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A multiple-fidelity computational framework is presented for designing energetically efficient flapping wings. The goal of this design process is to achieve specific aerodynamics characteristics, namely prescribed lift and thrust coefficients at low energetic cost. The wing kinematics (flapping frequency, flapping amplitude) and the resulting optimal wake-circulation distribution (wake circulation/vorticity strength and wake shape) are determined using a low-order, wake-only, energetics method. A quasi-inverse, doublet-lattice method is used to determine the detailed flapping-wing geometry corresponding to the optimal wake. Because of the efficiency of potential flow approximations, the wake-only method and quasi-inverse, doublet-lattice method tools can explore a large number of wing kinematics and wing-morphing strategies efficiently; however, only a handful of those wing designs will achieve the desired …