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Numerical assessment of negative spring on sparOWC based in the IVV method
Stefano Roveda
Thesis to obtain the Master of Science Degree in
Energy Engineering and Management
Supervisors: Prof. António José Nunes de Almeida Sarmento
Eng. Miguel Ângelo Macau Pais Rebelo Vicente
Examination Committee
Chairperson: Prof. José Alberto Caiado Falcão de Campos
Supervisor: Prof. António José Nunes de Almeida Sarmento
Member of the Committee: Prof. Luís Manuel de Carvalho Gato
December 2016
ii
Acknowlegements
Many were the people who supported me throughout this work and my university studies in
general, and I would like to thank all of them.
I would �rst like to express my gratitude to my advisor Miguel, for everything he taught me and
for all the e�ort he put in guiding me through the learning process of this master thesis.
Furthermore, I would like to thank my supervisor Antonio, for introducing me to the topic as
well as for his precious advice and remarks on my work.
I would like to thank all of my colleagues with which I was lucky enough to share this phase of
my life. In particular Patricio and Sebastian during my bachelor, Adriano and Christian during
my master.
Finally, I must express my very profound gratitude to my parents Rita and Alessandro, for
providing me with their constant support throughout my years of study, allowing me to focus on
my academic life and not worrying about anything else.
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Abstract
Wave power is largely unpredictable and the fact that the energy output of Wave Energy Con-
verters (WECs) depends on the match between wave frequency and natural frequency of the
system makes the adaptability one of the main issues of this technology. The negative spring
concept is proposed as a possible solution, providing a shift in the resonance frequency that can
be regulated depending on the sea state conditions.
This study concerns a negative spring implementation using the Immersed Variable Volume
(IVV) method on the Spar Buoy WEC. The system was �rst modelled analytically and then
implemented numerically. In order to do that, the values of hydrodynamic coe�cients were ob-
tained by running simulations with the software WAMIT and then used to resolve the system
dynamics with MATLAB codes. Whilst these steps help evaluating the feasibility and the chal-
lenges of the breakthrough technology, in order to capture the real potential added-value brought
by its implementation, reference design cases were established for each study case.
The goal of having a reference design is to measure the impact of the breakthrough by comparing
it against a design which is not provided with such technology. The analysis was carried out
for two di�erent cases: single WEC and array of devices; both of them were studied using the
same approach. As a result of the work, constraints in the method were found and discussed and
feasibility considerations were provided .
Keywords: Negative Spring - Resonance Frequency - OWC - Renewable Energy - Wave Energy
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Resumo
A energia das ondas é em grande parte imprevisível. O facto de que a produção dos dispositivos
de energia das ondas depende do ajustamento da frequência natural do sistema à frequência
das ondas faz da capacidade de adaptação um dos principais problemas desta tecnologia. O
conceito de mola negativa é proposto como uma solução possível, proporcionando uma mu-
dança na frequência de ressonância, que pode ser regulada em função das condições do estado
do mar. Este estudo diz respeito a uma aplicação da mola negativa, utilizando o método do
volume imerso variável numa coluna de água oscilante �utuante (spar buoy OWC). O sistema
foi modelado primeiro analiticamente, e em seguida implementado numericamente. Para tal, os
coe�cientes hidrodinâmicos foram obtidos através do software WAMIT, e depois, com códigos
Matlab, resolveu-se a dinâmica do sistema.
O objectivo do trabalho foi avaliar a viabilidade e os desa�os desta inovação tecnológica,
a �m de compreender o potencial benefício da sua implementação. Foi de�nido um design
de referência, por forma a medir o impacto da inovação, comparando-o com um dispositivo
desprovido desta tecnologia. A análise foi realizada para dois casos: dispositivo único isolado e
parque de dispositivos. Como resultado do trabalho, foram encontradas algumas limitações do
método,e foram apresentadas considerações sobre a sua viabilidade.
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Contents
1 Introduction 1
1.1 Thesis presentation and structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 WEC state of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Oscillating Water Column state of the art . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Fixed-structure OWCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Breakwater-integrated OWCs . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.3 Floating-structure OWCs . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.4 Floating structure WECs with interior OWC . . . . . . . . . . . . . . . . 7
1.3.5 Multi-OWC devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Linear WEC mathematical formulation 9
2.1 Negative spring e�ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 IVV concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Single WEC analysis 15
3.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.1 IVV negative spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.2 Analysis of the resonance frequency . . . . . . . . . . . . . . . . . . . . . 20
3.2 Modi�ed geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Closed bottom case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.2 WAMIT outputs and implementation . . . . . . . . . . . . . . . . . . . . 33
3.4 Open bottom case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 WEC array analysis 41
4.1 Array concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Analitical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Volume as function of the resonance frequency shift . . . . . . . . . . . . 45
4.3 Numerical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5 Conclusions 51
5.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
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List of Figures
1 classi�cation of WECs by distance to shore . . . . . . . . . . . . . . . . . . . . . 4
2 classi�cation of WECs by size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 classi�cation of WECs by working principle [3] . . . . . . . . . . . . . . . . . . . 5
4 system representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 IVV concept scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6 Volume required to obtain a certain resonance shift . . . . . . . . . . . . . . . . . 22
7 Asymptotes for di�erent ratios between �oater and bottom chamber areas . . . . 24
8 scheme of the OWC with quoted dimensions . . . . . . . . . . . . . . . . . . . . . 26
9 closed bottom design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10 open bottom design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
11 Added mass for proper modes and cross mode for tuned, closed-bottom-case . . . 34
12 Damping for proper modes and cross mode for tuned, closed-bottom-case . . . . 34
13 excitation forces on the two elements for tuned, closed-bottom-case . . . . . . . . 35
14 WEC body's and water column free surface's RAO . . . . . . . . . . . . . . . . . 36
15 Heave RAO for OWC with extra walls but �xed water surface patch . . . . . . . 37
16 added mass of the proper modes for body, column and chamber . . . . . . . . . . 38
17 damping coe�cient of the proper modes for body, column and chamber . . . . . 38
18 added mass of the cross modes between body, column and chamber . . . . . . . . 39
19 damping coe�cient of the cross modes between body, column and chamber . . . 39
20 Heave RAO for WEC body, central column free surface and bottom chamber free
surface with IVV e�ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
21 array layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
22 heave RAO for one device of the array with and without IVV implementation . . 47
23 numerical results of volume needed for a given peak frequency shift . . . . . . . . 48
24 comparison between analitical and numerical methods for identifying the volume
needed for a certain frequency shift . . . . . . . . . . . . . . . . . . . . . . . . . . 49
25 representation of moonpool cylinder . . . . . . . . . . . . . . . . . . . . . . . . . viii
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List of Tables
1 WAMIT input �les . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
3 WAMIT output �les . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
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1 Introduction
1.1 Thesis presentation and structure
The thesis content is based on the work carried out during the internship at WavEC - O�shore
Renewables in Lisbon, Portugal.
WavEC - O�shore Renewables is a private non-pro�t association founded in 2003; it has three
type of activities: applied research, consultancy and pro-bono activities.
The R&D activities and consulting services expand to Monitoring and Technology, Numerical
Modelling, Economy and Industry, Marine Environment and Public Policy, while the work for
the public good includes the dissemination and promotion of opportunities associated with the
early development of energy marine renewable [1].
The thesis work is part of a bigger project, namedWETFEET and funded by Horizon 2020, whose
coordinator is the WavEC itself. The goal of the project is to propose technological innovations
to identify disruptive innovations for wave energy technologies. These innovations will be applied
as a case study to two existing concepts of WECs, namely the Spar Buoy oscillating water column
(OWC) and the Symphony.
Whilst these objectives will help evaluating the feasibility and the challenges of each breakthrough
individually, this might not be su�cient to capture the real potential added-value brought by their
implementation. For this reason, common e�ort to resolve this issue was initiated by establishing
reference design cases. The goal of having reference design cases is to measure the impact of each
breakthrough by comparing against a design typically employing more conventional systems or
processes. [2] This work regards the technological innovation of the negative spring concept, in
particular, by implementing the method of the Immersed Variable Volume (IVV). The object of
this study is to evaluate the potential of this method as a negative spring that couinterbalanges
the very high stifness of the hydrostatic spring. Such an improvement would consist controllable
shifts in the resonance frequency, which means enhanced adaptability of the technology to the
resource and subsequent energy production. The IVV method is implemented on the Spar Buoy
Wave Energy Converter (WEC). The in�uence of the IVV negative spring is considered both for
an isolated single WEC and for the case of an array of multiple WECs.
The problem was approached according to the following steps:
1. Mathematical formulation with development of the analytical model
2. Identi�cation of a reference geometry
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3. Computation of hydrodynamic parameters via simulation using the software WAMIT
4. Numerical model implementation in MATLAB
5. Numerical results on the IVV and conclusions
The approach was the same for both single WEC and array cases.
During the development of the thesis work the candidate had the opportunity to understand and
apply the following:
• compilation of library �les using Intel FORTRAN in order to modify softwares
• calculation of physical quantities using WAMIT
• engineering approach of sensitivity analysis and problem solving
• mathematical modelling of a physical phenomenon
• numerical implementation of the mathematical model using MATLAB
• critical interpretation of results in di�erent forms
• professional working environment
1.2 WEC state of the art
Wave energy technology can �nd a father in the person of Yoshio Masuda (1925- 2009), a former
Japanese navy o�cer who carried out studies in the �eld since the 1940s. Despite the �rst trials
of construction of the �rst wave-driven devices had been not successful, the global interest in
wave energy, along with tall the other forms of renewables, considerably raised in conjunction
with the oil crisis of 1973. Starting from that year important research programs began running
in Europe, leading to the �rst international conferences and journal papers now considered as
landmarks, with the United Kingdom and Norway covering a prominent position in the scene. In
the following years, until the early 1990s, the activity in Europe remained mainly at the academic
level; however this work was fundamental for a further development of the �eld of interest, since
complex di�raction and radiation phenomena make wave energy absorption a hydrodynamic pro-
cess di�cult to theoretically understand and model. The studies conducted during that period
laid the foundations for the next era's progresses. In 1991 the condition of the wave energy �eld
was suddenly improved in term of interest and visibility by the European Commission's decision
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of including it in their R&D program in renewable energies. In fact, the �rst project started the
following year and since then many teams were formed and actively operating in Europe, with
the European Commission funding more than thirty projects after the �rst one in 1992, some of
them with the form of coordination activities. The European Commission also sponsored a series
of conference, namely the European Wave Energy Conferences and lately including tidal energy
as well. The conferences take place once every two years, alternating with the equally biennial
International Conference on Ocean Energy, focused on commercial, economic and environmental
issues.
In 2001, the International Energy Agency established an Implementing Agreement on Ocean
Energy Systems, called IEA-OES. The goal of the agreement is to facilitate and co-ordinate
ocean energy research, development and demonstration through international co-operation and
information exchange. Surveys of ongoing activities in wave energy worldwide can be found in
the IEA-OES annual reports. [3]
Since wave power is largely unpredictable, to have an assessment of the energy resource is funda-
mental for the strategic planning of its utilization and for the design of wave energy devices. [4]
Starting from the countries that wave developed energy technology �rst, such as the United
Kingdom, the characterization of the wave resource took place until the results led to the cre-
ation of the European Wave Energy Atlas, or WERATLAS, which remains a basic tool for wave
energy planning in Europe. [5, 6]
Unlike wind energy technology, which is largely deployed in the form of the three-blades mill,
wave technology has not reached yet a level of maturity such to allow the identi�cation of the
best design. For this reason the number of di�erent prototypes is more than one hundred and
increasing due to new projects and concepts.
The technology is overall at a relatively early stage because of various reasons: model testing in
wave basin, still essential, is expensive and time-consuming; moreover the �nal stage prototypes
have to be tested in the open sea, so they have to be full-scale devices in order to correctly
match with the wavelength. Therefore the development in this �eld is held back by the high
costs of constructing, deploying, and testing large prototypes, also the maintenance is an issue,
with environmental conditions that sometimes can be very rough.
Several methods have been proposed to classify wave energy systems, according to various pa-
rameter; the following �gures illustrate schematic classi�cations, by location, size and working
principle, and provide some examples of actually deployed WECs
3
Figure 1: classi�cation of WECs by distance to shore
Figure 2: classi�cation of WECs by size
4
Figure 3: classi�cation of WECs by working principle [3]
1.3 Oscillating Water Column state of the art
1.3.1 Fixed-structure OWCs
After the inclusion of wave energy in the European R&D program on renewable energies, basic
studies led to the design and construction of two full scale �xed-structure OWC plants, both
equipped with Wells turbines. The �rst, completed in 1999 and still running, was installed in
Pico , Azores, Portugal, with rated power of 400 kW. [7] The second, completed in 2000 and
rated 500 kW, was installed in Islay, Scotland, UK. [8]
Higher power output plants su�ered from deployment operation problems. This is the case of
5
the OWC named OSPREY and Green Wave, constructed by the Australian company Oceanlinx,
both rated 1 MW and both wrecked, in 1995 and 2014 respectively. With smaller plants the risk
of such operation can be overcome more easily; an example is the 500 kW rated OWC installed
at Yongsoo, o� the coast of Jeju Island, South Korea.
It is well known since the early studies that the wave energy absorption process can be enhanced
with protruding walls in the waves direction, in order to have an extended chamber structure,
shaped as a harbor or a collector. [9,10]. This concept was already adopted by some early OWC
prototypes, such in Toftestallen, Norway and Trivandrum, India. In 2005, with the same purpose,
the Australian company Energetech developed a parabolic-shaped collector for their nearshore
prototype at Port Kembla, Australia.
For OWCs deployed so far on the shoreline the access on foot is easy and the special conditions
at their location are not easily replicable elsewhere. For these reasons they can be considered as
demonstration prototypes and some of them have been used as experimental infrastructures.
1.3.2 Breakwater-integrated OWCs
In a �xed-structure OWC the most critical issue from the economical, constructional and opera-
tional points of view, is the design and construction. For this reason the integration of the WEC
into a breakwater is an appealing solution, since the construction costs are shared, as well as the
access for construction, which guarantees easier operation and maintenance routines.
This concept was implemented the �rst time in the harbour of Sakata, Japan, in 1990 [11] and
lately, in 2008, at the port of Mutriku, in Basque Country, Spain [12]; the plant presents 16
chambers, each of them equipped with a Wells turbine rated 18.5 kW.
A new design of OWC is being integrated into the breakwater of the harbor of Civitavecchia,
Italy: the cross-section is U-shaped, long in the wave crest direction and narrow in the fore-aft
direction. In this way the total length of the water column can be increased without placing the
opening too far below the sea surface [13].
1.3.3 Floating-structure OWCs
One of the �rst �oating OWC ever studied was the Backward Bent Duct Buoy (BBDB) [14], a
�oating device with an L shaped OWC positioned with its back facing the waves, a buoyancy
caisson-type module, an air chamber and an air turbine driving an electrical generator.
This technology has been studied both theoretically and with experimentation [15�19] along
the years by several countries, such Japan, South Korea, USA, China, India and many others
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in Europe like Ireland. Between 2008 and 2011 the OE Buoy, a 1:4th-scale BBDB OWC, was
tested in Galway Bay, Ireland [20]. Another �oating OWC was The Mighty Whale, rated 110
kW and deployed in Mie Prefecture, Japan, in 1998. The device was studied for several years
and presented three Wells turbine equipped air chambers located at the front, side by side, and
buoyancy tanks. Mk3 is the name of another �oating OWC, deployed from February to May 2010
o� Port Kembla, Australia, by the Australian company Oceanlinx. The WEC was a one-third-
scale grid-connected model of the 2.5MW full-scale OWC device, with eight OWC chambers.
Together with the BBDB, the axisymmetric �oating OWC, consisting of a relatively long vertical
tube, open at both ends and attached to a �oater, is a concept which has been considered since
the early days of wave energy conversion. This device, called OWC spar-buoy, was taken into
account in the earliest journal papers devoted to the theoretical modelling of WECs [21,22] and is
analyzed by McCormick in his pioneer book [23]. The concept was adopted in the production of
several types of wave-powered navigation buoys [24,25] and it was also considered for larger scale
energy production [26]. A much more recent device of the same type, initially studied at WavEC
and later at IST, the same but the �oater is design to oscillate signi�cantly at a frequency that
di�ers from the water column. [27]. Thanks to its simplicity the OWC spar-buoy is considered
to be a very interesting option for further development
1.3.4 Floating structure WECs with interior OWC
In some new type of OWC that have been proposed, the water column is not connected to the
outer sea water, but rather enclosed in the �oating structure; the advantage of this design is
having the air turbine protected from the corrosive and mechanical e�ects of sea water.
The U-Gen device is an example of WEC adopting this concept: it consists of an asymmetric
�oater with an interior U-shaped tank partially �lled with water and two lateral air chambers
connected by a duct where a Wells turbine is installed. The pitching motion of the OWC forces
the air through the duct. A 1:16th-scale model, was tested in 2010, in Brest, France [28].
Another pitching device with what can be considered as an interior OWC consists of a buoyant
tethered submerged circular cylinder which pitches around an axis below its center [29]. Within
the body of the cylinder an annular tank is �lled with �uid. A sloshing motion of the �uid
induced by the pitch forces air through a turbine connecting chambers above the two isolated
internal free surfaces.
An axisymmetric WEC with an internal OWC which was recently proposed consists in an air-
�lled box that can be �xed to the sea bottom or �oating [30]. The interface between the enclosed
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air and the surrounding sea water moves and the box enclosed air volume is connected to the
atmosphere by an OWC whose walls are coaxial tubes.
1.3.5 Multi-OWC devices
Multiple OWCs have been studied not only integrated in a breakwater, but also for �oating
devices, where in some cases several air chambers share a single unidirectional conventional air
turbine.
The Seabreath, an attenuator-type WEC under development at Padova University, Italy [31],
comprises a set of rectangular chambers with open bottom connected by non-return valves to
two longitudinal ducts, at high and low pressure, which feed a conventional unidirectional air
turbine.
The LEANCON is another multi-OWC device tested at Aalborg University, in Denmark at a
1:40 scale. In the model OWCs are arranged in two rows under two beams with a V-shape
connection to each other. The turbine implementation is the same as in Seabreath [32]. A very
similar protoype was numerically simulated and model tested at the smaller scale of 1:50
A largely similar, although substantially smaller, device was numerically simulated and model
tested at scale 1:50 at the large oceanic basin of the Hydraulics and Maritime Research Centre,
located at University College Cork, Ireland [33] The absorption of wave energy by OWCs was
proposed to be used for reducing the hydroelastic responses of very large �oating structures
(VLFS) [34].
8
2 Linear WEC mathematical formulation
The following section has the goal to describe the dynamics of a OWC WEC. This work focuses
on the speci�c case of only-heaving devices: all the analysis was carried out considering motion
in just one degree of freedom, that is heave.
For the sake of simplicity the description shall take into account linear waves, with irrotational
and inviscid �uid.
The general equation of motion according to Newton is
mz(t) = Fh(t) + Fex(t) (1)
Where:
• z is the acceleration along the vertical direction
• Fh(t) is total force produced by the water and depending on intrinsic characteristic of the
body, like the geometry. It includes both hydrostatic and hydrodynamic forces
• Fex(t) is the total external force. It typically includes moorings and loads induced by the
Power Take O� (PTO)
• m is the body mass
By considering a harmonic motion and the linear wave theory it is possible to decompose each
term into its space and time components, being the force vector considered as rotating on the
complex plane. Therefore all the forces acting on the device are described by
• A frequency dependent complex amplitude M(ω)
• A sinusoidal dependence eiωt
The amplitude is a constant value, which is multiplied by the phasor term that is time dependent.
The e�ective amplitude is then given by the projection of the rotating vector on the real axes,
so the displacement is
z(t) = Re{Z(ω)eiωt} (2)
and consequently
z(t) = Re{iωZ(ω)eiωt} (3)
z(t) = Re{−ω2z(ω)eiωt} (4)
9
Since all the terms can be expressed as the amplitude times the phasor eiωt, the latter can
be removed from both sides of equation and the analysis reduced to the complex amplitudes
only. Under the assumption of linear waves this operation is possible and eliminates the time
dependence of the object of study, which is considered in this way to be in the frequency domain.
In the frequency domain the equation of motion is then
−ω2mZ(ω) = Fh(ω) + Fex(ω) (5)
The linearity of the model makes possible to decompose the force term Fh into hydrostatic and
hydrodynamic components.
Fhd denotes the hydrodynamic part of the force acting on the WEC captor while Fhs the hy-
drostatic part due to gravity and buoyancy that tends to bring the body back to its equilibrium
position.
Fh(ω) = Fhd(ω) + Fhs(ω) (6)
The hydrodynamic force can be further decomposed into wave excitation Fe and wave radi-
ation Fr.
The potential �ow hydrodynamic force results from the integration of the dynamic pressure,
determined from the Bernoulli equation. Without considering second-order terms, the pressure
is
pe = −ρ(∂φ
∂t
)(7)
where ρ is the water density and φ the potential �ow, de�ned under the conditions of irrotational
and incompressible �uid as 5φ = ~u. Therefore, the linear hydrodynamic force on a �oating body
results to be
Fhd =
∫phd~ndS = ρ
∫∂φ
∂t~ndS (8)
where ~n denotes the unit vector normal to the body surface and S the surface itself. The de-
composition of the velocity potential allows writing the complex amplitude of the hydrodynamic
force as
~Fhd = ~Fe + ~Fr = iωρ
∫(φ0 + φs)~ndS + iωρ
∫zϕ~ndS (9)
Equation (9) is written considering only one degree of freedom as previously speci�ed. Fe is due
the e�ect of the incident waves on the body assumed to be �xed in its average position, while
Fr is related to the change in momentum of the �uid caused by the motion of the body. φ0 and
φs are the potential �ows corresponding to the two components of the wave excitation force Fe
10
: the Froude-Krylov force, , and the scatter or di�raction excitation term:
~Fe = ~FFK + ~Fs = ßωρ
∫φ0~ndS + iωρ
∫φS~ndS (10)
The second term can be considered as a sort of correction of the �rst one, since φ0, does not take
the body into account and describes the motion of an undisturbed wave �eld. φs corresponds
to the wave �eld that is scattered by the body which is �xed in space. If the dimensions of
the �oating body are much smaller than the incoming wavelength, the di�raction term can be
neglected. In the case of point absorber, such is the object of this work, the Froude-Krylov force
may represent a reasonable approximation to the vertical excitation force on the surface-piercing
�oating body.
The radiation force Fr is caused by the water displaced in the vicinity of the body when this
moves in the absence of incident waves. It can be written as
~Fr = −iω ~Zz (11)
where, using an electric theory analogy, Z denotes the radiation impedance, which, according to
the second term in (9), is given by
~Z = −iωρ∫ϕ~ndS (12)
The concept of impedance in AC circuits includes the e�ects of resistance, related to power
dissipation, and reactance, related to energy storage in components like inductors and capacitors.
Impedance is thought as a complex quantity, in which the real part represents the resistive e�ect
and the imaginary part represents the reactive e�ect.
~Z = −iωρ∫ϕ~ndS = R+ iX (13)
In this case the real part of the impedance Z, that is R, represents the so called hydrodynamic
damping coe�cient, which refers to a damping e�ect related with the energy transmitted to the
water by the body oscillations that gradually moves away from the body and is generally indicated
as B. On the other hand, the imaginary part X represents the radiation reactance which refers
to the di�erence between the average added kinetic energy, related to the velocity of the water
displaced, and the average added gravitational potential energy, related to the deformation of
the water surface when water is lifted from troughs to crests. The energy stored in the water
�ows into the mechanical system itself and back out again into the surrounding water (reactive
11
e�ect). The radiation reactance X(ω) is frequently written as ωA, where A represents the added
mass coe�cient, that is, an inertia increase due to the water displaced in the body vicinity when
the body moves.
Therefore, the impedance (13) may be rewritten as
Z = R+ iωA (14)
Accordingly, the hydrodynamic radiation force of a �oating body is given by
Fr = −iωBz + ω2Az (15)
From the last expression it is clear that the �rst term of the radiation is a damping force,
proportional to the captor velocity, and the second a inertial force, proportional to the captor
acceleration. If equation (15) was multiplied by the body velocity then it would express the
energy transfer between the body and the water. The damping term would be proportional to
the square of the velocity (with non-zero mean) and the added mass term to the product of its
velocity and displacement (with zero mean). The non-zero mean means that there is a non-zero
average transmission of energy from the body to the water due to the damping term. The zero
mean means that there is a zero net balance associated with the added mass term
While the hydrodynamic force, related to the potential �ow, refers to a change in the body's
conditions (on one hand it gets hit by the wave, on the other hand it reacts to the wave) the
hydrostatic force belongs to the restoring type, that is it works towards bringing the body back
to the equilibrium position.
The hydrostatic force Fhs derives from the integration of the hydrostatic pressure on the body;
when the oscillations are relatively small compared to the body itself, such is the case of linear
waves theory and of this entire work, the linearization of the theory is accurate and the approx-
imation plausible. In this way the force can be expressed as proportional to the displacement
and so the complex amplitude results as
Fhs = −Gz (16)
where the coe�cient of the hydrostatic spring sti�ness is
G = ρgS (17)
Where ρ g and S again refer to respectively water density, gravity acceleration and cross surface
cross surface of the body at its undisturbed position by the water freee-surface. According to the
12
formulae of hydrodynamic and hydrostatic forces, the oscillatory motion of the WEC is clearly
described as typical a mass-damper-spring model.
The external forces are related essentially to PTO and moorings, although some WECs have
also end-stop mechanisms in order to brake the energy absorber at the end of its run and so to
have a smooth operation without damages.
The PTO force is usually strongly non-linear because of the complex control system associated
to them; however, in order to keep the principle of superposition valid, it has to be linearized.
In its linear form the PTO force is composed by two parts: one damping term proportional to
the velocity and a spring term proportional to the displacement
FPTO = −iωRPTOz −GPTOz (18)
The same goes for moorings' force, which can be considered as linear under the assumption of
slack moorings, and small motion of the body. What follows is that the moorings provide a
restoring force as well
Fm = −Gmz (19)
As a result of the description of all the previous terms, equation (1) can be written as
z[−ω2 (m+A) + ρgS +GPTO +Gm + iω(B +BPTO)
]= iωρ
∫φ0ndS (20)
(20) represents the equation of motion of a heaving WEC and it will be taken as reference in the
rest of the work. The left hand side terms which are multiplied for the displacement de�ne the
total impedance.
2.1 Negative spring e�ect
As explained in the previous sections one e�ect of the negative spring is related to the change in
resonance frequency of the WEC. Taking (20) as reference it is possible to analyze the already
mentioned e�ect by identifying resonance conditions: the latters imply that the real part of the
total impedance is null. Therefore the condition for the resonance can be written as
−ω2(m+A) + ρgS +K = 0 (21)
13
where K represents the sti�ness of all the external restoring forces such as PTO or moorings.
Thus, the resonance frequency is given by
ω0 =
√ρgS +K
m+A(22)
The peak frequency may shift according to changes in the mass of the device or its cross
area, but these values are set at the design stage and, in general, it is not possible to vary them
during operation. In order to control changes in the resonance frequency, and hence to have
the WEC to operate optimally for various sea states, one could add a −Kns term alongside
the system to reduce the resonance frequency and to modify its value: this is a negative spring
e�ect. The reason why we want to reduce the resonance frequency is related to the fact that
small devices resonate at higher frequencies than the majority of real-case sea states; the im-
plementation of a negative spring would allow to deploy smaller devices and thus reduce the costs.
2.1.1 IVV concept
The negative spring concept considered in this study is the method of the Immersed Variable
Volume (IVV), proposed by Prof. Antonio Sarmento and developed at WavEC [35].
The IVV method was conceived to be implemented on the Spar Buoy model, considered in many
other projects of the WavEC; therefore its in�uence subsists for heave motion, that is, the main
degree of freedom of the aforementioned WEC.
It consists in a volume of air located underwater and connected to the WEC. The air is located
inside a chamber open on the bottom or enclosed in a variable volume reservoir or enclosed in a
variable volume reservoir: as the OWC main body heaves, the free water surface follows with a
similar motion, compressing and expanding the air, which changes in volume.
The negative spring e�ect is given by the compressibility of air: in fact during the downwards
movement of the oscillation, the air gets compressed and so its volume decreases. Less volume
leads to less buoyancy of the entire WEC system, which tends to sink even more. On the other
hand, in correspondence with the upward phase of the heave motion, the volume of air increases
and pushes the structure further up. This behavior describes clearly the negative spring nature
of the phenomenon, since the forces generated by the variation of the volume head on the same
direction of the displacement, so it enhances the motion from the equilibrium position instead of
opposing it, like a spring's restoring force would normally do. A more accurate way to look at the
phenomenon is to consider the di�erence in pressure between the outside and the inside of the
14
air chamber: in fact the pressure increase inside the air chamber generates a force which opposes
to the one described above and it has to be taken into account for a correct understanding of
the technology's potential [36].
The change of the underwater air volume is due to both OWC structure and water free surface
heave. When the two sinusoidal motions present a certain phase between each other, the force of
the negative spring modi�es the peak frequency at which the WEC reaches resonance. The IVV
allows in this way to control the Response Amplitude Operator (RAO) of the device by shifting
the resonance frequency to a lower value; thanks to this e�ect it is possible to adapt the response
to di�erent sea states and so to exploit the resource at best.
The in�uence of the IVV e�ect on a Spar Buoy WEC will be analyzed in detail in the next
chapters, following the study's structure previously described.
3 Single WEC analysis
3.1 Modeling
As introduced in the previous section, the object of the study is the e�ect of an IVV negative
spring on the resonance condition for an OWC Spar Buoy. The IVV concept is based on the
variation of an air volume located inside a chamber on the bottom part of the OWC, the expected
consequence of the implementation is a shift in the peak frequency from lower to higher periods.
In the �rst chapter a linear heaving OWC was analitically modeled and the formulation of a
general negative spring was provided.
In this chapter the work proceeds by modelling the dynamics of a complete IVV integrated
device, both analitically and numerically with the support of softwares. The WEC has been
modeled using the piston mode description of the OWC [37].
Figure 4: system representation
In this representation the free surface inside
the bottom air chamber is assumed completely
�at and moving just in one degree of freedom:
heave. With this approximation it is possi-
ble to maintain the equations of motion as for
two rigid bodies, respectively �oater and wa-
ter column free surface, simplifying the entire
15
model.
The greatest advantage of this method comes
during the numerical implementation, when,
using a panel method-based software such as
WAMIT, the free surface can be considered as
a unique patch with heaving motion.
The case under study corresponds to a heaving
point absxorber, where the water column di-
ameter is much smaller than the characteristic
length of the waves: all the degrees of freedom
other than heave have a minimum contribu-
tion on the overall displacement of the free surface and they can be neglected. It is therefore
reasonable to adopt the piston mode for the WEC's modelling.
By adopting the piston mode for the surfaces (the top and IVV) it is necessary to build a system
of equations similar to the equation of motion presented in the previous chapter, in order to take
into account the new degrees of freedom.
With reference to �gure 4, the system of equation is de�ned as follows:
X1
[−ω2 (m+A11) + iωB11 + ρgSf
]+X2
[ω2A12 + iωB12
]+
X3
[−ω2A13 + iωB13
]+Ns = AwFe1 (23)
X2[−ω2A22 + iωB22 + ρgSc] +X1[−ω2A12 + iωB12] +X3[−ω2A23 + iωB23] = AwFe2 (24)
X3[−ω2A33 + iωB33 + ρgSb] +X1[−ω2A13 + iωB13] +X2[−ω2A23 + iωB23] = AwFe3 (25)
Where
• X1 is the vertical displacement of the structure
• X2 is the vertical displacement of the top water column
• X3 is the vertical displacement of the bottom chamber free surface (in the IVV)
• Fe1 is the excitation force acting on the body
• Fe2 is the excitation force acting on the top water column
• Fe3 is the excitation force acting on the bottom chamber free surface
16
• A11 and B11 are respectively added mass and damping coe�cient of the main body
• A22 and B22 are respectively added mass and damping coe�cient of the water column
17
• A33 and B33 are respectively added mass and damping coe�cient of the bottom chamber
free surface
• A12 and B12 are respectively the added mass and the damping coe�cients of the main
body a�ecting the motion of the water column. They are equal to A21 and B21
• A23 and B23 are respectively the added mass and the damping coe�cients of the water
column a�ecting the motion of the bottom free surface. They are equal to A32 and B32
• A13 and B13 are respectively the added mass and the damping coe�cients of the main
body a�ecting motion of the bottom free surface. They are equal to A31 and B31
• Sc is the cross section of the water column
• Sb is the cross section of the bottom chamber
After a �rst modelling of the case under study, analytical investigation has been carried out
in order to �nd out the adequate air volume of the bottom chamber which would lead to the
appropriate negative spring force, Ns.
3.1.1 IVV negative spring
This analysis starts by assuming a submerged cylinder with its top at a depth h open in its
bottom and closed on the top, partially �lled with air.
Figure 5: IVV concept scheme
18
The cylinder may move up and down from its undisturbed position. The net vertical force
on the cylinder is given by
F (t) = [p2(t)− p1(t)]Sb −W (26)
where Sb is the cross section of the cylinder and W its weight. Assuming quasi-static pressure
variations, the pressure at the top and inner parts of the top surface of the cylinder are given
respectively by
p1(t) = ρwgh1(t) (27)
p2(t) = ρwg[h1(t) +H(t)] (28)
From which results that the vertical force on the cylinder is given from (26) as
F (t) = ρwgH(t)Sb −W (29)
Rewriting
H(t) = H0 +H ′(t) (30)
where H0 is the undisturbed air height and H ′(t) its time-variation, we obtain
F (t) = (ρwgSbH0 −W ) + ρwgSbH′(t) (31)
which shows that the time dependent vertical force is due to the change in buoyancy, as SbH′(t)
is the internal volume change. The total volume is given by
V (t) = V0 + v′(t) (32)
where V0 id the undisturbed volume and
v′(t) = SbH′(t) (33)
To compute v′(t) we use the isentropic relation for the air pressure change inside the cylinder
p02Vγ0 = p2(t)V γ(t) (34)
where p02 is the undisturbed internal air pressure. We can also write
p2(t) = p02 + p′(t) (35)
V (t) = V0 + v′(t) (36)
19
from which (34) can be linearized to give
v′(t) = − V0γp0
p′(t) (37)
from (28) is clear that
p2(t) = ρwg[h10 + h′1(t) +H0 +H ′(t)] (38)
from which results that
p′(t) = ρwg[h′1(t) +H ′(t)] (39)
using (33) and (37) we may write
H ′(t)Sb = − V0γp0
ρwg[h′1(t) +H ′(t)] (40)
which leads to
H ′(t) = −ρwgV0
γp0Sb
1 + ρwgV0
γp0Sb
h′1(t) (41)
or
H ′(t) = −(γp0SbρwgV0
+ 1)−1h′1(t) (42)
Since h′1(t) = −z(t) where z(t) is vertical upward motion, it results that the negative spring
coe�cient is
α =ρwgSb
1 + γp0SbρwgV0
(43)
3.1.2 Analysis of the resonance frequency
Regarding the equation of motion of the structure with the implementation of the IVV method,
the resonance frequency is computed as
ω =
√ρgS − αm+A
(44)
whereas the resonance frequency without the IVV e�ect is given by
ω0 =
√ρgS
m+A(45)
If we want the structure to resonate at ω
ω
ω0=
√ρwgS − αρwgS
(46)
and so
α = ρwgS
[1−
(ω
ω0
)2]
(47)
20
using (43) it results
Sb
1 + γp0SbρwgV0
= S
[1−
(ω
ω0
)2]
(48)
1 +γp0SbρwgV0
=SbS
[1−
(ω
ω0
)2]
(49)
SbV0
=ρwg
γp0
SbS
[1−
(ω
ω0
)2]− 1 (50)
Hb = V0/Sb =γp0ρwg
1
SbS
[1− ( ωω0
)2]− 1
(51)
Considering the reference geometry of a standard OWC, the bottom would be at the depth of
around 20 m deep and thus the inside air pressure p0=3 105. Since isoentropic relation are
considered the air pressure must refer to the absolute pressure. Being the ocean water density
ρw = 1025 [kg/m3] and the air isoentropic coe�cient γ = 1.4 (51) becomes
Hb =V0Sb
= 401
1− ( ωω0)2 − 1
(52)
Hb = 401− ( ωω0
)2
1− 1 + ( ωω0)2
(53)
Hb = 40
[(ω0
ω
)2− 1
](54)
if we assume that Hb = 5 m is a reasonable value (this is the height of the IVV air chamber) we
�nd that (ω0
ω )2 = 1,125 and so ω0 = 1.06. This allows typically to increase the resonance period
by 6%.
One solution that allows to exploit more air volume without increasing the dimensions of the
bottom chamber is to connect the latter with the top �oater by using pipes: in this way the air
in the top �oater would contribute to the negative spring as well, while the geometric parameters
would remain the same.
If the solution above is implemented half of the volume is split between the upper and bottom
bodies, resulting in a total volume twice bigger, for H0 = 5m it results that (ω0
ω )2 = 1,25 and
TT0
= 1.11, which is an increase of 11%, still very limited to be of interest due to the extra cost
of the lower body.
In order to have consistent results, the air volume connected to the negative spring has to be
obtained by considering the same area of the reference geometry for both top �oater and bottom
chamber.
21
In order to have a better grasp on the size of the air volume necessary for a certain peak frequency
shift, the following plot was obtained from (51).
Figure 6: Volume required to obtain a certain resonance shift
The graph presents an asymptote that separates positive values of volumes from negative
ones. In order to correctly interpret this evidence the formula is re arranged to have the shift in
frequency as a function of the volume.
(ω
ω0
)=
√√√√1−SbSf
γp0SbρgV0
+ 1(55)
Equation (55) allows to look more in detail to the extreme of the volume domain.
limV0→0
(ω
ω0
)=
√1−
SbSf
∞=√
1− 0 (56)
For a volume that tends to zero it is obtained what was expected, that is the frequency shift is
equal to 1. Depending the negative spring e�ect from the air volume inside the bottom chamber,
if the former is very little, the same goes for the latter and the shift of the resonance frequency
results negligible.
22
On the other hand, if the volume tends to in�nite
limV0→∞
(ω
ω0
)=
√1−
SbSf
0 + 1=
√1− Sb
Sf(57)
As the air volume increases the shift becomes more consistent until it gets close to the value of√1− Sb
Sf, which means that (44) becomes
ω =
√ρgSf − ρgSbm+A
(58)
By comparing (44) and (58),it is evident how the term that describes the negative e�ect in this
limit case is α = ρgSb, which describes a force similar to the hydrostatic restoration force on the
�oater. This force could be called a pure negative spring force, not a�ected by the change in
device vertical position. The physical interpretation of this behaviour is that the bigger the air
volume gets, the more compressible is the air and so the less the structure vertical position a�ects
the air pressure. For an in�nite air volume there is no change in pressure at all and the e�ect of
the negative spring appear as above. This is an ideal situation where as a result of a decrease in
volume due to a change in the vertical position of the body, the pressure does not increase; to
pass this limit means to be in conditions where the air would react to compression by decreasing
its pressure, which is a behavior whithout physical meaning: this justify the negative volume
passed the asymptote. Another evidence from (57) is that if Sb = Sf the asymptote appears for
ωω0
= 0 and the trend of the air volume could look like exponential. The same consideration can
be made for Sb > Sf , when the asymptote is located in the negative part of the x axes. Each
example provided is represented in the �gure below.
23
Figure 7: Asymptotes for di�erent ratios between �oater and bottom chamber areas
24
From �gure 6 it can be easily noticed that the air volume has a size which does not justify
the implementation and even connecting top and bottom chambers in order to access to twice
the volume, the dimensions are still prohibitive.
Moreover, form the exponential behavior in the �gure it is possible to see how the trend becomes
fairly �at for less radical shifts.
This means that the size of air volume is huge not just for the target shift of 0.5, but it remains
prohibitive for minor changes in frequency.
In light of the considerations shown up to this point it results that a negative spring e�ect
using the IVV is not suitable for the condition taken into account and the proposed usage and
it can provide only very small shifts of resonance.
We will see in the next chapter that the possibility to connect di�erent wave energy converters
to a central volume in an array may help to solve this problem.
3.2 Modi�ed geometry
The reference geometry used to build the geometry in WAMIT consists in two concentric cylin-
ders, where the inner one stands as a �uid chamber. As every structure simulated in WAMIT, the
entire geometry is considered to be underwater. The WAMIT software as well as the reference
geometry are accurately described in the annex. From the reference geometry, the OWC was
built as an ensemble of cylinders of di�erent dimensions. The structure consists in:
Top �oater: like the reference geometry, the upper part is at the same level of the water free surface,
therefore not computed in WAMIT, which considers only underwater patches.
Central column: this is the part related to the energy production in a �oating OWC WEC. In fact, the
heaving motion of the water inside the column makes the air to pass through a turbine,
connected to the electrical generation. In this study neither PTO nor turbine were modeled
and the air chamber was supposed open at the top. With this con�guration the pressure on
the water free surface inside the central column is always equal to the atmospheric value,
that is, there is no pressure variation.
Bottom chamber: the part of the structure related to the IVV, the air inside the chamber change in volume
(and so in pressure) in response to the excitation force and the heave of the whole body.
This variation provides the negative spring e�ect.
25
Figure 8: scheme of the OWC with quoted dimensions
The dimensions quoted in the �gure were named to be consistent with the formulation. For the
sake of clarity, they are listed below:
• Rf is the top �oater radius
• Rc is the central column radius
• Hf is the top �oater draft
• Hc is the distance between the lower surface of the bottom �oater and the upper cover of
the bottom chamber
• Rb is the radius of the bottom chamber
• Hb is the draft of the bottom chamber
26
The thick blue lines represent water free surfaces.
The dimensions listed above have been included in the geometry input �le in order to be read
by geomxact.dll. The WAMIT subroutine reads the values and save them with speci�ed names,
then it calculates derived dimensions, which would contribute in building the geometry together
with the input data. For the sake of clarity, the WAMIT subroutines with all dimensions were
inserted in the annex. On each patch a pair of parametric coordinates (u; v) are used to de�ne
the position. The parametric coordinates are normalized so that they vary between ± 1 on the
patch.
Any physically relevant body surface can be represented by an ensemble of appropriate patches,
where the Cartesian coordinates of the points on each patch are de�ned by the mapping functions
x = X(u; v); y = Y (u; v); z = Z(u; v). This is the fundamental manner in which the body
surface is represented for the higher order option of WAMIT.
Some parts of the OWC are �lled with water inside; since the thickness of the column's wall is
negligible for this study, the corresponding patch has been considered facing the �uid both inside
and out. The higher-order method can be used to analyze bodies which consist of elements with
zero thickness and the patches representing these elements are referred to as `dipole patches'.
Dipole patches are represented in the same manner as the conventional body surface. Since both
sides of the dipole patches adjoin the �uid, the direction of the normal vector is irrelevant. On
the dipole patches, the unknowns are the di�erence of the velocity potential. A positive di�erence
is de�ned to act in the direction of the normal vector.
A symmetry plane can be used when there are at thin elements represented by dipole patches on
the plane of symmetry. To analyze bodies with zero-thickness elements, the corresponding dipole
patches are identi�ed in the con�guration �les using the parameter NPDIPOLE. The indices of
the dipole patches are de�ned by including one or more lines starting with `NPDIPOLE=', fol-
lowed by the indices or ranges of indices of the dipole patches.
The new OWC geometry was tested in two steps:
1. With the bottom chamber closed in the lower part: in this case there is no negative spring
e�ect as there is no pressure variation. These con�guration was used to �nd a geometry
dataset to be used as new reference, the adopted procedure will be described afterwards.
2. With the bottom chamber open below: this is the case where there is an extra free surface
27
heaving, which brings along a sinusoidal pressure variation and so the negative spring e�ect.
The results of the simulation were then compared with the closed chamber case in order
to de�ne the consequences of IVV implementation.
Two di�erent new subroutines have been developed in order to simulate the two geometry se-
tups, respectively. Intel Fortran has been used in order to modify the GEOMXACT �le: a new
code had been written with the new subroutines and then converted into �.dll� �le by using the
compiler.
3D representations of closed and open bottom designs are provided below.
28
Figure 9: closed bottom design
29
Figure 10: open bottom design
3.3 Closed bottom case
The OWC was built with 9 patches:
• Outer side of top chamber
• Lower part of top chamber
• Inner side of top chamber
• Water free surface
30
• Central column (dipole)
• Upper part of bottom chamber
• Outer side of bottom chamber
• Inner side of bottom chamber
• Lower part of bottom chamber
This subroutine has been used to de�ne a zero-case scenario, representing an OWC device with
no IVV implemented that resonates under default conditions.
In order to test the implementation of the technology on a real device, real conditions have to
be taken into account. For this reason a scatter diagram referring to the north coast of Portu-
gal has been chosen. A range in wave frequency have been selected where the majority of the
waves occurs. That goes from a corresponding period of 6.5 s until 12.5 s. In order to prove its
e�ectiveness, the negative spring is expected to make the peak frequency shift from an extreme
to the other, that is from the lowest period to the highest and vice versa.
3.3.1 Tuning
The starting point has been decided to be the highest wave frequency (lowest period) of 1 rad/s,
corresponding to the period of 6.5 s and make the shift up to 0.5 , corresponding to the period
of 12.5 s.
If the negative spring contribution can change the resonance frequency through the whole range,
then it would be possible to control the e�ect and reach resonance for each frequency in between
the two chosen extreme values. This can be done by reducing the submerged air volume of the
IVV bottom chamber using a system of valves.
In order to proceed with the test of the negative spring e�ectiveness previously described, the de-
vice shall already resonate in the initial conditions. Therefore, a tuning process has been carried
out by sensitivity analysis on the geometry of the WEC, aiming to �nd geometry con�gurations
for the body to resonate at 6.5 s.
The geometric parameters identi�ed are:
1. Top chamber radius
31
2. Top chamber draft
3. Central column radius
4. Distance between lower part of top chamber and upper surface of bottom chamber
5. Radius of bottom chamber
6. Draft of bottom chamber
Looking at the peak frequency formula it is already clear that the most in�uent parameter is
the top chamber radius; in fact it a�ects the wet cross section, that is the top chamber cross
area, with squared proportion. The others parameters a�ect the mass by contribution to the
total volume. The bottom chamber dimension have a minor in�uence n the excitation force and
radiation coe�cient because of the increased depth, which damps the dynamic forces.
The tuning process has been iterative. Every dimension was de�ned starting from a �rst guess
value obtained from examples of real full scale OWC prototypes. Then the parameters were
made vary around the initial value, while keeping the other dimensions �xed. In this way various
geometry layouts were identi�ed to resonate around the set frequency.
Among the tuned geometries obtained, layouts with smaller top �oater radii were discarded.
These characteristic leads to less restoring forces connected to the wet cross section and therefore
to higher heave motion at peak frequency. Behaviors with very large resonance RAOs are not
consistent, because a heave motion much bigger than the top �oater draft would describe a
not realistic OWC jumping out of the water. The geometry chosen to carry on the study with
consisted in:
• Rf = 5 m
• Hf = 3.5 m
• Rc = 1 m
• Hc = 14 m
• Rb = 5 m
• Hb = 3.5 m
Despite the supplied geometry guarantees the model to resonance at the indicated frequency, it
is important to notice that when the IVV is added to the system it comes along with two extra
32
walls, which act as an extension of the bottom air chamber lateral surface in order to allow the
water free surface to heave with no costraints. This feature, although necessary, changes the
geometry characteristic and thus the dynamic of the system, which would resonate in di�erent
conditions. In light of this consideration, to have a correct insight of the IVV in�uence on the
WEC dynamics, the open bottom case must be compared not with the closed bottom case taken
into account for the �rst tuning, but with a similar OWC presenting the aforementioned extended
walls of the chamber, with a �xed water surface within. In this way the only di�erence between
the two cases would be the variation of the immersed volume and so the negative spring force.
3.3.2 WAMIT outputs and implementation
WAMIT has been run along a range of frequencies speci�ed in the POT �le: the simulation
started from 0.07 rad/s doing 80 steps of 0.07 rad/s, covering the spectrum until 2 rad/s. Even
though the suggested range surpass by far the designed area of interest, a wide spectrum was
chosen in order to allow to identify any possible incongruence or irregularity in the hydrodynamic
coe�cient behavior.
Added mass and damping were obtained from the numeric WAMIT output �frc.1� while the
excitation force calculated from Haskind relation and di�raction potential were respectively in
�frc.2� and �frc.3�. A plot of the datasets in function of frequency was used as a feedback to check
if the WAMIT simulation had occurred successfully: the trend had to show neither relevant noise
nor inconsistencies.
The plots of hydrodynamic coe�cients as well as the excitation forces are displayed below. The
mutual actions between two parts of the system, such main body and column, are referred as
cross modes, while the independent actions on one single part are referred as proper modes.
33
Figure 11: Added mass for proper modes and cross mode for tuned, closed-bottom-case
Figure 12: Damping for proper modes and cross mode for tuned, closed-bottom-case
34
Figure 13: excitation forces on the two elements for tuned, closed-bottom-case
WAMIT outputs were then used to implement the analytical dynamic model. The code
procedure operates as following:
• De�nition of the WAMIT output �le names and physical constants
• Acquisition of the geometry data from WAMIT geometry �le
• Acquisition of hydrodynamic coe�cients and forces from WAMIT output �les
• Removal of hydrodynamic coe�cients corresponding to in�nite and zero frequencies
• Re-arrangement of WAMIT datasets in matrices
• Generation of inertia and hydrostatic matrices
• Solution of equations of motion
35
Figure 14: WEC body's and water column free surface's RAO
The �gure above shows the trend of the Response Amplitude Operator(RAO) in heave, repec-
tively in black for the motion of the central column's water free surface and blue for the body
structure. As a consequence of the tuning process, the system peack frequency corresponds to
the required value, that is 1 rad/s.
3.4 Open bottom case
After de�ning a starting point geometry with the tuning process, the IVV e�ect was introduced
by simulating the bottom chamber open below. In this case there is an extra water free surface,
which means three degrees of freedom and 3x3 matrices. The geometry subroutine used to model
this con�guration has two extra patches than the previous one, that is 11 patches in total. The
lower part of the bottom chamber is now considered as another heaving free surface, while two
circular walls were added. The two patches represents the external and internal surfaces of the
bottom chamber part �lled with water, thus being de�ned as dipoles.
As explained in the previous paragraph, the new geometry was used keeping the water free
surface inside the bottom chamber as a rigid patch, that is considering it part of the main body
and so without any proper motion.
36
Figure 15: Heave RAO for OWC with extra walls but �xed water surface patch
As expected, the peak frequency is no longer the one for the zero case geometry, but is the one
to be compared with the IVV implemented case. At this point, a new subroutine was created in
the NEWMODES �le, in order to take into account the new extra degree of freedom; introducing
the new generalized mode with the index j=8, similarly to what had been done previously for
the �rst free surface.
The results of applying the IVV e�ect on the reference geometry are displayed below.
37
Figure 16: added mass of the proper modes for body, column and chamber
Figure 17: damping coe�cient of the proper modes for body, column and chamber
38
Figure 18: added mass of the cross modes between body, column and chamber
Figure 19: damping coe�cient of the cross modes between body, column and chamber
39
Figure 20: Heave RAO for WEC body, central column free surface and bottom chamber free
surface with IVV e�ect
By comparing �gure 20 with �gure 15 it can be noticed that the peak frequencies are almost
identical. This evidence means that an IVV air volume with size similar to the top �oater one
brings a shift in frequency almost null. This conclusion is con�rmed by the analytical relation
between IVV air volume and shift in frequency previously analyzed.
40
4 WEC array analysis
4.1 Array concept
It was concluded in the previous section that, to obtain an interesting shift in the resonance
frequency through the IVV method, the necessary volume of the bottom chamber was too large,
and therefore the method becomes unfeasible for a single WEC. In this section a possible solution
to overcome the aforementioned volume problem is studied.
The solution presented is to consider a common air volume shared by multiple devices: each
bottom chamber can be connected through a pipe to an extra air container. The container
can then provide the necessary large volume to match the frequency shift requirements. In this
way the cost of the extra volume is divided by all devices of the array. Furthermore, the extra
structure can be designed in such a way so it can be part of the mooring system. The object
of the section is therefore the analysis of the IVV method implemented in devices arranged in
array, each of them connected to a common air volume.
The behavior of a device within the array mentioned above is di�erent from the single WEC
case; in fact the common volume connects every WEC between each other. This means that
the immersed volume variation of a given device depends on the heaving motion of every device
and every bottom chamber internal free surface, since the air pressure is the same in every
individual chamber. It is assumed that the air can �ow between the chambers and the extra
common structure instantaneously and without energy dissipation. The combination between
the wavelength and the location of the various WECs in the array (i.e., the distance between each
other along the wave propagation direction), might lead to a di�erence in phase of the respective
heave motions. This e�ect was carefully studied.
41
Figure 21: array layout
4.2 Analitical modeling
Consider an array of N WECs as the one shown in �gure 10. Each bottom structure air chamber
is connected to an external structure, which is shared by all WECs.
If the WECs are located along the wave direction at xn, the instantaneous vertical position
of the di�erent WECs can be approximated by
h′n = h′ sin(ωt+ kxn) (59)
42
assuming the air pressure is the same in every bottom chamber and in the additional volume at
each instant. Decomposing the instantaneous air pressure as a �uctuation around an equilibrium
value, p(t) = p0+p′A,where p0 is the average value and p′ the �uctuation and with the assumption
that p′A/p0 << 1 , the linearized form of the isentropic air evolution can be written as
p′A = −γp0V0
Sb
N∑n=1
H ′n (60)
Where:
• γ is the air isoentropic coe�cient
• p0 is the equilibrium pressure
• V0 is the total common volume (chambers and extra structure)
• Sb is the cross section of the bottom chamber of each device
• H ′n is the change in internal height of the air volume at each bottom chamber.
Note that Sb∑Nn=1H
′n represents the total volume variation.
Assuming quasi-static pressure variation, the water pressure at the chamber's water-air interface
is given by
p(t) = ρg(hn +Hn) = ρg(Hn0 +H ′n + hn0 + h′n) (61)
Where the subindex ”0” means average positions and the index ' to time �uctuations with respect
to the mean. Thus
pA′ = ρg(H ′n + h′n) (62)
which is the same for every WEC ‘n′. For a mathematical convenience, this expression can be
rewritten as
p′A = ρg1
N
N∑n=1
(H ′n + h′n) (63)
Substituting (63) in (60) one may write
N∑n=1
H ′n = − ρg
ρg + γp0NV0
Sb
N∑n=1
h′n (64)
and �nally, introducing (64) in (63) and using (59) it is possible to obtain the expression
p′A(t) =1
Nρg + V0
γp0Sb
h′f(t) (65)
43
where f(t) =∑Nn=1 sin(ωt + kxn) is the element which introduces the di�erence between the
modeling of a single WEC and an array. The argument of the sinus for each device consists in a
sinusoidal time dependent term and a space dependent term which represents the phase between
each WEC. As mentioned in the �rst chapter, this study is performed in the frequency domain.
For each frequency, the summation of the N sinus is a sinus with the same frequency, but with
a di�erent amplitude, here designated as fmax. So in order to proceed with the formulation it is
necessary to identify the amplitude, fmax. In the light of the above, (65) becomes
pA = 1Nρg
+V0
γp0Sb
h′fmax (66)
where fmax is the maximum value of f(t).
Similarly to the single WEC case, the net negative spring force in each WEC is
Fns = ρgSbH (67)
where:
• ρ is the sea water density
• g is the acceleration of gravity
• Sb is the cross section of the bottom air chamber of each device
• H is the height of air volume in each single air chamber, obtained from the di�erence
between the pressures on top of the chamber and on the internal water free surface
To use the relation (64) to further obtain (67) would result in a poor approximation; in fact H
extracted in this way would correspond to a mean value of∑Nn=1H
′n, which is only valid if all
the devices have the same motion at the same time. The latter condition applies just when all
the WECs are in phase between each other. For these reasons the further analysis of (67) must
consider the negative spring net force directly from the quasi-static pressure formulas.
Fns = pA − p1 (68)
Where:
• pA is the pressure inside the air chamber, equal for all devices and responsible for the force
acting on the inner part of the upper surface of the bottom chamber
• p1 is the hydrostatic pressure on the top of the air chamber, outside the chamber itself,
de�ned as ρghn
44
Substituting (66) in (68) it becomes
Fns = ρgSb
[1− fmax
N + ρgV0
γp0Sb
]hn (69)
4.2.1 Volume as function of the resonance frequency shift
With the same procedure as in the case of the single WEC, the expression for the ratio between
the resonance frequencies respectively with and without the IVV method results to be:
ω
ω0=
√ρgSf − αρgS
(70)
Using (70) and the expression of the negative spring term given by (69), it is then possible to
write the volume V0 as a function of the shift in resonance frequency(ωω0
), in a similar way to
what was done for the single WEC:
V0 =γp0SbN
ρg
1− fmax
SfSb
[1−
(ωω0
)2]− 1
(71)
From (71) it is possible to tell the value of the common volume for a certain shift in peak
frequency. Similarly to the single WEC case, it is already possible from (71) to identify and
describe an asymptotic trend of the air volume that occurs for a frequency ratio given by the
following equation:
ω
ω0=
√√√√1− SbSf
(1− fmax
N + ρgV0
γp0Sb
)(72)
The asymptotes are reached for the same conditions of a single WEC
limV0→∞
(ω
ω0
)=
√1− Sb
Sf(73)
However, for an array there is another condition for which the asymptotic value of frequency
ratio is reached
limfmax→0
(ω
ω0
)=
√1− Sb
Sf(74)
4.3 Numerical modeling
The layouts of an array of devices connected to a common air tank can be various. Some examples
of possible air containers and dispositions that were thought of are:
45
• Central air volume with circular OWC disposition
• Elongated air volume with lateral OWC disposition on the sides
• Air volume aligned with OWCs, positioned below the devices
The array layout that was chosen to be analyzed consists of a cylindrical air container aligned
with the wave propagation direction, and the WECs placed along the container on both sides.
Figure 21 illustrates schematicly the array con�guration.
Both the number of WECs and the distance between each other have an impact on the heave
RAO of each WEC, through the e�ect of fmax in the chambers air pressure, see equation (66),and
also because of the hydrodynamic interactions between the incident wave and the devices and
between di�erent devices. The array modeled during this work considers 4 WECs. The WECs
are displaced every 30 meters along the wave propagation direction. Similarly to the single WEC
case, the cross modes between main body, water column and bottom chamber were neglected.
For the array case also the cross modes between each device were equally neglected. The only
interaction between the devices considered is the pressure, which is the same in each bottom
chamber and in the extra tank.
The same approach of the single WEC case was used to model the array of devices: the geometric
parameters were input in WAMIT, as well as the location of the bodies, given by the position
of their centres. Each WEC in the array has been modeled according to the following geometric
parameters:
• Rf = 6 m
• Hf = 3.0 m
• Rc = 1.75 m
• Hc = 14 m
• Rb = 5 m
• Hb = 3.5 m
The code used to perform the post-processment of the WAMIT output follows the same steps of
the single WEC case, with the appropriate number of equations of motion, which for a 4 WEC
array is 12. The de�nition of the negative spring term is also di�erent, as can be seen in the
expressions provided in the previous subsection.
46
The �gure below shows an example of change in heave RAO due to the IVV implementation for
the mentioned array. The extra volume considered in the exampe is Vextra = 5 103m3, so the
total volume consists in V0 = Vextra +N ∗Vbody, where Vbody is the air volume inside of each top
�oater connetted to the bottom chamber and the volume of air in the bottom chamber itself.
Figure 22: heave RAO for one device of the array with and without IVV implementation
It can be notice that, despite some inacuracy due to the numerical computations, the heave
RAO goes to 1 when the frequency goes to 0; this is an expected result which supports the
validity of the post-processment code. The inacurancies are expected to come from the introduc-
tion of the hydrodynamic values computed through WAMIT simulations of the array. In fact if
there is no IVV implemented each device can be considered as stand-alone, while a simulation of
multiple bodies connected between each others is much more complex and therefore more lickely
to present inacurancies.
In order to have a better grasp on the volume necessary to have a certain shift in resonance
frequency, equation (71) can be implemented with the chosen value of Sb and Sf , as well as
47
fmax, since the latter depends on the distance between the devices along the wave direction.
The same relation was obtained as well from processing the numerical results of WAMIT sim-
ulations, in the following manner. In order to identify the trend of the volume in function of
the peak frequency ratio, the MATLAB script implementing the equations of motion was run
in loop, swiping through a vector of various volumes. The latter was built starting from V0 = 0
and increasing. For each loop the equation of motions were applied and thus the heave RAO
corresponding to a speci�c volume of the vector was de�ned. The peak frequency was identi�ed
and saved into a vector, then every elements of this vector were divided by the �rst one, cor-
responding to the resonance frequency for V0 = 0, that is the case without any negative spring
e�ect.
The method explained above identi�es a series of points representing the trend of V0 in function
of ω/ω0, which is shown in the �gure below.
Figure 23: numerical results of volume needed for a given peak frequency shift
48
The following �gure represents the comparison between the results in �gure 23 and the trend
obtained from relation (71)
Figure 24: comparison between analitical and numerical methods for identifying the volume
needed for a certain frequency shift
The asymptotic behavior of the volume is respected in both plots and �gure 24 validates the
results of the numerical method, since they follow the same trend of the analytical one. The
analytical solution given by (71) shows negative volumes for low resonance frequency ratios that
obviously do not have any physical sense. To understand what these negative volumes mean, let
us �rst consider the meaning of having an in�nity volume to attaing, in the case of the array
under consideration, a resonance frequency ratio of around 0.58. An air reservoir with in�nit
volume means that the air pressure will not change with the heaving motion of each WEC, and
so that the water internal free-surface in the bottom chambers of the array would be always
at the same level. This gives the lower resonance frequency that can be attained for the array
49
under consideration. What equation (71) shows is that resonance at lower frequencies would
imply that the air itself would behave like a negative spring: the higher the air pressure the
higher the volume, which we know is not possible for air or any other �uid. This is the meaning
of the negative volume that comes out from (71).
From the results shown above an evidence is that there is a limit to the resonance frequency
reduction for each array shape (where the required air volume goes to in�nity). Away from this
limit, the common volume becomes very practical. From the engioneering perspective the main
challange is the connection of air between the devices and the central air chambers.
50
5 Conclusions
The IVV method consists to equip an OWC with an air chamber located on the bottom of the
structure. Being the chamber open below and the air compressible, the air volume varies with
the heave motion of the WEC. This variation generates a force that opposes to the hydrostatic
spring acting as a negative spring. The e�ect of the IVV force on the system dynamics is to bring
a shift in the resonance conditions towards lower frequencies; by controlling the air volume (and
the air compressibility, which reduces for smaller air volumes) it is therefore possible to control
the system resonance and improve its adaptability to various sea states.
The system was modeled both analytically and numerically and it was found out that, in order
to have interesting change in resonance frequency, the dimension of the air volume would need to
be very large and so not justify the implementation of IVV on one OWC. The reason is that the
key factor for the IVV negative spring e�ectiveness is the compressibility of air and with small
volumes the air behaves almost as an incompressible �uid; the e�ect of compressibility becomes
tangible with very large volumes, that are di�cult to implement on a single device.
The analysis was then carried out on a case of array, with the goal to overcome the problem of
the volume sizing. In fact the array solution allows to reduce the central air volume due to the
di�erent heaving phases of the di�erent devices in the array. This central volume is connected
to the di�erent arrays by �exible air ducts. The system had to be described with another model
than the single WEC case since the phase di�erence between the devices introduces additional
terms and further complexity. A speci�c case with 4 WECs 30 meters far from each other along
the wave direction was taken into account. The size of the air volume considered as common
for an array of OWCs is more acceptable and guarantee a more interesting shift in the peak
frequency of each device. Another consideration that can be addressed to the air container is
that without being directly integrated into the WEC's body it can be built of a cheaper material,
or can consists in an in�atable structure; in this way the costs could be further reduced and the
feasibility of the implementation increased.
The IVV method is a�ected by a second limitation which is not due to air compressibility. In
fact, while the size of the volume deals with the problem of a �uid not compressible enough, a
di�erent issue arises when the air acts as fully compressible: which means its pressure does not
change with the reduction of the bottom volumes in the di�erent WECs due to their heaving
motions.This situation describes the limit case of an in�nite volume, in which in response to a
volume change the pressure does not change. The force associated to the negative spring in this
51
case is similar to the hydrostatic restoring force on the �oater. It causes a shift in resonance
frequency proportional to the ratio between the cross areas of the bottom chamber and the top
�oater. As a consequence of that, for �oater cross surfaces larger than the bottom chamber ones,
the shift in frequency can not go further a speci�c asymptotic value. On the other hand, if the
�oater radius is equal or smaller the chamber radius, this constraint does not occur.
5.1 Future work
In a future investigation on the IVV topic, the focus should be addressed to the array case.
Di�erent WEC geometries shall be taken into account in order to match reference designs that
are interesting for real sea state cases and asymptotic frequencies that do not constrain the
system once the negative spring is implemented.
It would be interesting also to vary the number of devices into the array and the distance between
each other, which means di�erent array layouts. Those tests should follow a further analytical
analysis on the term of fmax and help to have a better grasp on its in�uence in the equations of
motion.
If the obtained results would be positive, then a more complex modeling could be necessary,
eventually considering non-linear models. For a more precise analysis the cross e�ect, as well as
the shadowing e�ect between the devices shall be calculated.
The air container should be modeled and computed in WAMIT and integrated into the array.
The shape of the devices can also be re�ned in order to have better hydrodynamic behavior. At
that point it would be interesting to simulate turbines in each device and calculated the e�ect
that the IVV has directly on the energy output; following with a techno-economic analysis.
52
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iv
Annex A: Description of WAMIT
WAMIT is a radiation/di�raction panel program developed by the Massachusetts Institute of
Technology for the analysis of the interaction of surface waves with various types of �oating
and submerged structures, that can be represented using either the traditional low-order panel
method or a higher-order method based on B-splines, which can be more versatile.
The quantities computed by WAMIT are:
1. Hydrostatic data. These include the volume, coordinates of the center of buoyancy and
the matrix of hydrostatic and gravitational restoring coe�cients. All the hydrostatic data
are expressed as surface integrals over the mean body submerged surface Sb by applying
Gauss' divergence theorem.
2. Added mass and damping coe�cients
3. Excitation force coe�cients. These can be calculated either using the Haskind relations or
from direct integration of hydrodynamic pressure.
4. Body motions in waves. This is the Response Amplitude Operator (RAO), computed by
two alternative procedures, corresponding to the two ways to compute the excitation forces.
5. Hydrodynamic pressure.
6. Free surface elevation.
7. Velocity vector in the body and in the �uid domain.
8. Mean drift force and moment.
Cartesian coordinates are used to de�ne the body geometry; the latter, as well as motions and
forces, are de�ned in relation to the body coordinates (x; y; z), which can be di�erent for each
body if multiple bodies are analyzed. The z-axis must be vertical, and positive upwards. If
planes of symmetry are de�ned for the body, the origin must be on these planes of symmetry.
The global coordinates (X; Y;Z) are de�ned with Z = 0 in the plane of the undisturbed free
surface, and the Z- axis positive upwards.
v
Inputs
In order to run the software it is necessary to provide appropriate input �les. WAMIT consists
of two subprograms, POTEN and FORCE, which normally are run in sequence. Their function
are described below:
POTEN It solves the radiation and di�raction velocity potentials (and source strengths)
on the body surface for the given modes, frequencies and wave headings. It takes the most of
the computational e�ort for each single WAMIT run.
FORCE It computes global quantities such the hydrodynamic coe�cients, motions, and
�rst and second-order forces. The subprogram also evaluate velocities and pressures on the body
surface, as well as additional �eld data like velocities and pressures at speci�c positions in the
�uid domain and wave elevations on the surface.
As stated above, the heavier subprogram is POTEN: it is possible then to use several times the
intermediate output data from the POTEN run, called �P2F� without launching it again and
just varying the parameter desired in the �nal output.
Filename Usage Description
�lename.pot POTEN Potential Control File
�lename.gdf POTEN Geometric Data File
�lename.frc FORCE Force Control File
gdf.spl POTEN Spline Control File
fnames.wam POTEN/FORCE Filenames list
break.wam POTEN Optional �le for runtime breakpoints
con�g.wam POTEN/FORCE Con�guration �le
�lename.cfg POTEN/FORCE Con�guration �le
gdf.ms2 POTEN Multi Surface geometry �le
gdf.csf FORCE Control surface geometry �le
gdf.bpi FORCE Speci�ed points for body pressure
frc.rao FORCE External RAO �le
Table 1: WAMIT input �les
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Output
The �rst three input �les are required in all cases. The others are required in some cases or are
optional. The following table lists the output �les which are produced by each single run:
Filename Program Description
pot.p2f POTEN P2F File (binary data for transfer to FORCE)
errorp.log POTEN Error Log File
errorf.log FORCE Error Log File
wamitlog.txt POTEN/FORCE Log �le of inputs
frc.out FORCE Formatted output �le
frc.num FORCE Numeric output �les
gdf pan.dat POTEN Panel data �le
gdf pat.dat POTEN Patch data �le
gdf.pnl FORCE Panel data �le
gdf.hst FORCE Hydrostatic data �le
rgklog.txt POTEN/FORCE MultiSurf log �le
frc.fpt FORCE List of �eld points
gdf.idf POTEN Interior free-surface panels
gdf.bpo FORCE Speci�ed points for body pressure
gdf low. gdf POTEN Low-order GDF �le
gdf csf.dat FORCE Control surface data �le
gdf low.csf FORCE Low-order control surface �le
Table 3: WAMIT output �les
Reference Geometry
The geometry used in WAMIT during the tests had been built on to a previous reference ge-
ometry already available in the WAMIT tests database. The reference is test 17: �cylinder with
moonpool�. The geometry consists in two concentric cylinders, where the inner one stands as a
�uid chamber, as shown in the �gure. The inner chamber of �uid, referred to as a `moonpool',
is open at the bottom of the cylinder to the external �uid domain. The top of the moonpool is
a free surface with atmospheric pressure.
The whole structure can be de�ned by describing just a quadrant as the software can replicate
vii
the shape over the two planes of symmetry xz and yz as explained above.
The geometry is represented analytically by the subroutine CYLMP (IGDEF=-7). In TEST17
three patches are used to represent the outer side r=RADIUS, the annular bottom z =-DRAFT,
and the inner side r=RADMP. The free surface inside the moonpool is part of the physical free
surface, and the appropriate free-surface boundary condition is satis�ed by the Green function.
The side of one quadrant is omitted in the �gures below, to show more clearly the bottom and
the side of the moonpool.
The free surface is modeled using the piston mode method, considering the surface heaving while
remaining �at; the pressure distribution on the free surface patch is constant.
Figure 25: representation of moonpool cylinder
The software takes the information about how to generate the geometries from a dynamic
library �le �.dll�, which has to be present in the folder where the executable �le is launched. The
same measure is necessary for the modes source �les, which provides the commands to simulate
the physics of the various models. Both �le must have speci�c names, respectively �geomxact� and
�newmodes�. If a body which is not included in the examples above can be described explicitly
by analytic formulae (either exactly or to a suitable degree of approximation) a corresponding
subroutine can be added to the GEOMXACT.F �le. Reference can be made to the source
�le GEOMXACT.F and to the subroutines already provided to understand the appropriate
procedures for developing new subroutines. Users of WAMIT cannot modify the source code
in general. However GEOMXACT has been separated from the rest of the source code, and
compiled separately as a dll (dynamic link library) to be linked to the rest of the executable code
viii
at run time. Thus users of the PC-executable code can modify or extend GEOMXACT for their
own applications.
The same goes for NEWMODES if the user wants to add subroutines to the generalized modes'
code.
The right subroutine for the geometry generation is taken as input in the geometric data �le
with extension �.gdf�, as well as the number of patches, that must equal the one in the de�ned
subroutine. The two parameter that provides these information are called respectively NPATCH
and IGDEF. The subroutine corresponding to the determined mode is similarly speci�ed in the
con�guration �le. Two optional con�guration �les may be used to specify various parameters and
options in WAMIT. The �rst con�guration �le is assigned the reserved name con�g.wam, while
the second �lename is speci�ed by the user, with the extension `.cfg'. Both �les are opened and
read, if they exist, and the con�guration parameter can be included in either the �rst or second
�le. The �rst �le is used for parameters which are the same for most or all applications and the
second �le for parameters which depend on the speci�c run. The parameter addressing to the
mode is IGENMDS: if di�erent than zero, use a DLL �le containing the subroutine NEWMODES.
This option can be used with both the low- and higher-order options. In every run of WAMIT
performed in this study IGNEMDS was included in the .cfg �le.
The geometric data �le also provides the values of the parameter used as input by the geometry
source DLL �le. In the geomxact subroutine CYLM, patch number 4 is de�ned as the circular
disc of radius RADMP in the plane Z=0. However allowance must be made for the motions of the
actual free surface relative to the body. This is done by de�ning appropriate generalized modes,
which are nonzero only on patch 4. The most important mode is a vertical translation, assigned
here in the subroutine �le NEWMODES.F with the index j = 7. Even if a pitch rotation of the
liquid (j = 8) could be also included to provide a more general de�ection of the free surface,
the adopted piston mode method neglects other motion outside heave. Therefore, only the
extra degree of freedom number 7 has been added to the system, also due to the fairly smaller
impact of the other degree of freedom to the model simulation. This added generalized mode,
physically analogous to heave but de�ned relative to the body, is introduced via the subroutine
MOONPOOL FS in NEWMODES.F.
ix
