Civil Engineering Reference
In-Depth Information
5.3 THE GALERKIN FINITE ELEMENT METHOD
The classic method of weighted residuals described in the previous section
utilizes trial functions that are global; that is, each trial function must apply over
the entire domain of interest and identically satisfy the boundary conditions. Par-
ticularly in the more practical cases of two- and three-dimensional problems
governed by partial differential equations, “discovery” of appropriate trial func-
tions and determination of the accuracy of the resulting solutions are formi-
dable tasks. However, the concept of minimizing the residual error is readily
adapted to the finite element context using the Galerkin approach as follows. For
illustrative purposes, we consider the differential equation
d
2
y
d
x
2
+
f
(
x
)
=
0
a
≤
x
≤
b
(5.8)
subject to boundary conditions
y
(
a
)
y
b
(5.9)
The problem domain is divided into
M
“elements” (Figure 5.4a) bounded by
M
=
y
a
y
(
b
)
=
+
1
values
x
i
of the independent variable, so that
x
1
=
x
a
and
x
M
+
1
=
x
b
to
n
1
(
x
)
1
0
x
1
x
2
x
3
x
4
x
5
n
2
(
x
)
1
0
x
1
x
2
x
3
x
4
x
5
n
3
(
x
)
1
0
x
1
x
2
x
3
x
4
x
5
n
4
(
x
)
1
1
2
3
M
x
1
(
x
a
)
x
2
x
3
x
M
x
M
1
0
x
1
(
x
b
)
x
2
x
3
x
4
x
5
(a)
(b)
Figure 5.4
(a) Domain
x
a
≤
x
≤
x
b
discretized into
M
elements. (b) First four trial functions. Note the overlap
of only two trial functions in each element domain.
