## 4 SEA Modeling
### SEA Equations and Hybrid FE-SEA Periodic Theory  
In SEA, a complex vibroacoustic structure is represented as an assembly of coupled subsystems that can receive, store, dissipate and transfer energy. By adopting a statistical description of the local dynamic properties of each subsystem, it is possible to predict the overall average response of a complex system across a broad frequency range. The basic SEA equations express the energy balance in the dynamic response model, which comprises a set of subsystems described by their gross geometric form and dynamic material properties. For a structure made of \(n\) subsystems, for every subsystem \(i\), the energy conservation can be written as [17]:  
\[P_{i,in}=P_{i,diss}+\sum_{j=1,\
eq i}^{n}P_{ij}, \tag{5}\]  
\(P_{i,in}\) is the input power in subsystem \(i\) that is known. The dissipated power in subsystem \(i\) (\(P_{i,diss}\)) and the power transferred from subsystem \(i\) to subsystem \(j\) (\(P_{ij}\)) are modeled as:  
\[P_{i,diss}=\omega\eta_{ii}\langle E_{i}\rangle, \tag{6}\]  
\[P_{ij}=\omega(\eta_{ij}\langle E_{i}\rangle-\eta_{ji}\langle E_{j}\rangle) \tag{7}\]  
where \(\eta_{ii}\) is the damping loss factor (DLF) (or internal loss factor) and \(\eta_{ij}\) is the coupling loss factor (CLF) between subsystem \(i\) and subsystem \(j\). The brackets \(\langle\rangle\) denote spatial the average of the energies and will be omitted hereafter for simplicity. The coupling loss factors \(\eta_{ij}\) and \(\eta_{ji}\) are related by the expression:  
\[N_{i}\eta_{ij}=N_{j}\eta_{ji}, \tag{8}\]  
where \(N_{ij}\) is the modal density of subsystems \(i,j\). Equation (7) can then be written as:  
\[P_{ij}=\omega\eta_{ij}N_{i}\Bigg{(}\frac{E_{i}}{N_{i}}-\frac{E_{j}}{N_{j}} \Bigg{)}. \tag{9}\]  
By substituting Equations (6) and (9) in the energy conservation Equation (5), the following set of linear equations is obtained:  
\[\omega \begin{bmatrix}N_{1}\sum_{j}\eta_{1j}&-N_{1}\eta_{12}&\cdots&-N_{1}\eta _{1n}\\ -N_{2}\eta_{21}&N_{2}\sum_{j}\eta_{2j}&\cdots&-N_{2}\eta_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ -N_{n}\eta_{nl}&-N_{n}\eta_{n2}&\cdots&N_{n}\sum_{j}\eta_{nj}.\end{bmatrix} \begin{bmatrix}E_{1}/N_{1}\\ E_{2}/N_{2}\\ \vdots\\ E_{n}/N_{n}\end{bmatrix} = \begin{bmatrix}P_{inj,1}\\ P_{inj,2}\\ \vdots\\ P_{inj,n}\end{bmatrix} . \tag{10}\]  
Therefore, knowing the injected power and the loss factors, the energies \(E_{i}\) of the subsystems can be determined. The traditional approach to SEA modeling typically involves the description of a complex system in terms of a number of simpler connected subsystems. These SEA subsystems are usually chosen in a fixed formulation library, in order to fit the real geometry as closely as possible. However, traditional SEA codes often have somewhat limited libraries of subsystems that are based on analytical derivations of wave propagation and scattering in simplified cross-section. Such cross-sections are adequate for simple systems; however, they are often not general enough to rigorously describe the vibroacoustic behavior of complex structures. The results presented in this article were obtained with Wave6 software, where a statistical energy analysis formulation using finite element and periodic structure theory, called hybrid FE-SEA periodic theory, was implemented [21; 30; 32]. One of the main advantages of this formulation is the ability to compute some of the parameters involved in the SEA method, such as the coupling loss factors and modal density for complex structures. These parameters are usually calculated analytically only for simple structures [17; 33] or need the development of experimental techniques [34; 35]. The hybrid FE-SEA method offers a way to use deterministic and statistical subsystems in the same model by coupling them at their boundaries. Each 2D SEA subsystem is described by a unit cell modeled with structural finite elements. Periodic boundary conditions are then applied to the edges and corners of the cell, and phase constant surfaces are obtained by solving an algebraic eigenvalue problem.