16. Summary of results

The comparison in each of the models to the analytical reference (limited to point \(G\)) is satisfactory.

The non-regression of the results is ensured by testing the movements in two nodes of the edge, \(H\) and \(I\). Since these nodes occupy slightly different positions depending on the models, one should not try to compare them between models.

This test in curved geometry highlights the usefulness of certain pairing parameters such as:

• smoothing: this is a process of modifying the normals that are used to write the contact conditions. Although sphere meshes are symmetric, in practice, there is always a slight offset at the last point of the contact surface. The smoothing of the normals allows symmetry to be restored here. This parameter is also useful for improving the geometric convergence of models using coarse meshes (this is the case of models \(\mathrm{2D}\) in this test).

• Nodal matching or the average of normals (master-slave): rarely used in general, these two pairing options, such as smoothing, make it possible to regain the symmetry of the problem.

Modeling A where contact is replaced by a unilateral connection also highlights the very important role played by geometric updating on the precision of the constraints. Indeed, this modeling makes it possible to remove the geometric nonlinearity of this problem, so it provides a reference solution. Thus, when we compare the results obtained on the same problem \(\mathrm{3D}\) by the models A, I and J, we note that:

• the displacement values obtained by modeling I show differences of up to 4% compared to the two other models

• The A and J models find an identical Hertz pressure, while the I model finds a 10% lower pressure

• in modeling I the geometric criterion is satisfied to within 5% while it is satisfied to 1% in modeling J

It therefore seems important to satisfy the geometric criterion as best as possible in order to obtain the best possible precision on the stress values.