# Stochastic Natural Vibration Analyses of Free-Form Shells
## 4 Results and Discussion
### Shell Model 1: A Negative-Gaussian Free-Form Shell  
#### 4.1.1 Model Generation  
The shell model in this example is as shown in Figure 2, which cover a rectangle plane section with a dimension of \(8\times 12m\). Both long-span edges are simply supported, while the two short-span edges are free. The material properties and the design shell thickness are as shown in Table 2. The material is assumed to be linear elastic.  
The shape is generated by the proposed optimization approach. NURBS functions are adopted to represent the geometric shape, with control points that are shown in Figure 2a, which forms a \(5\times 3\) grid. As aforementioned, the weight factors have small effects on the NURBS surface [6]. Therefore, the weight factors of all control points are chosen randomly from the range 0-1.0 as 0.5 and not changed during the optimization process. Both the degrees of the basis functions in the two directions are selected to be 2. The optimization aims at minimizing the strain energy under a given uniformly distributed vertical load \(q=3.5\mathrm{kN}/\mathrm{m}^{2}\). The strain energy is solved by a self-coded program, in which triangular shell element is used with 768 elements and 425 nodes.  
The gradient-based algorithm is used to solve the optimization problem. A cylindrical shell with the height of 4 m is given as an initial guess, as shown in Figure 2b. In this example, the control point on the long-span edges are fixed meaning the location and shape of the two edges are not changing during the optimization process. Consequently, the total number of optimization variables is 5.  
The variation of the objective function during the optimization process is depicted in Figure 3 to show the convergence process, in which the vertical coordinate indicates the ratio of the strain energy of the \(i\)th optimization step to that of the initial guess. It can be seen that the strain energy decreases significantly during the first 50 optimization steps. After the 50 th step, the strain energy decreases slowly and then converges at the 400 th step. Through the optimization, the strain energy decreases to 40% of that of the original design.  
In Figure 4, the initial, 20 th, 200 th, and 400 th (optimal) shapes are selected to show the shape variation during optimization process. It is observed that, via the first 20 optimization steps, the shape changes from a cylinder to a saddle to earn larger stiffness. Since the 20 th step, the shape remains  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Elastic Modulus (MPa)** & **Poisson’s Ratio** & **Thickness (mm)** & **Density (kg/m\({}^{3}\))** \\ \hline \(3.0\times 10^{4}\) & 0.617 & 140 & 2500 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material parameters and shell thickness.  
Figure 2: Layout of control points, mesh and initial guess shape of model 1: (**a**) Layout of control points and mesh, (**b**) Initial guess shape.  
saddle-shaped, and the surface curvature increases overall. The shape change is too slight to be observed after the 200 th optimization step.  
Prior to the stochastic analysis, the deterministic analysis of free vibration is implemented on shells with respect to several selected optimization steps, and the first three natural frequencies are plotted in Figure 5. It can be seen that, all the first three natural frequencies significantly change during the optimization process and increase by 158.0% (1 st), 61.1% (2 nd), and 111.5% (3 rd), respectively.