Civil Engineering Reference
In-Depth Information
Level of subdivision = 2 > 1
Figure 3.11
Level of subdivision between neighbouring cells.
To take the one-level Quadtree decomposition properly as a mesh refinement problem of
transition quadrilateral elements, the types of quadrilateral and their node-labelling scheme
have to be identified, as shown in Figure 3.12. In the most general situation, there are only
six types of quadrilaterals in a transition quadrilateral mesh satisfying the one-level refine-
ment restriction, in which the eight-node element can be divided into four four-node ele-
ments once it is formed.
As for the neighbouring relationship of the transition FE mesh, a four-node quadrilateral
can have four neighbours along the edges 1-2, 2-3, 3-4 and 4-1, respectively. In case no
neighbour exists on an edge, i.e. the edge is only connected to one single quadrilateral, then
it is a boundary edge. Quadrilateral elements with more nodes can join with more neigh-
bours; for instance, the seven-node quadrilateral can have as many as seven neighbours,
which are numbered sequentially on edges 1-5, 5-2, 2-6, 6-3, 3-7, 7-4 and 4-1. As usual,
the number zero can be used to stand for no neighbour (boundary edge), and a positive
integer represents a valid neighbour to that edge. In subdividing a transition quadrilateral
element, depending on the element type, the number of nodes added may vary, and the
neighbouring relationship has to be updated according to the types of quadrilaterals con-
nected to the element.
Let's take a look at how a quadrilateral is subdivided in a mesh and how the adjacency rela-
tionship of the relevant elements could be updated. A patch of four four-node quadrilaterals
4
3
4
3
4
3
6
1
2
1
5
2
1
5
2
Four-node
Five-node
Six-node
4
6
3
4
7
3
4
7
3
6
8
6
1
5
2
1
5
2
1
5
2
Six-node
Seven-node
Eight-node
Figure 3.12
Family of transition quadrilateral elements.
