Optimal Strategies to Steer and Control Water Waves

TL;DR

The paper addresses reducing hydrodynamic loads on floating structures by surrounding them with active and passive wave-manipulating devices. It formulates a PDE-constrained optimal-control problem in the frequency domain, coupling the velocity potential $\phi$, surface elevation $\eta$, and rigid-body motion $\mathbf{X}$ through boundary conditions on $\Gamma_f$, $\Gamma_c$, and $\Gamma_g$, and solves the resulting system with a finite-element method. Three control strategies are considered: an active approach that prescribes a boundary pressure $u$ on $\Gamma_c$ and two passive devices—a floating membrane with parameters $T$ and $m$, and a thin plate with flexural stiffness $B$—whose properties are optimized to minimize body motion. The method yields first-order optimality conditions (adjoint systems) and demonstrates significant reductions in oscillations for a floating sphere and for a semi-submersible wind turbine, indicating practical potential for wave-load mitigation and design guidance for future cloaking or energy-harvesting applications.

Abstract

In this paper, we propose a novel approach for controlling surface water waves and their interaction with floating bodies. We consider a floating target rigid body surrounded by a control region where we design three control strategies of increasing complexity: an active strategy based on controlling the pressure at the air-water interface and two passive strategies where an additional controlled floating device is designed. We model such device both as a membrane and as a thin plate and study the effect of this modelling choice on the performance of the overall controlled system. We frame this problem as an optimal control problem where the underlying state dynamics is represented by a system of coupled partial differential equations describing the interaction between the surface water waves and the floating target body in the frequency domain. An additional intermediate coupling is then added when considering the control floating device. The optimal control problem then aims at minimizing a cost functional which weights the unwanted motions of the floating body. A system of first-order necessary optimality conditions is derived and numerically solved using the finite element method. Numerical simulations then show the efficacy of this method in reducing hydrodynamic loads on floating objects.

Figures (5)

• Figure 1: top (a) and side (b) views of the computational domain. The floating body is a sphere with mass density half than water, the green region represents the control surface.
• Figure 2: Amplitude (a) and phase (b) of the optimal pressure on $\Gamma_c$; the improvement of the oscillations $\{\mathbf{X}\}$ is summarized in table \ref{['tab:oscillation reduction']}.
• Figure 3: Membrane properties in terms of optimal controls u (a) and v (b); (c) objective over iteration. The improvement of the oscillations $\{\mathbf{X}\}$ is summarized in table \ref{['tab:oscillation reduction']}.
• Figure 4: Plate properties in terms of optimal controls u (a) and v (b); (c) objective over iteration. The improvement of the oscillations $\{\mathbf{X}\}$ is summarized in table \ref{['tab:oscillation reduction']}.
• Figure 5: (a) The shape of the floating wind turbine VolturnUS-S considered in the study. (b) and (c) show respectively amplitude and phase of the pressure applied; (d) and (e) surface mass and tension of the membrane; (f) costs over iterations for the cases of membrane and plate control; (g) and (h) surface mass and flexural stiffness of the plate.