## 4 Applications
### Impeller  
The second example analyzes the response of an impeller, as shown in Figure 8a, to demonstrate the efficiency and accuracy advantages of the proposed method. The material of the impeller is carbon steel, with density and Young's modulus being \(\rho\) and \(E\), respectively. A shell element model is established as shown in Figure 8b, with a thickness \(t\). Considering the unpredictability of material properties and manufacturing processes, \(\rho\), \(E\), and \(t\) are treated as independent interval variables. The tangential and longitudinal degrees of freedom at the nodes on the impeller sleeve are constrained, as illustrated in Figure 8b. The angular velocity of the structure, denoted as \(\omega\), is also an uncertain-but-bounded parameter. This example also considers three uncertainty cases and the median values of all uncertain parameters are shown in Table 5.  
To observe the displacement variation in the impeller, two nodes were selected as measurement points, as shown in Figure 8b. The upper and lower bounds of the displacement and rotation at the measurement points were computed for different uncertainty levels. Case 1 sets the uncertainty level at 10% for all interval parameters, while cases 2 and 3 extend the uncertainty level to 20% and 30%, respectively. The bounds of displacements and rotations obtained using the PM are compared with those of SDAM and STM, and the solutions obtained by SM are used as references. The bounds and errors for the nodal displacements and rotations (\(x\)-component of the displacement of node 1 \(u_{1}\) and \(z\)-component of the rotation of node 2 \(\theta_{z,2}\)) calculated with different methods for each uncertainty case are given in Tables 6 and 7. In addition, the upper and lower bounds of the nodal displacements and rotations obtained using the different methods are plotted in Figure 9. In Figure 10, the overall structural displacement bound cloud maps for the impeller obtained using the PM at the 30% uncertainty level are given. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Parameters** & **Density** & **Young’s Modulus** & **Thickness** & **Angular Velocity** \\ \hline Interval median & \(\rho^{C}=7.89\times 10^{-9}\) t/mm\({}^{3}\) & \(E^{C}=2.1\times 10^{5}\) MPa & \(t^{C}=12\) mm & \(\omega^{C}=3.14\times 10^{4}\) rad/s \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Median values of interval parameters of impeller structure. Figure 8: Impeller structure: (**a**) geometric model; (**b**) finite element model. As shown in Tables 6 and 7 and Figures 9 and 10, the second numerical example shows that the PM is still accurate and efficient for impeller structure. (1) The response bounds obtained for SDAM, STM, and the PM have a tendency to increase with the level of uncertainty rising. When the uncertainty level reaches 30%, the maximum errors for SDAM are \(-119.06\%\) and those for STM are \(-127.33\%\). However, the error of the results obtained using the PM does not exceed 1%. Although the PM only retains the first-order term of the Taylor expansion, its accuracy is higher than STM, which retains the second-order term. There are two main reasons for this: Firstly, the PM utilizes a modified Neumann series, which retains higher-order terms of the Neumann series. Secondly, the PM divides the interval into subintervals, reducing the uncertainty level during the solving process. The reason why the PM is more accurate than SDAM is that SDAM only expands at the midpoint of the interval to approximate the response at the combination point, while the PM solves within the subinterval domains of each expansion route. (2) PM is still suitable for finite element models with high levels of uncertainty and multiple parameters and can maintain good efficiency. The computational effort of SM increases exponentially with the number of parameters, which makes it limited when the number of interval parameters is large.