This paper presents an approach for deriving the continuum sensitivity of superconducting systems operating at critical current densities and an optimization method based on the continuum sensitivity. In the sensitivity problem, the superconducting systems is represented by a variational state equation, wherein the magnetic permeability depends on the magnetic field, which is transformed from a state equation with a field-dependent source. The design sensitivity is derived using the material derivative concept of continuum mechanics and the adjoint variable method. The adjoint system has a material property represented as a symmetric tensor that contains the sensitivity of the current density with respect to the magnetic field. The design sensitivity is represented in the analytical form of a surface integral on the interface between the superconducting material and its surroundings, which depends on the sensitivity of the current density. The optimization scheme is constructed based on the continuum design sensitivity. In the design optimization, the level set method is used to express the shape variation of the superconducting materials. The numerical example of infinite solenoids demonstrates that the design sensitivity provides an accurate design solution considering the critical current condition. In addition, the design example of a magnetic resonance imaging solenoid shows that the derived design sensitivity has the inherent ability for attaining the compact design by treating the input current of a superconducting system as a critical condition.

本文提出了一种推导在临界电流密度下运行的超导系统的连续谱灵敏度的方法，以及一种基于连续谱灵敏度的优化方法。在灵敏度问题中，超导系统由变分状态方程表示，其中磁导率取决于磁场，磁场是从具有场相关源的状态方程转换而来的。设计灵敏度是使用连续介质力学的材料导数概念和伴随变量方法推导出来的。伴随系统的材料属性表示为对称张量，其中包含电流密度相对于磁场的敏感性。设计灵敏度以超导材料与其周围环境之间界面上的表面积分的分析形式表示，这取决于电流密度的灵敏度。优化方案是基于连续体设计灵敏度构建的。在设计优化中，采用水平集方法来表达超导材料的形状变化。无限螺线管的数值示例表明，考虑到临界电流条件，设计灵敏度提供了准确的设计解决方案。此外，磁共振成像螺线管的设计实例表明，通过将超导系统的输入电流视为临界条件，推导出的设计灵敏度具有实现紧凑设计的内在能力。