## 2 Material and Methods
### Analysis from the Mechanical Engineering Standpoint
#### 2.2.5 Discretization  
To proceed with the preliminary steps before analysis, it was crucial to perform the discretization of the model. This entailed dividing the model into multiple elements, each defined by nodes. Consequently, the analysis variables were computed at these nodes, with interpolation between them to determine the values within each element. The accuracy of this interpolation was contingent upon the number of nodes within each element; a higher number of nodes, corresponding to smaller element sizes, yielded a solution that more closely approximated reality. This aspect is crucial for the mesh quality, as these elements must be sufficiently small to capture the structural behavior's details, yet not so small as to render the computational problem intractable. For a better understanding of this stage, a brief introduction is set out below.  
The elements can be of various types, such as tetrahedral, hexahedral, or prismatic, contingent on the model geometry and the analysis requirements. Each element possesses a shape function that delineates the interpolation of displacements within the element from the nodal displacements. These shape functions are polynomials ensuring the continuity of displacements throughout the discretized domain.  
Consequently, the governing differential equations of the system's behavior were discretized into a system of algebraic equations derived through the application of the least squares principle or the Galerkin method. For each element, a stiffness matrix was constructed, which correlated nodal displacements with nodal forces. The global stiffness matrix of the system was then obtained by assembling the stiffness matrices of all individual elements [31].  
Once the global stiffness matrix was assembled and the boundary conditions and loads were applied, the system of linear equations was solved. For linear static analysis, the system is:  
\[f=Ku \tag{1}\]  
where \(f\) represents the vector of applied forces, \(K\) denotes the global stiffness matrix, and \(u\) is the vector of unknown nodal displacements.  
The results were then analyzed using post-processing software, which evaluates displacements, stresses, and strains within the model to ensure adherence to design and safety criteria. The equilibrium equations for a continuous system can be formulated as partial differential equations:  
\[\
abla\sigma+f=0 \tag{2}\]  
where \(\sigma\) represents the stress tensor and \(f\) denotes the applied force vector; it can be solved from the preceding equation. This is feasible because the global stiffness matrix \(K\) is derived by integrating the shape functions and material properties over the finite elements [32].  
In this study, quadratic interpolation elements were used, specifically parabolic-type elements (tetrahedra) (Figure 6), to achieve results as close as possible to reality. The software automatically generates a coarser mesh in larger areas and a finer mesh in smaller areas, allowing the grid to better fit the geometry of the part and improving the accuracy of the results. An average element size of 0.0025 m, a maximum element growth ratio of 1.1, a maximum rotation angle of 65\({}^{\circ}\), and a minimum element size of 0.001 m were defined. This configuration provided greater control over the mesh in contact areas. Following automatic meshing, the mesh was refined in specific areas where problems were anticipated, to facilitate a more precise study. In this research, the analysis was performed with a mesh consisting of 4,579,216 elements and 6,815,049 nodes.  
Following this brief overview of FEM and its relevance to the problem, we can observe the automatic discretization generated by the software (Figure 7).