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ee256-08-lecture09

ee256-08-lecture09 - 1 EE256 Numerical Electromagnetics...

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1 EE256 Numerical Electromagnetics H. O. #14 Marshall 24 June 2008 Summer 2008 LECTURE 9 9.1 TOTAL-FIELD/SCATTERED-FIELD FORMULATION IN THREE DIMENSIONS The total-field/scattered field formulation was described in Lecture #8 for the two-dimensional case, re- quiring two correction terms for the electric field components (for the TM case) located at the corners of the FDTD grid. It is important to note that in a three dimensional grid, only a single correction term is required for most of the components, except for the electric field components lying along the edges of the three dimensional FDTD grid. The single correction terms for each of the E and H components for the three dimensional case are given in Section 5.8 of the textbook. Accordingly, we focus our attention here on the electric field components along the edges. The vicinity of one of the corners ( i = i 0 , j = j 0 , k = k 0 ) of the three dimensional FDTD grid is shown in Figure 9.1. Here, the field components that reside inside or at the edges of the Region 1 FDTD grid are indicated with solid-filled arrows, while those that reside outside this region and inside Region 2 are indicated with empty arrowheads. Consider, for example, the E tot z i 0 ,j 0 ,k 0 +1 / 2 component, which is a total-field component, which is sur- rounded by two total-field components, H tot x i 0 ,j 0 +1 / 2 ,k 0 +1 / 2 and H tot y i 0 +1 / 2 ,j 0 ,k 0 +1 / 2 , and two scattered- field components, H scat x i 0 ,j 0 - 1 / 2 ,k 0 +1 / 2 and H scat y i 0 - 1 / 2 ,j 0 ,k 0 +1 / 2 . Thus its updating equation should be: E n +1 z i 0 ,j 0 ,k 0 +1 / 2 = E n +1 z i 0 ,j 0 ,k 0 +1 / 2 | {z } Normal FDTD update (Eqn . [5 . 1b]) - Δ t Δ x H inc y n +1 / 2 i 0 - 1 / 2 ,j 0 ,k 0 +1 / 2 + Δ t Δ y H inc x n +1 / 2 i 0 ,j 0 - 1 / 2 ,k 0 +1 / 2 [9.1] Note that the updating equation is similar for all the other E z components along the edge of Region 1, namely E tot z i 0 ,j 0 ,k>k 0 . Expressions similar to [9.1] can be written in a straightforward manner for the other electric field com- ponents at the edges of Region 1, namely E tot x i>i 0 ,j 0 ,k 0 and E tot y i 0 ,j>j 0 ,k 0 . We note from Figure 9.1 that all other electric field components located on the faces (but not on the edges) of the Region 1, require only a single correction term. For example, consider E tot x i 0 +1 / 2 ,j 0 ,k 0 +1 , which is surrounded by three total-field components, H tot y i 0 +1 / 2 ,j 0 ,k 0 +1 / 2 , H tot z i 0 +1 / 2 ,j 0 +1 / 2 ,k 0 +1 , and H tot y i 0 +1 / 2 ,j 0 ,k 0 +3 / 2 and one scattered-field component H tot z i 0 +1 / 2 ,j 0 - 1 / 2 ,k 0 +1 . Thus its update equation
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