# Numerical-Experimental Analysis to Optimize the Geometry of a Prosthetic Abtment
## 1 Introduction
A preparation taper between 5\({}^{\circ}\) and 22\({}^{\circ}\) can be considered clinically acceptable [11; 18]. In general, molar teeth require a minor taper preparation than premolar or anterior teeth to achieve an adequate resistance form [19]. In the preparation of molar crowns, the minimum height of the preparation wall should ideally range between 3.5 and 4 mm. The study by [PERSON] et al. demonstrated that increasing the taper of the preparation and rounding off the axial angles lead to a reduction in resistance. Further, the presence of grooves with walls perpendicular to the direction of the applied forces can improve the resistance form [20]. Therefore, U-shaped grooves provide greater resistance than V-shaped ones [21; 22]. In the case of a particularly inclined taper, a taper restriction of the cervical aspect was found to be more effective than other grooves [23]. Recently, it was found that the morphology of the abtument plays an important role in safeguarding the gingival margin. In fact, by creating a vertical connection between the emerging reconstruction and the crown through an appropriate finishing line of the abtument, the dual purpose of protecting the gum and providing a vertical stop for the distribution of functional loads is achieved [24]. The abtument marginal designs currently available are shown in Table 1. We can identify the following: knife-edge, feather-edge, chamfer, and shoulder [25; 26]. One of the main limitations of conventional clinical approaches found in the literature is that they typically assess the patient's response only after the application of a conservative technique. This approach is inherently subjective and fails to provide a comprehensive analysis of the stress state. While clinical testing remains essential, the proposed methodology seeks to minimize load transfer under consistent external conditions--suchas abutment quality, cement type, and other factors--thereby enhancing the mechanical safety of the abutment tooth/prosthesis system. To achieve this goal, the paper focuses on determining the optimal geometry for the abutment-prosthesis coupling by developing a new, more efficient design. This design was derived from an initial mathematical study and subsequently validated through numerical simulations. The experimental validation was accomplished by constructing a lower molar model at a 10:1 scale. The mathematical study was based on a modified theory concerning the localization of the geometric shapes of the prosthetic abutment [27]. Specifically, the mathematical relationships were tailored to the unique geometry of each individual tooth using a specialized calculation algorithm. The structure of this paper is as follows: Section 2 outlines the key geometric definitions for both the abutment and prosthesis. Section 3 presents an analytical study focused on the abutment geometry and the corresponding stress distribution, particularly emphasizing the calculation of minimum loads, including tensile forces. In Section 4, these analytical results are validated through Finite Element Analysis (FEA). Section 5 details the experimental validation using a 10:1 scale model, where the load transfer at the abutment-cement-prosthesis interface is measured. Finally, Section 6 provides a comparison between numerical and experimental findings, confirming a strong agreement between the two approaches, followed by the conclusions.