As explained at the end of Chapter 7, the computational vibroacoustic equation could be solved directly ω by ω, but, as explained in Chapter 1, it is recommended to use a reduced-order computational model, which is constructed as follows. The strategy used for constructing the reduced-order computational model consists in using the projection basis constituted of:

1. • The acoustic modes of the acoustic cavity with fixed boundary and without wall acoustic impedance. Two cases are considered. The first one is a closed acoustic cavity (the internal pressure varies with variation of the volume of the cavity), for which the boundary value problem has been defined in Section 6.2. For an almost closed cavity with a nonsealed wall, which are often encountered (the internal pressure does not vary with a variation of the volume of the cavity), an adapted procedure, derived from the closed cavity case, will be presented. The second one concerns an internal cavity filled with an acoustic liquid with a free surface for which the boundary value problem has been defined in Section 6.3.

2. • The elastic structural modes of the structure, taking into account a quasi-static effect of the internal acoustic fluid on the structure. The constitutive equation of the structure corresponds to an elastic material (see Eq. (5.35)) and consequently, the stiffness matrix of the structure has to be taken for ω = 0.

This chapter is devoted to the description of the equations in terms of the pressure field and the associated boundary conditions for the internal dissipative acoustic fluid of the vibroacoustic system. A wall acoustic impedance can be taken into account. In the last section, the case of a free surface for a compressible liquid is considered.

EQUATIONS IN THE FREQUENCY DOMAIN

As introduced in Section 2.3, the fluid is assumed to be homogeneous, compressible, and dissipative. In the reference configuration, the fluid is supposed to be at rest. The fluid is either a gas or a liquid and gravity effects are neglected (see Andrianarison and Ohayon, 2006, to take into account both gravity and compressibility effects for an inviscid internal fluid). Such a fluid is called a dissipative acoustic fluid. Generally, there are two main physical dissipations. The first one is an internal acoustic dissipation inside the cavity due to the viscosity and the thermal conduction of the fluid. These dissipation mechanisms are assumed to be small. In the model proposed, we consider only the dissipation due to the viscosity. This correction introduces an additional dissipative term in the Helmholtz equation without modifying the conservative part. The second one is the dissipation generated inside the wall viscothermal boundary layer of the cavity and is neglected here. We then consider only the acoustic mode (irrotational motion) predominant in the volume.

In this book, we are interested in the analysis of vibroacoustic systems, which are also called structural acoustic systems or fluid-structure interactions for compressible fluid (gas or liquid). Vibroacoustics concerns noise and vibration of structural systems coupled with external and/or internal acoustic fluids. Computational vibroacoustics is understood as the numerical methods solving the equations of physics corresponding to vibroacoustics of complex structures. Complex structures are encountered in many industries for which vibroacoustic numerical simulations play an important role in design and certification, such as the aerospace industry (aircrafts, helicopters, launchers, satellites), automotive industry (automobiles, trucks), railway industry (high speed trains), and naval industry (ships, submarines), as well as in energy production industries (electric power plants).

Since we are interested in the analysis of general complex structural systems in the sense of computational methods defined here, we do not consider analytical or semianalytical methods devoted to structures with simple geometry, asymptoticmethods mainly adapted to the high-frequency range (statistical energy analysis, diffusion of energy, etc.) and approaches that imply them. Concerning the latter, the coupling of the local dynamic equilibrium equation (finite element method) and power balances (implemented in the spirit of the statistical energy analysis) have been analyzed in Soize (1998); Shorter and Langley (2005); Cotoni et al. (2007).