Shape optimization of sea-crossing bridge piers for wave load reduction: A three-dimensional approach and mitigation mechanism

This study presents a novel framework for optimizing the three-dimensional hydrodynamic shape of bottom-fixed sea-crossing bridge piers to minimize wave-induced force. The proposed method integrates a Non-Uniform Rational B-Splines-based parametric modeling approach for representing the pier geometry with a frequency-domain Boundary Element Method for efficiently calculating wave forces. The optimization is driven by the Generalized Pattern Search algorithm to explore optimized shapes under various geometric constraints. A series of case studies involving different non-dimensional scales (χ), defined as the ratio of the structure's characteristic size to the wavelength, are conducted. The effect of two key geometric constraints, i.e., cross-sectional area and aspect ratio is systematically investigated. Results indicate that for small-scale structures (χ = 0.1–0.3), cross sections near the water surface tend to elongate in the wave propagation direction, while large-scale structures (χ ≥ 0.5) evolve into brim-like geometries. The mechanism reveals that the optimized shape mitigates the wave loads by lowering the diffraction force amplitude for small scales and by increasing both the amplitude and phase difference of the diffraction force relative to the Froude–Krylov force for large scales. The optimized shapes achieve wave load reduction rates ranging from 5% to over 50%, demonstrating the effectiveness of the optimization framework. The optimized shapes effectively suppress the amplitude of the wave field.