[en] To find the optimal coils for stellarators, nonlinear optimization algorithms are applied in existing coil design codes. However, none of these codes have used the information from the second-order derivatives. In this paper, we present a modified Newton method in the recently developed code FOCUS. The Hessian matrix is calculated with analytically derived equations. Its inverse is approximated by a modified Cholesky factorization and applied in the iterative scheme of a classical Newton method. Using this method, FOCUS is able to recover the W7-X modular coils starting from a simple initial guess. Results demonstrate significant advantages.

No abstract available

[en] The confinement of plasmas by magnetic fields with nonaxisymmetric shaping can be degraded or destroyed by the breakup of the magnetic surfaces through effects that are intrinsic to the shaping. An efficient perturbation method of determining this drive for islands was developed and applied to stellarator equilibria.

[en] A Vlasov-Maxwell equilibrium for a charged particle bunch is given in the beam frame by the distribution function that is a function of the single-particle Hamiltonian f=f(H), where in an axisymmetric cylinder H=p²/2m+κ_{perpendicular} r²/2+κ_zz²/2+qφ(r,z), the kinetic energy is p²/2m, κ_{perpendicular} and κ_z are the external focusing coefficients in the transverse and longitudinal directions, and φ is the electrostatic potential determined self-consistently from Poisson's equation ∇²φ=-4πq∫d³pf(H). The self-field potential φ introduces a coupling between the otherwise independent r and z motions. Under quite general conditions, this leads to chaotic particle motion. Poisson's equation is solved using a spectral method in z and a finite-difference method in r, and a Picard iteration method is used to determine φ self-consistently. For the thermal equilibrium distribution f=Aexp(-H/T), the single-particle trajectories display chaotic behavior. The properties of the chaotic trajectories are characterized

[en] The presence of error fields has been shown to degrade plasma confinement and drive instabilities. Error fields can arise from many sources, but are predominantly attributed to deviations in the coil geometry. In this paper, we introduce a Hessian matrix approach for determining error field sensitivity to coil deviations. A primary cost function used for designing stellarator coils, the surface integral of normalized normal field errors, was adopted to evaluate the deviation of the generated magnetic field from the desired magnetic field. The FOCUS code [Zhu et al., Nucl. Fusion 58(1):016008 (2018)] is utilized to provide fast and accurate calculations of the Hessian. The sensitivities of error fields to coil displacements are then determined by the eigenvalues of the Hessian matrix. A proof-of-principle example is given on a CNT-like configuration. We anticipate that this new method could provide information to avoid dominant coil misalignments and simplify coil designs for stellarators.

[en] Error fields are predominantly attributed to inevitable coil imperfections. Controlling error fields during coil fabrication and assembly is crucial for stellarators. Excessively tight coil tolerance increases time and cost, and, in part, led to the cancellation of the National Compact Stellarator Experiment and delay of W7-X. In this paper, we improve the recently proposed Hessian matrix method to rapidly identify important coil deviations. Two of the most common figures of merit, magnetic island size and quasi-symmetry, are analytically differentiated over coil parameters. By extracting the eigenvectors of the Hessian matrix, we can directly identify sensitive coil deviations in the order of the eigenvalues. The new method is applied to the upcoming Chinese First Quasi-axisymmetric Stellarator configuration. Important perturbations that enlarge n/m  =  4/11 islands and deteriorate quasi-axisymmetry of the magnetic field are successfully determined. The results suggest each modular coil should have separate tolerance and some certain perturbation combinations will produce significant error fields. By relaxing unnecessary coil tolerance, this method will hopefully lead to a substantial reduction in time and cost. (paper)

[en] Finding an easy-to-build coils set has been a critical issue for stellarator design for decades. Conventional approaches assume a toroidal ‘winding’ surface, but a poorly chosen winding surface can unnecessarily constrain the coil optimization algorithm, This article presents a new method to design coils for stellarators. Each discrete coil is represented as an arbitrary, closed, one-dimensional curve embedded in three-dimensional space. A target function to be minimized that includes both physical requirements and engineering constraints is constructed. The derivatives of the target function with respect to the parameters describing the coil geometries and currents are calculated analytically. A numerical code, named flexible optimized coils using space curves (FOCUS), has been developed. Applications to a simple stellarator configuration, W7-X and LHD vacuum fields are presented. (paper)

[en] The recent formulations of multi-region relaxed magnetohydrodynamics (MRxMHD) have generalized the famous Woltjer-Taylor states by incorporating a collection of “ideal barriers” that prevent global relaxation and flow. In this paper, we generalize MRxMHD with flow to include Hall effects, and thereby obtain the partially relaxed counterparts of the famous double Beltrami states as a special subset. The physical and mathematical consequences arising from the introduction of the Hall term are also presented. We demonstrate that our results (in the ideal MHD limit) constitute an important subset of ideal MHD equilibria, and we compare our approach against other variational principles proposed for deriving the partially relaxed states.

[en] It is demonstrated that there exists a plausible evolution of the discharge from the vacuum state to the desired high beta state with the self-consistent bootstrap current profile. The discharge evolution preserves stability and has adequate quasi axisymmetry along this trajectory. The study takes advantage of the quasi-axisymmetric nature of the device to model the evolution of flux and energy in two dimensions. The plasma confinement is modeled to be consistent with empirical scaling. The ohmic circuit, the plasma density, and the timing of the neutral beam heating control the poloidal flux evolution. The resulting pressure and current density profiles are then used in a three-dimensional optimization to find the desired sequence of equilibria. In order to obtain this sequence, active control of the helical and poloidal fields is required. These results are consistent with the planned power systems for the magnets