  This paper is devoted to the theoretical and numerical study of an optimal
design problem in high-temperature superconductivity (HTS). The shape
optimization problem is to find an optimal superconductor shape which minimizes
a certain cost functional under a given target on the electric field over a
specific domain of interest. For the governing PDE-model, we consider an
elliptic curl-curl variational inequality (VI) of the second kind with an
L1-type nonlinearity. In particular, the non-smooth VI character and the
involved H(curl)-structure make the corresponding shape sensitivity analysis
challenging. To tackle the non-smoothness, a penalized dual VI formulation is
proposed, leading to the G{\^a}teaux differentiability of the corresponding
dual variable mapping. This property allows us to derive the distributed shape
derivative of the cost functional through rigorous shape calculus on the basis
of the averaged adjoint method. The developed shape derivative turns out to be
uniformly stable with respect to the penalization parameter, and strong
convergence of the penalized problem is guaranteed. Based on the achieved
theoretical findings, we propose 3D numerical solutions, realized using a level
set algorithm and a Newton method with the Nedelec edge element discretization.
Numerical results indicate a favourable and efficient performance of the
proposed approach for a specific HTS application in superconducting shielding.
