# Compensating Data Shortages in Manufacturing with Monotonicity Knowledge
## 3 Applications in Manufacturing
### Laser Glass Bending  
A first application example is laser glass bending. In the industrial standard process of glass bending [43], a flat glass specimen is placed in a furnace with furnace temperature \(T_{\mathrm{f}}\), and then the heated specimen is bent at a designated edge driven by gravity. As an additional feature, a laser can be added to the industrial standard process in order to specifically heat up the critical region of the flat glass around the bending edge, and thus to speed up the process and achieve smaller bending radii [44; 45]. The laser can generally scribe in arbitrary directions. In the process considered here, however, the laser path is restricted to three straight lines parallel to the bending edge. While the middle line defines the bending edge, the two outer lines are at a fixed distance \(\Delta_{\mathrm{I}}/2=5.75\) mm in each direction to it. The laser spot moves along this path in multiple cycles with the number of cycles denoted by \(n_{\mathrm{c}}\). The scribing speed and the power of the laser are kept constant. A mechanical stop below the bending edge guarantees that the bending angle does not exceed \(90^{\circ}\). An illustration of the laser glass bending process is shown in Figure 1.  
The goal of the glass bending process considered here is to obtain bent glass parts with a prescribed bending angle. In order to achieve such a pre-defined bending angle, the process operator has to find suitable combinations of the two process parameters \(T_{\mathrm{f}}\) and \(n_{\mathrm{c}}\), which is usually done based on experience. A more systematic approach, however, is to set up an appropriate model of the bending angle \(y:=\beta\) as a function of the process variables  
\[x:=(x_{1},x_{2}):=(T_{\mathrm{f}},n_{\mathrm{c}})\in X, \tag{15}\]  
where \(X\subset\mathbb{R}^{2}\) is the rectangular set with the bounds specified in Table 1. In particular, such a model should allow sufficiently precise real-time predictions in order to support the process operator in quickly searching the parameter space \(X\) for optimal process parameter settings. Since SIAMOR models are polynomial by construction, they perfectly satisfy this real-time requirement. In contrast, a repeated evaluation of finite-element simulation models of the glass-bending process is too time-consuming to be of any practical use in quickly exploring the parameter space. As generating experimental training data from the real process is cumbersome, a two-dimensional finite-element model was set up to generate data numerically. The simulation of the process was based on a coupled thermo-mechanical problem with finite deformation. Since the CO\({}_{2}\) laser used in the process operates in the opaque wavelength spectrum of glass, the heat supply was modeled as a surface flux into the deforming sheet. In this two-dimensional setting, the heat was assumed to be deposited instantaneously along the thickness direction and also instantaneously on all three laser lines. Radiation effects were ignored, and heat conduction inside the glass was described by the classical Fourier law with the heat conductivity obtained experimentally via laser flash analysis. In view of the relevant relaxation and process time scales for the applied temperature range, the mechanical behavior of the glass was described by a simple Maxwell-type visco-elastic law. The deformation due to gravity is heavily affected by the pronounced temperature dependence of the viscosity above the glass transition, which was described in our models using the Williams-Landel-Ferry approximation [46]. The generation of the simulated data was conducted using the commercial finite-element code Abaqus\({}^{\heartsuit}\). It was used to create a training dataset comprising 25 data points sampled on a 2D rectangular grid.