## 3 Numerical Examples
### Shape Optimization  
In the example, the span of the structure is 30 m, the height is 5 m, and the shape is shown in Figure 3. In order to make the optimized structure have better symmetry, \(1/4\) of the structure is taken as the optimization area, and the initial control points are triangular meshes, as shown in Figure 3b. After the initial control points are symmetrical along the hypotenuse, a \(6\times 6\) rectangular control point mesh is obtained. The coordinates of the initial control points are shown in Table 1. After generating the _Bezier surface_ according to the rectangular control points, the points with the same \(x\) and \(y\) coordinates as each control point are found on the surface, and their coordinates are used as the node coordinates of the structure. Currently, the node coordinates of \(1/4\) of the structure are obtained (the green area in Figure 3b). The remaining node coordinates are obtained by symmetry along the \(x\)-axis (the blue area in Figure 3b) and the \(y\)-axis (the red area in Figure 3b) of these points. The design variables are the control points other than the boundary constraint points shown in Figure 3b, among which the 6 th, 11 th, 15 th, 18 th, 20 th, and 21 st points are constraint points. When the structure is iterated, the coordinates of the control points will change, resulting in changes in the shape of the surface. Figure 3: Schematic of the initial structure shape and initial control points. (**a**) Isometric view; (**b**) Top view; (**c**) Side view. The structural members were simulated using the _beam188_ unit in Ansys (Pittsburgh, PA, USA), with an _elastic modulus_ of E = 2.1 \(\times\) 10\({}^{11}\) Pa, a _Poisson's ratio_ of v = 0.3, a _density_ of \(\rho\) = 7850 kg/m\({}^{3}\), and a member cross-section of a circular ring with inner and outer radii of 0.1 m and 0.11 m, respectively. The _mass21_ unit was used to apply concentrated mass to the structure; the support was a fixed hinge, and displacement constraints were applied in three directions. According to the provisions of the _Code for Seismic Design of Buildings_[22], in the earthquake dynamic time history analysis, the representative value of the gravity load on the structural roof is recommended to be the standard value of the constant load plus 50% of the standard value of the live load. The dead load is the dead weight of the material. The dead weight of the load-bearing components is considered in Ansys through density and does not need to be added separately. Here, only the dead weight of the additional components (such as roof trusses, etc.) needs to be considered, which is taken as 0.43 kN/m\({}^{2}\) (including 0.3 kN/m\({}^{2}\) of glass roof and 0.13 kN/m\({}^{2}\) of steel roof trusses); the live load is mainly the wind load and roof load, which is taken as 2.64 kN/m\({}^{2}\) (including 2 kN/m\({}^{2}\) of accessible roof and 0.64 kN/m\({}^{2}\) of wind load), which is equivalent to a roof mass of 180 kg/m\({}^{2}\) and converted to the node. After conversion, the mass of each node is 1338.84 kg. A vertical static load of 10 kN/m\({}^{2}\) is applied to the structure, and it is equivalent to the node load. The total size remains unchanged, and the shape of the structure is optimized with vertical strain energy as the optimization target. The upper and lower limits of the optimization variables in the horizontal direction are set to \(\pm\)1 m, and the height range of the structure is 0-15 m. The convergence condition is that the difference between the two objective function values is less than the default value (1 \(\times\) 10\({}^{-6}\)), and the structure converges after 197 iterations. Figure 4 shows the structural shape of a typical iteration step in the optimization process. Figures 5 and 6 are, respectively, the von Mises stress cloud diagrams of the initial structure and the optimized structure at the same optimized load.