## 2 Materials and Methods
### Geometric Algorithm to Evaluate the Thickness of the Stretched Sheet  
Once the stretched sheet was simulated, the aim was to measure its thickness in many sections; the used software allowed measuring of the distance between two nodes of each element, although not the minimum distance that may be measured perpendicularly to the profile tangent (see Figure 3). Therefore, a geometric algorithm was developed to be applied to the simulated sheet in order to obtain the minimum value of the thickness in different sections along the deformed profile.  
The numerical model produced a set of points on the two semi-profiles of the two sides of simulated stretched sheet. Each semi-profile was fitted by a polynomial equation (first step in Figure 4), such as:  
\[x=a_{i}y^{i}+a_{i-1}y^{i-1}+\ldots+a_{1}y^{1}+a_{0} \tag{3}\]  
where \(i=50\) and was sampled through 61 points (where 0 was on the symmetry axis).  
A geometric algorithm was applied to calculate the distance between each couple of points on the two semi-profiles of the inner and outer surfaces, considering the following couple of facing profiles on the inner and outer surfaces:  
\[x_{out}=a_{i_{out}}y^{i}+a_{i-1_{out}}y^{i-1}+\ldots+a_{1_{out}}y^{1}+a_{0_{ out}} \tag{4}\]  
\[x_{in}=a_{i_{in}}y^{i}+a_{i-1_{in}}y^{i-1}+\ldots+a_{1_{in}}y^{1}+a_{0_{in}} \tag{5}\]  
For each point \(P(x_{0_{out}},y_{0_{out}})\) of the profile, the equation of the tangent line to the curve (4) at the point \(P\) was calculated:  
\[x_{tang}=f^{\prime}(y)\cdot y+q \tag{6}\]  
where \(f^{\prime}(y_{0_{out}})\) is the first derivation of Equation (4) at the point \(P\), and \(q\) is equal to:  
\[q=f(y_{0})-f^{\prime}(y)\cdot y_{0} \tag{7}\]  
Figure 3: Distance and minimum distance among inner and outer numerical profiles.  
Therefore, the line perpendicular to Equation (6) at the point \(P\) was calculated as:  
\[\mathrm{x}_{perp}=-\frac{1}{f^{\prime}(y)}\cdot y+q \tag{8}\]  
Its intersection with the profile (5) was calculated as the point \(P^{\prime}\left(x_{0_{in}},y_{0_{in}}\right)\).  
The distance between the two points \(P\) and \(P^{\prime}\) was calculated to evaluate the thickness of the stretched sheet as:  
\[thickness(x_{0},y_{0})=\sqrt{\left(x_{0_{out}}-x_{0_{in}}\right)^{2}+\left(y_{0 _{out}}-y_{0_{in}}\right)^{2}} \tag{9}\]  
The same reasoning was carried out point by point along the two profiles (4) and (5) to obtain the thickness trend.  
Figure 5 represents the trend of the sheet thickness obtained by the developed numerical model with three values of the friction coefficient: 0, 0.1 and 0.2. Figure 5 shows how the value of the friction coefficient influences the thickness trend, mainly in the region where the contact between sheet and punch is direct. In the absence of friction (\(\mu=0\)), the sheet metal material flowed undisturbed over the punch; the result was a thickness trend that decreased from the edge towards the component's symmetry axis. In the presence of friction (\(\mu=0.1\) and \(\mu=0.2\)) the stretching of the sheet was restrained by the presence of the punch, thus determining a thinning of the sheet in a region away from the axis of symmetry. Figure 4c is only an example, which cannot be considered for a comparison with the experimental results; the numerical simulation does not contain a model of material damage.  
Figure 4: Flowchart of the developed geometric algorithm applied to numerical data.