# Time-Domain Aeroelasticity Analysis by a Tightly Coupled Fluid-Structure Interaction Methodology
## 5 Fluid-Structure Coupling Procedure  
In an aeroelasticity problem, the flexible structure vibrates in response to the unsteady aerodynamic force, and in turn, the vibration affects the flow field around the structure. By discretizing the fluid and structural dynamic governing equations with the same pseudo time-stepping method, a tightly coupled fluid structure interaction method is established to analyze the aeroelasticity problem in the time domain.  
Figure 1 depicts the solution procedure of the tightly coupled FSI methodology. At the beginning of the solution, the flow field and the structure properties are initialized with given parameters. Then in each physical time step, there are several sub-iteration times between the unsteady fluid dynamic equations and structural dynamic equations. Within each physical time step, the unsteady CFD solver predicts the aerodynamic loads, according to which the structural displacements are determined. At the end of each iteration, volume mesh is updated based on the structural displacements for the next physical time step until the stop criterion is satisfied.  
The surface mesh points in fluid model and structural nodes usually don't coincide at the fluid-structure interface, hence a mapping algorithm is required in the coupled method. The algorithm transfers aerodynamic loads from the aerodynamic surface to the structural nodes, and the predicted displacement on the structural nodes are interpolated to the surface mesh points of the fluid mesh. Various algorithms, including finite-plate spline (FPS) [11], infinite-plate spline (IPS) [12], multiquadric-biharmonic (MQ) [13; 14], inverse isoparametric mapping (IIM) [15], and non-uniform B-spline (NUBS) [16], have been developed for exchanging information on the interface of fluid and structure. The IPS method is extensively used in programs such as MSC/NASTRAN, ENS3 DAE, CFL3 DAE. An RBF-based mapping algorithm [17], which is a three-dimensional extension of the IPS method, is employed for interface mapping here:  
\[\Big{\{}\mathbf{S}_{f}\Big{\}}=[\mathbf{G}]\{\mathbf{S}_{s}\} \tag{16}\]  
\[\{\mathbf{F}_{s}\}=[\mathbf{G}]^{T}\Big{\{}\mathbf{F}_{f}\Big{\}} \tag{17}\]  
where the subscripts \(f\) and \(s\) stand for variables in fluid model and structural model, respectively, and the interpolation matrix \([\mathbf{G}]\) is determined as follows:\[[G]=\left[A_{f_{s}}\right]\cdot\left[C_{ss}^{-1}\right] \tag{18}\] \[[A_{fs}]=\left[\begin{array}{cccccc}1&x_{f_{1}}&y_{f_{1}}&z_{f_{1}}& \phi_{f_{1}s_{1}}&\cdots&\phi_{f_{1}s_{\rm ss}}\\ 1&x_{f_{2}}&y_{f_{2}}&z_{f_{2}}&\phi_{f_{2}s_{1}}&\cdots&\phi_{f_{2}s_{\rm ss}} \\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 1&x_{f_{nf}}&y_{f_{nf}}&z_{f_{nf}}&\phi_{f_{nf}s_{1}}&\cdots&\phi_{f_{nf}s_{ \rm ss}}\\ \end{array}\right]\] \[[C_{ss}]=\left[\begin{array}{cccccc}0&0&0&0&1&\cdots&1\\ 0&0&0&0&x_{s_{1}}&\cdots&x_{s_{\rm ss}}\\ 0&0&0&0&y_{s_{1}}&\cdots&y_{s_{\rm ss}}\\ 0&0&0&0&z_{s_{1}}&\cdots&z_{s_{\rm ss}}\\ 1&x_{s_{1}}&y_{s_{1}}&z_{s_{1}}&\phi_{s_{1}s_{1}}&\cdots&\phi_{s_{1}s_{\rm ss}} \\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 1&x_{s_{\rm ss}}&y_{s_{\rm ss}}&z_{s_{\rm ss}}&\phi_{s_{1}s_{\rm ss}}&\cdots& \phi_{s_{\rm ss}s_{\rm ss}}\\ \end{array}\right]\]  
where \(\phi\) is the basis function, \(nf\) is the total number of the surface mesh and \(ns\) is structure nodes number.