## 4 Numerical Experiments
### 10-Bar Planar Truss Experiment  
To further verify the effectiveness of the KR-UCT algorithm, another comparison with the UCT algorithm is carried out under the same design domain, constraints, and load cases for the 10-bar planar truss experiment. The design domain is shown in Figure 7 and the details of specified essential nodes are given in Table 6. Material property settings are summarized in Table 7. The constraint combination in this experiment is \((g_{1},g_{2},g_{3},g_{4},g_{5})\), and the parameters or formulas are given in Table 8. The goal of this experiment is to find the truss layout of minimum weight under constraints.  
Figure 6: Heuristic distribution of node candidate sets in the KR-UCT algorithm.  
Figure 7: Design domain of 10-bar planar truss.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **UCT [20]** & **KR-UCT** \\ \hline weight (kg) & 1695.89 & 1463.44 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison results of the 17-bar experiment.  
In order to make the comparison in a more detailed manner, this paper uses a series of node number settings (maxp) varying from 6 to 9. The comparison results are summarized in Table 9. The generative design results are shown in Figure 8.  
In the structural design practice, the global optimal and unique solution does not always satisfy the architects' design intuition. Usually, they need a variety of different designs with high structural performance. The proposed algorithm has the ability to generate multiple competitive solutions with significantly different layouts. In KR-UCT, the three types of action in the decision-making process can expand the solution space. Note that each decision trajectory does not affect the other. This can fully explore the solution space and ensure the independence of every decision trajectory. Further, the Monte Carlo method can bring randomness to action-making steps in the decision tree; therefore, KR-UCT can generate various competitive truss design solutions under the same design requirements (design objectives, constraints, algorithm parameters). Figure 9 shows several truss layouts obtained by the KR-UCT algorithm when the total number of nodes is set to 8. The 12 layouts illustrated in Figure 9 have a weight ranging from 1852.46 kg to 2306.61 kg.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Essential Node** & **Node Location (mm)** & **Node Label** \\ \hline \((1)\) & \((0,\,0)\) & Pinned support \\ \((2)\) & \((0,\,9144)\) & Pinned support \\ \((3)\) & \((9144,\,0)\) & Loading \((0,-444,800\,\mathrm{N})\) \\ \((4)\) & \((18,288,\,0)\) & Loading \((0,-444,800\,\mathrm{N})\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Essential node coordinates in the 10-bar experiment.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Constraints** & **Parameters** & **Settings** \\ \hline \(\mathrm{g_{1}}\) & \(\mathrm{X_{min};X_{max}}\) & 0 mm; 18,288 mm \\ \(\mathrm{Y_{min};Y_{max}}\) & 0 mm; 9144 mm \\ \(\mathrm{g_{2}}\) & \(\mathrm{A_{min};A_{max}}\) & 0.6452 cm\({}^{2}\); 400 cm\({}^{2}\) \\ \(\mathrm{g_{3}}\) & \(\sigma_{min};\sigma_{\max}\) & 172.3 MPa; 172.3 MPa \\ \(\mathrm{g_{4}}\) & \(\delta_{\max}\) & 50.8 mm \\ \(\mathrm{g_{5}}\) & \(\sigma_{\mathrm{buckle}}\) & \(\pi^{2}\mathrm{EI}/\left(\mathrm{Al}^{2}\right)\) \\ \hline \hline \end{tabular}
\end{table}
Table 8: Constraint parameter settings in the 10-bar experiment.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithms** & **[PERSON] et al. [9]** & **[PERSON] et al. [21]** & **UCT [20]** & **KR-UCT** \\ \hline \multirow{3}{*}{Number of nodes (maxp)} & 6 & 2223.5 & 2217.54 & 2223.10 & 2153.87 \\  & 7 & N/A & N/A & 2255.72 & 1968.72 \\ \cline{1-1}  & 8 & N/A & N/A & 2280.93 & 1852.46 \\ \cline{1-1}  & 9 & N/A & N/A & 2339.62 & 2122.05 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Minimum weights (kg) for the different number of nodes in the 10-bar planar truss experiment.