A boundary conformal Discontinuous Galerkin method for electromagnetic field problems on Cartesian grids

The thesis presents a novel numerical method based on the high order Discontinuous Galerkin (DG) method for three dimensional electrostatic and electro-quasistatic field problems where materials are of very complex shape and may move over time. A well-known example is water droplets oscillating on the surface of high voltage power transmission line insulators. The electric field at the surface of the insulator causes the oscillation of the water droplets. The oscillation, in turn, triggers partial discharges which have damaging effects on the polymer insulation layers of high voltage insulators. The simulation of such phenomena is highly complex from an electromagnetic point of view. Most numerical methods which are applied to such field problems use conforming meshes where the elements are fitted exactly to the material geometry. This implies that the element interfaces conform to the material boundaries or material interfaces. In general, the generation of conforming meshes is computationally expensive. Furthermore, when dealing with materials that move over time, conforming meshes need to be adapted to the changing material geometry at each point in time. To avoid the often computationally costly generation and adaption step of conforming meshes, the numerical method proposed in this thesis operates on a single fixed structured Cartesian mesh. First, field problems with non-moving materials are considered. To obtain accurate simulation results on field problems with complex-shaped materials, an additional approach, namely the cut-cell discretization approach, is applied. The cut-cell discretization approach subdivides the elements at material boundaries or interfaces into smaller sub-elements which are referred to as cut-cells. The approach is embedded into the Discontinuous Galerkin (DG) method for standard Cartesian meshes since the DG method allows for high order approximations and offers a great flexibility for additional approaches. Since the mesh is not fitted to the material geometry, geometrically small cut-cells might emerge. Therefore, two supplementary approaches, the adaptive approximation order method and the cell merging method are proposed which enable an accurate approximation even on geometrically small cut-cells. Furthermore, a DG hybridization is presented which lowers the number of degrees of freedom in domains where the high number of DG degrees of freedom is not necessary to obtain accurate results. The numerical method comprising all above mentioned approaches is labelled as boundary conformal DG (BCDG) method. In a second step, the BCDG method is extended to field problems where materials move over time. We refer to this approach as extension of the BCDG (EBCDG) method. The EBCDG method adapts to the moving materials by recalculating only the cut-cells at each point in time while the underlying Cartesian grid is kept fixed. Therefore, no computationally expensive mesh adaption or mesh generation steps are needed. The BCDG and the EBCDG method are applied to numerical examples of electrostatic (ES) and electro-quasistatic (EQS) field problems. First, numerical results of the BCDG method on a verification example of a cylindrical capacitor filled with two dielectric layers are shown. A convergence study and a comparison study illustrate the high accuracy of the BCDG method with respect to the number of degrees of freedom. Finally, the EBCDG method is applied to an example of a water droplet oscillating artificially on the insulation layer of a high voltage insulator. A convergence study demonstrates that even on a coarse mesh a high resolution of the potential and electric field solution can be achieved.