References

Background and canonical references for VMEC and related equilibrium methods:

  1. S. P. Hirshman and J. C. Whitson, “Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria,” Physics of Fluids 26 (1983).

  2. S. P. Hirshman, W. I. van Rij, and P. Merkel, “Three-dimensional free boundary calculations using a spectral Green’s function method,” Computer Physics Communications 43 (1986).

  3. P. Merkel, “Solution of stellarator boundary value problems with external currents,” Nuclear Fusion 27 (1987).

  4. VMEC2000 reference documentation and wout file format notes (VMEC/LIBSTELL distribution and Princeton VMEC resources).

  5. VMEC++ numerics notes (local copy): vmecpp/docs/the_numerics_of_vmecpp.pdf.

  6. VMEC++ Fourier basis implementation note (local copy): vmecpp/docs/fourier_basis_implementation.md.

  7. VMEC2000 solver core (residuals, bcovar, preconditioner): STELLOPT/VMEC2000/Sources/General/funct3d.f and STELLOPT/VMEC2000/Sources/General/bcovar.f.

  8. VMEC2000 time-step control and restart logic: STELLOPT/VMEC2000/Sources/TimeStep/evolve.f and STELLOPT/VMEC2000/Sources/TimeStep/restart.f.

  9. VMEC2000 diagnostic scalars and Mercier stability: STELLOPT/VMEC2000/Sources/Input_Output/eqfor.f and STELLOPT/VMEC2000/Sources/Input_Output/mercier.f.

  10. A. H. Glasser, J. M. Greene, and J. L. Johnson, “Resistive instabilities in general toroidal plasma configurations,” Physics of Fluids 18(7), 875-888 (1975).

  11. M. Landreman and R. Jorge, “Magnetic well and Mercier stability of stellarators near the magnetic axis,” Journal of Plasma Physics 86(5), 905860510 (2020), arXiv:2006.14881.

  12. VMEC++ solver/restart structure and parity-relevant control flow: vmecpp/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec.cc.

  13. VMEC++ output-quantity and near-axis extrapolation notes: vmecpp/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.cc.

  14. P. Kim, R. Jorge, and W. Dorland, “The On-Axis Magnetic Well and Mercier’s Criterion for Arbitrary Stellarator Geometries,” Journal of Plasma Physics 87(4), 905870409 (2021), arXiv:2011.07416.

  15. J. Schilling et al., “Magnetohydrodynamic equilibrium and stability properties of the Infinity Two fusion pilot plant,” Journal of Plasma Physics 90(6), 905900615 (2024), Appendix B.

  16. J. Schilling et al., “VMEC++: The Numerics of VMEC,” arXiv:2502.04374 — hot restart, JSON input schema, zero-crash policy, and the wout validation methodology adopted here.

  17. C. S. Skene and K. J. Burns, “Fast automated adjoints for spectral PDE solvers,” arXiv:2506.14792 — adjoints reusing the forward spectral machinery; the template for the implicit-differentiation module.

  18. M. Blondel et al., “Efficient and Modular Implicit Differentiation,” NeurIPS 2022 (jaxopt) — the implicit-function-theorem custom_vjp formulation used for equilibrium gradients.

Confinement objectives and optimization:

  1. M. Landreman and E. Paul, “Magnetic fields with precise quasisymmetry for plasma confinement,” Physical Review Letters 128, 035001 (2022), arXiv:2108.03711 — the two-term quasisymmetry ratio residual and the precise-QA/QH configurations (Confinement physics: quasisymmetry, omnigenity, stability).

  2. A. Goodman et al., “Constructing precisely quasi-isodynamic magnetic fields,” Journal of Plasma Physics 89(5), 905890504 (2023), arXiv:2211.09829 — the constructed-QI target implemented by QIResidual.

  3. J. R. Cary and S. G. Shasharina, “Omnigenity and quasihelicity in helical plasma confinement systems,” Physics of Plasmas 4, 3323 (1997) — the bounce-integral formulation of omnigenity.

  4. D. Dudt et al., “Magnetic fields with general omnigenity,” Journal of Plasma Physics 90(1), 905900120 (2024), arXiv:2305.08026 — omnigenity optimization in a differentiable (DESC) framework.

  5. A. Redl et al., “A new set of analytical formulae for the computation of the bootstrap current and the neoclassical conductivity in stellarators,” Physics of Plasmas 28, 022502 (2021) — the Redl bootstrap closure.

  6. M. Landreman, S. Buller, and M. Drevlak, “Optimization of quasi-symmetric stellarators with self-consistent bootstrap current and energetic particle confinement,” Physics of Plasmas 29, 082501 (2022), arXiv:2205.02914 — the self-consistent bootstrap iteration reproduced in examples/optimization/*_bootstrap_selfconsistent.py.

  7. R. Jorge, A. Goodman, M. Landreman, J. Rodrigues, and F. Wechsung, “Single-stage stellarator optimization: combining coils with fixed boundary equilibria,” Plasma Physics and Controlled Fusion 65, 074003 (2023), arXiv:2302.10622 — the combined plasma–coil objective J = J_plasma + w_coils J_coils and the two-stage vs single-stage comparison protocol used by the single-stage examples.

  8. R. Jorge, A. Giuliani, and J. Loizu, “Simplified and flexible coils for stellarators using single-stage optimization,” arXiv:2406.07830 (2024) — cold-start single-stage optimization with staged Fourier-mode release.

  9. F. Wechsung et al., “Precise stellarator quasi-symmetry can be achieved with electromagnetic coils,” PNAS 119(13), e2202084119 (2022) — coil regularization set (length, curvature, coil–coil distance) and the normalized max |B·n|/|B| reporting convention.