# Research areas

## Boundary Element Methods

To calculate the resonant frequencies of superconducting cavity resonators for particle accelerators, one needs numerical methods capable of achieving accuracies that push established techniques to their limit. This is mostly due to the lacking precision of the geometry description. We investigate a new approach to the solution of Maxwell’s eigenvalue problem that combines building blocks of state-of the-art numerical methods.

The geometry of the resonators is described with **Non-Uniform Rational B-Splines** (NURBS), which can, contrary to the usual triangulations, exactly describe the geometry. NURBS became popular within the framework of **Isogeometric Analysis** (IGA) and have proven themselves already in the context of finite element methods. Because the creation of a volumetric representation of this kind requires a lot of manual effort, and the boundary data of the required format are already given by CAD systems, we combined the isogeometric approach with a **Boundary Element Method** (BEM).

Thanks to modern compression techniques and preconditioning, BEM is an important alternative to a finite element method. The algebraic eigenvalue problems generated by the boundary element method are nonlinear. A novel contour integral method is used for their solution.

The resulting IGA-BEM code has been published in the library BEMBEL.

## Differential Forms

Classical electrodynamics has been taught since the time of Oliver Heaviside by using vector analysis. Heaviside formulated Maxwell's theory in this form in the 1880s. Later, Henri Poincaré, Élie Cartan, and other mathematicians began to establish differential geometry and algebraic topology. They pioneered a new and more powerful framework for describing electrodynamics.

In recent decades, the electromagnetic research community has made great scientific advances in the adoption of geometric methods. Particularly important results are:

- the understanding of the de Rham complex (after Georges de Rham)
- the application of homology and co-homology
- the separation of topological and metric structures
- the recognition of the Hodge operator’s significance (after William Hodge)

We may depict electrodynamics in a concise diagram (Tonti 2013, d: exterior derivative, ∗: Hodge operator), see Figure 1.

Whitney forms (after Hassler Whitney) and, more generally, discrete differential forms, provide a tool to preserve the essential properties of electrodynamics in spaces of finite dimension. Many advances have been made in understanding the convergence of discrete solutions towards those of the continuous theory.

Among other things, ongoing research seeks to further develop these findings for discretization on dual grid systems and in the context of isogeometric analysis.

## Relativity in Engineering

Field problems involving moving bodies are often challenging to solve. An additional difficulty arises from misguided intuition. There are relatively simple examples in which the electric and magnetic fields combined with non-uniform motion can lead to paradoxical interpretations (Schiff 1939).

Such difficulties can be overcome by associating generalized observers with accelerating and deforming bodies. Generalized observers are described by fiber bundles over space-time. The underlying mathematics is that of a gauge theory on a fiber bundle, which is equipped with a so-called Ehresmann connection. This knowledge has not yet found its way into relativity in engineering, let alone into the design of codes for the simulation of electromagnetic fields.

In numerical methods, space and time are usually discretized separately, after a fixed observer has been introduced to determine the decomposition of space-time. A larger class of problems can be tackled if the discretization acts directly on space-time and is chosen locally. For example, an asynchronous time-stepping scheme may be introduced in which each spatial element is assigned its own time step, thereby increasing the simulation’s robustness and efficiency.

Typical applications include scattering from rotating bodies (e.g. helicopter blades) and optical rotation sensors based on the Sagnac effect. The Sagnac effect, on a larger scale, plays an important role in the design of the global positioning system GPS.