## 4 Mathematical Model
These functions are also well-suited for describing physical phenomena that have asymptotic behavior, meaning they tend towards a limiting value as the independent variable approaches infinity or zero. \[f(x)=\frac{b_{0}+b_{1}x+b_{2}x^{2}}{c_{0}+c_{1}x+c_{2}x^{2}} \tag{38}\]  
where \(b_{0}\), \(b_{1}\), \(b_{2}\), \(c_{0}\), \(c_{1}\), and \(c_{2}\) are adjustable coefficients. Rational functions stand out for their ability to accommodate data with intricate variations, owing to the flexibility provided by their form. This adaptability arises from the fact that both the numerator and the denominator of a rational function are polynomials, and their coefficients can be adjusted. Such adjustability enables these functions to more precisely fit experimental data compared to other functions with fixed mathematical structures. Moreover, rational functions are particularly adept at modeling physical phenomena characterized by asymptotic behavior. In these scenarios, as the independent variable trends towards infinity or zero, the function approaches a finite limit, making rational functions suitable for describing such behavior. Employing rational functions for data fitting allows for a more nuanced representation of the underlying physical or phenomenological principles. The coefficients of the polynomial terms in the function can be interpreted in a physical context, offering deeper insights into the system's behavior. To determine the optimal coefficients \(b_{i}\) and \(c_{i}\) for the rational function, least squares regression analysis is commonly used. This involves minimizing the sum of the squares of the differences between the observed data points and the values produced by the function. By taking partial derivatives of the error function \(E_{r}\) with respect to each coefficient and setting them to zero, we obtain a system of linear equations. Solving these equations yields the coefficients that minimize \(E_{r}\). Figure 6: Variation of bulk modulus of heterogeneous material as a function of contrast for a volume fraction value equal to 25%. Techniques such as matrix algebra or Gaussian elimination can be utilized to solve this system of linear equations. With the optimal coefficients determined, the rational function can then be used to predict dependent variable values for any given independent variable within the data range. By comparing these predictions to the actual data, the fit's accuracy can be assessed, and adjustments to the coefficients can be made as necessary. Figure 7 illustrates the fitting of various moduli using the quadratic rational function. These curves visually demonstrate how the values of the moduli vary with the parameter and help to establish the relationships between contrast and material properties. Table 3 presents the coefficients \(b_{i}\) and \(c_{i}\) for the fitting functions associated with different volume fraction values. These coefficients exhibit variations depending on the volume fraction, indicating changes in the material behavior as the volume fraction increases. Analyzing these coefficients provides valuable insights into the characteristics of the material and its response to the volume fraction. By utilizing these coefficients obtained from the fitting function, one can predict the material's behavior by substituting the input value with a desired contrast. Figure 7: Prediction of bulk, shear, and thermal conductivity as a function of volume fraction with varying contrast using polynomial and exponential curve fitting. An intriguing observation arises from analysis of the curve and parameter table. It becomes apparent that the preceding equation successfully satisfies the imposed boundary condition. Specifically, when the contrast value is set to \(1\) (\(C=1\)), the material's behavior aligns with that of the matrix phase. This correlation is clearly manifested in the calculated value of \((b_{0}+b_{1}+b_{2})/(c_{0}+c_{1}+c_{2})\), which closely approximates the known value of \(K_{m}\) (matrix bulk modulus) at 83.33.