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Computing the shape gradient of stellarator coil complexity with respect to the plasma boundary

Part of: Focus on Fusion

Published online by Cambridge University Press:  22 April 2021

Arthur Carlton-Jones Open the ORCID record for Arthur Carlton-Jones [Opens in a new window] ,
Elizabeth J. Paul Open the ORCID record for Elizabeth J. Paul [Opens in a new window]  and
William Dorland Open the ORCID record for William Dorland [Opens in a new window]
Arthur Carlton-Jones*
Affiliation:
University of Maryland, College Park, MD20742, USA
Elizabeth J. Paul
Affiliation:
Department of Astrophysical Sciences, Princeton University, Princeton, NJ08544, USA
William Dorland
Affiliation:
University of Maryland, College Park, MD20742, USA
*
†Email address for correspondence: [email protected]
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Abstract

Coil complexity is a critical consideration in stellarator design. The traditional two-step optimization approach, in which the plasma boundary is optimized for physics properties and the coils are subsequently optimized to be consistent with this boundary, can result in plasma shapes which cannot be produced with sufficiently simple coils. To address this challenge, we propose a method to incorporate considerations of coil complexity in the optimization of the plasma boundary. Coil complexity metrics are computed from the current potential solution obtained with the REGCOIL code (Landreman, Nucl. Fusion, vol. 57, 2017, 046003). While such metrics have previously been included in derivative-free fixed-boundary optimization (Drevlak et al., Nucl. Fusion, vol. 59, 2018, 016010), we compute the local sensitivity of these metrics with respect to perturbations of the plasma boundary using the shape gradient (Landreman & Paul, Nucl. Fusion, vol. 58, 2018, 076023). We extend REGCOIL to compute derivatives of these metrics with respect to parameters describing the plasma boundary. In keeping with previous research on winding surface optimization (Paul et al., Nucl. Fusion, vol. 58, 2018, 076015), the shape derivatives are computed with a discrete adjoint method. In contrast with the previous work, derivatives are computed with respect to the plasma surface parameters rather than the winding surface parameters. To further reduce the resolution required to compute the shape gradient, we present a more efficient representation of the plasma surface which uses a single Fourier series to describe the radial distance from a coordinate axis and a spectrally condensed poloidal angle. This representation is advantageous over the standard cylindrical representation used in the VMEC code (Hirshman & Whitson, Phys. Fluids, vol. 26, 1983, pp. 3553–3568), as it provides a uniquely defined poloidal angle, eliminating a null space in the optimization of the plasma surface. In comparison with previous spectral condensation methods (Hirshman & Breslau, Phys. Plasmas, vol. 5, 1998, p. 2664), the modified poloidal angle is obtained algebraically rather than through the solution of a nonlinear optimization problem. The resulting shape gradient highlights features of the plasma boundary that are consistent with simple coils and can be used to couple coil and fixed-boundary optimization.

Keywords

fusion plasmaplasma confinement
Type
Research Article
Information
Journal of Plasma Physics , Volume 87 , Issue 2 , April 2021 , 905870222
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Copyright
Copyright © The Author(s), 2021. 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 (3T), 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
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.CrossRefGoogle Scholar
Boozer, A. H. 2000 Stellarator coil optimization by targeting the plasma configuration. Phys. Plasmas 7 (8), 3378.CrossRefGoogle Scholar
Delfour, M. C. & Zolésio, J.-P. 2011 Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization. SIAM.CrossRefGoogle Scholar
Dewar, R. L. & Hudson, S. R. 1998 Stellarator symmetry. Physica D 112 (1-2), 275.CrossRefGoogle Scholar
Drevlak, M. 1998 Automated optimization of stellarator coils. Fusion Technol. 33 (2), 106.CrossRefGoogle Scholar
Drevlak, M., Beidler, C. D., Geiger, J., Helander, P. & Turkin, Y. 2018 Optimisation of stellarator equilibria with ROSE. Nucl. Fusion 59 (1), 016010.CrossRefGoogle Scholar
Farouki, R. T. 1986 The approximation of non-degenerate offset surfaces. Comput.-Aided Geom. Des. 3 (1), 15.CrossRefGoogle Scholar
Geraldini, A., Landreman, M. & Paul, E. J. 2021 An adjoint method for determining the sensitivity of island size to magnetic field variations. J. Plasma Phys. arXiv:2102.04497.Google 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.Google Scholar
Helander, P., Drevlak, M., Zarnstorff, M. & Cowley, S. C. 2020 Stellarators with permanent magnets. Phys. Rev. Lett. 124 (9), 095001.CrossRefGoogle ScholarPubMed
Henneberg, S. A., Hudson, S. R., Pfefferlé, D. & Helander, P. 2020 Combined plasma-coil optimization algorithms. J. Plasma Phys. arXiv:2012.09278.Google Scholar
Hirshman, S. P. & Breslau, J. 1998 Explicit spectrally optimized Fourier series for nested magnetic surfaces. Phys. Plasmas 5 (7), 2664.CrossRefGoogle Scholar
Hirshman, S. P. & Meier, H. K. 1985 Optimized Fourier representations for three-dimensional magnetic surfaces. Phys. Fluids 28 (5), 1387.CrossRefGoogle Scholar
Hirshman, S. P. & Whitson, J. C. 1983 Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria. Phys. Fluids 26 (12), 3553–3568.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., Dewar, R. L, Hole, M. J. & McGann, M. 2011 Non-axisymmetric, multi-region relaxed magnetohydrodynamic equilibrium solutions. Plasma Phys. Control. Fusion 54 (1), 014005.CrossRefGoogle Scholar
Hudson, S. R., Monticello, D. A., Reiman, A. H., Boozer, A. H., Strickler, D. J., Hirshman, S. P. & Zarnstorff, M. C. 2002 Eliminating islands in high-pressure free-boundary stellarator magnetohydrodynamic equilibrium solutions. Phys. Rev. Lett. 89 (27), 275003.CrossRefGoogle ScholarPubMed
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
Imbert-Gerard, L.-M., Paul, E. J. & Wright, A. 2019 An introduction to symmetries in stellarators. arXiv:1908.05360.Google Scholar
Jameson, A., Martinelli, L. & Pierce, N. A. 1998 Optimum aerodynamic design using the Navier-Stokes equations. Theor. Comput. Fluid Dyn. 10 (1-4), 213.CrossRefGoogle Scholar
Kress, R. 1989 Linear Integral Equations, chap. 5. Springer.Google 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. & Boozer, A. H. 2016 Efficient magnetic fields for supporting toroidal plasmas. Phys. Plasmas 23 (3), 032506.CrossRefGoogle Scholar
Landreman, M. & Paul, E. J. 2018 Computing local sensitivity and tolerances for stellarator physics properties using shape gradients. Nucl. Fusion 58 (7), 076023.CrossRefGoogle Scholar
Landreman, M. & Zhu, C. 2021 Calculation of permanent magnet arrangements for stellarators: A linear least-squares method. Plasma Phys. Control. Fusion 63 (3), 035001.CrossRefGoogle Scholar
Merkel, P. 1987 Solution of stellarator boundary value problems with external currents. Nucl. Fusion 27 (5), 867.CrossRefGoogle Scholar
Othmer, C. 2008 A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows. Intl J. Numer. Meth. Fluids 58, 861.CrossRefGoogle 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 a 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. Jr. 2020 b Gradient-based optimization of 3D MHD equilibria. arXiv:2012.10028.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
Qu, Z., Pfefferlé, D., Hudson, S. R., Baillod, A., Kumar, A., Dewar, R. L. & Hole, M. J. 2020 Coordinate parametrization and spectral method optimisation for Beltrami field solver in stellarator geometry. Plasma Phys. Control. Fusion 62, 124004.CrossRefGoogle Scholar
Sauer, T. 2012 Numerical Analysis. Pearson.Google Scholar
Shohet, J. L., Anderson, D. T., Anderson, F. S. B. & Talmadge, J. N. 1991 The University of Wisconsin-Madison Torsatron/Stellarator laboratory program FY 1991–1993 annual progress report.CrossRefGoogle Scholar
Spong, D. A., Hirshman, S. P., Berry, L. A., Lyon, J. F., Fowler, R. H., Strickler, D. J., Cole, M. J., Nelson, B. N., Williamson, D. E., Ware, A. S., et al. 2001 Physics issues of compact drift optimized stellarators. Nucl. Fusion 41 (6), 711.CrossRefGoogle Scholar
Strickler, D. J., Berry, L. A. & Hirshman, S. P. 2003 Integrated plasma and coil optimization for compact stellarators. Tech. Rep.Google Scholar
Zarnstorff, M. C., Berry, L. A., Brooks, A., Fredrickson, E., Fu, G. Y., Hirshman, 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.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, 016008.CrossRefGoogle Scholar
Zhu, C., Zarnstorff, M. C., Gates, D. A. & Brooks, A. 2020 Designing stellarators using perpendicular permanent magnets. Nucl. Fusion 60, 076016.CrossRefGoogle Scholar