3D Finite Element Modeling:

A comparison of common element types and patch test

verificationBY: 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 classLinear 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 elementsShape Functions (N matrix)Shape Function Derivatives (B Matrix)Gauss points and weights•Determining which element face was on a given boundary

Linear Tetrahedral Boundary Surface Element Solid Volume

Element

Tri-Linear Hexahedral Boundary Surface Element Solid Volume

Element

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

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.

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.

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

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

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!