## 8 Reduced-Order Vibroacoustic Computational Model
### Construction of the Reduced-Order Computational Model  
Using the projection basis \(\left[\,\mathcal{U}\,\right]\) defined by Equation (35) for the structure, and the projection basis \(\left[\,\mathcal{P}\,\right]\) defined by Equations (46), (47), or (51) for the acoustic fluid, the projection of Equation (23) yields the reduced-order computational model of order \(N_{S}\ll n_{S}\) and \(N\ll n\), which is written as  
\[\mathbb{U}(\omega)=\left[\mathcal{U}\right]\mathbf{q}^{S}(\omega)\,, \tag{52}\]  
\[\mathbb{P}(\omega)=\left[\mathcal{P}\right]\mathbf{q}(\omega)\,. \tag{53}\]  
The complex vectors \(\mathbf{q}^{S}(\omega)\) and \(\mathbf{q}(\omega)\) of dimension \(N_{S}\) and \(N\) are the solution of the following equation  
\[\left[A_{\text{FSI}}(\omega)\right]\left[\begin{array}{c}\mathbf{q}^{S}( \omega)\\ \mathbf{q}(\omega)\end{array}\right]=\left[\begin{array}{c}\mathbf{f}^{S}( \omega)\\ \mathbf{f}(\omega)\end{array}\right]\quad, \tag{54}\]  
in which the complex matrix \(\left[A_{\text{FSI}}(\omega)\right]\) is defined by  
\[\left[\begin{array}{cc}\left[A^{S}(\omega)\right]-\omega^{2}[A_{\text{BEM}} (\omega/c_{\text{E}})]&\left[\,\mathcal{C}\,\right]\\ \omega^{2}\left[\,\mathcal{C}\,\right]^{T}&\left[A(\omega)\right]+\left[A^{Z} (\omega)\right]\end{array}\right]. \tag{55}\]  
In Equation (55), the symmetric \(\left(N_{S}\times N_{S}\right)\) complex matrix \(\left[A^{S}(\omega)\right]\) is defined by  
\[\left[A^{S}(\omega)\right]=-\omega^{2}[M^{S}]+i\omega\left[D^{S}(\omega) \right]+\left[K^{S}(\omega)\right], \tag{56}\]  
in which \([M^{S}]\), \([D^{S}(\omega)]\), and \([K^{S}(\omega)]\) are positive-definite symmetric \(\left(N_{S}\times N_{S}\right)\) real matrices such that  
\[\left[D^{S}(\omega)\right]=\left[\mathcal{U}\right]^{T}\left[\mathbb{D}^{S}( \omega)\right]\left[\mathcal{U}\right]\quad,\quad\left[K^{S}(\omega)\right]= \left[\mathcal{U}\right]^{T}\left[\mathbb{K}^{S}(\omega)\right]\left[\mathcal{ U}\right]. \tag{57}\]  
The symmetric \(\left(N\times N\right)\) complex matrix \(\left[A(\omega)\right]\) is defined by  
\[\left[A(\omega)\right]=-\omega^{2}[M]+i\omega\left[D(\omega)\right]+\left[K \right], \tag{58}\]  
in which \([M]\), \([D(\omega)]\), and \([K]\) are symmetric \(\left(N\times N\right)\) real matrices. For a closed (sealed wall) acoustic cavity, matrix \(\left[K\right]\) is positive and not invertible with rank \(N-1\), while for an almost-closed (non-sealed wall) acoustic cavity or for a liquid with a free surface (see Section 4.3), matrix \(\left[K\right]\) is positive and invertible. Matrix \(\left[M\right]\) is positive and invertible. The diagonal \(\left(N\times N\right)\) real matrix \(\left[D(\omega)\right]\) is written as  
\[\left[D(\omega)\right]=\tau(\omega)\left[K\right], \tag{59}\]  
in which \(\tau(\omega)\) is defined by Equation (6). The \(\left(N_{S}\times N\right)\) real matrix \(\left[C\right]\) is written as  
\[\left[C\right]=\left[\mathcal{U}\right]^{T}\left[\mathbb{C}\right]\left[ \mathcal{P}\right]. \tag{60}\]  
The symmetric \(\left(N\times N\right)\) complex matrix \(\left[A^{Z}(\omega)\right]\) is such that  
\[\left[A^{Z}(\omega)\right]=\left[\mathcal{P}\right]^{T}\left[\mathbb{A}^{Z}( \omega)\right]\left[\mathcal{P}\right], \tag{61}\]  
and finally, the symmetric \(\left(N_{S}\times N_{S}\right)\) complex matrix \(\left[A_{\text{BEM}}(\omega/c_{\text{E}})\right]\) is given by  
\[\left[A_{\text{BEM}}(\omega/c_{\text{E}})\right]=\left[\mathcal{U}\right]^{T} \left[\mathbb{A}_{\text{BEM}}(\omega/c_{\text{E}})\right]\left[\mathcal{U} \right]. \tag{62}\]  
The given forces are written as  
\[\mathbf{f}^{S}(\omega)=\left[\mathcal{U}\right]^{T}\mathbb{F}^{S}(\omega)\quad, \quad\mathbf{f}(\omega)=\left[\mathcal{P}\right]^{T}\mathbb{F}(\omega)\,. \tag{63}\]