Civil Engineering Reference
In-Depth Information
β < 0
β = 1
β = 0
β = 0.5898
β = 0.4
β = 0.36
45
º
75
º
60
º
Figure 3.74
β
values of typical quadrilaterals.
3.9.3.2 Merging of triangles to form quadrilaterals
Triangular elements can be converted to quadrilateral elements by removing the diagonal
between a pair of adjacent triangles. The resulting mesh consists of both quadrilaterals and
triangles, and their proportion is governed by a user-specified parameter γ. The diagonal
removal and the forming of quadrilateral elements can be carried out by the following steps.
1.
Retrieval of unique lines (edges) from the triangular mesh
The list of edges and the two triangles connecting to each edge are needed to set up the
priority sequence in the removal of diagonals between adjacent triangles. An efficient
method to retrieve edges from a triangular mesh has been discussed in Section 2.5.4,
and with little care, the triangles attached to each edge can also be identified.
2.
Removal of diagonals
Before the removal of edges, a check is made on the removability and the β value of
each edge. In general, one can only remove those lines shared by two triangles, and
these removable edges will be referred to as diagonals. Hence, at the beginning of the
merging process, all boundary segments have to be flagged so as not to be removed. To
determine which diagonal is to be removed, the β* value of each diagonal is computed,
which is equal to the β value of the diagonal minus the β values of the neighbouring
diagonals. The diagonal having the greatest β* value will be removed, and a quadri-
lateral element will be generated. If no more edges can be removed and the β values of
the diagonals are less than the prescribed γ value, the merging process terminates.
3.
Updating the mesh
Whenever a quadrilateral is generated, the mesh has to be updated following steps i to
iii:
i. Suppose that line AC is to be removed as shown in Figure 3.75. The two triangles
ABC and ACD are identified and deleted from the mesh, whereas a quadrilateral
element ABCD is added to the mesh.
ii. The edges associated with the quadrilateral, i.e. the four edges of the quadrilateral
AB, BC, CD and DA, have to be flagged so that these edges will not be removed as
the merging process continues.
iii. The β* values of the neighbouring edges are updated; at most eight lines will be
affected, as shown in Figure 3.75. Return to step 2 for the creation of more quads.
The DT of 1000 randomly generated points is shown in Figure 3.76a. The merging algo-
rithm is applied to the triangular mesh with γ = 0.9, and the resulting mesh is shown in Figure
3.76b. It is seen that with a high γ value of 0.9, very few quadrilaterals closed to rectangles are
