# Stochastic Natural Vibration Analyses of Free-Form Shells
## 4 Results and Discussion
### Shell Model 2: A Free-Form Shell Consisting of Two Connected Positive--Gaussian Dones  
#### 4.2.1 Model Generation  
Figure 11 shows the shell of this example. The two long-span edges are simply supported and the other edges are free. The material properties are same to that in Table 2. The shape is generated by the proposed optimization approach. NURBS functions are adopted to represent the geometric shape. The control points are located, as shown in Figure 11a, which forms a \(5\times 3\) grid. Weight factors and the degrees of the basis functions are determined as similar to that in Example 1. The optimization aims at minimizing the strain energy under applied load, which is also determined as the uniformly distributed vertical load of \(q=3.5\) kN/m\({}^{2}\). The triangular shell element is used with 1024 elements and 561 nodes.  
Figure 11: Layout of control points, mesh and the initial geometry of Model 2: (**a**) Layout of control points and mesh; and, (**b**) Initial guess shape.  
Figure 10: Normalized sensitivity of uncertainty factors of Model 1.  
As presented in Section 2, the gradient method is used to solve the optimization problem. The initial shape of this model is set up as a free-form shape, which consists of two connected positive -Gaussian domes. Similar to Example 1, the control point on the long-span edges is fixed, meaning that the location and shape of these two edges are not changing during the optimization process. The total number of optimization variables is 5, which is same to Example 1.  
Figure 12 shows the convergence process of Model 2, which is slower than Model 1, since the shape is relatively complicated. The optimization converges at the 1000 th step with a 65% decrease of strain energy. Figure 13 shows the changes of geometric shape of Model 2 during the optimization process. It can be seen that, since the control points on the edges are fixed, the general shape remain similar to the initial shape, which consists of two connected positive-Gaussian surfaces. However, the curvature increases overall.  
Prior to the stochastic analysis, the deterministic analysis of free vibration is implemented on shell models with respect to several selected optimization steps, and Figure 14 depicts their first three natural frequencies. It is observed that, different from Model 1, both the first and third natural frequencies decrease through the optimization, while only the second natural frequency increases. It is also noted that the change of all the frequencies are slight. For example, the relative change of the first natural frequency is only 11.9%, which is significantly smaller than Model 1. This can be explained by the fact that, in this case, the stiffness is mainly increased by the increase of curvature, which results in the increase of mass. The natural frequencies are affected by both stiffness and mass, and they consequently show a slight change. The natural frequencies decrease when the increase of mass plays a more significant role than stiffness.  
Figure 12: The convergence process of Model 2.  
Figure 13: Shape variation of Model 2 during the optimization process: (**a**) Initial guess; (**b**) 100 th step; and, (**c**) 1000 th step (optimal).