Developing Closed-Form Equations of Maximum Drag and Moment on Rigid Vegetation Stems in Fully Nonlinear Waves

Coastal wetlands act as natural buffers against wave energy and storm surges. In the course of energy dissipation, vegetation stems are exposed to wave action, which may lead to stem breakage. An integral component of wave attenuation modeling involves quantifying the extent of damaged vegetation, which relies on determining the maximum drag force ($F_{D\max }$) and maximum moment of drag ($M_{D\max }$) experienced by vegetation stems. Existing closed-form theoretical equations for $M_{D\max }$ and $F_{D\max }$ are only valid for linear and weakly nonlinear deep water waves. To address this limitation, this study first establishes an extensive synthetic dataset encompassing 256,450 wave and vegetation scenarios. Their corresponding wave crests, wave troughs, $M_{D\max }$, and $F_{D\max }$, which compose the dataset, are numerically computed through an efficient algorithm capable of fast computing fully nonlinear surface gravity waves in arbitrary depth. Seven dominant wave and vegetation related dimensionless parameters that impact $M_{D\max }$ and $F_{D\max }$ are discerned and incorporated as input feature parameters into an innovative sparse regression algorithm to reveal the underlying nonlinear relationships between $M_{D\max }$, $F_{D\max }$ and the input features. Sparse regression is a subfield of machine learning that primarily focuses on identifying a subset of relevant feature functions from a feature function library. Leveraging this synthetic dataset and the power of sparse regression, concise yet accurate closed-form equations for $M_{D\max }$ and $F_{D\max }$ are developed. The discovered equations exhibit good accuracy compared with the ground truth in the synthetic dataset, with a maximum relative error below 6.6% and a mean relative error below 1.4%. Practical applications of these equations involve assessment of the extent of damaged vegetation under wave impact and estimation of $M_{D\max }$ and $F_{D\max }$ on cylindrical structures.