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Recent studies have demonstrated the potential of hyperbolic paraboloid (hypar), a doubly curved geometry, in coastal engineering applications. Predicting pressure distribution, critical for subsequent finite element analysis, on such novel three-dimensional structures require Computational Fluid Dynamics (CFD) simulations, which are computationally intensive. To address this challenge, the current study develops an artificial neural network (ANN) surrogate to predict pressure distributions on hypar free-surface breakwaters (FSBWs) under solitary wave loading. Using Smoothed Particle Hydrodynamics (SPH) as the CFD tool, simulations generate the supervised learning dataset, where inputs are the hypar warping Rn, breakwater draft dr, and wave height H. The targets consist of two 30×30 pressure maps at wave arrival (hydrostatic) and peak, together with the wave rise time {P(t0), P(tpeak), Δt=tpeak−t0}. Three architectures, FNN, CNN, and DeepONet, are trained with homoscedastic uncertainty loss weighting, each at two parameter sizes (~50k and ~500k). Results for training and testing show that all models achieve low errors, with models with ~50k parameters found to be sufficient, and scaling to ~500k yields some generalization improvement. Further reducing the parameters (~5k) degrades accuracy for all models, with DeepONet proven most robust to parameter size reduction. Overall, this study introduces a novel SPH-ANN workflow for predicting wave pressures on hypar FSBWs, where inference on new samples occurs in a few milliseconds per sample, delivering orders-of-magnitude speedups relative to running new SPH simulations. This computational efficiency enables rapid design iteration and optimization of hypar FSBWs, facilitating their potential deployment in coastal defense. 

This study investigates the potential of hyperbolic paraboloid (hypar) shapes for enhancing wave attenuation and structural efficiency in Free-Surface Breakwaters (FSBW). A decoupled approach combining Smoothed Particle Hydrodynamics (SPH) and Finite Element Method (FEM) is employed to analyze hypar-faced FSBW performance across varying hypar warping values and wave characteristics. SPH simulations, validated through experiments, determine wave attenuation performance and extract pressure values for subsequent FEM analysis. Results indicate that hypar-faced FSBW produces increased wave attenuation compared to traditional flat-faced designs, particularly for shorter wave periods and smaller drafts. Furthermore, hypar surfaces exhibit up to three times lower principal stresses under wave loading compared to the flat counterpart, potentially allowing for thinner surfaces. The study also shows that peak-load static stress values provide a reasonable approximation for preliminary design, with less than 6% average difference compared to dynamic analysis results. In summary, this research presents hypar-faced FSBW as a promising alternative in coastal defense strategies, offering effective wave attenuation and structural efficiency in the context of rising sea levels and increasing storm intensities. 

This project aims to advance the fundamental knowledge of wave-shell structure interaction by means of discovering efficient hydrodynamic thin-shell structural forms that have broad application to coastal structures. Physical experiments are conducted to study how specific structural parameters and wave characteristics influence wave-induced loads, in order to identify structural geometries that reduce wave forces while maintaining structural performance. The hazards of interest are coastal hazards, including storm surge, high tides, and elevated water levels during extreme weather events. The project includes two experimental programs conducted in the water flumes at Stony Brook University and Oregon State University, both using scaled models of different structural configurations. The models are exposed to waves with varied heights, periods, and steepness, and positioned at different elevations—semi-submerged, still water level, and elevated—to simulate a range of hazard scenarios. The project contains measurement results of wave forces, wave elevations, fluid velocities, hydrodynamic pressures, etc. The data can benefit coastal engineers, designers, and researchers by enabling reproducibility of experiments, validation of computational models, and development of data-driven tools for predicting wave–structure interaction. It also supports the creation of optimized structural forms for future coastal defense systems, offering a resource that can generate new studies and improve resilience strategies in coastal engineering. 

Abstract We prove that all Galerkin truncations of the 2d stochastic Navier–Stokes equations in vorticity form on any rectangular torus subjected to hypoelliptic, additive stochastic forcing are chaotic at sufficiently small viscosity, provided the frequency truncation satisfies$$N\ge 392$$ $N\ge 392$. By “chaotic” we mean having a strictly positive Lyapunov exponent, i.e. almost-sure asymptotic exponential growth of the derivative with respect to generic initial conditions. A sufficient condition for such results was derived in previous joint work with Alex Blumenthal which reduces the question to the non-degeneracy of a matrix Lie algebra implying Hörmander’s condition for the Markov process lifted to the sphere bundle (projective hypoellipticity). The purpose of this work is to reformulate this condition to be more amenable for Galerkin truncations of PDEs and then to verify this condition using (a) a reduction to genericity properties of a diagonal sub-algebra inspired by the root space decomposition of semi-simple Lie algebras and (b) computational algebraic geometry executed by Maple in exact rational arithmetic. Note that even though we use a computer assisted proof, the result is valid for all aspect ratios and all sufficiently high dimensional truncations; in fact, certain steps simplify in the formal infinite dimensional limit. 

We consider the top Lyapunov exponent associated to a dissipative linear evolution equation posed on a separable Hilbert or Banach space. In many applications in partial differential equations, such equations are often posed on a scale of nonequivalent spaces mitigating, e.g., integrability (L^p) or differentiability (W^{s,p}). In contrast to finite dimensions, the Lyapunov exponent could a priori depend on the choice of norm used. In this paper we show that under quite general conditions, the Lyapunov exponent of a cocycle of compact linear operators is independent of the norm used. We apply this result to two important problems from fluid mechanics: the enhanced dissipation rate for the advection diffusion equation with ergodic velocity field; and the Lyapunov exponent for the 2d Navier–Stokes equations with stochastic or periodic forcing.