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.

中文翻译：

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组合等离子线圈优化算法

讨论了用于设计仿星器的等离子体-线圈组合优化方法，并提出了一种计算多区域弛豫磁流体动力学（MRxMHD）的自由边界平衡的新方法。四种不同类别的仿星器优化，其中两种是新方法，是固定边界优化、广义固定边界优化、准自由边界优化和自由边界（线圈）优化。这些是使用 MRxMHD 能量泛函、Biot-Savart 积分、线圈惩罚泛函和虚拟套管积分及其导数来描述的。所提出的自由边界平衡计算与现有方​​法在边界值问题的提出方式上有所不同，对于新方法，似乎没有相关的能量最小化原则，因为出现了非对称泛函。我们建议使用光谱-Galerkin 方法解决这个问题的弱公式，这会将自由边界平衡计算减少到与固定边界计算相当的东西。在我们对组合等离子体线圈优化算法的讨论中，我们强调了稳定性矩阵的重要性。