Thrusters Benjamin - - PowerPoint PPT Presentation

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University of Michigan Plasmadynamics and Electric Propulsion Laboratory Data-driven Closure for Fluid Models of Hall Thrusters Benjamin Jorns University of Michigan Princeton University ExB Workshop University of Michigan Plasmadynamics

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University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Data-driven Closure for Fluid Models of Hall Thrusters Benjamin Jorns University of Michigan Princeton University ExB Workshop

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University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

The Hall effect thruster for space propulsion

𝑭 π‘ͺ

𝐹 𝐹 𝐢

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SLIDE 3

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Problem of anomalous electron transport

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University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Problem of anomalous electron transport

πœ–π‘œπ‘— πœ–π‘’ + 𝛼 βˆ™ π‘œπ‘—π’—π‘— = 0 Ion continuity Ion momentum πœ– π‘›π‘—π‘œπ‘—π’—π‘— πœ–π‘’ + 𝛼 βˆ™ π‘›π‘—π‘œπ‘—π’—π‘—π’—π‘— = π‘Ÿ π‘œπ‘—π‘­ βˆ’ πœ‰π‘—π‘›π‘— 𝒗𝑗 βˆ’ 𝒗𝑓 Ohm’s Law Electron Energy Current conservation 0 = 𝛼 βˆ™ π‘Ÿπ‘œπ‘“ 𝒗𝑓 βˆ’ 𝒗𝑗 πœ‰π‘“π‘›π‘“π‘œπ‘“π’—π’‡ = βˆ’π‘Ÿπ‘œπ‘“πΉ βˆ’ 𝛼𝑄

𝑓 βˆ’ π‘Ÿπ‘œπ‘“π’—π’‡ Γ— 𝐢

Closed set of classical equations that can be evaluated with standard techniques 3 2 π‘œπ‘“ πœ–π‘ˆ

𝑓

πœ–π‘’ = βˆ’π‘Ÿπ‘œπ‘“π‘­ βˆ™ 𝒗𝑓 βˆ’ 𝛼 βˆ™ 5 2 π‘œπ‘“π‘ˆ

𝑓𝒗𝑓

+ 𝑹𝒇 + 3 2 π‘ˆ

𝑓𝛼 βˆ™ π‘œπ‘“π’—π‘“

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University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Problem of anomalous electron transport

πœ–π‘œπ‘— πœ–π‘’ + 𝛼 βˆ™ π‘œπ‘—π’—π‘— = 0 Ion continuity Ion momentum πœ– π‘›π‘—π‘œπ‘—π’—π‘— πœ–π‘’ + 𝛼 βˆ™ π‘›π‘—π‘œπ‘—π’—π‘—π’—π‘— = π‘Ÿ π‘œπ‘—π‘­ βˆ’ πœ‰π‘—π‘›π‘— 𝒗𝑗 βˆ’ 𝒗𝑓 Ohm’s Law Electron Energy Current conservation 0 = 𝛼 βˆ™ π‘Ÿπ‘œπ‘“ 𝒗𝑓 βˆ’ 𝒗𝑗 πœ‰π‘“π‘›π‘“π‘œπ‘“π’—π’‡ = βˆ’π‘Ÿπ‘œπ‘“πΉ βˆ’ 𝛼𝑄

𝑓 βˆ’ π‘Ÿπ‘œπ‘“π’—π’‡ Γ— 𝐢

Ie/ Id ~ 0.1%

Electron cross-field current from evaluating classical equations 3 2 π‘œπ‘“ πœ–π‘ˆ

𝑓

πœ–π‘’ = βˆ’π‘Ÿπ‘œπ‘“π‘­ βˆ™ 𝒗𝑓 βˆ’ 𝛼 βˆ™ 5 2 π‘œπ‘“π‘ˆ

𝑓𝒗𝑓

+ 𝑹𝒇 + 3 2 π‘ˆ

𝑓𝛼 βˆ™ π‘œπ‘“π’—π‘“

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University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Problem of anomalous electron transport

πœ–π‘œπ‘— πœ–π‘’ + 𝛼 βˆ™ π‘œπ‘—π’—π‘— = 0 Ion continuity Ion momentum πœ– π‘›π‘—π‘œπ‘—π’—π‘— πœ–π‘’ + 𝛼 βˆ™ π‘›π‘—π‘œπ‘—π’—π‘—π’—π‘— = π‘Ÿ π‘œπ‘—π‘­ βˆ’ πœ‰π‘—π‘›π‘— 𝒗𝑗 βˆ’ 𝒗𝑓 Ohm’s Law Electron Energy 3 2 π‘œπ‘“ πœ–π‘ˆ

𝑓

πœ–π‘’ = βˆ’π‘Ÿπ‘œπ‘“π‘­ βˆ™ 𝒗𝑓 βˆ’ 𝛼 βˆ™ 5 2 π‘œπ‘“π‘ˆ

𝑓𝒗𝑓

+ 𝑹𝒇 + 3 2 π‘ˆ

𝑓𝛼 βˆ™ π‘œπ‘“π’—π‘“

Current conservation 0 = 𝛼 βˆ™ π‘Ÿπ‘œπ‘“ 𝒗𝑓 βˆ’ 𝒗𝑗 πœ‰π‘“π‘›π‘“π‘œπ‘“π’—π’‡ = βˆ’π‘Ÿπ‘œπ‘“πΉ βˆ’ 𝛼𝑄

𝑓 βˆ’ π‘Ÿπ‘œπ‘“π’—π’‡ Γ— 𝐢

Ie/ Id ~ 0.1%

Actual cross-field current from evaluating equations 1000 x higher!

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SLIDE 7

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Problem of anomalous electron transport

πœ–π‘œπ‘— πœ–π‘’ + 𝛼 βˆ™ π‘œπ‘—π’—π‘— = 0 Ion continuity Ion momentum πœ– π‘›π‘—π‘œπ‘—π’—π‘— πœ–π‘’ + 𝛼 βˆ™ π‘›π‘—π‘œπ‘—π’—π‘—π’—π‘— = π‘Ÿ π‘œπ‘—π‘­ βˆ’ πœ‰π‘—π‘›π‘— 𝒗𝑗 βˆ’ 𝒗𝑓 Ohm’s Law Electron Energy 3 2 π‘œπ‘“ πœ–π‘ˆ

𝑓

πœ–π‘’ = βˆ’π‘Ÿ πœπ‘— 𝑛𝑗 𝑭 βˆ™ 𝒗𝑓 βˆ’ 𝛼 βˆ™ 5 2 π‘œπ‘“π‘ˆ

𝑓𝒗𝑓

+ 𝑹𝒇 + 3 2 π‘ˆ

𝑓𝛼 βˆ™ π‘œπ‘“π’—π‘“

Current conservation 0 = 𝛼 βˆ™ π‘Ÿπ‘œπ‘“ 𝒗𝑓 βˆ’ 𝒗𝑗

πœ‰π‘“π‘›π‘“π‘œπ‘“π’—π’‡ = βˆ’π‘Ÿπ‘œπ‘“πΉ βˆ’ 𝛼𝑄

𝑓 βˆ’ π‘Ÿπ‘œπ‘“π’—π’‡ Γ— 𝐢 βˆ’π‘œπ‘“π‘›π‘“ πœ‰π΅π‘‚ 𝒗𝒇,

Ie/ Id ~ 0.1%

Actual cross-field current from evaluating equations 1000 x higher! Need to introduce ad hoc factor β€œAnomalous collision frequency ”

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University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Problem of anomalous electron transport

πœ–π‘œπ‘— πœ–π‘’ + 𝛼 βˆ™ π‘œπ‘—π’—π‘— = 0 Ion continuity Ion momentum πœ– π‘›π‘—π‘œπ‘—π’—π‘— πœ–π‘’ + 𝛼 βˆ™ π‘›π‘—π‘œπ‘—π’—π‘—π’—π‘— = π‘Ÿ π‘œπ‘—π‘­ βˆ’ πœ‰π‘—π‘›π‘— 𝒗𝑗 βˆ’ 𝒗𝑓 Ohm’s Law Electron Energy 3 2 π‘œπ‘“ πœ–π‘ˆ

𝑓

πœ–π‘’ = βˆ’π‘Ÿ πœπ‘— 𝑛𝑗 𝑭 βˆ™ 𝒗𝑓 βˆ’ 𝛼 βˆ™ 5 2 π‘œπ‘“π‘ˆ

𝑓𝒗𝑓

+ 𝑹𝒇 + 3 2 π‘ˆ

𝑓𝛼 βˆ™ π‘œπ‘“π’—π‘“

Current conservation 0 = 𝛼 βˆ™ π‘Ÿπ‘œπ‘“ 𝒗𝑓 βˆ’ 𝒗𝑗 Need to introduce ad hoc factor

Ie/ Id ~ 10%

Anomalous friction term promotes additional cross-field current

πœ‰π‘“π‘›π‘“π‘œπ‘“π’—π’‡ = βˆ’π‘Ÿπ‘œπ‘“πΉ βˆ’ 𝛼𝑄

𝑓 βˆ’ π‘Ÿπ‘œπ‘“π’—π’‡ Γ— 𝐢 βˆ’π‘œπ‘“π‘›π‘“ πœ‰π΅π‘‚ 𝒗𝒇,

β€œAnomalous collision frequency ”

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SLIDE 9

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Problem of anomalous electron transport

πœ–π‘œπ‘— πœ–π‘’ + 𝛼 βˆ™ π‘œπ‘—π’—π‘— = 0 Ion continuity Ion momentum πœ– π‘›π‘—π‘œπ‘—π’—π‘— πœ–π‘’ + 𝛼 βˆ™ π‘›π‘—π‘œπ‘—π’—π‘—π’—π‘— = π‘Ÿ π‘œπ‘—π‘­ βˆ’ πœ‰π‘—π‘›π‘— 𝒗𝑗 βˆ’ 𝒗𝑓 Ohm’s Law Electron Energy 3 2 π‘œπ‘“ πœ–π‘ˆ

𝑓

πœ–π‘’ = βˆ’π‘Ÿ πœπ‘— 𝑛𝑗 𝑭 βˆ™ 𝒗𝑓 βˆ’ 𝛼 βˆ™ 5 2 π‘œπ‘“π‘ˆ

𝑓𝒗𝑓

+ 𝑹𝒇 + 3 2 π‘ˆ

𝑓𝛼 βˆ™ π‘œπ‘“π’—π‘“

Current conservation 0 = 𝛼 βˆ™ π‘Ÿπ‘œπ‘“ 𝒗𝑓 βˆ’ 𝒗𝑗 Need to introduce ad hoc factor

Ie/ Id ~ 10%

Anomalous friction term promotes additional cross-field current Problem: introducing ad-hoc term opens set of equations (too many unknowns)

πœ‰π‘“π‘›π‘“π‘œπ‘“π’—π’‡ = βˆ’π‘Ÿπ‘œπ‘“πΉ βˆ’ 𝛼𝑄

𝑓 βˆ’ π‘Ÿπ‘œπ‘“π’—π’‡ Γ— 𝐢 βˆ’π‘œπ‘“π‘›π‘“ πœ‰π΅π‘‚ 𝒗𝒇,

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SLIDE 10

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Problem of anomalous electron transport

πœ–π‘œπ‘— πœ–π‘’ + 𝛼 βˆ™ π‘œπ‘—π’—π‘— = 0 Ion continuity Ion momentum πœ– π‘›π‘—π‘œπ‘—π’—π‘— πœ–π‘’ + 𝛼 βˆ™ π‘›π‘—π‘œπ‘—π’—π‘—π’—π‘— = π‘Ÿ π‘œπ‘—π‘­ βˆ’ πœ‰π‘—π‘›π‘— 𝒗𝑗 βˆ’ 𝒗𝑓 Ohm’s Law Electron Energy 3 2 π‘œπ‘“ πœ–π‘ˆ

𝑓

πœ–π‘’ = βˆ’π‘Ÿ πœπ‘— 𝑛𝑗 𝑭 βˆ™ 𝒗𝑓 βˆ’ 𝛼 βˆ™ 5 2 π‘œπ‘“π‘ˆ

𝑓𝒗𝑓

+ 𝑹𝒇 + 3 2 π‘ˆ

𝑓𝛼 βˆ™ π‘œπ‘“π’—π‘“

Current conservation 0 = 𝛼 βˆ™ π‘Ÿπ‘œπ‘“ 𝒗𝑓 βˆ’ 𝒗𝑗 Need to introduce ad hoc factor

Ie/ Id ~ 10%

Anomalous friction term promotes additional cross-field current Problem: introducing ad-hoc term opens set of equations (too many unknowns)

πœ‰π‘“π‘›π‘“π‘œπ‘“π’—π’‡ = βˆ’π‘Ÿπ‘œπ‘“πΉ βˆ’ 𝛼𝑄

𝑓 βˆ’ π‘Ÿπ‘œπ‘“π’—π’‡ Γ— 𝐢 βˆ’π‘œπ‘“π‘›π‘“ πœ‰π΅π‘‚ 𝒗𝒇,

We need a functional form for πœ‰π΅π‘‚(π‘ˆ

𝑓, π‘œπ‘“, . . )

that depends on classical fluid parameters

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SLIDE 11

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: first-principles

Ԧ 𝐺

𝐡𝑂 = βˆ’π‘œπ‘“π‘›π‘“ πœ‰π΅π‘‚π’—π‘“

*N. Gascon, M. Dudeck, and S. Barral, PoP, vol. 10, no. 10, 2003 † J. M. Fife and M. Martinez-Sanchez/ IEPC-95-24 ‑ M. A. Cappelli, C. V. Young, E. Cha, and E. Fernandez, PoP, vol. 22, no. 11, 2015.

  • T. Lafleur, S. D. Baalrud, and P. Chabert, PoP, vol. 23, no. 5, 2016 .
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SLIDE 12

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: first-principles

Wall Interactions* Bohm Diffusion† Instabilities‑

πœ‰π΅π‘‚= 1 𝐿 πœ•π‘‘π‘“ πœ‰π΅π‘‚= 𝛼 βˆ™ π‘£π‘—π‘œπ‘“π‘ˆ

𝑓

π‘›π‘“π‘‘π‘‘π‘œπ‘“v𝑒𝑓 πœ‰π΅π‘‚= 𝛾 π‘ˆ

𝑓

πœ‰π΅π‘‚= 1 𝐿 πœ•π‘‘π‘“ v𝑒𝑓 𝑑𝑑

2

*N. Gascon, M. Dudeck, and S. Barral, PoP, vol. 10, no. 10, 2003 † J. M. Fife and M. Martinez-Sanchez/ IEPC-95-24 ‑ M. A. Cappelli, C. V. Young, E. Cha, and E. Fernandez, PoP, vol. 22, no. 11, 2015.

  • T. Lafleur, S. D. Baalrud, and P. Chabert, PoP, vol. 23, no. 5, 2016 .

Ԧ 𝐺

𝐡𝑂 = βˆ’π‘œπ‘“π‘›π‘“ πœ‰π΅π‘‚π’—π‘“

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SLIDE 13

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: first-principles

Wall Interactions* Bohm Diffusion† Instabilities‑

πœ‰π΅π‘‚= 1 𝐿 πœ•π‘‘π‘“ πœ‰π΅π‘‚= 𝛼 βˆ™ π‘£π‘—π‘œπ‘“π‘ˆ

𝑓

π‘›π‘“π‘‘π‘‘π‘œπ‘“v𝑒𝑓 πœ‰π΅π‘‚= 𝛾 π‘ˆ

𝑓

πœ‰π΅π‘‚= 1 𝐿 πœ•π‘‘π‘“ v𝑒𝑓 𝑑𝑑

2

*N. Gascon, M. Dudeck, and S. Barral, PoP, vol. 10, no. 10, 2003 † J. M. Fife and M. Martinez-Sanchez/ IEPC-95-24 ‑ M. A. Cappelli, C. V. Young, E. Cha, and E. Fernandez, PoP, vol. 22, no. 11, 2015.

  • T. Lafleur, S. D. Baalrud, and P. Chabert, PoP, vol. 23, no. 5, 2016 .

Ԧ 𝐺

𝐡𝑂 = βˆ’π‘œπ‘“π‘›π‘“ πœ‰π΅π‘‚π’—π‘“

Closure models from first-principles are potentially predictive Models have to date have had limitations, yielding qualitative agreement over only limited range of conditions Possible that reality is too complicated or models or too reduced fidelity

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SLIDE 14

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: first-principles

Wall Interactions* Bohm Diffusion† Instabilities‑

πœ‰π΅π‘‚= 1 𝐿 πœ•π‘‘π‘“ πœ‰π΅π‘‚= 𝛼 βˆ™ π‘£π‘—π‘œπ‘“π‘ˆ

𝑓

π‘›π‘“π‘‘π‘‘π‘œπ‘“v𝑒𝑓 πœ‰π΅π‘‚= 𝛾 π‘ˆ

𝑓

πœ‰π΅π‘‚= 1 𝐿 πœ•π‘‘π‘“ v𝑒𝑓 𝑑𝑑

2

*N. Gascon, M. Dudeck, and S. Barral, PoP, vol. 10, no. 10, 2003 † J. M. Fife and M. Martinez-Sanchez/ IEPC-95-24 ‑ M. A. Cappelli, C. V. Young, E. Cha, and E. Fernandez, PoP, vol. 22, no. 11, 2015.

  • T. Lafleur, S. D. Baalrud, and P. Chabert, PoP, vol. 23, no. 5, 2016 .

Ԧ 𝐺

𝐡𝑂 = βˆ’π‘œπ‘“π‘›π‘“ πœ‰π΅π‘‚π’—π‘“

Closure models from first-principles are potentially predictive Alternative: empirical form for collision frequency Models have to date have had limitations, yielding qualitative agreement over only limited range of conditions Possible that reality is too complicated or models or too reduced fidelity

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SLIDE 15

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: empirical estimate

Experiment

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SLIDE 16

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: empirical estimate

πœ‘π΅π‘‚

31

πœ‘π΅π‘‚

32

πœ‘π΅π‘‚

33

πœ‘π΅π‘‚

21

πœ‘π΅π‘‚

22

πœ‘π΅π‘‚

23

πœ‘π΅π‘‚

11

πœ‘π΅π‘‚

12

πœ‘π΅π‘‚

13

Experiment

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SLIDE 17

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: empirical estimate

πœ‘π΅π‘‚

31=1

πœ‘π΅π‘‚

32=5

πœ‘π΅π‘‚

33=.1

πœ‘π΅π‘‚

21=100

πœ‘π΅π‘‚

22=29

πœ‘π΅π‘‚

23=50

πœ‘π΅π‘‚

11 =100

πœ‘π΅π‘‚

12 =3

πœ‘π΅π‘‚

13 =89

Experiment Iteration #1

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SLIDE 18

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: empirical estimate

πœ‘π΅π‘‚

31=1

πœ‘π΅π‘‚

32=5

πœ‘π΅π‘‚

33=.1

πœ‘π΅π‘‚

21=100

πœ‘π΅π‘‚

22=29

πœ‘π΅π‘‚

23=50

πœ‘π΅π‘‚

11 =100

πœ‘π΅π‘‚

12 =3

πœ‘π΅π‘‚

13 =89

Experiment Model Iteration #1

slide-19
SLIDE 19

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: empirical estimate

πœ‘π΅π‘‚

31=5

πœ‘π΅π‘‚

32=1.

πœ‘π΅π‘‚

33=18

πœ‘π΅π‘‚

21=26.

πœ‘π΅π‘‚

22=42

πœ‘π΅π‘‚

23=0.5

πœ‘π΅π‘‚

11 =16

πœ‘π΅π‘‚

12 =17

πœ‘π΅π‘‚

13 =3

Experiment Iteration #2

slide-20
SLIDE 20

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: empirical estimate

πœ‘π΅π‘‚

31=5

πœ‘π΅π‘‚

32=1.

πœ‘π΅π‘‚

33=18

πœ‘π΅π‘‚

21=26.

πœ‘π΅π‘‚

22=42

πœ‘π΅π‘‚

23=0.5

πœ‘π΅π‘‚

11 =16

πœ‘π΅π‘‚

12 =17

πœ‘π΅π‘‚

13 =3

Experiment Model Iteration #2

slide-21
SLIDE 21

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: empirical estimate

πœ‘π΅π‘‚

31=1

πœ‘π΅π‘‚

32=5.

πœ‘π΅π‘‚

33=3

πœ‘π΅π‘‚

21=100.

πœ‘π΅π‘‚

22=3.

πœ‘π΅π‘‚

23=0.5

πœ‘π΅π‘‚

11 =1

πœ‘π΅π‘‚

12 =8.

πœ‘π΅π‘‚

13 =2.

Experiment Iteration #3

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SLIDE 22

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: empirical estimate

πœ‘π΅π‘‚

31=1

πœ‘π΅π‘‚

32=5.

πœ‘π΅π‘‚

33=3

πœ‘π΅π‘‚

21=100.

πœ‘π΅π‘‚

22=3.

πœ‘π΅π‘‚

23=0.5

πœ‘π΅π‘‚

11 =1

πœ‘π΅π‘‚

12 =8.

πœ‘π΅π‘‚

13 =2.

Experiment Model Iteration #3

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SLIDE 23

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: empirical estimate

πœ‘π΅π‘‚

31=1

πœ‘π΅π‘‚

32=5.

πœ‘π΅π‘‚

33=3

πœ‘π΅π‘‚

21=100.

πœ‘π΅π‘‚

22=3.

πœ‘π΅π‘‚

23=0.5

πœ‘π΅π‘‚

11 =1

πœ‘π΅π‘‚

12 =8.

πœ‘π΅π‘‚

13 =2.

Model Iteration #3

  • Yields excellent agreement with

experimental results for a given

  • perating condition
  • Collision frequency is specified
  • empirically. Only applicable for data

set used for validation

slide-24
SLIDE 24

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: empirical estimate

πœ‘π΅π‘‚

31=1

πœ‘π΅π‘‚

32=5.

πœ‘π΅π‘‚

33=3

πœ‘π΅π‘‚

21=100.

πœ‘π΅π‘‚

22=3.

πœ‘π΅π‘‚

23=0.5

πœ‘π΅π‘‚

11 =1

πœ‘π΅π‘‚

12 =8.

πœ‘π΅π‘‚

13 =2.

Model Iteration #3

  • Yields excellent agreement with

experimental results for a given

  • perating condition
  • Collision frequency is specified
  • empirically. Only applicable for data

set used for validation

  • To date, empirical models have not

been predictive

slide-25
SLIDE 25

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Closures for anomalous collision frequency: empirical estimate

πœ‘π΅π‘‚

31=1

πœ‘π΅π‘‚

32=5.

πœ‘π΅π‘‚

33=3

πœ‘π΅π‘‚

21=100.

πœ‘π΅π‘‚

22=3.

πœ‘π΅π‘‚

23=0.5

πœ‘π΅π‘‚

11 =1

πœ‘π΅π‘‚

12 =8.

πœ‘π΅π‘‚

13 =2.

Model Iteration #3 Hypothesis: we can use empirical data to generate a functional form, πœ‰π΅π‘‚(π‘ˆ

𝑓, π‘œπ‘“, . . )

  • Yields excellent agreement with

experimental results for a given

  • perating condition
  • Collision frequency is specified
  • empirically. Only applicable for data

set used for validation

  • To date, empirical models have not

been predictive

slide-26
SLIDE 26

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Model from regression

πœ‘π΅π‘‚

31

πœ‘π΅π‘‚

32

πœ‘π΅π‘‚

33

πœ‘π΅π‘‚

21

πœ‘π΅π‘‚

22

πœ‘π΅π‘‚

23

πœ‘π΅π‘‚

11

πœ‘π΅π‘‚

12

πœ‘π΅π‘‚

13

ΰ΅― (πœ‘π΅π‘‚

31 , π‘ˆ 𝑓 31 , π‘œπ‘“ 31, . . .

ΰ΅― (πœ‘π΅π‘‚

32 , π‘ˆ 𝑓 32 , π‘œπ‘“ 32, . . .

ΰ΅― (πœ‘π΅π‘‚

33 , π‘ˆ 𝑓 33 , π‘œπ‘“ 33, . . .

ΰ΅― (πœ‘π΅π‘‚

23 , π‘ˆ 𝑓 23 , π‘œπ‘“ 23, . . .

) (πœ‘π΅π‘‚

22 , π‘ˆ 𝑓 22 , π‘œπ‘“ 22, . . .

ΰ΅― (πœ‘π΅π‘‚

13 , π‘ˆ 𝑓 13 , π‘œπ‘“ 13, . . .

) (πœ‘π΅π‘‚

12 , π‘ˆ 𝑓 12 , π‘œπ‘“ 12, . . .

) (πœ‘π΅π‘‚

11 , π‘ˆ 𝑓 11 , π‘œπ‘“ 11, . . .

) (πœ‘π΅π‘‚

21 , π‘ˆ 𝑓 21 , π‘œπ‘“ 21, . . .

Each point from empirical model yields data point

slide-27
SLIDE 27

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Model from regression

πœ‘π΅π‘‚

31

πœ‘π΅π‘‚

32

πœ‘π΅π‘‚

33

πœ‘π΅π‘‚

21

πœ‘π΅π‘‚

22

πœ‘π΅π‘‚

23

πœ‘π΅π‘‚

11

πœ‘π΅π‘‚

12

πœ‘π΅π‘‚

13

ΰ΅― (πœ‘π΅π‘‚

31 , π‘ˆ 𝑓 31 , π‘œπ‘“ 31, . . .

ΰ΅― (πœ‘π΅π‘‚

32 , π‘ˆ 𝑓 32 , π‘œπ‘“ 32, . . .

ΰ΅― (πœ‘π΅π‘‚

33 , π‘ˆ 𝑓 33 , π‘œπ‘“ 33, . . .

ΰ΅― (πœ‘π΅π‘‚

23 , π‘ˆ 𝑓 23 , π‘œπ‘“ 23, . . .

) (πœ‘π΅π‘‚

22 , π‘ˆ 𝑓 22 , π‘œπ‘“ 22, . . .

ΰ΅― (πœ‘π΅π‘‚

13 , π‘ˆ 𝑓 13 , π‘œπ‘“ 13, . . .

) (πœ‘π΅π‘‚

12 , π‘ˆ 𝑓 12 , π‘œπ‘“ 12, . . .

) (πœ‘π΅π‘‚

11 , π‘ˆ 𝑓 11 , π‘œπ‘“ 11, . . .

) (πœ‘π΅π‘‚

21 , π‘ˆ 𝑓 21 , π‘œπ‘“ 21, . . .

Each point from empirical model yields data point Maybe there is a function, πœ‰π΅π‘‚ π‘ˆ

𝑓, π‘œπ‘“, . . . , that fits the data

slide-28
SLIDE 28

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Model from regression

πœ‘π΅π‘‚

31

πœ‘π΅π‘‚

32

πœ‘π΅π‘‚

33

πœ‘π΅π‘‚

21

πœ‘π΅π‘‚

22

πœ‘π΅π‘‚

23

πœ‘π΅π‘‚

11

πœ‘π΅π‘‚

12

πœ‘π΅π‘‚

13

ΰ΅― (πœ‘π΅π‘‚

31 , π‘ˆ 𝑓 31 , π‘œπ‘“ 31, . . .

ΰ΅― (πœ‘π΅π‘‚

32 , π‘ˆ 𝑓 32 , π‘œπ‘“ 32, . . .

ΰ΅― (πœ‘π΅π‘‚

33 , π‘ˆ 𝑓 33 , π‘œπ‘“ 33, . . .

ΰ΅― (πœ‘π΅π‘‚

23 , π‘ˆ 𝑓 23 , π‘œπ‘“ 23, . . .

) (πœ‘π΅π‘‚

22 , π‘ˆ 𝑓 22 , π‘œπ‘“ 22, . . .

ΰ΅― (πœ‘π΅π‘‚

13 , π‘ˆ 𝑓 13 , π‘œπ‘“ 13, . . .

) (πœ‘π΅π‘‚

12 , π‘ˆ 𝑓 12 , π‘œπ‘“ 12, . . .

) (πœ‘π΅π‘‚

11 , π‘ˆ 𝑓 11 , π‘œπ‘“ 11, . . .

) (πœ‘π΅π‘‚

21 , π‘ˆ 𝑓 21 , π‘œπ‘“ 21, . . .

Maybe there is a function, πœ‰π΅π‘‚ π‘ˆ

𝑓, π‘œπ‘“, . . . , that fits the data

πœ‰π΅π‘‚= 𝑑1π‘ˆ

𝑓 + 𝑑2 π‘œπ‘“ 2 + 𝑑3ui

We do not know a priori what the functional form should be It is almost impossible to guess from inspection: there are 30 variables to choose from

slide-29
SLIDE 29

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Model from regression

πœ‘π΅π‘‚

31

πœ‘π΅π‘‚

32

πœ‘π΅π‘‚

33

πœ‘π΅π‘‚

21

πœ‘π΅π‘‚

22

πœ‘π΅π‘‚

23

πœ‘π΅π‘‚

11

πœ‘π΅π‘‚

12

πœ‘π΅π‘‚

13

ΰ΅― (πœ‘π΅π‘‚

31 , π‘ˆ 𝑓 31 , π‘œπ‘“ 31, . . .

ΰ΅― (πœ‘π΅π‘‚

32 , π‘ˆ 𝑓 32 , π‘œπ‘“ 32, . . .

ΰ΅― (πœ‘π΅π‘‚

33 , π‘ˆ 𝑓 33 , π‘œπ‘“ 33, . . .

ΰ΅― (πœ‘π΅π‘‚

23 , π‘ˆ 𝑓 23 , π‘œπ‘“ 23, . . .

) (πœ‘π΅π‘‚

22 , π‘ˆ 𝑓 22 , π‘œπ‘“ 22, . . .

ΰ΅― (πœ‘π΅π‘‚

13 , π‘ˆ 𝑓 13 , π‘œπ‘“ 13, . . .

) (πœ‘π΅π‘‚

12 , π‘ˆ 𝑓 12 , π‘œπ‘“ 12, . . .

) (πœ‘π΅π‘‚

11 , π‘ˆ 𝑓 11 , π‘œπ‘“ 11, . . .

) (πœ‘π΅π‘‚

21 , π‘ˆ 𝑓 21 , π‘œπ‘“ 21, . . .

Maybe there is a function, πœ‰π΅π‘‚ π‘ˆ

𝑓, π‘œπ‘“, . . . , that fits the data

πœ‰π΅π‘‚= 𝑑1π‘ˆ

𝑓 + 𝑑2 π‘œπ‘“ 2 + 𝑑3ui

We do not know a priori what the functional form should be It is almost impossible to guess from inspection: there are 30 variables to choose from Solution: use machine learning to regress data

slide-30
SLIDE 30

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Regression with machine learning

*I. G. Mikellides and I. Katz, Phys. Rev. E vol. 86, no. 4, pp. 1–17, 2012.

slide-31
SLIDE 31

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Regression with machine learning

Generate datasets from empirically validated codes 7 operating conditions from 4 different thrusters from Hall2De*: 700 data points

*I. G. Mikellides and I. Katz, Phys. Rev. E vol. 86, no. 4, pp. 1–17, 2012.

slide-32
SLIDE 32

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Regression with machine learning

Generate datasets from empirically validated codes 7 operating conditions from 4 different thrusters from Hall2De*: 700 data points Prepare datasets for regression 8 normalized lengthscales, velocities, and frequencies

*I. G. Mikellides and I. Katz, Phys. Rev. E vol. 86, no. 4, pp. 1–17, 2012.

slide-33
SLIDE 33

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Regression with machine learning

Generate datasets from empirically validated codes 7 operating conditions from 4 different thrusters from Hall2De*: 700 data points Prepare datasets for regression Apply ML regression algorithm 8 normalized lengthscales, velocities, and frequencies

*I. G. Mikellides and I. Katz, Phys. Rev. E vol. 86, no. 4, pp. 1–17, 2012. Image credit: M. Quade, Phys Rev. E. no 1. 2016

DataModeler symbolic regression from Evolved Analytics

slide-34
SLIDE 34

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Symbolic regression Pareto front

Models for πœ‰π΅π‘‚

slide-35
SLIDE 35

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Symbolic regression Pareto front

Models for πœ‰π΅π‘‚

slide-36
SLIDE 36

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Symbolic regression Pareto front

Models for πœ‰π΅π‘‚

Simple but poor fit to data

slide-37
SLIDE 37

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Symbolic regression Pareto front

Models for πœ‰π΅π‘‚

Simple but poor fit to data Complex and

  • verfits data
slide-38
SLIDE 38

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Symbolic regression Pareto front

Models for πœ‰π΅π‘‚

Simple but poor fit to data Complex and

  • verfits data

Compromise model

slide-39
SLIDE 39

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Symbolic regression Pareto front

Models for πœ‰π΅π‘‚

Simple but poor fit to data Complex and

  • verfits data

Models for analysis drawn from β€œknee”

slide-40
SLIDE 40

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Comparison of models to training data

Empirical data from one training dataset Normalized frequency on channel centerline

  • B. Jorns., PSST, 27 (10), 2018
slide-41
SLIDE 41

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Comparison of models to training data

Empirical data from one training dataset Normalized frequency on channel centerline

Note: model collision frequency independent of position

  • B. Jorns., PSST, 27 (10), 2018
slide-42
SLIDE 42

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Comparison of models to training data

Response plot of model from Pareto front

  • B. Jorns., PSST, 27 (10), 2018
slide-43
SLIDE 43

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Comparison of models to training data

Response plot of model from Pareto front

Correspondence over four orders of magnitude shows promise of ML regression

  • B. Jorns., PSST, 27 (10), 2018
slide-44
SLIDE 44

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Comparison of models to training data

Response plot of model from Pareto front

Correspondence over four orders of magnitude shows promise of ML regression Possible issue: model overfits data Is model from ML regression predictive?

slide-45
SLIDE 45

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Predictive capability of model

Data from thruster not included in training dataset

Note: model collision frequency independent of position

Empirical data from test dataset

slide-46
SLIDE 46

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Predictive capability of model

Data from thruster not included in training dataset

Note: model collision frequency independent of position

Empirical data from test dataset

Agreement not as critical here

slide-47
SLIDE 47

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Predictive capability of model

Response plot of ML model to test data

Even though ML model is fit to other data, it can predict collision frequency in new thruster and operating condition

slide-48
SLIDE 48

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Comparison of ML to first-principles models

First-Principles Model I

slide-49
SLIDE 49

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Comparison of ML to first-principles models

First-Principles Model I First-Principles Model II

slide-50
SLIDE 50

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Comparison of ML to first-principles models

First-Principles Model I First-Principles Model II Data-driven Model III ML model has best correspondence and predictive capability of proposed closures

slide-51
SLIDE 51

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Comparison of ML to first-principles models

ML model has best correspondence and predictive capability of proposed closures Test dataset Training dataset

slide-52
SLIDE 52

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Comparison of ML to first-principles models

ML model has best correspondence and predictive capability of proposed closures Test dataset Training dataset Success of data-driven model invites a number of questions Fundamentally, is this giving up on physics? Can any physical insight emerge from it? Practically, can this be used for predictive models?

slide-53
SLIDE 53

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Comparison of ML to first-principles models

ML model has best correspondence and predictive capability of proposed closures Test dataset Training dataset Success of data-driven model invites a number of questions Fundamentally, is this giving up on physics? Can any physical insight emerge from it? Practically, can this be used for predictive models?

slide-54
SLIDE 54

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Physical insight

Pareto front of models From these models, are there are any variables that are more common than others?

slide-55
SLIDE 55

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Physical insight

Pareto front of models From these models, are there are any variables that are more common than others? Frequency of variable appearance in best models

slide-56
SLIDE 56

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Physical insight

Pareto front of models From these models, are there are any variables that are more common than others? Frequency of variable appearance in best models Ion drift and Hall drift dominant variables

slide-57
SLIDE 57

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Physical insight

Pareto front of models From these models, are there are any variables that are more common than others? Frequency of variable appearance in best models Ion drift and Hall drift dominant variables Search for a first-principles mechanism that depends on these parameters Electron cyclotron drift instability one example

slide-58
SLIDE 58

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Physical insight

Pareto front of models From these models, are there are any variables that are more common than others? Frequency of variable appearance in best models Ion drift and Hall drift dominant variables Search for a first-principles mechanism that depends on these parameters Electron cyclotron drift instability one example

ML results can guide physical investigation of underlying physical processes

slide-59
SLIDE 59

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Comparison of ML to first-principles models

ML model has best correspondence and predictive capability of proposed closures Test dataset Training dataset Success of data-driven model invites a number of questions Fundamentally, is this giving up on physics? Can any physical insight emerge from it? Practically, can this be used for predictive models? Potentially, with more data But the parameter space is wide!

slide-60
SLIDE 60

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Generating additional data on transport in Hall thrusters

slide-61
SLIDE 61

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory

Summary

  • Fluid models are attractive tool for modeling Hall effect thrusters
  • Need to account for known anomalous electron transport in these models with a type of

closure: typically anomalous effects represented with scalar collision frequency (or mobility)

  • Data-driven, ML methods can be employed to find functional form for this anomalous

collision frequency

  • Predictions from ML results yield

– Improved results compared to first-principles models for anomalous collision frequency – ML algorithm also yields physical insight into dominant terms governing transport

  • ML is a promising path forward for closing anomalous electron transport problem.

Predictive capability has applications ranging from predictive design to qualification through analysis.

  • On-going challenges include

– Extrapolation – Data-generation