## 3 Results and Discussion  
Tables 2 and 3 show the results of the optimization process that was carried out to achieve more uniform thickness values of the formed hemisphere. In these tables, the \(\rm{s_{p}/s_{e}}\) ratio is used to evaluate the thickness uniformity. Table 2, built for \(\rm{H=1}\) and \(\rm{x=0.90}\) mm, shows how, with decreasing \(\rm{y}\) value, the \(\rm{s_{p}/s_{e}}\) ratio tends to increase up to 0.669 for \(\rm{y=0}\). This value is higher than that obtained for a blank with a constant initial thickness (\(\rm{s_{p}/s_{e}=0.508}\)) by 31.7%. Therefore, the value 0 represents the optimal condition for the \(\rm{y}\) parameter. Table 3, drawn up for \(\rm{H=1}\) and \(\rm{y=0}\), shows that as \(\rm{x}\) decreases, the \(\rm{s_{p}/s_{e}}\) ratio increases by achieving the highest values. However, the numerical simulation shows that the maximum equivalent strain, for \(\rm{x=0.85}\) and \(\rm{x=0.80}\) mm, is placed on the edge of the sheet, unlike what happens for the other conditions, where the maximum equivalent strain is placed on the pole of the sheet. Therefore, to avoid the sheet breaking on the edge, only the simulations that produce a maximum equivalent strain on the pole of the sheet were considered to reproduce processes feasibly. This consideration shifts the optimal value of the parameter \(\rm{x}\) from 0.85 mm and 0.8 mm to 0.9 mm. In the end, to achieve more uniform thicknesses in the formed hemisphere made of superplastic AZ31 magnesium alloy, the optimal values of the geometric \(\rm{x}\) and \(\rm{y}\) parameters are 0.90 mm and 0 mm respectively; they involve a value of \(\rm{s_{p}/s_{e}}\) ratio equal to 0.669. \begin{table}
\begin{tabular}{c c c c c} \hline x = 0.9 [mm] & \(\rm{s_{p}}\) [mm] & \(\rm{s_{e}}\) [mm] & Max Equivalent Strain & \(\rm{s_{p}/s_{e}}\) \\ \hline y = 0 & 0.437 & 0.653 & at pole & 0.669 \\ \hline y = 10 & 0.427 & 0.643 & at pole & 0.664 \\ \hline y = 15 & 0.420 & 0.639 & at pole & 0.657 \\ \hline y = 20 & 0.413 & 0.634 & at pole & 0.651 \\ \hline x = 1.0 y = 0 & & & & \\ constant thickness & 0.385 & 0.757 & at pole & 0.508 \\ \hline \end{tabular}
\end{table} 
Table 2: FEM optimization analysis for different values of y parameter. Figure 4: Flowchart of the proposed numerical approach. The simulation evidence shows how a blank with a linear non-constant thickness from 1 mm at the pole to 0.9 mm at the edge involves an \(\rm{s_{p}}/\rm{s_{e}}\) ratio equal to 0.669 which is 31.7% greater than that due to a blank with a constant thickness of 1 mm (0.508). This is because a smaller thickness of material at the edge involves an increase of the strain rate at the edge that makes the material flow plastically from the edge to the pole. This plastic flow of material happens while the material at the pole, free from constraints due to the die, is characterized by a high value of strain rate that makes the material flow plastically from the pole to the edge. These two flows imply a more uniform thickness from the pole to the edge of the resulting product. To validate the obtained optimization values, some experimental free-forming tests were carried out at a constant pressure of 0.4 MPa on two circular blanks in superplastic AZ31 magnesium alloy. The first one was a disc-shaped blank with a constant thickness of 1.0 mm, while the second was a circular blank with a variable initial thickness, that is, its thickness varies linearly from the axis of symmetry, where it is equal to 1.0 mm, up to the edge where it reaches a value of 0.90 mm, according to the optimal value obtained during the previous design stage. Three replicates for each type of blank were considered. The blank with a variable initial thickness was manufactured by removing chips from a sheet with a constant initial thickness of 1.0 mm. In this way, the machined blank had a volume of 4618 mm\({}^{3}\) which is about 8% lower than that of the blank with a constant initial thickness (5026 mm\({}^{3}\)).