# Probabilistic Sensitivity Analysis of Wear Property for MEMS Gas Bearing
## 2 Numerical Method
### Solution of the Lubrication Equation Considering the Rarefaction Effect
The pressure interpolation function was introduced to obtain the pressure value at a certain node in a finite element by the product of node pressure and local coordinates:  
\[P=\sum_{i=1}^{4}P_{i}^{e^{2}}\Phi_{i}=\mathbf{\Phi}^{T}\mathbf{P}_{I}^{e^{2}}. \tag{12}\]  
By using the Green formula and substituting Equation (12) into the lubrication equation, the FEM equation of the element gives:  
\[\left\{\iint\limits_{\Omega^{e}}\left[\frac{H^{3}}{2}\widetilde{Q}(D)\frac{ \partial\mathbf{\Phi}}{\partial\varphi}\frac{\partial\mathbf{\Phi}^{T}}{\partial \varphi}+\frac{H^{3}}{2}\widetilde{Q}(D)\frac{\partial\mathbf{\Phi}}{\partial \psi}\frac{\partial\mathbf{\Phi}^{T}}{\partial\psi}\right]\!\!d\varphi d\psi\right\} \mathbf{P}_{I}^{e^{2}}+\left\{\iint\limits_{\Omega^{e}}-\Lambda H\frac{\partial \mathbf{\Phi}}{\partial\varphi}\mathbf{\Phi}^{T}d\varphi d\psi\right\}\mathbf{P}_{I}^{e}=0 \tag{13}\]  
The isoparametric element is introduced and the form of interpolation function \(\mathbf{\Phi}\) is taken for Equation (13). The element interpolation function in Equation (14) is the dimensionless form of standard quadrilateral element interpolation function, where \(-1\leq\xi\leq 1\), \(-1\leq\eta\leq 1\). \[\left\{ \begin{array}{l}\Phi_{1}=\frac{1}{4}(1-\xi)(1-\eta)\\ \Phi_{2}=\frac{1}{4}(1+\xi)(1-\eta)\\ \Phi_{3}=\frac{1}{4}(1+\xi)(1+\eta)\\ \Phi_{4}=\frac{1}{4}(1-\xi)(1+\eta)\\ \end{array} \right.. \tag{14}\]  
Assembling the coefficient matrices of all the finite element equations in the whole solution domain, Equation (15) can be obtained:  
\[\mathbf{A}\mathbf{P}^{2}+\mathbf{B}\mathbf{P}=0 \tag{15}\]  
where \(\mathbf{A}\) and \(\mathbf{B}\) are the total coefficient matrices. The Newton-Raphson method was employed to solve the nonlinear equations:  
\[\mathbf{P}^{(k+1)}=\mathbf{P}^{(k)}-\mathbf{\left[\mathbf{F}^{\prime}(\mathbf{P}^{(k)})\right]}^{- 1}\mathbf{F}(\mathbf{P}^{(k)}) \tag{16}\]  
\[\mathbf{F}^{\prime}(\mathbf{P}^{(k)})=2\mathbf{P}\circ\mathbf{A}+\mathbf{B} \tag{17}\]  
where \(\mathbf{F}(\mathbf{P})\) is the implicit function of the dimensionless pressure, the Hamiltonian \"\(\circ\)\" represents the multiplication of the corresponding items in the two matrices. The distribution of the pressure field  
Figure 5: Rectangular solution domain and mesh model. can be obtained by iteration. Then, the Simpson integral for Equation (18) was adopted to obtain the load capacity of the gas bearing. \[\left\{ \begin{array}{l}W_{x}\\ W_{y}\end{array} \right\}=\int_{-\frac{L}{\mathcal{D}}}^{\frac{L}{\mathcal{D}}} \int_{0}^{2\pi}\left(P(\rho,\psi)-1\right)\left\{ \begin{array}{l}\cos(\varphi -\theta)\\ \sin(\varphi-\theta)\end{array} \right\}d\varphi d\psi. \tag{18}\]  
The solution of the transient lubrication equation can be obtained by the partial derivative method based on the static solution. The rarefied flow term was introduced into the transient lubrication equation to obtain the eight dynamic coefficients. Assuming that a small perturbation is applied to the journal, the coordinate (\(\varepsilon_{0}\), \(\theta_{0}\)) represents the static equilibrium position of the bearing, and the coordinate (\(\varepsilon\), \(\theta\)) represents the transient position of the journal at a certain moment, where \(\varepsilon\) is the eccentricity ratio and \(\theta\) is the attitude angle. Then, the location of the journal gives:  
\[\left\{ \begin{array}{l}\varepsilon=\varepsilon_{0}+E_{0}e^{i\Omega\tau}\\ \theta=\theta_{0}+\Theta_{0}e^{i\Omega\tau}\end{array} \right. \tag{19}\]  
where \(E_{0}\) is the perturbation amplitude of \(\varepsilon\), \(\Theta_{0}\) is the perturbation amplitude of \(\theta\). The perturbation pressure and the film thickness can also be expressed in this form. It can be seen from Equation (9) that the dimensionless Poiseuille flow coefficient can be regarded as a function of the pressure \(P\) and film thickness \(H\). Taylor expansion was performed at (\(P_{0}\), \(H_{0}\)), and the linear term was retained. Then, the linearized coefficient gives:  
\[