## 4 Numerical Examples and Discussions
For the lower tree-like structure (Figure 6b), the local axial forces are all pressure, indicating that the tree-like support generation method proposed in Section 2.2 can meet the requirement that loads converge and transfer to supports step by step along the branches under the premise of reasonable setting of support areas, but the axial forces of different branches at the same level are greatly different. After optimization, the upper surface is uplifted as a whole in terms of visual effect, but the adjustments in different areas are slightly different. Corresponding to the tension and compression stress distribution of the surface, as shown in Figure 6c, the adjustment of the shape value points and the intersection points of the shell rods not only strengthens the overall arch effect but also strengthens the catenary effect of the local tension area. Due to the change in branch angle caused by the adjustment of nodes, the local axial force of the branches in the same order tends to be uniform, and the load distribution of the members is more reasonable. To compare and analyze the influence of different design variable adjustment schemes on optimization results, this paper includes design variable adjustment schemes II-IV as shown in Table 2. Except for the type of nodes to be adjusted, all other conditions (initial structure, load, boundary conditions, and material parameters) are the same as in Scheme I.  
Figure 6: The structural mechanical properties. (**a**) Tensile or compressive stress distribution of surface (Initial); (**b**) Axial forces of branching structure (Initial); (**c**) Tensile or compressive stress distribution of surface (Final); (**d**) Axial forces of branching structure (Final). Three groups of performance indicators are tracked to understand the changes in strain energy and the proportion of each component during the optimization process:  
1. \(\text{C}_{i}/\text{C}_{0}\): Strain energy change rate, measuring the change in structural strain energy;
2. \(\text{C}_{1}^{\text{S}}/\text{C}_{0}^{\text{S}}\) and \(\text{C}_{1}^{\text{B}}/\text{C}_{0}^{\text{B}}\): Strain energy change rate in the shell and tree-like structures respectively, measuring the change in strain energy of each component in the hybrid structure;
3. \(\text{C}_{i}^{\text{S,m}}/\text{C}_{i}^{\text{S}}\) and \(\text{C}_{i}^{\text{B,m}}/\text{C}_{i}^{\text{B}}\): Measures the proportion of membrane strain energy and axial force-induced strain energy in the \(i\)-th step of the free-form surface shell and tree-like structure, respectively. where \(\text{C}_{0}\) is the initial strain energy of the structure, \(\text{C}_{i}\) is the strain energy of the \(i\)-th step structure, \(\text{C}_{0}^{\text{S}}\) is the initial strain energy of the free-form surface shell, \(\text{C}_{i}^{\text{S}}\) and \(\text{C}_{i}^{\text{S,m}}\) are the strain energy and membrane strain energy, respectively, of the \(i\)-th step free-form surface shell, \(\text{C}_{0}^{\text{B}}\) is the initial strain energy of the tree-like structure, and \(\text{C}_{i}^{\text{B}}\) and \(\text{C}_{i}^{\text{B,m}}\) are the strain energy and axial force-induced strain energy, respectively, of the \(i\)-th step tree-like structure. From the strain energy change rate and component proportion curves corresponding to each scheme in Table 2, it can be seen that the structural strain energy decreases significantly during the optimization process; the decrease rate shows a trend of gradually converging  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**No.