Finite Element Analysis
of Time Dependent
Problems
MANE 6960 – ADVANCED TOPICS IN FINITE ELEMENT
ANALYSIS
THOMAS PROVENCHER
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Introduction:
This analysis contains three different nonlinear problem, one where a cylinder of steel is pulled
beyond its yield point, another where warm fluid is cooled by the pipe carrying it, and finally one
where molten copper within a mold solidifies. The tension test represents the stress, strain, and
displacement of the rod when it begins to plastically deform. The fluid within a pipe model will
analyze the temperature and fluid velocity properties. The copper mold model will analyze the
temperature of the copper over time as it cools and solidifies, transferring it’s latent heat to the
surrounding mold.
Formulation and Solution for the Tension Test:
The Tension test sample was represented within COMSOL Multiphysics using a 2D
axisymmetric model to reduce the computational complexity. One rectangle was created to
represent the final geometry of the test sample. The lower surface of the rectangle was given a
zero displacement boundary condition and the upper surface was pulled upwards with a load of
800 MPa. Various meshes were chosen for the test sample ranging from extremely coarse to
extremely fine. The meshes were derived from COMSOL’s physics based mesh creator and
were constructed of Lagrange quadratic elements. Figure 1 shows the model geometry and the
finer mesh.
Figure 1: Jominy test model geometry and mesh
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The equation governing the tensile stress within the model is shown below:
𝛻 ∗ 𝜎 = 𝐹𝜐
The variational formulation is shown below:
(𝑢′ , 𝜐′) = (𝑓, 𝜐)
In total, 5 meshes were run to determine if the mesh had any effect on the model. The model
was run in two ways, one where the material was only allowed to elastically deform, and the
other where the rod was allowed to plastically deform with a Young’s modulus 1/100th of its
normal value. Figure 2 shows the stress and strain within the rod when meshed at a normal mesh
density. Table 1 shows the stress, strain, and displacement of the rod for all of the mesh
densities analyzed.
Figure 2: Tensile sample stress and strain
Table 1: Stress, strain, and displacement for the tensile sample
Plastic and Elastic Stress, Strain, and Displacement of an HY100 Steel Round Bar in Tension
Mesh
Stress
Stress
Strain
Strain
Displacement Dispalcement
Density
Elastic
Plastic
Elastic
Plastic
Elastic
Plastic
Extremely
8.00E+08 8.00E+08 4.00E-03 8.95E-03
4.00E-04
8.95E-04
Fine
Finer
8.00E+08 8.00E+08 4.00E-03 8.95E-03
4.00E-04
8.95E-04
Normal
8.00E+08 8.00E+08 4.00E-03 8.95E-03
4.00E-04
8.95E-04
Coarser
8.00E+08 8.00E+08 4.00E-03 8.95E-03
4.00E-04
8.95E-04
Extremely
8.00E+08 8.00E+08 4.00E-03 8.95E-03
4.00E-04
8.95E-04
Coarse
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Table 1 shows that the mesh had no effect on the results provided by the analysis. This indicates
that the model was so simple that COMSOL was able to apply basic theoretical models which
made the results unaffected by the mesh. The results also indicate that the bar’s displacement
more than doubled when the deformation was allowed to be plastic and not elastic.
Formulation and Solution for the Fluid in a Pipe Model:
The fluid flow problem was represented within COMSOL Multiphysics using a 2D axisymmetric
model to reduce the computational complexity. Two rectangles were created and laid next to
each other to represent the fluid and the pipe wall. A no-slip condition was added to the contact
area between the water and the pipe. The outer surface of the pipe was held at 298 K while the
water’s input temperature and velocity were 343 K and 0.001m/s. A zero pressure output flow
condition was assigned to the top pipe rectangle surface. Various meshes were chosen for the
model ranging from extremely coarse to extremely fine. The meshes were derived from
COMSOL’s physics based mesh creator and were constructed of Lagrange quadratic elements.
Figure 3 shows the model geometry and the finer mesh.
Figure 3: Carburization test model geometry and mesh
The equation governing the tensile stress within the model is shown below:
𝜌𝐶𝑝 ∗ 𝛻𝑇 = 𝛻 ∗ (𝐾𝛻𝑇)
The variational formulation is shown below:
(𝑢′ , 𝜐′) = (𝑓, 𝜐)
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In total, 5 meshes were run to ensure the results were mesh independent. Figure 4 shows the
velocity and temperature 3D plots of the fluid when meshed at an extremely fine mesh density.
Figure 5 shows the velocity and temperature fluid profiles as the fluid exits the pipe. Table 2
shows the centerline fluid velocity and temperature at the exit.
Figure 4: Fluid velocity and temperature 3D plots
Figure 5: Fluid velocity and temperature exit profiles
Table 2: Carburization values of the two points
Maximum Outlet Velocity and Temperature
Mesh Density
Velocity (m/s) Temperature (K)
Extremely Fine
0.00199
340.46492
Finer
0.00199
340.31701
Normal
0.00198
339.66898
Coarser
0.00194
336.69445
Extremely Coarse
0.00188
334.21126
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Table 2 shows that the greater mesh densities provided by the finer and extremely fine meshes
yield credibility to the model. The lack of temperature or velocity change between those two
meshes indicates it has reached a steady solution. Both the velocity and temperature profiles
yielded the classic “U” shape characteristic of laminar flow in a pipe.
Formulation and Solution for the Copper in a Mold:
The copper in a mold problem was represented within COMSOL Multiphysics using a 2D model
to reduce the computational complexity. Two rectangles were created, one the dimension of the
outside of the mold, and the other of the mold cavity. The iron mold and molten copper were
given initial temperatures of 293.15 K and 1400 K, respectively. Various meshes were chosen
for the model ranging from normal to extremely fine. The meshes were derived from
COMSOL’s physics based mesh creator and were constructed of Lagrange quadratic elements.
Figure 6 shows the model geometry and the finer mesh.
Figure 6: Carburization test model geometry and mesh
The equation governing the tensile stress within the model is shown below:
𝜕𝑢 𝜕 2 𝑢 𝜕𝑢
=
+
=0
𝜕𝑡 𝜕𝑥 2 𝜕𝑦
The variational formulation is shown below:
(𝑢̇ (𝑡), 𝜐) + 𝐴(𝑢(𝑡), 𝜐) = (𝑓(𝑡), 𝜐)
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Four meshes were run to a maximum time interval of 0.01 seconds ensure the results were mesh
independent. Figure 7 shows the temperature map of the copper and iron mold immediately
before and after the copper has finished solidifying using the extremely fine mesh. Figure 8
shows the temperatures at the center of the copper and at the corner between the copper and iron
over time as the copper cooled and solidified. Table 3 lists the Figure 8 maximum temperatures.
Figure 7: Copper and iron mold temperatures just before and after the copper finished solidifying
Figure 8: Temperature over time plot of the center and outer corner of the copper
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Table 3: Maximum Temperatures
at of the copper just before and after the copper finished solidifying
Maximum Temperatures at of the copper at Different Times
Mesh Density
4 seconds (K)
5.5 seconds (K)
Extremely Fine
1360
1221
Extra Fine
1360.2
1223.7
Finer
1360.6
1169.5
Normal
1370
1168.3
Table 3 shows a sudden temperature change at 5.5 seconds between the finer and extra fine mesh
densities. This was caused by a sudden change in the final time when the copper finished
solidifying as in the normal and finer models, the copper solidified just before 5 seconds and in
the extra fine and extremely fine models it did so just after 5 seconds. The temperature at 4
seconds however remained very steady as the mesh density increased and the temperatures at 5.5
seconds are also quite close indicating the model has stabilized.
Conclusions:
The COMSOL models were able to provide reliable results for all three of the nonlinear
problems. The plastic strain model showed that the software has an inherent theoretical solver
for simple plastic strains as the mesh density had no effect on its final outputs. The fluid in a
pipe problem benefitted well from the increased mesh refinement but provided satisfactory
results with more modest meshes. The fluid velocity and temperature exhibited the characteristic
“U” shaped profile laminar flows are known for. The molten copper and iron mold model
showed that a greater mesh density can sometimes yield unexpected results. The time where all
of the copper had finished solidifying increased but with further refinement the model seemed to
stabilize.
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