## 2 Materials and Methods  
Finite element modeling is a computational technique that divides complex geometric structures into smaller, interconnected elements to analyze mechanical behavior under applied forces. In orthodontic applications, this method enables prediction of stress distribution, deformation patterns, and tooth movement by solving mathematical equations that describe the mechanical properties of dental tissues and appliances. The simulation process involves creating a three-dimensional digital model, assigning material properties to each component, applying boundary conditions and forces, and calculating the resulting displacements and stresses throughout the structure. All finite element analyses in this study were performed using ANSYS 2024R software (ANSYS Inc., Canonsburg, PA, USA). The initial geometric data were obtained using CBCT imaging (PaX-Zenith 3D, Vatech Co., Seoul, Republic of Korea) from a patient with normal occlusion. The study protocol was approved by the Institutional Review Board (IRB) of Pusan National University Dental Hospital (IRB No. 2025-05-014). The finite element model included the alveolar bone, maxillary teeth, and periodontal ligament (PDL). The PDL was modeled with a uniform thickness of 0.25 mm along the outer surface of each root. The CA was constructed with a consistent thickness of 0.5 mm, extending from the outer surface of the crown and tailored according to the design specifications of each experimental group. A horizontal rectangular attachment was designed on the buccal surface of the maxillary canine with dimensions of 2 mm (height) \(\times\) 3 mm (width) \(\times\) 1 mm (thickness). Through its experimental design, this study aimed to evaluate biomechanical effects of varying the buccolingual width of the maxillary first premolar extraction space on the bodily movement of the maxillary canine during CA treatment. To achieve this, four types of CA designs incorporating beam-shaped structures of different widths within the extraction space were developed, and the corresponding finite element models were constructed (Figure 1):  
* Group 0: design with 0 mm beam width (no beam) in the extraction space. * Group 1: CA design with 1 mm beam width in extraction space. * Group 2: CA design with 2 mm beam width in the extraction space. * Group 3: CA design with 3 mm beam width in the extraction space. To conduct finite element analysis on the constructed three-dimensional model, the mechanical properties of each material component were defined. The teeth, maxilla, attachments, and clear aligners were all modeled as homogeneous, isotropic, and linearly elastic materials. Based on previous studies [16, 19, 20], [PERSON]'s modulus and Poisson's ratio for each material were applied, and a simplified linear elastic model was adopted for the periodontal ligament (PDL) (Table 1). \begin{table}
\begin{tabular}{l r r} \hline \hline
**Material** & **Young’s Modulus (MPa)** & **Poisson’s Ratio** \\ \hline Alveolar bone & 13,700 & 0.3 \\ Teeth & 19,600 & 0.3 \\ Clear aligner & 528 & 0.36 \\ Composite attachment & 12,500 & 0.36 \\ Periodontal ligament & 0.69 & 0.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of materials used in the finite element models. To quantify tooth displacement and directional movement, a global coordinate system was established to define the \(x\)-, \(y\)-, and \(z\)-axes in three-dimensional space:  
* \(X\)-axis: Defined as the intersection of the coronal and occlusal planes; on the axial plane, the positive direction points toward the patient's right side. * \(Y\)-axis: Oriented perpendicular to the \(x\)-axis on the occlusal plane; the positive direction points toward the patient's anterior. * \(Z\)-axis: Perpendicular to both the \(x\)- and \(y\)-axes; the positive direction is oriented toward the root of the tooth. Tooth movement simulation was conducted using the finite element method (FEM) to model the gradual displacement of the teeth and the PDL resulting from alveolar bone remodeling [19; 20].