# Interval Analysis of Vibro-Acoustic Systems by the Enclosing Interval Finite-Element Method
## 4 Numerical Applications in Vibro-Acoustic Coupling Problem
### Numerical Analysis of a 2D Interior-Car Vibro-Acoustic Coupling Model  
Few researchers have paid attention to the 2D uncertain vibro-acoustic coupling problem in previous literature [26], especially for the actual model with wide analytical band. The first application is implemented in 2D vibro-acoustic coupling system with uncertain-but-bounded parameters. As the prediction of a car's interior-acoustic field is available as a benchmark problem in many uncoupled acoustical studies with uncertainties [8; 21], it is valuable and innovative to characterize response of the 2D car's acoustic field under the influence of the uncertain structural coupling. Figure 7 shows a 2D interior-acoustic field of a car, whose top face is attached to the simple support Euler beam (space frame [36]), and the remaining boundary of the acoustic field is supposed to be rigid. The vibro-acoustic coupling system is excited by a unit point load \(\mathbf{F}_{s}\) on the midpoint of the beam. The beam is made by steel with material parameters \(E=210\) Gpa, \(\rho_{s}=7800\) kg/m\({}^{3}\), \(\mu=0.3\), and the cross-section parameters \(A=1\times 10^{-5}\) m\({}^{2}\) and \(I=8.3\times 10^{-11}\) m\({}^{4}\). The cavity is filled with air with a density of \(\rho_{f}=1.21\) kg/m\({}^{3}\), and sound speed of \(c=343\) m/s. The number of elements and nodes corresponding to beam are 12 and 13, respectively, and to the acoustic domain become 501 and 579, respectively. Considering the unpredictability of structural parameters, the Young's modulus \(E\) and density \(\rho_{s}\) are assumed to be uncertain-but-bounded variables; with the preset uncertain level \(\alpha=0.05\), the uncertainties can be rewritten as  
\[\begin{split} E^{I}&=E^{c}+[-\alpha,\alpha]\times E ^{c}\\ \rho_{s}^{I}&=\rho_{s}^{c}+[-\alpha,\alpha]\times \rho_{s}^{c}\end{split} \tag{38}\]  
Assuming the uncertain parameters defined above are independent, the Monte Carlo method, the TN-MIFFEM, and the SMW-IPFEM are all used to compare with the proposed enclosing-IFEM, in terms of recovering the 2D vibro-acoustic coupling field. The Monte Carlo method is simulated with 10,000 samples by the deterministic FEM, shown as the gray lines. Subsequently, based on these methods, from 0 to 100 Hz by the interval of 10 Hz, both the lower and upper bounds of the sound pressure at receiving points, R1 for front-passenger and R2 for back-passenger, are plotted in Figures 8 and 9, respectively. Additionally, the crisp bounds of sound pressure along the bottom line at a fixed frequency of \(f=90\) Hz are presented in Figure 10.  
From Figures 8-10, the solution sets by the enclosing-IFEM perfectly coincide with the results of the Monte Carlo method, as the complement of them can be neglected. It illustrates that the enclosing-IFEM can really confine the dependency phenomenon to avoid overestimation in 2D uncertain vibro-acoustic coupling prediction. However, for the non-conservative approximation by neglecting high-order terms, neither the TN-MIPFEM nor the SMW-IPFEM can eliminate the effect of non-conservativeness. Furthermore, the solution sets of these two perturbation-based methods are similar, except that the interval results near the resonant frequency are slightly broadened.  
Figure 7: 2D interior-car vibro-acoustic model.  
Figure 8: The sound pressure at the point of R1.  
Figure 10: The sound pressure along the bottom-line at frequency 90 Hz.  
Figure 9: The sound pressure at the point of R2.  
Table 1 gives the computational time by different methods; although the fastest method is the TN-MIPFEM, the difference between the enclosing-IFEM and TN-MIPFEM is not significant. On the contrary of Monte Carlo simulation method, all the inferred IFEMs' computational time is appropriate and acceptable, even to the longest SMW-IPFEM.