# Gaussian Process Surrogates for Modeling Uncertainties in a Use Case of Forging Superalloys
## 4 Case Study on Forging Superalloys
### FEM Simulation  
The considered use case, preforming an Inconel 625 superalloy billet on a forging press machine, is observed by utilizing a corresponding FEM simulation. Manufacturing process related FEM inputs \(Z^{(j)}\), i.e., different velocity profiles of the upper die over time, are modeled by the forging press GP. Inputs for the forging press GP are \(x_{1}\), \(x_{2}\) and \(t\), such that \(Z^{(j)(t)}=Z^{(j)(t)}(x_{1}^{(j)},x_{2}^{(j)},t)\). All 16 possible combinations for manufacturing process related FEM inputs are shown in Table 2. Billet related inputs \(x_{p}^{(j)}\) that are shared with our proposed surrogate model and FEM simulation are diameter, height and temperature, such that \(x_{p}^{(j)}=(d^{(j)},h^{(j)},\theta^{(j)})\). One possible billet configuration is shown in Figure 7 and possible billet parameters for different configurations are shown in Table 3. We define the radius of the rounded edges to be constant 10 mm across all configurations. We observe in total 27 different billets. Connecting manufacturing process related combinations with different billets, we obtain 432 combinations, i.e., \(j\in\{1,\dots,432\}\). For evaluation of the uncertainty propagation, we randomly draw \(z^{(i)(j)}\) with \(i\in\{1,\dots,20\}\) from each distribution \(Z^{(j)}\), i.e., 20 FEM simulations are performed for each input setting. Thus, a total of 8640 FEM simulation results are generated for our experiments. Selected FEM output variables for our surrogate model are the final diameter and height of the preformed billet, such that \(y^{(i)(j)}=(d^{(i)(j)}_{final},h^{(i)(j)}_{final})\) and \(Y^{(j)}=(\mu(d^{(j)}_{final}),\mu(h^{(j)}_{final}),\sigma_{al}(d^{(j)}_{final }),\sigma_{al}(h^{(j)}_{final}))\)  
Figure 6: Exemplary forging press characteristics \(Z^{(j)}\) represented by mean and 95% credibility interval of \(v_{die}\) over \(t\) with (**a**) low deviation, (**b**,**c**) moderate deviation and (**d**) high deviation. \begin{table}
\begin{tabular}{l l l l} \hline \hline  & \multicolumn{2}{c}{**Configuration Data**} \\ \hline Diameter \(d\) & 220 & 240 & 260 \\ Height \(h\) & 200 & 210 & 220 \\ Temperature \(\theta\) & 900 & 1000 & 1100 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Key parameters for billet configurations \(x_{p}^{(j)}\), values in mm. In respect to the final diameter \(d_{final}\), we calculate the empiric mean by \(\mu(d_{final}^{(j)})=\frac{1}{20}\sum_{i=1}^{20}d_{final}^{(i)(j)}\) and aleatoric standard deviation by \(\sigma_{al}(d_{final}^{(j)})^{2}=\frac{1}{20}\sum_{i=1}^{20}(d_{final}^{(i)(j )}-\mu(d_{final}^{(j)}))^{2}\). The calculations are analogous with respect to \(h_{final}\). Thus, we obtain a dataset with 432 instances described by six input features and four output features. For our running example, input features are clutch pressure, rotation speed, initial billet diameter, initial billet height, initial billet temperature and aggregated manufacturing process uncertainties obtained from data, i.e., the aggregated output of the forging press GP \(\bar{\Sigma}_{al}(Z^{(j)})=\sum_{t=1}^{T}\Sigma_{al}(Z)^{(j)(t)}\). Output features are the mean of the final billet diameter, the mean of the final billet height, the aleatoric uncertainty of the final billet diameter and the aleatoric uncertainty of the final billet height. The problem is defined as a 2D axisymmetric simulation task to utilize symmetries and make efficient use of computational resources. We utilize isotropic elasto-plastic Inconel 625 material behavior from literature. The Young's modulus is temperature-dependent and the yield stress depends on plastic strain, strain-rate and temperature. We set contact properties to tangential behavior with isotropic directionality and a friction coefficient of 0.3 between the billet and upper and lower forging tool, which means that we assume lubricated hot forging conditions.