###### Abstract  
This work deals with the design of the thickness of an AZ31 alloy blank, which is a superplastic magnesium material, to manufacture a hemisphere with a uniform final thickness. The finite element technique was used for the design process. The superplastic free-forming manufacturing was simulated for a part whose initial thicknesses were made to vary through two independent design parameters to obtain a linear thickness decrease from the pole to the end of the blank to form. This is because a linear thickness decrease is easily obtained through a machining process. The optimized blank, that is, the blank with a non  
The design of the superplastic forming processes involves the finite element method to determine the optimal load curve, that is, the curve which, by correlating the forming pressure to the forming time, can deform the sheet through the optimal values of strain rates [6,7]. Since forming times are very long (in the order of tens of minutes), researchers have recently oriented towards faster hot sheet metal forming processes by using the blow forming process with a decreased forming temperature and an increased pressure, thus the superplastic conditions arise [8,9]. To obtain a hemisphere, the superplastic forming process of a disc with a constant initial thickness was extensively studied in the past [10]. These studies show that the manufactured hemisphere presents thickness gradients from the axis of symmetry to the edge of the sheet and, therefore, non-uniform distribution of thicknesses. This thickness non-uniformity is as much greater as smaller as the value assumed by the strain rate sensitivity index, m [10,11]. To yield the final distribution of the thicknesses as uniformly as possible, two approaches can be followed:  
* Use multiphase superplastic forming;
* Use blanks with an initial variable profile of thicknesses. Multi-phase superplastic forming generally involves the first step of sheet forming to obtain a pre-shape that is afterwards submitted to a final forming step to achieve the final product. In this way, it is possible to obtain a more uniform final distribution of the thicknesses, but the technique involves long forming times because it requires different forming steps [12, 13, 14]. Conversely, the forming times are reduced if a blank with an initial variable profile of the thicknesses are used. In [15, 16, 17], how to prepare the blank with an initial variable profile of the thicknesses was not investigated. Moreover, the optimization of the initial profile of the blank thicknesses gives rise to profiles that follow laws that are difficult to actually achieve (such as parabolic laws or with discontinuous variations). In [11], the influence of a sheet with an initial variable profile of thicknesses, obtained by machining an aluminum alloy sheet with a constant initial thickness, on the material formability was investigated for a hot gas sheet metal forming process. This work aims to optimize the initial variable profile of blank thicknesses to make more uniform the thickness of the product manufactured through a free-forming process. The free-forming process constitutes the first step of a superplastic forming process. Furthermore, it represents a suitable process to characterize the behavior of the material. A more uniform thickness in a formed product allows the avoidance thinning that may involve early breaks of the part; therefore, it leads to greater strength of the product. The starting point was a disc with a radius of 40 mm and a constant thickness of 1 mm made of a superplastic AZ31 magnesium alloy. It was submitted to gas blow forming, such as in [18], where the authors studied the microstructure of the material once formed by starting from  
Figure 1: Scheme of the blow forming process. a constant thickness. They looked for a profile involving a linear thickness decrease that is easily obtained by a machining process.