## 7 Vibroacoustic Computational Model
### Matrix Equation of the Computational Model  
Let \(\mathbb{U}(\omega)\) be the complex vector of the \(n_{S}\) degrees-of-freedom (DOFs), which are the values of \(\mathbf{u}(\omega)\) at the nodes of the finite element mesh of domain \(\Omega_{S}\). For the internal acoustic fluid, let \(\mathbb{P}(\omega)\) be the complex vectors of the \(n\) DOFs, which are the values of \(p(\omega)\) at the nodes of the finite element mesh of domain \(\Omega\). The complex matrix equation of the computational model is written as  
\[\left[\mathbb{A}_{\text{FSI}}(\omega)\right]\ \left[\begin{array}{c} \mathbb{U}(\omega)\\ \mathbb{P}(\omega)\end{array}\right]=\left[\begin{array}{c}\mathbb{F}^{S}( \omega)\\ \mathbb{F}(\omega)\end{array}\right]\quad, \tag{23}\]  
in which the complex matrix \([\mathbb{A}_{\text{FSI}}(\omega)]\) is defined by  
\[\left[\begin{array}{cc}[\mathbb{A}^{S}(\omega)]-\omega^{2}[\mathbb{A}_{ \text{\tiny{BEM}}}(\omega/c_{\text{\tiny{E}}})]&[\mathbb{C}]\\ \omega^{2}\,[\mathbb{C}]^{T}&[\mathbb{A}(\omega)]+[\mathbb{A}^{Z}(\omega)] \end{array}\right], \tag{24}\]  
in which \([\mathbb{C}]^{T}\) is the transposed matrix of \([\mathbb{C}]\). In Equation (24), the symmetric \((n_{S}\times n_{S})\) complex matrix \([\mathbb{A}^{S}(\omega)]\) is defined by  
\[[\mathbb{A}^{S}(\omega)]=-\omega^{2}[\mathbb{M}^{S}]+i\omega\left[\mathbb{D}^ {S}(\omega)\right]+[\mathbb{K}^{S}(\omega)]\,, \tag{25}\]  
in which \([\mathbb{M}^{S}]\), \([\mathbb{D}^{S}(\omega)]\), and \([\mathbb{K}^{S}(\omega)]\) are symmetric \((n_{S}\times n_{S})\) real matrices, which represent the mass matrix, the damping matrix, and the stiffness matrix of the structure. Matrix \([\mathbb{M}^{S}]\) is positive and invertible (positive definite), and matrices \([\mathbb{D}^{S}(\omega)]\) and \([\mathbb{K}^{S}(\omega)]\) are positive and not invertible (positive semidefinite) due to the presence of six rigid body motions (free-free structure). The symmetric \((n\times n)\) complex matrix \([\mathbb{A}(\omega)]\) is defined by \[[\mathbb{A}(\omega)]=-\omega^{2}[\mathbb{M}]+i\omega\,[\mathbb{D}(\omega)]+[ \mathbb{K}]\,, \tag{26}\]  
in which \([\mathbb{M}]\), \([\mathbb{D}(\omega)]\), and \([\mathbb{K}]\) are symmetric \((n\times n)\) real matrices. Matrix \([\mathbb{M}]\) is positive and invertible, and matrices \([\mathbb{D}(\omega)]\) and \([\mathbb{K}]\) are positive and not invertible with rank \((n-1)\). In the case of a free surface \(\Gamma_{0}\) in the internal acoustic cavity (see Section 4.3), the boundary condition defined by Equation (10) is added, and consequently the corresponding matrix \([\mathbb{K}]\) is positive definite, and is thus invertible. The internal fluid-structure coupling matrix \([\mathbb{C}]\) is related to the coupling between the structure and the internal fluid on the internal fluid-structure interface, and is a \((n_{S}\times n)\) real matrix that is only related to the values of \(\mathbb{U}\) and \(\mathbb{P}\) on the internal fluid-structure interface. The wall acoustic impedance matrix \([\mathbb{A}^{Z}(\omega)]\) is a symmetric \((n\times n)\) complex matrix depending on the wall acoustic impedance \(Z(\mathbf{x},\omega)\) on \(\Gamma_{Z}\), and which is only related to the values of \(\mathbb{P}\) on boundary \(\Gamma_{Z}\). The boundary element matrix \([\mathbb{A}_{\text{BEM}}(\omega/c_{\text{E}})]\)--which depends on \(\omega/c_{\text{E}}\)--is a symmetric \((n_{S}\times n_{S})\) complex matrix that is only related to the values of \(\mathbb{U}\) on the external fluid-structure interface \(\Gamma_{E}\). This matrix is written as  
\[[\mathbb{A}_{\text{BEM}}(\omega/c_{\text{E}})]=-\rho_{\text{E}}\,[\mathbb{N}] ^{T}\,[B_{\Gamma_{E}}(\omega/c_{\text{E}})]\,[\mathbb{N}]\,, \tag{27}\]  
in which \([B_{\Gamma_{E}}(\omega/c_{\text{E}})]\) is the full symmetric \((n_{E}\times n_{E})\) complex matrix defined in Appendix B, and where \([\mathbb{N}]\) is a sparse \((n_{E}\times n_{S})\) real matrix related to the finite element discretization.