## 3 Finite Element Calculations
### Mesh Generation  
For the successful execution of finite element (FE) simulations, generating a mesh that accurately represents the analyzed system's geometry is essential. In our study, we employed a free mesh consisting of tetrahedral elements (C3D10) with complete integration to mesh the microstructures, as illustrated in Figure 2. This mesh type is particularly adept at modeling complex geometries, making it an ideal choice for the intricate structures present in our composite material analysis. To ascertain the minimum number of finite elements needed for accurately meshing an inclusion, we refined the mesh density and monitored the convergence of macroscopic properties. For this purpose, a 3D microstructure containing 200 particles with a volume fraction of 25% was selected as the model system. Throughout the process, various levels of mesh refinement were applied while maintaining the microstructure's integrity. The strategy involved increasing the mesh density by reducing the size of the finite elements. Consequently, a greater number of finite elements were utilized to mesh the system. Through this process, we examined the convergence of macroscopic properties to determine the optimal number of finite elements required for precise meshing of a single particle. In our approach, complete integration was employed utilizing four Gauss integration points per element. This technique facilitated the accurate computation of the microstructure's macroscopic properties. The use of a free mesh composed of tetrahedral elements enabled effective modeling of the microstructure and a thorough analysis of property convergence. Accurate mesh generation is fundamental for conducting finite element simulations as it significantly influences the results' precision. Optimizing mesh density is therefore crucial for the accurate representation of the system. Additionally, studying the convergence of macroscopic properties is essential to establish the ideal number of finite elements for accurate results. Figure 3, which illustrates the convergence of the bulk modulus and the thermal conductivity in relation to the number of finite elements used for meshing the microstructure, offers crucial insights into determining the optimal mesh density for precise simulations. The figure clearly shows that initially there is a significant decrease in the two moduli as the mesh becomes finer. This trend occurs because finer meshing, involving an increased  
Figure 2: Complete integration of tetrahedral elements (C3D10) in a free mesh configuration. number of finite elements, leads to a more accurate representation of the microstructure. However, beyond a certain point, as the number of finite elements continues to grow, the bulk modulus reaches a state of stabilization. This observation suggests that beyond this point, further refinement of the mesh does not contribute substantially to the improvement in accuracy. The convergence analysis revealed that the optimal number of finite elements for achieving accurate results in terms of the bulk modulus is approximately 205,070, while for thermal conductivity, the optimal number is around 180,950. Consequently, we will conduct the calculations using the specifically identified optimal number of finite elements corresponding to the bulk modulus. This figure applies to a volume containing 200 particles, each with a volume fraction of 25%. At this level, the bulk modulus reaches convergence, indicating that further increases in the number of finite elements do not significantly impact the accuracy of the results. Additionally, the analysis shows that the difference in bulk modulus values between simulations using 205,070 and 398,500 elements is less than 1%, reinforcing the notion that further mesh refinement beyond this point does not substantially enhance accuracy. Consequently, for all subsequent simulations, a mesh density of about 1025 elements per particle was selected.