- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Efficient computation of stellarator coils with an augmented Lagrangian optimization method

Pedro F. Gil1, Weiping Li2,3, Julianne Stratton3, Alan A. Kaptanoglu3, and Eve V. Stenson1

PDF

Phys. Rev. E 114, 025202 – Published 7 August, 2026

DOI: https://doi.org/10.1103/838f-vbq3

Abstract

Finding feasible coils for stellarator fusion devices is a critical challenge of realizing this concept for future power plants. Current design efforts struggle to navigate the highly nonconvex optimization landscape, spend considerable resources scanning the parameter space, and may produce suboptimal coils. In this work, we present an augmented Lagrangian approach to tackle the ill-posed problem of coil optimization. We illustrate its effectiveness and versatility by generating coils for five stellarators with very different symmetries and magnetic-field shaping. In all cases, we find Pareto-optimal coil solutions that in various ways outperform published coil sets.

View figure in article

Physics Subject Headings (PhySH)

See Also

Stellarator Coils for Future Fusion Reactors via an Augmented Lagrangian Approach

Pedro F. Gil, Weiping Li, Julianne Stratton, Alan A. Kaptanoglu, and Eve V. Stenson
Phys. Rev. Lett. 137, 065101 (2026)

Article Text

References (75)

  1. F. C. Schuller, Disruptions in tokamaks, Plasma Phys. Control. Fusion 37, A135 (1995).
  2. G. Jahns, M. Soler, B. Waddell, J. Callen, and H. Hicks, Internal disruptions in tokamaks, Nucl. Fusion 18, 609 (1978).
  3. P. Helander, C. D. Beidler, T. M. Bird, M. Drevlak, Y. Feng, R. Hatzky, F. Jenko, R. Kleiber, J. H. E. Proll, Y. Turkin, and P. Xanthopoulos, Stellarator and tokamak plasmas: A comparison, Plasma Phys. Control. Fusion 54, 124009 (2012).
  4. G. Grieger, W. Lotz, P. Merkel, J. Nührenberg, J. Sapper, E. Strumberger, H. Wobig, R. Burhenn, V. Erckmann, U. Gasparino, L. Giannone, H. J. Hartfuss, R. Jaenicke, G. Kühner, H. Ringler, A. Weller, F. Wagner, the W7‐X Team, and the W7‐AS Team, Physics optimization of stellarators, Phys. Fluids B 4, 2081 (1992).
  5. T. Klinger, C. Baylard, C. Beidler, J. Boscary, H. Bosch, A. Dinklage, D. Hartmann, P. Helander, H. Maßberg, A. Peacock, T. Pedersen, T. Rummel, F. Schauer, L. Wegener, and R. Wolf, Towards assembly completion and preparation of experimental campaigns of Wendelstein 7-X in the perspective of a path to a stellarator fusion power plant, Fusion Eng. Des. 88, 461 (2013).
  6. F. Anderson, A. Almagri, D. Anderson, P. Matthews, J. Talmadge, and J. Shohet, Helically symmetric experiment, (HSX) goals, design and status, Fusion Technol. 27, 273 (1995).
  7. S. P. Hirshman and J. C. Whitson, Steepest‐descent moment method for three‐dimensional magnetohydrodynamic equilibria, Phys. Fluids 26, 3553 (1983).
  8. J. Nührenberg and R. Zille, Quasi-helically symmetric toroidal stellarators, Phys. Lett. A 129, 113 (1988).
  9. M. Landreman and E. Paul, Magnetic fields with precise quasisymmetry for plasma confinement, Phys. Rev. Lett. 128, 035001 (2022).
  10. D. W. Dudt and E. Kolemen, DESC: A stellarator equilibrium solver, Phys. Plasmas 27, 102513 (2020).
  11. L.-M. Imbert-Gérard, E. J. Paul, and A. M. Wright, An Introduction to Stellarators: From Magnetic Fields to Symmetries and Optimization (Society for Industrial and Applied Mathematics, Philadelphia, 2024); https://epubs.siam.org/doi/pdf/10.1137/1.9781611978223.
  12. V. Erckmann, H.-J. Hartfuss, M. Kick, H. Renner, J. Sapper, F. Schauer, E. Speth, F. Wesner, F. Wagner, M. Wanner, A. Weller, and H. Wobig, The W7-X project: Scientific basis and technical realization, in 17th IEEE/NPSS Symposium Fusion Engineering (Cat. No.97CH36131) (Institute of Electrical and Electronics Engineers (IEEE), Piscataway, NJ, 1997), Vol. 1, pp. 40–48.
  13. R. Strykowsky, T. Brown, J. Chrzanowski, M. Cole, P. Heitzenroeder, G. Neilson, D. Rej, and M. Viol, Engineering cost & schedule lessons learned on NCSX, in 2009 23rd IEEE/NPSS Symposium on Fusion Engineering (IEEE (Institute of Electrical and Electronics Engineers), New York, NY, 2009), pp. 1–4.
  14. H.-S. Bosch, R. Brakel, T. Braeuer, V. Bykov, P. van Eeten, J.-H. Feist, F. Füllenbach, M. Gasparotto, H. Grote, T. Klinger, H. Laqua, M. Nagel, D. Naujoks, M. Otte, K. Risse, T. Rummel, J. Schacht, A. Spring, T. Sunn Pedersen, R. Vilbrandt, et al., Final integration, commissioning and start of the Wendelstein 7-X stellarator operation, Nucl. Fusion 57, 116015 (2017).
  15. F. Wechsung, A. Giuliani, M. Landreman, A. Cerfon, and G. Stadler, Single-stage gradient-based stellarator coil design: Stochastic optimization, Nucl. Fusion 62, 076034 (2022).
  16. A. Giuliani, Direct stellarator coil design using global optimization: Application to a comprehensive exploration of quasi-axisymmetric devices, J. Plasma Phys. 90, 905900303 (2024).
  17. R. Wu, T. Kruger, and C. Swanson, Planar coil optimization for the Eos stellarator using sparse regression, Plasma Phys. Control. Fusion 67, 035019 (2025).
  18. R. Jorge, A. Giuliani, and J. Loizu, Simplified and flexible coils for stellarators using single-stage optimization, Phys. Plasmas 31, 112501 (2024).
  19. Y. Suzuki, J. Huang, N. Wang, and Y. Ding, Design of simple stellarator using tilted toroidal field coils, Fusion Eng. Des. 173, 112843 (2021).
  20. L. Fu, E. J. Paul, A. A. Kaptanoglu, and A. Bhattacharjee, Global stellarator coil optimization with quadratic constraints and objectives, Nucl. Fusion 65, 026045 (2025).
  21. M. Drevlak, Optimization of heterogeneous magnet systems, 12th International Stellarator Workshop, Madison, WI (1999).
  22. N. Pomphrey, L. Berry, A. Boozer, A. Brooks, R. Hatcher, S. Hirshman, L.-P. Ku, W. Miner, H. Mynick, W. Reiersen, D. Strickler, and P. Valanju, Innovations in compact stellarator coil design, Nucl. Fusion 41, 339 (2001).
  23. D. J. Strickler, L. A. Berry, and S. P. Hirshman, Designing coils for compact stellarators, Fusion Sci. Technol. 41, 107 (2002).
  24. T. Brown, J. Breslau, D. Gates, N. Pomphrey, and A. Zolfaghari, Engineering optimization of stellarator coils lead to improvements in device maintenance, in 2015 IEEE 26th Symposium on Fusion Engineering (SOFE) (IEEE (Institute of Electrical and Electronics Engineers), Piscataway, NJ, 2015), pp. 1–6.
  25. J. Lion, J.-C. Anglès, L. Bonauer, A. Bañón Navarro, S. Cadena Ceron, R. Davies, M. Drevlak, N. Foppiani, J. Geiger, A. Goodman, W. Guo, E. Guiraud, F. Hernández, S. Henneberg, R. Herrero, C. Hintze, H. Höchter, J. Jelonnek, F. Jenko, R. Jorge, et al., Stellaris: A high-field quasi-isodynamic stellarator for a prototypical fusion power plant, Fusion Eng. Des. 214, 114868 (2025).
  26. D. T. Anderson, J. M. Canik, C. C. Hegna, and C. M. Mowry, A comprehensive, unified baseline physics design for the type one energy stellarator fusion pilot power plant, ‘Infinity Two', J. Plasma Phys. 91, e65 (2025).
  27. D. Gates, S. Aslam, B. Berzin, P. Bonofiglo, A. Cote, D. Dudt, E. Flom, D. Fort, A. Koen, T. Kruger, S. Kumar, M. Martin, A. Ottaviano, S. Pasmann, P. Romano, C. Swanson, L. Tang, E. Winkler, and R. Wu, Stellarator fusion systems enabled by arrays of planar coils, Nucl. Fusion 65, 026052 (2025).
  28. F. Volpe, Renaissance Fusion Technologies, https://renfusion.eu/technology (2023), accessed: 1 October 2023.
  29. J. Miyazawa and T. Goto, Development of steady-state fusion reactor by Helical Fusion, Phys. Plasmas 30, 050601 (2023).
  30. A. Goodman, K. Camacho Mata, S. Henneberg, R. Jorge, M. Landreman, G. Plunk, H. Smith, R. Mackenbach, C. Beidler, and P. Helander, Constructing precisely quasi-isodynamic magnetic fields, J. Plasma Phys. 89, 905890504 (2023).
  31. D. W. Dudt, A. G. Goodman, R. Conlin, D. Panici, and E. Kolemen, Magnetic fields with general omnigenity, J. Plasma Phys. 90, 905900120 (2024).
  32. J. L. Velasco, I. Calvo, F. J. Escoto, E. Sánchez, H. Thienpondt, and F. I. Parra, Piecewise omnigenous stellarators, Phys. Rev. Lett. 133, 185101 (2024).
  33. M. Landreman and P. J. Catto, Omnigenity as generalized quasisymmetry, Phys. Plasmas 19, 056103 (2012).
  34. A. G. Goodman, P. Xanthopoulos, G. G. Plunk, H. Smith, C. Nührenberg, C. D. Beidler, S. A. Henneberg, G. Roberg-Clark, M. Drevlak, and P. Helander, Quasi-isodynamic stellarators with low turbulence as fusion reactor candidates, PRX Energy 3, 023010 (2024).
  35. R. Nies, E. J. Paul, D. Panici, S. R. Hudson, and A. Bhattacharjee, Exploration of the parameter space of quasisymmetric stellarator vacuum fields through adjoint optimisation, J. Plasma Phys. 90, 905900620 (2024).
  36. M. Landreman, Mapping the space of quasisymmetric stellarators using optimized near-axis expansion, J. Plasma Phys. 88, 905880616 (2022).
  37. M. Landreman, B. Medasani, F. Wechsung, A. Giuliani, R. Jorge, and C. Zhu, SIMSOPT: A flexible framework for stellarator optimization, J. Open Source Softw. 6, 3525 (2021).
  38. P. F. Gil, W. Li, J. Stratton, A. A. Kaptanoglu, and E. V. Stenson, companion paper, Stellarator coils for future fusion reactors via an augmented lagrangian approach, Phys. Rev. Lett. 137, 065101 (2026).
  39. P. F. Gil, W. Li, J. Stratton, A. Kaptanoglu, and E. V. Stenson, Coilsets and Scripts from Augmented Lagrangian Methods for Stellarator Coils [Dataset], Zenodo, 2026, https://doi.org/10.5281/zenodo.18497939.
  40. P. Merkel, Solution of stellarator boundary value problems with external currents, Nucl. Fusion 27, 867 (1987).
  41. M. Landreman, An improved current potential method for fast computation of stellarator coil shapes, Nucl. Fusion 57, 046003 (2017).
  42. M. Landreman, B. Medasani, and C. Zhu, Stellarator optimization for good magnetic surfaces at the same time as quasisymmetry, Phys. Plasmas 28, 092505 (2021).
  43. R. Conlin, P. Kim, D. W. Dudt, D. Panici, and E. Kolemen, Stellarator optimization with constraints, J. Plasma Phys. 90, 905900501 (2024).
  44. A. R. Conn, N. I. M. Gould, and P. L. Toint, Trust-Region Methods, MPS/SIAM Series on Optimization (Society for Industrial and Applied Mathematics, Philadelphia, 2000).
  45. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer Science & Business Media, New York, NY, 2006).
  46. D. C. Liu and J. Nocedal, On the limited memory BFGS method for large scale optimization, Math. Program. 45, 503 (1989).
  47. C. Zhu, S. R. Hudson, Y. Song, and Y. Wan, New method to design stellarator coils without the winding surface, Nucl. Fusion 58, 016008 (2018).
  48. A. A. Kaptanoglu, A. Wiedman, J. Halpern, S. Hurwitz, E. J. Paul, and M. Landreman, Reactor-scale stellarators with force and torque minimized dipole coils, Nucl. Fusion 65, 046029 (2025).
  49. S. Hurwitz, M. Landreman, and T. M. Antonsen, Efficient calculation of the self-magnetic field, self-force, and self-inductance for electromagnetic coils, IEEE Trans. Magn. 60, 1 (2024).
  50. R. Robin and F. A. Volpe, Minimization of magnetic forces on stellarator coils, Nucl. Fusion 62, 086041 (2022).
  51. S. Guinchard, S. R. Hudson, and E. J. Paul, Including the vacuum energy in stellarator coil design, Plasma Phys. Control. Fusion 67, 035028 (2025).
  52. S. Hurwitz, M. Landreman, P. Huslage, and A. Kaptanoglu, Electromagnetic coil optimization for reduced Lorentz forces, Nucl. Fusion 65, 056044 (2025).
  53. Z. S. Hartwig, R. F. Vieira, B. N. Sorbom, R. A. Badcock, M. Bajko, W. K. Beck, B. Castaldo, C. L. Craighill, M. Davies, J. Estrada, et al., VIPER: An industrially scalable high-current high-temperature superconductor cable, Supercond. Sci. Technol. 33, 11LT01 (2020).
  54. Z. Zhao, P. Moore, and L. Chiesa, Structural modeling of REBCO VIPER cable for high-field magnet applications, IEEE Trans. Appl. Supercond. 32, 1 (2022).
  55. N. Riva, R. Granetz, R. Vieira, A. Hubbard, A. Pfeiffer, P. Harris, C. Chamberlain, Z. Hartwig, A. Watterson, D. Anderson, et al., Development of the first non-planar REBCO stellarator coil using VIPER cable, Supercond. Sci. Technol. 36, 105001 (2023).
  56. F. Wechsung, M. Landreman, A. Giuliani, A. Cerfon, and G. Stadler, Precise stellarator quasi-symmetry can be achieved with electromagnetic coils, Proc. Natl. Acad. Sci. USA 119, e2202084119 (2022).
  57. F. Najmabadi, A. R. Raffray, S. I. Abdel-Khalik, L. Bromberg, L. Crosatti, L. El-Guebaly, P. R. Garabedian, A. A. Grossman, D. Henderson, A. Ibrahim, T. Ihli, T. B. Kaiser, B. Kiedrowski, L. P. Ku, J. F. Lyon, R. Maingi, S. Malang, C. Martin, T. K. Mau, B. Merrill, et al., The ARIES-CS compact stellarator fusion power plant, Fusion Sci. Technol. 54, 655 (2008).
  58. D. Spong, J. Harris, A. Ware, S. Hirshman, and L. Berry, Shear flow generation in stellarators—configurational variations, Nucl. Fusion 47, 626 (2007).
  59. A. Wiedman, S. Buller, and M. Landreman, Coil optimization for quasi-helically symmetric stellarator configurations, J. Plasma Phys. 90, 905900307 (2024).
  60. C. D. Beidler, H. M. Smith, A. Alonso, T. Andreeva, J. Baldzuhn, M. N. A. Beurskens, et al., Demonstration of reduced neoclassical energy transport in Wendelstein 7-X, Nature (London) 596, 221 (2021).
  61. P. Helander and J. Nührenberg, Bootstrap current and neoclassical transport in quasi-isodynamic stellarators, Plasma Phys. Control. Fusion 51, 055004 (2009).
  62. P. Helander, Theory of plasma confinement in non-axisymmetric magnetic fields, Rep. Prog. Phys. 77, 087001 (2014).
  63. H. E. Mynick, N. Pomphrey, and P. Xanthopoulos, Optimizing stellarators for turbulent transport, Phys. Rev. Lett. 105, 095004 (2010).
  64. J. Proll, G. Plunk, B. Faber, T. Görler, P. Helander, I. McKinney, M. Pueschel, H. Smith, and P. Xanthopoulos, Turbulence mitigation in maximum-J stellarators with electron-density gradient, J. Plasma Phys. 88, 905880112 (2022).
  65. P. Helander, J. Geiger, and H. Maaßberg, On the bootstrap current in stellarators and tokamaks, Phys. Plasmas 18, 092505 (2011).
  66. C. Beidler, G. Grieger, F. Herrnegger, E. Harmeyer, J. Kisslinger, W. Lotz, H. Maassberg, P. Merkel, J. Nuhrenberg, F. Rau, et al., Physics and engineering design for Wendelstein VII-X, Fusion Technology 17, 148 (1990).
  67. CEA Irfu, Tests on the Wendelstein 7-X Stellarator Coils, Institut de Recherche sur les lois Fondamentales de l'Univers (Irfu), CEA Saclay, https://irfu.cea.fr/dacm/en/Phocea/Vie_des_labos/Ast/ast_visu.php?id_ast=3012.
  68. T. S. Pedersen, M. Otte, S. Lazerson, et al., Confirmation of the topology of the Wendelstein 7-X magnetic field to better than 1:100,000, Nat. Commun. 7, 13493 (2016).
  69. S. P. Gerhardt, J. N. Talmadge, J. M. Canik, and D. T. Anderson, Measurements and modeling of plasma flow damping in the Helically Symmetric eXperiment, Phys. Plasmas 12, 056116 (2005).
  70. J. M. Canik, D. T. Anderson, F. S. B. Anderson, C. Clark, K. M. Likin, J. N. Talmadge, and K. Zhai, Reduced particle and heat transport with quasisymmetry in the Helically Symmetric Experiment, Phys. Plasmas 14, 056107 (2007).
  71. B. Jang, R. Conlin, and M. Landreman, Exponential spectral scaling: Robust and efficient stellarator boundary optimization via mode-dependent scaling, arXiv:2509.16320.
  72. E. Rodríguez, E. Paul, and A. Bhattacharjee, Measures of quasisymmetry for stellarators, J. Plasma Phys. 88, 905880109 (2022).
  73. C. G. Albert, S. V. Kasilov, and W. Kernbichler, Symplectic integration with non-canonical quadrature for guiding-center orbits in magnetic confinement devices, J. Comput. Phys. 403, 109065 (2020).
  74. C. G. Albert, S. V. Kasilov, and W. Kernbichler, Accelerated methods for direct computation of fusion alpha particle losses within, stellarator optimization, J. Plasma Phys. 86, 815860201 (2020).
  75. T. M. Schuett and S. A. Henneberg, Exploring novel compact quasi-axisymmetric stellarators, Phys. Rev. Res. 6, L042052 (2024).

Outline

Information

Phys. Rev. E 114, 025202– Published 7 August, 2026
  • Received 10 October 2025
  • Accepted 14 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation