## 2 Materials and Methods
In this case, the Standard connection was based on the contacting surfaces observed in the cross-section plane while the Partial connection was simulated considering only cervical contact at the implant platform level (without contact at the connection level). To simulate the difference between the Standard and Partial abutment-implant connections, we modified the contact definitions in the computer-aided engineering (CAE) software rather than altering the CAD geometry. The Partial configuration represented a \"third-party\" abutment with similar overall geometry but insufficient contact with the implant walls. To replicate this condition, we reduced the number of contacting faces in the contact definition, ensuring that the abutment did not fully engage with the implant walls as observed in microscopy. This approach represented the mechanical behavior of an improperly fitting abutment without artificially changing its dimensions. For the Bone-Level implant, the Partial contact in the cervical portion, calculated using the Area command in CAD, was 31 square mm, while the Standard connection covered 73 square mm, representing 42.47% of the total area. In contrast, at the Tissue Level, the cervical area was 64 square mm, and the total area was 74 square mm, representing 86.49% of the total area. An equal number of tangent faces between solids was established to facilitate analysis convergence. An initial mesh consisting of tetrahedral elements was automatically generated, and a thorough check ensured the absence of any mesh marked as obsolete by the software before proceeding to the final mesh refinement (Figure 5). For meshing, the convergence test was based on 150,320 nodes and 72,430 elements. The mesh used tetrahedral elements with 10 nodes (Tet-10). The mesh quality parameters were as follows: an element size of 0.4 mm, quality of 0.78 \(\pm\) 0.12, aspect ratio of 1.56 \(\pm\) 0.45, average maximum corner angle of 93.50\({}^{\circ}\), and average skewness of 0.25 \(\pm\) 0.09. The inflation option was applied to ensure a smooth transition between the geometries. Rigid body behavior was standardized with dimensionally reduced settings to optimize the model's accuracy. Mesh control was determined using a 10% convergence test to minimize its impact on the results of the mathematical calculation [18]. The load was based on the trigonometric identity to calculate the resulting force. In this study, the resultant force was set at 300 N (Figure 6). The oblique loading and exposed threads were used to simulate the ISO14801 worst-case scenario, positioning the implant's long axis at a 30\({}^{\circ}\) angle in the loading direction. The current boundary conditions considered a fixation at the neck of the implant, where it interfaced with the cortical bone, the primary region that bears the majority of the load, especially during axial loading scenarios. Two models, the Tissue-Level (TL) and Bone-Level (BL) ones, were created, each featuring scenarios with both Partial and Standard contacts between the abutment and implant (Figure 7). Figure 5: (**A**) Mesh generated for Tissue-Level model and (**B**) Bone-Level model. Figure 6: Boundary conditions simulated in the present study. The blue area shows the fixation and the red arrow shows the loading.