In this chapter it has been shown how boundaries in each dimensional setting can be represented by two data structures. The square as a set of straight line panels and the sphere represented by triangles and axisymmetric cone panel are used throughout the remainder of this series to demonstrate the boundary element methods. In the subsequent work in this series the two data structures that represent the relevant structure are passed as array parameters to the subroutines. By setting the validation parameter LVALID=.TRUE. in the subroutines a check is also made to ensure that the boundaries are closed in the BEM.
The subdivision of the square and the sphere into triangles enables us to simulate general Helmholtz problems in two- and three- dimensions. However, the axisymmetric elements are uniform when rotated about the z-axis. Conical elements should only be used when the Helmholtz field as well as the surface is known to be axisymmetric.
In the 2D example of the square in Figure 2.3, the panels are of uniform length. For the sphere illustrated in Figure 2.4 the triangles are of approximately equal size. As a general rule, the element sizes for general two- and three-dimensional boundaries should be as close to uniform as possible across the boundary. Moreover, in the three-dimensional case, the triangles should not deviate too far from the equilateral shape. For axisymmetric surfaces, the lengths of the generator of the elements should be as close to uniform size as possible. If the validation parameter is set LVALID=.TRUE. then the input panels are checked to ensure that their sizes are reasonably uniform and the input boundary is checked to ensure it does not contain sharp angles.
In some cases it is wise to overrule the general guidelines of the previous paragraph. For example if a boundary has an intricate shape in a localised area it would be beneficial to use more elements in that region. In other cases it may be known that the potential is strongly varying in some areas of the surface - for example in the neighbourhood of a sharp corner the potential can be singular - and in these regions it is often beneficial to increase the number of elements.
The approximation methods used are very simple, but each is sufficient to approximate the boundaries in each class of domain. For less straightforward geometries, the boundary would be better represented by curved panels but these are not considered in this series.