Chapter 6
The Shell Element Method

The surfaces in exterior problems may not be that of a substantive body such as a sphere but may be a thin shell-like open surface. Often the width of the shell is much less than the size of a reasonable panel. For example a metallic conducting square plate of dimensions 1m × 1m × 0.001m in a three-dimensional dielectric. It could require panels with sides of 0.01m to describe the solution on the upper and lower surfaces (400 triangles), but such panels would be much too large to model the edge of the square. Alternatively, if panels with sides of size 0.001m are used around the edge of the square then an extra 8000 triangles would be requires around the edge of the square. Having done that the triangles that make up the shell(s) are of disproportionate sizes and using such a mesh would probably result in numerical errors. However if a uniform mesh is used then 40000 triangles would be needed to describe the upper and lower surfaces. the whole surface would require 48000 triangles.

A two-dimensional illustration of the sort of (open) surface that we are interested in is shown in figure 6.1. For modelling purposes it is generally better to assume that such a surface is infinitessimally thin; having no width at all. The shell represents a discontinuity of potential or its derivative.

Figure 6.1. An open 2D surface.

There is an alternative way of applying the boundary element method to the problem of computing the solution in the domain surrounding such a surface. It involves completing the surface as illustrated in figure 6.2 and solving the resulting interior and exterior problems simultaneously; coupling the numerical solutions across common boundaries. However, this involves introducing a fictitious surface (the dotted line in figure 6.2). This increases the number of elements that are required to represent a problem.

Figure 6.2. Preparation of the completed surface for the traditional BEM.

In this chapter the problem of solving the Laplace equation

Ñ2 f = 0
in the region exterior to an open surface G is addressed. The field is allowed to be discontinuous ot the shell surface; the potential generally takes different values on the upper and lower surfaces of G f+(q) and f-(q) (q Î G). Also the derivative of the potential with respect to the normal n (in the - to + surface direction) also generally takes different values on either side of G, denoted v+ = [(f+)/( n)] and v- = [(f-)/( n)]. In order to describe the shell conditions, is also helpful to introduce further shell functions which represent the averages and differences of the potential and its normal derivative across G: d(p) = f+(p) - f-(p), F(p) = 1/2 ( f+(p) + f-(p)), n(p) = v+(p) - v-(p), V(p) = 1/2 ( v+(p) + v-(p)). The shell conditions are structured as follows

a(p) d(p) + b(p) n(p) = f(p)
(91)
and

A(p) F(p) + b(p) V(p) = F(p)
(92)
for p Î G.

Shell elements enable us to solve the Laplace equation directly through representing the open surface alone by a set of panels. The method is based on an integral equation reformulation of the Laplace equation exterior to a thin shell. Only the open surface itself needs to be described as a set of panels with the potential and its normal derivative allowed to take different values at either side of each panel.

The subroutine LSEM3 has been applied to the simulation of electrostatic capacitors in Kirkup [19].

6.1  Integral Equation Formulation

In this section the integral equation formulations of the Laplace field surrounding a set of shells is stated. Further details on the derivation of these equations are given in Warham [25]. However in the formulations that follow the possibility of an incident field is also included.

The potential in the domain is related to the conditions on the shell by the following equation

f(p) = f*(p)+{M d}G(p) - {L n}G(p)      (p Î E).
(93)
For points on the shell we have the following equations

F(p) = f*(p) +{M d}G(p) - {L n}G(p)      (p Î G) ,
(94)

V(p) = v*(p) +{N d}G(p) - {Mt n}G(p)      (p Î G) .
(95)
where f* is the incident field and

V*(p) = f*(p)/ np    (p Î G).

Given the shell conditions d and n can be obtained on the shell by solving (94) and (95). Once approximations to these functions have been obtained, the value of f for any point in the domain can be obtained from equation (93).


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