Civil Engineering Reference
In-Depth Information
computed by subtracting from the β value the β values of its neighbouring edges. Thus, the
β* value is an overall measure on the value of the diagonal and the influence of its removal
on neighbouring edges. The diagonal that bears the highest β* value will be the first to be
removed. After the removal of a diagonal and the generation of a quadrilateral, the mesh as
well as the β* values of the neighbouring edges have to be updated. Removal of diagonals
continues as long as there is a diagonal whose β value is greater than the prescribed γ value.
For a given triangulation, more quadrilateral elements will be produced for a small γ value
and vice versa; and in the extreme case that γ = 1, no diagonal will be removed. The genera-
tion process stops when no more edges can be removed, or the β values of all edges fall below
γ. The resulting mesh will be a mixed mesh of triangular and quadrilateral elements.
Definition
Recall the dimensionless α-quality of a triangle,
signed area
sumofedgesquared
CA
×
CB
α(
ABC
)
=
=
23
2
2
2
CA
+
AB
+ BC
3.9.3.1 Distortion coefficient
β
of a quadrilateral
Four triangles can be obtained by cutting the quadrilateral ABCD along the diagonals AC
and BD, as shown in Figure 3.73. Let α
1
, α
2
, α
3
and α
4
be the α values of the triangles arrang-
ing in a descending order of magnitude, i.e.
{α
1
, α
2
, α
3
, α
4
} = {α(ABC), α(ACD), α(ABD), α(BCD)} and α
1
≥ α
2
≥ α
3
≥ α
4
Definition
The distortion coefficient β of quadrilateral ABCD is given by
αα
αα
34
12
β
(ABCD) =
β would lie between 0 and 1 for a convex (valid) quadrilateral. For rectangles, β attains
a maximum value of 1, whereas for quadrilaterals degenerated into triangles, β approaches
0. The higher the β value, the better the shape of the quadrilateral relative to a rectangle.
Hence, quadrilaterals can be preferably generated by removing diagonals having large β
values. The β values of some typical quadrilateral are shown in Figure 3.74.
D
C
A
B
Figure 3.73
Quadrilateral ABCD.
