# Finite Element Analysis of Mandibular Anterior Teeth with Healthy, but Reduced Periodontium
## 2 Materials and Methods
### Stages of periodontal ligament 3D modeling
This analytical model is the basis for the development of the finite element analytical model, and its predictive variables are the basis of the numerical model's simulation performance. The analytical model must therefore capture this phenomenon of deformation under the action of external forces. It will contain, in this case, the definition relations of the normal unitary effort and the specific deformations. In order to be able to individualize the behavior of a certain material under the action of external loadings, a constitutive or material law must also be included in the analytical model. For us, this is [PERSON]'s law, which shows that, in the case of an axially stressed solid material, as long as the external forces do not exceed a certain limit (flow limit), the unit forces are directly proportional to the specific deformations. Material properties were assigned to each component of the assembly in order to perform a finite element analysis. Therefore, a specific material was used for each type of tissue. The value of E, the modulus of elasticity (Young's modulus), the breaking strength, the maximum deformation and the Poisson's ratio were all taken into consideration for our research. The modulus of elasticity is the characteristic of the material and is unique for each type of material (Table 1). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Young’s Modulus** & **Poisson’s Ratio** \\ \hline PDL & \(7.1\times 10^{4}\) GPa [13] & 0.4 \\ Bone & \(140\) GPa [14] & 0.3 \\ Tooth & \(20.3\) GPa [14] & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties. Figure 7: Illustration of the 3 ligaments (from left to right—the central incisor, the lateral incisor and the canine). 6. The operating rules for each element were established in the next step. Thus, a fixing constraint (clamp) was established for the fixed body, the mandibular bone. A clamp was used on the mesial side and one on the distal side. The tooth was considered mobile and the Generative Structural Analysis (GSA) module was used to apply a Fastened Connection constraint. For the PDL, a _Fastened Connection_ constraint in the GSA module was also applied. The next step proceeded with the construction of the model for finite element analysis. This was done by applying boundary conditions and establishing fixed surfaces. For this, the _Analysis and Simulation/Generative Structural Analysis_ workbench was opened. Through this module, the 3D model was further discretized and the conditions of demand and degrees of freedom were imposed. For the surfaces that were in contact, boundary conditions were imposed through the General Analysis Connection tool. This is a very powerful tool that can be used to connect any part of an assembly on an overall model, with or without a handling point. This tool allows any type of geometry to be connected, and it is a general way of connecting two components. The _Fastened Connection_ rigid assembly connection tool was used to assign specific characteristics for simulation. Its purpose was to model a fixed connection between two bodies with a common boundary and the association of a geometric assembly constraint, or a General Analysis Connection type connection. Finally, there were 6 connections between the contact surfaces of the following components: central incisor and central incisor ligament; central incisor ligament and bone; lateral incisor and lateral incisor ligament; lateral incisor ligament and bone; canine and canine ligament; canine ligament and bone. After the creation of these connections, the surfaces that were considered fixed were constrained with the _Clamp_ tool. These constrained surfaces were the two lateral surfaces, the mesial and the distal surfaces of the mandibular bone. In establishing the values for the discretization network of previously built 3D models, the following parameters were used (Table 2):  
1. Size--choosing or changing the size of finite elements;
2.