# Numerical-Experimental Analysis to Optimize the Geometry of a Prosthetic Abtment
## 4 Numerical Simulation
### FEA Boundary Conditions and Meshing  
To simulate the compression test, a quasi-static multi-step force-controlled calculation was adopted. The CAD design of the 10:1 scale model is composed of two solid parts representing, respectively, the abutment and the crown. To ensure a similarity ratio \(E^{*}=1\) in the numerical simulation of the scale model and subsequently in the experimental test, materials with elastic modulus and Poisson ratio close to those of the original materials of the abutment (dentin) and crown (dental ceramic) were selected. Table 4 shows the values of the mechanical characteristics of the crown and abutment materials most commonly found in the literature [35; 36; 37; 38]. In the scale model, the material used for the abutment was the Pb-Bi alloy (76.7% Lead-23.3% Bismuth), having a Young modulus and Poisson ratio of 21 GPa and 0.3, respectively [39], while the material used for the crown was the ZA-8 alloy (91.6% Zinc-8.4% Aluminum), having a Young modulus and Poisson ratio of 85.5 GPa and 0.29, respectively [40]. Table 4 provides the Poisson ratio and Young modulus values commonly recognized for ceramics and dentin, and the central values of the intervals presented in this table have been assumed. In the developed numerical model, these values were scaled to take into account the model theory, and therefore, the real value was adjusted with a factor of 10/1, and therefore, the behavior of the dentin would be simulated with a Young modulus equal to 210 GPa. For the experimental validation, a Pb-Bi alloy was instead adopted, having a Young modulus equal to 85 GPa, a material with characteristics close to those in the theory. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Material** & **Poisson Ratio** &
\begin{tabular}{c} **Young Modulus** \\ **[GPa]** \\ \end{tabular} \\ \hline Crown & Dental ceramic & 0.28–0.29 & 65–95 \\ \hline Abutment & Dentin & 0.3 & 16–25 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Main material properties of the coupling materials. For having the same value in the FEA and in the experimental test, a Young modulus equal to 85 GPa for the abutment was adopted for both analyses. According to the statements in Section 3, the heaviest condition on the crown-abutment coupling was analyzed (Figure 14); in order to do that, the application of a load with an inclination of 10\({}^{\circ}\) to the normal direction to the occlusal plane was imposed. This inclination was achieved through a Pb-Bi alloy basement integral to the abutment having a 10\({}^{\circ}\) angle to the horizontal direction. The mesh of the CAD model was made according to the Voronoi-Delaunay algorithm with 16 Jacobian points per element. The mesh counted a total of 178,626 nodes and 122,032 elements. A selective set of mesh controls allows to perform a finer meshing on the in-contact parts' surfaces. For the cusp surfaces, the size of elements was set at 1.2 \(\pm\) 0.24 mm, while for the lateral surfaces of the abutment, the size of elements was set at 3.2 \(\pm\) 0.64 mm. Figure 14 shows mesh elements described in the text. A compatible mesh with no penetration option between the in-contact surfaces was performed to simulate a perfect coupling of the abutment and the crown. Boundary conditions consisted of fixed constraints set to the basement, while a force was imposed vertically to the upper face of the crown. Lastly, the compressive force was simulated with a linear trend and rate of 1 kN/s from 0 kN to its maximum value. In particular, the maximal compressive force simulated was 90 kN, corresponding to the maximal heavy chewing force under the similarity ratio adopted in Section 4.1 (\(F^{\star}\) = 100). A further FEA highlighted the effectiveness of the luting cement on the same geometric configuration (shown in Figure 14). In this numerical simulation, a 0.5 mm thick layer was added to represent the luting cement between the coupling surfaces of crown and abutment.