Coil complexity is a critical consideration in stellarator design. The traditional two-step optimization approach, in which the plasma boundary is optimized for physics properties and the coils are subsequently optimized to be consistent with this boundary, can result in plasma shapes which cannot be produced with sufficiently simple coils. To address this challenge, we propose a method to incorporate considerations of coil complexity in the optimization of the plasma boundary. Coil complexity metrics are computed from the current potential solution obtained with the REGCOIL code (Landreman, Nucl. Fusion, vol. 57, 2017, 046003). While such metrics have previously been included in derivative-free fixed-boundary optimization (Drevlak et al., Nucl. Fusion, vol. 59, 2018, 016010), we compute the local sensitivity of these metrics with respect to perturbations of the plasma boundary using the shape gradient (Landreman & Paul, Nucl. Fusion, vol. 58, 2018, 076023). We extend REGCOIL to compute derivatives of these metrics with respect to parameters describing the plasma boundary. In keeping with previous research on winding surface optimization (Paul et al., Nucl. Fusion, vol. 58, 2018, 076015), the shape derivatives are computed with a discrete adjoint method. In contrast with the previous work, derivatives are computed with respect to the plasma surface parameters rather than the winding surface parameters. To further reduce the resolution required to compute the shape gradient, we present a more efficient representation of the plasma surface which uses a single Fourier series to describe the radial distance from a coordinate axis and a spectrally condensed poloidal angle. This representation is advantageous over the standard cylindrical representation used in the VMEC code (Hirshman & Whitson, Phys. Fluids, vol. 26, 1983, pp. 3553–3568), as it provides a uniquely defined poloidal angle, eliminating a null space in the optimization of the plasma surface. In comparison with previous spectral condensation methods (Hirshman & Breslau, Phys. Plasmas, vol. 5, 1998, p. 2664), the modified poloidal angle is obtained algebraically rather than through the solution of a nonlinear optimization problem. The resulting shape gradient highlights features of the plasma boundary that are consistent with simple coils and can be used to couple coil and fixed-boundary optimization.

中文翻译：

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计算仿星器线圈复杂度相对于等离子体边界的形状梯度

线圈复杂性是仿星器设计中的一个关键考虑因素。传统的两步优化方法，其中等离子体边界针对物理特性进行优化，然后线圈被优化以与该边界一致，可能导致等离子体形状不能用足够简单的线圈产生。为了应对这一挑战，我们提出了一种将线圈复杂性考虑纳入等离子体边界优化的方法。线圈复杂度度量是根据使用 REGCOIL 代码 (Landreman,核。融合, 卷。57, 2017, 046003)。虽然此类指标以前已包含在无导数固定边界优化中（Drevlak等。,核。融合, 卷。59, 2018, 016010)，我们使用形状梯度 (Landreman & Paul,核。融合, 卷。58, 2018, 076023)。我们扩展 REGCOIL 以计算这些度量关于描述等离子体边界的参数的导数。与之前关于绕组表面优化的研究一致（Paul等。,核。融合, 卷。58, 2018, 076015)，形状导数是用离散伴随方法计算的。与之前的工作相比，导数是根据等离子体表面参数而不是绕组表面参数计算的。为了进一步降低计算形状梯度所需的分辨率，我们提出了等离子体表面的更有效表示，它使用单个傅里叶级数来描述距坐标轴的径向距离和光谱凝聚的极向角。这种表示优于 VMEC 代码 (Hirshman & Whitson,物理。流体, 卷。26, 1983, pp. 3553–3568)，因为它提供了一个独特定义的极向角，消除了优化等离子体表面的零空间。与以前的光谱浓缩方法相比（Hirshman & Breslau，物理。等离子, 卷。5, 1998, p. 2664），修正的极向角是通过代数获得的，而不是通过解决非线性优化问题。由此产生的形状梯度突出了等离子体边界的特征，这些特征与简单的线圈一致，可用于耦合线圈和固定边界优化。