## 2 Materials and Methods  
The finite element analysis was carried out starting from a circular disc with a radius of 40 mm and a thickness of 1.0 mm. The material of the disc was an AZ31 magnesium alloy whose chemical composition by weight is Mg-3% Al1% Zn. The material has superplastic performances at a temperature of 440 \({}^{\circ}\)C [19]. Under these conditions, the hardening law follows the power law of the type:  
\[\overline{\sigma}=\mathrm{C}\overline{\varepsilon}^{\mathrm{n}}\cdot\frac{ \mathrm{\Sigma}^{\mathrm{m}}}{\overline{\varepsilon}^{\mathrm{m}}}, \tag{1}\]  
In Equation (1), \(\overline{\sigma}\) is the equivalent stress, \(\overline{\varepsilon}\) is the equivalent strain, \(\overline{\varepsilon}\) is the equivalent strain rate while m, n, and C (i.e., the strain rate sensitivity index, the hardening index, and the strength coefficient respectively) are material constants. These constants have a value of m = 0.407, n = 0.12, C = 289.9 MPa respectively, that were experimentally identified, as reported in [19], and used to simulate different superplastic forming processes using the AZ31 alloy [20]. These values characterize a material affected by a little strain-hardening [20]. The material constants were obtained by free forming tests performed at different pressure levels (0.275 MPa and 0.4 MPa) by using the inverse analysis technique to minimize the differences between the numerical data and the experimental measurements [19].  
Both the use of an analytical model [21] and the experimental data [22] showed that a free-forming process of a sheet with an initial constant thickness produces a trend of the final thickness, s, along the arc length, l, varying according to a parabolic relationship such as:  
\[\mathrm{s}(\mathrm{l})=\mathrm{s}_{\mathrm{p}}+\frac{\mathrm{s}_{\mathrm{e}}- \mathrm{s}_{\mathrm{p}}}{\mathrm{L}^{2}}\mathrm{l}^{2}, \tag{2}\]  
where the subscripts e and p refer to the edge and the pole of the hemisphere respectively, while L is the arc length.  
Subsequently, in [11], the correlation between the thickness at the edge, \(\mathrm{s}_{\mathrm{e}}\), and the thickness at the pole, \(\mathrm{s}_{\mathrm{p}}\), was established through the same simple analytical model for both an initial blank with a constant thickness and an initial blank with a variable thickness. However, this result is affected by the use of an analytical model with simplifying hypotheses. Therefore, this work used the finite element analysis to optimize the initial profile of blank thicknesses to manufacture a hemisphere in superplastic AZ31 magnesium alloy (whose geometry is characterized by H = h/a = 1 with a = initial radius of the disc and h = height of the hemisphere) with a final distribution of the thicknesses as uniformly as possible.  
The simulation of the free-forming test, starting from an initial variable profile of thicknesses, was carried out using the non-linear finite element software package MARC(r) by MSC. The rigid-plastic flow formulation was used for the superplastic forming analysis. Due to the axisymmetric nature of the problem, an axisymmetric simulation model was used. The scheme of the simulated free-forming test is shown in Figure 2. The disc was discretized through two rows of 128 continuous 4-node iso-parametric elements commonly used for axisymmetric applications. The nodes on the symmetry axis are constrained in such a way that they cannot move orthogonally to this axis. Moreover, to simulate the presence of a blank holder, the nodes placed on the periphery of the blank were constrained. The die is modeled as a rigid body. The disc is deformed by a gas at a constant pressure that is applied uniformly to the upper edges of the disc.