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

TL;DR

This paper develops a theoretical framework and numerical methods for optimizing superconductor shapes in high-temperature superconductivity, addressing complex H(curl)-elliptic variational inequalities with non-smooth features.

Contribution

It introduces a penalized dual VI formulation enabling shape sensitivity analysis and derives a stable shape derivative for the optimization process.

Findings

Effective 3D numerical solutions using level set and Newton methods

Stable shape derivatives with respect to penalization parameters

Successful application to superconducting shielding design

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 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.