## 3 Process Simulation
### Finite Element Method-Based Process Model  
The axial compression forming process, employed to decrease the workpiece height while increasing its diameter, cannot be described in detail due to confidentiality constraints. However, the following overview outlines the origin of the dataset used in this study. The process is first modeled using the Finite Element Method (FEM). The simulation was developed using Simufact forming 2021 software from Hexagon AB, which employs an implicit finite element solver. To simplify the model, a two-dimensional, rotationally symmetric approach was chosen. This approach is permissible due to the rotationally symmetric forming in the process investigated. Figure 1a depicts the setup of the simulation model. The simulation model consists of all parts that interact with the workpiece. Accordingly, the punch, the workpiece, the die, and the ejector are considered. The punch, die, and ejector are simulated as rigid bodies, as the yield strength of the tool materials is significantly higher than that of the workpiece material, in order to reduce simulation time. In particular, to account for the machine stiffness \(k\), a spring element is included for the punch. The spring constant is considered an input parameter, since changes may occur due to different machines, lubrication conditions, or temperatures. A vertical movement of the punch realizes the kinematics of the forming process. The vertical motion is therefore not only influenced by the stroke \(h\) but also by the machine stiffness \(k\). The position of the dieand the ejector are fixed in place. The elastic-plastic material behavior of the C14C (1.0401) workpiece is taken from the Simufact forming material database [25]. The initial mesh of the workpiece is shown in Figure 1b. The minimum mesh size \(e_{\text{min}}\) of the hexahedral mesh in the area of forming is \(e_{\text{min}}=0.1\) mm. During the simulation, remeshing was taken into account to avoid excessive distortion of individual elements. The strain of the elements triggers remeshing. In addition to the machine stiffness \(k\) and the stroke \(h\), the initial tool temperature \(T_{\text{tool\_0}}\) and the initial flange thickness \(b_{\text{FL\_0}}\) were used as input parameters. As output parameters, the flange thickness \(b_{\text{FI}}\), the flange radius \(r_{\text{FI}}\), and twelve forming forces evenly distributed over the stroke \(h\) were considered. Experimental forming tests validate the simulation model using a tool provided by the ESW GmbH on a Schuler HPX400 from Schuler Group GmbH. An initial tool temperature \(T_{\text{tool\_0}}\) corresponding to room temperature was ensured by allowing sufficient cooling times between each experiment. The initial flange thickness, \(b_{\text{FI\_0}}\), is measured using an outside micrometer. Three measurements are averaged for each sample. Two different strokes, \(h_{1}\) and \(h_{2}\), are used. For both strokes, twenty repeats are conducted for statistical validation. The results of the first stroke, \(h_{1}\), are used to calibrate the machine stiffness, \(k\). The second stroke \(h_{2}\) is higher than the first stroke \(h_{1}\) by \(\Delta h=0.1\) mm. The results corresponding to the second stroke \(h_{2}\) are used to validate the simulation model (see Figure 1c; cf. Table 1). The deviation \(e\) between the simulation results and experimental data is found to be less than 1%--i.e., \(e<1\%\). Based on this minimal deviation, the simulation model is considered sufficiently accurate for use in further analysis. A full factorial design of experiments with three levels was employed to generate a database, yielding 81 data points. Bayesian optimization of a Gaussian Process Regression (GPR) model was then used to identify additional simulation points. Figure 1d shows the deviation of the predicted flange thickness (GPR) from the simulated flange thickness (FEM) \(\Delta b_{\text{FI}}\) and the coefficient of determination \(R^{2}\) of the GPR model over the cycles of Bayesian optimization.