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Direct computation of magnetic surfaces in Boozer coordinates and coil optimization for quasisymmetry

Published online by Cambridge University Press:  13 July 2022

Andrew Giuliani*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Florian Wechsung
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Georg Stadler
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Antoine Cerfon
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Matt Landreman
Affiliation:
Institute for Research in Electronics & Applied Physics, University of Maryland, College Park, MD 20742, USA
*
Email address for correspondence: giuliani@cims.nyu.edu

Abstract

We propose a new method to compute magnetic surfaces that are parametrized in Boozer coordinates for vacuum magnetic fields. We also propose a measure for quasisymmetry on the computed surfaces and use it to design coils that generate a magnetic field that is quasisymmetric on those surfaces. The rotational transform of the field and complexity measures for the coils are also controlled in the design problem. Using an adjoint approach, we are able to obtain analytic derivatives for this optimization problem, yielding an efficient gradient-based algorithm. Starting from an initial coil set that presents nested magnetic surfaces for a large fraction of the volume, our method converges rapidly to coil systems generating fields with excellent quasisymmetry and low particle losses. In particular for low complexity coils, we are able to significantly improve the performance compared with coils obtained from the standard two-stage approach, e.g. reduce losses of fusion-produced alpha particles born at half-radius from $17.7\,\%$ to $6.6\,\%$. We also demonstrate 16-coil configurations with alpha loss ${<}1\,\%$ and neoclassical transport magnitude $\epsilon _{\text {eff}}^{3/2}$ less than approximately $5\times 10^{-9}$.

Type
Research Article
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Albert, C.G., Kasilov, S.V. & Kernbichler, W. 2020 a Accelerated methods for direct computation of fusion alpha particle losses within stellarator optimization. J. Plasma Phys. 86, 815860201.CrossRefGoogle Scholar
Albert, C.G., Kasilov, S.V. & Kernbichler, W. 2020 b Symplectic integration with non-canonical quadrature for guiding-center orbits in magnetic confinement devices. J. Comput. Phys. 403, 109065.CrossRefGoogle Scholar
Bader, A., Anderson, D.T., Drevlak, M., Faber, B.J., Hegna, C.C., Henneberg, S., Landreman, M., Schmitt, J.C., Suzuki, Y. & Ware, A. 2021 Modeling of energetic particle transport in optimized stellarators. Nucl. Fusion 61, 116060.CrossRefGoogle Scholar
Boozer, A.H. 1981 Plasma equilibrium with rational magnetic surfaces. Phys. Fluids 24 (11), 19992003.CrossRefGoogle Scholar
Boozer, A.H. 2005 Physics of magnetically confined plasmas. Rev. Mod. Phys. 76, 10711141.CrossRefGoogle Scholar
Brown, T., Breslau, J., Gates, D., Pomphrey, N. & Zolfaghari, A. 2015 Engineering optimization of stellarator coils lead to improvements in device maintenance. In 2015 IEEE 26th Symposium on Fusion Engineering, pp. 1–6. doi: 10.1109/SOFE.2015.7482426.CrossRefGoogle Scholar
Dewar, R., Gibson, A. & Hudson, S. 2010 Unified theory of ghost and quadratic-flux-minimizing surfaces. J. Plasma Fusion Res. 9.Google Scholar
D'haeseleer, W.D., Hitchon, W.N., Callen, J.D. & Shohet, J.L. 2012 Flux Coordinates and Magnetic Field Structure: A Guide to a Fundamental Tool of Plasma Theory. Springer Science & Business Media.Google Scholar
Drevlak, M. 1998 Automated optimization of stellarator coils. Fusion Technol. 33 (2), 106117.CrossRefGoogle Scholar
Garren, D.A. & Boozer, A.H. 1991 Existence of quasihelically symmetric stellarators. Phys. Fluids B 3 (10), 28222834.CrossRefGoogle Scholar
Giuliani, A., Wechsung, F., Cerfon, A., Stadler, G. & Landreman, M. 2022 Single-stage gradient-based stellarator coil design: optimization for near-axis quasi-symmetry. J. Comput. Phys. 459, 111147.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., Hudson, S.R., Pfefferlé, D. & Helander, P. 2021 Combined plasma–coil optimization algorithms. J. Plasma Phys. 87 (2).CrossRefGoogle Scholar
Hirshman, S.P. & Whitson, J.C. 1983 Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria. Phys. Fluids 26 (12), 35533568.CrossRefGoogle Scholar
Klinger, T., Baylard, C., Beidler, C., Boscary, J., Bosch, H., Dinklage, A., Hartmann, D., Helander, P., Maßberg, H., Peacock, A., et al. 2013 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 Engng Des. 88 (6), 461465.CrossRefGoogle Scholar
Kruger, T.G., Zhu, C., Bader, A., Anderson, D.T. & Singh, L. 2021 Constrained stellarator coil curvature optimization with FOCUS. J. Plasma Phys. 87 (2), 175870201.CrossRefGoogle Scholar
Landreman, M. 2017 An improved current potential method for fast computation of stellarator coil shapes. Nucl. Fusion 57 (4), 046003.CrossRefGoogle Scholar
Landreman, M., Medasani, B., Wechsung, F., Giuliani, A., Jorge, R. & Zhu, C. 2021 SIMSOPT: a flexible framework for stellarator optimization. J. Open Source Softw. 6, 3525.CrossRefGoogle Scholar
Landreman, M. & Paul, E. 2022 Magnetic fields with precise quasisymmetry for plasma confinement. Phys. Rev. Lett. 128, 035001.CrossRefGoogle ScholarPubMed
Landreman, M. & Sengupta, W. 2018 Direct construction of optimized stellarator shapes. Part 1. Theory in cylindrical coordinates. J. Plasma Phys. 84 (6), 905840616.CrossRefGoogle Scholar
Landreman, M. & Sengupta, W. 2019 Constructing stellarators with quasisymmetry to high order. J. Plasma Phys. 85 (6).CrossRefGoogle Scholar
Landreman, M., Sengupta, W. & Plunk, G.G. 2019 Direct construction of optimized stellarator shapes. Part 2. Numerical quasisymmetric solutions. J. Plasma Phys. 85 (1), 905850103.CrossRefGoogle Scholar
Merkel, P. 1987 Solution of stellarator boundary value problems with external currents. Nucl. Fusion 27 (5), 867871.CrossRefGoogle Scholar
Neilson, G.H., Gruber, C.O., Harris, J.H., Rej, D.J., Simmons, R.T. & Strykowsky, R.L. 2010 Lessons learned in risk management on NCSX. IEEE Trans. Plasma Sci. 38 (3), 320327.CrossRefGoogle Scholar
Nemov, V.V., Kasilov, S.V., Kernbichler, W. & Heyn, M.F. 1999 Evaluation of $1/\nu$ neoclassical transport in stellarators. Phys. Plasmas 6, 4622.CrossRefGoogle Scholar
Nocedal, J. & Wright, S. 2006 Numerical Optimization. Springer Science & Business Media.Google Scholar
Paul, E., 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
Singh, L., Kruger, T.G., Bader, A., Zhu, C., Hudson, S.R. & Anderson, D.T. 2020 Optimization of finite-build stellarator coils. J. Plasma Phys. 86 (4), 905860404.CrossRefGoogle Scholar
Spong, D.A. 2015 3D toroidal physics: testing the boundaries of symmetry breaking. Phys. Plasmas 22, 055062.CrossRefGoogle Scholar
Strickler, D.J., Berry, L.A. & Hirshman, S.P. 2002 Designing coils for compact stellarators. Fusion Sci. Technol. 41 (2), 107115.CrossRefGoogle Scholar
Strickler, D.J., Hirshman, S.P., Spong, D.A., Cole, M.J., Lyon, J.F., Nelson, B.E., Williamson, D.E. & Ware, A.S. 2004 Development of a robust quasi-poloidal compact stellarator. Fusion Sci. Technol. 45 (1), 1526.CrossRefGoogle Scholar
Strykowsky, R.L., Brown, T., Chrzanowski, J., Cole, M., Heitzenroeder, P., Neilson, G.H., Rej, D. & Viol, M. 2009 Engineering cost & schedule lessons learned on NCSX. In 2009 23rd IEEE/NPSS Symposium on Fusion Engineering, pp. 1–4. doi: 10.1109/FUSION.2009.5226449.CrossRefGoogle Scholar
Tröltzsch, F. 2010 Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Graduate Studies in Mathematics, vol. 112. American Mathematical Society.Google Scholar
Wechsung, F., Giuliani, A., Landreman, M., Cerfon, A. & Stadler, G. 2021 a Single-stage gradient-based stellarator coil design: stochastic optimization. Nucl. Fusion 62 (7), 076034.CrossRefGoogle Scholar
Wechsung, F., Landreman, M., Giuliani, A., Cerfon, A. & Stadler, G. 2022 Precise stellarator quasi-symmetry can be achieved with electromagnetic coils. Proc. Natl Acad. Sci. USA 119 (13), e2202084119.CrossRefGoogle ScholarPubMed
Yu, G., Feng, Z., Jiang, P. & Fu, G. 2022 Existence of an optimized stellarator with simple coils. J. Plasma Phys. 88 (3), 905880306.CrossRefGoogle Scholar
Zhu, C., Hudson, S.R., Song, Y. & Wan, Y. 2017 New method to design stellarator coils without the winding surface. Nucl. Fusion 58 (1), 016008.CrossRefGoogle Scholar