## 3 Results and Discussion  
### Thermal Calculation Results and Analysis  
Figures 13-15 show the numerical analysis results of the heat flux density contour maps in the \(x\)-direction for three typical model modes. The numerical simulations analyzed the thermal performance of the heat-conducting units within a range of VRs from 39.00 to 1.00. The typical models reflect the variation of heat flux density values of insulating materials with the VR of GPPs. Figure 13 shows the thermal analysis results of the model with GPP-cement VRs. The results indicate that within the specified range of VRs, the VR significantly affects thermal performance, with the average heat flux density rapidly increasing as the VR decreases. Figure 14 presents the thermal analysis results of the model with GPP-GM VRs. Within the specified range of VRs, the VR has a minor impact on thermal performance. Figure 15 displays the thermal analysis results of the model with GPP-SF VRs. Within the specified range of VRs, the VR affects the TC, with the impact being smaller than that of the previous VR. Through numerical simulation of the randomly distributed GCIM, the results of the thermal analysis were obtained. By extracting the thermal flux density values of 27,000 units,  
The results from Tables 2-4 demonstrate that in the three models, the VR of GPPs exhibits a consistent trend in its impact on the TC of the GCIM. With an increase in the VR of GPPs, the average heat flux density of the thermal units gradually decreases, leading to a gradual reduction in the TC of the GCIM, resulting in an enhancement of the insulation performance. Within the specified range of VR variations, the VR has the greatest influence on the GPP-cement VR model and the least influence on the GPP-GM VR model. The variation in VR tends to stabilize in the later stage. Therefore, typical models with four later-stage VRs are selected. The FE results of the typical models in the three modes are compared with the theoretical calculations, as shown in Tables 5-7. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Group** & **1** & **2** & **3** & **4** \\ \hline VRs & \(\Psi_{1}/\Psi_{2}=39.00\) & \(\Psi_{1}/\Psi_{2}=4.00\) & \(\Psi_{1}/\Psi_{2}=1.86\) & \(\Psi_{1}/\Psi_{2}=1.00\) \\ Average heat flux density (W/m\({}^{2}\)) & 4.4900 & 5.6020 & 6.6676 & 7.8092 \\ Average TC (W/m\(\cdot\)K) & 0.0449 & 0.0560 & 0.0667 & 0.0781 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The average heat flux density value and average TC of the GPP-SF VR model. The results in Tables 5-7 indicate that the SPAM has higher computational accuracy compared to the traditional series model. Furthermore, the error of the theoretical calculation model is obtained by comparing it with the FE results, as shown in Tables 8-10. The results demonstrate that the error of SPAM is significantly reduced compared to the series model. \begin{table}
\begin{tabular}{c c c c c} \hline
**Group** & **1** & **2** & **3** & **4** \\ \hline VRs & \(\Psi_{1}/\Psi_{2}=39.00\) & \(\Psi_{1}/\Psi_{2}=19.00\) & \(\Psi_{1}/\Psi_{2}=12.33\) & \(\Psi_{1}/\Psi_{2}=9.00\) \\ SPAM & 0.04093 & 0.04163 & 0.04241 & 0.04332 \\ FE & 0.03981 & 0.04255 & 0.04557 & 0.04875 \\ Series model & 0.04924 & 0.05651 & 0.06378 & 0.07105 \\ \hline \end{tabular}
\end{table} 
Table 6: The TC of the GPP-GM VR model (Unit: W/m\(\cdot\)K). \begin{table}
\begin{tabular}{c c c c c} \hline
**Group** & **1** & **2** & **3** & **4** \\ \hline VRs & \(\Psi_{1}/\Psi_{2}=39.00\) & \(\Psi_{1}/\Psi_{2}=19.00\) & \(\Psi_{1}/\Psi_{2}=12.33\) & \(\Psi_{1}/\Psi_{2}=9.00\) \\ SPAM & 0.04217 & 0.04281 & 0.04337 & 0.04408 \\ FE & 0.04490 & 0.04642 & 0.04793 & 0.04948 \\ Series model & 0.06245 & 0.06444 & 0.06642 & 0.06841 \\ \hline \end{tabular}
\end{table} 
Table 7: The TC of the GPP-SF VR model (Unit: W/m\(\cdot\)K).