Enhancing flapping foil propulsion through adaptive stiffness scheduling via deep reinforcement learning

Bio-inspired flapping propulsion offers advantages in maneuverability, noise reduction, and adaptability, but its performance is governed by strongly nonlinear fluid–structure interactions between unsteady kinematics and structural compliance. For flexible propulsors with tunable stiffness, it remains unclear how the stiffness level and its within-cycle variation should be designed to improve the propulsive performance under prescribed flapping kinematics. Here, we study a two-dimensional flexible heaving foil at Re = 1100, coupled to a deep reinforcement learning controller that modulates the non-dimensional bending stiffness Kb(t). Three reward weights (β = 0.2, 0.3, and 0.45) are employed to bias the learning objective toward thrust enhancement, a balanced trade-off, or efficiency improvement. Compared with the constant-stiffness baselines, the thrust-oriented setting can boost thrust while maintaining relatively high propulsive efficiency. Relative to the stiffer baseline (Kb = 2.5), the thrust-oriented policy increases thrust by up to 44%. By contrast, the efficiency-oriented setting delivers a smaller thrust gain but achieves the largest efficiency improvement, with up to a 76% increase in propulsive efficiency relative to the same stiffer baseline. Across β, thrust-oriented solutions operate at a higher overall stiffness level than efficiency-oriented ones. For the high-stiffness group (β = 0.2 and 0.3), increasing stiffness around stroke reversal suppresses the deformation lag that occurs when the foil is overly compliant, so the instantaneous attitude and pressure loading project more effectively onto the thrust direction. This reversal-stage stiffening also helps preserve the load by preventing excessive bending from weakening the pressure-driven thrust contribution. In contrast, softening near mid-stroke allows controlled camber to develop, which can enhance favorable pressure projection while avoiding the large reactive loads and sharp power spikes associated with constant-stiffness foils. These findings provide a compact control principle for bio-inspired propulsors: phase-aware stiffness modulation can improve the thrust–power trade-off without altering the prescribed kinematics.