## 2 Methodology  
### Governing Equations of the Aeroelastic System  
The aeroelastics is one of coupled fluid-structure interaction problems. As introduced in the reference [49], its general governing equations of the aeroelastic system include three fields to account for the evolution of fluid, structural, and computational meshes, respectively. Their coupled partial differential equations are described as follows:  
\[\frac{\partial(J\mathbf{w})}{\partial t}\Big{|}\xi+J\
abla_{x}\cdot\left(F( \mathbf{w})-\frac{\partial x}{\partial t}\mathbf{w}\right)=J\
abla_{x}\cdot R (\mathbf{w}) \tag{1}\]  
\[\rho_{s}\frac{\partial^{2}\mathbf{u}_{s}}{\partial t^{2}}-div\left(\sigma_{s} \bigg{(}\varepsilon_{s}(\mathbf{u}_{s}),\frac{\partial\varepsilon_{s}}{ \partial t}(\mathbf{u}_{s})\bigg{)}\right)=b \tag{2}\]  
\[\tilde{\rho}\frac{\partial^{2}x}{\partial t^{2}}-div\left(\widetilde{\sigma}( \widetilde{\epsilon}(x))\right)=0 \tag{3}\]  
Equation (1) is the ALE conservative form of the Navier-Stokes equations where \(x(t)\) is the time-dependent displacement of a fluid grid point, and \(\xi\), as well as \(J=\det(dx/d\xi)\), denotes its position in a reference coordinate. Vector \(\mathbf{w}\) is the convective variable in the fluid domain, and F and R represent the convective and diffusive ALE fluxes. Depending on the type of considered physical problem, such as compressibility and viscosity, diverse fluid governing equations are developed. Limited by the transonic flutter problem, the viscous Newtonian flow in the compressible regime is governed by Navier-Stokes equations. If necessary, the large eddy simulation or RANS is employed to augment the compressible Navier-Stokes equations by solving transport equations of additional turbulent variables. Rather, the unsteady aerodynamic forces are computed by the compressible Euler equations if viscosity is neglected. In the elastodynamic Equation (2), \(\mathbf{u}_{s}\) and \(\rho_{s}\) denote the displacement and density of the structure, \(\sigma_{s}\) and \(\epsilon_{s}\) are the tensors for stress and strain, and b indicates the body forces. By the finite element method, Equation (2) is solved and is capable of expressing the nonlinearity of structure with satisfactory accuracy. In fact, under the assumption of small displacement amplitudes, the harmonic motion method is a common alternative to alleviate the intensive computation of the structure equations. Modal structural equations are adopted here to achieve satisfactory accuracy with acceptable computational burden. Equation (3) represents the dynamics of moving computational meshes. Similar to the elastodynamic equation, it is treated as a pseudo-structural system. Since the technology is out of interest in this work, the inverse-volume spring method is applied directly to treat the mesh deformation. Figure 1 displays that the statement of boundary conditions at the dynamic interface is necessary, which couples tightly the aforementioned Equations (1)-(3) and closes the complete governing equations of the aeroelastic system. First, Equations (1) and (3) are directly coupled. Then, under the supposition that \(u_{f},p,\sigma_{f}\) represent the ALE displacement, pressure, and viscous stress tensor, respectively, the constraint conditions at the interface between fluid and structure are expressed as  
\[\sigma_{s}\cdot\mathbf{n}=-p\mathbf{n}+\sigma_{f}\cdot\mathbf{n},\text{ {on} }\Gamma \tag{4}\]  
\[\frac{\partial u_{s}}{\partial t}=\frac{\partial u_{f}}{\partial t},\text{ {on} }\Gamma \tag{5}\]  
where \(\Gamma\) denotes the fluid-structure interface boundary, and \(\mathbf{n}\) is the corresponding normal vector. On the interface boundary \(\Gamma\), Equation (4) describes the tractions equilibrium between the structure and fluid domain. Additionally, the compatibility of velocity among the structure and fluid is guaranteed by Equation (5).