## 3 Numerical Simulations of an Aircraft Tire Rolling on a Steel Drum
### Mesh Sensitivity and Computational Time (CPU)  
In order to optimize the simulation computational time, different mesh densities with 51, 101 and 181 circumferential elements for the tire and 200 elements for the drum were analyzed, (Figure 15a-c). Obviously, the mesh density plays a very considerable role for the accuracy of the tangential forces in the contact zone. A finer mesh permits a very accurate estimation of the contact area, of the grip forces and the temperature distribution, all generated by frictional contact. A sensitivity study on the contact area depending on the element size is presented here after (Figure 16a). The maximum element size has been chosen bearing in mind the small amount of load applied during the acceleration period and the circular shapes of the tire and the drum. As it can be seen, the contact is realized on a small surface. A closer look shows that the mean estimated contact area converges to the measured one. The tire does not present any groove self-contact (contrary to that observed in [22] in another context) or lateral deformations, which is the case for higher loads. Figure 14: (a) The tire position on the drum at skidding angles at 0\({}^{\circ}\) and 6\({}^{\circ}\), (b) numerical simulation, tire/drum set-up. Figure 13: Exponential “softened” pressure-overclosure relationship. Figure 16b represents the computational time as function of the number of elements, for the loading step. As it can be expected, choosing a higher number of elements will dramatically increase the computational time (CPU). The following numerical simulations are done with 51 continuum solid elements. Figure 17 illustrates a numerical simulation of a tire modeled with 51 elements. Figure 17a shows the 2-D section of the tire before and after inflation at the nominal pressure. Figure 17b represents 3-D sections of the tire with a preloading of l\({}_{1}\) kN, Figure 17c shows the tire loaded with l\({}_{2}\) = 10 \(\times\) l\({}_{1}\) kN and Figure 17d illustrates the tire section loaded with l\({}_{3}\) = 25 \(\times\) l\({}_{1}\) kN. The model realized with 51 circumferential elements seems to be convenient for loading less than 300 kN. It is clear that in the case of a loading greater than 300 kN, the option of 101 or 181 elements seems to be necessary, knowing that the computational time is still a considerable factor to take into account. Figure 16: (**a**) Variation of contact area as a function of the number of elements, (**b**) computational time (CPU) as a function of the number of elements. Figure 17: Numerical simulation, tire modeled with 51 elements (**a**) 2-D, deflated and inflated tire, (**b**) 3-D inflated tire pre-loaded with l\({}_{1}\) kN, (**c**) inflated tire loaded with l\({}_{2}\) = 10 \(\times\) l\({}_{1}\) kN, (**d**) inflated tire loaded with l\({}_{3}\) = 25 \(\times\) l\({}_{1}\) kN. Figure 15: Different mesh options: Tire modeled using: (**a**) 51 elements, (**b**) 101 elements, (**c**) 181 elements. Numerical Simulation of an Aircraft Tire Rolling on a Steel Drum, Loading with 200 kN, Velocity up to 114 km/h, Skidding Angle = 6\({}^{\circ}\)  
Figure 18a shows the global temperature evolution of tire tread at the end of testing. Figure 18b represents the experimental data for several frames of the thermal evolution of the tire tread for the third step, recorded for a period of 1.5 s of rolling. The present numerical model does not take into account neither the wear phenomenon nor the thermal exchange between the tire and the drum. Figure 19 illustrates the thermal evolution of the tire tread loading with 200 kN, rolling with a velocity up to 114 km/h with skidding angle of 6\({}^{\circ}\). The experimental and numerical data are compared in the next figure. Figure 20 illustrates the temperature distribution along the cross section of the tire tread.