Abstract
Combined plasma–coil optimization approaches for designing stellarators are discussed and a new method for calculating free-boundary equilibria for multiregion relaxed magnetohydrodynmics (MRxMHD) 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 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.
Highlights
The design space for stellarators is larger than that of tokamaks because stellarators exploit three-dimensional (3-D) magnetic fields, by which it is meant that there is no continuous symmetry, whereas tokamaks are notionally axisymmetric (Helander 2014)
If we were to proceed with the approach of specifying BT,n on D, some type of ‘self-consistent’ iteration, for example, must be implemented to determine the BT,n that satisfies the matching condition, namely that BT,n − BP,n[BT |S ] = BE,n, where BP,n[BT |S ] may be considered to be a linear, non-local operator acting on the tangential total field on the plasma boundary BT|S, which is only known after the equilibrium has been computed
We began by summarizing all the functional derivatives of the multiregion relaxed magnetohydrodynmics (MRxMHD) energy,16 the coil-penalty and the virtual-casing integral needed for a combined plasma–coil optimization
Summary
The design space for stellarators is larger than that of tokamaks because stellarators exploit three-dimensional (3-D) magnetic fields, by which it is meant that there is no continuous (e.g. rotational) symmetry, whereas tokamaks are notionally axisymmetric (two-dimensional) (Helander 2014). This comes at the cost of computing the coil geometry or some approximation of it at every stage of the fixed-boundary plasma optimization Another approach for the combined plasma–coil design is the direct coil optimization using a free-boundary equilibrium code (Hudson et al 2002; Strickler, Berry & Hirshman 2002b).
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