Civil Engineering Reference
In-Depth Information
1 and vary ξ to generate two sets of five contour lines. The intersection of these two sets of
contour lines will produce the required sub-domains, as shown in Figure 3.2.
As for triangular domains, that is, regions bounded by three straight lines and/or curved
edges, the triangular element interpolations have to be used. The interpolation functions for
linear triangular element T3 are given by
H
1
= L
1
, H
2
= L
2
, H
3
= L
3
,
where L
1
, L
2
and L
3
are the area co-ordinates, as shown in Figure 3.3.
The interpolation functions for the quadratic triangular element T6 are given by
H
1
= L
1
(2L
1
− 1),
H
2
= L
2
(2L
2
− 1),
H
3
= L
3
(2L
3
− 1),
H
4
= 4L
2
L
3
, H
5
= 4L
3
L
1
, H
6
= 4L
1
L
2
A point
p
in the reference domain will divide up the triangle into three zones by joining
the point with the corners of the triangle, as shown in Figure 3.3. The area co-ordinates of
point
p
are given by the ratios of the areas of the three zones so formed to the total area of
the triangle.
Δ
Δ
Δ
Δ
Δ
Δ
1
2
3
L
=
,
L
=
,
L
=
,
where
ΔΔ ΔΔ
= ++
1
2
3
1
2
3
x
7
η
2 × 2 square
x
3
x
4
x
8
4 × 4 mesh
x
6
ξ
x
H
x
1
x
2
x
5
Figure 3.2
Q8 curved element divided into sub-elements.
L
2
x
3
x
2
x
1
x
x
5
x
4
H
∆
1
x
∆
3
x
2
p
x
3
∆
2
x
1
L
1
x
6
L
3
Figure 3.3
Triangular mesh by finite element interpolation.
