Combined plasma-coil optimization algorithms

Combined plasma-coil optimization approaches for designing stellarators are discussed and a new method for calculating free-boundary equilibria is proposed. Four distinct categories of stellarator optimization, two of which are novel approaches, are the fixed-boundary optimization, the generalized fixed-boundary optimization, the quasi free-boundary optimization, and the free-boundary (coil) optimization. These are described using the multi-region relaxed magnetohydrodynmics (MRxMHD) energy functional, the Biot-Savart integral, the coil-penalty functional and the virtual casing integral, and their derivatives. The proposed free-boundary equilibrium calculation differs from existing methods in how the boundary-value problem is posed, and for the new approach it seems that there is not an associated energy minimization principle because a non-symmetric functional arises. We propose to solve the weak formulation of this problem using a spectral-Galerkin method, and this will reduce the free-boundary equilibrium calculation to something comparable to a fixed-boundary calculation. In our discussion of combined plasma-coil optimization algorithms, we emphasize the importance of the stability matrix.