Simple-to-build quasi-isodynamic stellarator

Find a stellarator plasma boundary whose magnetic field is close to precisely quasi-isodynamic, a property that confines fast particles well, while keeping the field smooth enough to be made by simple coils set far from the plasma. Such fields have historically needed intricate coils, so this second ConStellaration problem asks whether good confinement and buildability can coexist.

Task

A candidate is a stellarator-symmetric boundary given by Fourier coefficients $R_{mn}$, $Z_{mn}$ and a number of field periods $N_{\mathrm{fp}}$, exactly as in the geometric problem; the forward model computes its vacuum equilibrium with VMEC++. The problem is

$$\max\ \tilde L_{\nabla B}\quad\text{subject to}\quad A\le10,\quad \lvert\tilde\iota\rvert\ge0.25,\quad \log_{10}\mathit{QI}\le-4,\quad \Delta\le0.2,\quad \epsilon_{\max}\le5.$$

Here $\tilde L_{\nabla B}$ is the smallest magnetic-field gradient scale length on the boundary (Kappel et al., 2024), normalized by $a/N_{\mathrm{fp}}$ with $a$ the minor radius; $\mathit{QI}=\frac{1}{4\pi^2}\iint r_{QI}^2\,d\theta\,d\phi$ is the residual measuring the departure from a precisely quasi-isodynamic field (Goodman et al., 2023); $\Delta=(B_{\max}-B_{\min})/(B_{\max}+B_{\min})$ is the mirror ratio on the boundary; $A$ is the aspect ratio, $\tilde\iota$ the edge rotational transform per field period and $\epsilon_{\max}$ the maximum elongation. A design is feasible when every normalized violation $(c_i-c_i^*)/\lvert c_i^*\rvert$ is at most $0.01$. The score is $\min(1,\tilde L_{\nabla B}/20)$ for a feasible design and $0$ otherwise; higher is better.

Background

A quasi-isodynamic (QI) field is omnigenous, so trapped particles have no net radial drift on average, and its field-strength contours close poloidally, which makes the net toroidal plasma current vanish. Wendelstein 7-X demonstrated the benefits of approximately QI fields, and QI is a target for several power-plant designs. A long gradient scale length means the field can be produced by coils placed further away with gentler bends. Cadena et al. (2025) posed this problem to test the belief that QI fields require complex coils. The aspect-ratio, elongation and mirror-ratio bounds stop optimizers from reaching QI the easy way, through large, stretched or strongly mirrored configurations.

Data and evaluation

• Instances: one fixed problem. The dataset of about 158,000 QI-like boundaries and the public leaderboard results can serve as starting points.
• Evaluation code: constellaration==0.2.6, problems.SimpleToBuildQIStellarator().evaluate(boundary), with VMEC++ at high fidelity and the QI computation switched on. The leaderboard uses exactly this release. A submission is one boundary as SurfaceRZFourier JSON.
• Hardware: none assumed; the score does not depend on run time.

What is known

• Paper baseline (Cadena et al., 2025): the augmented-Lagrangian NGOpt method reached $\tilde L_{\nabla B}=8.61$, a score of 0.431, after 34 hours on 96 vCPUs; SciPy's trust-constr and COBYQA ended infeasible.
• Leaderboard progress: 0.6284 (user phanerozoic, May 2026), 0.6361 (davidkh, July 2026), and the best on 2026-10-20, 0.65773 (canyon-ant, September 2026), which corresponds to $\tilde L_{\nabla B}\approx13.15$.
• Several top entries, including the best, use most of the 1% feasibility tolerance (largest normalized violation about 0.009), so a design that meets the constraints exactly may score slightly lower.

What would count as progress

• A boundary scoring above 0.65773 under constellaration==0.2.6; the check is one forward-model evaluation.
• A design within 0.01 of the best score that meets every constraint with no tolerance at all.
• A coil set for a high-scoring boundary, showing that the larger gradient scale length really buys simpler coils (for example, larger coil-plasma distance at fixed field error).

References

1. S. A. Cadena et al. ConStellaration: A dataset of QI-like stellarator plasma boundaries and optimization benchmarks. NeurIPS 2025 Datasets and Benchmarks Track. arXiv:2506.19583
2. J. Kappel, M. Landreman, D. Malhotra. The magnetic gradient scale length explains why certain plasmas require close external magnetic coils. Plasma Physics and Controlled Fusion 66, 025018 (2024). doi:10.1088/1361-6587/ad1a3e
3. A. G. Goodman et al. Constructing precisely quasi-isodynamic magnetic fields. Journal of Plasma Physics 89, 905890504 (2023). doi:10.1017/S002237782300065X
4. J. Schilling. The numerics of VMEC++. 2025. arXiv:2502.04374

Source

Imported from ConStellaration simple-to-build QI problem, curated by Proxima Fusion. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.