论文标题
由H（Curl） - 纤维化变异不等式控制的超导体的形状优化
Shape optimization for superconductors governed by H(curl)-elliptic variational inequalities
论文作者
论文摘要
本文致力于高温超导性（HTS）中最佳设计问题的理论和数值研究。形状优化问题是找到一个最佳的超导体形状，该形状可最大程度地减少特定感兴趣域的电场上给定目标下的一定成本功能。对于管理PDE模型，我们考虑了第二种具有L1型非线性的椭圆形卷曲 - 折曲变化不等式（VI）。特别是，非平滑VI字符和所涉及的H（卷曲）结构使相应的形状灵敏度分析具有挑战性。为了解决非平滑度，提出了惩罚的双VI公式，从而导致相应的双变量映射的g {–} teaux的可不同性。该属性使我们能够根据平均伴随方法来得出成本功能的分布式形状导数。相对于惩罚参数，开发的形状导数证明是均匀稳定的，并且可以保证惩罚问题的强烈收敛。基于所达到的理论发现，我们提出了3D数值解决方案，使用级别集算法和具有Nedelec Edge元素离散化的牛顿方法实现。数值结果表明，在超导屏蔽中，针对特定HTS应用的建议方法的表现有利有效。
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{â}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.
