## 4 Basic Principle of Thermal Parameter Inversion  
### Parameter Selection  
The hydration heat of cement is a vital factor for the variation in internal thermal stress. During the calculation process of the temperature field, the adiabatic temperature rise \(\theta\) of the concrete is customarily utilized as the principal parameter. Adiabatic temperature rise is the temperature growth that is directly brought about by all the heat released by the adequate hydration process of cementitious materials in concrete when there is no heat loss in the ideal state of completely isolating the external heat exchange [25]. The calculation of adiabatic temperature rise depends on a series of thermal characteristic parameters of concrete. It is often expressed in hyperbolic form, and the expression is as follows:  
\[\theta(\tau)=\theta_{0}\frac{\tau}{n+\tau} \tag{11}\]  
Within the formula, \(\theta(\tau)\) is the adiabatic temperature rise function of concrete; \(\theta_{0}\) is the final adiabatic temperature rise of concrete; \(\tau\) is the age of concrete; \(n\) is the parameter of temperature rise law. Although the results obtained by the hyperbolic formula demonstrate a high degree of coincidence with the experimental data, the calculation process is relatively complicated. In view of this, [PERSON] proposed a straightforward and efficient exponential approach to calculate the adiabatic temperature rise of concrete. The expression is as follows:  
\[\theta(\tau)=\theta_{0}s(1-e^{-m_{1}\tau})+\theta_{0}(1-s)(1-e^{-m_{2}\tau}) \tag{12}\]  
The parametric relationship of Equation (12) is as follows:  
\[m_{1}=\frac{1.45}{n},m_{2}=0.10m_{1} \tag{13}\]  
In Equations (12) and (13), \(m_{1},m_{2}\) represent the hydration heat dissipation coefficient of concrete. As constants without units, they dominate the slope change of the temperature rise curve in the primary stage and the advanced stage during the evolution of the concrete's temperature. It means that they affect the rate of concrete temperature rise; \(s\) is a coefficient related to the concrete material. Most of the \(s\) values in the practical application were 0.60. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **ALO** & **SOA** & **SCSO** \\ \hline F1 & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) \\ F2 & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) \\ F3 & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) \\ F4 & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) & \(8.83\times 10^{-5}\) \\ F5 & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) \\ F6 & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) \\ F7 & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) & \(3.38\times 10^{-4}\) \\ F8 & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) & \(1.40\times 10^{-4}\) \\ F9 & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) & \(1\) \\ F10 & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) & \(0.002\) \\ F11 & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) & \(1\) \\ F12 & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) & \(8.86\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: \(p\)-values of the Wilcoxon rank sum test for different algorithms on functions F1–F12. In this paper, the combined exponential formula that is applicable for operation is adopted as the expression for the adiabatic temperature rise of concrete during the parameter inversion calculation process. The inversion coefficient \(m_{1}\) is transformed by Equation (13) to obtain the parameter \(n\). The expression of the final result is formulated as a hyperbolic form that can be easily organized.