## 4 Numerical Experiments  
### Proof of Concept  
As mentioned, the most significant difference between UCT and KR-UCT is whether the design variables are continuous values or not. Serving as a proof of concept, the 17-bar truss experiment is used to demonstrate the design workflow of the KR-UCT algorithm. This experiment illustrates the design of a long cantilever truss in the design domain. The design domain of the 17-bar truss and the details of specified essential nodes [21] are shown in Figure 2 and Table 2. The settings of material properties are given in Table 3. The constraint combination in this experiment is \((g_{1},g_{2},g_{3},g_{4},g_{5})\), and the parameters or formulas are summarized in Table 4. The purpose of this experiment was to find the truss layout of minimum weight under constraints.  
Figure 2: Design domain of the 17-bar planar truss.  
There are three action sets for KR-UCT to choose sequentially. In the first action set, nodes are chosen from the candidate node set and added to the structure. In UCT, the design domain needs to be uniformly discretized, whereas, in KR-UCT, only the initial number of actions needs to be set. All initial actions are generated randomly from the design domain. In the second action set, several bars are added to the truss until it passes all the constraints. In the third action set, KR-UCT assigns optimal cross-sectional areas to the generated bars. In UCT, the candidate cross-sectional area set (\([A_{\min},A_{\max}]\)), also needs to be discretized. In KR-UCT, only the initial number of actions needs to be set. All initial actions are generated randomly from the optional cross-sectional area set.  
Figure 3 shows the optimal truss layout in this experiment. The weight of the optimal truss layout is 1463.44 kg. Note that a bar in red/blue color indicates that it is in tension/compression, respectively. Figure 4 shows its construction process, which depicts how KR-UCT makes decisions to build a truss. In the UCT algorithm, the node candidate set in every step of the adding-node stage is uniformly discretized, as shown in Figure 5; Figure 6 shows the heuristic distribution of node candidate sets in the KR-UCT algorithm in order to prove the effectiveness of kernel regression. It is found that, in each step of adding a node to the truss, a large number of optional actions are gathered around the location of the final selected point. This indicates that the selection of optional node location has heuristic significance.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Constraints** & **Parameters** & **Settings** \\ \hline \(\mathrm{g}_{1}\) & \(\mathrm{X}_{\min};\mathrm{X}_{\max}\) & 0 mm; \(10,160\) mm; \\ \(\mathrm{Y}_{\min};\mathrm{Y}_{\max}\) & 0 mm; \(2540\) mm \\ \(\mathrm{g}_{2}\) & \(\mathrm{A}_{\min},\mathrm{A}_{\max}\) & 0.6452 cm; \(200\) cm\({}^{2}\) \\ \(\mathrm{g}_{3}\) & \(\mathrm{\sigma}_{\min},\mathrm{\sigma}_{\max}\) & 334.6 MPa; 334.6 MPa \\ \(\mathrm{g}_{4}\) & \(\mathrm{\delta}_{\max}\) & 50.8 mm \\ \(\mathrm{g}_{5}\) & \(\mathrm{\sigma}_{\mathrm{buckle}}\) & \(\mathrm{\pi}^{2}\mathrm{EI}/\mathrm{L}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Constraint parameter settings in the 17-bar experiment.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Essential Node** & **Node Location (mm)** & **Node Label** \\ \hline \((1)\) & \((0,\,0)\) & Pinned support \\ \((2)\) & \((0,\,2540)\) & Pinned support \\ \((3)\) & \((10,160,\,0)\) & Loading (0, \(-444\),800 N) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Essential node coordinates in the 17-bar experiment.  
Figure 3: Optimal truss layout generated in the 17-bar experiment.  
Figure 4: Action-taking process for constructing an optimal truss in the 17-bar experiment.  
Figure 5: Uniform distribution of node candidate set in the UCT algorithm.