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Since the current in a resistor is just the conductance times the voltage difference between the two ends, it is somewhat easier to illustrate how to use fixed currents in the finite difference algorithm. Consider the problem discussed in Section 6.1, that of the effective resistance between two nodes of a lattice, in 2-D or 3-D. There the problem was solved by fixing the voltage difference between the two nodes, and then computing the current in and out the two nodes. The same problem can also be done by fixing the current in and out at the two nodes. The gradient of the energy at a node is just the negative of the current at that node. If we add 1 and −1 to the gradients at the two nodes before relaxation occurs, this will be like forcing a fixed current at these nodes. The voltages will relax, so as to force the total gradient to become zero, but with the current at the nodes fixed at its original value, since there are no other current or field sources in the problem. The same solution to the 2-D problem, with some slight round-off difference, was found as in Section 6.1. The same technique could then be used if fixed current boundary conditions, instead of periodic fixed voltage conditions, were desired. The currents at the boundary could be fixed using this method.
In the finite element programs, the negative gradient of the elastic energy with respect to a nodal displacement is just the force at that node. In the same way, in the subroutine ENERGY, after the gradient is first computed, a fixed value can be added to the gradient at a particular node, to simulate a fixed force at that node. If the subroutine ENERGY is then run again, after LDEMB conjugate gradient cycles in DEMBX, this addition should be renewed before going back into the conjugate gradient routine.
Usually the effect of fixed stresses is considered, not the effect of fixed forces. In this case, if for example a pressure p is to be put on a surface, the way one does this is to consider each pixel face, and put a force of p/4 on each node of each face, in 3-D. The edge length of the pixels is one. That way, nodes that are shared by four pixel faces will end up with p, nodes that are only shared by two pixel faces will have p/2, and nodes that are only on one pixel face, will have p/4. This takes care of overcounting. Consider four (2 x 2) pixel faces in a plane, containing 9 nodes. The total area is four, if the pixels have unit length. If a force p was put on each node, then the total pressure on the face would be 9P/4, not p. However, if the method suggested is used, then the middle node will have a force p, the four side nodes will each have a force p/2, and the four corner nodes will each have a force p/4, for a total pressure of 4 p/4 = p. If the pixels are in a 1 x 4 arrangement, then the four corner nodes will have a force p/4, and the six middle nodes will each have a force of p/2, for a total pressure of 4 p/4 = p.
This use of fixed pressures can be tested using eq. (76), Mackenzie's exact result for an isotropic porous solid filled with a fluid at pressure p. If we consider a porous material with a dilute concentration of spherical holes, we can combine eq. (64) for the bulk modulus of such a system with Mackenzie's result to get an exact prediction for the strain expected at a pressure p. Alternatively, we can directly compute, using ELAS3D.F, the effective bulk modulus of the porous solid. For a 253 unit cell, containing a single spherical hole of diameter 11 pixels, when the solid frame bulk modulus was 30.0825, the effective bulk modulus of the porous material was 27.3909 as computed by ELAS3D.F. When p = 0.1, the prediction from Mackenzie's result is ε = 1.08882 x 10−4. The numerical result, using the algorithm described above, was ε = 1.08888 x 10−4, a difference of only 0.006%. So this algorithm works very well.
One warning: often pores will go across the boundary when periodic boundary conditions are used. In this case, this algorithm does not seem to work very well [41]. An alternate method can be to enclose the porous solid with a "skin" of effective material with elastic properties such that the bulk modulus of the composite system is unchanged from that of the porous solid alone. This algorithm does work on interior pores. When a skin is placed around the material, all pores become interior pores.
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