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In the finite difference electrical programs, one must be careful in defining the average current in a pixel. In d dimensions, there are 2d bonds coming into a node, with 2d currents to consider. The most obvious way to define the average current in a pixel is to average the current in the two x-bonds, the two y-bonds, and the two z-bonds, and thus obtain the three components of the average current vector in the pixel. Subroutine CURRENT in the finite difference programs computes the total current for the whole image by summing over all the pixel currents. Variables cur1, cur2, and cur3 are the local average currents in a pixel.
The middle images of Figure 13 shows the same problems as described in the previous section but now for a finite difference solution. The current maps are very similar, with similar small anomalies at the inclusion boundary. To the eye, there is very little difference between the finite difference and the finite element current maps. Recall from Fig. 8 and Table 10 that in this range of inclusion to matrix conductivity ratios, 0.1 to 10, the finite element and finite difference intrinsic conductivities agreed rather well.
The bottom images of Fig. 13 show the equivalent maps for an exact solution to the same problem, without periodic boundary conditions. The exact solution is described below, in Section 7.6. Most of the variation of the current is near to the inclusion, so that it is reasonable to compare infinite matrix analytical and periodic boundary numerical solutions. The finite
difference and finite element solutions are seen, by comparing the different parts of Fig. 13, to at least qualitatively give accurate solutions of the conductivity problem. The analysis of the fields via their distribution in histogram form will show better the small differences between the exact and numerical solutions.
Next: Finite element elastic Up: Making and analyzing Previous: Finite element electrical