# Reduced-Order Model Based on Neural Network of Roll Bending
## 3 Finite Element Model
### Simulation Results of the Sheet Convexity  
It can be assumed that the shape of the sheet in the output plane of the rolling mill is identical to the shape of the surface of the work roll. Therefore, the convexity of the sheet can be calculated as the difference between the vertical coordinates of the rolling surface in the center and the edge of contact with the sheet.  
Some results obtained in 2D FEM simulations with a rolling force of 35,000 kN and without anti-bending force are shown in Figure 6a. The plots are the dependences of changes in the convexity of the sheet on its width at different convexities of the work and backup rolls. The sheet convexity calculated for various rolling and anti-bending forces is shown in Figure 6b. These results were obtained for flat rolls (\(c_{w}=c_{b}=0\) mm). It can be seen that flat rolls do not guarantee the achievement of a flat sheet shape, even with a high anti-bending force (\(Q=2500\) kN). The convex shape of the sheet was observed in all rolling cases shown in Figure 6b.  
Then the FEM model was used to calculate the sheet convexity for various rolling conditions:  
* Rolling force \(F\): 0, 12,500, 25,000, 37,500 and 50,000 kN;
* Anti-bending force \(Q\): 0, 625, 1250, 1875 and 2500 kN;
* Convexity of work \(c_{w}\) and backup \(c_{b}\) rolls: 0, 0.1; 0.2; 0.3; 0.4 and 0.5 mm;
* Sheet width \(b\): 1500 to 3500 mm every 200 mm.  
In total, 3575 results of sheet convexity calculations were obtained for various combinations of parameters. The 3575 results are combinations of 5 values of rolling force \(F\), 5 values of anti-bending force \(Q\), 13 combinations of convexities, and 11 variants of  
Figure 6: Dependencies of changes in the convexity of the sheet on its width: (a) different rolls’ convexities (rolling force \(F=35,000\) kN); (b) different rolling and anti-bending forces.  
sheet width \(b\); (5 \(\times\) 5 \(\times\) 13 \(\times\) 11 = 3575). Examples of the results obtained are presented in Figure 7, where the dependencies of the sheet convexity on its width are presented. From these plots, it can be seen that each parameter affects the convexity of the sheet in different ways, which allowed us to think that choosing one pattern for the common effect of all factors is almost impossible. This requires the use of more complex methods to calculate phenomena such as the convexity of sheet metal. One of the possibilities is the use of an artificial NN, which is used as a ROM and described in the following section.  
The results are collected in a common dataset for the training, testing, and verification of the NN. The main limitation of the model and/or dataset is the connection to a given rolling mill.  
Variations in roll diameters that are in an acceptable range for the mill can be omitted.  
The elastic properties of the roll materials can be taken into account if they are significantly different from the modeled ones.  
Another essential assumption of the model is that the convexity of the rolls and the sheet is an approximate quadratic function. That is true for the initial shape of the rolls, but this shape changes during the campaign due to the thermal expansion and wear of the rolls. The influence of these factors is usually reduced by planning the campaign; that is, by choosing the order of rolling sheets of different widths.