  We are interested in the design of stellarators, devices for the production
of controlled nuclear fusion reactions alternative to tokamaks. The confinement
of the plasma is entirely achieved by a helical magnetic field created by the
complex arrangement of coils fed by high currents around a toroidal domain.
Such coils describe a surface called "coil winding surface" (CWS). In this
paper, we model the design of the CWS as a shape optimization problem, so that
the cost functional reflects both optimal plasma confinement properties,
through a least square discrepancy, and also manufacturability, thanks to
geometrical terms involving the lateral surface or the curvature of the CWS. We
completely analyze the resulting problem: on the one hand, we establish the
existence of an optimal shape, prove the shape differentiability of the
criterion, and provide the expression of the differential in a workable form.
On the other hand, we propose a numerical method and perform simulations of
optimal stellarator shapes. We discuss the efficiency of our approach with
respect to the literature in this area.
