## 3 Finite Element Models  
### Structural Finite Element Model  
A Finite Element Model of the HB-2 structure was built using Abaqus, as shown in Figure 6. The maximum element length on the envelope was 5 mm. The two parts (bottom and envelope) were tied together. The first modes, in the (5 Hz-2500 Hz) frequency range, were computed. The dynamics equation was projected onto this modal basis of size \(N_{m}=93\). The modal frequency transfer functions are defined by:  
\[H_{k}(\omega)=\left(m_{k}\left({\omega_{k}}^{2}-2j\xi_{k}\omega_{k}\omega+ \omega^{2}\right)\right)^{-1} \tag{2}\]  
where, for every mode \(k\), \(\omega_{k}\) is the proper angular frequency, \(m_{k}\) is the modal mass, and \(\xi_{k}\) is the modal damping coefficient. This modal damping coefficient was set as 0.3% for all modes. This constant value was chosen to fit the first experimental data obtained in the reverberant chamber. An additional experiment dedicated to the identification of the Coupling Loss Factors of the SEA method (see Section 4.2) confirmed this modeling  
Figure 5: The HB-2 structure in the horizontal configuration in the reverberation chamber and, behind the window, the scanning laser vibrometer. hypothesis: the average modal damping of the first structural modes wa 0.3%. The proper vector is denoted \(\Phi_{\mathbf{k}}\). When the air volume surrounding the structure is not modeled, no scattering effect is considered. The dynamic pressure loading on the external surface of the structure is defined by the controlled mean PSD in the reverberant chamber as shown in Equation (1). The cross power spectral density is known analytically for a diffuse field in an open air volume:  
\[S_{p}(\mathbf{x_{1}},\mathbf{x_{2}},\,\omega)=S_{p}(\omega).\text{sinc}\Big{(} \frac{\omega\|\mathbf{x_{1}}-\mathbf{x_{2}}\|}{c}\Big{)} \tag{3}\]  
where \(c\) is the speed of sound. The computation of the dynamic response of the structure to that dynamic load can be directly performed. This requires the computation of the cross spectral modal forces (or joint modal acceptances) that are defined by:  
\[F_{k_{1}k_{2}}(\omega)=\sum_{\mathbf{x_{1}}\in\mathcal{S}}\sum_{\mathbf{x_{2}} \in\mathcal{S}}S_{p}(\mathbf{x_{1}},\mathbf{x_{2}},\,\omega)\big{(}\Phi_{k_{ 1}}(\mathbf{x_{1}})^{*}\,\mathbf{d}\mathbf{x_{1}}\big{)}\big{(}\Phi_{k_{2}}( \mathbf{x_{2}})^{*}\,\mathbf{d}\mathbf{x_{2}}\big{)} \tag{4}\]  
\(\big{(}\Phi_{k_{1}}(\mathbf{x_{1}})^{*}\,\mathbf{d}\mathbf{x_{1}}\big{)}\) is the projection of the 3D vector \(\Phi_{k_{1}}(\mathbf{x_{1}})\) on the locally oriented surface in point \(\mathbf{x_{1}}\). Each evaluation of \(F_{k_{1}k_{2}}(\omega)\) requires a double integration over the external surface \(\mathcal{S}\), which is computationally expensive. The computation method used in this study is implemented in the vibroacoustics simulation software Wave6. The computation of the cross-spectral modal forces is equivalent to the computation of the acoustic dynamic stiffness matrix [30]. Instead of computing this matrix directly in the modal basis, an intermediate basis is used. This basis is a wavelet discretization of the surface. The method is detailed in [31]. Once the acoustic dynamic stiffness matrix has been evaluated on this wavelet basis, it can be easily projected onto any mode shape, at any frequency, which enhances the computational efficiency of the method. This first method was used to compute the vibroacoustic response of the HB-2 structure without any acoustic phenomena (scattering or internal acoustic resonances). The vibroacoustic model of the system is then improved by modeling the surrounding air volumes using a Boundary Element Model. A second model was implemented in which only the external air volume is modeled, and thereby the scattering effects are simulated. A third model is then implemented in which both the external and internal air volumes are modeled, so that the scattering effects and internal acoustic resonance phenomena are simulated simultaneously.