ME 422 Winter 2007-2008
Do all 5 problems
Name
Problem
Problem
Problem
Problem
Problem
TOTAL
1
2
3
4
5
(30)
(30)
(10)
(20)
(10)
(100)
1
Problem 1 (30 points): Suppose that there is a 4-node quadrilateral element for which
J=
Find
∂N1
∂x
and
∂N1
.
∂y
"
1
(3 + t)
4
1
(1 + s)
4
0
1
2
#
Recall that
J
−1
1
=
detJ
"
2
∂y
∂t
− ∂x
∂t
− ∂y
∂s
∂x
∂s
#
Problem 2 (30 points): Consider the 4-node finite element shown below. Find K23 . Set
up all integrals completely, but do not integrate.
y
(-1,0)
2
3 (-1,-1)
(0,0)
1
x
4
(0,-1)
3
Problem

3 (10 points): Suppose that we solve 100 linear equations in 100 unknowns, with
1


 0 


 0 



F =
 . . The answer is x, a vector with 100 values in it.


 . 


 . 


0

1. Now we solve the same problem with F =
(a) The same as the original x.












2
0
0
.
.
.
0







.





What is x?
(b) Every entry is twice the original x.
(c) Every entry is half the original x.
(d) You can’t know until you solve the equations.
(e) Other (explain)

2. Next we solve the same problem with F =












0
1
0
.
.
.
0







.





What is x?
(a) Shift every entry of the original x down by one.
(b) Shift every entry of the original x up by one.
(c) The same as the original x.
(d) You can’t know until you solve the equations.
(e) Other (explain)
4
Problem 4 (20 points): Recall that if we have a fin with convection off its sides, the
governing differential equation is
d
dT
kA
dx
dx
!
= hC(T − T∞ )
(There is no internal heat generation.) If we use a single 2-node element to discretize the
problem, the weak form for this becomes
Z
x2
x1
kA
dT dT
dx =
dx dx
Z
x2
x1
hCT (T − T∞ )dx
Now suppose that h, C, k, and A are all constant. The contribution to the system of
equations from the left hand side of the weak form is a stiffness matrix:
kA
Le
"
1 −1
−1 1
#
What are the contributions to the system of equations from the right hand side of the weak
form? Set up all integrals completely, but do not integrate.
5
Problem 5 (10 points): The Helmholtz equation is often used to find the vibration modes
of cavities, including exhaust systems. In 1-D the governing differential equation for the
problem is
d2 p
+ k2p = 0
dx2
with k 2 a constant, and p the pressure in the cavity. Suppose that p(0) = 0 and p(L) = 0.
What is the weak form for the problem? (Hint: the weighting function would be called p
since the unknown is p.)
6