## 5 The Second-Round Optimal Design of Crucial Structural Dimensions
### FSI Modeling of Aerostatic Thrust Bearing  
To calculate the performance of aerostatic thrust bearing with consideration of elastic deformation of the thrust plate, a two-way FSI modeling method [23] of aerostatic thrust bearing is adopted, as illustrated in Figure 6. It consists of the FEM model of the thrust plate and the FEM model of the aerostatic thrust bearing. The elastic deformation of the thrust plate can be calculated by the FEM model of the thrust bearing. The pressure distribution of the air film can be acquired by the FEM model of the aerostatic thrust film. Since the modeling method of the aerostatic thrust bearing has already been detailed in the previous section, it is not detailed again in this section, and only the modeling method of the thrust plate is explained below. Considering the structure feature of the thrust plate, a two-dimensional FEM model is built in this section to save calculation time, and the shell element is employed to simulate the bending deformation of the thrust plate. It is usually to simulate the thin-walled to moderately thick-walled structures, which can acquire excellent  
Figure 6: The FSI model of the aerostatic thrust bearing. (**a**) The FEM model of aerostatic thrust bearing; (**b**) the pressure distribution of air film; (**c**) the pressure load applied on thrust plate; (**d**) the FEM model of thrust plate; and (**e**) the structure deformation of the thrust plate. high computational efficiency. As shown in Figure 6d, the inner edge of the thrust plate is constrained by fixed support, and the symmetrical boundary condition is adopted to further save calculation time. The air film pressure is applied on the bearing surface of the thrust plate. The governing equation for the FEM model of the thrust plate can be written as below. \[[K]\{u\}=\{F\} \tag{19}\]  
where \(\{u\}\) and \([K]\) represent the displacement vector and stiffness matrix, respectively. \(\{F\}\) donates the pressure load applied on the bearing surface of the thrust plate. According to the Mindin-Reissner shell theory, the stiffness matrix of the shell element can be acquired by Equation (20) [24]. \[\mathbf{K}^{c}=\frac{h_{e}^{3}}{12}\int\int\mathbf{B}_{b}^{T}\mathbf{D}_{b} \mathbf{B}_{b}dxdy+h_{e}\int\int\mathbf{B}_{s}^{T}\mathbf{D}_{s}\mathbf{B}_{s}dxdy \tag{20}\]  
where \(h_{e}\) is the thickness of the shell element. \(\mathbf{D}\) is the elastic matrix as follows:  
\[\mathbf{D}_{b}=\frac{E}{1-\mu^{2}}\left[ \begin{array}{ccc}1&\mu&0\\ \mu&1&0\\ 0&0&(1-\mu)/2\end{array} \right],\mathbf{D}_{s}=\left[ \begin{array}{cc}\kappa G &0\\ 0&\kappa G\end{array} \right] \tag{21}\]  
where \(E\) denotes the elasticity modulus. \(\mu\) is the Poisson ratio. \(\kappa\) denotes the shear correction factor. \(G\) is the shear modulus. \(\mathbf{B}\) can be calculated by Equations (22) and (23). \[\mathbf{B}_{b}=\left[ \begin{array}{ccc}\mathbf{B}_{b1}&\mathbf{B}_{b2}&\ldots &\mathbf{B}_{n}\end{array} \right],\mathbf{B}_{s}=\left[ \begin{array}{ccc} \mathbf{B}_{s1}&\mathbf{B}_{s2}&\ldots&\mathbf{B}_{sn}\end{array} \right] \tag{22}\]  
\[\mathbf{B}_{bi}=\left[ \begin{array}{ccc}0&0&\frac{\partial N_{i}}{\partial x} \\ 0&-\frac{\partial N_{i}}{\partial y}&0\\ 0&-\frac{\partial N_{i}}{\partial x}&\frac{\partial N_{i}}{\partial y}\end{array}  \right],\mathbf{B}_{si}=\left[ \begin{array}{ccc}\frac{\partial N_{i}}{ \partial y}&-N_{i}&0\\ \frac{\partial N_{i}}{\partial y}&0&-N_{i}\end{array} \right] \tag{23}\]  
where \(N\) is the shape function of rectangular elements. \[N_{i}=(1+\xi_{i}\xi)(1+\eta_{i}\eta)/4(i=1,2,3,4) \tag{24}\]  
where \(\xi_{i}\) and \(\eta_{i}\) are the coordinates of nodes in the local coordinate system of the shell element. A crucial part of the two-way FSI modeling is to build the bidirectional data exchange interface between the FEM model of the aerostatic thrust bearing and the FEM model of the thrust plate.