## 2 Materials and Methods  
### Simulation Model Development  
A maxillary bone model was developed using cone beam computed tomography (CBCT) data from a healthy adult patient with no craniofacial anomalies. CBCT images were acquired using a Planmeca ProMax 3D Mid system (Planmeca, Helsinki, Finland) with a slice thickness of 0.1 mm and were retrieved from the archives of the Department of Radiology, Faculty of Dentistry, Ankara Yildirim Beyazt University. The DICOM data were segmented in 3D Slicer software (version 4.11.20210226, available at [[https://www.slicer.org](https://www.slicer.org)]([https://www.slicer.org](https://www.slicer.org)), accessed on 15 September 2023) to create a three-dimensional anatomical model, which was exported in STL format for finite element analysis (FEA).  
The model included all maxillary teeth from the right second molar to the left second molar. The periodontal ligament (PDL) was modeled as a 0.2 mm layer between the roots and the surrounding alveolar bone. Gingival mucosa was generated by projecting the outer surface of the cortical bone. Two aligner geometries were designed: the first featured a trimline ending at the gingival margin, while the second had a trimline extending 2 mm above the gingiva. Both aligners were modeled with a uniform thickness of 0.5 mm, consistent with previous studies [14; 17].  
All components--including teeth, cortical bone, gingiva, aligners, and attachments--were assumed to be homogeneous, isotropic, and linearly elastic materials. The mechanical properties assigned to these materials are summarized in Table 1[15; 21]. In contrast, the PDL was modeled as a nonlinear hyperelastic material using a second-order Ogden constitutive model, which more accurately reflects the physiological behavior of soft periodontal tissues under orthodontic forces [16; 22; 23; 24]. The material parameters employed in the Ogden model were based on values previously reported by [PERSON] et al. [22]. Specifically, the model incorporated two terms with corresponding constants: u1 = 0.00554 MPa, \(\alpha_{1}\) = 0.25, and u2 = 0.11 MPa, \(\alpha_{2}\) = 0.1153. Additionally, the compressibility parameters were defined as D1 = 0.1216 MPa and D2 = 0.97 MPa. These constants were used to simulate the nonlinear mechanical behavior of periodontal tissues in the finite element model. However, time-dependent viscoelastic properties of the PDL were not incorporated into the model.  
A nonlinear static analysis was performed using the implicit solver in LS-DYNA. The primary source of nonlinearity in this analysis stemmed from the material models. However, additional factors such as geometric and contact nonlinearities may also have contributed to the overall nonlinear behavior of the system.  
Four finite element models were generated based on the type and placement of attachments on the upper first and second molars. These models included the following: no attachment (NA), a vertical attachment (VA), a 25\({}^{\circ}\) angled vertical attachment (25 VA), and a double horizontal attachment (DHA). Additionally, two trimline configurations were considered: the M0 group featured a trimline ending at the gingival margin, while the M2 group included a trimline extending 2 mm above the gingiva. This design matrix resulted in a total of eight models (Figure 1).  
Vertical and 25\({}^{\circ}\) angled vertical attachments were modeled as single rectangular blocks with dimensions of 3 \(\times\) 2 \(\times\) 1 mm, consistent with typical clinical designs reported in recent finite element studies [10; 14]. Double horizontal attachments were modeled as two individual blocks per tooth (placed mesially and distally), each measuring 2 \(\times\) 2\(\times\) 1 mm to accommodate the buccal surface anatomy of molars. All attachments were assumed to behave as rigid bodies, given the high stiffness of composite resin materials.