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Coil complexity is a critical consideration in stellarator design. Thetraditional two-step optimization approach, in which the plasma boundary isoptimized for physics properties and the coils are subsequently optimized to beconsistent with this boundary, can result in plasma shapes which cannot beproduced with sufficiently simple coils. To address this challenge, we proposea method to incorporate considerations of coil complexity in the optimizationof the plasma boundary. Coil complexity metrics are computed from the currentpotential solution obtained with the REGCOIL code (Landreman 2017 Nucl. Fusion57 046003). We compute the local sensitivity of these metrics with respect toperturbations of the plasma boundary using the shape gradient (Landreman & Paul2018 Nucl. Fusion 58 076023). We extend REGCOIL to compute derivatives of thesemetrics with respect to parameters describing the plasma boundary. In keepingwith previous research on winding surface optimization (Paul et al. 2018 Nucl.Fusion 58 076015), the shape derivatives are computed with a discrete adjointmethod. In contrast with the previous work, derivatives are computed withrespect to the plasma rather than the winding surface parameters. To furtherreduce the required resolution, we present a more efficient representation ofthe plasma surface using a single Fourier series to describe the radialdistance from a coordinate axis and a spectrally condensed poloidal angle. Thisrepresentation is advantageous over the standard cylindrical representationused in the VMEC code (Hirshman & Whitson 1983 The Physics of Fluids 263553-3568), as it provides a uniquely defined poloidal angle, eliminating anull space in the optimization of the plasma surface. The resulting shapegradient highlights features of the plasma boundary consistent with simplecoils and can be used to couple coil and fixed-boundary optimization.

Computing the shape gradient of stellarator coil complexity with respect to the plasma boundary