# Uncertainty Analysis Based on Kriging Meta-Model for Acoustic-Structural Problems
## 5 Numerical Examples
### Simplified Car Interior with Flexible Plates  
In this section, we considered a vibro-acoustic finite element model of a simplified automobile passenger compartment. The finite element model presented in Figure 11 was modeled in ANSYS software [11]. For modeling the acoustic cavity, FLUID30 elements were used. These are 3D elements having 8-nodes with 4 degrees of freedom per node: 3 translational displacements in the \(x\), \(y\), and \(z\) directions, and a pressure. For structure plates, SHELL181 elements were adopted which are 2D quadrilateral elements. A unit harmonic force in the frequency range 0-500 Hz is applied perpendicularly at the center of the firewall plate. The fluid inside the cabin is the air whose properties are: its density is 1.21 kg/m\({}^{3}\) and the sound speed is 343 m/s.  
The material characteristics of the structure are presented in Table 3.  
Table 5 presents the different uncertain parameters chosen for this example.  
The sound pressure level inside the cavity is approximated using Kriging model. The Kriging estimator is constructed through 50 Latin hypercube sampling points. An additional 50 sampling points are generated in order to validate the efficiency of the surrogate models. The accuracy of the approximation was verified by the cross-validation as shown in Figure 12.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Panel** & **Thickness Value (mm)** \\ \hline firewall (\(t_{1}\)) & 0.8 \\ bulkhead (\(t_{2}\)) & 0.8 \\ roof (\(t_{3}\)) & 0.7 \\ floor (\(t_{4}\)) & 0.9 \\ windshield (\(t_{5}\)) & 5 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Panel thicknesses.  
Figure 11: 3D model of the passenger cavity.  
After the validation of the Kriging surrogate model, a sampling set with the number of \(N=10^{6}\) for uncertain parameters (Table 5) is generated, then the MCS is performed using the Kriging approximation Figure 13.  
It is clear that the presence of uncertainties affects the mean of the sound pressure level with small variation. Indeed, the maximum value of the SPL mean at frequency \(\sim\)280 Hz increased by 4.8 dB as shown in Figure 13. It should be noted that the deterministic FEM computing time for one sampling is approximate 10 min. Furthermore, the computational cost of the training points is equal to: \(50\times 10=500\) min. So, if the uncertainty analysis is performed using MCS based on FEM, the computational time will be taken about \(10\times 10^{6}\) min, while the MCS based on Kriging models strongly reduces the computational cost (about 108 s).  
Figure 12: Cross-validation results for SPL using Kriging.  
Figure 13: Variation of the SPL mean with the frequency.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & **Mean Value** & **Cov** & **Distribution Type** \\ \hline \(E_{s}\) (Pa) & 2.1 \(\times\) 10\({}^{11}\) & 0.05 & Normal \\ \(E_{g}\) (Pa) & 6.2 \(\times\) 10\({}^{10}\) & 0.05 & Normal \\ \(\rho_{a}\) (kg/m\({}^{3}\)) & 1.21 & 0.05 & Normal \\ \(t_{1}\) (mm) & 0.8 & 0.02 & Uniform \\ \(t_{5}\) (mm) & 5 & 0.02 & Uniform \\ \hline \hline \end{tabular}
\end{table}
Table 5: Characteristics of uncertain parameters.