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Examples of histograms

Consider the same 200 x 200 system, with a 41-pixel diameter circular inclusion (phase 2) in the middle of the matrix (phase 1), as was used in Fig. 13. Histograms for the average current per pixel will be generated using both finite difference and finite element methods, and will be compared to the exact result, which uses the analytical solution of the problem.

This analytical solution is simple to generate. Using polar coordinates, (r,θ), the potential outside the circle is −Er cos θ + B cos θ / r, and the potential inside the circle is −Ar cos θ, where A and B are unknown constants and the applied field E is in the x-direction only. By requiring that the potential and the normal current be continuous at the r = R boundary (the exterior of the circle), the coefficients A and B are determined via these two equations. The x component of the current is then:

One problem in comparing the exact solution to the numerical solutions is that the exact solution assumes an infinite matrix, while the numerical solutions use periodic boundary conditions. For purposes of comparison, the currents from the exact solution at the pixel centers, constricted to a 200 x 200 area around the inclusion, will be used to create the exact solution histogram. An electric field of magnitude unity was applied in the x-direction. For all three computations, there will be 40,000 numbers, the magnitude of the current in the x-direction for each pixel. The same range of bins, and bin sizes, were chosen for all three (finite element, finite difference, exact) so as to make the histograms more comparable. The exact solution could of course have been used to generate a histogram with infinite resolution.

In the first example, the isotropic conductivities were: σ₁ = 1, σ₂ = 10. Figure 14 shows all three histograms plotted from a minimum current of 0.2 to a maximum current of 2.0. The finite element and finite difference solutions actually had a few (10-20) pixels that had higher currents than this, but as they would not show up on a graph, they have not been plotted. These come from digital boundaries, and are not relevant to such a comparison. The finite difference and finite element histograms are essentially the same, and are slightly different from the exact result. Part of this is due to the digital boundary of the circular inclusion, which leads to inaccuracies in the currents near the surface. Part of the disagreement is also due to the fact that the computations used periodic boundary conditions, while the exact solution is for an infinite matrix.

In the second example, the isotropic conductivities were σ₁ = and σ₂ = 0.1. Figure 15 shows all three histograms plotted from a minimum current of 0.1 to a maximum current of 2.0. All currents were within these bounds. The finite difference and finite element histograms are essentially the same, and are slightly different from the exact result. Again, part of this is due to the digital boundary of the circular inclusion, which leads to inaccuracies in the currents near the surface. Similarly to Fig. 14, part of the disagreement between the analytical and numerical histograms is also due to the fact that the computations used periodic boundary conditions, while the exact solution was for an infinite matrix.

The results for Figs. 14 and 15 make it clear that current distributions can differ, but in such a way that averages (first order moments) are nearly the same. However, higher moments will differ more. This is why the error on the effective conductivity (1st moment) was always less than the error in the average field squared (2nd moment) in the checkerboard example given in Table 8.