## 4 Discussion
The observed force reduction throughout treatment reflects the stress relaxation phenomenon inherent in thermoplastic materials, representing a characteristic response to the progressive mismatch between CA geometry and evolving tooth positions [28, 29, 30, 31, 32]. However, both the rate and pattern of this force reduction varied significantly with the beam width because the beam geometry fundamentally altered thedynamic force transmission characteristics throughout the treatment sequence. An optimal force range exists for different types of tooth movement. Studies by [PERSON] et al. have established 70-120 g (approximately 0.70-1.20 N) as the optimal force range for bodily movement [33; 34; 35; 36]. In the groups in the present study, the initial Stage 1 values represent the maximum forces occurring upon appliance insertion. As previously mentioned, the stage numbers cannot be directly equated with temporal progression. Nevertheless, when considered as a sequential process corresponding to tooth movement, the forces following the initial appliance insertion (Stages 1-2) remain within the optimal range for bodily movement through Stages 3-10 and gradually diminish thereafter. For bodily tooth movement, a specific moment-to-force (M/F) ratio is required to direct the force through the center of resistance or counteract the moments generated by force application at the center of resistance. When a force is applied to the crown, an inevitable moment occurs that causes tooth tipping. In Group 0, the M/F ratio increased continuously from an initial value of 2.594 to 13.693 at Stage 50, while other groups achieved final values of 13.775, 13.776, and 14.003, respectively. The combination of forces and moments transmitted to the canine across all experimental groups did not achieve the ideal M/F ratio for pure bodily tooth movement. Although Group 3 generated larger initial forces, the final tooth displacement was marginally smaller than that of Group 0. This indicates that the tooth movement pattern manifested as root uprighting following initial crown tipping, rather than achieving a true 'bodily movement' force system from treatment initiation. This sequential movement pattern suggests a controlled tipping-and-uprighting mechanism rather than a pure translation: the deformation patterns of the CA varied significantly according to the beam width implemented in the extraction space. Notably, compared to the other configurations, the 3 mm beam width design demonstrated minimal deformation in the gingival margin area of the maxillary central incisor. This reduced deformation can be attributed to the ability of the beam structure to locally enhance the structural rigidity of the aligner within the extraction space region. The presence of the beam directly influenced the mechanical properties of the aligner material. The stiffness of the aligner materials plays a crucial role in transmitting the necessary light forces while maintaining a high elastic limit to prevent permanent deformation [17; 18; 37]. The 3 mm beam width design significantly enhanced the aligner's deformation resistance capabilities. CA deformation results directly from the applied stress and the material's inherent deformation resistance. Consequently, the reduced deformation in specific regions indicates either a lower applied stress in that area or enhanced deformation resistance under stress conditions. The 3 mm beam width design increases the local stiffness, enabling the aligner to maintain its intended geometry while transmitting more controlled forces to the adjacent teeth. This enhanced structural performance aligns with previous research findings demonstrating that strategic 'extraction space area' modifications can facilitate localized control and minimize unwanted deformation patterns [37]. Finite element analysis is a valuable tool for evaluating biomechanical effects of orthodontic appliance designs. The selection of finite element analysis as the primary evaluation method was based on several critical considerations.