## 1 Introduction  
Industry 4.0 aims to digitalize the manufacturing process to improve working conditions, to create new business models and to increase the productivity and the product quality of the plants. It tends to advanced production systems, i.e., interconnected and modular systems, that allow flexibility and performance. A fundamental aspect of the digitalized manufacturing processes is the simulation of the manufacturing process in order to develop its virtual representation, known as digital twin, whose purposes may be monitoring, and control [1]. The simulation purpose is to reproduce the behaviour of physical objects in a virtual space in order to plan or optimize products or production plants. Algorithms to elaborate the simulated data in order to improve the control of the manufacturing process are very important, and they need to be developed. Sheet metal forming is a widely used process to manufacture parts with a high production rate and a low cost. A metal sheet is formed once it is put between a punch and a die, whose geometries are negative to one another. To assess the quality of the metal sheet forming process, the material formability, the spring-back that changes the dimensions of the part and the minimization of the -dimensional variations due to the lubrication, the material properties, and the thickness variation all need to be taken into account. A framework is discussed [PERSON] et al. [2], that allows to simulate, through a finite element model, the influence of these process parameters on the sheet-stamping process in order to organize and filter raw data. The use of materials lighter than steel, for example aluminium and magnesium alloys, reduces the component weight and, in some fields such as the automotive industry, increases the fuel economy. However, these metal alloys have a reduced formability when compared with that of steel and produce greater problems in terms of both spring-back and sheet metal breakage. To overcome these manufacturing problems, some authors used hydraulic actuators to control the force on the blank-holder [3], to reduce processing times [4] and to prevent defects [5; 6]. The complexity of sheet metal forming is connected with the large plastic deformations involved in the process. The geometry of the dies, the friction in the sheet metal-die interface, and the force on the blank-holder mainly influence the process. An unsuitable evaluation of these manufacturing parameters can reduce the quality of the part being worked, because it gives rise to wrinkles, tears, excessive spring-back and sheet thinning. Traditionally, to design a sheet metal forming process trial-and-error methods were used exclusively. This approach was expensive and time-consuming. Nowadays, finite element analysis, able to simulate the forming process and to take into account all the manufacturing parameters, allows a faster and less expensive design than the traditional method. The reliability of the results, costs, benefits, and times required to perform numerical simulations for industrial examples of sheet metal forming that predict fracture, final sheet thickness, wrinkling, loads, spring-back and residual stresses in the product was evaluated by [PERSON] et al. [7]. Finite element analysis allows the consideration of the effects of both the spring-back and the material formability in plastic deformation processes [8]. The sheet formability is typically due to the stretching of the sheet over a punch. Complex shapes may be planned by finite element analysis and obtained, such as double curved surfaces [9; 10]. A simple method to predict the spring-back of an aeronautical panel is presented by [PERSON] and [PERSON] [11].The deformation of small circles marked on the sheet surface is measured, whereas there is the tear, and the forming limit curve is tracked. Forming limit diagrams (FLD) are widely used as a forecasting method to avoid damage of sheet material in plastic deformation processes. Different models to determine the forming limit curves were presented by [PERSON] et al.