We consider Maxwell eigenvalue problems on uncertain shapes with perfectly conducting TESLA cavities being the driving example. Due to the shape uncertainty the resulting eigenvalues and eigenmodes are also uncertain and it is well known that the eigenvalues may exhibit crossings or bifurcations under perturbation. We discuss how the shape uncertainties can be modelled using the domain mapping approach and how the deformation mapping can be expressed as coefficients in Maxwell's equations. Using derivatives of these coefficients and derivatives of the eigenpairs, we follow a perturbation approach to compute approximations of mean and covariance of the eigenpairs. For small perturbations these approximations are faster and more accurate than sampling or surrogate model strategies. For the implementation we use an approach based on isogeometric analysis, which allows for straightforward modelling of the domain deformations and computation of the required derivatives. Numerical experiments for a three-dimensional 9-cell TESLA cavity are presented.

翻译：我们以完美导电TESLA腔体作为驱动示例，考虑不确定形状下的马克斯韦尔特征值问题。由于形状不确定性，所得特征值与特征模态亦具有不确定性，且众所周知特征值在扰动下可能呈现交叉或分岔现象。本文探讨了如何利用域映射方法对形状不确定性进行建模，以及如何将形变映射表达为马克斯韦尔方程中的系数。通过使用这些系数的导数与特征对的导数，我们采用摄动方法计算特征对均值与协方差的近似值。对于小扰动，这些近似值比抽样或代理模型策略更快且更精确。在实现过程中，我们采用基于等几何分析的方法，该方法能够直接对域形变进行建模并计算所需导数。最后，我们展示了三维9腔TESLA腔体的数值实验。