## 2.2 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' = å_{j = 1}^{n} 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
Laplace 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
L2LC , L3LC and L3ALC 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 Laplace 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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