# Adjoint approach to calculating shape gradients for 3D magnetic confinement equilibria

## TL;DR

This paper introduces an efficient adjoint method for calculating shape gradients in 3D magnetic confinement equilibria, significantly reducing computational effort for optimization and sensitivity analysis.

## Contribution

It presents a novel adjoint approach leveraging self-adjoint properties of MHD equations to compute shape gradients more efficiently in 3D toroidal equilibria.

## Key findings

- Reduces the number of equilibrium computations by factors of O(N)
- Applicable to shape changes of flux surfaces and coils
- Validated with multiple figures of merit

## Abstract

The shape gradient quantifies the change in some figure of merit resulting from differential perturbations to a shape. Shape gradients can be applied to gradient-based optimization, sensitivity analysis, and tolerance calculation. An efficient method for computing the shape gradient for toroidal 3D MHD equilibria is presented. The method is based on the self-adjoint property of the equations for driven perturbations of MHD equilibria and is similar to the Onsager symmetry of transport coefficients. Two versions of the shape gradient are considered. One describes the change in a figure of merit due to an arbitrary displacement of the outer flux surface; the other describes the change in the figure of merit due to the displacement of a coil. The method is implemented for several example figures of merit and compared with direct calculation of the shape gradient. In these examples the adjoint method reduces the number of equilibrium computations by factors of $\mathcal{O}(N)$, where $N$ is the number of parameters used to describe the outer flux surface or coil shapes.