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This paper is devoted to the theoretical and numerical study of an optimaldesign problem in high-temperature superconductivity (HTS). The shapeoptimization problem is to find an optimal superconductor shape which minimizesa certain cost functional under a given target on the electric field over aspecific domain of interest. For the governing PDE-model, we consider anelliptic curl-curl variational inequality (VI) of the second kind with anL1-type nonlinearity. In particular, the non-smooth VI character and theinvolved H(curl)-structure make the corresponding shape sensitivity analysischallenging. To tackle the non-smoothness, a penalized dual VI formulation isproposed, leading to the G{\^a}teaux differentiability of the correspondingdual variable mapping. This property allows us to derive the distributed shapederivative of the cost functional through rigorous shape calculus on the basisof the averaged adjoint method. The developed shape derivative turns out to beuniformly stable with respect to the penalization parameter, and strongconvergence of the penalized problem is guaranteed. Based on the achievedtheoretical findings, we propose 3D numerical solutions, realized using a levelset algorithm and a Newton method with the Nedelec edge element discretization.Numerical results indicate a favourable and efficient performance of theproposed approach for a specific HTS application in superconducting shielding.

Shape optimization for superconductors governed by H(curl)-elliptic variational inequalities