Abstract:
This paper investigates methods for improving solutions to Maxwell's equations when using the Time Domain Vector Finite Element Method (TDVFEM), a technique that is promising for its use of unstructured triangular grids but is sensitive to grid quality. The study demonstrates that the accuracy of the solution and the computational effort required are directly linked to the condition of the grid, specifically how close its triangular elements are to being equilateral. Through analytical dispersion analysis, the authors show that using an equilateral grid makes the TDVFEM fourth-order accurate in space, significantly reducing numerical anisotropy and phase velocity errors compared to standard Finite Difference Time Domain methods. To achieve this on complex geometries, the paper examines several grid pre-conditioning techniques, including Laplacian smoothing, edge swapping, and a novel grid energy minimization method designed to overcome the limitations of the other two. Computational experiments on rectangular and coaxial cylindrical cavities confirm that these pre-conditioning techniques produce more equilateral grids, leading to more accurate calculations of resonant frequencies and a significant reduction in the computational time needed to solve the system's linear equations at each time step.
