Shape optimization with respect to eigenvalues of a cavity plays an important role in the design of new resonators or in the optimization of existing ones. In our paper, we propose a gradient-based optimization scheme, which we enhance with closed-form shape derivatives of the system matrices. Based on these, we can compute accurate derivatives of eigenvalues, eigenmodes and the cost function with respect to the geometry, which significantly reduces the computational effort of the optimizer. We demonstrate our work by applying it to the 9-cell TESLA cavity, for which we tune the design parameters of the computational model to match the design criteria for devices in realistic use cases. Since eigenvalues may cross during the shape optimization of a cavity, we propose a new algorithm based on an eigenvalue matching procedure, to ensure the optimization of the desired mode in order to also enable successful matching along large shape variations.

翻译：腔体特征值的形状优化在新型谐振器设计及现有谐振器优化中具有重要作用。本文提出一种基于梯度的优化方案，该方案结合了系统矩阵的闭式形状导数。基于此，我们可精确计算特征值、特征模态及代价函数对几何参数的导数，从而显著降低优化器的计算开销。通过将方法应用于9单元TESLA腔体，我们展示了其实用性：通过调整计算模型的设计参数，使器件在真实应用场景中满足设计准则。针对腔体形状优化过程中可能出现的特征值交叉问题，我们提出一种基于特征值匹配过程的新算法，确保对目标模式进行有效优化，即便在剧烈形状变化条件下仍能实现稳定的模式匹配。