## 4 Conclusions  
In this paper, a tightly coupled ALE-LBFS with the mode decomposition method is presented to predict the 2D and 3D flutter phenomenon in compressible flows. The vertex-centered ALE-LBFS is applied to the unstructured mesh to solve for the flow field and the mode decomposition method is applied for the solution of structural deformations. The exchange of information between the flow field and solid field at the interface is carried out by using the RBF interpolation. A tightly coupled temporal scheme is used to advance the governing equations of the system. The performances of the proposed method are summarized as follows.  
(a) The ALE-LBFS on unstructured dynamic mesh achieves good accuracy in the cases of the 2D forced pitching of the NACA0012 airfoil, the 3D uCRM-13.5 wing, and the BSCW wing with harmonic motions. When tightly coupled with a structure solver, it is qualified to solve aeroelastic problems, regardless of whether the dynamic of the structure is self-executed or controlled manually. The dynamic changes in pressure coefficient distribution \(C_{p}\) around the wing surface are obtained. Compared with the published results, it achieves good agreement.  
(b) The ALE-LBFS coupled with the mode decomposition method has been successfully predicting the flutter phenomenon of the 2D airfoils and 3D wings. The so-called transonic dip captured the critical flutter velocities at different Mach numbers compared with the  
Figure 26: Displacement in x-, y-, z-directions at the probe of (**a**) the leading edge and (**b**) the trailing edge for Ma = 0.499 and \(V_{f}^{*}=0.44\).  
published data. The damped response, divergent response, and limit cycle oscillation (LCO) are demonstrated as well.  
Witnessing the satisfactory performance, it seems that the ALE-LBFS coupled with the mode decomposition method presented in this work has a high potential and can be used as a powerful alternative to analyze flutter with more complex flow situations and structural configurations. It is a reliable and accurate numerical tool and lays the methodical and technical foundation for active flutter control as well.  
Conceptualization, Y.W. and T.G.; methodology, Y.W., T.G. and F.W.; software, Y.W. and T.G.; validation, Y.W. and T.G.; formal analysis, Y.W. and T.G.; investigation, T.G.; resources, Y.W. and T.G.; data curation, Y.W. and T.G.; writing--original draft preparation, T.G.; writing--review and editing, Y.W. and F.W.; visualization, Y.W. and T.G.; supervision, Y.W.; funding acquisition, Y.W. and F.W. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the National Natural Science Foundation of China (Grant Nos. 12272178), Aeronautical Science Foundation of China (Grant No. 20220012052004), Key Laboratory of Computational Aerodynamics, AVIC Aerodynamics Research Institute, and the Spring Sunshine Program of the Ministry of Education of China.  
Not applicable.  
Data Availability Statement: Data will be made available on request.  
This work is also partially supported by the High-Performance Computing Platform of Nanjing University of Aeronautics and Astronautics.  
The authors declare no conflict of interest.