## 5 Conclusions  
In this article, we introduced a novel homogenization method centered around curve-fitting techniques. This method presents notable advantages in terms of time and resource efficiency, particularly when compared to conventional numerical or analytical methods. The development of our model comprised several key stages.  
Initially, we collected 3D microstructures featuring 200 spherical inclusions while varying the contrast between the phases and their volume fractions. Employing finite element homogenization, we calculated various moduli, including the bulk modulus, shear modulus, and thermal conductivity. The resultant database was then leveraged to construct a mathematical model through a three-step process. The first step involved plotting module curves as functions of the volume fraction and contrast and enabled us to ascertain the shapes and limits of each curve and gain insight into the behavior of the modules. Subsequently, in the second step, we mathematically synthesized and integrated these modules to create a unified model that accurately predicts the values of different moduli. This involved analyzing the interrelations between the modules and the specific parameters to identify patterns and trends. Finally, we formulated mathematical equations or functions that effectively encapsulate and represent these relationships.  
This methodical development process culminated in a robust mathematical model capable of precisely predicting different moduli based on variations to the volume fraction and contrast. The model offers a comprehensive understanding of the interplay between these parameters and the modules and thus serves as an invaluable tool for efficient and accurate predictions for similar heterogeneous material systems.  
The results obtained affirm the efficacy of the chosen mapping functions and parameters. The functions are in compliance with the imposed boundary conditions and align with the first- and second-order analytical bounds. This alignment underscores the model's success at capturing the fundamental characteristics and behaviors of the modules in question and validates its application in the field of material science and engineering.  
Methodology, T.K., T.M. and A.A.; Software, N.B.-L.; Validation, H.B.; Writing-original draft, H.B.; Supervision, T.K., T.M. and A.A. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author. The data are not publicly available due to ethical and legal restrictions.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Coefficient** & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{c}\) & \(\mathbf{d}\) & \(\mathbf{e}\) & \(\mathbf{f}\) & \(\mathbf{g}\) & \(\mathbf{h}\) & \(\mathbf{k}\) \\ \hline Bulk & 229.9 & 108.3 & 66.35 & 242.6 & \(-\)5268.4 & 7370.9 & 11.69 & \(-\)220.2 & \(-\)324.07 \\ \hline Shear & 182.75 & 60.65 & 20.71 & \(-\)142.5 & 2905.9 & \(-\)817.47 & \(-\)6.94 & 123.02 & 334.16 \\ \hline Thermal & 194.1 & 66.68 & 14.27 & \(-\)103.91 & 5525.81 & \(-\)3917.67 & \(-\)5.44 & 265.67 & 1049.5 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The optimal coefficients of the generalized fitting function obtained for the bulk modulus, shear modulus, and thermal conductivity.