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), doi:10.1063/1.861224.

  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.

  19. A. H. Boozer, “Plasma equilibrium with rational magnetic surfaces,” Physics of Fluids 24, 1999 (1981), doi:10.1063/1.863297 — the Boozer coordinates in which quasisymmetry and the fast-ion proxies are stated.

  20. R. Sanchez et al., “Ballooning stability optimization of low-aspect-ratio stellarators,” Plasma Physics and Controlled Fusion 42, 641 (2000), doi:10.1088/0741-3335/42/6/303 — the ballooning-optimization companion to the COBRA formulation used by vmex.core.stability.

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, doi:10.1103/PhysRevLett.128.035001 — the two-term quasisymmetry ratio residual (the residual inside eq. 1) 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 ConstructedQIResidual; the lightweight QIResidual is a level-set surrogate.

  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, C. Angioni, E. Belli, and O. Sauter, “A new set of analytical formulae for the computation of the bootstrap current and the neoclassical conductivity in tokamaks,” Physics of Plasmas 28, 022502 (2021), doi:10.1063/5.0012664 — the Redl bootstrap closure (eqs. 10-16, 19-21).

  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, doi:10.1063/5.0098166 — the self-consistent bootstrap iteration reproduced in benchmarks/*_bootstrap_selfconsistent.py; the compact finite-beta optimization workflows are examples/optimization/*_optimization_bootstrap.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.

  10. K. Unalmis et al., “Spectrally accurate, reverse-mode differentiable bounce-averaging algorithm and its applications,” Journal of Plasma Physics (2026), doi:10.1017/S0022377826101652 — endpoint-regularized quadrature and the independent DESC oracle for vmex.core.bounce.

  11. E. Rodríguez, P. Helander, and A. G. Goodman, “The maximum-J property in quasi-isodynamic stellarators,” Journal of Plasma Physics 90, 905900212 (2024), doi:10.1017/S0022377824000345 — the signed outward \(\partial\mathcal J_\parallel/\partial s<0\) condition implemented by MaximumJResidual.

  12. E. Rodríguez and G. G. Plunk, “Near-axis quasi-isodynamic database,” Journal of Plasma Physics (2026), doi:10.1017/S0022377826101688 — defines the bounce-time-weighted Maxwellian maximum-J fraction \(f_J\), which is distinct from a uniform resolved-orbit count.

  13. D. A. Spong and J. H. Harris, “New QP/QI symmetric stellarator configurations,” Plasma and Fusion Research 5, S2039 (2010), doi:10.1585/pfr.5.S2039 — motivates quasi-poloidal symmetry as a practical precursor to poloidally closed-contour quasi-isodynamic configurations.

  14. R. Conlin, P. Kim, D. W. Dudt, D. Panici, and E. Kolemen, “Stellarator Optimization with Constraints,” Journal of Plasma Physics 90, 905900501 (2024), arXiv:2403.11033 — hard shaping constraints and augmented-Lagrangian methods for stellarator design.

  15. B. Jang, R. Conlin, and M. Landreman, “Exponential Spectral Scaling: Robust and Efficient Stellarator Boundary Optimization via Mode-Dependent Scaling,” arXiv:2509.16320 (2025) — the direct full-spectrum variable scaling exposed by use_ess=True.

  16. D. Panici, B. Jang, R. Conlin, D. Dudt, Y. G. Elmacioglu, and E. Kolemen, “Deflation Techniques for Stellarator Equilibrium and Optimization,” arXiv:2602.09957 (2026) — a systematic route to distinct minima when a single deterministic continuation path is insufficient.

  17. H. Chen et al., “Direct Optimization of Stellarator Omnigenity from the Second Adiabatic Invariant,” arXiv:2608.02418 (2026) — recent direct bounce-action optimization complementary to constructed geometric QI targets.

  18. V. V. Nemov, S. V. Kasilov, W. Kernbichler, and M. F. Heyn, “Evaluation of \(1/\nu\) neoclassical transport in stellarators,” Physics of Plasmas 6, 4622–4632 (1999), doi:10.1063/1.873749 — defines the effective-ripple transport measure reported by NEO as \(\epsilon_{\mathrm{eff}}^{3/2}\).

  19. V. V. Nemov et al., “Poloidal motion of trapped particle orbits in real-space coordinates,” Physics of Plasmas 15, 052501 (2008), doi:10.1063/1.2912456 — defines the \(\Gamma_c\) fast-ion proxy (eq. 61) evaluated by GammaC.

  20. J. L. Velasco et al. and the W7-X Team, “A model for the fast evaluation of prompt losses of energetic ions in stellarators,” Nuclear Fusion 61, 116059 (2021), doi:10.1088/1741-4326/ac2994 — the \(\Gamma_c\) organization used here (eq. 16) and the approximately linear prompt-loss relation of eqs. 20-21.

  21. P. Helander, Reports on Progress in Physics 77, 087001 (2014), doi:10.1088/0034-4885/77/8/087001 — the review behind the statement that the two-term residual vanishes exactly for a quasisymmetric \(|B|\).

  22. O. Sauter, C. Angioni, and Y. R. Lin-Liu, Physics of Plasmas 6, 2834 (1999), doi:10.1063/1.873240 (erratum Physics of Plasmas 9, 5140 (2002), doi:10.1063/1.1517052) — the collisionality formulas (eqs. 18b-18e) that the Redl closure reuses in vmex.core.bootstrap.

  23. Y. R. Lin-Liu and R. L. Miller, Physics of Plasmas 2, 1666 (1995), doi:10.1063/1.871315 — the trapped-fraction groundwork underlying the Sauter and Redl fits.

  24. A. Bader et al., “Stellarator equilibria with reactor relevant energetic particle losses,” Journal of Plasma Physics 85, 905850508 (2019), doi:10.1017/S0022377819000680 — energetic-particle losses measured against proxy values.

  25. A. Bader et al., “Modeling of energetic particle transport in optimized stellarators,” Nuclear Fusion 61, 116060 (2021), doi:10.1088/1741-4326/ac2991 — documents the imperfect correlation between \(\Gamma_c\) and simulated energetic-particle losses.

Numerical implementation:

  1. JAX documentation, “The Autodiff Cookbook” — the wide-Jacobian cost model (jacfwd versus jacrev) used to select jacrev for the local three-surface raw-force blocks, https://docs.jax.dev/en/latest/notebooks/autodiff_cookbook.html.

  2. X. S. Li, “An Overview of SuperLU: Algorithms, Implementation, and User Interface,” ACM Transactions on Mathematical Software 31, 302–325 (2005) — globally pivoted sparse factorization used for the equilibrated block-banded radial solve, doi:10.1145/1089014.1089017.

  3. D. Panici, R. Conlin, D. W. Dudt, K. Unalmis, and E. Kolemen, “The DESC stellarator code suite. Part 1: Quick and accurate equilibria computations,” Journal of Plasma Physics 89, 955890303 (2023), doi:10.1017/S0022377823000272 — Eqs. (32)–(34b) define the volume-averaged relative force error \(\langle|\mathbf F|\rangle / \langle|\nabla p|\rangle\) reported beside the pointwise certificate, and its vacuum-safe companion that divides by the volume-averaged magnetic pressure gradient instead.