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General aspects and node labelling scheme

 

The theory on which the finite element programs described in this manual are based is very simple. The essential idea is that a variational principle exists for the linear elastic and linear electrical conductivity problems. For a given microstructure, subject to applied fields or other boundary conditions, the final voltage or elastic displacement distribution is such that the total energy stored in the elastic case, or the total energy dissipated, in the electrical conductivity case, is extremized, such that the gradient of the energy with respect to the variables of the problem (voltage or elastic displacement) is zero. The variable En is used for both of these cases, and is called an energy in all the following text, even though, in the case of electrical conductivity, it is really an energy dissipation per unit time or power. To minimize En, a function of many variables um, the various partial derivatives must equal zero,

for all values of m. In all the programs, the sum of the squares of all elements of the gradient vector, whose m'th element is just the partial derivative in eq. (1), is determined during the solution or relaxation process. The solution of the problem is considered to be reached when this sum is less than a given small value, so that the condition in eq. (1) is approximately satisfied for all m. This value, denoted gtest in all the programs, should be chosen small enough so that the answers obtained, the currents or stresses in the pixels, are no longer changing significantly with further relaxation.

A labelling scheme must be defined for a single element or pixel in order to derive the finite element equations for that pixel. "Pixel" and "element" will be used interchangeably throughout this manual. In the finite element method, each node attached to a corner of a pixel (8 in 3-D, 4 in 2-D) has a separate label within that pixel. Since in the finite element method, the energy is defined within each pixel using only the nodes attached to that pixel, it is important to be able to refer easily to these attached nodes. The (i,j,k) label for a pixel, which gives its position in a three-dimensional lattice, is the same as the (i,j,k) label for the node numbered 1 in the pixel. Table 1 gives this single-pixel labelling scheme, in terms of the Δi, Δj, and Δk positions of the nodes with respect to (i,j,k). Figure 1 shows this labelling scheme graphically, and also defines the coordinate system used. The (i, j,k) axes coincide with the (x,y,z) axes, respectively.

  


Figure 1: Graphic view of cubic pixel = tri-linear finite element, showing the 1-8 labels of the vertices. The i,j,k axes coincide with the x,y,z axes.

  


Table 1: Finite element (fem) labels for within a single pixel labelled (i,j,k). The Δi, etc. values are with respect to the node labelled (i,j,k).

The basic derivations for the electric case and the elastic case will now be reviewed, with the pixel length taken to be unity. Most books on finite elements will have much of this derivation. It is given here in order to clarify the structure of the programs described in this manual. In all the discussion below, r and s run from 1 to 8, and indicate the node of the finite element being considered, while p and q run from 1 to 3, and indicate the Cartesian coordinate (1 = x, 2 = y, 3 = z) of vector quantities. The convention is used throughout the rest of the manual that if a subscript is repeated, it is assumed to be summed over.


Next: Electrical conductivity Up: Finite element theory Previous: Finite element theory