# Tolerance Modeling and Analysis Considering Form Defects for Spaceborne Array Antenna
## 2 Tolerance Analysis Considering Form Errors
### Tolerance Sampling Method Based on Variance Separation  
The main process of the sampling method was illustrated by taking the reflection plate in Figure 1a as example as well. The performance of the antenna is determined by the position of the elements, which is defined by Equation (13). \[y=f(E)=\sum_{i=1}^{m}\sum_{j=1}^{n}a_{ij}\exp\left\{\left[\int x\left(u\cdot ex _{ij}+v\cdot ey_{ij}+w\cdot ez_{ij}\right)\right]\right\} \tag{13}\]  
Figure 5: (**a**) The model of flatness. The blue points represent the discrete horn elements, the black points represent the measurement points, and the red points represent the bolt holes on the plate. The whole NURBS surface represents the realistic reflection plate, while the red rectangle borders define the tolerance zone of the flatness. (**b**) The top view of the flatness model. To analyze the influence of the flatness \(p\) on the performance \(y\), the sample of flatness \(p\) must be generated first based on Equation (12), then the sample of measurement grid \(A_{kl}(x_{kl},y_{kl},z_{kl})\) can be generated based on Equation (10), then the sample of elements \(\left(\omega_{ij},ey_{ij},ez_{ij}\right)\) can be calculated according to Equation (9), and finally, the sample of performance \(y\) can be calculated and the analysis can be achieved. This is a double-loop procedure. By introducing the variance separation method into Equation (13), the original double loop process was simplified to a single loop process. For any value \(p\) of the flatness \(P\), the PDF of the z-directional coordinate \(z_{kl}\) is determined by \(f_{z_{kl}}(z_{kl}|p)\), the corresponding CDF is defined as \(F_{z_{kl}}(z_{kl}|p)\). Supposing that:  
\[u_{kl}=F_{z_{kl}}(z_{kl}|p),\ u_{kl}\sim U(0,1) \tag{14}\]  
Then according to Equations (2) and (3), one can derive that  
\[z_{kl}=F_{z_{kl}}^{-1}(u_{kl},p) \tag{15}\]  
According to Equations (5) and (6), the z-directional coordinate \(z_{kl}\) can be rewritten as follow. \[z_{kl}=F_{z_{kl}}^{-1}\Big{(}F_{\omega_{kl}}(\omega_{kl},p)\Big{)} \tag{16}\]  
where  
\[F_{\omega_{kl}}(\omega_{kl})=F_{z_{kl}}(z_{kl}|p) \tag{17}\]  
Since \(z_{kl}\) and \(\omega_{kl}\) are both normally distributed variables, then based on Equations (16) and (17) one can derive that:  
\[z_{kl}=\frac{1}{6}p\omega_{kl} \tag{18}\]  
For ease of expression, the calculation of the element position based on the measurement point (Equations (7)-(9)) is abbreviated as Equation (19). \[ez_{ij}=g(z_{11},\cdots z_{kl},\cdots z_{KL}) \tag{19}\]  
Finally, Equation (13) can be rewritten as follow. \[ \begin{array}{l}y=f\Big{(}ez_{11},\cdots ez_{ij},\cdots ez_{mn}\Big{)}\\ ez_{ij}=g\Big{(}F_{z_{11}}^{-1}(F_{\omega_{11}}(\omega_{11},p)),\cdots F_{z_{ kl}}^{-1}\Big{(}F_{\omega_{kl}}(\omega_{kl},p)\Big{)},\cdots F_{z_{KL}}^{-1} (F_{\omega_{KL}}(\omega_{KL},p))\Big{)}\end{array}  \tag{20}\]  
Based on Equation (20), the single loop sampling method is defined clearly. First, the sufficient sample of variable \(\omega_{kl}\) and \(P\) was generated, and the size of the sample is \(S\), then the corresponding output \(y\) was calculated. Finally, the whole sample of output and input variables was achieved as shown in Equation (21). \[\left[ \begin{array}{ccccc}\omega_{11}^{1}&\cdots&\omega_{KL}^{1}&p^{1}&y^{ 1}\\ \vdots&\ddots&\vdots&\vdots&\vdots\\ \omega_{11}^{s}&\cdots&\omega_{KL}^{s}&p^{s}&y^{s}\\ \vdots&\ddots&\vdots&\vdots&\vdots\\ \omega_{11}^{s}&\cdots&\omega_{KL}^{S}&p^{s}&y^{s}\\ \end{array} \right] \tag{21}\]  
The single loop process is much efficient than the original double loop process. Here is the comparison of the sample size needed in the Monte Carlo simulation by taking the reflection plate in Figure 1a as example. Supposing the simulation precision is \(\varepsilon\), the confidence constant is \(c\). **(1) Double loop process:** First, the sample of the flatness must be generated, \(\{p_{1},\cdots p_{k},\cdots p_{k}\}\), recording the sample size as \(K\).