4.1. Performance of the approach when a uniform mesh is used for the whole domain of the contact problem

The performance of the approach when a uniform pressure element mesh is used for the whole potential contact area is illustrated in this section. To do so, the contact problems defined by cases of study I and II are solved under several configurations of the approach, in which Luni has been varied, keeping Luni ¼ Lmax and wmax ¼ 0 (setting 1 in Section 3.1). Figure 10a–c show examples of the resulting contact area and pressure element mesh that have been obtained for case of study I under this setting of the approach. The computational cost of the proposed

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The contact pressure distributions along the principal axes of the contact area of the solutions shown in Figure 10a–c are shown in Figure 11a. As expected, it can be observed that as the pressure element mesh is refined (by increasing the value selected for Luni), the results obtained

Using this configuration of the approach, a mesh containing 4Luni pressure elements is used, regardless of the nature of the contact problem to be solved. Under these circumstances, the computational cost is proportional to 42<sup>∙</sup>Luni , and the factor of proportionality is the number of finite dimensions taken into account in the contact problem (as explained in Section 2.3). In consequence, for any value of Luni, the computational cost of the algorithm to solve case of

In this section, the performance of the proposed approach when adaptive refinement is performed outside the true contact area is illustrated. To do so, the contact problems defined by cases of study I and II are solved under several configurations of the approach, in which Luni and Lmax have been varied, keeping Luni < Lmax and wmax ¼ 0 (setting 2 in Section 3.1). Figures 10d and 12a show examples of the resulting contact area and pressure element mesh

The results obtained in these cases show that the accuracy in which the contact problem is solved is independent of the value selected for Luni. For any given value of Lmax, the same contact pressure distributions as the ones obtained with a uniform pressure element mesh for the whole domain have been obtained (shown in Figure 11a), regardless of the value selected for Luni. This implies that the variation of the pressure element mesh outside the true contact

On the other hand, comparing the computational cost of the solutions shown in Figure 10c (uniform mesh) and 10d (adaptive refinement outside the true contact area), it can be observed

study II will always be greater than the computational cost to solve case of study I.

4.2. Performance of the approach when adaptive mesh refinement is performed

that have been obtained for cases of study I and II under this setting of the approach.

area does not have any impact on the solution of the contact problem.

approach to solve the case of study I is also shown for each configuration.

Figure 11. Contact pressure distribution for CoSI under several configurations of the approach.

by the proposed approach converge toward the reference solution.

outside the true contact area

Figure 10. Axisymmetric representation of the resulting contact area and pressure element mesh obtained for CoSI under several configurations of the approach.

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Figure 11. Contact pressure distribution for CoSI under several configurations of the approach.

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

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

The performance of the approach when a uniform pressure element mesh is used for the whole potential contact area is illustrated in this section. To do so, the contact problems defined by cases of study I and II are solved under several configurations of the approach, in which Luni has been varied, keeping Luni ¼ Lmax and wmax ¼ 0 (setting 1 in Section 3.1). Figure 10a–c show examples of the resulting contact area and pressure element mesh that have been obtained for

Figure 10. Axisymmetric representation of the resulting contact area and pressure element mesh obtained for CoSI under

within setting 2), and 4.3 (for configurations within setting 3).

are obtained for this case using a validated finite element model.

domain of the contact problem

74 Contact and Fracture Mechanics

several configurations of the approach.

4.1. Performance of the approach when a uniform mesh is used for the whole

case of study I under this setting of the approach. The computational cost of the proposed approach to solve the case of study I is also shown for each configuration.

The contact pressure distributions along the principal axes of the contact area of the solutions shown in Figure 10a–c are shown in Figure 11a. As expected, it can be observed that as the pressure element mesh is refined (by increasing the value selected for Luni), the results obtained by the proposed approach converge toward the reference solution.

Using this configuration of the approach, a mesh containing 4Luni pressure elements is used, regardless of the nature of the contact problem to be solved. Under these circumstances, the computational cost is proportional to 42<sup>∙</sup>Luni , and the factor of proportionality is the number of finite dimensions taken into account in the contact problem (as explained in Section 2.3). In consequence, for any value of Luni, the computational cost of the algorithm to solve case of study II will always be greater than the computational cost to solve case of study I.

#### 4.2. Performance of the approach when adaptive mesh refinement is performed outside the true contact area

In this section, the performance of the proposed approach when adaptive refinement is performed outside the true contact area is illustrated. To do so, the contact problems defined by cases of study I and II are solved under several configurations of the approach, in which Luni and Lmax have been varied, keeping Luni < Lmax and wmax ¼ 0 (setting 2 in Section 3.1). Figures 10d and 12a show examples of the resulting contact area and pressure element mesh that have been obtained for cases of study I and II under this setting of the approach.

The results obtained in these cases show that the accuracy in which the contact problem is solved is independent of the value selected for Luni. For any given value of Lmax, the same contact pressure distributions as the ones obtained with a uniform pressure element mesh for the whole domain have been obtained (shown in Figure 11a), regardless of the value selected for Luni. This implies that the variation of the pressure element mesh outside the true contact area does not have any impact on the solution of the contact problem.

On the other hand, comparing the computational cost of the solutions shown in Figure 10c (uniform mesh) and 10d (adaptive refinement outside the true contact area), it can be observed that an important reduction of the computational cost is achieved by increasing the difference between Lmax and Luni. Similar tendencies are observed for case of study II, where the reductions of computational cost are even more remarkable due to the presence of finite dimensions.

gradient is small, without a significant loss of accuracy when describing the contact pressure

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The obtained results show that the accuracy of the approach to predict the size of the true contact area does not depend on the value selected for wmax, since the same values are obtained regardless of the value selected for this parameter. This is because when wmax < 1, the accuracy in which the border of the contact area is computed depends only on the value selected for

Finally, comparing the computational cost of the solutions shown in Figure 10d and e (and Figure 12a and b), it can be observed that a further reduction of the computational cost can be achieved by specifying values of wmax > 0. Although this reduction is not as important as the one achieved by maximizing Lmax � Luni (discussed in Section 4.2), it still can help to reduce the

A new semi-analytical approach has been developed to solve frictionless elastic contact problems using adaptive mesh refinement. Starting from a coarse initial uniform mesh (whose density is defined by the parameter Luni), a mesh refinement is performed based on two different criteria: (i) the maximum allowed rate of change of a physical magnitude (the contact pressure), defined by the parameter wmax and (ii) the maximum degree of mesh refinement,

The configuration of the approach is defined by a unique combination of values for Luni, Lmax, and wmax. The performance of the proposed approach has been illustrated with several cases of study solved under different configurations of the approach, and the obtained results enable

i. When Luni ¼ Lmax, a uniform mesh is used to solve the contact problem, regardless of the value selected for wmax. Under this configuration, it can be observed that the obtained results converge toward the reference solution as Luni is increased. However, an exponential growth of the computational cost is produced as the pressure element mesh is refined.

ii. When Luni < Lmax and wmax ¼ 0, adaptive mesh refinement is performed outside the true contact area. Under these circumstances, it can be observed that the computational cost of the approach is reduced by maximizing Lmax � Lini, while the accuracy of the solution

iii. In last place, when Luni < Lmax and wmax > 0, adaptive mesh refinement is performed both inside and outside the true contact area. Under these circumstances, it can be observed that a further reduction of the computational cost can be achieved. However, a loss of accuracy can be expected in the prediction of the contact pressure distribution as wmax is

distribution.

Lmax, as stated in Section 3.2.

5. Conclusions

computational cost of the approach.

defined by the parameter Lmax.

remains unaffected.

A further discussion on this topic can be found in Ref. [11].

increased.

us to draw the following conclusions:
