NLFK4ALL: An Open-Source Demostration Toolbox for Computationally Efficient Nonlinear Froude-Krylov Force Calculations

Abstract

Accurate modelling of wave-structure interaction is essential for a successful and reliable design of offshore structures and their ancillary systems. However, the fidelity of mathematical models is challenged when nonlinearities become significant, i.e. when the floater operates in severe sea states and/or it shows large dynamic responses compared to the incoming wave. This is normally the case for wave energy converters (WECs) because large motions are desirable for better power extraction, while conventional offshore structures are usually designed to prevent large responses and ensure stability. Fast computation is a further mandatory requirement for mathematical models that are used for design purposes. This is particularly true for wave energy applications, since extensive parametric studies are needed to minimize the cost of energy, i.e. increase power extraction capabilities while limiting capital and operational expenditures. Furthermore, model-based control strategies, essential for substantial increase of the WEC performance, require representative mathematical models. The requirements of accuracy and low computational time are usually contrasting, so an appropriate compromise should be defined. This paper proposes a nonlinear Froude-Krylov (NLFK) force model for axisymmetric floaters. The symmetry of the geometry (common for WECs) is exploited to achieve a low computational time (about real-time computation). Furthermore, since NLFK forces are the main nonlinearities for such devices, the obtained accuracy is higher than linear models. An open-source Matlab demonstration toolbox is introduced, called NLFK4ALL (doi: 10.5281/zenodo.3544848), which provides a ready-to-use implementation of the NLFK approach in six degrees of freedom. Accuracy and computational aspects of the method are here discussed, such as the complexity of the analytical description of the intersection between free surface elevation and the floater, and the impact of tolerances on the convergence of the numerical integration.

Modelling of Parametric Resonance for Heaving Buoys with Position-Varying Waterplane Area

“…First, to investigate the influence of the novel parametric excitation term (introduced in Section 2.2.2), the results of the analytical model are compared against the results from a model that only extends the conventional linear model with a nonlinear hydrostatic restoring force term. Next, the performance of the analytical model is compared and verified against a high-fidelity benchmark produced herein by the NLFK model implemented using an open source toolbox [100]. The WEC geometry is introduced in Section 3.1, and the derivation of the analytical model for this particular geometry is presented in Section 3.2.…”

“…To calculate the Froude-Krylov force F FK z in the time domain simulations, the opensource NLFK4ALL MATLAB toolbox developed by Giorgi [100,103] was utilised. The Froude-Krylov force F FK z (z, t, ω) was calculated every time step using NLFK4ALL toolbox, and the equation of motion was integrated using the fourth order Runge-Kutta method.…”

Optimal control of wave energy systems considering nonlinear Froude–Krylov effects: control-oriented modelling and moment-based control

“…As discussed in Sect. 1, while the approach proposed in [35,36,39], implemented in the open-source toolbox Nlfk4all [32,34], is able to achieve real-time (numerical) computation of both static and dynamic FK effects, the methodology proposed assumes a 'frequency-by-frequency' decomposition of the associated pressure field, producing a mathematical description which is, in general, not compatible with state-of-the-art energy-maximising control design/synthesis procedures, whose formulation virtually always requires a closed-form of (at least) the input-output description of the WEC dynamics 2 .…”

“…After selecting a suitable multisine signal f id for the generation of η, and a sufficiently large initial value k 0 , the matrices Ξ k and Υ k are constructed iteratively, using information on the (supplied) input η = f id , and the numerically computed device motion z, and total static force y st . Note that the latter two variables can be readily computed via any numerical nonlinear FK solver, such as the Nlfk4all toolbox [32,34] (which is explicitly considered in this paper within the case study presented in Sect. 5).…”

“…Though highly efficient numerical schemes have been presented in the literature of hydrodynamic WEC modelling, the assumptions adopted are often restrictive, and produce mathematical representations which are not entirely suitable for control design/synthesis. For instance, a particularly numerically efficient approach to the computation of nonlinear FK effects, is that proposed in [35,36,39], implemented via the open-source toolbox Nlfk4all [32,34]. While such an approach is effectively able to achieve real-time computation of FK forces, the methodology proposed assumes a 'frequency-by-frequency' decomposition of the associated pressure field, an assumption which renders a mathematical description generally incompati-ble with model-based control design and synthesis procedures, which require a closed-form of (at least) the input-output description of the WEC dynamics [20].…”