# Numerical Simulation on Corneal Surface Behavior Applying Luminous Beam Levels
## 2 Materials and Methods
### Numerical Requirements  
The eyeball is continuously under different beam levels of energy. Nowadays, every single person uses devices or tools that are sources of light and that emit luminous brightness, which can cause injuries or blindness. To develop the numerical simulation, it was necessary to consider the eyeball and the corneal surface where the luminous beams impact; different common cases were considered, such as sunlight, the screen of a smartphone, and beams of light from devices (e.g., high brightness luminaries or laser light); depending on the case, different compression loads are exerted on the structural surface. Once the detailed design was completed successfully, it was exported to a computer program where the pre-processing, numerical configuration, solution, and post-processing were carried out; the FEM was applied, high-order tetrahedral elements were considered to the whole model, and the irradiance of the light to the cornea was at a 200 mm distance. The model in Figure 2 shows a cross-section of the eyeball and one study case representation.  
The developed model was considered for the numerical analysis. It is essential to follow the methodology proposed. After the reconstruction of the eyeball's 3D model, the  
Figure 1: Reconstruction of a human eyeball and its parts; isometric cross-section view.  
Figure 2: Anterior parts of the eyeball and the smartphone study case representation.  
next step was to export the model in STL format to the computer program MATLAB[6]; in this program, a 3D scenario was created to perform the numerical solution. The model consists of 17,131 faces, with a weight of 587 kb. Then, a tetrahedral mesh was applied to the model, and high-order tetrahedral finite elements were applied. There are 10 nodes per element: 4 angular nodes (vertices) and 6 nodes between each vertex. Figure 3 shows the corneal surface (area to be analyzed) in tetrahedral elements.  
This analysis only considers the corneal surface as the element of study, considering its mechanical properties, such as its percentage of transparency (70%) and convex shape. In the MATLAB[6] program student version, the boundary conditions were applied on specific nodes to fix the cornea's contour (474 points), and nodes of movement (2223 points) on the corneal surface were assigned to the study case. The structure of the cornea is composed of collagen fibers; the human cornea has the highest penetration compared to domestic animals such as rabbits or pigs. The modulus of elasticity in the human cornea is approximately 2.29 MPa with a variation of \(\pm\)0.06 MPa (in ages from 18 to 30). Thus, the average is from 0.13 MPa to 0.43 MPa because the younger the person, the lower the value, while elderly people have higher values [28]. In the anterior part of the model, the cornea is considered as a deformable material, while the posterior parts of the visual organ were taken as anchorage elements to achieve the numerical simulation for this study; in Figure 4, the anchorage nodes and the movement nodes were configured in the whole model.  
Models are used to describe mechanical behavior and obtain stress and strain patterns applied to organic tissues like the human cornea. The Ogden model is regarded as a  
Figure 4: Boundary conditions are assigned on the model, blue dots as support nodes and yellow dots as movement nodes.  
Figure 3: Tetrahedral mesh on the corneal surface.  
suitable model for hyperelastic material behavior and depicts strain energy \(\mathbf{W}\) considering the main extensions: \(\lambda_{1},~{}\lambda_{2},~{}\lambda_{3}\). The results of stress tests are obtained with an acceptable approximation using this mathematical model. The appropriate expression is:  
\[\mathbf{W}=\sum_{p=1}^{n}\frac{u_{p}}{\alpha_{p}}.\left(\lambda_{1}^{ap}+\lambda_{1 }^{ap}+\lambda_{1}^{ap}-3\right)\]