Chris Bickford
Introduction to Finite Elements
HW #4; Problem 3
Part a) MAPLE:
Both equations and boundary conditions satisfy the Laplace equation. When evaluating
these functions, it is possible to determine the exact solution. Using Maple, it can be
shown that the value of a surface integration over the specified domain is zero for the
initial function and 0.500 for the stream function (exact solution). A more detailed
summary is shown in the Maple sheets attached.
Part b) COMSOL FEA CODE:
COMSOL software was used to perform a static 2-dimensional finite element analysis
(FEA) to obtain several approximate solutions for problem (3). These approximations
were solved for using different two different mesh density settings in the physics
controlled mesh option (Normal and Finer) and three element types (Linear, Quadratic,
Cubic). The results are summarized in . All approximations are very close to zero and
improve as the mesh becomes more refined and the order of the element is increased.
Table 1 COMSOL FEA Solutions: Surface Integration Values
Surface
Integration of
COMSOL
Approximation
Element Basis
Function Order
Exact Solution = 0
Mesh Density Setting
Normal
Finer
1
-2.04E-06
-7.83E-07
2
-4.95E-17
4.29E-16
3
1.64E-15
-
Figure 1 Normal Mesh Setting: Linear Element Type
Part c) ABAQUS FEA CODE:
ABAQUS 6.14 software was used to perform finite element analysis (FEA) to obtain an
approximate solution for problem (3). A 2-dimensional static heat transfer analysis was
performed assuming the thermal conductivity is 1 and applied analytical boundary
conditions in accordance with the problem statement. Quadratic elements were used
with a mesh density of 256 elements. Figure 2 is fringe plot which shows an
approximation for the value u(x,y) over the specified domain.
Figure 2 Two-Hundred Fifty-Six (256) Elements: Quadratic Element Type