Application of the Shape Gradient and Hessian Matrix methods to compute the Sensitivities of Magnetic Islands to parameter perturbations in Permanent Magnet Stellarators

We applied methods for quantifying sensitivities of error fields to shape deviations in stellarator coils to novel permanent magnet stellarators. Using permanent magnets to produce stellarator fields offers a promising alternative to coil complexity. In the modular coil approach, small deviations in coil shape can have detrimental effects on particle and energy confinement. Permanent magnet stellarators are, however, also sensitive to source perturbations, and as a result, a detailed analysis of the possible field perturbations that can be caused by permanent magnets is necessary for device construction. We applied the shape gradient [Landreman & Paul, Nuclear Fusion 58(7), 076023 (2018)] and Hessian matrix methods [Zhu et al. Plasma Physics and Controlled Fusion, 60(5), 054016 (2018)] to permanent magnet stellarators to analyze the impact of perturbations to permanent magnet parameters including position, orientation, and magnetic moment magnitude on the production of error fields. We used the JAX automatic differentiation toolkit in Python to compute analytic derivatives of error field quantities. These methods were applied to study the sensitivity of the resonant perturbation metric, an indicator of the width of magnetic islands in the stellarator field. The shape gradient and Hessian matrix methods were applied to analyze the sensitivity of this error field quantity to perturbations of permanent magnet parameters in the MUSE permanent magnet stellarator and the PM4Stell permanent magnet project under development at PPPL. Implications for construction methods and precision requirements will be discussed.