Dealing with corners and changing geometry - The Boundary Element Method with Programming - page 323

12.2 CORNERS AND EDGES

A number of schemes for dealing with the problem of sharp corners, where the solution
for t is not unique have been proposed in the past.

t

t

r

Region II

Region I

Figure 12.1 Example of a multi-region problem with corners

The following are some methods that have been suggested:
x Numerically round off the corner by using an average outward normal, i.e., an
average of all normal vectors of elements connecting to the node. This is not really
correct, as the geometry of the element should be rounded off too.

x The unknown values of t are computed by extrapolation from the nodes adjacent to
the corner node 1 . This method is not difficult to implement but its accuracy would
greatly depend on the size of the boundary elements adjacent to the corner.

x Use of auxiliary equations 2,6 based on stress symmetry and on the differential
equation of equilibrium to compute extra values of t . Another method to solve the
problem is based on derivations of potentials or displacements on each side of the
corner to get the appropriate number of equations for solving multi valued flow or
tractions 3 .
x Use discontinuous elements 4 introduced in section 3.7.2. Here t is actually not
computed right at the corner but slightly inside the element. Therefore two sets of t
may be computed at each side of the corner.

The approaches which add auxiliary equations to the system of equation have been
implemented and tested 6 , and it has been found that with a careful implementation they