*3.1. Measurement Results of the Real Dimple Dimensions and Definition of Dimensionless Dimple Parameters*

The aim of the real dimple dimensions measurement procedure, which has been previously described in Section 2.3, is the identification of the effective texture geometry. While the real dimple diameter and depth reveal larger values compared to the nominal dimensions, the dimple distances are smaller than the nominally specified values, compare Tables 1 and 3.

In addition to the diameter, distance, and depth dimple parameters, the dimensionless aspect ratio and textured area (equal to area density), which are often referred in literature [36–38], are further specified in Table 3. The aspect ratio is the quotient of the dimple depth and the dimple diameter and varies between 0.04 and 0.11 for the analysed textures. The area density is equal to textured area percentage. It is calculated by the quotient of the textured area *Atextured* and the nominal circular contact area *Anominal*, which are both visualised in Figure 5.

**Figure 5.** (**a**) Red marked nominal circular contact area *Anominal* between an untextured rubber specimen and the counter surface and (**b**) contact area of a textured sample. The textured area *Atextured* is indicated by the black circular dimples, the untextured area that is in direct contact with the counter surface is coloured in blue.

The area density of the studied textures varies between 9% and 55%. Despite the fact that the three different normal forces *FN* analysed result in three different nominal contact areas *Anominal*, the percentage textured area is independent of the nominal contact area *Anominal*, since more dimples come into contact with increasing normal force *FN*, compare also Table 2b.

#### *3.2. Reduced Order Modelling on Friction Coefficient Data*

The two ROM functions (please refer to Equations (1) and (3)), obtained for untextured and textured friction coefficient datasets, converged to a stable solution, using only 2 and 17 terms, respectively. In order to assess the correctness and precision of the ROM results, the predicted value is compared to the corresponding experimental one, as ideally, both should be the same. The prediction lines are plotted, for both untextured and textured ROM models, in Figure 6a,b.

From Figure 6a,b it is possible to conclude that the ROM prediction is extremely accurate, being the obtained ROM models' standard deviations *σuntext* = 0.0014 and *σtext* = 0.0012 for untextured and textured data, respectively, and both regression lines (dashed black) match the goal prediction line (solid red).

Both ROM models were validated using the k-fold cross validation technique (see Section 2.4). The results obtained for the *R*<sup>2</sup> are shown in Table 4a,b for untextured and textured data, respectively.

**Figure 6.** ROM prediction for (**a**) untextured and (**b**) textured friction coefficient data. The blue dots are obtained by evaluating the ROM with the experimental data, the dashed black line is the linear regression that fits the data, while the solid red line represents the ideal ROM result, where the ROM evaluation on each experimental friction value returns the same value.



Table 4a,b show that for both ROM models the results of the validation are very precise and that there is always an excellent correlation between the input data, i.e., untextured or textured friction coefficient data (available under [24]), and the ROM prediction.
