## 3 Results  
In this study, three finite element simulations were carried out by varying the MAD design to analyze the stress distribution and deformation of teeth and PDLs. Tooth movement and pain are the biomechanical response of the stress that develops in teeth and PDLs under the application of a load. Therefore, to evaluate the teeth that are more loaded and the areas that are prone to be compressed and stretched, results in terms of von Mises stress and deformation of teeth and PDLs were presented. (Figure 2). Regarding the analysis of stress on the periodontal ligaments, significant differences emerge between Orthoapnea(tm) (Grupo Dental Ortoglu, Malaga, Spain), Herbst(tm) (Somnomed, Sydney, Australia), and Somnodent(tm) (Somnomed, Sydney, Australia). In the case of Orthoapnea(tm), which has an anterior reverse rod activation mechanism, the stresses on the individual teeth develop mainly on the coronal portion, while apically, the stresses are minor. The most involved teeth are the lower and upper incisors, with the PDLs exhibiting a maximum expression of strength in correspondence of the anterior cervical margin reaching 4.26 kPa (kilopascal). The teeth of the posterior sectors (premolars and molars) are less stressed. The stresses are slightly uneven in the various portions of the ar  
Figure 2: Three models of mandibular advancement devices: OrthoApnea (**a**), Herbst (**b**), Somnodent (**c**). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mesh Size** & **Nodes/Elements** & **Von Mises Stress** & **Elapsed Time** & **Memory Used** \\
**(mm)** & & **(MPa)** & **(hour)** & **(GB)** \\ \hline
0.26 & 631,362/3,317,272 & \(8.07\times 10^{-3}\) & 3 & 18.972 \\
0.20 & 1,073,911/5,706,491 & \(4.26\times 10^{-3}\) & 22 & 33.212 \\
0.15 & 1,928,454/10,404,873 & \(4.48\times 10^{-3}\) & 36 & 70.366 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mesh convergence study. HerbstTM is a bilateral telescopic activation device. It presents periodontal stresses that are concentrated in the most coronal portion of the teeth similarly to OrthoapneaMT. The main difference with the first device is that tensions caused by HerbstTM are much more uniform in the arches. The most stressed teeth are the upper and lower premolars and molars (but the differences are not significant while compared with other teeth). The PDLs are mainly affected by stress in the medio-distal cervical region with a peak of pressure that reaches 3.56 kPa. SomnodentTM is activated with two lateral propulsion screws and this case also presents concentrated stresses, especially in the coronal portion of the teeth. Stresses are concentrated in the periodontal ligaments of the second molars and, with a minor component, the anterior teeth of both arches. The stress mainly affects the medio-distal cervical sides of PDLs and the lingual sides of mandibular PDLs associated with molars. The maximum pressure achieved is about 3.27 kPa. SomnodentTM is the device that generates the least stresses among the three considered in the study. In Figure 4, the stress analysis of dental elements reflects periodontal tensions. In OrthoapneaTM, the most loaded teeth are the upper and lower incisors. The most affected dental portions are the incisal margins with a maximum value of 600 kPa. HerbstTM has less intense stresses that are concentrated mainly in the maxillary molars and the mandibular premolars. The maximum load value reached is 302 kPa. SomnodentTM has stresses concentrated on the upper molars and mandibular second premolars in correspondence of the splint's wings. The highest pressure is 287 kPa. The analyses also allowed us to obtain images, from the occlusal view, representing the stresses and deformations on periodontal ligaments (Figure 3) and teeth (Figure 4). Figure 3: Contour plots. Periodontal ligaments (PDLs) stress distribution (kPa), frontal and occlusal view. A lateral representation of the arches shows the deformation vectors to which the individual dental elements of both arches are submitted to (Figure 5).