## 3 Development and Validation of CFD/CSD Coupled Solvers
### Verification of the CFD/CSD Co-Simulation  
In order to verify the accuracy of transonic unsteady aerodynamic solutions and facilitate their application in the study of aeroelasticity problems, a series of flutter experiments were conducted around the AGARD 445.6 wing in the transonic wind tunnel at NASA Langley Center. In this paper, flutter simulation was carried out for AGARD445.6 wings, and the validity of the CFD/CSD coupling simulation method was verified by comparing the simulation results with the flutter test data of AGARD445.6 wing. The main attributes of AGARD 445.6 are as follows: the mass of the model is 1.8628 kg, the aspect ratio is 1.6525, the root-tip ratio is 0.6576, the sweep Angle of the 1/4 string is 45\({}^{\circ}\), the airfoil of each station along wingspan is NACA 65A004, and the experimental angle of attack (AOA) is set to zero. Firstly, according to the geometric size in Figure 3 and the airfoil information of NACA 65A004, the geometric model of AGARD445.6 was established in the 3D modeling software, which served as the basis for the subsequent structural modeling and aerodynamic modeling. Then, in order to obtain the necessary structural modal information for CFD/CSD coupling simulation calculation, the STP format wing model was imported into the finite element software MSC.PATRAN and the structure model of AGARD445.6 wing were established for modal calculation, as shown in Figure 4 below. Based on the assumption of homogenization, the constructed structural model adopts three-dimensional Hex8-type grid elements, with 3080 elements and 4845 structural nodes. The material directions of the elements are divided into longitudinal and lateral directions,  
Figure 4: FEM model of AGARD 445.6 Wing. Figure 3: Geometric parameters of AGARD 445.6 Wing. corresponding to the x and y axes of the local coordinate system in the figure, respectively. The material properties of the model are as Table 1:  
By transferring the above structural model to MSC.NASTRAN solver, the first several modal information of the wing structural model can be obtained, including natural frequency, mode shape, etc. According to relevant studies, the difference between the flutter velocity calculated by the first four modes and the fourteen modes is less than 1%. Therefore, in the aeroelastic analysis of AGARD45.6, only the first four-order natural modes of the structure should be retained for the subsequent modal superposition method to solve the dynamic response of the structure. In this paper, the results of the first four-order modal natural frequencies of the AGARD445.6 wing calculated by MSC.NASTRAN were compared with the measured experimental values, as shown in the table below. It can be seen that the calculated values have a good agreement with the experimental values, indicating that the constructed structural model can effectively reflect the structural dynamic characteristics of the real wing. In addition, the mode shapes of the first four modes were obtained by MSC.NASTRAN are 1 order bending, 2 order torsion, 3 order bending and 4 order torsion, respectively. The experimental values of natural frequencies of the first four modes are compared with the calculated values in this paper, as shown in Table 2.  
The aerodynamic model of AGARD45.6 was established in ICEM. The flow field mesh and the wing surface grid is shown in Figure 5. The size of the flow field calculation domain is 14 \(\times\) 10 \(\times\) 6 m, and there are six surfaces forming the outer boundary of the flow field. The boundary condition of the side face connected with the wing is set as the wall surface, and the remaining five surfaces are set as the far field. \begin{table}
\begin{tabular}{c c} \hline \hline
**Material Properties** & **Value** \\ \hline Longitudinal Young’s modulus & 3.1511 Gpa \\ Lateral Young’s modulus & 0.4162 Gpa \\ Poisson’s ratio & 0.31 \\ Density & 381.98 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material Properties of AGARD 445.6 Wing. Figure 5: CFD mesh of AGARD 445.6 Wing.