The finite difference time domain method in two and three dimensions

Introduction In the previous chapter, the basic concepts of the finite difference time domain method were introduced via a one-dimensional example. We will briefly reprise the issues one must attend to when doing an FDTD simulation, as follows. An FDTD mesh (or grid) must be created for the problem. (This is trivial in 1D, requires a little thought in 2D, and becomes quite a major problem in 3D.) This mesh must be fine enough – i.e. Δ s must be no more than perhaps one-tenth of the minimum wavelength (i.e. maximum frequency) of interest (Δ s represents the spatial step size; quite often, Δ x , Δ y and Δ z are chosen equal and Δ s is used as shorthand for this). The time step Δ t must satisfy the Courant limit (but be as close to this as possible to minimize dispersion). Boundary conditions (the source and load resistors in our 1D example) must be specified. An appropriate signal shape (e.g. differentiated Gaussian) with suitable time duration for the desired spectral content must be chosen. Also, in general, its spatial position must be specified. (In the transmission line example, it was fixed as the source voltage generator.) In this chapter, we will study the FDTD method in two and three dimensions. Firstly, we will develop a 2D simulator for a problem of scattering in free space. Following this, a very important development, the perfectly matched layer absorbing boundary condition, will be discussed and implemented. This is followed by a brief discussion of the extension to three dimensions. We conclude the chapter with a discussion of the use of CST MICROWAVE STUDIO TM , a commercial electromagnetics simulation package which includes an FDTD solver.