Civil Engineering Reference
In-Depth Information
the result is
∂
∂
1
8
a
(1
1
8
a
(1
x
=
−
s
)(1
+
t
)(
2
−
1
)
+
+
s
)(1
+
t
)(
3
−
4
)
1
8
a
(1
1
8
a
(1
+
−
s
)(1
−
t
)(
6
−
5
)
+
+
s
)(1
−
t
)(
7
−
8
)
(6.74)
Referring to Figure 6.19, observe that, if the gradient of the field variable in the
x
direction is constant,
∂
/∂
x
C
,
the nodal values are related by
2
=
1
+
∂
∂
=
d
x
=
1
+
C
(2
a
)
x
3
=
4
+
∂
∂
d
x
=
4
+
C
(2
a
)
x
(6.75)
6
=
5
+
∂
∂
d
x
=
5
+
C
(2
a
)
x
7
=
8
+
∂
∂
d
x
=
8
+
C
(2
a
)
x
Substituting these relations into Equation 6.74, we find
∂
∂
1
8
a
[(1
x
=
−
s
)(1
+
t
)(2
aC
)
+
(1
+
s
)(1
+
t
)(2
aC
)
+
(1
−
s
)(1
+
t
)(2
aC
)
+
(1
+
s
)(1
−
t
)(2
aC
)]
(6.76a)
which, on expansion and simplification, results in
∂
∂
x
≡
C
(6.76b)
Observing that this result is valid at any point (
r
,
s
,
t
) within the element, it
follows that the specified interpolation functions indeed allow for a constant gra-
dient in the
x
direction. Following similar procedures shows that the other partial
derivatives also satisfy the completeness condition.
6.8 ISOPARAMETRIC FORMULATION
The finite element method is a powerful technique for analyzing engineering
problems involving complex, irregular geometries. However, the two- and three-
dimensional elements discussed so far in this chapter (triangle, rectangle, tetra-
hedron, brick) cannot always be efficiently used for irregular geometries.
Consider the plane area shown in Figure 6.20a, which is to be discretized via a
mesh of finite elements. A possible mesh using triangular elements is shown in
Figure 6.20b. Note that the outermost “row” of elements provides a chordal ap-
proximation to the circular boundary, and as the size of the elements is decreased
(and the number of elements increased), the approximation becomes increasingly
