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The shape gradient is a local sensitivity function that provides the changein a figure of merit associated with a perturbation to the shape of the object.The shape gradient can be used for gradient-based optimization, sensitivityanalysis, and tolerance calculations. However, it is generally expensive tocompute from finite-difference derivatives for shapes which are described bymany parameters, as is the case for stellarator geometry. In an accompanyingwork (Antonsen et al. 2019), generalized self-adjointness relations areobtained for MHD equilibria. These describe the relation between perturbedequilibria due to changes in the rotational transform or toroidal currentprofiles, displacements of the plasma boundary, modifications of currents inthe vacuum region, or the addition of bulk forces. These are applied toefficiently compute the shape gradient of functions of magnetohydrodynamic(MHD) equilibria with an adjoint approach. In this way, the shape derivativewith respect to any perturbation applied to the plasma boundary or coil shapescan be computed with only one additional MHD equilibrium solution. Wedemonstrate that this approach is applicable for several figures of merit ofinterest for stellarator configuration optimization: the magnetic well, themagnetic ripple on axis, the departure from quasisymmetry, the effective ripplein the low-collisionality $1/\nu$ regime ($\epsilon_{\text{eff}}^{3/2})$ (Nemovet al. 1999), and several finite collisionality neoclassical quantities.Numerical verification is demonstrated for the magnetic well figure of meritwith the VMEC code (Hirshman & Whitson 1983) and for the magnetic ripple withmodification of the ANIMEC code (Cooper et al. 1992). Comparisons with thedirect approach demonstrate that to obtain agreement within several percent,the adjoint approach provides a factor of $\mathcal{O}(10^3)$ in computationalsavings.

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