Civil Engineering Reference
In-Depth Information
η
2 × 2 square
x
3
m × n mesh
x
4
ξ
x
n
H
x
1
x
2
m
Figure 3.1
Quadrilateral mesh by finite element interpolation.
where
x
i
= (x
i
, y
i
), i = 1,4, are the nodal points at the corners, and H
1
= H
1
(ξ, η) ξ, η ∈ [−1,
1], are interpolation functions of the Q4 element, which are given by
1
4
1
4
H
=−−
(
11
ξη ξη
)(
),
H
= +−
(
11
)(
),
1
2
1
4
1
4
H
=++
(
11
ξ
)(
η
),
H
4
=−+
(
11
ξ
)(
η
)
3
The grid lines dividing the domain into smaller quadrilaterals can be constructed by
setting a co-ordinate to constant and varying the other co-ordinate between −1 and 1. For
example, a grid line along the ξ-co-ordinate can be generated by setting η = constant and
varying ξ between −1 and 1. Similarly, to generate a grid line along the η-co-ordinate, sim-
ply set ξ = constant and vary η between −1 and 1. The intersection of these two sets of co-
ordinate lines will produce an m × n mesh of quadrilateral elements, where m and n are,
respectively, the number of subdivisions along the ξ and η directions, as shown in Figure 3.1.
A quadrilateral domain composed of boundary segments of parabolic curves can be rep-
resented by means of higher-order quadrilateral FEs, named by Zienkiewicz and Phillips
(1971) as super-elements. In terms of the quadratic Serendipity element Q8, a curved bound-
ary quadrilateral domain is given by
x
∑
H
ii
i1
8
, where
x
i
are the nodal points at the cor-
ners and on the mid-sides of a curved edge, and Hi
i
= H
i
(ξ, η) ξ, η ∈ [−1, 1] are interpolation
functions of the Q8 element given by
=
x
1
4
1
4
H
=−−−−− =+−−−
(
11
ξηξη
)(
)(
1
),
H
2
(
11
ξηξη ,
)(
)(
1
)
1
1
4
1
4
H
=+++−
(
11
ξηξη
)(
)(
1
),
H
= −+−−
(
11
ξηηξ1
)(
)(
),
3
4
1
2
1
2
2
2
H
=+−
(
1
ξ η ηξ
)(
1
),
H
=
(
1
−
)(
1
+
),
5
6
1
2
1
2
2
2
H
=
(
1
−
ξ η ηξ
)(
1
+ =− −
),
H
8
(
1
)(
1
).
7
Again, the grid lines can be generated by setting one co-ordinate constant and varying the
other co-ordinate between −1 and 1. For instance, to divide the original curved domain into
4 × 4 = 16 sub-domains, set ξ at −1, −0.5, 0, 0.5, 1, and vary η then set η at −1, −0.5, 0, 0.5,
