## 3 Development and Validation of CFD/CSD Coupled Solvers
### Structure/Fluid Interface Interpolation  
The fluid-solid interface load/displacement transfer is a bidirectional information transfer on the interface between the fluid computing domain and the solid computing domain. Due to the difference in topological form and distribution density between the nodes of fluid mesh and the structural FEM node, the aerodynamic load calculated by CFD on the fluid nodes needs to be transformed into the aerodynamic load on the FEM nodes of the structural model in a solid domain to calculate the structural dynamics through the transfer matrix in the coupling calculation process; on the other hand, the displacement of the FEM nodes obtained by the structural dynamics solver should be transformed into the displacements of the fluid nodes at this time step through the transfer matrix, so as to update the fluid motion boundary required by CFD calculation. From a mathematical point of view, displacement and load transfer at a fluid/solid interface is an interpolation problem. In this paper, global radial basis functions and monomial basis functions are selected to construct interpolation functions. The radial basis function takes the distance from the target point \(y\) to the node \(x_{i}\) as its independent variable. As a classical global radial basis function, thin plate spline is widely used in function fitting, and its expression is:  
\[\phi_{i}(y)=d_{i}^{2}lnd_{i},(i=1,2,\ldots,ns), \tag{1}\]  
where \(ns\) is the number of nodes in the domain of definition. The introduction of the monomial basis function can improve the interpolation accuracy and computational stability. The linear monomial basis of three-dimensional space is:  
\[p(y)^{T}=[1\ x\ y\ z],\ m=4 \tag{2}\]  
where \(m\) is the number of elements in the monomial basis function. The interpolation function \(u^{h}(y)\) is expressed by thin-plate spline and monomial basis as:  
\[u^{h}(y)=\sum_{i=1}^{N}a_{i}\varphi_{i}(y)+\sum_{j=1}^{m}b_{j}p_{j}(y)=\Phi^{ T}(y)a+p^{T}(y)b \tag{3}\]  
where \(N\) is the number of structural nodes in the domain, \(a_{i}\) and \(b_{j}\) are undetermined coefficients, \(\varphi_{i}\) are thin plate splines function and \(p_{j}\) is a monomial basis function. The corresponding array is defined as:  
\[ \begin{array}{c}\Phi(y)=[\phi_{1}(y),\phi_{2}(y),\ldots,\phi_{N}(y)]^{T},\\ p(y)=[p_{1}(y),p_{2}(y),\ldots,p_{m}(y)]^{T},\\ \text{a}=[a_{1},a_{2},\ldots,a_{N}]^{T},\\ \text{b}=[b_{1},b_{2},\ldots,b_{N}]^{T},\end{array}  \tag{4}\]  
The interpolation function is forced through the structure nodes in its definition of the domain by Equation (5)  
\[\sum_{i=1}^{N}a_{i}\varphi_{i}(y)+\sum_{j=1}^{m}b_{j}p_{j}(y)=u_{k},k=1,2, \ldots,N, \tag{5}\]  
In order to ensure the uniqueness of monomial base interpolation, additional constraints are imposed:  
\[\sum_{i=1}^{N}a_{i}\varphi_{i}(y)=0,j=1,2,\ldots,m, \tag{6}\]  
Written in matrix form:  
\[B \begin{bmatrix}a\\ b\end{bmatrix} = \begin{bmatrix}u\\ 0\end{bmatrix} , \tag{7}\]where:  
\[B= \begin{bmatrix}A&P\\ p^{T}&0\end{bmatrix}  \tag{8}\] \[A= \begin{bmatrix}\Phi^{T}(x_{1})\\ \Phi^{T}(x_{2})\\ \vdots\\ \Phi^{T}(x_{N})\end{bmatrix} = \begin{bmatrix}\phi_{1}(x_{1})&\phi_{1}(x_{1})& \cdots&\phi_{1}(x_{1})\\ \phi_{1}(x_{2})&\phi_{1}(x_{2})&\cdots&\phi_{1}(x_{2})\\ \vdots&\vdots&\vdots&\vdots\\ \phi_{1}(x_{2})&\phi_{1}(x_{2})&\cdots&\phi_{1}(x_{2})\end{bmatrix} \] \[P= \begin{bmatrix}\mathbb{P}^{T}(x_{1})\\ \mathbb{P}^{T}(x_{2})\\ \vdots\\ \mathbb{P}^{T}(x_{N})\end{bmatrix} = \begin{bmatrix}P_{1}(x_{1})&P_{2}(x_{1})& \cdots&P_{m}(x_{1})\\ P_{1}(x_{2})&P_{2}(x_{2})&\cdots&P_{m}(x_{2})\\ \vdots&\vdots&\vdots&\vdots\\ P_{1}(x_{2})&P_{2}(x_{2})&\cdots&P_{m}(x_{N})\end{bmatrix} \]  
Because \(\phi_{i}(x_{j})=\phi_{j}(x_{i})\), is a symmetric matrix. The matrix \(a\) and \(b\) can be solved from Equation (7) and substituted into Equation (3), thus:  
\[u^{n}(y)= \begin{bmatrix}\Phi^{T}(y)&\mathbb{P}^{T}(y)\end{bmatrix} B^{-1} \begin{bmatrix}u\\ 0\end{bmatrix} = \begin{bmatrix}\Phi^{T}(y)&\mathbb{P}^{T}(y)\end{bmatrix}