Efficient Methods for Parameterized Eigenvalue Problems in Electromagnetism

This thesis addresses the development of efficient methods for analyzing the parameterized electromagnetic eigenvalue problem (EVP). This problem describes the characteristic oscillation states of the electromagnetic field within bounded structures. In this work, we investigate the EVP in cavities. In the context of electromagnetism, the so-called Maxwell EVP provides insights into the resonant frequencies and corresponding electromagnetic field distributions (eigenmodes) of the cavities. When analyzing the resonant frequencies and eigenmodes and optimizing them in the design process, the parametric variation of the cavity geometry, in particular, poses a central challenge. Particularly when the EVP has to be solved frequently, as is the case with iterative design and optimization tasks, conventional approaches lead to considerable computational costs. This motivates the development of efficient computational methods. To understand the operating principle of cavities, desired and parasitic modes must be reliably identified. The method proposed in this thesis involves progressively deforming the cavity to a reference geometry for which an intuitive mode classification is available. Further, eigenmodes and resonant frequencies depend sensitively on shape deviations. The corresponding trajectories of the resonant frequencies may cross under such geometric changes. Therefore, their trajectories must also be taken into account during the analysis. For this purpose, the eigenmodes are tracked along the deformation using a suitable tracking algorithm. In addition, solving the EVP with many degrees of freedom (DoFs) is computationally expensive, making methods that rely on numerous evaluations increasingly inefficient. A model order reduction (MOR) is therefore developed to reduce the computational effort. This approach must be adapted to the Maxwell EVP with geometry-dependent parameters, ensuring that spurious solutions appearing in the reduced basis (RB) are reliably excluded. Also, for classical sampling-based uncertainty quantification (UQ) methods, it is essential to ensure the consistency of the eigenmodes across the parameter variations. Hence, we develop a shape UQ framework that computes statistical measures in closed-form using derivatives of resonant frequencies and eigenmodes with respect to geometric deformations. The proposed methods are demonstrated using the practically relevant TeV-Energy Superconducting Linear Accelerator (TESLA) cavity as an application example. The tracking-based classification of eigenmodes proves to be robust and allows for an intuitive and reliable eigenmode classification. By employing MOR, the efficiency of the tracking process is significantly enhanced, enabling the construction of the reduced system and subsequent tracking to outperform computations on the full system in terms of efficiency. To determine sensitivities in closed-form, derivatives are calculated based on the geometry representation in computer-aided design (CAD). This representation also enables the direct calculation of higher-order sensitivities. Notably, the application of these derivatives in UQ for small geometric deformations, such as those arising from manufacturing imperfections, shows substantial efficiency gains compared to sampling-based approaches. Thereby, a UQ analysis, which would otherwise require several hours on a server computer, can be completed in minutes on a standard laptop.