# Wing Geometry and Kinematic Parameters Optimization of Flapping Wing Hovering Flight
## 7 Discussion  
Due to the synergistic influence of WGP and WKP on the aerodynamic performance of wing planform of hovering insect and thus eventual power consumption in linear or nonlinear coupled way by \(AR\) and \(Re\), any parameter set obtained by sole WGP or WKP optimization is not the global optimal one in global design variable space. This has been verified by the results listed in Table 4. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **For Original** & \multicolumn{2}{c}{**For Optimal WGP with 2D**} & \multicolumn{2}{c}{**For Optimal**} & \multicolumn{2}{c}{**For Combined Optimal WGP and WKP**} \\  & **WGP and WKP** & **(Left)** & **and** 3D (Right) & **C\({}_{\text{Plans}}\) (a)** & **of WKP** & **with 2D (Left) and 3D (Right) & \(C_{\text{Plans}}\) (a)** \\ \hline \(AR\) & 3.7643  & 2.9029 & 2.9506 & 3.7643  & 2.9417 & 3.0411 \\ \(Re\) & 172.8926  & 157.6046 & 168.8122 & 171.7340 & 186.1461 & 237.6147 \\ \(Lloyd\)-1 & 0.2890 & 3.7 \(\times\) 10\({}^{-8}\) & 52.7 \(\times\) 10\({}^{-8}\) & 1.6 \(\times\) 10\({}^{-8}\) & 3.0 \(\times\) 10\({}^{-8}\) & 1.9 \(\times\) 10\({}^{-8}\) \\ \(\left<{}^{1}F_{\text{water}}\right>/\left<{}^{1}F_{\text{water}}\right>_{\text{m}}\) & 0.8834 & 0.8855 & 0.8960 & 1.1241 & 1.1089 & 1.0173 \\ \(\left<{}^{2}\text{water}\right>/\left<{}^{1}F_{\text{water}}\right>\) & 8.4327 & 5.8237 & 5.7343 & 1.4711 & 3.7811 & 2.4905 \\ \(P\)* (Wkg\({}^{-1}\)) & 33.0097  & 20.4839  & 21.1160  & 9.1074 & 4.7867 & 6.7516 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The estimated values of hovering fruit fly and optimization results. The values listed in Table 4 are acquired by substituting the original WGP and WKP of hovering fruit fly and optimized parameters from five cases of optimization into the power density model without embedding hybrid-GA. It is interesting to find that the linear constraints (\(AR\)) and nonlinear constraints (\(Re\)) for five cases of optimization do not manifest substantial variation. The power density obtained by combined optimal WGP and WKP is lower than that from sole WGP or WKP optimization, which may be due to the influence of strong coupling constraint of \(AR\) and \(Re\) on global optimal parameters. Moreover, the value of power density obtained by solely optimal WKP is less than the one reported by [PERSON] and [PERSON] [PERSON] and [PERSON] (2018), which might be due to two factors. The first one is that the stroke plane deviation angle has been ignored here. The second factor must be the crucial one to improve the estimation of power density. That is a much more reasonable assumption about non-dimensional chordwise distribution of COP for translational and rotational circulation force introduced into current extended quasi-steady aerodynamic model to estimate the rotational circulation power consumption. This assumption is different from the one made by [PERSON] and [PERSON]. In their work, they also assume that the translational and rotational circulation forces have a similar chordwise distribution of COP, but the chordwise location was assumed to be fixed at the middle point of each strip element. This does not conform to the empirical formula fitted from experimentally measured data [47; 48]. Thus, their result about the power density might be over-estimated except including the power consumption from stroke plane deviation motion. All of the power density values for optimzation of WGP or/and WKP are much lower than the estimating value of approximate steady hovering flight of fruit fly (Table 4). It is worth noting that the difference value between the ratio of lift-to-weight and one (\(LtoW\)-1) is limited by relative toleration of \(10^{-8}\) (Table 4), which is larger than the one used by [PERSON] and [PERSON] [PERSON] and [PERSON] (2018). We choose this larger relative toleration just by considering the calculation cost. Frankly speaking, the lower this difference value is prescribed, the better the condition of nonlinear equality constraint is satisfied.