Civil Engineering Reference
In-Depth Information
3
4
x
3
x
4
x
5
y
(3)
(
x
4
)
y
(4)
(
x
4
)
d
y
(3)
d
x
d
y
(4)
d
x
x
4
x
4
Figure 5.5
Two elements
joined at a node.
terms are not, in general, zero. Nevertheless, in the assembly procedure, it is
assumed that, at all interior nodes, the gradient terms appear as equal and oppo-
site from the adjacent elements and thus cancel unless an external influence acts
at the node. At global boundary nodes however, the gradient terms may be spec-
ified boundary conditions or represent “reactions” obtained via the solution
phase. In fact, a very powerful technique for assessing accuracy of finite element
solutions is to examine the magnitude of gradient discontinuities at nodes or,
more generally, interelement boundaries.
EXAMPLE 5.5
Use Galerkin's method to formulate a linear finite element for solving the differential
equation
d
2
y
d
x
2
d
y
d
x
−
x
+
4
x
=
0
1
≤
x
≤
2
subject to
y
(1)
=
y
(2)
=
0
.
■
Solution
First, note that the differential equation is equivalent to
d
d
x
x
d
y
d
x
−
4
x
=
0
which, after two direct integrations and application of boundary conditions, has the exact
solution
3
ln 2
x
2
y
(
x
)
=
−
ln
x
−
1
For the finite element solution, the simplest approach is to use a two-node element for
which the element solution is assumed as
x
−
x
1
x
2
−
x
1
y
2
where
y
1
and
y
2
are the nodal values. The residual equation for the element is
x
2
x
2
−
x
x
2
−
x
1
y
1
+
y
(
x
)
=
N
1
(
x
)
y
1
+
N
2
(
x
)
y
2
=
N
i
d
d
x
x
4
x
d
x
d
y
d
x
−
=
0
i
=
1, 2
x
1
