## 3 Experimental Validation and Results Discussion
The corresponding profile on the inner surface was identified in order to obtain a couple of profiles, one on the inner and one on the outer sides of the sheet, along the horizontal and vertical directions. Therefore, each profile was fitted by a polynomial equation [26], such as that of Equation (3), for the horizontal profiles and Equation (4) was used for the vertical profiles:  
\[x=b_{j}z^{j}+b_{j-1}z^{j-1}+\ldots+b_{1}z^{1}+b_{0} \tag{10}\]  
where \(j=50\). Then, a set of 121 points were sampled on each polynomial equation by imposing a value of \(y\) or \(z\), starting from 0 on the symmetry axis and sampling 60 points on both the right and on the left of the symmetry axis, and finding the corresponding values of \(x\) in Equations (3) and (10). The average of the \(x\)-coordinate of the points at the same distance from the symmetry axis was carried out in order to obtain one semi-profile on each side of the stretched sheet, constituted of 61 points by considering the 0 on the symmetry axis, to compare with that due to the numerical model. The same reasoning from Equation (4) to Equation (9) was applied to the two obtained sets of 60 points on the two sides of the manufactured stretched sheet in order to calculate the numerical thickness distribution. Figure 11a shows the experimental profiles (inner and outer) of the deformed sheets for a stroke of the punch up to 10 mm, 20 mm and breaking. Inner and outer profiles of the stretched sheet were measured (see Figure 11a) to obtain the thickness along the sheet (see Figure 11b). Figure 11b reports the experimental trend of the sheet thickness for the three different forming steps. Figure 11b underlines how the minimum value of the sheet thickness moves away from the symmetry axis of the spherical surface; it is due to the presence of friction between the sheet and punch. In fact, friction slows the sliding of the material on the punch by moving the minimum thickness value away the axis of symmetry as increasing as the stroke of the punch. This demonstrates the correct working of the proposed geometric algorithm. Figure 12 shows the comparison between the two sheet thickness trends due to the numerical model and the experimental tests. It should be noted that a greater discrepancy between the results is where contact occurs between sheet and punch. In particular, the numerical thickness variation due to a friction coefficient \(\mu=0.1\) is the nearest to that obtained experimentally. Table 1 shows the maximum value of the percentage difference in absolute value \(\big{|}\big{(}th_{num}-th_{hexp}\big{)}/th_{exp}\big{|}*100\) between the thickness values due to numerical (\(th_{num}\)) and experimental (\(th_{ex}\)) approaches along the 61 points on the inner and outer surfaces of the stretched sheet. This percentage difference ranges between a minimum of 4%, when \(\mu=0.1\) and the punch stroke is equal to 20 mm, and a maximum of 9%, when \(\mu=0\) and the punch stroke is equal to 20 mm. This maximum value is probably due to the model adopted to describe the plastic behaviour of the material. However, the difference between numerical and experimental results is so low that it can be considered negligible. Figure 13 shows the curves related to the thickness results versus the distance from the symmetry axis due to traditional node-to-node method and the proposed geometric algorithm. The difference between the two methods depends on the dimensions of the mesh used to discretize the part (fine or coarse), the deformation depth and the friction coefficient. \begin{table}
\begin{tabular}{|p{113.8 pt} p{113.8 pt} p{113.8 pt}|} \hline
**Punch Stroke** & **Percentage Difference** 1 (_th\({}_{mm}-\)th\({}_{exp}\)_)/_th\({}_{exp}\)_1 \(\times\) 100 (\%)_ \\ \hline  & \(\mu\) = 0 & \(\mu\) = 0.1 & \(\mu\) = 0.2 \\ \hline \(\approx\)10 mm & 2.25 & 2.21 & 2.18 \\ \hline \(\approx\)20 mm & 8.69 & 4.01 & 5.37 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison between numerical and experimental values of thickness.