Chris Bickford
Introduction to Finite Elements
HW #5; Problem 2
MAPLE:
Using Maple software, the exact solution of the transient response was obtained. The final temperature at the center of the domain u(0.5 , 3600*72) is 45.0697. See the attached Maple sheets for more details.
Part b) ABAQUS FEA CODE:
ABAQUS 6.14 software was used to perform finite element analysis (FEA) to obtain an approximate solution for problem (2). A 2-dimensional transient heat transfer analysis was performed assuming no boundary conditions on the top (x , 0.5) and bottom (x, -
0.5) edges, which effectively makes the problem one-dimensional. Four analyses were performed using two element types (linear and quadratic) and two mesh densities (144
and 288 elements). Figure 1 and Figure 2 show the temperature distribution for the
coarse and fine mesh densities, respectively. Table 1 compares the FEA results with
the exact solution. The results show that both elements performed well at the given mesh densities.

Figure 1 One-Hundred Forty-four (144) Elements: Linear Element Type
Figure 2 Two-Hundred Eighty-eight (288) Elements: Quadratic Element Type

Temperatue at x = 0.5
30
25
20
15
10
5
0
50
45
40
35
0
Linear Elem (144)
Linear Elem (288)
Quadratic Elem (144)
Quadratic Elem (288)
50000 100000 150000
Time
200000 250000
Figure 3 Plot: Time History of Temperature at x = 0.5
Exact Soln
[t = 3600*72]
45.0697
Element
Densities
144
288
Table 1 Final Temperature (t = 3600*72) at x = 0.5
Linear Elements
Value (x=0.5)
44.8471
44.88
% Error
0.49%
0.42%
Quadratic Elements
Value (x=0.5)
44.8911
44.8912
% Error
0.40%
0.40%