Grid dependency test on Bevel gear assembly

Grid Dependency Test

`Grid Dependency Test\', \'Grid Independency Test\' and \'Mesh Refinement Test\' are the synonyms and they all mean the same. While performing Finite Element Analysis, one of the most common question is finite element mesh quality. How can one know if the mesh is good enough for the analysis? Mesh density is a significant metric used to control accuracy (element type and shape also affect accuracy). Assuming no singularities are present, a high-density mesh will produce results with high accuracy. However, if a mesh is too dense, it will require a large amount of computer memory and long run times, especially for multiple-iteration runs that are typical of nonlinear and transient analyses.
The most fundamental method of checking the mesh quality is to refine a mesh until a critical result such as Equivalent stress does not change significantly with each refinement (i.e. it converges). This is Grid dependency test, where, analysis begins with a coarse mesh and it is refined to finer mesh for each run until the results do not change significantly and depend of the grid size anymore. As the grid or mesh size approaches zero, the approximate solution is almost equal to the actual solution. The aim of this test is to find the optimum mesh size for the solution, where it is neither to coarse to produce inaccurate results, neither too fine, where it will consume a lot of computing effort and time.
There are different mesh refinement techniques are used which has advantages and disadvantages and some depend on the experience and technical expertise of the FEA analyst. These techniques include- Reducing the element size, Increasing the element order, Global Adaptive mesh refinement, local adaptive mesh refinement, manually adjusting the mesh, Time adaptive mesh refinement, wavelength adaptive mesh refinement, etc.

Grid Dependency Test on Bevel Gears
To perform the test, the simulation is run on three different element sizes- 6mm, 5mm, and 4mm. The good practice is to run the simulation at default mesh size and then refine it for the successive runs.

It is a good and necessary practice to remove the unwanted regions and simplify the CAD geometry before the analysis as shown above. This ensures that computing resources are conserved, and less time consumed to run the simulation.

The material assigned is Structural Steel

General Set-up
First, for the sake of simplicity, the gears are named as \'Small\' and \'Big\' according to their sizes. Then, contact between all the tooth faces of the both the gears is defined with following settings-
Type Frictional

Behavior Symmetrical

Formulation Augmented Lagrange

Interface Treatment Adjust to touch

Augmented Lagrange

In the analysis, a relation must be established between two surfaces to prevent them from passing through each other.

Augmented Lagrange or Pure Penalty is preferred for non-linear solid body contact faces.

Augmented Lagrange: FNormai = kNormai X penetration + λ

Higher the contact stiffness KN, lower the penetration Xp. The extra term A, this method is less sensitive to the magnitude of the contact stiffness, and solution results are accurate.

Then two revolute joints are defined, one for each gear with a connection type of \'Body-to-ground\'.

Analysis settings

Joints Load

Big Gear

Small Gear

Case1 6mm

mesh at 6mm element size and a face sizing of 1.75mm on the teeth.

Mesh Quality

Mesh Matrix

Total Deformation:-

Animation:-https://youtu.be/fDAKg1jhMAw

Equivalent Elastic strain:-

Animation:-https://youtu.be/QYd8SzykY2o

equivalent stress:-

Animaton:-https://youtu.be/URH-V0h5YTU

Case2

5mm and face sizing of 1.75mm on the teeth.

Mesh

Mesh Quality

Mesh Matrix

Total Deformation:-

Animation:-https://youtu.be/WFfv4m_f87I

Equivalent Elastic Strain:-

Amimation:-https://youtu.be/LRwxaP65PpE

Equivalent Stress:-

Animation:-https://youtu.be/tdPo43ELceg

Case3

4mm and face sizing of 1.75mm on the teeth.

Mesh

Mesh Quality

Mesh Matrix

Total Deformation:-

Animation:-https://youtu.be/OWBrcDQ8ylo

Equivalent Elastic Strain:-

Animation:-https://youtu.be/T8Z053finRw

Equivalent Stress

Animation:-https://youtu.be/InM1QvEQOOU

Discussion
The Grid dependency test performed for three cases, where global mesh size used was 6mm, 5mm, and 4mm while the face sizing was kept at 1.75 mm in all the cases. The following are the results for the three cases.

Element Size	Total Deformaton	Equivalent Elastic Strain	Equivalent Stress
(mm)	(mm)	(mm/mm)	(Mpa)
6mm	47.873	2.69E-05	3.8479
5mm	47.873	2.23E-05	3.8161
4mm	47.873	3.01E-05	4.4479

It is observed that no change is obtained for the Total Deformation for all the cases and remained at 47.87 mm (Max).

As the element size was reduced from coarse to fine, it can be observed that the number of Nodes and Elements increased as shown in the following table.

Mesh Size	NODES	ELEMENTS
6	24123	12966
5	24671	13359
4	25511	13818

As expected from the Grid dependency test, as the mesh was changed from coarse to fine, the accuracy in the results improved. It can be seen from the first table that there was a significant improvement in results for both Equivalent Stress and Equivalent Strain from 6mm to 5mm.

It is also observed that when the mesh was made finer to 4mm, big change in Equivalent Stress result was observed, which means that Grid Independency has reached and further the results are not dependent on Grid size or element size anymore and further reducing element size is unnecessary.

If the mesh size is further reduced, it will only increase computational time. Therefore, it can be said that the optimal mesh size is 4mm.