Including the vacuum field energy in stellarator coil design

Being ``three-dimensional'', stellarators have the advantage that plasma currents are not essential for creating rotational-transform; however, the external current-carrying coils in stellarators are usually not planar. Reducing the inter-coil electromagnetic forces acting on strongly shaped 3D coils while preserving the favorable properties of the ``target'' magnetic field is a design challenge. In this work, we recognize that the inter-coil ${\mathbf j} \times {\mathbf B}$ forces are the gradient of the vacuum magnetic energy, $\displaystyle E := \frac{1}{2\mu_0}\int_{R^3} \!\!\! B^2 \, dV$. We introduce an objective functional, ${\cal F}\equiv \Phi_2 + \omega E$, built on the usual quadratic flux on a prescribed target surface, $\displaystyle \Phi_2 := \frac{1}{2}\int_{\cal S} ( {\mathbf B} \cdot {\mathbf n} )^2 \, dS$, and the vacuum energy, where $\omega$ is a weight penalty. The Euler-Lagrange equation for stationary states is derived, and numerical illustrations are computed using the SIMSOPT code \cite{simsopt}.