Civil Engineering Reference
In-Depth Information
8
16
50
2
5
100
Figure 2.19
Dividing edges into segments.
2.4.9.2 Element size is specified along the edge
Apart from the nodal values specified at the ending nodes, if in addition the nodes have to be
distributed along the edge following a specified spacing function, then a more sophisticated
method has to be devised to generate nodes in compliance with the required node spacing.
A set of new nodes has to be generated along the segments on the boundary
B
, such that
the node distribution along the boundary segments complies with the given node spacing
function ρ prior to MG:
New nodesbased on
ρ
B
=
{,
i
i
=
1
,
N
}
→= =
Γ
{,
Li
1
,
N
b
}
B
i
where N
B
is the number of segments on the original domain boundary, and N
b
is the number
of segments after node generation along the segments in
B
.
The domain boundary
B
is first decomposed into straight lines composed of one or more
boundary segments, each of which is bounded by two consecutive sharp corners (turns)
found on boundary
B
. The task of boundary discretisation is to generate intermediate nodes
along straight edges retrieved from
B
in compliance with the given node spacing function.
A simple method assuming a linear node spacing variation between nodes is adopted for
generating intermediate nodes on a straight line with n nodes and (n − 1) segments, as shown
in Figure 2.20. In order to avoid any complicated node generation process involving integra-
tion and iteration (Frey 1987), an explicit formula that allows the position of new nodes to
be calculated one by one along the straight line is employed.
With the exception of corner nodes and nodes where there is a sharp change of direction,
nodes are all regenerated along each straight edge of
B
consisting of one or more line seg-
ments according to the given node spacing function. Depending on the given node spacing
function, the number of segments N
b
of the domain boundary after node regeneration may
ρ
ρ
n
ρ
1
ρ
2
Figure 2.20
Linear node spacing variation between nodes.
