Civil Engineering Reference
In-Depth Information
or if both these two sub-fronts consist of an even number of segments, quadrilateral
ABCL can be formed. Otherwise, triangle CBR will be considered. In case R divides
the merging front into sub-fronts having an odd number of segments, go to step v.
iv. Triangle ABC divides the merging front into two sub-fronts SFL on the left and SFR on
the right, as shown in Figure 3.79b. As there are always an even number of segments
on a merging front, excluding segment AB, one sub-front should have an even num-
ber of segments and the other sub-front an odd number of segments. To maintain an
even number of segments on each boundary loop, the triangle selected to merge with
triangle ABC is the one lying on the sub-front having an odd number of segments, as
shown in Figure 3.79b.
v. The triangle CBR is split into three triangles by inserting a point at its centroid D, as
shown in Figure 3.79c. Diagonal BC between triangles ABC and CBD is removed to
create quadrilateral ABDC, and the merging front is split into two sub-fronts of even
number of segments.
3.9.4.3 Updating merging front
Whenever a quadrilateral is constructed, the merging front has to be updated in a way simi-
lar to the update in the AFT, as described in Section 3.6.2. The segments that have been
used to form the sides of the new quadrilateral element will be deleted from the merging
front, and new edges created in the merging process will be added to the merging front, as
shown in Figure 3.79.
3.9.4.4 Complete conversion to quadrilateral mesh
Steps i to v are repeated to convert triangles into quadrilaterals until the number of segments
in the merging front is reduced to zero. By this time, all the triangular elements are con-
verted to quadrilaterals. It is remarked that by selecting a triangle for merging with the base
triangle as described in steps iii to v, all the merging fronts and sub-fronts will, at any time,
remain as closed loops of an even number of line segments. The flowchart for the merging
of a triangular mesh into a quadrilateral mesh is shown in Figure 3.80.
3.9.5 Mesh quality enhancement
Some of the quadrilaterals produced by the mesh merging algorithm may be distorted, and
elements with internal angles greater than or equal to 180° may be generated. Hence, mesh
quality enhancements are required to bring the mesh quality up to an acceptable level.
Mesh quality improvement can be achieved by geometrical node shifting or by topological
modification on the ways elements are connected (Canann et al. 1998). Mesh enhancement
by node shifting and topological modification will be discussed in detail in Chapter 6, and
in the following, some simple topological operations pertinent to quadrilateral meshes will
be discussed.
In most cases, the information in judging whether a swap would improve the overall qual-
ity of the mesh is the number of elements connected to a node. This information has to be
prepared beforehand, and an algorithm has already been presented in Section 2.5.3 to find
out the elements connected to each node of the mesh. The number N, which is the number
of elements surrounding a given node, indicates the degree of uniformity of the quadrilateral
mesh. When N equals to four for most of the interior nodes, the mesh is highly uniform.
However, if N is much greater or less than four, severe distortion will result. Therefore, it is
