4.6 Finite Difference Time-Domain Method in Three Dimensions

Clearly the most general application of the FDTD, and the one of most interest in the calculation of RCS for realistic targets, is the three-dimensional formulation. The development of the update equations in three dimensions follows the same procedure as for one and two dimensions. The details have been worked out in several references.2 ^,7

A three-dimensional Yee cell is shown in Fig. 4.39. As in the case for two-dimensions, we use the following shorthand notation:

• Grid dimensions: ( ? x, ? y, ? z)

• Time step: ? t

• Nodes indices: (i, j, k) ? (i ? x, j ? y, k ? z)

• Notation for field samples: E _x (i ? x, j ? y, k ? z, n ? t) ? (i, j, k,)

Fig. 4.39: Yee cell for three-dimensional implementation of the FDTD.

The E components are defined in the middle of edges, whereas the B components are at the centers of the faces, denoted by integers plus one-half (for example i + 1/2). With this notation the magnetic field update equations are2

The electric field update equations are

where

and ? = x, y, or z. The three-dimensional formulation is the basis of several commercially available software packages.