ME 422 Winter 2005-2006
Do all 5 problems
Name
Problem
Problem
Problem
Problem
Problem
TOTAL
1
2
3
4
5
(5)
(25)
(15)
(20)
(35)
(100)
1
Problem 1 (5 points): Suppose we have an element whose Jacobian is
J=
"
2 0
0 1
#
Which of the following elements might it be?
a)
1
2
b)
1
4
c)
1
2
d)
2
4
e) None of the above. (Draw the correct one.)
2
Problem 2 (25 points): Suppose that there is a 4-node quadrilateral element for which
J=
"
1
(3 + t)
4
1
(1 + s)
4
0
1
2
#
Find K13 . Set up all integrals completely but do not integrate. Recall that
J−1
1
=
detJ
"
3
∂y
∂t
− ∂x
∂t
− ∂y
∂s
∂x
∂s
#
Problem 3 (15 points): Consider the 4-node finite element shown below. Find x(s, t).
y
(-1,0)
2
(0,0)
1
x
4
(0,-1)
3 (-2,-2)
4
Problem 4 (20 points): Consider the three-node one-dimensional conduction element
shown below. Unlike ordinary isoparametric elements, this one has node three at r = − 12 .
What would the shape functions (N1 , N2 , N3 ) be for this element?
1
r=-1/2
2
r=-1
3
r=+1
5
Problem 5 (35 points): Suppose that we have a cylinder with heat flux on the outside
surface and convection to a known temperature Ta on the inside surface. There is no internal
heat generation. If we use a single 2-node element to discretize the problem, the weak form
for this problem becomes
Z
x2
x1
kx
dT dT
dx + hx1 (T1 − Ta )T 1 − Qx2 T 2 = 0
dx dx
Heat Flux Q
x2
x
x1
Convection
1
2
Find the global system of equations that comes from this discretization and weak form.
6
7