Search for: All records

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. 

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.