2.1  Subdivision of the Boundary into Panels


Let S be the original boundary and DS'j ( for j = 1,2,..,n) be the panels that represent an approximation to S in the boundary element method. If S' = åj = 1n DS is the surface described by the complete set of panels then S' is the approximation to S (that is S' » S). The representation of the boundary in this way is the first step in the discretisation of the integral operators that occur in the boundary integral formulation of the Helmholtz equation. Since every panel (in each particular dimensional space) has a similar characteristic form, the integration over each panel can be generalised. This function is carried out by the subroutines H2LC , H3LC and H3ALC for the two-, three- and axisymmetric three dimensional problems respectively and these subroutines form the core modules for each of the interior, exterior and modal Helmholtz problems. A full description of the methods employed in the discretisation of the operators by these subroutines is given in Chapter 3.

The representation of the boundary by a set of characteristic panels enables us to easily define the boundary using a data structure. For example an ordered list of the coordinates of the vertices of the approximating polygon in Figure 2.1 defines the boundary. As an illustration, the boundaries of the test problems are explicitly stated in this manual. The characteristic panels that are used in 2D, 3D and axisymmetric problems are illustrated in Figure 2.2.

Fig 2.2. The straight line, planar triangle and truncated cone panels.

Part of the numerical error in the boundary element solution will be a result of the approximation of the boundary. A better boundary approximation and a smaller numerical error will generally arise if the boundary is represented by curved panels. The methods described in this work apply to only the simplest panels, straight line panels for two-dimensional boundaries, planar triangles for general three-dimensional boundaries and truncated conical panels for axisymmetric problems, as shown in Figure 3.2. In this manual we consider the representation of the boundary S in terms of these panels in each dimensional space. The boundary can be expressed by two data structures in each case; the first enumerating the vertices and storing their coordinates, the second lists the individual panel by indicating the two or three vertices that define each panel.


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