Chris Bickford
Introduction to Finite Elements
HW #4; Problem 4
MAPLE:
Using Maple, the exact solution is obtained through separation of variables using the
Fourier-Bessel coefficients. The exact solution at the center of the domain (0,0.5) is
0.38391. The surface integral of the exact solution (0≤ r ≤ 1, 0 ≤ z ≤ 1) is 0.33047. The
maple sheets are attached.
Part a) COMSOL FEA CODE:
COMSOL software was used to perform a 2-dimensional axisymmetric finite element
analysis (FEA) to approximate the distribution of u(r , z) over the specified domain. The
physics controlled mesh option was used and two mesh settings were analyzed. Figure
1 shows the temperature distribution for the coarse mesh setting and Figure 2 shows
the temperature distribution for the finer mesh setting. Table 1 shows that the
approximations from both mesh settings are very close to the exact solution.
Table 1 Temperature Values at (0,0.5) for Both Mesh Settings in COMSOL
Mesh Setting
Exact Soln = 0.3839
Temperature at (0,0.5)
Coarse
Finer
0.3855
0.3842
Figure 1 Coarse Mesh Setting: Linear Element Type
Figure 2 Finer Mesh Setting: Linear Element Type
Part b) ABAQUS FEA CODE:
ABAQUS 6.14 software was used to perform finite element analysis (FEA) to obtain an
approximate solution for problem (4). A 2-dimensional axisymmetric static heat transfer
analysis was performed assuming the thermal conductivity is 1, the boundary
temperature is 0 at r = 1 and z = 0, and boundary temperature is 1 at z = 1. Figure 3
shows a fringe plot of the approximate temperature distribution. The model uses 256
quadratic elements and predicts the value at (0,0.5) to be 0.383911, which is very close
to the exact solution.
Figure 3 Two-Hundred Fifty-Six (256) Elements: Quadratic Element Type