Page 209 - Basic Structured Grid Generation

198 Basic Structured Grid Generation

Equating these expressions for area, it is easy to show that the aspect ratio of the
triangle may be expressed in terms of the lengths of the sides as
abc
A.R. = . (8.5)
8(s − a)(s − b)(s − c)
For an equilateral triangle it follows that A.R. = 1. Clearly, the nearer the three
vertices of a triangle approach to lying on a straight line, the closer will s approach to
the values a, b,or c, and the larger the value of the aspect ratio will become. In that
sense the aspect ratio is a measure of the ‘skinniness’ of a triangle.
The following algorithm makes use of the above ideas:
Step 1 – Generate an initial triangular grid using boundary data points and the
Bowyer-Watson algorithm.
Step 2 – Apply Rule 1 to eliminate all triangles having an area larger than 1.5 times
the area of the equilateral triangle formed by the largest boundary segment between
two data sets. This step will eliminate all large triangles.
Step 3 – Apply a combination of Rules 2 and 3 to eliminate triangles with large
circumcircle radius and high aspect ratio. It may be convenient to leave alone triangles
near the boundary which are small but have a high aspect ratio (by applying Rule 2
only). The criterion for high aspect ratio is empirical. Often a critical aspect ratio of
1.5 is used; this would be the aspect ratio of isosceles triangles with apex angles of
about 24 or 104 degrees. This criterion can be applied to eliminate all triangles of high
aspect ratio, or until all triangles with high aspect ratio have an area of less than twice
the area of an equilateral triangle with sides equal to the minimum boundary-point
spacing.
However, when this algorithm is adopted, three situations may arise when point
insertion at circumcentres of the selected triangles must be rejected, namely:
Case 1 – The circumcentre of the selected triangle does not lie within the solution
domain.
Case 2 – The circumcentre of the selected triangle is too close to the boundary of
the solution domain.
Figure 8.12 shows an example of a solution domain with inner and outer boundaries
and a selection of boundary points (circled). The insertion of point B leads to a triangle
ABC of high aspect ratio. However, the circumcentre P of ABC lies outside the solution
domain and must be rejected. As an example of Case 2, insertion of point D leads to
a triangle EDA which may be of large area. The circumcentre Q is, however, close to
the boundary. The criterion for rejection often taken is that the distance of the point
from the boundary is less than one-third of the length of the nearest boundary segment.
In this case rejection would apply if QN < 1 EA.
3
Control of grid-density A drawback of the above method is that the grid-density of
triangulation over the whole (planar) domain is controlled by the grid-density along the
boundary of the domain. As a result, the size of triangles far from the boundary may
turn out to be unsatisfactory for the numerical solution of partial differential equations
in the domain.
To overcome this problem, it may be convenient to introduce a function of position
f(x) which specifies the required value of a characteristic dimension of a triangle, say
its circumradius, at a location x in the domain. For a triangle with circumradius R and