Non-Parametric Kinematic Optimization of Flapping Foil Propulsion Using a Discrete Adjoint Method

Optimizing flapping-foil kinematics for underwater propulsion is challenging due to strong temporal coupling and nonlinear fluid–structure interactions. Most existing approaches rely on parameterized motion profiles, which restrict the accessible design space. A non-parametric kinematic optimization framework based on the discrete adjoint method is developed, enabling direct optimization of time-resolved motions without predefined functional forms. A Morison-based low-order hydrodynamic model, calibrated against Computational Fluid Dynamics (CFD), is employed for efficient evaluation within a validated regime. Results show that optimized motions substantially enhance propulsion performance over conventional sinusoidal motions, yielding non-sinusoidal, high-efficiency kinematics. In thrust-maximization cases, the optimized kinematics achieve a 50.29% increase in mean thrust by redistributing heave and pitch amplitudes and timing. Under balanced thrust–power conditions, the optimized motions consistently outperform sinusoidal counterparts. In power-minimization cases, a “generator-like” regime emerges, indicating a reversal of net energy transfer enabled by the non-parametric formulation. These results demonstrate that non-parametric optimization provides enhanced design flexibility and improved propulsion performance, offering a practical framework for biomimetic underwater propulsion design.