Page 208 - Basic Structured Grid Generation

Unstructured grid generation 197

then be placed at their circumcentres and re-triangulation carried out until the grid is
satisfactory.
The following rules for the selection of forming triangles may be used:

1. The triangle with the largest area is selected. This rule works well when the initial
triangulation is based on a set of boundary data points that are distributed in a fairly
uniform manner. However, when this does not apply, the rule when used on its own
does not generally produce good grids.
2. The triangle with the largest circumcircle radius is selected. This rule has the desir-
able feature that it will eliminate triangles with obtuse angles, together with triangles
that are ‘skinny’ and have small area (so that they would be missed by Rule 1).
3. Obtuse-angled and skinny triangles are targeted directly. To this end a criterion for
recognizing skinny triangles is required. One such is provided by consideration of
the aspect ratio of a triangle.

The aspect ratio A.R. of a triangle ABC (Fig. 8.11) is defined as R/2r,where R is
the radius of the circumcircle and r is the radius of the inscribed circle. A standard
formula for the area of a triangle is

area = s(s − a)(s − b)(s − c) (8.1)
where
1
s = (a + b + c). (8.2)
2
The following formulas are also straightforward to establish:

1
area = (a + b + c)r = sr (8.3)
2
and (making use of the Sine Rule for triangles)

abc
area = . (8.4)
4R

C

R

b a
r

B
A c

Fig. 8.11 Circumcircle and inscribed circle of triangle ABC.