0 - 521 - 19069 - 5 - Numerical Electromagnetics : The FDTD Method

With the continued growth of computing power, modeling and numerical simulation has grown immensely as a tool for understanding and analyzing just about any problem in science. Where in the mid-twentieth century, detailed analyses were required to get any meaningful insight out of complex problems, today we can simply plug the governing differential equations into a computer, the results of which can provide an immense amount of information, which is of course complementary to theoretical analyses. The growth of computing power has brought with it a smorgasbord of modeling methods, applicable in any number of fields. The problem, then, is knowing when to use which method. In electromagnetic problems, which are of interest to us in this book, there are quite a number of useful numerical methods, including the Method of Moments, Finite Volume methods, Finite Element methods, and Spectral methods, just to name a few. The FDTD method, however, grew to become the method of choice in the 1990s, for a number of reasons. First, it has always had the advantage of being a very simple method; we shall see in Chapter 3 that the derivation of difference equations is very straightforward. However, before the 1990s, the FDTD method was hindered by the need to discretize the simulation space on sub-wavelength scales, with time steps commensurately small. Hence, any reasonable problem would require a large amount of computer memory and time. Since the 1990s, however, with the growth of computing power, the FDTDmethod has taken off. As an example, a typical 3D problem would require, at minimum, 100 grid cells in each dimension, or 106 grid cells total. With a minimum of six fields to compute (three components each of the electric field and magnetic field ), and 2 bytes per value (for 16-bit resolution), we require 12MB of memory. As for computation time, our simulation might require 1,000 time steps. Each of six equations will have four additions and two multiplications (at minimum, for the free-space algorithm in Chapter 4) at each of the 1 million grid cells, for ∼36 billion operations over the time of our simulation. In 1990, 12 MB of memory and 36 billion operations was a significant calculation; today, you could quite easily run this simulation on your mobile phone.1