## 2 DNSGA Algorithm
### Algorithm Validation  
Five classical multi-objective algorithms, including VEGA, SPEA2, EMOEA, NSGAII and DNSGA, are run 30 times independently under two classical test functions (Table 1) from the ZDT problems proposed by [PERSON] et al. (2018) to test the performance of the presented DNSGA.  
Figure 1: DNSGA algorithm workflow chart.  
The hypervolume ratio (HVR) and generational distance (GD) are recorded, and statistical results are shown in Table 2 and the box plots are shown in Figures 2 and 3. On the one hand, the HVR metric measures the volume covered by the obtained PF for problems where all objectives are to be minimised. A larger value of the metric indicates better diversity. On the other hand, the GD metric indicates how far the tested PF is from the real PF. A low GD value reflects good convergence performance.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & **Max** & **Min** & **Mean** & **Standard Deviation** \\ \hline  & & \multicolumn{3}{c}{ZDT2-HVR} \\ VEGA & 0.3828 & 0.2293 & 0.31139000 & 0.043664290 \\ SPEA2 & 0.6513 & 0.4933 & 0.611976667 & 0.048765225 \\ EMOEA & 0.6263 & 0.4835 & 0.512056667 & 0.024596811 \\ NSGAII & 0.6436 & 0.4859 & 0.615383333 & 0.038940014 \\ DNSGA & 0.8502 & 0.8033 & 0.83826000 & 0.012841796 \\ \hline  & & \multicolumn{3}{c}{ZDT3-HVR} \\ VEGA & 0.5610 & 0.3502 & 0.457723333 & 0.051953901 \\ SPEA2 & 0.7978 & 0.7716 & 0.79000000 & 0.006022347 \\ EMOEA & 0.7957 & 0.7715 & 0.784923333 & 0.006618493 \\ NSGAII & 0.7988 & 0.7883 & 0.793970000 & 0.002069485 \\ DNSGA & 0.8473 & 0.8323 & 0.842230000 & 0.003388623 \\ \hline  & & \multicolumn{3}{c}{ZDT2-GD} \\ VEGA & 0.3498 & 0.1197 & 0.206133333 & 0.055084906 \\ SPEA2 & 0.0644 & 0.0243 & 0.032513333 & 0.007023804 \\ EMOEA & 0.0991 & 0.0066 & 0.046606667 & 0.019133303 \\ NSGAII & 0.0678 & 0.0192 & 0.033546667 & 0.008564412 \\ DNSGA & 0.0493 & 0.0283 & 0.043323333 & 0.004708976 \\ \hline  & & \multicolumn{3}{c}{ZDT3-GD} \\ VEGA & 0.1526 & 0.0572 & 0.104370000 & 0.024940384 \\ SPEA2 & 0.0066 & 0.0008 & 0.001750000 & 0.001029158 \\ EMOEA & 0.0053 & 0.0008 & 0.001863333 & 0.001091630 \\ NSGAII & 0.0022 & 0.0008 & 0.001293333 & 0.000248909 \\ DNSGA & 0.0125 & 0.0044 & 0.006093333 & 0.001383940 \\ \hline  & & \multicolumn{3}{c}{ZDT3-GD} \\ VEGA & 0.1526 & 0.0572 & 0.104370000 & 0.024940384 \\ SPEA2 & 0.0066 & 0.0008 & 0.001750000 & 0.001029158 \\ EMOEA & 0.0053 & 0.0008 & 0.001863333 & 0.001091630 \\ NSGAII & 0.0022 & 0.0008 & 0.001293333 & 0.000248909 \\ DNSGA & 0.0125 & 0.  
As shown in Table 2, Figures 2 and 3, for test functions the ZDT2 and the ZDT3, the mean values of the HVR metrics for DNSGA are significantly higher than that of other classical multi-objective algorithms, and the standard deviations of the HVR metrics for DNSGA are also relatively lower than other multi-objective algorithms, indicating that the population diversity performance of DNSGA is superior to others evolution algorithms.  
For the ZDT2, the mean value and standard deviation of the GD metrics for DNSGA do not differ much from the other algorithms. However, for the ZDT3, the mean value of the GD metrics for DNSGA lower than VEGA and higher than other algorithms, and the standard deviation of the GD metrics for DNSGA do not differ much from the other algorithms.  
To solve the defects of NAGAII algorithm, the proposed DNSGA algorithm trades off the population diversity and the convergence. By the comparison of HVR and GD metric, although the convergence of DNSGA is slightly inferior to that of other classical multi-objective optimization algorithms, DNSGA is superior to other algorithms in population diversity. For the multi-objective aerodynamic optimization problem of complex operating conditions for FC, we put more emphasis on the performance of population diversity.