## 2 Numerical Methods and Target Model
### Vibro-Acoustics Model  
#### 2.3.1 Finite Element Method  
The equation of motion of the structure excited by an external force \(F\) can be written in the following form [23; 24]:  
\[[M]\{\ddot{x}\}+[C]\{\dot{x}\}+[K]\{x\}=\{F\}, \tag{21}\]  
where \(\ddot{x},\ \dot{x},\ \text{and}\ x\) are the acceleration, velocity, and displacement, respectively, and \(M,\ C\), and \(K\) are the mass, damping, and stiffness matrices of the structure, respectively. The equation of motion shows the relationship between pressure and displacement. An acoustic wave equation can be derived by the equation of state, which indicates the relationship between density and pressure as follows:  
\[\
abla^{2}p-\frac{1}{c^{2}}\frac{\partial^{2}p}{\partial t^{2}}=0, \tag{22}\]  
Figure 1: Illustrative slanted Dirac cone separating incompressible and compressible parts in three-dimensional periodogram: \(U_{0}=110\) km/h and \(c_{0}=343\) m/s. \[c^{2}=\frac{\kappa}{\rho}, \tag{23}\]  
where \(p\) and \(c\) are the sound pressure and the speed of sound in air, and \(\rho\) and \(\kappa\) are the fluid density and bulk modulus. The FEM is the most appropriate numerical technique for the dynamic analysis of this kind of vibroacoustic system. The vibroacoustic problems are most commonly described in an Eulerian formulation, in which the fluid is described by a single scalar function, such as the acoustic pressure, and the structural components are described by a displacement vector. Using Galerkin weighted residuals, expressing \(p=\left[N_{f}\right]\{p_{i}\}\), and using \(\left[N_{f}\right]^{T}\) as the weighting function to minimize the residuals, we obtain  
\[\int_{V}N_{f}^{T}\bigg{(}\frac{1}{\kappa}\ddot{p}-\frac{1}{\rho_{0}}\
abla \cdot\
abla p\bigg{)}dV=\int_{V}\dot{Q}_{s}\delta(\mathbf{x}-\mathbf{x}_{s})dV, \tag{24}\]  
where \(Q_{s}\) and \(\delta\) are the volume velocity per unit volume and Kronecker's delta, respectively. Because of the Gauss divergence theorem, Equation (24) can be expressed in the form  
\[ \begin{array}{c}\frac{1}{\kappa}\int_{V}N_{f}^{T}\ddot{p}dV+\frac{1}{\rho_{ 0}}\int_{V}\
abla N_{f}^{T}\cdot\
abla pdV-\frac{1}{\rho_{0}}\int_{S}N_{f}^{T }\
abla p\cdot n_{f}dS\\ \\ =\int_{V}N_{f}^{T}\dot{Q}_{s}\delta(\mathbf{x}-\mathbf{x}_{s})dV.\end{array}  \tag{25}\]  
The boundary conditions are set to  
\[\
abla p\cdot n_{f}=-\rho_{0}\ddot{u}_{f_{n}}, \tag{26}\]  
\[\ddot{u}_{f_{n}}\cdot n_{f}=\frac{\
abla p\cdot n_{f}}{\rho_{0}}=\frac{\dot{p }}{Z}, \tag{27}\]  
where \(Z\) is the specific acoustic impedance. Using \(p=\left[N_{f}\right]\{p_{i}\}\) and substituting Equations (26) and (27) into Equation (25), we obtain  
\[\left[M_{f}\right]\{\ddot{p}\}+\left[B_{f}\right]\{\dot{p}\}+\left[K_{f} \right]\{p\}+\left[A_{f}\right]\{\ddot{u}_{f}\}=\Big{\{}P_{f}\Big{\}}, \tag{28}\]  
where \(\left[M_{f}\right]\) is the compressibility matrix, \(\left[K_{f}\right]\) is the inverse mass or mobility matrix, \(\left[A_{f}\right]\) is the area matrix, \(\left[B_{f}\right]\) is the impedance matrix, and \(\Big{\{}P_{f}\Big{\}}\) is the acoustic source vector. In the coupled vibroacoustic analysis, the fluid pressure on the structure boundary causes surface traction on the structure as follows:  
\[\{F_{s}\}=\int_{S}N_{s}^{T}\{\varphi\}dS, \tag{29}\]  
where \(N_{s}\) is a shape function for the structure, and substituting \(\varphi=-pn_{s}\) into Equation (29) gives  
\[\{F_{s}\}=-\int_{S}N_{s}^{T}p\{n_{s}\}dS=\int_{S}N_{f}^{T}N_{f}\{n_{s}\}dS\{p \}=-[A]\{p\}. \tag{30}\]  
The structure equation of motion then becomes  
\[[M_{s}]\{\ddot{u}_{s}\}+[B_{s}]\{\dot{u}_{s}\}+[K_{s}]\{u_{s}\}=\{P_{s}\}-[A] \{p\}, \tag{31}\]  
where  
\[[A]=\int_{S}N_{s}^{T}N_{f}\{n_{s}\}dS\text{ or }[A]=-\int_{S}N_{s}^{T}N_{f} \Big{\{}n_{f}\Big{\}}dS, \tag{32}\]  
where \(N_{s}\) and \(N_{f}\) are structure and fluid shape functions, respectively, and \(n_{s}=-n_{f}\).