3.2. Final remarks

In contact problems, there are different physical magnitudes that can be observed to perform the adaptive mesh refinement, having each one its advantages and disadvantages. In this work, the observed magnitude to perform the adaptive mesh refinement is the contact pressure (λ<sup>i</sup> ¼ pi ), and the mesh refinement is performed based on the discrete gradient of the contact pressures. The main advantages of choosing the gradient of the contact pressure as refinement criterion instead of any other derived magnitude are:


However, it must be taken into account that the contact pressure distribution function is not differentiable in the border of the contact area. In consequence, according to Eq. (17), the discrete rate of change of the contact pressure between an element Δ<sup>i</sup> that is within the contact area (pi > 0) and of an adjacent element Δ<sup>j</sup> that is outside of the contact area (pj ¼ 0) is always wj,i ¼ 1. Therefore, if a value lower than 1 is specified for wmax, the refinement strategy will refine the mesh at the boundary of the contact area until the maximum degree of mesh refinement will be reached at the border of the true contact area.

The topology of the resulting pressure element mesh, inside and outside the true contact area, depends on the configuration of the proposed approach, which is defined by a unique combination of the three input parameters:


The possible configurations of the approach, and their effect on the resulting pressure element mesh, are categorized intro three different settings:


From the nine steps of the main algorithm, step A5 is the most time consuming. For this reason, the computational cost of the approach can be defined by the number of influence coefficients that are calculated to solve the contact problem, which can be determined using Eq. (16).
