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For a steady state conductivity problem, where the currents are steady in time, the charge conservation equation,
becomes simply
or, in regions of constant conductivity,
| σ∇2V = 0 | (29) |
where V is the position dependent potential or voltage. Between phases having different conductivities, the boundary conditions are that the current normal to the interface and the potential are continuous.
Consider a region of uniform conductivity, and a point (i,j,k) in this region, in the middle of a pixel. If the voltage at (i,j,k) is u(i,j,k), then, to second order in the pixel dimensions, the potential at i ± 1, j ± 1, or k ± 1 is just
where Δx = ± a, Δy = ± b, Δz = ± c, and a, b, and c are the dimensions of the pixel, in some units. Then the finite difference form of Laplace's equation, in terms of the 1-d label m and the six nearest neighbors, can be obtained by adding together the above six equations in pairs and solving for the 2nd derivative terms. The first derivatives cancel out, the gradient squared is formed from the sum of the second derivatives, and Laplace's equation in finite difference form, at each node m, becomes
where Σm,n is the conductance of the bond connecting node m to its nearest neighbor with nearest neighbor label n, the sum is over the n = 1,6 nearest neighbors, and for convenience, both sides of eq. (33) have been multiplied by the volume of the pixel, abc. The values of Σm,n are Σm,1 = Σm,4 = σ bc/a, Σm,2 = Σm,5 = σ ac/b, and Σm,3 = Σm,6 = σ ab/c, with neighbors (1,2,3) in the (x,y,z) directions, and neighbors (4,5,6) in the (−x,−y,−z) directions. As has been pointed out previously [10], this equation is formally identical to solving Kirchoff's laws of zero net current into each node for this conductance lattice, where the conductances are those of a pixel of material with a given conductivity tensor in a given direction. Therefore formally the length of the pixel appears in the conductance. Note that the finite difference programs are set up for cubic pixels, with a = b = c = 1. They can be easily modified for a rectangular parallelipiped pixel, with three arbitrary dimensions. When setting up the conductances, these lengths matter, as the conductances of a given conductivity pixel are different in directions that have a different dimension.
There is an equation like eq. (33) for each node m in the system. Putting all these equations together, a global equation can be written: Amn un = 0. If we build a quadratic form out of this matrix A, ½ uAu, then this form is extremized when its gradient with respect to u is zero: Amn un = 0, which is the same as the set of equations we are trying to solve. The formal problem being solved is then the minimization of the quadratic form ½ uAu, where the vector u is the voltage vector, and A is a sparse matrix composed of the conductances of all the bonds in the problem. In the finite difference programs, the matrix A is implicitly stored in the vectors gx, gy, and gz. The storage requirements for the finite difference electric programs is much less than those for the finite element programs, and so in this case, the global matrix A is stored. The result of A multiplied times an arbitrary vector can be generated using the vectors gx, gy, and gz, which store the conductances of the problem in the x, y, and z directions respectively. The subroutine PROD does this matrix multiplication, while maintaining periodic boundary conditions and the applied field (see Section 3.3). If there were not the constraint of an applied field, then of course the minimization of a quadratic form such as ½uAu would be trivially given by u = 0.
Next: Boundary conditions Up: Finite difference theory Previous: General descriptioncomparison
