## 3 Results
### Finite Element Analysis  
At a maximum activation of 4 mm per side in the transversal axis, the _Double-arch TPA_, with two arches of 1 mm diameter each, generated the highest forces (3.1 Newton (N) per side). The _Triple-loop-arch TPA_ and the _S-bow TPA_, both with a round diameter of 1.2 mm and 1.5 mm, generated 2.5 and 1.7 N per side. The _Omega-loop TPA_ with an oval diameter of 1 \(\times\) 2 mm and a big loop generated 1.3 N per side, the lowest force. In the sagittal and vertical direction, negligible forces less than 0.01 N were found.  
The applied and resulting moments were assessed at CR and PF at the inner surface of the shell, palatal to the dental crown of the upper right molar. Overall, the highest moments were found around the meso-distal axis representing a buccal _tipping_.  
At the maximum activation of 4 mm per side, at CR, moments ranged from 2.1 Nmm (_Double-arch TPA_) to \(-6.6\) Nmm (_S-bow TPA_), and, at PF, the moments ranged from \(-9\) Nmm (_Omega-loop TPA_) to \(-29\) Nmm (_Triple-loop-arch TPA_). The _rotatory_ moments around the vertical axis reached a maximum of 6 Nmm (_Triple-loop-arch TPA_) at CR and \(-5.4\) Nmm (_Omega-loop TPA_) at PF. The _angulation_, i.e., moments around the oro-vestibular axis, were very small for all treatment appliance with absolute values less than 0.5 Nmm. Figure 5a, b summarizes the _buccal tipping_ at PF and CR for the investigated TPAs.  
The FEA enabled calculation of the von Mises stresses for each TPA variant. Up to an activation of 3 mm per side, the local von Mises stresses ranged from 51 to 128 MPa and sometimes slightly exceeded the yield point from PEEK of 120 MPa. Upon further activation of the TPAs up to 4 mm, three appliances produced local von Mises stress clearly above the yield point (_S-bow TPA_ and _Triple-loop-arch TPA_: 154 MPa and _Double-arch TPA_: 165 MPa). The von Mises stresses for the different TPA variants are visualized in Figure 6 for a bilateral activation of 4 mm. It can be seen that the highest material stresses occurred at the interface between the TPA and the shells (marked in red-orange).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & & & & & & & & \\ \cline{2-13} \multirow{2}{*}{Activation (mm)} & \multicolumn{4}{c}{**Fores**} & \multicolumn{4}{c}{**Moments at the Point of**} & \multicolumn{4}{c}{**Moments at the Center**} & \multicolumn{4}{c}{_von_} & \multirow{2}{*}{**Fores**} & \multicolumn{4}{c}{**Moments at the Center**} \\  & & & & & & & & & & & & & & & & & & &The displacement in the sagittal direction differed depending on the TPA geometry: Anterior movement was seen for the _Double-arch TPA_ and the _Omega-loop TPA_ (1.3 mm and 0.5 mm), whereas distal displacement was seen for the _Triple-loop-arch TPA_ (\(-\)0.5 mm). A  
Figure 5: (a,b) FEA, resulting moments in Newton millimeter (Nmm) for buccal tipping per molar 16 (dashed line) and 26 (continuous line) measured by transversal activation of 0.5 to 4 mm per side; (a) at the point of force application; (b) at the center of resistance.  
Figure 6: FEA, graphical illustration of the von Mises stress at a maximum of 4 mm bilateral activation per side. The side scale indicates the corresponding numerical data (here: megapscals) to the color gradient, blue shows the lowest stress and red the highest stress (areas marked in red-orange).  
different pattern was seen for the _S-bow TPA_ where one side moved to the anterior, and the other in a distal direction (1.5 mm and \(-\)2 mm). In the apical direction, movements ranged from \(-\)1.5 mm (_Omega-loop TPA_) to \(-\)3.9 mm (_S-bow TPA_).  
Figure 7 summarizes the deformation pattern of the four TPAs in all three dimensions. It can be seen that apical displacement was most pronounced for the _S-bow TPA_ and the _Double-arch TPA_, reaching 3.9 mm and 2.7 mm, respectively.