| In this research, several improvements to the FDTD method are implemented by recasting it as a mixed Whitney-form finite-element method. This step permits the application of numerous advancements developed by the finite element community, such as non-uniform nested refinement schemes and higher order Nedelec-type basis functions. The first improvement is h-refinement, a method for abutting disjoint grids with distinct edge lengths (commonly referred to as "subgridding"). This overcomes one of the major disadvantages of the FDTD method: its restriction to uniform sample resolution regardless of physical local feature size. With this improvement, field singularities can be modeled with greater fidelity and stair-casing error can be reduced for geometries with curved features for local instead of global cost. Another improvement explored herein is the development of higher order accurate FDTD-like methods, by replacing the collections of Yee cells with "Lobatto cells" which support mixed order Lagrange polynomial finite elements. Unlike traditional high order finite-difference schemes, the proposed Lobatto cell maintains high order accuracy at material interfaces and metal wall boundaries. These methods have superior geometrical fidelity and dispersion characteristics over the typical Yee method. Both the subgridding and high order improvements inherit the Yee's schemes famous robustness by construction: they are provably energy conserving, electric charge conserving and magnetic charge conserving. When integrated in time with the leapfrog method, both can also be proven conditionally stable for some nonzero timestep. Additional work will present an algorithm for interfacing grids with completely discontinuous basis functions, and investigate more exotic multirate leapfrogging schemes. Both of these concepts are crucial for general adaptive hp-refinement, where abutting grids might be refined in unpredictable ways. All methods are verified on canonical problems in electromagnetism and compared to existing alternative algorithms. The ultimate aim, though beyond the scope of this work, is to develop automatic adaptive refinement capability for the FDTD method. This work lays the foundation for those developments. |
