## 4 Numerical Examples and Discussions  
In this section, a numerical example is presented to verify the proposed method. The necessity of prioritizing the enhancement of overall structural performance as the objective in the optimization process of branch-supported free-form surface structure is illustrated through the discussion. Finally, the effectiveness of the optimization is demonstrated through a case study of a multi-tree-supported free-form surface structure. Figure 4 shows an initial model of a branch-supported free-form surface structure. The structure spans a rectangular plane measuring 50.0 m \(\times\) 50.0 m. The upper structure is represented by a (2, 2)th-degree B-spline surface that interpolates a set of 6 \(\times\) 6 data points, and the lower structure is composed of the truck and two-level branching structure. The upper shell thickness \(t\) = 0.1 m, elastic modulus \(E_{\text{C}}\) = 3.0 \(\times\) 104 MPa, Poisson ratio \(\mu_{\text{C}}\) = 0.2, the lower tree-like structure members using solid circular section rod, the diameter of the 0 th-level branch \(D_{0}\) = 1.0 m, and the diameters of other level branches are scaled in a ratio of 0.7 from bottom to top, elastic modulus \(E_{\text{B}}\) = 2.1 \(\times\) 105 MPa, Poisson ratio \(\mu_{\text{B}}\) = 0.3. The vertical downward uniform load \(q\) = 3.0 kN/m\({}^{2}\) acts on the curved surface, and the nodes of the tree-like supporting structure are fixed. The optimization scheme of the structure is usually determined according to the architectural design requirements. In this example, it is assumed that all three types of nodes forming the structure need to be adjusted, as shown in Figure 5a (Scheme I). The value points of the curved surface are only adjusted vertically (in the z direction), the intersection points between the tree structure and the free-form surface are adjusted within the surface, and the nodes of the tree structure except the intersection points are adjusted on the xoy plane where the point is located, while the bottom supports are fixed without adjustment. With a small positive value \(\varepsilon\) = 0.05, Figure 5b shows the shape when the structure optimization tends to be stable under this adjustment scheme. Figure 4: Initial model of a branch-supported free-form surface structure. Figure 5: This is a figure. The morphological optimization of branch-supported free-form structures. (**a**) Optimization scheme I; (**b**) evolutionary result. In order to investigate the tension and compression distribution of the upper shell roof before and after optimization, the dimensionless index of Equation (29) is used to represent:  
\[\lambda_{\sigma}=\frac{\sigma_{1}^{\text{mid}}}{\sigma_{1}^{\text{mid}}+\left| \sigma_{3}^{\text{mid}}\right|} \tag{29}\]  
where \(\sigma_{1}^{\text{mid}}\) is the first principal stress of the shell, usually the tensile stress, and \(\sigma_{3}^{\text{mid}}\) is the third principal stress of the shell, usually the compressive stress. As can be seen from Equation (29), \(\lambda_{\sigma}\in[0,1]\), when this index is close to 0, it is considered that the part is mainly under pressure; when this index is close to 1, it is considered that the part is mainly under tension. For the lower tree support structure, the tensile and compressive conditions can be measured by the local axial force FN. Figure 6a-d, respectively, show the comparison between the tensile and compressive stress distribution of the upper freeform surface and the axial force distribution of the lower tree-like structure before and after optimization. It can be seen from Figure 6a that under the load set by this example, the upper freeform surface of the initial structure is mainly subjected to the film compressive stress, but the proportion of the film tensile stress in the local area, especially the area near the support, is larger.