## 2 Materials and Methods
### Numerical Simulation
The PA 6.6 and HDPE sheets had the same thickness (\(t\) = 3 mm) and did not undergo any pretreatment. Regarding lubrication, for the PA 6.6 sheets, we used a synthetic lubricant, recommended for the forming process of these types of polymeric materials, namely, Vascomill CSF 10, produced by Blaser Swisslube. This type of vegetable oil-based lubricant avoids the risk of contamination of the polymeric material and is environmentally friendly. The lubricant was applied to the surface that comes into contact with the punch. When processing polymeric materials by forming, the phenomenon of self-heating occurs. In the case of single-point incremental forming, this is more pronounced because the punch has a complex trajectory that leads to greater friction with the polymeric sheet. By ensuring the lowest possible roughness of the punch (\(Ra\) = 0.8 \(\upmu\)m), proper lubrication, and a larger vertical step (\(s\) = 1 mm), we ensured that the part temperature did not exceed 64 \({}^{\circ}\)C in the case of PA 6.6 and 56.3 \({}^{\circ}\)C in the case of PEHD. The type of finite element considered is a thin shell element, suitable for surface discretization. Both the tools (punch, die and blankholder) and the blank were discretized with this type of element. Shell elements have had a number of formulations over the years. The \(-16\) formulation appropriately modifies an existing standard assumed strain element, namely, the type 16 element (fully integrated shell), which is a fully integrated extension of the Belytschko-Lin-Tsay element. In the analyses presented in the paper, the \(-16\) formulation was used, as it leads to an acceptable accuracy under the conditions of a lower running time. In order to choose the type of material for each part that composes the forming model, the non-deformable solid parts (forming tools) had to be separated from the deformable part (blank). As far as the forming tools are concerned, there was no problem in choosing the type of material because Ls-Dyna R13.1.0 provides the user with a non-deformable type of material for which, in addition to the material properties, any constraints that may be required must also be determined when defining the material. This material model is called Rigid in the program. Thus, the following parameters were introduced for the die: the longitudinal modulus of elasticity, E = 210,000 MPa, the density \(\rho\) = 7.85 \(\times\) 10\({}^{-9}\) t/mm\({}^{3}\) and Poisson's ratio, \(\
u\) = 0.3. In addition, since the die is fixed throughout the entire forming process, all six degrees of freedom were cancelled. While the same material input data were entered for the punch, only the degrees of freedom related to the rotations of the punch were cancelled, thereby allowing for the 3 translational degrees of freedom. There is an issue concerning the cancellation of the rotational degrees of freedom because there are situations where the incremental forming process is carried out on a machining centre or a CNC milling machine, where the punch can rotate around its own axis (Oz axis). In the present paper, where the forming process will be performed on an industrial robot, the punch is unable to rotate around its own axis; therefore, the three rotational degrees of freedom were cancelled. The material data assigned to the blankholder are the same as for the die and the punch and, with regard to the degrees of freedom, all rotations and translations in the Ox and Oy directions were cancelled, only allowing for the translation in the Oz direction, which is the direction of the holding force. As for the choice of the material model for the blank, the deformable solid, two models were chosen from the Ls-Dyna R13.1.0 program library, which are recommended in the case of polymeric materials. Both material 24 and material 89 can be used to simulate the forming process of polymeric materials. Material model 24 is widely used in Ls-Dyna R13.1.0 due to its robust algorithms, computational efficiency, and support for load curve-based input.