Fluid-structure interactions with numerics for a smoothed augmented Lagrangian variational principle applied to a wave-energy device

We investigate the coupling of nonlinear water-wave motion to heaving buoy wave-energy dynamics [1] in the presence of an inequality constraint. Building on augmented Lagrangian variational principles (VPs) developed by Burman [2] and others, we impose constraints of the form G(q)≥0, where q involves system variables, through a Lagrange multiplier λ. The strict Kuhn–Karush–Tucker (KKT) conditions {λ G=0,G(q)≥0, λ≤0} are replaced by those smooth approximations of the involved function F(c G(q)−λ)=max(c G(q)−λ,0), with smoothing parament c>0, allowing explicit computation of the multiplier λ as (part of) a force. Our approach combines: (a) an Average Vector Field (AVF) energy-conserving time-stepping method, extended to water-wave systems with an auxiliary field, enforcing energy conservation in the discrete system; (b) a (novel) smooth relation λ(G) that regularises the KKT conditions by approximating the solution G=0 with λ≤0 and G>0 with λ=0 in the (λ,G)-plane, but leading to an implicit definition of the function F(c G(q)−λ). This framework has been implemented and tested in the finite-element environment Firedrake, leading to improved and surviving benchmarks for the problems: (i) a point particle under gravity bouncing off a rigid table, (ii) a particle moving in a rectangular (“billiard”) domain, and (iii) forced (Variational “Boussinesq” Model-type) nonlinear water waves in a horizontal channel causing buoy motion in a wave-enhancing contraction. The latter, finite-element, model supports design of a prototype wave-energy device for enhanced energy capture. More generally, this work aims to develop analytical and computational tools for finite-element coupling of nonlinear wave dynamics in fluid-structure interactions, here exemplified by the vertical (heave) motion of the buoy.

[1] O. Bokhove, A. Kalogirou and W. Zweers (2019) From Bore–Soliton–Splash to a new wave-to-wire wave-energy model. Water Waves 1, 217-258.
[2] E. Burman, P. Hansbo and M.G. Larson (2023) The augmented Lagrangian method as a framework for stabilised methods in computational mechanics. Archives of Computational Methods in Eng. 30, 2579–2604.