Boundary Integral Equations - The Boundary Element Method with Programming - page 122

The error in the temperature at segment 1 versus the number of elements (collocation
points) is plotted in Figure 5.14. It can be seen that the error falls below 1% for 24
elements. A plot of the heat flow in vertical direction along a horizontal line depending
on the number of segments is shown in Figure 5.15. The theoretical value of q y should
approach the value of 2.0 exactly on the boundary. It can be seen that as we get very
near to the boundary the values are significantly in error and that this error depends on
the element size adjacent to the interior point. As we will see later, this is typical of the
boundary element method and will be more pronounced when numerical integration is
used. However with the higher order elements introduced next we will see that results
exactly on the boundary can be computed with an alternative method. Figure 5.16 and
5.17 finally show the graphical display of the results as it may be produced by a
postprocessor. Figure 5.16 shows the contours of the temperature distribution whereas in
Figure 5.17 the flow vectors are depicted by arrows whose magnitude depends on the
value of heat flow. It can be seen that the temperature contours align normal to the
boundary as they should and that the flow vectors approach zero values at the bottom
and the top of the circular isolator.

Figure 5.17 Flow past a cylindrical isolator: flow vectors

5.7 CONCLUSIONS

In this chapter we have introduced the Trefftz and boundary integral equation methods.
Although we found that the Trefftz method is not suitable for general purpose
programming it can be used to demonstrate the basic principles involved, because of its
simplicity. A short program can be written and used for numerical experiments. As the