3D Finite Element
Modeling:
A comparison of common element
types and patch test verification
BY: RACHEL SORNA AND WILLIAM WEINLANDT
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
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…
3D FEM Code Attempt #2 – From Scratch
SUCCESS!
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
Linear Tetrahedral
Solid Volume Element
Boundary Surface Element
Tri-Linear Hexahedral
Solid Volume Element
Boundary Surface Element
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
Fixed Displacement Test
𝛿
𝜎= 𝐸
𝐿
Fixed displacement of .0001 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)
Moderate
(750/729)
Fine
(3000/3375)
3.6e-3
7.56e-4
1.06e-3
2.50e-5
3.06e-4
6.25e-6
Fixed Traction Test
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:
𝐹𝐿
𝛿=
𝐴𝐸
Table 2. Normalized L2 Norm Error values for tetrahedral and hexahedral
elements for varying degrees of mesh fineness.
Element Type
Mesh Fineness
Linear Tetrahedral
Coarse
(24/27 )
Moderate
(750/729)
Fine
(3000/3375)
1.06e-2
Tri-Linear
Hexahedral
3.69e-4
1.90e-3
4.62e-5
3.60e-5
1.60e-5
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
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.
Mesh Convergences
18000
16000
Maximum Equivalent Stress
14000
12000
10000
8000
ANSYS Tetrahedral
6000
MATLAB Tetrahedral
MATLAB Hexahedral
4000
ANSYS Hexahedral
2000
0
0
500
1000
1500
2000
2500
Number of Elements
3000
3500
4000
4500
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
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!
Future Potential Applications
Just Meshing Around!