## 10 Experimental Validation with a Complex Computational Vibroacoustic Model of an Automobile  
We present an experimental validation of the nonparametric probabilistic approach of uncertainties for a complex computational vibroacoustic model of an automobile [75; 110]. **Description of the vibroacoustic system**. The vibroacoustic system is an automobile of a given type with several optional extra, for which a single mean computational model is developed. The experimental variabilities are due to the manufacturing process and to the optional extra. The objective is to predict the booming noise for which the engine rotation is \([1500,4800]\) rpm (rotationsper minute), corresponding to the frequency band [50; 160] Hz, for which the input forces are applied to the engine supports, and for which the output observation is the acoustic pressure at a given point localized in the acoustic cavity. **Nominal computational model and stochastic reduced-order model**. The nominal computational model is a finite element model of the structure and of the acoustic cavity shown in Figure 2. The structure is modeled with 978,733 structural DOFs of displacement, and the acoustic cavity is modeled with 8139 acoustical DOFs of pressure. The structural reduced-order basis is such that \(N=1722\) and the acoustical reduced-order basis is such that \(N_{f}=57\). The hyperparameters of the stochastic reduced-order model are \(\delta=(\delta_{M},\delta_{D},\delta_{K})\) for the structure, \(\delta_{f}=\delta_{M_{f}}=\delta_{D_{f}}=\delta_{K_{f}}\) for the acoustic cavity, and \(\delta_{C}\) for the vibroacoustic coupling. **Experimental identification of hyperparameter \(\delta_{f}\)**. The acoustical input is an acoustic source placed inside the acoustic cavity. The acoustical measurements have been performed for \(\
u=30\) cars of the same type with different configurations corresponding to different seat positions, different internal temperatures, and different numbers of passengers. The acoustical pressures have been measured with \(\
u_{m}=32\) microphones distributed inside the acoustic cavity. For the statistical inverse problem that is required for performing the experimental identification of hyperparameter \(\delta_{f}\), the observation is the real-valued random variable \(U\) defined by  
\[U\!=\!\!\int_{\mathcal{B}}\mathrm{dB}(\omega)\,d\omega\quad,\quad\mathrm{dB}( \omega)=10\,\log_{10}\!\left\{\frac{1}{p_{\mathrm{ref}}^{2}}\,\frac{1}{\
u_{m }}\sum_{j=1}^{\
u_{m}}|P_{k_{j}}(\omega)|^{2}\right\}\,, \tag{99}\]  
in which \(P_{k_{1}}(\omega),\ldots,P_{\
u_{m}}(\omega)\) are the components of \(\mathbf{P}(\omega)\), which correspond to the observed DOFs, and which are computed with the SROM defined by Equations (64)-(69). Let \(u^{\mathrm{exp},1},\ldots,u^{\mathrm{exp},\
u}\) be the corresponding measurements for the \(\
u=30\) cars. The identification of hyperparameter \(\delta_{f}\) is performed by using the maximum likelihood method [87],  
\[\delta_{f}^{\mathrm{opt}}=\arg\max_{\delta_{f}}\mathcal{L}(\delta_{f})\quad, \quad\mathcal{L}(\delta_{f})=\sum_{\ell=1}^{\
u}\log_{10}(p_{U}(u^{\mathrm{ exp},\ell};\delta_{f}))\,. \tag{100}\]  
For \(\ell=1,\ldots,\
u\), the value \(p_{U}(u^{\mathrm{exp},\ell};\delta_{f})\) of the probability density function (pdf) of the random variable \(U\) is estimated with the kernel density estimation method by using the SROM for which the stochastic solver has been chosen as the Monte Carlo method with \(\
u_{s}=\) 2,000 realizations. **Experimental identification of hyperparameter \(\delta=(\delta_{M_{s}},\delta_{D_{s}},\delta_{K_{s}})\)**. The structural inputs are forces applied to the engine supports. The measurements of the accelerations in the structure have been performed for \(\
u=20\) cars of the same type. The random vector-valued observation is \(\mathbf{U}(\omega)=(U_{1}(\omega),\ldots,U_{6}(\omega))\) such that  
\[U_{j}(\omega)=\log_{10}(\omega^{2}\,|Y_{k_{j}}(\omega)|)\quad,\quad j=1,\ldots, 6\,, \tag{101}\]  
Figure 2: Finite element model of the structure (**left figure**).