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 j × B forces are the gradient of the vacuum magnetic energy, E := 1/2μ₀∫R³ \!\!\! B² \, dV. We introduce an objective functional, F≡ Φ₂ + ω E, built on the usual quadratic flux on a prescribed target surface, Φ₂ := 1/2∫S ( B · n )² \, dS, and the vacuum energy, where ω is a weight penalty. The Euler-Lagrange equation for stationary states is derived, and numerical illustrations are computed using the SIMSOPT code {simsopt}.