Civil Engineering Reference
In-Depth Information
Figure 3.67
Planar domain and its bounding box.
3.8.1 Quadtree partition of the bounding box
Similar to the classical approach, given the planar domain and the node spacing specifica-
tion, the first step of the enhanced scheme is to subdivide the rectangular bounding box into
quadrilaterals by means of the classical Quadtree decomposition. This can be achieved using
the recursive subdivision method described in Section 3.4, and the result of the Quadtree
partition of the bounding box is shown in Figure 3.67. It is seen that element concentrations
are found at the interior of the domain and around the interior boundary of the planar
domain.
3.8.2 Removal of quadrilaterals near domain boundary
The next step is to remove those quadrilaterals that are outside the bounded region or
too close to the domain boundary. Quadrilaterals that are too close to the boundary can
be removed simply by checking the distance to the boundary segments. Suppose that the
domain boundary consists of N
b
line segments; then quadrilateral Q will be removed if
min
,
(distancebetween Qand segmenti) izeofQ
<
i
=
1
Nb
where distance is the distance between the centre of Q and the i
th
line segment under con-
sideration, as discussed in Section 2.4.1, and the size of Q can be taken as the longest edge
of Q. Upon the removal of the quadrilaterals near the boundary by the distance check, the
remaining quadrilaterals are well away from the boundary line segments.
However, those quadrilaterals that are outside the domain boundary still have to be
removed. The Inside_or_Outside check provided in Section 2.4.11 can be applied here to
determine which quadrilaterals are to be retained. A more consistent treatment is to find
out the status of one quadrilateral and determine the patch of quadrilaterals by means of
the neighbouring relationship. Whichever method is used, the Quadtree partition within the
planar domain upon the removal of outside and nearby quadrilaterals is shown in Figure
3.68. It is observed that the quadrilaterals are all removed over regions of large element size,
showing that quadrilaterals cannot be easily fitted to an irregular boundary, and it is also
an inherent difficulty of Quadtree decomposition for a systematic and strategic positioning
and orientation of the grid with respect to the given domain.
