Читать книгу Electromagnetic Simulation Using the FDTD Method with Python - Dennis M. Sullivan - Страница 11

The time‐dependent Maxwell’s curl equations for free space are

(1.1a)

(1.1b)

E and H are vectors in three dimensions, so, in general, Eq. (1.1a) and (1.1b) represent three equations each. We will start with a simple one‐dimensional case using only E_x and H_y, so Eq. (1.1a) and (1.1b) become

(1.2a)

(1.2b)

These are the equations of a plane wave traveling in the z direction with the electric field oriented in the x direction and the magnetic field oriented in the y direction.

Taking the central difference approximations for both the temporal and spatial derivatives gives

(1.3a)

(1.3b)

In these two equations, time is specified by the superscripts, that is, n represents a time step, and the time t is t = Δt ⋅ n. Remember, we have to discretize everything for formulation into the computer. The term n + 1 means one time step later. The terms in parentheses represent distance, that is, k is used to calculate the distance z = Δx ⋅ k. (It might seem more sensible to use Δz as the incremental step because in this case we are going in the z direction. However, Δx is so commonly used for a spatial increment that we will use Δx.) The formulation of Eq. (1.3a) and (1.3b) assume that the E and H fields are interleaved in both space and time. H uses the arguments k + 1/2 and k − 1/2 to indicate that the H field values are assumed to be located between the E field values. This is illustrated in Fig. 1.1. Similarly, the n + 1/2 or n − 1/2 superscript indicates that it occurs slightly after or before n, respectively. Equations (1.3a) and (1.3b) can be rearranged in an iterative algorithm:

(1.4a)

(1.4b)

Notice that the calculations are interleaved in space and time. In Eq. (1.4a), for example, the new value of E_x is calculated from the previous value of E_x and the most recent values of H_y.

Figure 1.1 Interleaving of the E and H fields in space and time in the FDTD formulation. To calculate H_y, for instance, the neighboring values of E_x at k and k + 1 are needed. Similarly, to calculate E_x, the values of H_y at k + 1/2 and are needed.

This is the fundamental paradigm of the FDTD method (1).

Equations (1.4a) and (1.4b) are very similar, but because ε₀ and μ₀ differ by several orders of magnitude, E_x and H_y will differ by several orders of magnitude. This is circumvented by making the following change of variables (2):

(1.5)

Substituting this into Eq. (1.4a) and (1.4b) gives

(1.6a)

(1.6b)

Once the cell size Δx is chosen, then the time step Δt is determined by

(1.7)

where c₀ is the speed of light in free space. (The reason for this will be explained in Section 1.2.) Therefore, remembering that ε₀μ₀ = 1/(c₀)²,

(1.8)

Rewriting Eq. (1.6a) and (1.6b) in Python gives the following:

(1.9a)

(1.9b)

Note that the n, n + 1/2, or n − 1/2 in the superscripts is gone. Time is implicit in the FDTD method. In Eq. (1.9a), the ex on the right side of the equal sign is the previous value at n − 1/2, and the ex on the left side is the new value n + 1/2, which is being calculated. Position, however, is explicit. The only difference is that k + 1/2 and k − 1/2 are rounded to k and k − 1 in order to specify a position in an array in the program.

The program fd1d_1_1.py at the end of this chapter is a simple one‐dimensional FDTD program. It generates a Gaussian pulse in the center of the problem space, and the pulse propagates away in both directions as seen in Fig. 1.2. The E_x field is positive in both directions, but the H_y field is negative in the negative direction. The following points are worth noting about the program:

1 The Ex and Hy values are calculated by separate loops, and they employ the interleaving described above.

2 After the Ex values are calculated, the source is calculated. This is done by simply specifying a value of Ex at the point k = kc and overriding what was previously calculated. This is referred to as a hard source because a specific value is imposed on the FDTD grid.

Figure 1.2 FDTD simulation of a pulse in free space after 100 time steps. The pulse originated in the center and travels outward.