4. Numerical examples

ii. On the other hand, Li must be less than or equal to Lj, to ensure that the level difference between two adjacent elements does not differ more than one level in the quadtree

If these conditions are met, the pressure element Δ<sup>i</sup> is marked to be split by defining K<sup>i</sup> ¼ TRUE. The algorithm finishes when all the pressure elements have been evaluated,

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 pres-

contact pressures. The main advantages of choosing the gradient of the contact pressure as

i. On one hand, in this approach, the solution of the contact problem is found in terms of the contact pressure distribution. From the calculated contact pressure distribution, derived results are obtained. Since the accuracy of the derived results is dependent from the accuracy in which contact pressure distribution is calculated, it is important to obtain

ii. On the other hand, using the contact pressure distribution, instead of the derived results, as refinement criteria helps reducing the computational cost of the proposed approach,

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

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 combi-

The possible configurations of the approach, and their effect on the resulting pressure element

), and the mesh refinement is performed based on the discrete gradient of the

returning an array that contains the indices of those elements where K<sup>i</sup> ¼ TRUE.

refinement criterion instead of any other derived magnitude are:

an accurate description of the contact pressure distribution.

refinement will be reached at the border of the true contact area.

nation of the three input parameters:

i. The initial level of uniform mesh density, Luni. ii. The maximum degree of mesh refinement, Lmax.

mesh, are categorized intro three different settings:

because obtaining derived results implies additional calculations.

iii. The maximum allowed rate of change of the physical magnitude, wmax.

structure, avoiding unbalanced meshes.

3.2. Final remarks

72 Contact and Fracture Mechanics

sure (λ<sup>i</sup> ¼ pi

The performance of the proposed approach is illustrated in this section, considering its accuracy and computational cost. For such a purpose, two cases of study are considered:


The material of both indenters (CoSI and CoSII) and the plane is assumed to have a Young modulus of 70 GPa and a Poisson coefficient of 0:35. A total contact load FT ¼ 60 kN is considered.

In both cases, the root cell of the quadtree results in a 20 � 20 mm square. The spherical indenter has been considered as an elastic half-space. In contrast, two finite dimensions have been considered for the longitudinal direction of the cylindrical indenter, using the correction method described in Section 2.3.

Figure 9. Definition of the indenters for (a) case of study I and (b) case of study II.

The cases of study I and II are solved under several configurations of the proposed approach, selected from the three settings described in Section 3.2, and the performance of each configuration is discussed in Sections 4.1 (for configurations within setting 1), 4.2 (for configurations within setting 2), and 4.3 (for configurations within setting 3).

For each configuration, the computational cost of the approach to solve the contact problem is evaluated using Eq. (16). The accuracy of the approach is evaluated by comparing the obtained contact pressure distributions with reference solutions. For case of study I, the reference solution is determined using the analytical solution provided by the Hertz contact theory [10]. In contrast, since Hertz theory is no longer applicable for case of study II, reference results are obtained for this case using a validated finite element model.
