High Order Free Boundary MHD Equilibria in DESC

Daniel W. Dudt; Dario Panici; Egemen Kolemen; Jonathan Schilling; Rogerio Jorge; Rory Conlin

REVIEW 3 major objections 5 minor 26 references

High Order Free Boundary MHD Equilibria in DESC

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read DESC computes free-boundary ideal MHD equilibria with a high-order singular boundary integral method, avoiding the exterior Neumann problem and achieving lower boundary error than VMEC.

desk verdict First free-boundary DESC capability with a clean residual-minimizing formulation; the high-order quadrature claim is plausible but under-verified, and the paper deserves review with requests for standalone convergence and runtime tests. read the letter →

arxiv 2412.05680 v1 pith:WJ36AF74 submitted 2024-12-07 physics.plasm-ph cs.NAmath.NA

Rory Conlin , Jonathan Schilling , Daniel W. Dudt , Dario Panici , Rogerio Jorge , Egemen Kolemen This is my paper · ORCID

classification physics.plasm-phcs.NAmath.NA

keywords freeboundaryequilibriumidealMHDstellaratorsingularintegralspartitionofunityquadraturesheetcurrentvirtualcasingDESC

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper shows how to compute free-boundary ideal MHD equilibria in the DESC code without solving the exterior Neumann problem. The free-boundary conditions are recast as a residual objective, minimized over the plasma boundary shape and a surface-current potential, and the needed singular Biot-Savart integral over the boundary is evaluated with a partition-of-unity quadrature that is claimed to be 10th to 12th order accurate. Benchmarks against direct field-line tracing and against VMEC free-boundary calculations support the method: a precise quasisymmetric equilibrium is reproduced, a helical stellarator converges in a few iterations, and a finite-$\beta$ case with a nonzero edge pressure yields a sheet current of about 3.6 kA. For a W7-X-like equilibrium at $\beta=2\%$, the DESC boundary shape has roughly two to three times lower boundary-condition residual than the VMEC boundary. If these results hold, DESC gains a free-boundary capability that preserves the accuracy advantage it already has in fixed-boundary calculations.

What carries the argument

The load-bearing object is the partition-of-unity singular quadrature for the two-dimensional Biot-Savart integral (4.3). The integrand is split into a smooth part and a singular part supported near the evaluation point using $\chi(\rho)=e^{-36\rho^8}$; the smooth part is integrated with the trapezoidal rule, which converges exponentially on the periodic domain, and the singular part is integrated in polar coordinates, whose Jacobian $\rho$ cancels the $1/r$ singularity. DESC's adaptation replaces the original 12th-order Lagrange interpolation with Fourier interpolation, accounting for the shifted grid by a phase factor in the Fourier domain, so the interpolation is spectrally accurate and no metric quantities need be recomputed in polar coordinates. This quadrature supplies the external-field contribution at each optimization step, turning the free-boundary problem into minimization of the residual (4.4) over boundary shape and surface-current potential.

What would settle it

Evaluate the boundary integral (4.3) on a surface with a known analytic field (for example a circular torus with a uniform sheet current) and measure the error as the grid is refined; if the observed order is far below the claimed 10th to 12th, the accuracy advantage over VMEC is not established. A complementary check is to reconstruct the same W7-X-like equilibrium with an independent high-accuracy method and compare flux surfaces and boundary residuals to the DESC and VMEC results.

Watch

Extended reading notes

Core claim

The central claim is that the free-boundary equilibrium can be found by evaluating one singular boundary integral per iteration, the virtual-casing Biot-Savart integral $$B_{\rm out}=B_{\rm coil}+\frac{\mu_0}{4\pi}\int_{\mathcal{D}}\frac{[n\times(B_{\rm fixed}+\nabla\Phi)]\times(r-r')}{|r-r'|^3}\,$d^{2}$r',$$ instead of inverting the large linear operator of the exterior Neumann problem. The boundary conditions $B\cdot n=0$, $[[p+B^2/2\mu_0]]=0$, and $n\times[[B]]=\mu_0 K$ become the three rows of the residual objective (4.4), minimized over the boundary coefficients $(R_b,Z_b)$ and the current potential $\Phi$. The paper reports that this reproduces a precise quasisymmetric equilibrium in agreement with field-line tracing, converges an initially circular boundary to a helical stellarator in about four steps, and for a W7-X-like case at $\beta=2\%$ gives a boundary whose residual is two to three times smaller than VMEC's, attributed to the higher order boundary integral method. The same formulation also handles the sheet current that must exist when edge pressure is finite, as demonstrated by the $\sim 3.6$ kA surface current in a finite-$\beta$ helical stellarator test.

Load-bearing premise

The argument stands or falls on whether the partition-of-unity singular quadrature, as adapted with Fourier interpolation, actually retains high-order accuracy for the Biot-Savart integral on DESC's Fourier-parameterized boundary surfaces, and the paper does not report an independent convergence-order test in that setting.

Editorial extensions

If this is right

DESC free-boundary runs need only one singular surface integral per evaluation, so the exterior Neumann linear solve is removed from the equilibrium iteration.
Finite edge pressure is represented by a surface-current potential, so equilibria with sheet currents can be computed and the sheet current quantified, as in the $\sim3.6$ kA helical-stellarator test.
Because the boundary conditions are a minimization objective, the same code path can be extended to single-stage optimization in which plasma boundary and coil degrees of freedom vary together.
The reported two-to-three-fold lower boundary residual than VMEC on the W7-X-like case suggests free-boundary calculations can inherit the accuracy DESC already shows for fixed-boundary equilibria.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the quadrature really delivers 10th to 12th order accuracy on DESC surfaces, the same singular-integral machinery could also upgrade virtual-casing diagnostics and coil verification, not just equilibrium solves.
The residual-objective formulation invites a direct test: use the high-order integral to score boundary-condition residuals of equilibria produced by other solvers and see whether the error ordering persists across configurations and resolutions.
The explicit sheet-current potential may prove useful for modelling edge-localized current layers in finite-pressure devices, where the tangential-field jump is physically meaningful rather than a numerical artifact.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read this if you work on stellarator equilibria. It delivers the first free-boundary capability in DESC, built on a partition-of-unity singular quadrature, and the benchmarks are honestly described and mostly convincing. The vacuum QA case is validated by direct field-line tracing; the finite-beta sheet-current case shows residuals converging as resolution increases. That is a real capability the DESC ecosystem did not have.

What's new: casting the free-boundary conditions as a minimization over boundary shape and sheet-current potential, avoiding the exterior Neumann solve that VMEC/NESTOR does. Not a radically new idea—minimizing boundary-condition residuals is old—but this is the first implementation in DESC and it handles finite edge pressure with sheet currents, which NESTOR doesn't do explicitly. The code is open and the method section is clear.

Soft spots, in order. (1) The high-order claim is under-supported. The paper swaps Malhotra's Lagrange interpolation for Fourier interpolation and calls it spectral, but there is no standalone convergence test of the adapted Biot-Savart integral on DESC's Fourier surfaces. Figure 3's residual curve blends the equilibrium solve, boundary shape, current potential, and quadrature, so it doesn't isolate the quadrature order. The VMEC comparison's 2–3x lower residual is computed with the same unverified quadrature, so it's not independent evidence of superior accuracy. This is a real gap, but it doesn't sink the paper: the field-line-tracing agreement in Figure 1 gives end-to-end confidence that the method produces correct equilibria. (2) The efficiency claim is asserted, not measured—no wall-clock comparison against NESTOR or VMEC. (3) The MGRID interpolation is first-order, which muddies the order of the overall method in the helical benchmark; that's acknowledged in the text, though.

Bottom line: a solid, useful capability paper for the DESC user base and for anyone doing single-stage optimization. The mathematics is standard, the citation pattern looks fair, and the missing pieces are numerical evidence, not correctness. A serious referee should ask for a focused convergence test of the singular integral and a runtime comparison; both are doable. I would send it to review.

Referee Report

3 major / 5 minor

Summary. This paper extends the DESC fixed-boundary equilibrium code to free-boundary problems. The exterior field is represented as the sum of the coil field, a virtual-casing field obtained from the fixed-boundary solution, and a surface-current field parameterized by a potential Φ; the free-boundary conditions (normal-field balance, pressure balance, and sheet-current relation) are then enforced by minimizing a residual objective in the boundary shape and Φ. Singular Biot–Savart integrals are evaluated with a partition-of-unity polar quadrature adapted from Malhotra et al., using Fourier interpolation on DESC's boundary grid. Results are shown for a vacuum quasi-axisymmetric equilibrium compared with direct field-line tracing, a finite-edge-pressure helical stellarator with sheet current and residual convergence, and a W7-X-like β=2% case compared with free-boundary VMEC. The paper concludes that DESC's high-order method is more accurate than VMEC.

Significance. If the accuracy claims are substantiated, this is a valuable capability: it avoids an explicit exterior Neumann solve and the associated large linear system inversion, handles finite edge pressure and sheet currents naturally, and is structured to fit into single-stage plasma–coil optimization. The open-source implementation and the effort to benchmark against field-line tracing and VMEC are strengths. However, the central high-order claim is not yet directly supported: the paper validates the coupled free-boundary solver, not the singular quadrature in its new Fourier-interpolated form, and the VMEC comparison uses the very method whose accuracy is at issue. The significance is therefore contingent on the additional verification requested below.

major comments (3)

[§5, Eqs. (5.1)–(5.6)] The central claim of high-order accuracy rests on the partition-of-unity singular quadrature retaining its 10th–12th order behavior when the original Lagrange interpolation is replaced by Fourier interpolation on DESC's boundary grid. The paper states that this interpolation is spectrally accurate and that derived quantities need not be recomputed, but it provides no convergence study of the singular integral itself. Figure 3 shows the full free-boundary residual decreasing with spectral resolution, which conflates the boundary representation, the equilibrium solve, and the quadrature behavior. I request a direct test, for example using a manufactured toroidal surface with a known Biot–Savart integral or a known harmonic field, reporting the error in Eq. (4.3) as a function of grid resolution and quadrature order, to establish the order of the adapted scheme.
[§6, Figure 4 and final paragraph] The reported 2–3x lower residual for DESC over VMEC is computed with the same high-order boundary integral method that is the subject of the paper. If the quadrature has a systematic bias that favors DESC's Fourier boundary representation, this comparison would not establish superior physical accuracy. An independent error metric is needed, such as field-line tracing error, a known analytic free-boundary equilibrium, or residual evaluation with an independent method such as NESTOR. As written, the superiority claim is not independently corroborated.
[§5 and §6, MGRID-based examples] The paper itself states that MGRID coil-field interpolation is only first-order accurate in the grid spacing. The helical stellarator example with sheet current (Figures 2 and 3) uses MGRID for the external field. This is inconsistent with the high-order claim unless the MGRID grid is fine enough that its interpolation error is negligible; no grid-resolution check for MGRID is shown. The vacuum QA benchmark uses direct filament Biot–Savart and is more convincing, but the sheet-current convergence demonstration should either use an accurate coil field or demonstrate that the MGRID error is subdominant at the reported resolutions.

minor comments (5)

[Eq. (5.5)] The displayed definition of ρ appears malformed: the factor s should presumably divide the normalized distance rather than multiply it. Please correct the equation or clarify the notation.
[References] The entry 'Drevlak & Lobsien 2022' is cited as 'private communication'; this should be replaced with a citable document, a preprint, or removed from the formal reference list.
[Figure 3] The vertical axis is labeled only as 'residual'; please specify the exact norm used (e.g., L2 norm of the boundary-condition vector), the normalization, and the discretization parameters held fixed while spectral resolution is increased.
[§6, initialization statement] The statement that 'results do not seem to depend on the choice of initialization' is anecdotal as written; a small scan over initial conditions or a more cautious phrasing would make the claim reproducible.
[Abstract and §7] The word 'efficient' in the abstract and conclusion is not supported by any timing or complexity data. A short runtime or cost comparison with VMEC/NESTOR for one of the benchmark cases would substantiate this claim.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the free-boundary solve minimizes the physical boundary-condition residual and is independently checked by field-line tracing.

full rationale

The paper's free-boundary method is a self-consistent solve of the MHD boundary conditions, not a circular derivation. Equation (4.4) defines an objective that penalizes the residuals of the physical conditions B·n = 0, pressure balance, and the sheet-current jump relation; minimizing that residual is a standard weighted-residual approach. The field Bfixed is obtained from a fixed-boundary DESC solve for a trial boundary, and Bout is constructed independently via Biot-Savart integrals over virtual-casing and sheet-current sources, so the ingredients are not defined in terms of the final equilibrium. No fitted parameter is relabeled as a prediction: the sheet-current potential Φ is an unknown solved as part of the equilibrium, and the reported 3.6 kA surface current is a solution output, not a predicted quantity. The high-order singular quadrature is imported from external work by Malhotra et al. and Ying et al., and the Fourier-interpolation adaptation is a numerical implementation choice, not an ansatz that smuggles in the conclusion. Self-citations to the DESC code suite describe prior code components and fixed-boundary accuracy results; they are not load-bearing for the new free-boundary formulation. Independent field-line tracing from the coil set (Figure 1) verifies the QA equilibrium, and the convergence study in Figure 3 checks the residual as spectral resolution is increased. The comparison with VMEC uses the residual of Equation (4.4) as a benchmark; although that residual is DESC's optimization objective, it is a physical boundary-condition error measure and VMEC is expected to satisfy the same conditions, so the comparison is a benchmark rather than a tautology. The main caveat, not a circularity, is that the VMEC 'more accurate' claim would be stronger with an independent error metric, but the central free-boundary capability does not reduce to its inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The method relies on standard MHD free boundary conditions and the virtual casing principle, plus the numerical accuracy of the partition-of-unity quadrature from Malhotra et al. No new physical entities are introduced. Several numerical parameters (support size, quadrature orders) are chosen by hand but are not fitted to data and do not affect the physics.

free parameters (2)

Partition of unity support size s = not stated
Appears in eq. (5.5) and controls the singular domain; chosen by hand, value not reported.
Polar quadrature orders (n_rho, n_omega) = not stated
Orders for Gauss-Legendre and trapezoidal rules in the singular integral; chosen by hand, not reported.

assumptions (4)

domain assumption Standard ideal MHD free boundary conditions (eqs. 3.1-3.3)
The objective (4.4) is built from these; the paper assumes they are the correct conditions for 3D ideal MHD with possible sheet currents.
domain assumption Virtual casing principle
Used in eq. (4.3) to compute the plasma field outside the boundary from a surface integral of n x B_fixed; standard in MHD equilibrium.
standard math Partition-of-unity singular quadrature accuracy (Malhotra et al.)
The paper relies on the claimed 10th to 12th order accuracy of the quadrature for the Biot-Savart kernel, and on the Fourier interpolation preserving that accuracy.
domain assumption Well-posedness of residual minimization
The paper assumes that minimizing the residual in (4.4) yields the unique physically correct equilibrium; no proof of uniqueness is given.

how reviews work

0 comments

Cite this review

Pith. "Pith review of High Order Free Boundary MHD Equilibria in DESC." pith.science (2026). https://pith.science/paper/WJ36AF74

@misc{pith2026241205680,
  author       = {Pith},
  title        = {Pith review of: High Order Free Boundary MHD Equilibria in DESC},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WJ36AF74}},
  note         = {Machine review of arXiv:2412.05680}
}

read the original abstract

In this work we consider the free boundary inverse equilibrium problem for 3D ideal MHD. We review boundary conditions for both fixed and free boundary solutions and under what circumstances a sheet current may exist at the plasma-vacuum interface. We develop an efficient and accurate algorithm for computing the residual of these boundary conditions and use it to compute free boundary equilibria in the DESC code both in vacuum and at finite plasma beta, with and without sheet currents.

Figures

Figures reproduced from arXiv: 2412.05680 by the authors.

**Figure 2.** Deformation of initially circular boundary shape into helical stellarator. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Convergence of boundary condition residual as function of spectral resolution [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Comparison of free boundary DESC and VMEC solutions for W7-X at [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 22 canonical work pages

[1]

, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...
[2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...
[3]

Conlin, Rory , Dudt, Daniel W , Panici, Dario & Kolemen, Egemen 2023 The desc stellarator code suite. part 2. perturbation and continuation methods . Journal of Plasma Physics 89 (3), 955890305

work page 2023
[4]

private communication

Drevlak, Michael & Lobsien, Jim-Felix 2022 VMEC free-boundary stellarator optimizations with ONSET . private communication

work page 2022
[5]

https://github.com/PlasmaControl/DESC

Dudt, Daniel , Conlin, Rory , Panici, Dario , Kolemen, Egemen , Unalmis, Kaya , Kim, Patrick & Schilling, Jonathan 2024 DESC . https://github.com/PlasmaControl/DESC

work page 2024
[6]

Physics of Plasmas 27 (10)

Dudt, DW & Kolemen, E 2020 Desc: A stellarator equilibrium solver . Physics of Plasmas 27 (10)

work page 2020
[7]

Journal of Plasma Physics 89 (2), 955890201

Dudt, Daniel W , Conlin, Rory , Panici, Dario & Kolemen, Egemen 2023 The desc stellarator code suite part 3: Quasi-symmetry optimization . Journal of Plasma Physics 89 (2), 955890201

work page 2023
[8]

Cambridge University Press

Freidberg, Jeffrey P 2014 ideal MHD\/ . Cambridge University Press

work page 2014

Show all 26 references

[9]

Cambridge University Press

Goedbloed, Hans , Goedbloed, JP , Keppens, Rony & Poedts, Stefaan 2019 Magnetohydrodynamics: Of Laboratory and Astrophysical Plasmas\/ . Cambridge University Press

2019
[10]

In APS Division of Plasma Physics Meeting Abstracts\/ , , vol

Hanson, James 2017 Surface currents on the plasma-vacuum interface in mhd equilibria. In APS Division of Plasma Physics Meeting Abstracts\/ , , vol. 2017 , pp. CP11--129

2017
[11]

In 58th Annual Meeting of the Division of Plasma Physics of the APS San Jos \'e , CA, USA

Hanson, James D 2016 Surface current on the plasma-vacuum interface in mhd equilibria. In 58th Annual Meeting of the Division of Plasma Physics of the APS San Jos \'e , CA, USA. Poster No. GP10\/ , , vol. 64

2016
[12]

Nuclear Fusion 49 (7), 075031

Hanson, James D , Hirshman, Steven P , Knowlton, Stephen F , Lao, Lang L , Lazarus, Edward A & Shields, John M 2009 V3fit: a code for three-dimensional equilibrium reconstruction . Nuclear Fusion 49 (7), 075031

2009
[13]

Journal of Plasma Physics 87 (2), 905870226

Henneberg, Sophia A , Hudson, Stuart R , Pfefferl \'e , David & Helander, Per 2021 Combined plasma--coil optimization algorithms . Journal of Plasma Physics 87 (2), 905870226

2021
[14]

, van RIJ, W.I

Hirshman, S.P. , van RIJ, W.I. & Merkel, P. 1986 Three-dimensional free boundary calculations using a spectral Green 's function method . Computer Physics Communications 43 (1), 143--155

1986
[15]

Hirshman, S. P. & Whitson, J. C. 1983 Steepest‐descent moment method for three‐dimensional magnetohydrodynamic equilibria . The Physics of Fluids 26 (12), 3553--3568 , publisher: American Institute of Physics

1983
[16]

Plasma Physics and Controlled Fusion 62 (8), 084002

Hudson, S R , Loizu, J , Zhu, C , Qu, Z S , Nührenberg, C , Lazerson, S , Smiet, C B & Hole, M J 2020 Free-boundary MRxMHD equilibrium calculations using the stepped-pressure equilibrium code . Plasma Physics and Controlled Fusion 62 (8), 084002

2020
[17]

Journal of Open Source Software 6 (65), 3525

Landreman, Matt , Medasani, Bharat , Wechsung, Florian , Giuliani, Andrew , Jorge, Rogerio & Zhu, Caoxiang 2021 Simsopt: A flexible framework for stellarator optimization . Journal of Open Source Software 6 (65), 3525

2021
[18]

Physical Review Letters 128 (3), 035001

Landreman, Matt & Paul, Elizabeth 2022 Magnetic fields with precise quasisymmetry for plasma confinement . Physical Review Letters 128 (3), 035001

2022
[19]

Journal of Computational Physics 397 , 108791

Malhotra, Dhairya , Cerfon, Antoine , Imbert-G \'e rard, Lise-Marie & O'Neil, Michael 2019 Taylor states in stellarators: A fast high-order boundary integral solver . Journal of Computational Physics 397 , 108791

2019
[20]

Plasma Physics and Controlled Fusion 62 (2), 024004

Malhotra, Dhairya , Cerfon, Antoine J , O’Neil, Michael & Toler, Evan 2020 Efficient high-order singular quadrature schemes in magnetic fusion . Plasma Physics and Controlled Fusion 62 (2), 024004

2020
[21]

Journal of Computational Physics 66 (1), 83--98

Merkel, Peter 1986 An integral equation technique for the exterior and interior neumann problem in toroidal regions . Journal of Computational Physics 66 (1), 83--98

1986
[22]

1988 Applications of the Neumann Problem to Stellarators: Magnetic Surfaces, Coils, Free-Boundary Equilibrium, Magnetic Diagnostics

Merkel, P. 1988 Applications of the Neumann Problem to Stellarators: Magnetic Surfaces, Coils, Free-Boundary Equilibrium, Magnetic Diagnostics . In Theory of Fusion Plasmas \/ (ed. A. Bondeson, E. Sindoni & F. Troyon ) , pp. 25--46

1988
[23]

& Strumberger, E

Merkel, P. & Strumberger, E. 2015 Linear mhd stability studies with the starwall code, arXiv:arXiv: 1508.04911

2015 arXiv
[24]

Panici, Dario , Conlin, Rory , Dudt, Daniel W , Unalmis, Kaya & Kolemen, Egemen 2023 The desc stellarator code suite. part 1. quick and accurate equilibria computations . Journal of Plasma Physics 89 (3), 955890303

2023
[25]

Master's thesis, Universtät Kiel, Kiel, Germany

Schilling, J 2018 Experimental MHD equilibrium analysis of magnetic configurations in the Wendelstein 7-X stellarator . Master's thesis, Universtät Kiel, Kiel, Germany

2018
[26]

Journal of Computational Physics 219 (1), 247--275

Ying, Lexing , Biros, George & Zorin, Denis 2006 A high-order 3d boundary integral equation solver for elliptic pdes in smooth domains . Journal of Computational Physics 219 (1), 247--275

2006

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

[3] [3]

Conlin, Rory , Dudt, Daniel W , Panici, Dario & Kolemen, Egemen 2023 The desc stellarator code suite. part 2. perturbation and continuation methods . Journal of Plasma Physics 89 (3), 955890305

work page 2023

[4] [4]

private communication

Drevlak, Michael & Lobsien, Jim-Felix 2022 VMEC free-boundary stellarator optimizations with ONSET . private communication

work page 2022

[5] [5]

https://github.com/PlasmaControl/DESC

Dudt, Daniel , Conlin, Rory , Panici, Dario , Kolemen, Egemen , Unalmis, Kaya , Kim, Patrick & Schilling, Jonathan 2024 DESC . https://github.com/PlasmaControl/DESC

work page 2024

[6] [6]

Physics of Plasmas 27 (10)

Dudt, DW & Kolemen, E 2020 Desc: A stellarator equilibrium solver . Physics of Plasmas 27 (10)

work page 2020

[7] [7]

Journal of Plasma Physics 89 (2), 955890201

Dudt, Daniel W , Conlin, Rory , Panici, Dario & Kolemen, Egemen 2023 The desc stellarator code suite part 3: Quasi-symmetry optimization . Journal of Plasma Physics 89 (2), 955890201

work page 2023

[8] [8]

Cambridge University Press

Freidberg, Jeffrey P 2014 ideal MHD\/ . Cambridge University Press

work page 2014