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It is simple to make a current map using the finite element programs ELECFEM2D.F or ELECFEM3D.F. In the subroutine CURRENT, the total average current is computed. In a small DO loop in the middle of the main DO loop, called DO 465, the average current for a given pixel is computed. The components of the average current in a pixel are called cur1, cur2, and cur3, for the x, y, and z directions. These variables can be written to a file, or perhaps only the magnitude of the current need be stored. Then an image, using some suitable scaling system (0-255 for gray scale, or a color system) can created using almost any kind of imaging software. The shareware program XV can also be used to convert the ASCII gray scale values back to a graphics format image.
In the finite element programs, the average current in a pixel is given in terms of the nodal voltages, using the average over the pixel of the conductivity tensor times the matrix in eq. (4). This matrix gives the local field in the pixel in terms of the nodal voltages, so the conductivity tensor times the local field gives the local current.
Figure 13 shows (top images), for the applied field in the horizontal direction, the horizontal current density, for two choices of the inclusion conductivity. The matrix always had a conductivity of 1, and the inclusion had either a conductivity of 10 (left) or 0.1 (right). The actual current densities have been scaled from 0 to 255 by assigning the value of 200 to the average current density. Any current density that scaled to a value over 255 was simply set to 255. Then a color scale was assigned according to this order: red, green, gray, and black (top to bottom, high to low current density in accompanying color bar). In the left top image of Fig. 13, the current tends to bunch up at the left and right of the inclusion, due to the normal current being continuous at the boundary. The current density is depleted at the top and bottom of the inclusion, as it is energetically favorable for the current to curve and go through the high conductivity inclusion.
Figure 13: Image of the horizontal current magnitudes, with the applied electric field
in the x-direction in all images. The inclusion is phase 2.
Left: σ2 = 10, Right: σ2 = 0.1,
and both images had
σ1 = 1.
Top: Finite element solution. Middle: finite difference solution.
Bottom: exact solution, no periodic boundary conditions.
Color bar shows high (red) to low (black) current scale.
The right top image of Fig. 13 shows a similar picture for an inclusion conductivity of 0.1. Notice now the current tends to bunch up at the top and bottom of the inclusion, as it is energetically more favorable to go around the low conductivity inclusion. Also, at the left and right of the inclusion, the current is low, because at the boundary the normal current must match the low current inside the inclusion.
In the top images of Fig. 13, one can notice a ring of miscolored pixels around the inclusion boundary. This is an artifact of having a digitally-rough approximation to a smooth boundary. Locally averaging the pixel intensities around the boundary would make the boundary much smoother, and give a better approximation to reality.
Next: Finite difference electrical Up: Making and analyzing Previous: General features
