## 2 Materials and Methods
### Model 2: Molar Unrighting Simulation  
A second CAD model was also developed using SolidWorks(r) software. Initially, a three-dimensional (3D) model of the mandible's right posterior teeth segment was reconstructed from a computed tomography (CT) scan image, obtained from the DICOM-image repository provided via osirix database \"[[https://www.osirix-viewer.com](https://www.osirix-viewer.com)]([https://www.osirix-viewer.com](https://www.osirix-viewer.com)) (accessed on 20 November 2023)\". The 3D CAD model was then converted into a 2D CAD representation by extracting its edges along a central plane intersecting the middle of the teeth. The periodontal ligament (PDL) was assigned a uniform thickness of 0.2 mm, an average value widely supported by anatomical literature [26]. While this simplification does not capture local anatomical variations--such as root concavities or patient-specific PDL irregularities--it provided a stable and practical framework for the iterative simulation process. Similarly, the dental geometry preserved the essential morphological features relevant to sagittal-plane movement, though finer anatomical details were not included. These modeling choices were intentional, as the study focused on capturing the overall biomechanical trends driving molar uprighting, rather than resolving localized stress concentrations, while still ensuring reliable results for the proposed iteration-to-time correlation method. The characteristics of this model were adapted from the previous literature [13]. The model consisted of a mandibular second molar tilted by 30\({}^{\circ}\) mesially, with the mandibular canine and first and second premolars acting as the anchorage teeth. The incorporated appliances included standard edgewise brackets of 0.022-inch slot size bonded to the anchorage teeth, a 4 mm wide edgewise molar buccal tube, and a segmental 0.016 \(\times\) 0.016-inch stainless steel wire connecting the anchorage teeth. In addition to a conventional molar uprighting spring made of 0.016 \(\times\) 0.016-inch stainless steel wire, with a 2 mm diameter helical loop and a long lever arm with a length of 22 mm. This long lever arm was projected anteriorly to lie passively 12 mm gingival to the anchorage wire between the first and second premolars, and was activated when raised and hooked onto the anchorage wire. The spring's short lever arm, as defined in this context, represented the arm engaging the molar buccal tube slot and was assumed cinched at the distal end of the buccal tube to prevent any possible sliding of the wire (Figure 4). A 2D plane stress mesh was generated in Abaqus/Standard software using exclusively linear quadrilateral elements (CPS4R) with reduced integration. A total of 461 two-node linear beam elements (B21) were used to model the uprighting spring as a slender 0.016\" \(\times\) 0.016\" stainless steel component. The use of quadratic beam elements did not provide any gain in terms of stress capture with a significant increase in computation time. Figure 4: Finite element (FE) representation of Model 2 for molar uprighting using a conventional uprighting spring. (**A**) FE mesh of the initial configuration before spring activation; (**B**) CAD representation of the configuration after spring activation. To better capture contact forces, a very fine mesh (0.05 mm) was applied to the portion of the wire inserted into the bracket. The contact between the wire and bracket was modeled using a friction coefficient of 0.28, as reported in previous studies [27; 28]. The mesh was also locally refined around the PDL and root surfaces to improve accuracy in regions with high stress gradients. A total of 4279 CPS4R elements were generated to model the molar PDL. The use of the Medial Axis Algorithm for mesh control enabled a symmetrical distribution of elements along both the bone-PDL and root-PDL interfaces.