## 2 Methodology
### Fluid-Solid Coupling  
In coupling Equations (8) and (37), two challenges arise. The former originates from the question of how often solutions of the fluid and structure solver are computed within one physical time step and how the relevant kinematic and dynamic quantities are exchanged. Depending on the frequency of the fluid-solid interaction, the coupling strategies can be distinguished as the loose-coupling method and the tight-coupling method. For the former, the coupling loop marches and is fixed per time step. The coupling constraints are weakly enforced in each time step because data only communicate after each discrete time instance. Thus, it is hard to guarantee the balance of forces and displacements between the fluid and solid domains at the interface. The favorite tight-coupling method relieves the restriction of the physical time step, which is small enough so that the coupling accuracy is guaranteed. Tight-coupling method exchanges data between fluid and structure several times within each physical time step, as described in Figure 5. These operations, which are equivalent to pre-estimation and correction iterations, eliminate time step delay and improve temporal accuracy. In Figure 5, the computational flowchart for the tight-coupling method displays the progress of one physical time step from \(t^{n}\) to \(t^{n+1}\). In the wake of structural response yielding from structure solver \(q^{n}\) at the time step \(t^{n}\), the new mesh is regenerated and grid velocity is calculated according to the updated structural positions. After solving fluid governing equations, the new flow state is obtained and the aerodynamic forces acting on the solid are solved. Then, the next position of the dynamic object is determined from the solutions of structural equations. For matching mesh, it is natural and easy to realize the above displacements and force transfer. However, fluid and structure meshes do not necessarily coincide in a node-to-node manner at the interface in most FSI problems. Typically, the fluid domain is discretized with finer levels of mesh resolution compared with the solid meshes. It constitutes the second challenge to couple the fluid and solid solver and means that some mapping between the fluid and solid solver has to be applied to interpolate necessary information between them. The general data transformation is in the form of  
\[\mathbf{h}=\mathbf{G}\mathbf{x} \tag{45}\]  
\[\mathbf{F}_{s}=\mathbf{G}^{T}\mathbf{F}_{a} \tag{46}\]  
where matrix \(\mathbf{x}\) represents the displacement for all elements of structural grids while \(\mathbf{h}\) represents the displacement for all vertexes accounting for the deformable boundary. \(\mathbf{G}\) is the connection matrix that relates and interpolates the elements on the structural grids to the aerodynamic grids. \(\mathbf{G}^{T}\) is the transposed matrix interpolating force load on the grid of the structure from loading on the aerodynamic grids. These mappings are guaranteed by the principle of virtual work. Here, the radial basis function (RBF) establishes the bridge to realize the communication of interface displacement and loading force between the fluid solver and the structure  
Figure 5: Computational flowchart for the tight-coupling method. solver. The interpolation function adapts the radial basis function corrected with a linear mapping related to the interpolation location as  
\[w(x,y,z)=a_{0}+a_{1}x+a_{2}y+a_{3}z+\sum_{i=1}^{N}\lambda_{i}r_{i}(x,y,z) \tag{47}\]  
where \(r_{i}(x,y,z)\) is the connection matrix relating the interpolating point \((x,y,z)\) with the \(i\)-th source point \((x_{i},y_{i},z_{i})\). Its mathematic expression can be defined as  
\[r_{i}(x,y,z)=\sqrt{(x-x_{i})^{2}+(y-y_{i})^{2}+(z-z_{i})^{2}} \tag{48}\]  
For interpolation using the function \(w(x,y,z)\), it is necessary to obtain all coefficients in Equation (47) first.