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Vol. 27（2022），127 ～ 137

Comparing BEM solvers for analyzing wave energy
converters
Vaibhav RAGHAVAN*1 and George LAVIDAS*1*1 Department of Hydraulic Engineering, Delft University of Technology
Mekelweg 5, Delft, 2600AA, Netherlands

Abstract
Wave energy converters (WECs) have the potential to significantly contribute to the energy transition, but
this strongly depends on their interactions with the resource. For accurate calculations of the power derived from
WECs, precise modelling of the wave-structure interaction is essential. The boundary element method (BEM)
based on the linear potential flow theory has yielded accurate results at low computational costs when compared tocomplex Computational Fluid Dynamics methods. Hydrodynamic Analysis of Marine Structures (HAMS) and
Capytaine are recently developed open-source BEM frequency domain solvers, originally created for large marine
structures. These solvers have since been utilized for studying wave energy converters albeit for very few converter
geometries. Owing to the implementation of parallelization in both HAMS and Capytaine, both these solvers could becapable for significantly lower computational costs as compared to the traditional BEM solvers such as Nemoh. This
research aims to compare hydrodynamic coefficients and computational costs in Nemoh, HAMS and Capytaine forvarious WEC geometries.
Key words : BEM, Nemoh, HAMS, Capytaine, Potential flow, Hydrodynamics

1. Introduction
Given its high energy density, ocean wave energy can provide at least twice as much electricity as globally producedin the world (Balitsky, 2019). Apart from the vastness of the resource, waves are more predictable, and available
throughout the year when compared with other forms of renewable energy. For the rapid development of the wave energy
industry, it is essential to develop efficient numerical tools which would offer the advantage of employing significantly
less expenses as compared to performing physical tests on scaled models using wave tank testing.
BEM is perhaps the most common numerical method for studying wave-structure interaction in the field of wave
energy with those employing the linear potential flow theory being the most popular. There are well establishedcommercial codes such as WAMIT (Lee, 1995) and ANSYS AQWA (2012) as well as open-source codes such as Nemoh(Babarit & Delhommeau 2015), Aquadyn and Aquaplus (Delhommeau, 1993) with WAMIT and Nemoh being the mostpopular. Capytaine (Ancellin & Dias, 2019) and HAMS (Liu, 2019) are two recently developed open-source solvers.
Open-source solvers for modelling wave-structure interaction could provide valuable options to meet the numericalchallenges within the field of ocean engineering, particularly the possibility of low computational effort with good
accuracy.
This research makes some comparisons between the open-source solvers Nemoh, HAMS and Capytaine for two
different types of WECs: a semi-immersed cylindrical Point Absorber (PA) and a semi-immersed Oscillating Surge Wave
Energy Converter (OSWEC). The compared parameters include hydrodynamic coefficients, exciting forces andcomputational efficiency.
Although the Cylindrical PA has been widely studied with BEM solvers as a simple benchmark case(Penalba, et al.,
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Vaibhav RAGHAVAN and George LAVIDAS

2017, Liu, et al., 2018, Liu, 2019, Sheng, et al. 2022), it has been chosen here to show an example of the capability of
HAMS to efficiently remove the so-called ‘irregular’ frequencies, which is also possible on the commercial solvers such
as WAMIT or Aquadyn. Additionally, the case of OSWEC is demonstrated as this has only been analyzed previouslywith Nemoh and HAMS.

2. BEM solvers
Nemoh is a Matlab/Fortran based BEM solver first released in 2014 and was originally developed by A. Babarit and
G. Delhommeau. It is one of the most popular open-source BEM solvers for wave-structure interaction calculations for
single body and multi-body interactions of floating rigid structures. HAMS was developed and released in 2019 by Y.
Liu as an open-source Fortran based BEM solver for large floating offshore structures. Over the years, it has gained
popularity for analysis of single floating structures, but still lacks the capability of solving multi-body wave-structureinteraction problems. Capytaine, an open-source Python based BEM solver, was first released in 2019 and is a modified
version of Nemoh developed my Matthieu Ancellin. The core routines of Nemoh, written in Fortran, were updated intoa modern coding style and the naïve solver linear solver in Nemoh has been replaced by state-of-the-art numpy and scipy
libraries. While Nemoh does not allow for parallelization of the calculations, both HAMS and Capytaine are capable ofparallelized calculations.
Looking deeper into the algorithms used within these solvers, all of them employ panel methods based on the linear
potential flow theory, which uses the Green’s function to solve the diffraction/radiation problem of the source distribution
on the surface of the body. Based on this, Boundary Integral Equations (BIE) are established using specific Green’s
function that satisfy the free surface condition and wave radiation condition at infinity. Nemoh and Capytaine employ aninterpolation based on a look-up table to solve for the wave part of the Green’s function (Sheng, et al. 2022) while HAMS
employs algorithms based on Newman’s approximation methods for solving the free surface Green’s function, thus
running these approximations for each individual run.
While Nemoh has been shown to provide good solutions for many problems in wave-structure interaction, it is stillsusceptible to ‘irregular frequencies’, which are purely numerical and arise from ill-conditioning in the BIE problems,
sometimes resulting in large underestimation or overestimation of hydrodynamic parameters at certain frequencies (Kelly,
et al., 2021). These are shown to coincide with the eigen frequencies of the hypothetical sloshing modes, which are
obtained from the internal Dirichlet problem (Marburg, et al. 2005) and numerically caused by the interaction of thewater-plane section of the floating bodies intersecting with the free-water surface (Liu, 2019). HAMS is capable oftackling this by discretizing the free surface within the body and solving an extended BIE problem (Lee, et al., 1996, Liu,2019).

3. Hydrodynamic analysis of WEC devices using BEM
In this section, the geometries of the examined two types of WECs are initially introduced. This is followed by thecomparison of the hydrodynamic coefficients, exciting forces and RAOs.

3.1 Semi-immersed cylindrical point absorber (PA)A cylindrical PA of height 3.0 m and radius of 3.0 m radius is considered. Its draft is 1.5 m, thus, it is modelled as atruncated cylinder. The center of gravity coincides with the origin of the global coordinate systems oxyz located at the
Mean Water Level (MWL), while the center of buoyancy is at 0.75 m below the MWL. Deep water condition isconsidered for the PA.
In order to remove the irregular frequencies in HAMS, an additional water-plane mesh needs to be provided as input
when running the simulation. Hence, two cases are considered here: Case (a) – Only hull, where, just the hull of thestructure is modelled and Case (b) – Hull and water plane, where both the hull and the interior water-plane are modelled
to enable the removal of irregular frequencies. The interior water-plane is modeled with an additional mesh applied at
the level of the MSL within the circumference of the cylindrical PA. The meshes for the two cases are shown in Figure 1
and Figure 2. The mesh in Figure 1 is used for the hull in Nemoh, HAMS and Capytaine. The mesh in Figure 2 is used
Date of Manuscript Acceptance 20xx.x.xx
E-mail of corresponding author: v.raghavan@tudelft.nl*

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129

for removal of irregular frequencies for HAMS.

Fig. 1

Mesh for the hull (purple) of the cylindrical PA asmodelled in Nemoh, HAMS and Capytaine

The hydrodynamic coefficients and exciting forces for Case (a) are shown in Figures 3, 4 and Figure 5 respectively. The
corresponding results for Case (b) are shown in Figures 6, 7 and 8 respectively. All 6 Degrees Of Freedom (DOFs) wereconsidered here. For brevity, results are only shown for heave degree of freedom. Exciting forces, F3, are normalized by
ρgAπR2, while added mass, A33, and radiation damping, B33, coefficients by ρR3 and ρωR3 respectively, where ρ=1025
kg/m3 is the water density, g the acceleration due to gravity, ω the frequency, and A is the unit wave amplitude.

Fig. 2Mesh for the hull (purple) and water plane (cyan) forthe cylindrical PA as modelled in HAMS. The global
ox (red) oy (green) oz (blue) axes are shown in (a).
The same are used in (b).

As seen in Figures 3, 4 and 5, the results of Nemoh, HAMS and Capytaine are generally close. The hydrodynamic
coefficients and exciting forces of Nemoh and Capytaine are very close, which reinforces the similarity in the backend
algorithms used. Slight deviation is observed for frequencies > 3 rad/s for the A 33 when comparing Nemoh/Capytaine

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Vaibhav RAGHAVAN and George LAVIDAS

with HAMS. Additionally, in Case (a) when not solving for irregular frequencies, a small jump is observed in HAMS
while a larger jump is observed for Nemoh/Capytaine at the ‘irregular frequency’ (close to 3 rad/s) for A33 and B33 forthe same case.

Fig. 3Fig. 4

Heave added mass of the cylindrical PA where only thehull mesh is used (irregular frequency highlighted
within the red circle)

Heave radiation damping of the cylindrical PA where
only the hull mesh is used (irregular frequency
highlighted within the red circle)

Comparing BEM solvers for analyzing wave energy converters

Fig. 5

Heave exciting force of the cylindrical PA where only
the hull mesh is used (irregular frequency highlighted
within the red circle)

Fig. 6

Heave adding mass of the cylindrical PA where the hull
mesh and water plane mesh are used for HAMS
(irregular frequency highlighted within the red circle)

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Vaibhav RAGHAVAN and George LAVIDAS

Fig. 7

Heave radiation damping of the cylindrical PA where
only the hull mesh and water plane mesh are used for
HAMS (irregular frequency highlighted within the redcircle)Fig. 8Heave exciting force of the cylindrical PA where only thehull mesh and water plane mesh are used for HAMS
(irregular frequency highlighted within the red circle)

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133

As seen in Figures 6, 7 and 8, the irregular frequency is suppressed in HAMS when comparing it with Nemoh and
Capytaine. In this case, a hull and water mesh is given as input for the simulation in HAMS. HAMS employs an extended
boundary integral equation which assumes that the potentials in the interior of the water plane are zero. This equation ishence used as an additional equation to the input boundary integral equations which are solved on the hull surface
simultaneously (Liu 2019).

3.2 Semi-immersed oscillating surge WEC (OSWEC)
The dimensions of the OSWEC were taken from Penalba et al. (2017) and correspond to the Oyster device. The
height of the device, h, is 12.0 m, its length is 20.0 m and its width is 2.0 m. The draft is considered equal to 10.0 m. The
OSWEC is bottom hinged. In order to emulate this condition in the analysis, the center of rotation was taken at the bottom (z = -10.0 m). The shallow water depth of 10.5 m was considered, to keep the bottom of the OSWEC close to thesea bottom. The mesh (hull) of the OSWEC model in HAMS is shown in Figure 9.
The hydrodynamic coefficients and exciting forces for the OSWEC are shown in Figures 10, 11 and 12 respectively.
Only pitching degree of freedom was considered here. Exciting forces, F5, are normalized by ρgAh3, while added mass,
A55, and radiation damping, B55, coefficients by ρh5 and ρωh5 respectively. From Figures 10, 11 and 12, it can be observedthat the results for all solvers are close. HAMS slightly differs in the pitch added mass, radiation damping and exciting
force when compared to Nemoh and Capytaine close to 0.75 rad/s.

Fig. 9Hull mesh of OSWEC

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Vaibhav RAGHAVAN and George LAVIDAS

Fig. 10

Pitch added mass for OSWEC

Fig. 10

Pitch radiation damping for OSWEC

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Comparing BEM solvers for analyzing wave energy converters

Fig. 10

Pitch exciting force for OSWEC

4. Comparison of Computational TimeFor making a fair comparison of the three solvers, the simulations were carried out in the same device. The device isa 64-bit laptop with 16 GB RAM, 4 cores and Intel i7-1185G7 processor of 3.00 GHz CPU.
The comparison of the computation effort is shown in Table 1. 80 frequencies between 0.2 and 4.2 rad/s wereconsidered for the analysis.
Table 1Comparison of computation time of the applied BEM solvers

WEC

SOLVERTHREADS

PA(HULL)

Nemoh
Capytaine
Capytaine
HAMS
HAMS
Nemoh
Capytaine
Capytaine
HAMS
HAMS
Nemoh
Capytaine
Capytaine
HAMS
HAMS

No parallelization
18
1
8No parallelization
18
1
8No parallelization
1
8
1
8

PA
(HULL+WAT
ER-PLANE
MESH
OSWEC

NO.OF
PANELS
2448
2448
2448
2448
2448
2448
2448
2448
2448+ 1584
2448+ 1584
716
716
716
716
716

DOF

TIME (s)

6
6
6
6
6
6
6
6
6
6
1
6
6
6
6

1900
920
394
323
140
1900
920
394
1800
810
354
235
78
110
50

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Vaibhav RAGHAVAN and George LAVIDAS

While Nemoh does not have the capability for parallelizing the calculation, both Capytaine and HAMS have thecapability to do this. HAMS uses OpenMP parallelization while Capytaine allows for two types of parallelization –through OpenMP and through joblib. When solving a single problem, matrix constructions and linear algebra operations
can be parallelized through OpenMP. This can be controlled using the environment variable OMP_NUM_THREADS ona Windows system. When solving multiple problems, joblib can be utilized in addition to OpenMP to run multiple jobs.
The calculations performed with 8 threads in Capytaine as seen in Table 1 were performed utilizing both OpenMP andjoblib with running 8 jobs simultaneously. Further research into an optimized way of utilizing OpenMP and joblib could
help make Capytaine faster. This was not done as part of this research.
From Table 1, it can be seen that HAMS is faster than Nemoh (about 15 times for the case of PA with only hullmesh) and Capytaine (2-3 times for the PA) when using the parallelization in both HAMS and Capytaine. Similar results
were observed for the OSWEC, albeit the multipliers were slightly lower. It should be noted that for the OSWEC case,the diffraction and radiation problem in Nemoh was solved only for pitch, while in the case of HAMS and Capytaine,
this was done for all DOFs. When irregular frequency removal is taken into account, HAMS takes 5 times longer for the
calculation as compared to Case(a). When comparing the results for the PA (Figures 3 to 8), it could be important tocheck if the water plane mesh is actually required with regard to the range of frequencies that are important to a certain
analysis.

5. Conclusions
This research makes some comparisons among the open-source solvers Nemoh, HAMS and Capytaine for two
different types of WECs: a semi-submerged cylindrical PA and a semi-submerged OSWEC. The compared parameters
include the hydrodynamic coefficients, excitation forces and the computational efficiency.
For the PA, two cases are highlighted here. Case (a) focuses on the simulation where only the hull mesh is used for
all the three solvers. Case (b) focuses on the simulation where the water plane mesh is added to the hull mesh for HAMS.
The water plane mesh is added particularly to remove irregular frequencies. When comparing the hydrodynamic
coefficients and exciting force for Case (a) and Case (b), it can be concluded that all three solvers are generally close.
For the PA case, slight difference is observed in the heave added mass for frequencies greater 3 rad/s between HAMS
and Nemoh/Capytaine. Furthermore, when comparing Case(a) and Case(b) for the PA, it can be observed that with Case(b), the irregular frequency is suppressed with HAMS as compared to Nemoh and Capytaine. HAMS employs an
extended boundary integral equation which assumes that the potentials in the interior of the water plane are zero. Thisequation is hence used as an additional equation to the input boundary integral equations which are solved on the hull
surface, thus suppressing the irregular frequencies. In the case of the OSWEC, all solvers were generally close. Slightdifferences in HAMS with respect to Nemoh/Capytaine were observed for the pitch added mass, radiation damping andexciting force close to 0.75 rad/s.
The last segment of this research compares the computational efficiency of the BEM solvers. HAMS is significantly
faster than Nemoh for all the simulations going up to 15 times for the case of the PA and 7 times for the OSWEC with
only the hull mesh. When comparing HAMS (with OpenMP) with Capytaine (with OpenMP and joblib), HAMS is atleast twice as fast. Further optimization of the combination of OpenMP and joblib could make Capytaine faster, but thatwas not done as part of this study. Considering the inclusion of the water plane mesh (Case (b)) in the case of the PA for
removing irregular frequencies effects, HAMS is about 5 times slower than Case (a). Hence, when considering the water
plane mesh, it is important to consider if this is really required, given the frequency of interest. If this holds true, aconvergence test to set the limit of the mesh size is suggested to obtain accurate results with less computational effort.
Within the domain of open-source solvers for wave-structure interactions, HAMS offers some unique advantages ascompared to Nemoh and Capytaine as seen. HAMS and Capytaine have the potential to become more valuable options
to meet the numerical challenges within the field of ocean engineering, particularly the possibility of low computational
effort with good accuracy.

6. Acknowledgements
The work was part of the PhD project that is funded by Delft University of Technology and the Marine Renewables
Energies Lab. The authors would also like to thank and acknowledge the discussions with and scripts shared by Markel

Comparing BEM solvers for analyzing wave energy converters

137

Penalba for using Nemoh. The authors would also like to thank and acknowledge the discussions with Yingyi Liu
regarding the usage of HAMS and discussions with Matthieu Ancellin regarding Capytaine.

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