Direct computation of magnetic surfaces in Boozer coordinates and coil optimization for quasi-symmetry

Abstract

We propose a new method to compute magnetic surfaces that are parametrized in Boozer coordinates for vacuum magnetic fields. We also propose a measure for quasisymmetry on the computed surfaces and use it to design coils that generate a magnetic field that is quasi-symmetric on those surfaces. The rotational transform of the field and complexity measures for the coils are also controlled in the design problem. Using an adjoint approach, we are able to obtain analytic derivatives for this optimization problem, yielding an efficient gradient-based algorithm. Starting from an initial coil set that presents nested magnetic surfaces for a large fraction of the volume, our method converges rapidly to coil systems generating fields with excellent quasi-symmetry and low particle losses. In particular for low complexity coils, we are able to significantly improve the performance compared to coils obtained from the standard two-stage approach, e.g. reduce losses of fusion-produced alpha particles born at half-radius from 17.7% to 6.6%. We also demonstrate 16-coil configurations with alpha loss < 1% and neoclassical transport magnitude 3/2 eff less than approximately 5 × 10 −9 .

Optimization of quasisymmetric stellarators with self-consistent bootstrap current and energetic particle confinement

“…For equal minor radius, field strength, and profiles of density and temperature, the new configurations have alpha particle confinement comparable to or even better than a tokamak. Similarly excellent confinement of alpha particles is seen for other β = 0 stellarators reported in recent months 2,46,47 .…”

Stochastic and a posteriori optimization to mitigate coil manufacturing errors in stellarator design

“…For the longer coils, L-BFGS is only able to reduce the gradient by about seven orders of magnitude and we gain an extra order of magnitude reduction using the Newton iterations. We note that poor conditioning of coil design problems as the coil length increases was also observed in [7,Fig. 4].…”

“…To quantify quasi-symmetry, we recall that a magnetic field is quasi-axisymmetric if the field strength is independent of the angle θ when expressed in terms of Boozer angles ϕ and θ [12]. Thus we can quantify the deviation from quasisymmetry as follows: for each perturbed coilset, with currents scaled so the average field strength is 1 T, we compute surfaces in Boozer angles for the magnetic field using [7], and perform a Fourier transform of the field strength. That is, we express |B(s, θ, ϕ)| = m,n B m,n (s) cos(mθ − nϕ).…”

“…As computing resources and simulation capabilities have increased over the last few decades, large-scale numerical optimization has proven to be the method of choice for the design of new stellarator experiments [11]. Numerical simulations of guiding-center trajectories have shown that the state-of-the-art stellarator designs obtained from these optimization efforts are now able to confine even highly energetic particles for long timescale with minimal losses -a key requirement for practical use as fusion reactors [4,7,16,26]. However, good performance of a stellarator can only be guaranteed if the coils are manufactured and placed with high accuracy [1,10,23], a requirement that has turned out to be a significant cost driver and risk factor for any plans to build a stellarator experiment [21,24].…”

Stellarator coil optimization supporting multiple magnetic configurations

“…In the future, similar work could be pursued that considers windowpane coils to add primarily poloidal flux or attempts to achieve more than one type of quasisymmetry with a single coil set. Other formulations could be examined that incorporate objectives ensuring quasisymmetry to high order off-axis [16], target integrability on several surfaces throughout the plasma volume, or eliminate the need for an integrability objective by ensuring quasisymmetry on a given flux surface [28]. Stochastic optimization techniques could also be used to create coils that perform robustly even in the face of manufacturing errors [29].…”