In this paper we consider the free-form optimization of eigenvalues in electromagnetic systems by means of shape-variations with respect to small deformations. The objective is to optimize a particular eigenvalue to a target value. We introduce the mixed variational formulation of the Maxwell eigenvalue problem introduced by Kikuchi (1987) in function spaces of (H(\operatorname{curl}; \Omega)) and (H^1(\Omega)). To handle this formulation, suitable transformations of these spaces are utilized, e.g., of Piola-type for the space of (H(\operatorname{curl}; \Omega)). This allows for a formulation of the problem on a fixed reference domain together with a domain mapping. Local uniqueness of the solution is obtained by a normalization of the the eigenfunctions. This allows us to derive adjoint formulas for the derivatives of the eigenvalues with respect to domain variations. For the solution of the resulting optimization problem, we develop a particular damped inverse BFGS method that allows for an easy line search procedure while retaining positive definiteness of the inverse Hessian approximation. The infinite dimensional problem is discretized by mixed finite elements and a numerical example shows the efficiency of the proposed approach.

翻译：本文研究通过微小形变引起的形状变分实现电磁系统中特征值的自由形式优化。目标是将特定特征值优化至目标值。我们引入Kikuchi (1987) 在函数空间(H(\operatorname{curl}; \Omega)) 与(H^1(\Omega))中提出的麦克斯韦特征值问题的混合变分形式。为处理该形式，需对这些空间进行适当变换，例如对(H(\operatorname{curl}; \Omega))空间采用Piola型变换。这使得问题可在固定参考域上结合域映射进行表述。通过特征函数的归一化处理获得解的局部唯一性。基于此，我们推导出特征值关于域变分的导数伴随公式。针对所得优化问题的求解，我们开发了一种特殊的阻尼逆BFGS方法，该方法在保持逆海森近似正定性的同时，可实现简便的线搜索过程。通过混合有限元对无限维问题进行离散化，数值算例验证了所提方法的有效性。