## 2 Materials and Methods  
### Numerical Simulation  
Simulations that can be performed in the plastic, nonlinear domain and have been implemented in commercial numerical simulation software are inverse analysis and explicit or implicit direct analysis, and optimization analyses. The inverse analysis starts with the final geometric shape of the part (in most cases, deep-drawn). Then, an analysis that does not take into account the forming tools (punch, die, and blankholder), the clearances between them, and the holding forces in the process is used to determine the shape and size of the blank before the plastic deformation process, as well as a presumptive state of the strains and thinning on both the deformed part and the blank. This analysis is useful in the case of forming processes because it allows one to obtain the shape and size of the blank not only based on geometrical considerations, by means of calculating the unfolded length, but also by taking into account the thinning or thickening occurring in the process, based on the law of constant volume. The direct analysis involves the modelling of all elements involved in the forming process, punch, die, blankholder and blank, as well as determining their initial positions so as to ensure the clearances between these elements. The kinematics of the punch or blank are then defined, according to the forming diagram, together with the force or pressure acting on the blankholder and the contacts and frictional forces between the elements participating in the forming process. Direct analyses can be implicit or explicit. Explicit dynamic analyses are used to calculate the unknown functions of a model at a time other than the current time. In contrast, an implicit dynamic analysis finds a solution for the unknown functions by solving an equation that includes both the current time and the time following the current time. This implicit analysis requires additional calculations and is more difficult to implement. Optimization analyses are intended for identifying the material or rheological parameters by combining an analysis with a physically conducted experiment and by imposing the condition that the results of both studies overlap, allowing for a specified coefficient of error. Regarding the type of analysis concerning the numerical simulation of the incremental forming process of polymeric materials, an explicit dynamic analysis was chosen due to the advantages it offers and the reduction in the total time required for the analysis. The program used for the analysis with the finite element method was Ls-Dyna R13.1.0. As far as the construction of the geometrical model of the part proposed for the simulation of the incremental forming process of polymeric materials is concerned, it can be performed either directly in the pre-processing module of the analysis program or in one of the dedicated CAD programs. It is worth mentioning that, since the shell elements were desired for the analysis, the volumes of the tools participating in the incremental forming process (punch, die, and blankholder) were not modelled, but only their outer surfaces, which will come into contact with the blank. Thus, the hemispherical-headed punch is modelled as a sphere with an outer diameter equal to the diameter of the active side of the punch, the die is represented by its outer surfaces, and the blankholder is modelled as a flat surface pressing directly on the blank. The blank is represented by a square surface with a side of 250 mm and is modelled along its middle fibre, i.e., at the middle of its thickness. In order to allow for the comparison of the results of the simulation using the finite element method with the ones obtained experimentally, the following dimensional data were chosen for the active elements: punch diameter, D\({}_{\text{P}}\) = 6 mm; die length, L\({}_{\text{d}}\) = 190 mm; die radius, R\({}_{\text{d}}\) = 5 mm; die corner radius, R\({}_{\text{corner}}\) = 10 mm; and blankholder length, L\({}_{\text{bl}}\) = 190 mm.