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Adjoint methods for quasi-symmetry of vacuum fields on a surface

Published online by Cambridge University Press:  21 January 2022

Richard Nies Open the ORCID record for Richard Nies [Opens in a new window] ,
Elizabeth J. Paul Open the ORCID record for Elizabeth J. Paul [Opens in a new window] ,
Stuart R. Hudson  and
Amitava Bhattacharjee
Richard Nies*
Affiliation:
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08543, USA Princeton Plasma Physics Laboratory, Princeton, NJ 08540, USA
Elizabeth J. Paul
Affiliation:
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08543, USA Princeton Plasma Physics Laboratory, Princeton, NJ 08540, USA
Stuart R. Hudson
Affiliation:
Princeton Plasma Physics Laboratory, Princeton, NJ 08540, USA
Amitava Bhattacharjee
Affiliation:
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08543, USA Princeton Plasma Physics Laboratory, Princeton, NJ 08540, USA
*
†Email address for correspondence: rnies@pppl.gov
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Abstract

Adjoint methods can speed up stellarator optimisation by providing gradient information more efficiently compared with finite-difference evaluations. Adjoint methods are herein applied to vacuum magnetic fields, with objective functions targeting quasi-symmetry and a rotational transform value on a surface. To measure quasi-symmetry, a novel way of evaluating approximate flux coordinates on a single flux surface without the assumption of a neighbourhood of flux surfaces is proposed. The shape gradients obtained from the adjoint formalism are evaluated numerically and verified against finite-difference evaluations.

Keywords

fusion plasmaplasma devicesplasma confinement
Type
Research Article
Information
Journal of Plasma Physics , Volume 88 , Issue 1 , February 2022 , 905880106
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Anderson, F.S.B., Almagri, A.F., Anderson, D.T., Matthews, P.G., Talmadge, J.N. & Shohet, J.L. 1995 The helically symmetric experiment, (HSX) goals, design and status. Fusion Technol. 27 (3 T), 273–277.CrossRefGoogle Scholar
Antonsen, T., Paul, E.J. & Landreman, M. 2019 Adjoint approach to calculating shape gradients for three-dimensional magnetic confinement equilibria. J. Plasma Phys. 85 (2), 905850207.CrossRefGoogle Scholar
Bader, A., Drevlak, M., Anderson, D.T., Faber, B.J., Hegna, C.C., Likin, K.M., Schmitt, J.C. & Talmadge, J.N. 2019 Stellarator equilibria with reactor relevant energetic particle losses. J. Plasma Phys. 85 (5), 905850508.CrossRefGoogle Scholar
Beidler, C., Grieger, G., Herrnegger, F., Harmeyer, E., Kisslinger, J., Lotz, W., Maassberg, H., Merkel, P., Nührenberg, J., Rau, F., et al. 1990 Physics and engineering design for wendelstein VII-X. Fusion Technol. 17 (1), 148–168.CrossRefGoogle Scholar
Boozer, A.H. 2019 Curl-free magnetic fields for stellarator optimization. Phys. Plasmas 26 (10), 102504.CrossRefGoogle Scholar
Burby, J.W., Kallinikos, N. & MacKay, R.S. 2020 Some mathematics for quasi-symmetry. J. Math. Phys. 61 (9), 093503.CrossRefGoogle Scholar
Delfour, M.C. & Zolésio, J.P. 2011 Shapes and Geometries. Advances in Design and Control. PUBNAMESociety for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Dewar, R., Hole, M., McGann, M., Mills, R. & Hudson, S. 2008 Relaxed plasma equilibria and entropy-related plasma self-organization principles. Entropy 10 (4), 621–634.CrossRefGoogle Scholar
Drevlak, M., Brochard, F., Helander, P., Kisslinger, J., Mikhailov, M., Nührenberg, C., Nührenberg, J. & Turkin, Y. 2013 ESTELL: a quasi-toroidally symmetric stellarator. Contrib. Plasma Phys. 53 (6), 459–468.CrossRefGoogle Scholar
Garren, D.A. & Boozer, A.H. 1991 Existence of quasihelically symmetric stellarators. Phys. Fluids B 3 (10), 2822–2834.CrossRefGoogle Scholar
Geraldini, A., Landreman, M. & Paul, E. 2021 An adjoint method for determining the sensitivity of island size to magnetic field variations. J. Plasma Phys. 87 (3), 905870302.CrossRefGoogle Scholar
Giuliani, A., Wechsung, F., Cerfon, A., Stadler, G. & Landreman, M. 2020 Single-stage gradient-based stellarator coil design: Optimization for near-axis quasi-symmetry. arXiv:2010.02033.CrossRefGoogle Scholar
Greene, J.M., Mackay, R.S. & Stark, J. 1986 Boundary circles for area-preserving maps. Physica D 21 (2–3), 267–295.CrossRefGoogle Scholar
Hall, L.S. & McNamara, B. 1975 Three-dimensional equilibrium of the anisotropic, finite-pressure guiding-center plasma: theory of the magnetic plasma. Phys. Fluids 18 (5), 552.CrossRefGoogle Scholar
Helander, P. 2014 Theory of plasma confinement in non-axisymmetric magnetic fields. Rep. Prog. Phys. 77 (8), 087001.CrossRefGoogle ScholarPubMed
Henneberg, S.A., Drevlak, M. & Helander, P. 2020 Improving fast-particle confinement in quasi-axisymmetric stellarator optimization. Plasma Phys. Control. Fusion 62 (1), 014023.CrossRefGoogle Scholar
Henneberg, S.A., Drevlak, M., Nührenberg, C., Beidler, C.D., Turkin, Y., Loizu, J. & Helander, P. 2019 Properties of a new quasi-axisymmetric configuration. Nucl. Fusion 59 (2), 026014.CrossRefGoogle Scholar
Hirshman, S.P., van RIJ, W.I. & Merkel, P. 1986 Three-dimensional free boundary calculations using a spectral Green's function method. Comput. Phys. Commun. 43 (1), 143–155.CrossRefGoogle Scholar
Hudson, S.R., Dewar, R.L., Dennis, G., Hole, M.J., McGann, M., von Nessi, G. & Lazerson, S. 2012 Computation of multi-region relaxed magnetohydrodynamic equilibria. Phys. Plasmas 19 (11), 112502.CrossRefGoogle Scholar
Hudson, S.R., Zhu, C., Pfefferlé, D. & Gunderson, L. 2018 Differentiating the shape of stellarator coils with respect to the plasma boundary. Phys. Lett. A 382 (38), 2732–2737.CrossRefGoogle Scholar
Ku, L.P. & Boozer, A.H. 2011 New classes of quasi-helically symmetric stellarators. Nucl. Fusion 51 (1), 013004.CrossRefGoogle Scholar
Landreman, M., Medasani, B. & Zhu, C. 2021 Stellarator optimization for good magnetic surfaces at the same time as quasisymmetry. Phys. Plasmas 28 (9), 092505.CrossRefGoogle Scholar
Landreman, M. & Paul, E. 2018 Computing local sensitivity and tolerances for stellarator physics properties using shape gradients. Nucl. Fusion 58 (7), 076023.CrossRefGoogle Scholar
Landreman, M. & Paul, E. 2021 Magnetic fields with precise quasisymmetry. arXiv:2108.03711.CrossRefGoogle Scholar
Malhotra, D., Cerfon, A., Imbert-Gérard, L.-M. & O'Neil, M. 2019 Taylor states in stellarators: a fast high-order boundary integral solver. J. Comput. Phys. 397, 108791.CrossRefGoogle Scholar
McGrath, P. 2016 On the Smooth Jordan Brouwer separation theorem. Am. Math. Mon. 123 (3), 292.CrossRefGoogle Scholar
Meiss, J.D. 1992 Symplectic maps, variational principles, and transport. Rev. Mod. Phys. 64 (3), 795–848.CrossRefGoogle Scholar
Mercier, C. 1964 Equilibrium and stability of a toroidal magnetohydrodynamic system in the neighbourhood of a magnetic axis. Nucl. Fusion 4 (3), 213–226.CrossRefGoogle Scholar
Nies, R. 2021 Dataset from: “Adjoint methods for quasisymmetry of vacuum fields on a surface”. Available at: doi:10.5281/zenodo.5248498.CrossRefGoogle Scholar
Nührenberg, J. & Zille, R. 1988 Quasi-helically symmetric toroidal stellarators. Phys. Lett. A 129 (2), 113–117.CrossRefGoogle Scholar
Paul, E. 2020 Adjoint methods for stellarator shape optimization and sensitivity analysis. arXiv:2005.07633.Google Scholar
Paul, E.J., Abel, I.G., Landreman, M. & Dorland, W. 2019 An adjoint method for neoclassical stellarator optimization. J. Plasma Phys. 85 (5), 795850501.CrossRefGoogle Scholar
Paul, E.J., Antonsen, T., Landreman, M. & Cooper, W.A. 2020 Adjoint approach to calculating shape gradients for three-dimensional magnetic confinement equilibria. Part 2. Applications. J. Plasma Phys. 86 (1), 905860103.CrossRefGoogle Scholar
Paul, E.J., Landreman, M. & Antonsen, T. 2021 Gradient-based optimization of 3D MHD equilibria. J. Plasma Phys. 87 (2), 905870214.CrossRefGoogle Scholar
Paul, E.J., Landreman, M., Bader, A. & Dorland, W. 2018 An adjoint method for gradient-based optimization of stellarator coil shapes. Nucl. Fusion 58 (7), 076015.CrossRefGoogle Scholar
Plunk, G.G. & Helander, P. 2018 Quasi-axisymmetric magnetic fields: weakly non-axisymmetric case in a vacuum. J. Plasma Phys. 84 (2), 905840205.CrossRefGoogle Scholar
Qu, Z.S., Pfefferlé, D., Hudson, S.R., Baillod, A., Kumar, A., Dewar, R.L. & Hole, M.J. 2020 Coordinate parameterisation and spectral method optimisation for Beltrami field solver in stellarator geometry. Plasma Phys. Control. Fusion 62 (12), 124004.CrossRefGoogle Scholar
Rodríguez, E. & Bhattacharjee, A. 2021 a Solving the problem of overdetermination of quasisymmetric equilibrium solutions by near-axis expansions. I. Generalized force balance. Phys. Plasmas 28 (1), 012508.CrossRefGoogle Scholar
Rodríguez, E. & Bhattacharjee, A. 2021 b Solving the problem of overdetermination of quasisymmetric equilibrium solutions by near-axis expansions. II. Circular axis stellarator solutions. Phys. Plasmas 28 (1), 012509.CrossRefGoogle Scholar
Rodríguez, E., Helander, P. & Bhattacharjee, A. 2020 Necessary and sufficient conditions for quasisymmetry. Phys. Plasmas 27 (6), 062501.CrossRefGoogle Scholar
Rodriguez, E., Paul, E. & Bhattacharjee, A. 2021 Measures of quasisymmetry for stellarators. arXiv:2109.13080.Google Scholar
Sengupta, W., Paul, E.J., Weitzner, H. & Bhattacharjee, A. 2021 Vacuum magnetic fields with exact quasisymmetry near a flux surface. Part 1. Solutions near an axisymmetric surface. J. Plasma Phys. 87 (2), 905870205.CrossRefGoogle Scholar
Sokołowski, J. & Zolésio, J.P. 1992 Introduction to shape optimization: shape sensitivity analysis. Springer Series in Computational Mathematics, vol. 16. Springer.CrossRefGoogle Scholar
Spitzer, L. 1958 The stellarator concept. Phys. Fluids 1 (4), 253.CrossRefGoogle Scholar
Walker, S.W. 2015 The shapes of things: a practical guide to differential geometry and the shape derivative. Advances in Design and Control. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Zarnstorff, M.C., Berry, L.A., Brooks, A., Fredrickson, E., Fu, G.-Y., Hirschman, S., Hudson, S., Ku., L.-P., Lazarus, E., Mikkelsen, D., et al. 2001 Physics of the compact advanced stellarator NCSX. Plasma Phys. Control. Fusion 43 (12A), A237–A249.CrossRefGoogle Scholar
Zhu, C., Hudson, S.R., Song, Y. & Wan, Y. 2018 New method to design stellarator coils without the winding surface. Nucl. Fusion 58 (1), 016008.CrossRefGoogle Scholar