*2.5. Reduced Order Modelling Data Pre-Processing for Friction Reduction*

The friction variations (in%) obtained with the introduction of a dimple texture on the rubber surface are evaluated using Equation (4) in order to quantify the surface textureinduced friction variations.

$$
\Delta\mu = 100 \frac{\mu\_{\text{untex}} - \mu\_{\text{tot},i}}{\mu\_{\text{untex}}} \% \alpha \tag{4}
$$

In Equation (4) textured cases are compared to their corresponding untextured result, so that a negative value suggests a friction increase and a positive one reveals a friction reduction.

The final Δ*μ* prediction was obtained using Equation (4), where the *μuntext* and the *μtext*,*<sup>i</sup>* values, predicted by the ROM models, were used, instead of using the experimentally measured results.

The two datasets, prepared and used in the statistical analysis, were created from two data samples of 48 data points each, using Python NumPy version 1.18.1 [34] and SciPy version 1.4.1 [35] libraries, as described in Sections 2.6 and 3.3.

#### *2.6. Statistical Analysis of Real Dimple Dimensions*

Due to manufacturing tolerances, the real dimple dimensions are not equal to the nominal ones. Experimental data were used to extract statistical information from 16 measurements of geometrical parameters of the textured surface, i.e., mean values and standard deviations for the three geometrical parameters were identified: dimple depth, diameter, and distance. The measuring procedure and mean values are described in Sections 2.3 and 3.1, respectively. By comparing the results to a foreseen normal distribution centred on the nominal value, having as standard deviation the value obtained from the different repetitions, it was possible to infer the type of probability density function (PDF) of each of the studied parameters.

Once the PDFs were known, the dimple depth was varied separately, while diameter and distance were varied together, as their variations are not independent and occur in the same plane, as defined by the textured area (Section 3.1). To do that, not all data were taken into account, but only those dimple parameters whose variations fell within the ROM's definition limits, minimum and maximum values in Tables 1 and 2, which resulted in 48 data points for both datasets, that were obtained when vertical, i.e., depth, and horizontal, i.e., diameter and distance, variations were applied.

The PDFs were hence created by means of a Python script and were used to generate the above mentioned datasets of 48 data points each, as it follows. Each original point was replicated 100 times where the nominal values were changed accordingly to their evaluated PDFs and a final amount of 4800 points were obtained. As mentioned before, the described approach was performed two times: one repeating depth values only and a second time replicating diameter and distance values together. Since dimple diameter and distance are dependent magnitudes, these were made to vary accordingly to one another as follows: at first a new diameter value was randomly generated through the known PDF, then the cumulative density function (CDF) was obtained for the specific diameter value and the corresponding distance value was calculated as the value holding the complement of the CDF in the distance PDF.

The new obtained dataset contains input parameters statistical fluctuations and was used to perform ROM predictions on the enlarged sets of points, where the dimple values experimental variations on dimple dimensions' nominal values are taken into account. The effects on friction due to the experimental texturing deviations were then analysed and a t-Student test was used for a statistical comparison of the obtained results to an ideal PDF around nominal surface texture values, as described in Section 3.4.
