Shape Optimization for Superconductors Governed by H(curl)-Elliptic Variational Inequalities

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