Hybrid Time-Domain Methods and Wire Models for Computational Electromagnetics

The Finite-Difference Time-Domain (FD-TD) method is the most commonly used time-domain method for solving the Maxwell equations. The FD-TD method pioneered by K.S. Yee 1966 is an explicit finite-difference scheme using central differences on a staggered Cartesian grid (Yee grid) and is second-order accurate in both space and time. It has been attractive for industrial users since the early 1980s because the basic method is relatively simple to program and the geometry handling is fairly straightforward. The main drawback of the FD-TD method is its inability to accurately model curved objects and small geometrical features. The Cartesian FD-TD grid leads to a staircase approximation of the geometry and parts smaller than an FD-TD cell might be neglected by the grid generator. We present three different methodologies to minimize this drawback of FD-TD but still benefit from its advantages. They are parallelization, hybridization with unstructured grids, and subcell models for thin wires. Parallel computers can solve very large FD-TD problems. This is illustrated by a lightning problem for a real aircraft where more than one billion FD-TD cells are used. The cell size is one inch which gives a very fine grained grid. This type of simulation is important for electromagnetic compatibility problems where the complexity of the geometry requires small cells to give accurate results. The idea behind our hybridization method is to use unstructured boundary fitted grids to resolve the geometrical features and Yee grid elsewhere. In this way we avoid the staircasing problems, and small parts can be resolved but still keep the efficiency of FD-TD. In two dimensions, our hybrid methods are second-order accurate and stable. This is demonstrated by extensive numerical experiments. In three dimensions, our hybrid methods have been successfully used on realistic geometries such as a generic aircraft model. The methods show super-linear convergence for a vacuum test case and almost second-order convergence for a perfect electric sphere. However, they are not second-order accurate. This is shown to be caused by the interpolation needed when sending values from the Yee grid to the unstructured grid. Stability issues are also discussed. The cross-section of thin wires are smaller than the Yee cells and hence subcell models for thin wires have been developed for FD-TD. We present a new model for arbitrarily oriented thin wires. Previously published models for FD-TD require the wire to be oriented along the edges of the grid and hence a staircasing error is introduced. The new model avoids these errors. Results are presented illustrating the superiority of the new thin-wire subcell model. ISBN 91-7283-058-1 • TRITA-NA-0106 • ISSN 0348-2952 • ISRN KTH/NA/R--01/06--SE