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

Abstract

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 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 nonsmooth VI character and the involved H(curl)-structure make the corresponding shape sensitivity analysis challenging. To tackle the nonsmoothness, a penalized dual VI formulation is proposed, leading to the Gateaux 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 three-dimensional numerical solutions, realized using a level set algorithm and a Newton method with the Nédélec edge element discretization. Numerical results indicate a favorable and efficient performance of the proposed approach for a specific HTS application in superconducting shielding.