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There will be times when one might not want to use the periodic boundary conditions that are built into the programs. It is simple to get rid of them. Simply use a unit cell that is (nx + 1) x (ny + 1) x (nz + 1), and set the i = nx + 1, j = ny + 1, and k = nz + 1 pixels to have zero conductivity or modulus, i.e. "air." In the finite element programs, this will effectively zero out the linear and quadratic term in the energy, which arise from the periodic boundary conditions. Terms that connect variables across the system will be multiplied by zero and thus contribute nothing. The program can then be run as is, but of course one must allow for the layer of air on the outside in the field averages. Also, one must put some kind of constraint on the boundary in order to have a non- zero result, since the applied electric field or strain came from the boundary conditions. One easy way to apply a uniform strain in an elastic problem is to use THERMAL3D.F, and allow every pixel to have the same eigenstrain. The system will then expand, but there will be internal relaxations because of the different moduli between phases. This is equivalent to applying an external strain [7]. One can also fix the displacements of the outermost pixels, to mimic the desired applied strain. In electrical problems, one can fix the voltage at one or more places. Other ways include putting a fixed force or current somewhere in the system, which is discussed next.
Usually when applying these programs to a real 2-D image, it will be desirable to remove periodic boundary conditions. Also, one will want to change an image, usually in gif or tif or jpeg or some other graphics format, into an image of phase labels. The hardest part is to convert the graphic image into an image of ASCII numbers. These number can then be easily read and converted into the appropriate phase labels, either outside the programs or within subroutine PPIXEL. The shareware program XV, easily obtainable on the Internet, has an output file format called pgm or ppm. This is an ASCII format of either gray scales from 0 to 255, or RGB triples, each ranging from 0 to 255. Simply read the image with XV, then output an ASCII file, which is then easily converted to phase labels using a separate small program.
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