ME 608
Numerical Methods in Heat, Mass, and Momentum Transfer
Sample Final Exam Problems
April 25, 2006
Instructor: J. Murthy
1. Consider 1D steady conduction on a uniform mesh of twelve cells, as shown in Fig.1. The domain length is
L
=
1
.
2
m
.
Cells 14 and 912 contain a solid of thermal conductivity
k
A
=
1
W
/
mK
while cells 58 contain a solid of thermal
conductivity
k
B
=
100
W
/
mK
. There is no source term. The boundary conditions are as shown. A student developing
a multigrid procedure decides to use three mesh levels as shown in the figure.
(a) Develop the discrete equation set for the three mesh levels using the algebraic multigrid procedure developed in
class.
(b) Starting with an initial guess of
T
=
300
K
everywhere, perform one Vcycle with
ν
1
=
0 (i.e., no GaussSeidel
sweeps on the down leg) and
ν
2
=
1 (i.e., one GaussSeidel sweep on the up leg) and show the resulting
T
solution.
1
2
3
4
5
6
7
8
9
10
11
12
Level 0
Level 1
Level 2
400K
300K
Figure 1: Computational Domain for Problem 1
2. Consider steady 2D conduction in a square domain of side
L
=
2
m
.
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 Fall '10
 NA
 Numerical Analysis, Momentum Transfer, J. Murthy, discrete momentum equations

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