# Multiple Thermal Parameter Inversion for Concrete Dams Using an Integrated Surrogate Model
## 3 Case Study
### Analysis and Discussion of Inversion Results
#### 3.4.1 Inversion Results Based on the ISM  
It can be seen in Figure 6 that the R\({}^{2}\) variation of the integrated agent model is slight when the sample size is greater than 60. Thus, the ISM with a training sample size of 60 is selected to replace the finite element model. The Jaya algorithm invokes the trained integrated agent model for inversion with a population size of 40 and a maximum iteration number of 200. The inversion results are shown in Table 4.  
Since the actual thermal parameter values are unknown on the construction sites, the reliability of its results cannot be directly verified. The common approach to this problem is to use FEM to obtain the calculated temperature based on the inversion results and evaluate the degree of agreement between this temperature and the monitored one. According to the assumptions mentioned earlier, the adiabatic temperature rise model ignores the influence of factors other than age on the hydration process, the homogenization treatment of concrete materials, etc. These factors are difficult to describe accurately in the temperature analysis of dams under construction, but will undoubtedly cause deviations between the calculated and monitored temperatures. To prevent overfitting, the global agreement of the temperature should be used to evaluate the reliability of the inversion results. The mean absolute error (MAE) is quantified in this article, and Figure 8 depicts a comparison of the calculated and monitored temperatures. From Table 4, it may be noted that the inversion results vary for the two concretes; therefore, it is necessary to invert the parameters by sub-regions for concrete. According to Figure 8, in the beginning, the heat dissipation rate of the pouring block surface and water pipe cooling is lower than the heat release rate of cement hydration, resulting in the accumulation of heat inside the cast block and causing the temperature to rise until it reaches the maximum value rapidly. As the heat release rate decays, the surface heat dissipation and water pipe cooling begin to dominate, leading to a gradual decrease in concrete temperature. It can be found that the overall variation of the calculated and monitored temperatures tends to be consistent, with stage deviations mainly concentrate in the early times, especially in the temperature rise phase, as shown in Figure 8e,f. The main reasons for these deviations between the two are related to the aforementioned calculation assumptions. The MAE of the C35 and C40 concrete pouring blocks were 0.32 \({}^{\circ}\)C and 0.31 \({}^{\circ}\)C, respectively, with maximum errors of 1.02 \({}^{\circ}\)C (15#-53) and 1.75 \({}^{\circ}\)C (15#-59). The agreement between their monitored and calculated temperatures was supervised in the inverse analysis for the above six pouring blocks, while the other pouring blocks were not. To further investigate the universality of the inversion results, one pouring block was selected from zone A and another from zone B, neither participating in the sample set generation. The degree of agreement between the calculated and monitoring results of each of these two blocks was then compared, as shown in Figure 9. The MAE of C35 and C40 were 0.59 \({}^{\circ}\)C and 0.73 \({}^{\circ}\)C, respectively. The maximum error of the C35 pouring block was 2.35 \({}^{\circ}\)C, which happened during the warming stage, after which the errors were within 1.5 \({}^{\circ}\)C. The maximum error of the C40 pouring block was 1.17 \({}^{\circ}\)C. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & \(\mathbf{a}\) & \(\mathbf{\theta}\) & \(\mathbf{m_{1}}\) & \(\mathbf{s}\) & \(\mathbf{\beta}\) \\  & \(\mathbf{m^{2}/d}\) & \({}^{\circ}\)C & & & \(\mathbf{kJ/(m^{2}\text{-d}\cdot^{\circ}\text{C})}\) \\ \hline C35 & 0.0987 & 25.57 & 0.60 & 0.6400 & 516.0 \\ C40 & 0.0625 & 27.45 & 0.75 & 0.6297 & 908.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Inversion results. Figure 8: The comparison of calculated and monitored temperatures.