## 3 Results and Discussion  
### Optimization Results and Optimal Wing Performance  
A total of six hundred design points were explored in the sample space for the combined optimization of wing geometric and kinematic parameters. Figure 8a shows the objective function values at these design points on a logarithmic scale. For those points where the obtained value of \(F\) is greater than 10, we marked them with hollow red diamonds and identified them as points that violate the constraints and suffer penalties. In contrast, those marked with green circles were recognized to be within feasible regions because the smaller objective function values were returned there. They account for about 52.6% of the design points, more than half of the total. Therefore, it is reasonable to believe that the sample space has been explored well and that the global optimum has been obtained so far. Table 2 lists the combined optimization results, including the optimal combination of those eight design variables. For optimal wing kinematics, the associated flapping and rotation angles at the root and tip are plotted as a function of normalized time in Figure 8b, where various phases of each half-stroke, such as translation, pronation, and supination, are annotated. The angle of attack gradually increases and decreases during supination and pronation, respectively. It is constant during the translational phase. In other words, this can be indicated by the rate of change of the geometric angle of attack over time. \(\dot{\theta}>0\) and \(\dot{\theta}<0\) represent supination and pronation, respectively. \(\dot{\theta}=0\) represents the translation phase. As shown in this figure, the rotation angle is characterized by a round-trapezoidal profile, with a relatively larger amplitude at the wingtip than at the root. The optimal wing twist angle is approximately 37 deg, and both supination and pronation take a fifth of the time of the half cycle. In terms of wing morphology, all wing geometric parameters are slightly smaller than their respective upper bounds. According to the statistical analysis of the aspect ratio of real hummingbirds [39], \(\dot{A}R=6.9m^{0.09}\), the estimated aspect ratio of a single wing (\(\dot{A}R/2\)) was calculated to be 4.68. That is very close to the current result, 4.41. A comparison study for wing models obtained through three different approaches was carried out, and the results were plotted in Figure 7. Compared with the simplified wing from a tiny hummingbird (see Figure 7a), the optimized wing in Figure 7c in this study has a larger area due to a higher loading requirement. The wing planform depicted in Figure 7b,c share similarities, and the former was derived based on a statistical analysis prediction for a specific target load. Figure 6: Summary of the adopted method to solve the optimization problem, and the flow chart of the optimization program. Moreover, the Reynolds number that controls the flow around the optimal wing was also examined. Its value is about 12,740, approaching the upper limit of the permissible Reynolds number (13,000). The coupling between geometric and kinematics parameters through Re plays a significant role in the search for the optimal value of each design parameter in the combined optimization. After these analyses, it was found that the optimal wing has the feature of large geometrical parameters, medium amplitude of the flapping angle, and low frequency. In a previous study on a hovering fruit fly's wing, [PERSON] et al. [47] used a modified quasi-steady aerodynamic model and hybrid genetic algorithm to obtain a similar result, although the dynamic twisting of the wing was ignored. Different from the current result, the optimal wing used a much higher frequency to produce enough lift at a lower Reynolds number. In a study of a hummingbird-like wing with passive twisting [39], the results from the optimization with morphological and kinematic parameters considered simultaneously also have a similar feature to that found in our study.