# Magneto-Fluid Dynamics Seminar

#### An adjoint approach for the shape gradients of three-dimensional magneto-hydrodynamic equilibria

Speaker: Elizabeth Paul, University of Maryland, College Park

Location: Warren Weaver Hall 905

Date: Tuesday, November 19, 2019, 11 a.m.

Synopsis:

Stellarators are a class of magnetic confinement devices without continuous toroidal symmetry. The design of modern stellarators often employs gradient-based optimization to navigate the high-dimensional spaces used to describe their geometry. However, computing the numerical gradient of a target function with respect to many parameters can be expensive. The adjoint method allows these gradients to be computed at a much lower cost and without the noise associated with finite differences. In addition to gradient-based optimization, the derivatives obtained from the adjoint method are valuable for local sensitivity analysis and tolerance quantification.

A continuous adjoint method has been developed for obtaining the derivatives of functions of the magneto-hydrodynamic (MHD) equilibrium equations with respect to the shape of the boundary of the domain or the shape of the electro-magnetic coils [1]. This approach is based on the generalization of the self-adjointness of the linearized MHD force operator. The adjoint equation corresponds to a perturbed force balance equation with the addition of a bulk force, rotational transform, or toroidal current perturbation. We numerically demonstrate this approach by adding a small perturbation to the non-linear VMEC [2] solution, obtaining an order $10^2-10^3$ reduction in cost in comparison with a finite difference approach.  Examples are presented for the shape gradient of the rotational transform and vacuum magnetic well, a proxy for MHD stability. The adjoint solution required for the magnetic ripple, a proxy for near-axis quasisymmetry, requires the addition of an anisotropic pressure tensor to the MHD force balance equation. This modification has been implemented in the ANIMEC [3] code. We furthermore demonstrate that this adjoint approach can be applied to compute shape gradients of two important figures of merit [4], the departure from quasisymmetry and the effective ripple in the low-collisionality neoclassical regime, but require the development of new equilibrium solvers. Finally, initial steps toward adjoint solutions with a linearized equilibrium approach will be presented.

[1] Antonsen, T.M., Paul, E.J. & Landreman, M. 2019 Adjoint approach to calculating shape gradients for three-dimensional magnetic confinement equilibria. Journal of Plasma Physics 85 (2).

[2] Hirshman, S.P. & Whitson, J.C. 1983 Steepest descent moment method for three-dimensional magnetohydrodynamic equilibria. Physics of Fluids 26 (12), 3553.

[3] Cooper, W.A., Hirshman, S.P., Merazzi, S. & Gruber, R. 1992 3D magnetohydrodynamic equilibria with anisotropic pressure. Computer Physics Communications 72 (1),1–13.

[4] Paul, E.J., Antonsen, T.M.,  Landreman, M., Cooper, W.A. Submitted to Journal of Plasma Physics.