## 2 Parameter Uncertainty Quantification Method
### Global Sensitivity Analysis  
The combination of the generalized chaotic polynomial and the Sobol global sensitivity analysis method was used to realize a fast calculation of the Sobol global sensitivity index. By combining the coefficients of the chaotic polynomial, the global sensitivity index of different random input variables can effectively obtained, and the first-order sensitivity index is obtained as:  
\[S_{X_{i}}=\frac{Var[E(Y|\xi_{i})]}{Var(Y)}=\frac{\sum\limits_{i\in I_{i}}\xi_{i }^{2}E\big{(}\varphi_{i}^{2}\big{)}}{\sum\limits_{i=1}^{m-1}\xi_{i}^{2}E\big{(} \varphi_{i}^{2}\big{)}} \tag{13}\]  
\(\xi_{i}\) represents the corresponding standard normal random variable \(x_{i}\), \(I_{i}\) represents the set of polynomials \(\varphi_{i}\) containing only the random variable \(\xi_{i}\), and \(\xi_{i}\) represents the coefficient of the polynomial \(\varphi_{i}\). \(E\big{(}\varphi_{i}^{2}\big{)}\) can be calculated by the following formula:  
\[E\big{(}\varphi_{i}  
in Figure 1. The length, width, and height of the mobile phone body are 13.65 cm, 6.94 cm, and 0.95 cm, respectively, which are basically the same as those of a commonly used mobile phone body, providing a certain reference value. Due to the precise modeling of the modeling software, the geometric shapes of the main components of the mobile phone were fully considered, and different components were assigned different electromagnetic parameters, as shown in Table 2.  
The electromagnetic model of the mobile phone antenna used in this paper is shown in Figure 2. The antennas used in the mobile phone model were PIFA antennas and patch antennas. The mobile communication module used patch antennas, and the antenna WIFI module used PIFA antennas. The system used two patch antennas located on the back cover of the mobile phone to realize the 3G frequency band, and the side antenna was used to realize the 4G frequency band. In order to cover the specific frequency band of 5G, the mobile phone also embedded an additional patch antenna. The RF system was based on two PIFA antennas. After modification and installation, it could operate in the 2.4 GHz and 5 GHz bands.  
With the mobile phone antenna model used in this paper, the return loss simulation results were obtained through simulation calculations.  
Figure 3 shows the S parameters (return loss) of these antenna modules, where S11 is the return loss of the 4G antenna, S22 and S33 are the return loss of the two 3G antennas, respectively, S44 and S55 are the return loss of WIFI antenna 1 and WIFI antenna 2, and S66 is the return loss of the 5G antenna. The results demonstrated the return losses were all lower than \(-10\) dB, which is in line with the requirements of the antenna design. Therefore,  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Component** & **Relative Permittivity** & **Conductivity** \\ \hline antenna & 1.00 & PEC \\ battery & 1.00 & PEC \\ battery jar & 5.00 & PEC \\ display screen & 4.80 & 0.01 \\ PCB & 1.00 & PEC \\ phone receiver & 1.00 & PEC \\ loudspeaker & 1.00 & PEC \\ speaker connector & 1.00 & PEC \\ inner shell & 2.30 & PEC \\ shell & 2.20 & PEC \\ camera & 1.90 & PEC \\ casing pipe & 3.00 & 0.01 \\ vibrator & 1.00 & PEC \\ \hline \hline \end{tabular} PEC: Perfect Conductor.  
\end{table}
Table 2: Electromagnetic parameters corresponding to different components of mobile phones.  
Figure 1: Mobile phone electromagnetic simulation model.  
these antennas can cover 2100 MHz in 3G, 2300 MHz in 4G, 3500 MHz in 5G, and 5200 MHz in WIFI and could be used for the SAR simulation.