Direct computation of magnetic surfaces in Boozer coordinates and coil optimization for quasisymmetry

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 quasisymmetric 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 quasisymmetry and low particle losses. In particular for low complexity coils, we are able to significantly improve the performance compared with 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 $\epsilon _{\text {eff}}^{3/2}$ less than approximately $5\times 10^{-9}$ .

Mapping the space of quasisymmetric stellarators using optimized near-axis expansion

“…This finding is consistent with the fact that previous optimizations for QA at (NCSX and ARIES-CS) have had significant imperfections in the symmetry, whereas excellent QA has been obtained at (Giuliani et al. 2022; Landreman & Paul 2022). When , QA solutions were found that satisfied all constraints, but they resembled configurations that were translated or rotated to break two-field-period symmetry.…”

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 offaxis [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].…”

Single-stage stellarator optimization: combining coils with fixed boundary equilibria

“…the magnitude of the normal component n of the magnetic field induced by the coils on the boundary S. This is in contrast to the current carrying surface approach where coils are restricted to lie on a particular surface (called the coil winding surface) which is employed in the NESCOIL [33], REGCOIL [34], ONSET [35], COILOPT [3], and COILOPT++ [36] codes. The FOCUS approach implemented in SIMSOPT has been shown to yield exceptionally low field errors and allow for simple coil shapes (see [11,37]) and is therefore the one used here. We target the minimization of the field error by defining as a cost function the quadratic flux quantity f QF , given by…”

“…Previous combined optimization methods have either relied directly or indirectly on free boundary equilibrium calculations [3][4][5][6][7], which often demand many iterations between an equilibrium solution, or these can only be used with vacuum configurations [8][9][10][11][12]. In [13] several combined optimization approaches were discussed, including the possibility of using fixed boundary equilibria and the corresponding penalty functionals.…”

“…As stated in [14], it was not until a combined plasma-coil optimization was performed that a family of consistent solutions (termed M45) that met engineering feasibility requirements and adequately reconstructed the plasma properties of the initial LI383 equilibrium was found. Another example where an a posteriori combined plasma-coil optimization led to major improvements in loss fraction and effective helical ripple with respect to an already found solution from a two-stage approach is described in [11], although this method is only applicable to quasisymmetric magnetic fields in vacuum. Such studies show the importance of combined plasma-coil optimization algorithms in the design of stellarator experiments.…”