Civil Engineering Reference
In-Depth Information
Four-node
elements
RL = 0
Five-node
elements
RL = 1
Six-node
elements
RL = 3
Seven-node elements
RL = 2
Figure 5.105
Transition quadrilateral elements.
analysis (Lo et al. 2010). In 2D, quadrilaterals of four to seven nodes have to be used so that
large elements can be connected to smaller elements without a change in shape, as shown in
Figure 5.105. The one-level restriction (1-LR) rule between adjacent elements will be enforced
to ensure a gradual change in element size, and the types of transition quadrilaterals will be
limited to only four, namely, four-, five-, six- and seven-node quadrilaterals.
5.8.14 Generation of transition quadrilateral mesh
In the 1-LR mesh for a 2D adaptive refinement, each edge of a 2D transition element can
interface with at most two adjacent elements, as shown in Figure 5.105. Neighbouring ele-
ments refer to elements that share at least two common nodes. Hence, elements sharing
a common edge or part of a common edge are neighbouring elements. Furthermore, the
refinement level (RL) of an element is the number of refinements done to attain the current
configuration or size of the element. Thus, in the initial mesh in which no subdivision has
been performed, RL = 0 for all the elements. As the refinement process goes on, when an
element is indicated to be refined, it is necessary to check the RLs of its neighbouring ele-
ments. If their values are equal to or greater than the element's own RL, the element can be
subdivided into smaller elements, and their RLs are increased by one. Otherwise, the ele-
ment cannot be subdivided until its neighbours are subdivided first. This RL check can be
done locally and is easily implemented in code by a recursive sub-routine.
A regular element, which refers to a four-node quadrilateral element, can only be divided
into four smaller regular elements by adding four mid-side nodes and a centroidal node,
as shown in Figure 5.105. Elements of different RLs are connected by transition elements,
which are quadrilateral elements of five, six and seven nodes with a different number of
mid-side nodes, as shown in Figure 5.105. The following is an example of mesh refinement
with transition quadrilaterals following the 1-LR refinement rule. Mesh i in Figure 5.106
shows the initial mesh of only four-node quadrilaterals. All their RLs are set to zeros. Now
we try to subdivide element 1. Before doing so, the RLs of its neighbouring elements, i.e.
elements 2, 3 and 4, are checked to ensure that the 1-LR mesh refinement rule is satisfied. By
dividing element 1, four new quadrilateral elements are generated, and their RLs are equal
to one, as shown in mesh ii of Figure 5.106. Elements 2 and 4 become transition elements of
five nodes, and yet their RL remains to be zero. In the next stage of refinement, element 5 is
supposed to be refined. The RLs of its neighbouring elements, i.e. elements 1, 2 and 6, are
