Fix winding-surface optimization example

[rogeriojorge]

This updates the winding-surface optimization example and adds a resolved comparison with REGCOIL and QUADCOIL.

Corrections

• include winding-surface Jacobians and quadrature weights in the induction matrix
• use close=False so periodic endpoints are not duplicated
• include the missing 4*pi^2 normalization
• prevent the entropy objective from improving by weakening the full operator
• add a nonlocal self-separation check alongside the area-element check
• use 64-bit JAX calculations

ESSOS peak-current Pareto objective

For every trial winding surface, ESSOS solves the regularized current-potential problem at three nearby peak-current limits:

$$ \Phi_i=\underset{\Phi}{\operatorname{argmin}}\left[f_B(\Phi)+\lambda_i f_K(\Phi)\right], \qquad K_{\max}(\Phi_i)=K_i, $$

where

$$ f_B=\int_{S_p}(\mathbf B\cdot\mathbf n)^2,dA, \qquad f_K=\int_{S_c}|\mathbf K|^2,dA =\int_{S_c}|\nabla_s\Phi|^2,dA, $$

and $K_i=(0.95,1.00,1.05)K_{\mathrm{target}}$. The winding surface minimizes

$$ J_{\mathrm{Pareto}}(S_c)=\frac{1}{3}\sum_{i=1}^{3} \frac{f_B(S_c;K_i)}{f_B(S_{c,0};K_i)}+J_{\mathrm{geometry}}. $$

The three samples cover a short section of the $f_B$--$K_{\max}$ trade-off curve, so the surface is not tuned to one regularization value. The geometry term preserves volume, controls the Fourier spectrum, retains at least 90% of the initial plasma clearance, and penalizes small Jacobian or nonlocal self-approach. This is a local three-point Pareto objective, not a construction of the complete Pareto front.

The optimization uses 32 points per angle per field period. Surface motion is capped at 2% of the plasma minor radius per active Fourier coefficient.

Results

For the included QA example at 7 MA/m, a separate 96x96 REGCOIL evaluation gives:

• sheet fB: 19.873 -> 2.712
• max |Bn|/B: 0.0569 -> 0.0267
• 40-filament fB: 20.331 -> 3.010
• dense Kmax: 6.989 MA/m

At 10 MA/m, the Pareto surface reduces resolved sheet fB by 27.6% relative to the normal offset for QA and by 88.2% for W7-X. W7-X filament fB is reduced by 88.1%. The QA filament result is worse despite its improved sheet result, so no universal coil-improvement claim is made.

The current-complexity comparison uses the same $f_K$ as the inner regularization. Relative to the normal offset, ESSOS Pareto changes $f_K$ by +0.24% for QA and +22.2% for W7-X. W7-X RMS current increases from 2.208 to 2.432 MA/m while both surfaces satisfy the same 10 MA/m peak limit. The field-error improvement therefore has a visible current-complexity cost.

The driver, numerical tables, all figures, and detailed notes are under examples/winding_surface_comparison/. QUADCOIL's W7-X solves did not converge and are marked failed rather than treated as valid results.

Eight-coil study

An additive study under examples/winding_surface_comparison_8_coils/ repeats the comparison with eight coils per half field period and adds Landreman-Paul QH. At 96x96, the QH Pareto iterate reduces sheet fB by 75.1% and filament fB by 73.6% relative to the normal offset. The QA case shows the limitation clearly: its sheet result improves, but filament fB is 15.7% worse with the reduced coil count. The QH Pareto outer solve stopped on a line-search failure, so it is reported as a valid non-converged iterate.

[rogeriojorge]

@dseidita1919 Doing my own tests, to see if a couple things might help:

• Adding winding and plasma Jacobian/quadrature weights as Landreman & Boozer mention
• Explicit close=False, since ESSOS’s close=True includes duplicate periodic endpoint (perhaps we should change this, as SIMSOPT does)
• Missing 4 pi^2 full-torus integration factor for minor radius and volume
• flot64 instead of float32
• some random way of checking self intersection that literature search found at https://arxiv.org/abs/1102.3642 and https://arxiv.org/abs/2107.01664
• there seems to be a loophole that all svd values can decrease while flattening, giving a worse solution but better entropy

[rogeriojorge]

The resolved comparison is now included in the PR under examples/winding_surface_comparison.

Main findings:

• At 10 MA/m, the ESSOS Pareto surface reduces the 96x96 sheet fB by 27.6% for Landreman-Paul QA and 88.2% for W7-X relative to a normal-offset surface.
• The W7-X filament result improves by 88.1% (5.909 to 0.703), with max |Bn|/B reduced from 0.260 to 0.109.
• The QA sheet improves, but its filament fB becomes worse. This is an important limitation and is reported explicitly.
• The entropy proxy is worse than the normal-offset baseline in both resolved sheet comparisons.
• ESSOS and REGCOIL current solves agree when evaluated at the same resolution. QUADCOIL agrees for QA after refinement; its W7-X solves did not converge and are marked failed.
• The standalone 7 MA/m example reduces validated sheet fB from 19.873 to 2.712. Its saved coefficients have dense Kmax = 6.989 MA/m.

Resolved sheet and filament metrics:

Resolved current potentials and coil contours:

Resolved winding surfaces and coils:

The remaining figures and CSV tables are in the same folder.

[rogeriojorge]

Added the requested Pareto definition and current-complexity comparison.

For each trial surface, the inner problems are

$$ \Phi_i=\arg\min_{\Phi}\left[f_B(\Phi)+\lambda_i f_K(\Phi)\right], \qquad K_{\max}(\Phi_i)=K_i, $$

with $K_i=(0.95,1.00,1.05)K_{\mathrm{target}}$ and

$$ f_K=\int_{S_c}|\mathbf K|^2,dA =\int_{S_c}|\nabla_s\Phi|^2,dA. $$

The outer objective minimizes the mean normalized $f_B$ at those three current limits, plus the explicit geometry controls. Sampling three nearby points makes the surface improve a local part of the $f_B$--$K_{\max}$ trade-off instead of one selected regularization value.

The new figure compares $f_K/f_{K,\mathrm{offset}}$ and $K_{\mathrm{rms}}=\sqrt{f_K/A_c}$ for all four winding-surface methods:

For QA, the Pareto surface has essentially the same complexity as the normal offset: $f_K$ changes by +0.24% and RMS current changes from 4.956 to 4.953 MA/m. For W7-X, $f_K$ increases by 22.2% and RMS current increases from 2.208 to 2.432 MA/m. Both still satisfy the same 10 MA/m peak-current limit. This makes the cost of the W7-X field-error improvement explicit.

The dimensional $f_K$ and RMS-current values are included in data/surface_validation_96.csv.

[rogeriojorge]

Added a second, additive study with eight coils per half field period. The original ten-coil results are unchanged. The new study also adds Landreman-Paul QH.

Main results from the independent 96x96 validation:

• QH: ESSOS Pareto reduces sheet fB by 75.1% and filament fB by 73.6% relative to the normal offset. Relative to the tested REGCOIL adjoint surface, the reductions are 65.1% and 63.1%.
• W7-X: filament fB remains 88.1% below the normal-offset result.
• QA: sheet fB improves by 27.6%, but filament fB is 15.7% worse. Going from ten to eight coils per half period raises QA filament error by about a factor of eight, so the present sheet objective is not enough for sparse coil sets.
• The QH Pareto run stopped after seven outer iterations on a line-search failure. Its saved surface is valid and feasible at 10 MA/m on the 96x96 grid, but it is labeled as a non-converged iterate. QUADCOIL exceeded six minutes and about 2.8 GB on QH without completing, so those entries are marked failed.

Resolved metrics:

Current potential and coil contours:

Winding surfaces and coils:

The reproducible driver, CSV tables, complexity plot, convergence plot, and remaining figures are in examples/winding_surface_comparison_8_coils/.