## 8 Reduced-Order Vibroacoustic Computational Model  
One possible method for constructing a reduced-order model would consist of choosing as a reduced-order basis the elastoacoustic modes (coupled vibroacoustic modes) of an associated conservative system that must be defined and which will be directly computed. The direct computation of these elastoacoustic modes can be very expensive for solving the eigenvalue problem of large-scale computational models, even if advanced algorithms based on Krylov methods are used. However, with such an approach, the nature of the couplings between the subsystems through the knowledge of the eigenfrequencies and the mode shapes of each subsystem can be difficult to analyze. In addition, the use of the elastoacoustic modes does not allow anymore for adapting the level of uncertainties as a function of the matrices of the coupled system (inertial forces, damping forces, stiffness forces in the structure, and equivalent forces in the internal dissipative acoustic fluid, internal fluid-structure coupling forces). This aspect is particularly important for the analyses in the medium-frequency range. Certainly, knowledge of the elastoacoustic modes can also be very useful for analyzing the consequences of the couplings. Their computation can be considerably decreased by using a reduced-order model such as the one proposed hereinafter, which avoids the expensive direct computation. Finally, if the direct computation of the elastoacoustic modes can be very useful for the cases for which there is a strong effect of the internal acoustic fluid on the structural modes (e.g., the case of a strong coupling such as a structure coupled with an internal liquid or of a lightweight structure coupled with and internal gas), the use of the reduced-order model presented in Section 8 is also very efficient (as explained in Section 8.1.2), because the structural modes are computed taking into account the quasistatic effect of the internal fluid (the added mass effect). Finally, in order to simplify the presentation of Section 8--which is devoted to the construction of the reduced-order model--we have considered the case for which the internal acoustic fluid is a gas and the case for which the internal acoustic fluid is a liquid in order to illustrate the two important cases of a weak coupling and of a strong coupling. Clearly, these two cases cover many coupling situations, such as a lightweight structure coupled with an internal gas for which a strong coupling can appear; in such a case, the formulation presented in Section 8.1.2 must be used. The construction of the reduced-order computational model requires appropriate projection bases to be introduced, as explained below.  
* _Projection basis for the structure._
* If the internal acoustic fluid is a gas, the projection basis can be chosen as the undamped elastic structural modes of the structure in vacuo for which the constitutive equation corresponds to elastic materials (see Equation (A5)), and consequently, the stiffness matrix has to be constructed for \(\omega\,=\,0\).
* If the internal acoustic fluid is a liquid (with or without free surface), for the structure, the projection basis can constructed as for a gas but by taking into account the effects of liquid's added mass.
* _Projection basis for the internal acoustic fluid._
* We consider the undamped acoustic modes of an internal acoustic cavity with fixed boundary (and rigid wall) and without wall acoustic impedance. Two cases must be considered: one for which the internal pressure varies with a variation of the volume of the cavity (a cavity with a sealed wall called a closed cavity), and the other one for which the internal pressure does not vary with the variation of the volume of the cavity (a cavity with a non-sealed wall, called an almost-closed cavity).