## 4 Mathematical Model  
Our database comprises 3D microstructures that serve as inputs to provide insight into the elastic and thermal behaviors of materials as outputs. We aim to utilize this database to construct a mathematical model capable of estimating various moduli based on two pivotal parameters: volume fraction and contrast. The essence of this approach is its ability to predict material properties directly from microstructural data. By employing a database replete with virtual microstructures and correlating this information with the mechanical and thermal properties of materials, we can formulate a model that streamlines and expedites the prediction process. The forthcoming mathematical model will incorporate the volume fraction and contrast as primary parameters. Volume fraction denotes the proportion of each component within the microstructure, while contrast quantifies the disparity in properties between these components. These parameters are instrumental for determining the macroscopic properties of materials. Our objective is to establish a precise mathematical relationship that links these parameters with the overall properties of the materials. In the initial phase of our study, we focus on characterizing the material behavior in terms of its volume fraction \(V_{f}\) and contrast \(C\). The objective here is to develop mathematical equations based on finite element calculations tailored for different values of \(V_{f}\) and \(C\) that have already been ascertained. To accomplish this, we employ curve fitting: an optimization technique that involves selecting the most suitable set of parameters for a function so that it aligns well with the observed data trends. Essential to this approach is the defining of a mapping function that links the input data (volume fraction or contrast) to the output data (material properties). Our goal is to identify parameters for a fundamental function that accurately models the relationship between the material's behavior and these variables, allowing for flexibility in form. This process will enable us to control the shape of the resulting curve through optimization, pinpointing the precise parameters of the function. In our analysis, we will use a two-dimensional fitting approach, where the x-axis represents the model inputs (either volume fraction or contrast) and the y-axis corresponds to the outputs: namely, the material's elastic and thermal properties. To execute curve fitting effectively, we initially define the mathematical form of the objective function and then proceed to identify its parameters that minimize error. This error is quantified as the discrepancy between the output values generated by the proposed mapping function and those derived from finite element calculations. By differentiating this error relative to the function's parameters, we can ascertain the optimal parameter settings. Once the parameters have been determined, the mapping function becomes a tool to interpolate or extrapolate new data points within the region of interest. This is achieved by feeding a sequence of input values into the mapping function, which allows us to graphically represent the variation in the output in relation to the input. For error minimization, we utilize the least squares minimization method [28; 29; 30]. This method is versatile and applicable to any form of mapping function since all functions can theoretically be expressed in polynomial terms. The use of this method ensures that the best fit for the data is achieved by taking into consideration the least cumulative squared difference between the model and the actual data. The equation for the error is written in the form:  
\[E_{r} =\sum_{i=1}^{n}E_{i}^{2}\] \[=\left(y_{1}-f(x_{1})\right)^{2}+\left(y_{2}-f(x_{2})\right)^{2} +\left(y_{3}-f(x_{3})\right)^{2}\ldots+\left(y_{n}-f(x_{n})\right)^{2} \tag{29}\] \[=\sum_{i=1}^{n}(y_{i}-f(x_{i}))^{2}\]  
where: \(n\): as the number of data points;  
\(x\): input data;\(y\): output data;  
\(f\): mapping function.