Grid Dependency Test- Bevel Gears

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 and following are properties for the material

	Density	Young’s Modulus	Poisson’s ratio	Yield Strength	Ultimate Tensile Strength
Structural Steel	7850 Kg m^-3	200 GPa	0.3	250 MPa	460 MPa

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: F_Normal = k_Normal X_Penetration + λ

Higher the contact stiffness K_N, lower the penetration X_p. The extra term λ, 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’.

For the analysis settings, following settings are applied-

Number of Steps	6
Auto- time setting	On
Defined by	Time
Initial time step	0.1s
Maximum Time Step	5e-2s
Minimum Time Step	0.2s

Large-deflection is turned ‘On’ as prompted during the analysis to overcome convergence issues.

Two Joint loads are defined. The small gear is given a clockwise rotation with 20° increment, up to 120°. The Big gear is given a Moment in anti-clockwise direction with 100Nm for each step.

For mesh, a local face sizing of 1.75mm on the teeth of both the gears is applied since that is the area of interest from where results will be obtained. This is carried out to obtain accurate results along with the global mesh size.

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

The following results were obtained for the three cases-

6mm- Case 1

Total Deformation- 47.87mm (Max)

Equivalent Stress- 4.83 MPa (Max)

Equivalent Elastic Strain 2.47e-5 mm/mm (Max)

5mm- Case 2

Total Deformation 47.87mm (Max)

Equivalent Stress 5.467 MPa Max

Equivalent Strain 3.0178e-5 mm/mm (Max)

4mm- Case 3

Total Deformation 4.87mm (Max)

Equivalent Stress 5.463 MPa (Max)

Equivalent Strain 3.35e-5 mm/mm (Max)

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	Equivalent Stress (Max)	% change	Equivalent Strain (Max)	% change	Total Deformation (Max)
6mm	4.83 MPa	-	2.47e-5 mm/mm	-	47.87 mm
5mm	5.467 MPa	13.04 %	3.017e-5 mm/mm	22.14%	47.87 mm
4mm	5.463 MPa	negligible	3.35e-5 mm/mm	16 %	47.87 mm

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.

	6mm Element Size	5mm Element Size	4mm Element Size
Nodes	25469	26965	27726
Elements	13891	14674	15013

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, the percentage improvement is 13.04% for Equivalent Stress and 22.14 % for Equivalent Strain.

It is also observed that when the mesh was made finer to 4mm, no 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.

Animations are attached.