An adjoint method for neoclassical stellarator optimization

Abstract

Received xx; revised xx; accepted xx)Stellarators are a promising route to steady-state fusion power. However, to achieve the required confinement, the magnetic geometry must be highly optimized. This optimization requires navigating high-dimensional spaces, often necessitating the use of gradient-based methods. The gradient of the neoclassical fluxes is expensive to compute with classical methods, requiring O(N ) flux computations, where N is the number of parameters. To reduce the cost of the gradient computation, we present an adjoint method for computing the derivatives of moments of the neoclassical distribution function for stellarator optimization. The linear adjoint method allows derivatives of quantities which depend on solutions of a linear system, such as moments of the distribution function, to be computed with respect to many parameters from the solution of only two linear systems. This reduces the cost of computing the gradient to the point that the finite-collisionality neoclassical fluxes can be used within an optimization loop.With the neoclassical adjoint method, we compute solutions of the drift kinetic equation and an adjoint drift kinetic equation to obtain derivatives of neoclassical quantities with respect to geometric parameters. When the number of parameters in the derivative is large (O(10 2 )), this adjoint method provides up to a factor of 200 reduction in cost. We demonstrate adjoint-based optimization of the field strength to obtain minimal bootstrap current on a surface. With adjoint-based derivatives, we also compute the local sensitivity to magnetic perturbations on a flux surface and identify regions where tight tolerances on error fields are required for control of the bootstrap current or radial transport. Furthermore, the solve for the ambipolar electric field is accelerated using a Newton method with derivatives obtained from the adjoint method.

Combined plasma–coil optimization algorithms

“…Recently, much work has been devoted to the development of adjoint methods and automatic differentiation to calculate derivatives of the plasma and coil properties, e.g. for the rotational transform, neoclassical transport and for the volume averaged β Paul et al 2018;Antonsen et al 2019;Paul et al 2019;Carlton-Jones et al 2020). These methods promise significant advantages for stellarator optimization.…”

“…A related direct coil optimization uses a near-axis analytic approximation to the equilibrium state (Giuliani et al 2020), which might be sufficient for large aspect-ratio configurations but perhaps not so for 'compact' stellarators. Also, recent work has exploited analytical gradient information and adjoint methods for a variety of optimization problems Paul et al 2018;Zhu et al 2018;Antonsen, Paul & Landreman 2019;Paul et al 2019;Carlton-Jones et al 2020).…”

Gradient-based optimization of 3D MHD equilibria

“…Adjoint methods have also been used to optimize the local magnetic field for neoclassical properties (Paul et al. 2019) and the coil winding surface for properties of the current potential (Paul et al. 2018).…”

“…Furthermore, we have developed adjoint methods for direct optimization of coil shapes for properties of an MHD equilibrium (Antonsen et al 2019;Paul et al 2020), although its application to optimization is not presented in this work. Adjoint methods have also been used to optimize the local magnetic field for neoclassical properties (Paul et al 2019) and the coil winding surface for properties of the current potential (Paul et al 2018).…”

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

“…With the application of an adjoint method, we furthermore eliminate the noise associated with the finite-difference step size. The adjoint method has recently been applied to several problems in stellarator design (Paul et al 2018(Paul et al , 2019(Paul et al , 2020aAntonsen, Paul & Landreman 2019;Giuliani et al 2020;Geraldini, Landreman & Paul 2021) and is commonly implemented in the fields of fluid dynamics and aerodynamic engineering (Jameson, Martinelli & Pierce 1998;Othmer 2008).…”