###### Abstract  
Due to the complexity of tailoring the wing flexibility and selecting favorable kinematics, the design of flapping wings is a considerably challenging problem. Therefore, there is an urgent need to investigate methods that can be used to design wings with high energy efficiency. In this study, an optimization model was developed to improve energy efficiency by optimizing wing geometric and kinematic parameters. Then, surrogate optimization was used to solve the design optimization model. Finally, the optimal design parameters and the associated sensitivity were provided. The optimized flapping wing, inspired by hummingbirds, features large geometrical parameters, a moderate amplitude of the flapping angle, and low frequency. With the spanwise twisting deformation considered in the parameterization model, the optimization solver gave an optimized wing with a pitching amplitude of approximately 39 deg at the root and 76 deg at the tip. According to the sensitivity analysis, the length of the wing, flapping frequency, and flapping amplitude are the three critical parameters that determine both force generation and power consumption. The amplitude of the pitching motion at the wing root contributes to lowering power consumption. These results provide some guidance for the optimal design of flapping wings. g 2023 13 5704 12  
calculation is simplified by treating the flapping wings as idealized rigid plates. With this design, it becomes more feasible to analyze the effects of shape and motion parameters on the performance of FWMAVs. Some studies have explored the impact of wing flexibility on lift and energy requirements by modeling the intricate fluid-structure interaction between the wing and the surrounding air. This approach has enabled researchers to uncover the contribution of both flexibility and inertia to FWMAV performance [15; 16; 17]. However, to effectively increase the hovering time, finding the best balance between these factors is still an important task from the perspective of developing a FWMAV but remains a largely trial-and-error challenge due to the lack of valid design principles [8]. Therefore, it is important for the design of the FWMAV to determine the design parameters of the flapping wings as soon as possible. In the initial stages of design, some methods can guide the selection of design parameters. For example, we often rely on the scaling law and empirical formula established for flying creatures or the micro vehicles developed previously to determine some parameters [18]. In addition, another method is engineering design through surrogate modeling [19], which has been applied successfully to some specific optimization problems in various engineering fields. Recently, numerical simulations and phenomenology-based aerodynamic modeling [20; 21; 22; 23] have been employed to directly relate the aerodynamic forces exerted on a flapping wing to its time-varying motion and geometry. The effects of the design parameters on the wing performance can be predicted quickly, which allows for combining it with the surrogate optimization method [24] to advance the design of the wing. Using a quasi-steady model instead of the more computationally costly method, such as the direct numerical simulation [15; 25; 26], allows us to perform optimization programs and detailed sensitivity analysis. Previously, some attempts were made to find the optimal design of a hovering flapping wing. Using the quasi-steady aerodynamic model, [PERSON] and [PERSON] [27] explored the optimal wing kinematics pattern to minimize the power density of the flapping wing during the hovering flight. They found that this optimization yielded kinematics that are qualitatively and quantitatively similar to the data observed previously on natural insects. Their further research [28; 29] concluded that the aerodynamic efficiency of hovering flapping flight with optimal kinematic parameters exceeds that of steady wing flight with the optimal angle of attack.