Triangular spatial elements

Thus an electric potential as calculated in Listing 13.13 can be used to evaluate the electric field within each of the triangular spatial elements and this can then snb-sequently be used in a second PDE calculation to evaluate a temperature change in a resistive element such as the square comer resistor. 

The FE solvers developed in this chapter have made use of some of the approximate matrix solution techniques developed in the previous chapter. However, most of the code is new because of the different basic formulation of the finite element approach. In this work the method of weighted residuals has been used to formulate sets of FE node equations. This is one of the two basic methods typically used for this task. In addition, the development has been based upon flie use of basic triangular spatial elements used to cover a two dimensional space. Other more general spatial elements have been sometimes used in the FE method. Finally the development has been restricted to two spatial dimensions and with possible an additional time dimension. The code has been developed in modular form so it can be easily applied to a variety of physical problems. In keeping with the nonlinear theme of this work, the FE analysis can be applied to either linear or nonlinear PDEs. 

The first and perhaps the most basic step in the FE approach is flie selection of finite elements to uniformly cover the spatial dimensions of a physical problem. While many possible shapes could be considered for the fundamental spatial elements, in practice, fairly simple shapes are typically used. For two spatial dimensions, the elements typically considered are triangular elements or general quadrilateral elements as illustrated in Figure 13.2. (The three dimensional case will be briefly considered in a subsequent section). Three triangular elements are shown in (a) - (c) with specified nodes of 3, 6 and 10 where nodes are identified by the solid points in the figure and are the points at which the solution variable is... 

The number of terms of a complete polynomial of any given degree will hence correspond to the number of nodes in a triangular element belonging to this family. An analogous tetrahedral family of finite elements that corresponds to complete polynomials in terms of three spatial variables can also be constructed for three-dimensional analysis. 

There are some constraints on the shape functions in order to have a consistent finite element formulation. Equation 8.12 demands that the shape function Nn be at least linear in the spatial coordinates. The simplest and most widely used shape function in our case is perhaps the piecewise linear interpolation function as described in Kennedy (1995). With this shape function, one can achieve C° continuity in pressure, that is, the pressure field variable is continuous at element interface, but its gradients are not. The pressure gradient field is piecewise constant over the elements and is discontinuous across element interfaces. Consequently, the resulting velocity and shear rate fields are not continuous across element boundaries. An example of triangular elements with higher order shape functions can be found in Hieber and Shen (1980). 

To accomplish this one needs to parameterize flic solution in terms of a finite set of solution values and then seek to minimize the functional with respect to the selected set of solution values. The set of solution values are taken as some set over each triangular element. The simplest sets of solution values to select are the three node point values as shown in Figure 13.2(a). However, a more complex set could be the six or ten values shown in Figure 13.2(b) and 13.2(c). After selecting the parameterizing set of values an interpolation method is then needed over the spatial domain. The conventional approach is to assume that the solution varies linearly between the triangular nodes and thus to use the shape function as derived in the previous section to approximate the spatial variation of the solution. When written in this form the solution becomes ... 

Figure 13.6. Spatial interpolation function around node k, composed of composite of local shape functions within adjoining triangular elements.

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