Coupled thermal structural finite elements analysis

1. Coupled Thermal-Structural
Finite Elements Analysis
Presented by:
Junyan He
Udayan Ghosh
Brett Reeder
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2. Motivation
Thermal stress arises in structures when the structures are held in some fixtures and
experience a temperature change. In practice, this poses danger as thermal stress can cause
unexpected failure of the structure.
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Figure 1. Window glass cracked due to thermal stress Figure 2. Rails buckled after extensive sun exposure
Therefore, calculating the thermal stress inside a structure is important to engineers
in design and verification process.

3. Solution Strategy
Two common types of solution strategies are used in coupled thermal-structural problems:
• Sequentially coupled, valid when the stress field does not influence the temperature field significantly (i.e.
elastic deformation)
• Fully coupled, preferred when temperature field is affected by stress field (i.e. plastic deformation, friction)
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Thermal BCs
and ICs
Nodal
temperatures
Mechanical
BCs
Stress field
Chart 1. Flow chart for sequentially coupled scheme
Thermal BCs
and ICs
Mechanical
BCs
Coupled thermal-mechanical
model
Stress field
Chart 2. Flow chart for fully coupled scheme

4. Implementation
In this project, both solution schemes are attempted to see the difference.
• A MATLAB script was develop to solve the problem using sequentially coupled scheme. It
uses distmesh package to generate a 2D mesh over the domain. It can model transient
behaviors using an implicit integration method and feature a full Newton’s solver to account
for the non-linearity in the temperature dependence of material properties.
• The FEniCS (A general purpose finite elements program) package is used to implement the
fully coupled scheme.
• Commercial code Abaqus was used to validate the solutions in MATLAB and FEniCS.
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5. MATLAB Script Validation
Problem Statement for validation:
A convergence study was conducted on the transient temperature response. The plate is initially at 25 °C and the left
boundary is held at 300 °C while all other boundaries are insulated. The material of the plate is copper.
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Figure 3. Comparison between FE solution of
different mesh sizes and the analytical solution
Table 1. Error norm at different element sizes
The error norm does not approach 0, which is unexpected. However, the overall agreement is satisfying.
Element Size
(m)
Error Norm Percentage
Decrease
0.08 565.9 -
0.04 551.4 -2.63%
0.02 550 -0.254%

6. MATLAB Script Validation
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Figure 4. Comparison between FE solution and
analytical solution at different time instances
The FE solution captures the trend
of temperature distribution but
underestimates the temperature
values. When compared with the
analytical solution at different time
instances, the agreement is
reasonable, with a mean error
norm of 550.

7. Problem Statement
The domain is a copper plate of dimensions 2x0.2x0.05m. Initially it is held at 25 °C and at
t=0, the top and bottom surfaces are held at 100 and 10 degrees C, respectively. The left and
right surfaces and insulated and held fixed. The process is modeled for 100s.
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Figure 5. Schematic Diagram of the Problem statement.
T(top) = 373 K
T(bottom) = 283 K
T(intial) = 298 K

8. Results From Sequentially Coupled MATLAB Script
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Figure 6. Temperature
distribution for
different mesh sizes
Element
Size (m)
Error
Norm
0.06 5.49
0.03 2.60
0.01 1.96
Table 2. Error norm at
different element sizes
Video 1. Temperature
distribution of the plate at t=100s
As the element size decreases, the error norm on temperature decreases and approaches 0. This is evidence that the
temperature solution converges to the analytical solution.

9. Results From Sequentially Coupled MATLAB Script
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Figure 7. Surface stress
for different mesh sizes
Table 3. Surface stress at
different element sizes
Video 2. Stress distribution of the
plate at t=100s
• As the element size decreases, the stress on both surfaces keeps increasing, showing a slow rate of convergence.
• An attempt was made to simulate with smaller element size, but the size of the stiffness matrix becomes huge and
was aborted due to long computational time.
• Therefore, it can’t be concluded that our MATLAB code produces a converged solution on stress.
Element
size (m)
Top
surface
stress
(GPa)
Bottom
surface
stress
(GPa)
0.06 1.31 0.164
0.03 1.40 0.227
0.01 1.46 0.278

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Abaqus Simulation
Convergence Study in Abaqus
• Element Type: Linear triangle
• Element Size: 0.04, 0.02, 0.01
• Symmetry was applied on right edge to reduce computational time
• Top plane stress values are considered for convergence study of Abaqus simulation
Figure 9. 2D model after meshing (element size 0.01); inset shows
closeup of triangular element mesh.
Figure 8. Convergence study of Abaqus
simulation

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Comparison With Abaqus Coupled Thermal-Structural Simulation
A convergence study has been performed on the Abaqus model. All results presented here are taken from the converged
mesh.
Figure 10. Temperature distribution at t=15sec. Figure 11. Stress distribution at t=15sec.
Figure 12. Temperature distribution at t=15s. Simulated
using MATLAB
Figure 13. Stress distribution at t=15s. Simulated using
MATLAB

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Comparison With Abaqus Coupled Thermal-Structural Simulation
Figure 14. Temperature distribution at t=100s. Figure 15. Stress distribution at t=100s.
Figure 16. Temperature distribution at t=100s. Simulated
using MATLAB
Figure 17. Stress distribution at t=100s. Simulated
using MATLAB

13. Comparison With Abaqus Coupled Thermal-Structural Simulation
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Table 5. Comparison between Abaqus and MATLAB simulation, t = 100s
Abaqus simulation MATLAB simulation Percent difference
Mid-line temperature
(Degrees C)
55.0 55.5 +1.84%
Top surface stress (GPa) 1.47 1.43 -2.72%
Bottom surface stress (GPa) 0.267 0.260 -2.62%
Mid-line stress (GPa) 0.592 0.503 -15.0%
Table 4. Comparison between Abaqus and MATLAB simulation, t = 15s
Abaqus simulation MATLAB simulation Percent difference
Mid-line temperature
(Degrees C)
31.5 31.7 +0.6%
Top surface stress (GPa) 1.43 1.33 -7.00%
Bottom surface stress (GPa) 0.285 0.261 -8.42%
Mid-line stress (GPa) 0.137 0.133 -2.92%

14. Conclusion
The agreement between the MATLAB simulation and Abaqus simulation is satisfying, with an average error of
5.2% at t=100s and 5.5% at t=15s.
Abaqus uses a fully coupled solution scheme to calculate the response. Despite the different solution scheme, both Abaqus and
MATLAB simulation yields very similar solutions.
This may look unreasonable at the first glance, but it can be explained.
The level of importance of the coupling effect is determined by the ratio of the isothermal modulus (M) and the adiabatic modulus
(A). [1]
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𝑴 =
𝑬(𝟏 − 𝝊)
(𝟏 + 𝝊)(𝟏 − 𝟐𝝊)
A = 𝑴 + (
𝑬𝜶
𝟏−𝝊
) 𝟐 𝑻 𝟎
𝝆𝑪
Equation 1. Isothermal Modulus Equation 2. Adiabatic Modulus
Using material properties of Copper, we found a ratio A/M of 1.014. This indicates that the coupling effect is weak and
therefore how the coupling is done in the simulation causes minimum difference in the final results.
[1] J. P. Carter, J. R. Booker, Computers & Structures 1989, 31, 73.

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Future Steps
• To compare results from FEniCS , MATLAB and Abaqus simulation.
• To improve the implementation of convection in the MATLAB script.
• To implement Quasi-Newton’s Method instead of Full-Newton’s Method to
increase computation speed.

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Questions?