Time-Domain Methods for the Maxwell Equations

The most widespread time-domain method for the numerical simulation of the Maxwell equations is the finite-difference time-domain method (FD-TD). It has been widely used for electromagnetic simulation, for instance in radar cross section computations and electromagnetic compatibility investigations. The FD-TD method is second-order accurate and very efficient for simple geometries. A major drawback with the FD-TD method is its inability to accurately handle curved boundaries. Such boundaries are approximated with so-called staircasing to fit into the Cartesian FD-TD grid. Staircasing introduces errors that destroy the secondorder accuracy of the FD-TD method. We present three different methodologies to tackle the errors caused by staircasing. They are parallelization, hybridization with unstructured grids, and regularization of material interfaces. By using parallel computers it is possible to lower the staircasing errors by using a grid with many cells. We examine the scale-up and speed-up properties of the FD-TD method and demonstrate that it can be used to solve gigantic problems. This is shown by a one-billion-cell computation of an aircraft. We also present a new hybridization strategy. We hybridize FD-TD with methods for unstructured tetrahedral grids. On the unstructured grid we use either an explicit finite volume method or an implicit finite element method, depending one the size of the smallest tetrahedron in the unstructured grid. The implicit method is used on grids with tetrahedra that are much smaller than the hexahedra in the FD-TD grid. Otherwise the explicit method is used. In two dimensions, our hybrid methods are second-order accurate and stable. This is demonstrated by extensive numerical experimentation. 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. However, they are not second-order accurate. This is shown to be caused by the interpolation when sending values from the FD-TD grid to the unstructured grid. Our hybrid methods have been implemented in a code package that is used in an industrial environment. The hybridization strategy is successful but can be expensive in terms of memory and arithmetic operations needed per cell in the grids. We present a new regularization procedure for material interfaces that restore second-order accuracy without adding any extra memory or arithmetic operations during the timestepping. By replacing the discontinuous material function with a properly chosen continuous function prior to the discretization, we can restore second-order accuracy. This is shown for a circular dielectric cylinder for the TMz polarization of the Maxwell equations. ISBN 91-7283-043-3 • TRITA-NA-0101 • ISSN 0348-2952 • ISRN KTH/NA/R--01/01--SE