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A digital image, by the fact that it is already divided into pixels, is easily adapted to discrete computational methods, like finite difference, finite element, and lattice Boltzmann methods. Since there is an underlying lattice, any known algorithm for lattice problems can be applied. A manual, available through the Internet, describes a collection of various programs [57] that apply finite difference and finite element methods to any 2-D or 3-D digital images. These programs can be used to compute a variety of material properties to compare with experiment. We discuss below how the different methods can be applied to two-phase pore-solid images, representing materials in which either the solid or the pores have a uniform property, and the other is zero. An example of the case where the solid is insulating and the pores are filled with a conductive fluid is VycorTM glass filled with a liquid metal [58]. On the other hand, we could have a conducting granular backbone, and insulating pores [59]. Similarly for elastic properties, the solid is assumed to have a uniform elastic modulus tensor while the pores have zero elastic modulus. For hydraulic permeability, the fluid can only flow in the pores.
Both finite element and finite difference methods are simply means of converting partial differential equations into a set of approximate algebraic equations. However, it is worth mentioning at this point some of the differences between the finite element and finite difference methods discussed in this chapter. The linear electrical conduction and linear elasticity problems can be formulated either directly as a set of linear partial differential equations, or indirectly, as an energy functional of partial derivatives that obeys a variational principle. We present finite difference methods for the electrical conduction case, and finite element methods for the elasticity case. The full Navier-Stokes equations are non-linear and do not have an associated variational principle [60]. The linearized forms of these equations, the Stokes equations, do have a variational formulation [61]. However, we present only finite difference methods for the fluid flow case.
In the different methods as they are presented in this chapter, there is also a difference as far as node placement. In a digital image, we want to use no more than one node per pixel, if possible, in order to conserve memory. Philosophically, this is also desirable since having more than one node per pixel would seem to imply that more information is available than is really present in the pixel structure. In a digital image, there are as many pixel corners as there are pixels or pixel centers, so a reasonable choice of node location would be either pixel corners or pixel centers. Of course, just as one can use any coordinate system to solve a physical problem, the node placement can be arbitrary as well. However, for example, one would not choose parabolic coordinates to solve a problem involving the surface of an ellipsoid, because ellipsoidal coordinates give much easier algebra to work with than would parabolic coordinates in this case. In the same way, for the finite difference and finite element methods, certain ways of choosing the node placement result in much simpler equations.