An adjoint method for neoclassical stellarator optimization
Abstract
Stellarators remain an enticing route to steadystate fusion power. However, to achieve the required confinement, the magnetic geometry must be highly optimized. This optimization requires navigating highdimensional spaces, often necessitating the use of gradientbased methods. The gradient of the neoclassical fluxes is expensive to compute with classical methods, requiring$O(N)$$flux computations, where$$N$$is the number of parameters. To reduce the cost of the gradient computation, we present an adjoint method for computing the derivatives of moments of the neoclassical distribution function for stellarator optimization. The linear adjoint method allows derivatives of quantities which depend on solutions of a linear system, such as moments of the distribution function, to be computed with respect to many parameters from the solution of only two linear systems. This reduces the cost of computing the gradient to the point that the finitecollisionality neoclassical fluxes can be used within an optimization loop. With the neoclassical adjoint method, we compute solutions of the drift kinetic equation and an adjoint drift kinetic equation to obtain derivatives of neoclassical quantities with respect to geometric parameters. When the number of parameters in the derivative is large ($$O(10^{2})$), this adjoint method provides up to a factor of 200 reduction in cost. We demonstrate adjointbased optimization of the field strength to obtain minimal bootstrap current on a surface. With adjointbased derivatives, we also compute the local sensitivity to magnetic perturbations on a flux surface and identify regions where tight tolerances on error fields are required for control of the bootstrap current or radial transport. Moreover, the solve for the ambipolar electric field is accelerated using a Newton method with derivatives obtained from the adjoint method.
 Authors:

 Univ. of Maryland, College Park, MD (United States). Inst. for Research in Electronics and Applied Physics
 Univ. of Maryland, College Park, MD (United States). Inst. for Research in Electronics and Applied Physics; Chalmers Univ. of Technology, Göteborg (Sweden)
 Publication Date:
 Research Org.:
 Univ. of Maryland, College Park, MD (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Fusion Energy Sciences (FES)
 Contributing Org.:
 National Energy Research Scientiﬁc Computing Center (NERSC)
 OSTI Identifier:
 1597696
 Grant/Contract Number:
 FG0293ER54197; FC0208ER54964
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Plasma Physics
 Additional Journal Information:
 Journal Volume: 85; Journal Issue: 5; Journal ID: ISSN 00223778
 Publisher:
 Cambridge University Press
 Country of Publication:
 United States
 Language:
 English
 Subject:
 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; fusion plasma; plasma confinement; plasma simulation
Citation Formats
Paul, Elizabeth J., Abel, Ian G., Landreman, Matt, and Dorland, William. An adjoint method for neoclassical stellarator optimization. United States: N. p., 2019.
Web. https://doi.org/10.1017/S0022377819000527.
Paul, Elizabeth J., Abel, Ian G., Landreman, Matt, & Dorland, William. An adjoint method for neoclassical stellarator optimization. United States. https://doi.org/10.1017/S0022377819000527
Paul, Elizabeth J., Abel, Ian G., Landreman, Matt, and Dorland, William. Fri .
"An adjoint method for neoclassical stellarator optimization". United States. https://doi.org/10.1017/S0022377819000527. https://www.osti.gov/servlets/purl/1597696.
@article{osti_1597696,
title = {An adjoint method for neoclassical stellarator optimization},
author = {Paul, Elizabeth J. and Abel, Ian G. and Landreman, Matt and Dorland, William},
abstractNote = {Stellarators remain an enticing route to steadystate fusion power. However, to achieve the required confinement, the magnetic geometry must be highly optimized. This optimization requires navigating highdimensional spaces, often necessitating the use of gradientbased methods. The gradient of the neoclassical fluxes is expensive to compute with classical methods, requiring$O(N)$flux computations, where$N$is the number of parameters. To reduce the cost of the gradient computation, we present an adjoint method for computing the derivatives of moments of the neoclassical distribution function for stellarator optimization. The linear adjoint method allows derivatives of quantities which depend on solutions of a linear system, such as moments of the distribution function, to be computed with respect to many parameters from the solution of only two linear systems. This reduces the cost of computing the gradient to the point that the finitecollisionality neoclassical fluxes can be used within an optimization loop. With the neoclassical adjoint method, we compute solutions of the drift kinetic equation and an adjoint drift kinetic equation to obtain derivatives of neoclassical quantities with respect to geometric parameters. When the number of parameters in the derivative is large ($O(10^{2})$), this adjoint method provides up to a factor of 200 reduction in cost. We demonstrate adjointbased optimization of the field strength to obtain minimal bootstrap current on a surface. With adjointbased derivatives, we also compute the local sensitivity to magnetic perturbations on a flux surface and identify regions where tight tolerances on error fields are required for control of the bootstrap current or radial transport. Moreover, the solve for the ambipolar electric field is accelerated using a Newton method with derivatives obtained from the adjoint method.},
doi = {10.1017/S0022377819000527},
journal = {Journal of Plasma Physics},
number = 5,
volume = 85,
place = {United States},
year = {2019},
month = {9}
}
Web of Science
Works referenced in this record:
Physics of the compact advanced stellarator NCSX
journal, November 2001
 Zarnstorff, M. C.; Berry, L. A.; Brooks, A.
 Plasma Physics and Controlled Fusion, Vol. 43, Issue 12A
Bootstrap current and parallel viscosity in the low collisionality regime in toroidal plasmas
journal, January 1989
 Shaing, K. C.; Crume, E. C.; Tolliver, J. S.
 Physics of Fluids B: Plasma Physics, Vol. 1, Issue 1
Ballooning stability optimization of lowaspectratio stellarators*
journal, May 2000
 Sanchez, R.; Hirshman, S. P.; Ware, A. S.
 Plasma Physics and Controlled Fusion, Vol. 42, Issue 6
Plasma Transport in Toroidal Confinement Systems
journal, January 1972
 Rosenbluth, M. N.
 Physics of Fluids, Vol. 15, Issue 1
Variational bounds for transport coefficients in three‐dimensional toroidal plasmas
journal, March 1989
 van Rij, W. I.; Hirshman, S. P.
 Physics of Fluids B: Plasma Physics, Vol. 1, Issue 3
A review of the adjointstate method for computing the gradient of a functional with geophysical applications
journal, November 2006
 Plessix, R. E.
 Geophysical Journal International, Vol. 167, Issue 2
The ambipolar electric field in stellarators
journal, April 1985
 Hastings, D. E.; Houlberg, W. A.; Shaing, K. C.
 Nuclear Fusion, Vol. 25, Issue 4
On the numerical computation of the minimumdrag profile in laminar flow
journal, November 1975
 Glowinski, R.; Pironneau, O.
 Journal of Fluid Mechanics, Vol. 72, Issue 02
Linearized model collision operators for multiple ion species plasmas and gyrokinetic entropy balance equations
journal, November 2009
 Sugama, H.; Watanabe, T. H.; Nunami, M.
 Physics of Plasmas, Vol. 16, Issue 11
The adjoint method in seismology
journal, August 2006
 Fichtner, A.; Bunge, H. P.; Igel, H.
 Physics of the Earth and Planetary Interiors, Vol. 157, Issue 12
Existence of quasihelically symmetric stellarators
journal, October 1991
 Garren, D. A.; Boozer, A. H.
 Physics of Fluids B: Plasma Physics, Vol. 3, Issue 10
A general solution of the rippleaveraged kinetic equation (GSRAKE)
journal, April 1995
 Beidler, C. D.; D'haeseleer, W. D.
 Plasma Physics and Controlled Fusion, Vol. 37, Issue 4
Adjoint and defect error bounding and correction for functional estimates
journal, November 2004
 Pierce, Niles A.; Giles, Michael B.
 Journal of Computational Physics, Vol. 200, Issue 2
Physics Design for ARIESCS
journal, October 2008
 Ku, L. P.; Garabedian, P. R.; Lyon, J.
 Fusion Science and Technology, Vol. 54, Issue 3
Stellarator bootstrap current and plasma flow velocity at low collisionality
journal, March 2017
 Helander, P.; Parra, F. I.; Newton, S. L.
 Journal of Plasma Physics, Vol. 83, Issue 2
Physics optimization of stellarators
journal, March 1992
 Grieger, G.; Lotz, W.; Merkel, P.
 Physics of Fluids B: Plasma Physics, Vol. 4, Issue 7
Development of a Robust QuasiPoloidal Compact Stellarator
journal, January 2004
 Strickler, Dennis J.; Hirshman, Steven P.; Spong, Donald A.
 Fusion Science and Technology, Vol. 45, Issue 1
Automated divertor target design by adjoint shape sensitivity analysis and a oneshot method
journal, December 2014
 Dekeyser, W.; Reiter, D.; Baelmans, M.
 Journal of Computational Physics, Vol. 278
Evaluation of 1/ν neoclassical transport in stellarators
journal, December 1999
 Nemov, V. V.; Kasilov, S. V.; Kernbichler, W.
 Physics of Plasmas, Vol. 6, Issue 12
Theory of plasma confinement in nonaxisymmetric magnetic fields
journal, July 2014
 Helander, Per
 Reports on Progress in Physics, Vol. 77, Issue 8
Generalized universal instability: transient linear amplification and subcritical turbulence
journal, July 2015
 Landreman, Matt; Plunk, Gabriel G.; Dorland, William
 Journal of Plasma Physics, Vol. 81, Issue 5
Multiscale gyrokinetics for rotating tokamak plasmas: fluctuations, transport and energy flows
journal, October 2013
 Abel, I. G.; Plunk, G. G.; Wang, E.
 Reports on Progress in Physics, Vol. 76, Issue 11
An adjoint method for gradientbased optimization of stellarator coil shapes
journal, May 2018
 Paul, E. J.; Landreman, M.; Bader, A.
 Nuclear Fusion, Vol. 58, Issue 7
A one shot method for divertor target shape optimization: A one shot method for divertor target shape optimization
journal, December 2014
 Dekeyser, Wouter; Reiter, Detlev; Baelmans, Martine
 PAMM, Vol. 14, Issue 1
The role of dissipation in the theory and simulations of homogeneous plasma turbulence, and resolution of the entropy paradox
journal, October 1994
 Krommes, John A.; Hu, Genze
 Physics of Plasmas, Vol. 1, Issue 10
Benchmarking of the monoenergetic transport coefficients—results from the International Collaboration on Neoclassical Transport in Stellarators (ICNTS)
journal, June 2011
 Beidler, C. D.; Allmaier, K.; Isaev, M. Yu.
 Nuclear Fusion, Vol. 51, Issue 7
Solution of drift kinetic equation in stellarators and tokamaks with broken symmetry using the code NEO2
journal, August 2016
 Kernbichler, W.; Kasilov, S. V.; Kapper, G.
 Plasma Physics and Controlled Fusion, Vol. 58, Issue 10
Rotation and neoclassical ripple transport in ITER
journal, August 2017
 Paul, E. J.; Landreman, M.; Poli, F. M.
 Nuclear Fusion, Vol. 57, Issue 11
Adjoint approach to calculating shape gradients for threedimensional magnetic confinement equilibria
journal, March 2019
 Antonsen, Thomas; Paul, Elizabeth J.; Landreman, Matt
 Journal of Plasma Physics, Vol. 85, Issue 2
Physics and Engineering Design for Wendelstein VIIX
journal, January 1990
 Beidler, Craig; Grieger, Günter; Herrnegger, Franz
 Fusion Technology, Vol. 17, Issue 1
Neoclassical transport in toroidal plasmas with nonaxisymmetric flux surfaces
journal, April 2015
 Belli, E. A.; Candy, J.
 Plasma Physics and Controlled Fusion, Vol. 57, Issue 5
Direct construction of optimized stellarator shapes. Part 1. Theory in cylindrical coordinates
journal, December 2018
 Landreman, Matt; Sengupta, Wrick
 Journal of Plasma Physics, Vol. 84, Issue 6
J* optimization of small aspect ratio stellarator/tokamak hybrid devices
journal, May 1998
 Spong, D. A.; Hirshman, S. P.; Whitson, J. C.
 Physics of Plasmas, Vol. 5, Issue 5
Electrostatic potential variations on stellarator magnetic surfaces in low collisionality regimes
journal, August 2018
 Calvo, Iván; Velasco, José Luis; Parra, Félix I.
 Journal of Plasma Physics, Vol. 84, Issue 4
On optimum design in fluid mechanics
journal, June 1974
 Pironneau, O.
 Journal of Fluid Mechanics, Vol. 64, Issue 1
Properties of a new quasiaxisymmetric configuration
journal, January 2019
 Henneberg, S. A.; Drevlak, M.; Nührenberg, C.
 Nuclear Fusion, Vol. 59, Issue 2
Physics design of a highbbeta quasiaxisymmetric stellarator
journal, December 1999
 Reiman, A.; Fu, G.; Hirshman, S.
 Plasma Physics and Controlled Fusion, Vol. 41, Issue 12B
Computing local sensitivity and tolerances for stellarator physics properties using shape gradients
journal, June 2018
 Landreman, Matt; Paul, Elizabeth
 Nuclear Fusion, Vol. 58, Issue 7
Optimisation of confinement in a fusion reactor using a nonlinear turbulence model
journal, April 2018
 Highcock, E. G.; Mandell, N. R.; Barnes, M.
 Journal of Plasma Physics, Vol. 84, Issue 2
Neoclassical bootstrap current and transport in optimized stellarator configurations
journal, October 1993
 Maassberg, H.; Lotz, W.; Nührenberg, J.
 Physics of Fluids B: Plasma Physics, Vol. 5, Issue 10
Physics of compact stellarators
journal, May 1999
 Hirshman, S. P.; Spong, D. A.; Whitson, J. C.
 Physics of Plasmas, Vol. 6, Issue 5
Consequences of TimeReversal Symmetry for the Electric Field Scaling of Transport in Stellarators
journal, April 1986
 Hirshman, S. P.; Shaing, K. C.; van Rij, W. I.
 Physical Review Letters, Vol. 56, Issue 16
Comparison of particle trajectories and collision operators for collisional transport in nonaxisymmetric plasmas
journal, April 2014
 Landreman, M.; Smith, H. M.; Mollén, A.
 Physics of Plasmas, Vol. 21, Issue 4
Single Particle Motion in Toroidal Stellarator Fields
journal, January 1967
 Gibson, A.
 Physics of Fluids, Vol. 10, Issue 12
Direct multiscale coupling of a transport code to gyrokinetic turbulence codes
journal, May 2010
 Barnes, M.; Abel, I. G.; Dorland, W.
 Physics of Plasmas, Vol. 17, Issue 5
Quasihelically symmetric toroidal stellarators
journal, May 1988
 Nührenberg, J.; Zille, R.
 Physics Letters A, Vol. 129, Issue 2
Steepestdescent moment method for threedimensional magnetohydrodynamic equilibria
journal, January 1983
 Hirshman, S. P.
 Physics of Fluids, Vol. 26, Issue 12
Design of the national compact stellarator experiment (NCSX)
journal, September 2003
 Nelson, B. E.; Berry, L. A.; Brooks, A. B.
 Fusion Engineering and Design, Vol. 6668
Plasma transport coefficients for nonsymmetric toroidal confinement systems
journal, September 1986
 Hirshman, S. P.; Shaing, K. C.; van Rij, W. I.
 Physics of Fluids, Vol. 29, Issue 9
Works referencing / citing this record:
Adjoint approach to calculating shape gradients for threedimensional magnetic confinement equilibria. Part 2. Applications
journal, January 2020
 Paul, Elizabeth J.; Antonsen, Thomas; Landreman, Matt
 Journal of Plasma Physics, Vol. 86, Issue 1