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Code-to-Code Nonlinear Hydrodynamic

Modelling Veriﬁcation for Wave Energy

Converters: WEC-Sim vs. NLFK4ALL

Giuseppe Giorgi, Markel Penalba, and Rui P.F. Gomes

Abstract—In the wave energy conversion ﬁeld, simu-

lation tools are crucial for effective converter and con-

troller design, but are often prone to become very case-

speciﬁc, in both structure and parameter selection. This is

due to majorly different working principles and diverse

importance of nonlinear effects, at times requiring ad-

hoc modelling approaches. To tackle this challenge, WEC-

Sim (Wave Energy Converter SIMulator) was born from

the National Renewable Energy Laboratory (NREL) and

Sandia National Laboratories, providing a unique simula-

tion platform for all WECs. Nonlinearities related to time-

varying wetted surface, especially important in ﬂoating

WECs, are included in WEC-Sim through a mesh-based

computation of nonlinear Froude-Krylov forces. Virtually

arbitrary geometries can be considered, thanks to the

discretized representation of wetted surfaces, at the price

of a signiﬁcant increase in computational burden. This

paper considers a time-effective alternative, implemented

in the open-source toolbox called NLFK4ALL, applicable to

the popular and wide family of axisymmetric ﬂoaters. The

Spar-buoy ﬂoating oscillating water column device is con-

sidered, particularly challenging due to a submerged vol-

ume composed of several different sections. The accuracy

of WEC-Sim and NLFK4ALL is veriﬁed by a preliminary

cross-comparison, using independent methods to compute

virtually same effects. Fixed-body numerical experiments

are used to quantify nonlinearities and compare not only

the accuracy, but also the computation burden. Results

show that both methods provide almost identical results,

although WEC-Sim doubles computational requirements.

Index Terms—Wave Energy Converters Nonlinear

Froude-Krylov force, Spar-buoy oscillating OWC, WEC-

Sim, NLFK4ALL.

I. INTRODUCTION

SIMULATION tools are essential for effective design

and development of power conversion systems.

However, the deﬁnition of a reliable and representa-

tive mathematical model for wave energy converters

(WECs) is especially challenging. In fact, there is a wide

variety of devices, based on substantially different

working principles and mounting various components,

often making the model quite case-speciﬁc. Moreover,

since an appropriate representation of nonlinear effects

This research was funded by the European Research Council (ERC)

under the European Unions Horizon 2020 research and innovation

programme under Grant Agreement No. 832140

Marine Offshore Renewable Energy Lab (MOREnergy Lab),

DIMEAS, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129,

Turin, Italy (e-mail: giuseppe.giorgi@polito.it).

Department of Mechanical and Manufacturing, Mondragon Uni-

versity, Loramendi 4 Apdo. 23, 20500, Arrasate, Spain (e-mail: mpe-

nalba@mondragon.edu).

IDMEC, Instituto Superior T´

ecnico, Universidade de Lis-

boa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal (e-mail:

ruigomes@tecnico.ulisboa.pt).

is likely to be required for achieving acceptably accu-

rate results, the complexity of mathematical models

rapidly escalates, at the expense of its transparency

and ﬂexibility [1]. In order to address such issues,

the National Renewable Energy Laboratory (NREL)

and Sandia National Laboratories developed the open

source high-level software in 2014 WEC-Sim (Wave

Energy Converter SIMulator) [2]. WEC-Sim is gaining

increasing popularity in the wave energy community,

both academic and industrial, since it provides a frame-

work for simulating virtually any WEC [3], including

different subsystems of the wave energy conversion

chain: from mooring systems [4] to power take-off

[5], including various nonlinear hydrodynamic effects

[6]. This paper focuses especially on the computation

of the nonlinear Froude-Krylov (NLFK) force, which

is one of the major and most impactful nonlinearity

in ﬂoating WECs [7]–[9], with effects on parametric

resonance [10], [11], mooring design [12] and model-

based controller design [13], [14]. NLFK forces are

due to the undisturbed pressure ﬁeld acting on the

instantaneous wetted surface of the ﬂoater, whereas

the linear approximation considers the mean wetted

surface as constant. Geometries of arbitrary complexity

require a mesh-based discretization of the wetted sur-

face, that can be handled in two alternative ways: either

a re-meshing routine is used, re-computing the mesh

discretization at each time step according to the instan-

taneous position of the ﬂoater with respect to the free

surface elevation; or a constant but very ﬁne mesh is

implemented, with a boolean decision on which panel

is either fully wet or fully dry [15]. Both approaches

may slow down the simulation signiﬁcantly, depend-

ing on the size of the spatial discretization. WEC-Sim,

with the purpose of being ﬂexible on the ﬂoater shape,

implements a mesh-based computation of NLFK forces

[16]. However, for axisymmetric geometries there is

an equivalent but computationally more efﬁcient ap-

proach based on an analytical formulation of the in-

stantaneous wetted surface, hence getting rid of the

computational bottleneck of the meshed wetted surface

[17]. An additional frequency-domain (FD) approach

was also suggested in [18], where nonlinear effects are

included using a projection of the dynamical equations

onto a basis of trigonometric functions.

Note that the popular family of point absorber WECs

is mainly composed of axisymmetric devices, mak-

ing such an assumption not too restrictive. Therefore,

the analytical approach, available in the open source

Matlab demonstration toolbox NLFK4ALL [19] and

11971-

Proceedings of the 14th European Wave and Tidal Energy Conference 5-9th Sept 2021, Plymouth, UK

ISSN 2309-1983 Copyright © European Wave and Tidal Energy Conference 2021

2

validated with experimental wave tank data [20], is

implemented in this paper, and compared with results

obtained with the nonlinear hydrodynamics features

of WEC-Sim. The existing Sparbuoy ﬂoating oscillating

water column (OWC) is used as a case study [21].

The reminder of the paper is organized as follows:

Sect. II introduces the device considered in this paper,

used to quantify the comparison between the two pro-

posed approaches for the NLFK calculation, WEC-Sim

and NLFK4ALL, respectively discussed in Sect. II-B

and Sect. II-C. Section III shows results, while some

ﬁnal remarks and conclusions are provided in Sect. IV.

II. THE SPAR-BUO Y OWC DE VI CE

The Spar-buoy OWC is a ﬂoating oscillating water

column, composed of a hollow ﬂoater opened at the

bottom, hence containing a water column within its

structure. As waves arrive, a relative motion between

the ﬂoater and the water column is generated, driving

the air pocket above the water column through a tur-

bine, extracting useful energy. Figure 1 shows a picture

of the prototype [22], tested at 1:16 scale. This device,

being of the spar buoy type, is prone to experience

the highly nonlinear phenomenon of parametric reso-

nance: when the excitation frequency is twice that of

the natural frequency of the pitching degree of freedom

(DoF), the system becomes parametrically unstable and

diverts part of the incoming energy to the rolling

DoF [23]. Such a behaviour, potentially detrimental for

power extraction efﬁciency and hindering survivability,

is due to time-variations of the wetted surface, so that

it can be successfully articulated by NLFK models. For

this reason, the Spar-buoy is an appropriate choice for

the case study presented in this paper.

Fig. 1. 1:16 scale prototype of the Spar-buoy OWC is a ﬂoating

oscillating water column [22].

A. Nonlinear Froude-Krylov force

Froude-Krylov forces are deﬁned as the integral of

the undisturbed pressure ﬁeld (p) over the wetted

surface of the ﬂoater. In the linear approximation, it

is assumed that the relative motion between the body

and the free surface is small, so that FK forces are com-

puted with respect to the mean wetted surface. On the

contrary, nonlinear FK force calculations are performed

with respect to the actual instantaneous wetted surface

(Sw(t)):

fF K =fg+ZZ

Sw(t)

pndS, (1a)

τF K =rg×fg+ZZ

Sw(t)

pr×ndS, (1b)

where fF K are nonlinear forces, τF K are torques, fg

is the gravity force, nis the unity vector normal to

the surface, ris the generic position vector, and rgis

the position vector of the centre of gravity. The undis-

turbed incident pressure ﬁeld of an uni-directional

regular wave travelling in the x-direction is deﬁned

as:

p(x, z, t) = −ρg z+acos (ωt −kx)cosh (k(z′+h))

cosh(kh),

(2)

where a,ω, and kare the wave amplitude, frequency,

and wavenumber, respectively, ρthe water density, g

the acceleration of gravity, hthe water depth, and z′the

vertical coordinate zmodiﬁed according to Wheeler’s

stretching [24]:

z′=hz+h

η+h−h(3)

where ηis the free surface elevation.

B. The WEC-Sim mathematical model

The WEC-Sim mathematical model is based on

a Simulink implementation [2], with speciﬁcally de-

signed blocks for typical components of wave energy

converters. Figure 2 shows a snapshot of the block

diagram for the Spar-buoy, with two hydrodynamic

bodies, respectively the ﬂoater and the OWC, between

which there is a translational power take-off (PTO)

unit converting the relative motion into electricity. The

system is then referenced to the global-frame through

a constraint block, which can be set to free motion (6-

DoF), planar motion (3-DoF), vertical motion (1-DoF),

or no motion (ﬁxed).

The PTO system of an OWC is an air turbine, which

converts the bidirectional air ﬂow induced by the OWC

motion inside the ﬂoater. The pressure drop across the

turbine can be simulated at model scale using an oriﬁce

plate [21]:

FP T O =8ρaA3

a

π2C2

dd4

0

˙zr|˙zr|(4)

where ρais the air density, Aais the cross-sectional area

of the air chamber, Cdis the discharge coefﬁcient (Cd=

21971-

FIRST-AUTHOR-SURNAME et al.: PAPER-TITLE 3

Fig. 2. Simulink block diagram in WEC-Sim.

0.6466 for the oriﬁce plate used in the experiments), d0

is the diameter of the oriﬁce plate, and ˙zris the relative

velocity between the water column and the buoy. Note

that FP T O acts on both the buoy and the water column,

but with opposite sign.

The NLFK calculations in WEC-Sim (NLFKs) are

performed based on a panel mesh, summing up all

the contribution of each mesh panel. The mesh used

in this study, shown in Fig. 3, has been used for both

the NLFK calculation and computation of FD hydro-

dynamic coefﬁcients via the boundary element method

(BEM) software WAMIT [25]. The computational time

and accuracy of the NLFKsapproach are highly de-

pendent on the number and size of panels, respectively.

The mesh considered in this study is composed of 5376

panels.

C. The NLFK4ALL mathematical model

While solving the integrals in (1) requires, in general,

mesh-based approaches, a computationally efﬁcient ap-

proach is available for axisymmetric bodies, exploiting

cylindrical coordinates (̺, ϑ)to achieve an analytical

representation of the wetted surface:

ˆx(̺, ϑ) = f(̺) cos ϑ

ˆy(̺, ϑ) = f(̺) sin ϑ

ˆz(̺, ϑ) = ̺

, ϑ ∈[−π, π)∧̺∈[̺1, ̺2](5)

where f(̺)is a generic function of the vertical coor-

dinate ̺, describing the proﬁle of revolution of the

axisymmetric body. Since it is convenient to deﬁne the

FK integrals in the body-ﬁxed frame of reference, the

pressure ﬁeld must be mapped from the global to the

body-ﬁxed frame. After the required adjustments, the

integral in (1a), for example, becomes:

fF K =RT

Θfg+ZZ

Sw(t)

P(ˆx, ˆy, ˆz)ndS =

=RT

Θfg+

π

Z

−π

̺2

Z

̺1

P(̺, ϑ) (e̺×eϑ)d̺ dϑ,

(6)

Fig. 3. Mesh discretization of the Spar-buoy device, used for both

the BEM code for the linear hydrodynamic curves, and for the NLFK

computation in WEC-Sim.

where RΘis the rotation matrix from body- to world-

frame, e̺and eϑare the unit vector along ̺and ϑ,

respectively. Note that when internal patches (facing

the water column) are considered, the sign of the nor-

mal vector in (6) should be reversed. The integral in (6)

is solved numerically, using a 2D-quadrature scheme

for trapezoidal integration [26]. An open source Matlab

demonstration toolbox for deﬁnition and computation

of nonlinear FK forces for axisymmetric ﬂoaters is

available at [19].

Fig. 4 shows, for an arbitrary displacement of the

buoy and wave ﬁeld, the conﬁguration in both the

world-frame (on the left) and the body-frame (on the

right), and the corresponding mapping of the free sur-

face elevation. Note that the mesh-like representation

in Fig. 4 has a mere visualization purpose, since the

surfaces are described analytically and no mesh is

needed.

Note that the considered geometry is rather complex,

with several changes of cross-sectional area. Twelve

different patches can be identiﬁed, namely 5 cylindrical

sections, 4 conical sections, 2 quarters of torus, and

a disk for the inner piston. This increases the over-

all computational time, since each patch requires an

independent formulation, hence raising the number

of integrals to be computed. However, it is worth

remarking that equally considering all patches is likely

to be unnecessary, since some patches are relatively

31971-

4

Fig. 4. Example of displaced buoy, in the world frame (on the left) and body-ﬁxed frame (on the right), with corresponding mapped wave

ﬁeld and its intersection with the buoy.

small and/or so deep that the dynamic pressure has

already signiﬁcantly decayed. Nevertheless, since the

purpose of this paper is comparison rather than com-

putational time minimization, no simplifying assump-

tion has been investigated.

III. RESULTS

While WEC-Sim naturally provides a framework for

time-simulations, the analytical-based NLFK computa-

tion algorithm (NLFKa) has been included into an in-

house code. The correctness of implementation is ver-

iﬁed via comparison of time-domain simulations (TD)

for small (linear) waves, independently computed with

WEC-Sim and the in-house code, compared with linear

frequency domain (FD) simulations. Figure 5 shows

the resulting response amplitude operator (RAO) in

surge, heave, and pitch, for a representative set of

wave periods (Tw) at a wave height (Hw) of 0.25m. A

perfect overlap is obtained between all models, giving

conﬁdence on the ﬂawless implementation of the time-

domain models under linear conditions.

Successively, in order to ease the comparison be-

tween NLFKaand NLFKs, results are generated for a

numerical excitation force experiment, i.e. by ﬁxing the

ﬂoater at the rest position, and sending monochromatic

waves, at different Twand Hw; the total excitation force

is measured, composed of NLFK and linear diffraction

components. While the proportionality between linear

forces and Hwis constant for all the the range of Hw,

nonlinear forces may show differences as Hwincreases.

A. Computational time

Quantifying the computational time has important

consequences on the applicability of either of the two

models for speciﬁc simulation purposes, ranging from

power production, response and load assessment, to

control and optimization. However, since both accu-

racy and computational effort depend on different

setup parameters for each model, a comprehensive

sensitivity analysis should be performed in order to

express a deﬁnitive, robust and solid statement on the

comparative performance.

Calculations are performed on a single core of a

standard laptop, with processor Intel(R) Core (TM)

i7-7500U CPU @ 2.70GHz and 8GB RAM. Since the

overall computational effort depends on the time-

advancing scheme and time step size (δt), the compu-

tational time is hereafter referred to a single time step

execution of the numerical excitation force experiment

(tCP U ). WEC-Sim results in a required 6 ms per δt,

which seems satisfactory. However, the excitation force

experiment is favourable to mesh-based approaches,

since the mesh is not displaced at each time step,

which may increase the computational time. Moreover,

with the current mesh discretization, the accuracy of

NLFKsis questionable in surge and pitch for longer

waves, as further discussed in Sect. III-B; therefore, a

more reﬁned mesh should be tested, implying higher

computational burden. Nevertheless, it can be assumed

with conﬁdence that the computational time for NLFKs

remains, at worst, below the tens of milliseconds per δt,

which is faster than previous mesh-based approaches

suggested in the literature, e.g. [15].

Conversely to the NLFKsapproach, the considered

case study of the Spar-buoy is particularly disadvan-

tageous to the NLFKaapproach. In fact, the compu-

tational time is linearly proportional to the number

of geometrical sections considered; therefore, NLFK

force computation for the Spar-buoy, composed of 12

patches, is likely to be about 3 times longer than the

time required for a simpler ﬂoater composed of just 4

patches (for example 2 cylinders connected via a trun-

cated cone). Nevertheless, the resulting computational

time is of about 3 ms per δt.

B. NLFK results

Figure 6 shows a representative time trace of the

total excitation force (Froude-Krylov and diffraction),

using the linear model (LFK) as a reference benchmark,

and the nonlinear models for comparison. Since they

41971-

FIRST-AUTHOR-SURNAME et al.: PAPER-TITLE 5

5 10 15

0

0.2

0.4

0.6 WAMIT (FD)

IN-HOUSE (TD)

WECSIM (TD)

5 10 15

0

1

2

3

5 10 15

0

0.5

1

1.5

2

2.5

Fig. 5. Response amplitude operators (RAO) in surge (left), heave (middle), and pitch (right), using linear frequency domain (FD) data, and

time-domain (TD) models with a small wave (height of 0.25m).

0 0.5 1

-5

0

5106

LFK

NLFKs

NLFKa

0 0.5 1

-1.5

-1

-0.5

0

0.5

1

1.5 106

0 0.5 1

-1.5

-1

-0.5

0

0.5

1

1.5 108

Fig. 6. Time trace over a single wave period of the total excitation force at Twof 7 s and Hwof 4 m.

are different approaches to compute the same quantity,

they should overlap. In Fig. 6 a short wave Twof 7 s

is considered, with Hwof 4 m, hence very steep and

nonlinear. It can be remarked that NLFKsand NLFKa

agree well, particularly in heave. Differences between

linear and nonlinear forces are negligible in surge and

pitch, while are remarkable in heave.

Figure 7 shows a long wave, Twof 14 s, at the same

height of 4 m. On the one hand, the perfect match

between NLFK approaches is preserved in heave; on

the other hand, NLFKspresents irregular oscillation

in surge and pitch due to numerical inaccuracies,

while NLFKaremains smooth. This can be ascribed

to a static mesh-discretization of the wetted surface,

not sufﬁciently reﬁned to describe variations of the

nonlinear forces smoothly. In fact, although very wob-

bly, the moving average of NLFKsfollows the NLFKa

trend closely. The numerical inaccuracies of NLFKs

are not present in heave because the portion of the

buoy intersecting the free surface elevation, i.e. where

the extension of wetted surface is varying in time, is

cylindrical, while all other sections are either fully-

submerged or fully-dry: the normals of a cylinder with

vertical axis are horizontal, hence perpendicular to the

heave direction, so they do no contribute to the build-

up of the total heave force; it follows that the total

vertical force depends on portions of the buoy that

are always fully-submerged, hence not affected by im-

proper deﬁnition of a mesh discretization. Conversely,

surge and pitch forces depend also on the top cylinder,

whose mesh quality inﬂuences the smoothness and

accuracy of the results.

Finally, a synthetic but comprehensive view on the

differences between linear and nonlinear excitation

force coefﬁcient is provided in Fig. 8, where ˆ

Fex is

deﬁned as the ratio between the force and the wave

height: in a linear case, all curves should overlap,

while differences may arise as nonlinearities increase.

In the plot, both the amplitude and the mean are

presented, where the amplitude is deﬁned as the semi-

difference between peak and trough. In the linear case,

the amplitude is the same as the peak value, while

the mean is zero. Figure 8 shows that, as Hwincreases

(colour code), the mean diverges from zero, especially

in heave. Conversely, differences in the amplitude seem

negligible.

51971-

6

0 0.5 1

-3

-2

-1

0

1

2

3106

LFK

NLFKs

NLFKa

0 0.5 1

-3

-2

-1

0

1

2

3106

0 0.5 1

-4

-2

0

2

4107

Fig. 7. Time trace over a single wave period of the total excitation force at Twof 14 s and Hwof 4 m.

5 10 15

-0.5

0

0.5

1

1.5

2

2.5 106

0.5

1

1.5

2

2.5

3

3.5

4

5 10 15

-2

0

2

4

6

8

10

12

105

0.5

1

1.5

2

2.5

3

3.5

4

5 10 15

-1

0

1

2

3

4

5

6107

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 8. Amplitude and mean values of the nonlinear excitation force at different wave heights. Dashed line corresponds to the linear case.

IV. CON CL US IO NS

This paper implements two alternative methods for

the computation of nonlinear Froude-Krylov forces,

one using a mesh-based approach in the popular WEC-

Sim software, the other using a mesh-less analytical

formulation of the wetted surface and numerical inte-

gration. As a case study, the Spar-buoy ﬂoating oscillat-

ing water column is used. An overall good agreement

is found between the two alternative methods, cross-

checking the underlying mathematical frameworks and

implementations. However, the mesh-based approach

shows inaccuracies in surge and pitch force computa-

tion, likely due to an improper selection of the mesh

discretization. The computational time, although de-

pending on various parameters and simulation setups,

appear to be in favour of the analytical formulation

(twice as fast as the mesh-based approach), despite of a

complex shape deﬁnition, also providing accurate and

smooth results.

In the future, this trends should be evaluated with

a more thorough sensitivity analysis for the mesh and

incorporating the dynamics of ﬂoating buoys with, at

least, 3 degrees-of-freedom.

ACKNOWLEDGEMENT

This research has received funding from the Eu-

ropean Research Council (ERC) under the European

Union’s Horizon 2020 research and innovation pro-

gram under Grant No. 832140.

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