# Gaussian Process Surrogates for Modeling Uncertainties in a Use Case of Forging Superalloys
## 4 Case Study on Forging Superalloys
### Forging Aggregate Characteristic  
We calculate the mean forming velocity values of an artificially designed forging process on the example of a forging screwpress by a self-designed parameterized curve in (15) that models the die velocity \(v_{die}\) in mm/s as a function of the process timestep \(t\) in _seconds_, clutch pressure \(x_{1}\) in _bar_ and rotation speed of the electric motor \(x_{2}\) in rpm, such that \(v_{die}=v_{die}(x_{1},x_{2},t)\). Where, \(x_{1}\) and \(x_{2}\) are two process-specific setting parameters, i.e. \(x_{m}=(x_{1},x_{2})\). \[v_{die}(x_{1},x_{2},t)=\kappa_{1}\cdot x_{1}\cdot x_{2}\cdot t^{2}-x_{2}\cdot x _{1}\cdot x_{2}\cdot t^{3} \tag{15}\]  
where \(\kappa_{1}=\frac{5}{3}\) mm\({}^{2}\)/kg and \(\kappa_{2}=\frac{5}{3}\) mm\({}^{2}\)/kgs are constants. We utilize a designed forging press specific GP with data generated by using (15) to model the mean and input dependent deviations in respect to the manufacturing process characteristic \(Z\) (i.e., \(Z\) represents the velocity profile of the forging die \(v_{die}\)). \(Z\) is defined by a distribution with mean \(m(Z)\) and aleatoric standard deviation \(\Sigma_{el}(Z)\). With respect to our use case, the forging press specific GP with output \(Z^{(j)}\) is at the very beginning of the uncertainty propagation analysis, see Figure 4. The inputs for the forging press GP are \(x_{m}=(x_{1},x_{2})\) and time increments \(t=\{0,\ldots,T\}\), where \(T\) represents the duration of the manufacturing process. The output of the forging press GP is \(Z\), such that \(Z\sim\mathcal{GP}(m(x_{m},t),k((x_{m},t),(x_{m},t)^{\prime}))\) with mean \(m(x_{m},t)\) and covariance function \(k((x_{m},t),(x_{m},t)^{\prime})\). Thus, we obtain for each time increment a distribution describing the velocity at time \(t\). The principle GP design for the forging press can be seen in Figure 5. As covariance function, \(k\) we found out that an RBF kernel is appropriate. Figure 4: Uncertainty Propagation Analysis: the characteristic \(Z^{(j)}\) of the manufacturing process is described by a distribution, since deviations occur when the process is repeated with identical \(x_{m}^{(j)}\). With \(M\) draws of \(z^{(i)(j)}\) out of the distribution \(Z^{(j)}\) as manufacturing process characteristic and to be manufactured part parameters \(x_{j}^{(j)}\), FEM simulations are executed to obtain targets \(y^{(i)(j)}\) that describe a distribution \(Y^{(j)}=(\mu^{(j)},\sigma_{el}^{(j)})\) with mean \(\mu^{(j)}\) and aleatoric standard deviation, i.e., uncertainty \(\sigma_{el}^{(j)}\) for given inputs \(x_{j}^{(j)}\) and \(x_{j}^{(j)}\). We utilize (15) and different input parameter combinations to generate training data for the forging press GP, see Table 1. In terms of time step size \(t\), we assume that each forging stroke lasts one second, and we model each stroke with a resolution of 100 time steps. To obtain different deviations connected to different \(x_{1}\) and \(x_{2}\) combinations, we use the underlying inference properties of GP regression and vary inter- and extrapolation tasks in respect to the input values for forging process representation, see Table 2.  
We define interpolation such that a value is within the training range (i.e., \(x_{1}\) equals 14 or 18 and \(x_{2}\) equals 55 or 65) and extrapolation such that a value is out of the training range (i.e., \(x_{1}\) equals 10 or 22 and \(x_{2}\) equals 45 or 75). Exemplary forging press characteristics can be seen in Figure 6, where Figure 6a shows low deviation because \(x_{1}\) and \(x_{2}\) are both lie within the range of training data, Figure 6b,c show moderate deviation because one of the process parameters is within and the other is outside the range of the training data and Figure 6d shows the highest deviation because both of the process-parameters lie outside the range of training data. Thus, our forging press GP represents a forging aggregate characteristics with uncertainties dependent on the inputs.