## 3 Finite Element Calculations  
Numerical homogenization, a key technique in numerical simulations, is employed to estimate the effective elastic and thermal properties of composite materials. Within this framework, the finite element (FE) method is used to model and analyze the behavior of the composite material. This material is composed of a cubic matrix reinforced with spherical inclusions.  
The linear elastic and thermal behavior of the matrix is characterized by fixed values of Young's modulus and Poisson's ratio, while the behavior of the inclusions is variable and dependent on the chosen contrast value. This contrast is defined as the ratio of material properties between the inclusion and the matrix.  
To estimate the effective properties of the composite material, we apply the numerical homogenization approach, which involves treating the composite as the smallest volume for property estimation, to homogenize the entire domain. The use of the Representative Volume Element (RVE) [23] or Unit Cell (UC) [24; 25; 26] is prevalent in this context as they provide effective and reliable material characterization. In our study, a cubic representative pattern is adopted as an RVE composed of 200 inclusions. Table 1 displays the various moduli characterizing the behavior of the matrix phase: Poisson ratio \(\
u\), Young's modulus \(E\), bulk modulus \(k\), shear modulus \(\mu\), and thermal conductivity \(\lambda\), which are essential for the numerical simulations.  
The behavior of the reinforcement in composite material is predominantly characterized by the contrast between the properties of the reinforcing phase and the matrix phase. This contrast can be articulated through various formulas tailored to the specific material properties under consideration. In our study, the contrast between the two phases is expressed in terms of the Young's modulus and thermal conductivity while maintaining a constant Poisson's ratio. Utilizing the contrast formula allows for recalibration of the reinforcement behavior, thereby facilitating an accurate estimation of the composite material's effective properties.  
To calculate the contrast between the two phases, the following formula can be used:  
\[C=\frac{E_{i}}{E_{m}}, \tag{2}\]  
where \(E_{i}\) is the [PERSON]'s modulus of the reinforcing inclusion and \(E_{m}\) is the [PERSON]'s modulus of the matrix phase.  
Similarly, the thermal behavior of both phases is linear. The contrast in thermal conductivity can be calculated using the following formula:  
\[C=\frac{\lambda_{i}}{\lambda_{m}}, \tag{3}\]  
where \(\lambda_{i}\) is the thermal conductivity of the reinforcing inclusion and \(\lambda_{m}\) is the thermal conductivity of the matrix phase.  
In this case, the contrast between the matrix and reinforcing phases has been defined using 10 different values: 10, 20, 50, 80, 100, 120, 150, 180, and 200. To determine the values of the bulk modulus \(k\) and shear modulus \(\mu\) of the composite material, which is considered  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(\
u\) & \(E\) (MPa) & \(k\) (MPa) & \(\mu\) (MPa) & \(\lambda\) (W/m.K) \\ \hline Matrix & 0.3 & 100 & 83.33 & 38.46 & 30 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Typical values of moduli describing the elastic and thermal behavior of the matrix.  
macroscopically isotropic, we can use the following formulas that take into account the values of [PERSON]'s modulus \(E\) and [PERSON]'s ratio \(\
u\):  
\[\mu=\frac{E}{2(1+v)}\quad\text{and}\quad k=\frac{E}{3(1-2v)} \tag{4}\]