Gradient-based optimization of 3D MHD equilibria

Abstract

Using recently developed adjoint methods for computing the shape derivatives of functions that depend on magnetohydrodynamic (MHD) equilibria (Antonsen et al., J. Plasma Phys., vol. 85, issue 2, 2019; Paul et al., J. Plasma Phys., vol. 86, issue 1, 2020), we present the first example of analytic gradient-based optimization of fixed-boundary stellarator equilibria. We take advantage of gradient information to optimize figures of merit of relevance for stellarator design, including the rotational transform, magnetic well and quasi-symmetry near the axis. With the application of the adjoint method, we reduce the number of equilibrium evaluations by the dimension of the optimization space ( ${\sim }50\text {--}500$ ) in comparison with a finite-difference gradient-based method. We discuss regularization objectives of relevance for fixed-boundary optimization, including a novel method that prevents self-intersection of the plasma boundary. We present several optimized equilibria, including a vacuum field with very low magnetic shear throughout the volume.

The DESC Stellarator Code Suite Part III: Quasi-symmetry optimization

“…Furthermore, finite differences approximations are very sensitive to the step size, which can make this approach inaccurate and unreliable. Recent progress has been made through the use of adjoint methods, as in the ALPOpt code 10 . The adjoint approach reduces the computation burden to only two equilibrium solutions and avoids the noise of numerical derivatives 11,12 .…”

Adjoint methods for quasi-symmetry of vacuum fields on a surface

“…to represent the plasma boundary) they can be prohibitively expensive computationally if the gradients are evaluated via finite differences. A more efficient way of obtaining gradient information is provided by adjoint methods, which were recently introduced in the stellarator optimisation field and have already found widespread application Paul et al 2018;Antonsen, Paul & Landreman 2019;Paul et al 2019Paul et al , 2020Giuliani et al 2020;Paul 2020;Geraldini, Landreman & Paul 2021;Paul, Landreman & Antonsen 2021).…”

“…Previous work (Antonsen et al 2019;Paul et al 2020Paul et al , 2021 applied adjoint methods to ideal magnetohydrostatic (MHS) equilibria, building in the assumption of integrability, i.e. the existence of a set of nested flux surfaces.…”

“…However, three-dimensional magnetic fields are generally not integrable due to the lack of continuous symmetry. Moreover, singularities arise at rational surfaces for linearised ideal MHS equilibria, making the computation of derivatives challenging (Paul et al 2021). To overcome these challenges, different equilibrium models can be considered, such as vacuum or force-free fields.…”

Unsupervised Discovery of Inertial-Fusion Plasma Physics using Differentiable Kinetic Simulations and a Maximum Entropy Loss Function

“…Analytic approaches have resulted in the development of adjoint methods for shape derivatives of functions that depend on magnetohydrodynamics (MHD) equilibria ( [13,14]). These methods have been used to perform optimization of stellarator design ( [15]). In other work, by using gradients obtained from analytic ( [16]) and automatic differentiation (AD) ( [17]), the FOCUS and FOCUSADD codes optimize coil shape.…”