Civil Engineering Reference
In-Depth Information
Figure 3.16
Templates to triangulate quadrilaterals.
3.4.3 Creation of internal points and elements
In the earlier classical application of the Quadtree meshing, the refined quadrilateral ele-
ments are subdivided into triangular elements to ensure compatibility across the element
boundary by means of a template, as shown in Figure 3.16. In the Quadtree mesh with one-
level restriction, there are 2
4
= 16 cases to be considered, which could be reduced to only 6
using various symmetries (Yerry and Shephard 1983). At the turn of the twenty-first cen-
tury, efficient transition quadrilateral elements were developed (Lo et al. 2005, 2010), and
the refined quadrilateral mesh is a valid FE mesh in its own right with full compatibility on
the element boundaries. The transition element technology allows the interior part to be
kept intact without any further treatment, and what remains to be done is to merge it with
the boundary segments in the final step.
3.4.4 Connection of the interior elements
with the boundary segments
In the traditional Quadtree meshing, quadrilaterals cut across by boundary segments are
divided into triangles, which is achieved in two steps. The first step is to determine all
the intersections between the boundary segments and the quadrilaterals in the Quadtree
partition. In the second step, a set of templates for the decomposition of a quadrilateral is
prepared to subdivide the intersected quadrilaterals into triangles. However, in general,
the number of intersections of a quadrilateral cannot be pre-determined, and some bound-
ary segments may also align in the principal directions of the Quadtree grid resulting in
very narrow pointed triangles. In order to reduce the number of cases of the intersection
to a manageable limit, intersection points have to be snapped onto the
4
positions of the
edges of the quadrilaterals. Some of the typical divisions of a quadrilateral into triangles
are shown in Figure 3.17, and for a detailed discussion of the templates and the sub-
sequent connections into triangles, readers can refer to the original work of Yerry and
Shephard (1983).
There are a number of drawbacks in the triangulation of quadrilaterals by means of
templates: (i) The boundary integrity may be jeopardised as intersection points have to be
snapped onto quarter points; (ii) determination of intersections is a tedious process prone
to numerical errors; (iii) the position and the orientation of the Quadtree grid with respect
to the given boundary segments are rather arbitrary; and (iv) narrow pointed triangles
Figure 3.17
Templates for division of quadrilateral at quarter points.
