## 4 Mathematical Model
Since we want to fit a polynomial, we can write \(f(x)\) as:  
\[ \begin{split} f(x)&=u_{0}+u_{1}x+u_{2}x^{2}+\ldots+u_ {n}x^{n}\\ &=u_{0}+\sum_{j=1}^{k}u_{j}x_{i}^{j}\end{split}  \tag{30}\]  
where: \(k\) is the order of the polynomial. By replacing\(f\) with this expression in the error equation, we obtain:  
\[E_{r}=\sum_{i=1}^{n}\left(y_{i}-\left(u_{0}+\sum_{j=1}^{k}u_{j}x_{i}^{j} \right)\right)^{2} \tag{31}\]  
The application of calculus is a well-established method for identifying the minimum of an error function. This involves taking the derivative of the error function with respect to each unknown coefficient, which yields an equation representing the function's slope. The critical points, where this slope equals zero, are indicative of either a minimum, a maximum, or a saddle point in the function. To pinpoint the minimum, we set the first derivative to zero and solve for the unknown coefficients. In our case, there are \(k\) unknowns, denoted as \(a_{0},a_{1},\ldots,a_{k}\). The process necessitates computing the derivative for each of these unknowns individually and thereby determining their optimal values to minimize the error function. This approach is crucial for precisely calibrating our model to reflect the actual behavior of the material based on the collected data. \[ \begin{split}\frac{\partial E_{r}}{\partial u_{0}}& =-2\sum_{i=1}^{n}\left(y_{i}-\left(u_{0}+\sum_{j=1}^{k}u_{j}x_{i}^ {j}\right)\right)\quad=0\\ \frac{\partial E_{r}}{\partial u_{1}}&=-2\sum_{i=1 }^{n}\left(y_{i}-\left(u_{0}+\sum_{j=1}^{k}u_{j}x_{i}^{j}\right)\right)x_{i} \quad=0\\ \frac{\partial E_{r}}{\partial u_{2}}&=-2\sum_{i=1 }^{n}\left(y_{i}-\left(u_{0}+\sum_{j=1}^{k}u_{j}x_{i}^{j}\right)\right)x_{i} ^{2}=0\\ \vdots\\ \frac{\partial E_{r}}{\partial u_{k}}&=-2\sum_{i=1 }^{n}\left(y_{i}-\left(u_{0}+\sum_{j=0}^{k}u_{j}x_{i}^{j}\right)\right)x_{i}^{ k}=0\\ \end{split}  \tag{32}\]  
These equations can be re-arranged into matrix form:  
\[\left[ \begin{array}{cccc}n&\sum x_{i}&\sum x_{i}^{2}&\cdots&\sum x_{i}^{k }\\ \sum x_{i}&\sum x_{i}^{2}&\sum x_{i}^{3}&\cdots&\sum x_{i}^{k+1}\\ \sum x_{i}^{2}&\sum x_{i}^{3}&\sum x_{i}^{4}&\cdots&\sum x_{i}^{k+2}\\ \vdots&\vdots&\vdots&\cdots&\vdots\\ \sum x_{i}^{k}&\sum x_{i}^{k+1}&\sum x_{i}^{k+2}&\cdots&\sum x_{i}^{k+k}\\ \end{array} \right]\left[ \begin{array}{c}u_{0}\\ u_{1}\\ u_{2}\\ \vdots\\ u_{k}\\ \end{array} \right]=\left[ \begin{array}{c}\sum(y_{i})\\ \sum(x_{i}y_{i})\\ \sum(x_{i}^{2}y_{i})\\ \vdots\\ \sum(x_{i}^{k}y_{i})\\ \end{array} \right] \tag{33}\]  
Hence, the solution of the matrix system is:  
\[\mathbf{U}=\mathbf{X}^{-1}\mathbf{B} \tag{34}\]  
In the end, we can derive the polynomial form that best fits the curve:  
\[y=U[0]+U[1]x+U[2]x^{2}+\ldots+U[k]x^{k} \tag{35}\]The Python library scipy offers an API that is adept at curve fitting to a dataset by identifying optimal parameters for a mapping function. This is achieved through the least squares minimization method, which efficiently finds the best fit for the given data. Modeling of Effective Elastic and Thermal Properties of Heterogeneous Material as a Function of Inclusion Volume Fraction  
Our initial focus in this study is to analyze how the elastic and thermal behavior of a heterogeneous material varies as a function of the volume fraction while maintaining a constant contrast value. This investigation entails examining the material's response to changes to the volume fraction. Volume fraction, in this context, quantifies the proportion of one phase in a two-phase material system. By manipulating the volume fraction and keeping the contrast constant, we gain insights into the dynamic behavior of the material. This approach enables us to identify the ideal volume fraction that results in the desired elastic and thermal properties of the material. Figure 4 visually illustrates the variation in the bulk modulus with different volume fractions.