3D Finite Element Modeling: A comparison of common element types and patch test verification
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3D Finite Element Modeling:
A comparison of common element types and patch test
verificationBY: RACHEL SORNA AND WILLIAM WEINLANDT
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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 classLinear Tetrahedral Tri-Linear Hexahedral
Understand and utilize the patch test as a means of verifying our code and element modeling
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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…
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3D FEM Code Attempt #2 – From Scratch
SUCCESS!
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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 elementsShape Functions (N matrix)Shape Function Derivatives (B Matrix)Gauss points and weights•Determining which element face was on a given boundary
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Linear Tetrahedral Boundary Surface Element Solid Volume
Element
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Tri-Linear Hexahedral Boundary Surface Element Solid Volume
Element
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Patch Test: Code and Element Verification
Block Dimensions:
Height: 5m
Width: 5m
Depth: 5m
Material Properties:
E: 200GPa
ν: .3
Two simple, yet effective tests were done: fixed displacement and fixed tractionTest Subject: A
simple Cube
Hexahedral Meshing
Tetrahedral Meshing
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Fixed Displacement Test
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
Fixed displacement of .0001 was applied in the negative Z-direction. The following analytical solution gives a stress value of 4MPa throughout the cube:
𝜎=𝛿𝐿 𝐸
Table 1. Normalized L2 Norm Error values for tetrahedral and hexahedral elements for varying degrees of mesh fineness.
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Fixed Traction Test𝛿=
𝐹𝐿𝐴𝐸
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
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.
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Application – A Complex Loading SchemeCounterclockwise 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.
ANSYS
Our Code
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Convergence PlotThe 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.
0 500 1000 1500 2000 2500 3000 3500 4000 45000
2000
4000
6000
8000
10000
12000
14000
16000
18000Mesh Convergences
ANSYS Tetrahedral
MATLAB Tetrahedral
MATLAB Hexahedral
ANSYS Hexahedral
Number of Elements
Max
imum
Equ
ival
ent S
tres
s
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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
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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!
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Future Potential Applications
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Just Meshing Around!