Adjoint approach to calculating shape gradients for three-dimensional magnetic confinement equilibria. Part 2. Applications

Abstract

The shape gradient is a local sensitivity function defined on the surface of an object which provides the change in a characteristic quantity, or figure of merit, associated with a perturbation to the shape of the object. The shape gradient can be used for gradient-based optimization, sensitivity analysis, and tolerance calculations. However, it is generally expensive to compute from finite-difference derivatives for shapes that are described by many parameters, as is the case for stellarator geometry. In an accompanying work (Antonsen et al. 2019), generalized self-adjointness relations are obtained for MHD equilibria. These describe the relation between perturbed equilibria due to changes in the rotational transform or toroidal current profiles, displacements of the plasma boundary, modifications of currents in the vacuum region, or the addition of bulk forces. These are applied to efficiently compute the shape gradient of functions of magnetohydrodynamic (MHD) equilibria with an adjoint approach. In this way, the shape derivative with respect to any perturbation applied to the plasma boundary or coil shapes can be computed with only one additional MHD equilibrium solution. We demonstrate that this approach is applicable for several figures of merit of interest for stellarator configuration optimization: the magnetic well, the magnetic ripple on axis, the departure from quasisymmetry, the effective ripple in the low-collisionality 1/ν regime ( 3/2 eff ) (Nemov et al. 1999), and several finite collisionality neoclassical quantities. Numerical verification of this method is demonstrated for the magnetic well figure of merit with the VMEC code (Hirshman & Whitson 1983) and for the magnetic ripple with modification of the ANIMEC code (Cooper et al. 1992). Comparisons with the direct approach demonstrate that in order to obtain agreement within several percent, the adjoint approach provides a factor of O(10 3 ) in computational savings.

Gradient-based optimization of 3D MHD equilibria

“…2019; Paul et al. 2020). Here the perturbation to the magnetic field due to a perturbation of the boundary is expressed as where the displacement vector satisfies on the boundary for a given normal perturbation to the surface and is the perturbation to the rotational transform profile that may arise due to the constraint of fixed .…”

“…The gradient of this objective function is obtained by computing an equilibrium with the addition of an anisotropic pressure tensor, with as described in Paul et al. (2020).…”

“…This technique has been demonstrated for computing the shape derivatives of several figures of merit relevant for stellarator design, including the magnetic well, rotational transform, and magnetic ripple (Paul et al. 2020). Each of these objective functions have been included in modern stellarator designs (Beidler 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.…”

“…This adjoint method results from a generalized self-adjointness property of the MHD force operator (Antonsen, Paul & Landreman 2019). This technique has been demonstrated for computing the shape derivatives of several figures of merit relevant for stellarator design, including the magnetic well, rotational transform, and magnetic ripple (Paul et al 2020). Each of these objective functions have been included in modern stellarator designs (Beidler et al 1990;Anderson et al 1995;Zarnstorff et al 2001;Henneberg et al 2019).…”