## 1 Introduction  
Despite the expanding clinical applications of clear aligner (CA) therapy, structural limitations in effectively achieving bodily tooth movement still remain [1, 2]. These limitations are primarily attributed to the low stiffness of the aligner material and the nature of the force system, which is typically applied near the center of the crown. Consequently, root movement is often insufficient, leading to increased tipping, commonly referred to as the \"Roller-Coaster Effect\" [3, 4]. This biomechanical shortcoming is particularly problematic in complex cases that require precise tooth control, such as during extraction space closure. The literature describes various strategies to address biomechanical limitations of CA therapies. One approach involves increasing aligner rigidity by modifying physical properties, including the development of multilayer materials with enhanced stiffness, the incorporation of reinforcing fibers, and the optimization of thermoplastic polymer compositions. For instance, studies have investigated polyethylene terephthalate glycol (PETG) and thermoplastic polyurethane (TPU) materials with varying elastic moduli to achieve optimal force delivery characteristics. Although these material modifications can reduce appliance deformation and promote more precise tooth movement, such changes may affect overall treatment dynamics, patient comfort, and cost-effectiveness, necessitating further research and cautious clinical implementation [5]. Other approaches involve the use of auxiliary appliances such as attachments, power ridges, intraoral elastics, and temporary anchorage devices (TADs) [6; 7; 8; 9; 10]. Although these adjuncts can enhance the magnitude and direction of orthodontic forces, they have several drawbacks, including difficulty in appliance removal, increased patient discomfort and pain, and unintended biomechanical effects on the adjacent non-attached teeth [11; 12]. Despite these improvements, the predictability and efficiency of tooth movement with CAs remains suboptimal. Therefore, continued efforts toward the biomechanical optimization of aligner design in conjunction with improved treatment protocols are essential for improving clinical outcomes [13]. Recent studies have demonstrated that specific structural design elements of CAs play critical roles in determining the direction and nature of tooth movement. These elements include aligner thickness variations that affect force magnitude and distribution, trimline design and gingival extension patterns that influence force application points, attachment configurations that enhance rotational and translational control, power ridges that create localized stress concentrations for targeted movements, and extraction space modifications such as pontic designs and structural reinforcements that affect force transmission across edentulous regions [3; 4; 13; 14]. In line with this research trend, an increasing number of investigations have focused on analyzing how different aligner designs influence tooth movement patterns, particularly within edentulous spaces. In a study comparing non-pontic, semi-pontic (half the anatomical pontic length), and anatomic pontic designs for the anterior movement of mandibular second molars, the semi-pontic design demonstrated the highest root-to-crown movement ratio and exerted the least extrusion force, thereby facilitating the most effective bodily movement [15]. Similarly, in a study examining the effects of non-pontic, full-pontic, half-pontic (half-width), and beam designs on the premolar extraction space during posterior canine movement, the beam design initially produced the greatest displacement. However, over time the beam design also resulted in the lowest overall displacement and inclination, confirming its superiority in controlling bodily movement [16].