# Computational Methodology for the Development of Wrinkled Tubes by Plastic Deformation
## 4 Results and Discussion
### Results of Von Mises Stresses  
The discretisation with an element size of 2 mm generated 1489 elements, while the 1.5 mm size generated 2594 elements. Finally, the 1 mm mesh generated 8078 elements. It can be observed that the die shows greater variation in discretisation due to its larger dimensions, while in the tube, due to its geometry, there were variations only in the number of elements. It is important to note that mesh refinement reduces element distortion in the simulation and produces more accurate results.  
After analysing the mesh sensitivity, deformation of the tube was observed and results were obtained regarding von Mises stresses, as shown in Figure 5. As expected, only the tube underwent deformation, given the higher stiffness imposed on the die to ensure its rigidity.  
Discretisation with 1 mm element sizes generated better results, both in the final deformation of the tube and in the maximum stresses applied during the simulated process. Therefore, subsequent analyses were conducted considering only this optimised model.  
According to [36], mesh refinements in numerical simulations present challenges such as computational cost, occurrence of distortions, and degeneration of elements. However, the mesh sensitivity study conducted in this work produced satisfactory results, as it highlighted the best outcomes and greater accuracy for the tube filling the die cavities during deformation.  
The values of the maximum pressure applied during the process were sensitive to mesh refinement, as expected [37].  
Figure 4: Numerical model of the tube-die, % expansion 2D asymmetry.  
Figure 5: Mesh and von Mises stress of the tube after deformation. Mesh size: (**a**) 2.0 mm, (**b**) 1.5 mm, (**c**) 1.0 mm.  
The result demonstrated satisfactory tube deformation, as the tube was deformed similarly in both cavities and flowed into the cavities effectively. This allowed us to obtain the wrinkled tube. However, as the goal was to optimise both the die geometry and the process parameters, the cavity was not filled to its full depth. So, for a reduction in this dimension and to better utilise the die geometry and facilitate the plastic conformation of the tube, more numerical tests were produced.  
Therefore, a new die dimension was modelled to ensure that the cavities were filled and to generate a more accurate and optimised result. The shape and dimensions of the internal cavity of the die are key factors in ensuring contact between the tube and the die during the tube-forming process, affecting the flow and deformation conditions of the materials [38]. In the new model, the cavity depth was set to 4 mm, justified as it is the closest integer value to the maximum displacement of the tube elements along the \(x\)-axis after deformation in both cavities. The results for von Mises stress, and horizontal and vertical displacements, are presented in Figure 7 and Table 2.  
Finally, this model ensured that the die cavity was filled after the full deformation of the tube and produced an optimised result for the die geometry and simulation parameters. For clearer visualisation of the results, Figure 8 presents: (a) a % expansion view of the assembly, and (b) a % expansion view of the model after tube deformation and filling of the die cavities.