University of California, Berkeley
ME 180, Engineering Analysis Using the Finite Element Method
Spring 2008
Instructor: T. Zohdi
Quiz 3 Solutions
Problem 1
Calculate the sti
ff
ness and force matrices corresponding to the problem
d
dx
c
(
x
+
1)
2
du
dx
!
+
3
u
=
4
e

x
in
Ω =
(0
,
L
)
,
u
=
37
on
Γ
u
=
L
,
du
dx
=

11
on
Γ
q
=
0
,
with a 1D mesh of 5 elements of arbitrary size.
Solution:
(15 points) The first step will be to derive the weak form. Since you should be familiar
with this by now, my derivation will be abbreviated:
Z
Ω
wR d
Ω =
0
,
Z
Ω
w
"
d
dx
c
(
x
+
1)
2
du
dx
!
+
3
u

4
e

x
#
d
Ω =
0
,
Z
Ω
"
d
dx
wc
(
x
+
1)
2
du
dx
!

dw
dx
c
(
x
+
1)
2
du
dx
#
+
3
wu

4
we

x
d
Ω =
0
,
Z
Ω

c
(
x
+
1)
2
dw
dx
du
dx
+
3
wu

4
we

x
d
Ω +
Z
Γ
q
wc
(
x
+
1)
2
du
dx
n
x
d
Γ =
0
,
Z
Ω
"

c
(
x
+
1)
2
dw
dx
du
dx
+
3
wu

4
we

x
#
d
Ω +
11
cw
(0)
=
0
.
Now, we recall that the solution and the weighing function over an element
e
are approximated
using the same interpolation functions, as
u
=
N
e
I
u
e
I
,
w
=
N
e
I
w
e
I
,
where the
N
e
I
are the element interpolation functions, and the
u
e
I
and
w
e
I
are the nodal values of
the solution and the weighing function for the element, respectively. Note that, since we don’t
have to do any calculations, we will ignore the isoparametric formulation, and just assume that
N
e
I
=
N
e
I
(
x
).
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 Spring '08
 ZOHDI
 Finite Element Method, Trigraph, dx dx

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