# Tolerance Modeling and Analysis Considering Form Defects for Spaceborne Array Antenna
## 3 Numerical Simulation Results  
The proposed tolerance analysis method was applied to an X-band spaceborne active phased array antenna, as shown in Figure 1. The whole array antenna comprises four subarrays. Each subarray consists of \(8\times 24\) elements. The base plate is affixed to the frame by 10 bolts, and the reflection plate is affixed to the base plate by 9 bolts. A total of 100,000 sets of samples were generated for the tolerance analysis, and one of them is illustrated in Figure 8.  
Figure 7: The parallel coordinates plot. The parallel coordinates plot of the samples is illustrated in Figure 9. For the sake of clarity, only 500 sets of results were drawn. The feasible design interval of each tolerance variable was divided into five intervals to analyze the failure possibility. The failure probability of the ith interval of a tolerance variable is defined as follow  
\[prob_{i}=\frac{N_{rad}^{i}}{N_{total}^{i}}\times 100\% \tag{31}\]  
where \(N_{rad}^{i}\) represents the number of the red lines passing through the ith interval, \(N_{total}^{i}\) and represents all the lines. Figure 8: The final state of the whole array from a set of the samples. Figure 9: The parallel coordinates plot of 500 sets of samples. The failure probability of each interval of the tolerance was illustrated in Figure 10. Results show that as the tolerance value increases, the probability of product failure increases. The maximum power density is most sensitive to the flatness of the base plate, the failure probability is 0.06% for Interval 1 and the failure probability increases to 3.25% rapidly for Interval 5. The maximum power density is not sensitive to the flatness of the frame, while the maximum power density is sensitive to the flatness of the reflection plates. The failure probability \(prob_{i}^{c}\) of the combination of the six tolerance variables was calculated, which reflects the interactive effect of the six tolerance variables on the maximum power density. \[prob_{j}^{c}=1-\frac{M_{blue}^{j}}{M_{total}^{j}}\times 100\% \tag{32}\]  
where \(M_{total}^{j}\) represents the total number of the set of samples passing the specific combination of intervals. For example, one of the combinations is illustrated in Figure 11. All the sets of samples passing the interval [3,1,5,2,3,2] satisfy the requirement, which means \(prob_{j}^{c}=0\%\). Figure 8.0 The failure probability of each interval of the tolerances. For cases that all the tolerances pass the strict interval, such as the interval combination [1,1,1,1,1,1], there is no need to perform the simulation, because it will lead to high costs. For cases that all the tolerances pass the loose interval, such as the interval combination [5,5,5,5,5], the failure probability must be high. Hence, only reasonable combinations are chosen, which are subjective because of the lack of accurate tolerance cost data. Finally, tolerance analysis results of three kinds of combinations are listed in Table 1 for the array antenna. The failure probability of these three kinds of combinations are very small, and the tolerances are not strict.