## 3 Results and Discussion
Therefore, the forming time measured for H = 1 (t = 1490 s) is reduced by 12% if it is compared to that needed to form a blank with a constant initial thickness (t = 1700 s). The free forming test was performed using the experimental equipment designed at the Technology and Manufacturing Systems Laboratory of the University of Cassino at Cassino, Italy. It has an interface, to acquire data, made up of a PC, a multimeter KEITHLEY 2700 and a power supply ATTEN TPR3003T--an interface to manage the test parameters and to translate the punch, a mold creeper, a punch, and a load cell LUNITEK FT; a crossbar actuated by a rotating screw jack and an electric motor. This equipment allows the submission of discs with a radius value of 40 mm to hot forming tests. Before carrying out the test, the die needs to be uniformly heated in such a way as to bring the disc to a temperature of 440 \({}^{\circ}\)C. This is possible using a system to heat and control the temperature of both the die and the sheet. To carry out the test, once the test temperature is reached, gas is introduced at the predetermined pressure. The control phase of the test involves recording the h-t curve (i.e., displacement at the apex of the dome h depending on the forming time t) through the use of a measurement laser. Moreover, when the forming process ended, the thickness of the deformed disc was measured using a Prismo Vast MPS coordinate measuring machine of Zeiss\({}^{\text{\textregistered}}\). The adopted measuring method of the sheet thickness is reported in [23]. Figure 5 shows the final thickness versus the distance from the symmetry axis for H = 1. It is possible to see that, by using a blank with initial variable thicknesses, the final thickness values are more uniform than those due to a blank with a constant initial thickness. Moreover, the curve due to a blank with a variable thickness shows that the thickness of the formed hemisphere at the pole is larger than that due to a blank with a \begin{table}
\begin{tabular}{c c c c c} \hline
**y = 0 [mm]** & **s\({}_{\text{p}}\) [mm]** & **s\({}_{\text{e}}\) [mm]** & **Max Equivalent Strain** & **s\({}_{\text{p}}\)/s\({}_{\text{e}}\)** \\ \hline x = 0.80 & 0.487 & 0.526 & at edge & 0.926 \\ \hline x = 0.85 & 0.460 & 0.609 & at edge & 0.755 \\ \hline x = 0.90 & 0.437 & 0.653 & at pole & 0.669 \\ \hline x = 0.95 & 0.410 & 0.706 & at pole & 0.581 \\ \hline x = 1.00 & & & & \\ constant thickness & 0.385 & 0.757 & at pole & 0.508 \\ \hline \end{tabular}
\end{table} 
Table 3: FEM optimization analysis for different values of x parameter. constant thickness. This means that the blank with a variable thickness will deform more than the blank with a constant thickness before breaking. This means that the blank with a variable thickness has greater formability than the blank with a constant thickness. In particular, from Figure 5, the \(\mathrm{s_{p}/s_{e}}\) ratio value (for experimental limits \(\mathrm{s_{e}}\) value is measured at a distance of 25 mm from the symmetry axis) obtained for a blank with variable initial thicknesses is 20% higher than that of a blank with a constant initial thickness. The experimental evidence shows that a blank with a non-constant thickness from 1 mm at the pole to 0.9 mm at the edge involves a weight reduction of the formed hemisphere weight of 8% and a forming time decrease of 12% in comparison with a blank with a constant thickness of 1 mm. Moreover, it involves an \(\mathrm{s_{p}/s_{e}}\) ratio 20% less than that due to a blank with a constant thickness of 1 mm. The difference between the percentage reductions of the \(\mathrm{s_{p}/s_{e}}\) ratio values obtained numerically and experimentally (31.7% and 20% respectively) is because the \(\mathrm{s_{e}}\) value was measured at a distance of 25 mm from the symmetry axis since it was not possible to experimentally measure that at the edge.