1. By: Rachel Sorna and William weinlandt
3D Finite Element Modeling: A comparison of common element types and patch test verification
By: Rachel Sorna and William weinlandt

2. Objectives
Develop a sound understanding of 3D stress analysis through derivation, construction, and implementation of our own 3D FEM Matlab Code.
Compare the accuracy of two different element types mentioned in class
Linear Tetrahedral
Tri-Linear Hexahedral
Understand and utilize the patch test as a means of verifying our code and element modeling

3. 3D FEM Code Attempt #1 – Conversion from 2DStressAnalysis Code
Can you guess what loading scheme created the deformation on the right?
Answer: Uniform traction of 1000N/m2 along the top boundary…

4. 3D FEM Code Attempt #2 – From Scratch
SUCCESS!

5. Major Parts of Code Construction
Addition of third dimensional variable, zeta
Manual generation of meshes and defining elements and corresponding nodes
3D Elasticity matrix
3D Transformation matrix
Defining solid elements and corresponding boundary surface elements
Shape Functions (N matrix)
Shape Function Derivatives (B Matrix)
Gauss points and weights
Determining which element face was on a given boundary

6. Linear Tetrahedral
Solid Volume Element
Boundary Surface Element

7. Tri-Linear Hexahedral
Solid Volume Element
Boundary Surface Element

8. Patch Test: Code and Element Verification
Two simple, yet effective tests were done: fixed displacement and fixed traction
Test Subject: A simple Cube
Material Properties:
E: 200GPa
ν: .3
Block Dimensions:
Height: 5m
Width: 5m
Depth: 5m
Hexahedral Meshing
Tetrahedral Meshing

9. Fixed Displacement Test
𝜎= 𝛿 𝐿 𝐸
Fixed displacement of was applied in the negative Z-direction. The following analytical solution gives a stress value of 4MPa throughout the cube:
Table 1. Normalized L2 Norm Error values for tetrahedral and hexahedral elements for varying degrees of mesh fineness.
Element Type
Mesh Fineness
Tetrahedral
Hexahedral
Coarse
(48/64)
3.6e-3
7.56e-4
Moderate
(750/729)
 1.06e-3
 2.50e-5
Fine
(3000/3375)
 3.06e-4
 6.25e-6

10. Tri-Linear Hexahedral
Fixed Traction Test
𝛿= 𝐹𝐿 𝐴𝐸
Fixed traction of 1000N/m2 was applied in the positive Z-direction. The following analytical solution gives a displacement value of 2.5e-8m on the top surface:
Table 2. Normalized L2 Norm Error values for tetrahedral and hexahedral elements for varying degrees of mesh fineness.
Element Type
Mesh Fineness
Linear Tetrahedral
Tri-Linear Hexahedral
Coarse
(24/27 )
1.06e-2
3.69e-4
Moderate
(750/729)
1.90e-3
4.62e-5
Fine
(3000/3375)
 3.60e-5
 1.60e-5

11. Application – A Complex Loading Scheme
Counterclockwise oriented 1000N shearing surface tractions applied to the X and Y faces produced the following ‘twisty’ cube. Comparison to the same loading scheme in ANSYS revealed that our model was valid for complex loading schemes as well.
Our Code
ANSYS

12. Convergence Plot
The following plot shows the convergence of von-Mises stress for tetrahedral and hexahedral elements performed using our code as well as ANSYS for the ‘twisty’ cube loading condition.

13. Current Limitations of Our Code
Matlab can only handle so many elements and thus degrees of freedom before it becomes impossibly slow or runs out of memory
Limitations of the variation in orientation of tetrahedral elements
Limited possible geometries
Linear elements only, no quadratic elements

14. Conclusions
Hexahedral elements appear to more accurately model simple linear deformation problems than tetrahedral elements – they also take far less time!
Tetrahedral elements can be useful for complex geometries and loading conditions (rounded surfaces, sharp corners, etc.)
For accurate meshing and modeling of complex multi-part geometries, mixed meshing, or using both tetrahedral and hexahedral, as well as other element types, is best!
Writing your own FEM Code is a long and arduous process, but once you are done, you have an unparalelled understanding of the fundamental concepts!

15. Future Potential Applications

16. Just Meshing Around!