# Numerical-Experimental Analysis to Optimize the Geometry of a Prosthetic Abtment
## 4 Numerical Simulation
### Numerical Analysis and Results  
The numerical analysis was useful to evaluate the response of the crown-abutment coupling to the heaviest clinical condition (90 kN); therefore, several simulations have been performed using the DoE of Table 5 that varies with the abutment taper angle. Figure 14: Areas subjected to tensile stress with no luting cement simulation (\(\alpha\) = 12\({}^{\circ}\)). In the geometric configuration shown in Figure 15, it is noted that the tensile stress state prevails on the sidewall opposite the load, as expected. Conversely, the compressive stress state prevails on the surface of the cusp and on the side where the load is applied (Figure 16). This confirms the results discussed in Section 3, according to which the aforementioned areas resist the dislodging rotation of the crown. From the comparison between Figures 15 and 16, the compression stress values measured on the cusp are greater than the tensile stress values on the sidewall. This proves the effectiveness of the cusp contribution to the axial resistance form. Table 5 shows the tensile and compressive stress output of FEA under 90 kN load. It is noted that, even in the worst case (\(\alpha\) = 36\({}^{\circ}\)), both values are lower than the related strengths of the types of cements shown in Table 6.  
The following is an estimate of the uncertainty in the FEA (Finite Element Analysis) when considering the uncertainty associated with the quality of the mesh and/or the parameters used to describe the mechanical behavior of the material, in accordance with the variation in calculation parameters according to the Taguchi method which was also used by [PERSON] et al. [41]. Numerical convergence is linked to mesh quality, and it is assumed that Table 5 is generated with the mesh generation parameters already optimized. Regarding the material properties in the calculation model, the uncertainty arises from the values of the Young modulus and Poisson ratio, as the variance of these parameters can lead to variations in the calculation results. Standard deviations were assumed to be equal to 1.148 GPa for the Young modulus and equal to 0.042 for the Poisson ratio of the Pi-Bi alloy [39]. With the mesh parameters being fixed, the FEA were performed using the extreme values (mean \(\pm\) standard deviation) of Young modulus and Poisson ratio for the Pi-Bi alloy across all the geometries included in Table 5, allowing for the estimation of the compounded uncertainty of the FEA results, as presented in the following table. A numerical convergence on elements was involved, with a ten-step repetition of the Finite Element Analysis (FEA) implementation, focusing on variations in mesh granularity. The general element length varied from 12 mm to 10 mm, while the numerical control for the elements at the contact surfaces of the abutment ranged from 2.80 mm to 0.5 mm, constrained by the available computational memory. Additionally, the P1 stress complement was evaluated for the cusp area, as depicted in Figure 16. The root mean square (RMS) value of P1 was assessed and compared across the various FEA configurations. Figure 17 resumes the P1 stress RMS value with respect to the number of elements of the mesh. The numerical calculation of the crown-abutment coupling was repeated considering all the configurations described in Section 2, imposing \(\alpha\) = 12\({}^{\circ}\) and applying the maximum chewing load (90 kN). Therefore, the following abutment shapes were analyzed, i.e., flat head, with cusps, with grooves, with cusps and shoulder connection, and with cusps and chamfer connection. The results of the stress state on the crown-abutment couplings are summarized in Figures 17 and 18 and are listed in Tables 7 and 8.