## 4 Numerical Experiments
### Three-Dimensional Cantilever Sundial Design  
This design case is a test example of the KR-UCT algorithm on space trusses design, which is adapted from the sundial bracket truss built in Paternoster Square, London, UK [37]. A cantilever space truss with a length of 4634 mm is needed as the sundial bracket. Due to the need to read the scale based on the shadow of the sundial tip, the design of the sundial bracket is very strict with the stiffness of the space truss. The purpose of this experiment is to find the minimum truss weight under constraints in the three-dimensional design domain. The design domain of this case is shown in Figure 14. This domain area only represents the essential node locations, and there is no mandatory geometric boundary on newly added nodes and bars. There are four fixed nodes in the design domain, among which nodes (1), (2), and (3) are pinned supported nodes, which are fixed on the wall in an isosceles triangle. Node (4) is the sundial tip, which is the loading node. It is fixed with a metal disk with holes of 300 mm in diameter and 6 mm in thickness. The load is 50 N. The details of specified essential nodes and material property settings are summarized in Tables 17 and 18. The constraint combination for this experiment is \((g_{1},g_{2},g_{3},g_{4},g_{6},g_{7})\) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **GA**[35] & **GP**[34] & **GA**[36] & **KR-UCT** \\ \hline weight (kg) & 60,329 & 45,404 & 46,222 & 44,566 \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Comparison results of the long-span bridge truss experiment. Figure 13: Optimal truss layout generated for long-span bridge truss experiment. \begin{table}
\begin{tabular}{c c} \hline \hline
**Material Properties** & **Settings** \\ \hline Young’s modulus & 201 GPa \\ Density & 7851.03 kg/m\({}^{3}\) \\ Yield strength & 248.8 MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Material property settings in the long-span bridge experiment. The parameters or formulas are expressed as follows: the allowable tension stress and the allowable compression stress \(\sigma_{c}\) are both \(0.6f_{y}\); the allowable node displacement is limited to 2 mm; the allowable bar slenderness ratios are specified to be 220 for tension bars and 180 for compression bars; the minimum and maximum length of bars are 0.03 m and 5 m; in addition to the conventional constraint model, this example also requires that the angle between any two connecting bars should not exceed 1 degree; the cross-section of the bars used in this paper is the section of cold-formed thin-wall welded round steel tube (GB50018-2002). There are 61 groups of cross-section sizes from d25t1.5 to d245t4.0. In terms of structural load, this design case also considers the influence of the self-weight of the truss structure. The self-weight of the structure is evenly distributed on the end node of bars. For the three different numbers of nodes varying from 8 to 10, the results generated by the KR-UCT algorithm for these three groups of experiments are illustrated in Figure 15. Two design results are presented for each group. Among the six generative design layouts, each layout has its own characteristics. In terms of weight, layout 3 is the lightest one, which receives the largest reward in the decision process. From the perspective of novelty, layout 4 may be an innovative design, which looks similar to a flapping wing. In architectural structural design, structural weight and shape novelty is often weighed and considered according to the user or designer of the buildings, without a clear, quantifiable index. For example, in this experiment, KR-UCT can generate completely different layout schemes based on deterministic indicators through different decision trajectories. The final design scheme can be determined by comparing and selecting from various schemes. To facilitate a reference for comparison by subsequent studies, this paper lists the design results of layout 3 from multiple perspectives, as shown in Figure 16.