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Included in this package of programs are finite difference programs for solving linear conductivity problems on general digital images in 2-D and 3-D, for D.C. and A.C. problems. There are some advantages and disadvantages when comparing finite difference and finite element computations. Regarding the programs described in this manual, the finite difference programs take less memory, as they do not need the integer variable ib, which stores neighbor information and is especially needed at the boundaries. This is because the finite difference programs handle the periodic boundary conditions in a different way, as will be discussed below. The finite difference programs in this manual could be written in an energy form, and make use of the second derivative matrix and the variational principle approach in the same way as the finite element formulation. They are not so written in this manual, however.
The way the finite difference algorithm handles boundary conditions between phases is also different from the finite element algorithm. Electrical problems can be handled quite readily with finite differences, but there are some difficulties with elastic problems. At boundaries at which a displacement or stress component takes on a certain value, it is straightforward to write finite difference equations for elastic problems that handle this kind of boundary, using non-centered difference equations. For example, a finite difference elastic code can be written for a porous material, where there is a single kind of solid phase, and so all solid-pore boundaries are zero stress boundaries [8]. Also, the case when the second phase is infinitely stiff, so that the displacement must be the same thoughout connected parts of the phase, can also be handled by finite differences. For a boundary where the displacement and normal force are continuous, however, it is difficult to see how to write a finite difference program that accurately takes both these conditions properly into account. However, two-phase elastic boundaries where both phases have finite moduli can be easily be handled by finite elements.
As for material properties, finite difference electrical conductivity programs can handle arbitrary diagonal conductivity tensors. It is possible to extend them to tensors having non-zero off-diagonal elements, although it makes phase-phase boundaries somewhat more trickier to handle, and requires 2nd and 3rd neighbor information, instead of just nearest neighbor information. Therefore the finite difference programs described in this manual have only diagonal conductivity tensors.
A major advantage of the finite difference approach that we have found is in percolation cases, where a conducting phase becomes sparsely connected. We have found that in simulating continuum percolation cases with digital images, assessing connectivity only through the nearest neighbors agrees rather well with the equivalent continuum calculations [9]. When using finite differences, the only current flow possible is through nearest neighbor connections. In finite elements, however, a node is connected electrically with its first, second, and third nearest neighbors. Therefore a phase could become disconnected in terms of first neighbors, and so disconnected acording to a percolation algorithm, and yet stay connected electrically, if a finite element scheme was being used. So in this case, the finite difference scheme is more physically realistic. If the resolution were infinite, then the result obtained would be the same for the finite difference and finite element methods. Since in a typical problem, and especially so for a sparsely connected problem, the resolution is not infinite, one needs to choose the method that will give more accurate results.
The finite difference programs in this package are: AC3D.F and AC2D.F, for complex (finite frequency or a.c.) problems in 3-D and 2-D, respectively, and DC3D.F and DC2D.F, for d.c. problems in 3-D and 2-D. They are all based on a general digital image, where each phase has a different (and frequency-dependent in the a.c. cases) conductivity. They are not based on a true energy formulation, but rather on a solution of Kirchkoff's laws for the resulting conductance network, which is equivalent to solving Laplace's equation, as is shown next.
Next: Derivation of finite Up: Finite difference theory Previous: Finite difference theory