174
Finite Element Analysis and Design
6.
Consider the following differential equation:
2
2
1
0,
0
1
(0)
0
1
x
d u
u
x
x
dx
u
du
dx
The solution is approximated as
2
1
2
( )
u x
c x
c x
Calculate the unknown coefficients using Galerkin method.
Compare
u
(
x
) and
du
(
x
)/
dx
with the exact solution:
u
(
x
) = 3.7 sin
x
–
x
by plotting the solution.
Solution:
Substitute the approximate solution and make the weighted residuals equal to zero
1
,
0
(
)
0,
1,2
xx
i
u
u
x
dx
i
Use integration by parts for the first term:
1
1
1
1
,
,
,
0
0
0
0
(
)
(
)
(
)
0
x
i
x
i x
i
i
u
u
dx
u
dx
x
dx
From the boundary term, we can recognize the essential and natural BCs.
The essential
boundary specifies
u
or
i
, while the natural boundary specifies
u
,
x
.
Implementing the
BCs and also substituting the approximate solution, we obtain the Galerkin equations:

This ** preview** has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*