Adrien Grellier

Vincent Degat

Siyimane Moyaine

Julie Hemmer

Hélène Clémot

Laetitia Pernod

Reda Jaafri Je Remercie Également Monsieur

Jean-Yves Hascoët

Madame Annie

Keywords: Opérations marines, Dynamique multicorps, Dynamique de câbles, Écoulement potentiel, Weak-scatterer, Interaction uide-structure, Découpe de maillages Marine operations, Multibody dynamics, Cable dynamics, Potential ow theory, Weak-scatterer, Fluid-structure interaction, Panel cutting method

En premier lieu, je souhaite remercier Messieurs René Huijsmann et Julien Salomon pour avoir accepté d'être les rapporteurs de cette thèse de doctorat. Leurs commentaires sur mon travail m'ont été précieux. Je remercie également Madame Erin Bachynski et Monsieur Boyin Ding pour avoir fait partie du Jury de cette thèse. J'ai bénécié tout au long de ce doctorat d'un excellent encadrement au sein du LHEEA. Je suis reconnaissant à Pierre Ferrant, directeur de thèse, ainsi qu'à Aurélien Babarit et François Rongère, tous deux encadrants, d'y avoir fortement contribué. Je tiens à remercier tout particulièrement Pierre Ferrant et Aurélien Babarit pour m'avoir permis de terminer ma thèse dans de bonnes conditions. Je sais gré à Aurélien Babarit d'avoir toujours relu mon travail avec précision et rapidité, de m'avoir donné l'opportunité de faire une visite dans un laboratoire japonais et d'avoir eu l'idée des essais en bassin. Je remercie François Rongère pour ses conseils, toujours enrichissants, notamment en mécanique multicorps et en informatique. Ce doctorat a été nancé via une convention CIFRE avec l'entreprise INNOSEA. Aussi, je remercie Hakim Mouslim et Maxime Philippe de m'avoir choisi pour réaliser cette thèse. Maxime Philippe puis Mattias Lynch ont encadré ce travail chez INNOSEA et je leur suis reconnaissant pour leurs conseils et leurs points de vue industriels ainsi que pour m'avoir laissé toute la liberté nécessaire pour mener à bien ce projet. Je souhaite également remercier toutes les personnes extérieures à l'encadrement qui m'ont apporté leur aide à divers moments au cours de ce doctorat. Je pense en particulier à Lucas Letournel et à Camille Chauvigné pour leur expertise sur le code WS_CN et pour leurs réponses à mes (nombreuses) questions. Je suis très reconnaissant à Félicien Bonnefoy pour avoir encadré et fortement contribué au projet des essais en bassin. Je remercie également Messieurs Alexandre Simos et Rafael Watai pour leur

III.1

Fully nonlinear model of a oating body (yellow) in waves at their exact elevation (solid blue line) and its wetted surface (hatched area) . . . . .

III.2

Weakly nonlinear model based on the weak-scatterer hypothesis of a oating body (yellow) in waves at their incident elevation (solid blue line) and its wetted surface (hatched area) . . . . . . . . . . . . . . . .

III.3

Body exact model of a oating body (yellow) in waves at their mean elevation (solid blue line) and its wetted surface (hatched area) . . . . .

III.4

Linear model of a oating body (yellow) in waves at their mean elevation (solid blue line) and its wetted surface (hatched area) . . . . . . . . . .

III.5

Example of the neighborhood . . . . . . . . . . . . . . . . . . . . . . . .

III.6

Example of an isoparametric line for a cylinder . . . . . . . . . . . . . .

III.7

The steps of the mesh generation of the whole domain with a oating The seven steps of the tight coupling between InWave and WS_CN.

The red, green and blue colors represent the C++, Python and Fortran languages. The arrows denote the communication between the modules.

The color of the arrows depends on the programming language which sends the data. Osmotic energy.

In 2017, 42.6 GWh were generated from these MRE devices. This represents 0.17 % of the global electricity production1 . 19 GW capacity have been installed so far 2 . Oshore wind contributed to 41.6 GWh of the electricity generation and 18.7 MW of the installed capacity, namely 98 % of the total. Thus, oshore wind is the most advanced and used marine energy. Between 2016 and 2017, the oshore wind installed capacity grew by 30

%. Industrial oshore wind farms are currently operational while other marine energies are still in their earliest stage.

Two types of oshore wind turbines exist (Figure 1):

Bottom-xed turbines, mainly used in shallow waters (up to 40 m) for near-shore sites. The foundations include three main designs:

• Monopiles;

• Jackets;

• Gravity-based.

Floating turbines, for wind farms in deep water. The advantages of such a technology are the more powerful and constant winds found further oshore, the increase in available sites and the reduction of visual impact. Turbines may be horizontalaxis or vertical-axis, this latter enabling to lower the center of gravity of the structure and becoming insensitive to wind direction. Only few oating wind turbines have been installed so far. Three main dierent oaters are considered:

• Semi-submersibles;

• Spars;

• Tension-leg platforms.

To increase the wind power extraction, turbines get taller and blades become longer.

Furthermore, wind farms are further oshore and so, subject to rough seas and strong winds. Consequently, oshore installations are more challenging and weather windows for marine operations are shorter and less frequent. Marine operations costs (installation, operations and maintenance) are signicant in oshore wind and have a large impact on the cost of electricity. This latter is split into two expenditures:

Capital expenditure or CAPEX : purchase or improvement of assets (turbine, foundation, sub-station, cables, installation, transportation to the site, etc.);

Operational expenditure or OPEX : ongoing costs to operate a product (land rental, insurance, taxes, operation, maintenance, etc.).

According to Crabtree et al. [START_REF] Crabtree | Wind energy: UK experiences and oshore operational challenges[END_REF], CAPEX accounts for 70 % of the cost of electricity, while OPEX represents 30 %. The reduction of both CAPEX and OPEX is necessary to reach the cost of the onshore wind industry and make the marine renewable energies competitive with other energy sources (coal, oil, gas, etc.). Including installation, the marine operations for oshore wind account for 30 % of the cost of electricity. The reduction of these costs goes through the improvement and the optimization of the marine operations, such as installations, maintenance and decommissioning operations [2]. The reduction of their risks, their duration and the increase of the available weather windows are also necessary. That is why, the study of marine operations (theoretically, experimentally and numerically) is a very important topic for the development of marine renewable energies.

(a) Bottom-xed 3 (b) Floating 4 Figure 1 Oshore wind turbines crane and setting it down on the sea bottom, its foundation or leaving at its stable-oating position (Figure 3);

Ooading operations: two vessels (for instance a FPSO 5 and a shuttle tanker) are connected to each other (to transfer the oil from the FPSO to the shuttle tanker) in tandem (in front of each other) (Figure 4) or side-by-side;

Floatover installation: a vessel transports the topsides (upper parts of oshore structures) to site, then the vessel places the topsides over the xed or oating substructures and lowers the topsides while sustaining the vessel position (Figure 5);

Loadout operations: transferring the cargo from the quay onto a barge using cranes, skid rails, trailer, etc. (Figure 6);

Laying operations: deployment of cables, umbilicals or power cables from a pipelaying vessel equipped with a chute (Figure 7);

Piling operations: oshore monopiles pushed into the seabed (Figure 8);

Lowering and lifting operations;

Etc.

The lowering and lifting operations are the goal of this PhD work, consequently they are presented in details in the next section. 5 Floating, Production, Storage and Ooading Figure 4 Tandem ooading operation between a FPSO and a shuttle tanker 8Figure 5 Floatover installation of a topside 9Figure 6 Loadout of a tension-leg platform using skid rails 10Figure 7 Power cable laying 11 . The buoys indicate the cable position.

Figure 8 Piling operation of a monopile 12 3 Lowering and lifting operations

Lowering and lifting operations are used to lower or lift an object into the sea. The lifting equipment is made of:

A vessel;

A crane on top of the vessel, with possibly several booms;

A hoisting cable connecting the crane to the hook;

A winch to wind or unwind the hoisting cable;

Slings (loops of material around the payload and connected to the hook) or riggings (ropes or chains connecting the payload to the hook);

The payload to be lowered or lifted.

Vessels may be jack-up vessels (Figure 9) or oating crane vessels (Figure 10) (monohull, semi-submersible or catamaran). Jack-up vessels are self-elevating platforms providing a stable position for installation operations.

Payloads may be:

A subsea template [4]; 12 https://www.delta.tudelft.nl A subsea manifold [START_REF] Nam | Experimental and numerical study on coupled motion responses of a oating crane vessel and a lifted subsea manifold in deep water[END_REF];

A ship;

A ROV (Remotely Operated Vehicle);

A wind turbine component (blade, nacelle, foundation) [START_REF] Ren | Development and application of a simulator for oshore wind turbine blades installation[END_REF];

A pre-installed wind turbine [START_REF] Ku | Dynamic response simulation of an oshore wind turbine suspended by a oating crane[END_REF];

A tidal turbine (Figure 10); A wave energy converter;

An anchor;

Etc.

An example of lifting operation is displayed in Figure 10 with a crane-mounted ship, a hoisting cable, riggings and a payload which is a tidal turbine.

A lowering operation is split into four steps [START_REF] Nam | Experimental and numerical study on coupled motion responses of a oating crane vessel and a lifted subsea manifold in deep water[END_REF][START_REF]Modelling and analysis of marine operations[END_REF]:

The payload is lifted o from the deck of the vessel and manoeuvred in the air before being lowered. Snap loads (high tension in the hoisting cable) or pendulum motion involving risks of collision may occur.

The hoisting cable is unwound and the cargo is lowered through the wave zone.

Hydrodynamic loads appear, in particular slamming loads (due to the hydrodynamic impact) which can cause snap loads. In case of lifting operation, slamming loads become water exit loads.

The payload is lowered deeply into the sea. The increase of the cable length involves a modication of the natural frequencies of the system. Vertical oscillations of the payload due to the vessel motion induced by the waves may be signicant.

Current (with possibly both time-dependent velocity and direction) needs to be taken into account and leads to a horizontal oset. The drift motion of the vessel may occur if a dynamic positioning system is not used.

Once the payload is close enough to the seabed, the nal step is the landing operation. In case of lifting operation, it is a retrieval operation. The impact of the object on the seabed must not lead to any damage. The accuracy of the payload position must be guaranteed.

Thus, before starting any lowering or lifting operation, every lifting equipment has to be checked to avoid overloads. This includes the crane capacity, the rigging design, the structural strength of the cargo or the seakeeping of the vessel. This analysis can be eciently support by models. Dierent branches of the physics are necessary:

Multibody dynamics: the oating crane forms an articulated body system (vessel, crane, cable, payload) in mechanical interactions (the motion of the vessel involves a motion of the payload through the cables, and conversely);

Cable dynamics;

Hydrodynamics: several bodies are subject to hydrodynamic loads (vessel, payload), hydrodynamic interactions and slamming loads;

Fluid-structure interaction: waves induce the motion of the vessel and the payload, in return the motion of those bodies modies the wave eld;

Control systems: for reducing the motion of the payload by controlling its position and the tension in the hoisting cable.

A sketch of these several physical elds is presented in Figure 11.

Figure 9 Jack-up vessel 13Figure 10 Lifting operation of the Sabella D10 tidal turbine 14Figure 11 Modeling of a lowering operation. The hoisting cable is in black. The thick arrows represent the uid-structure coupling and the double-headed thin arrows denote the mechanical interactions between bodies.

Note

Other methods exist to lower a payload without using a oating crane. For example, the pendulous installation method requires two vessels: a transportation vessel and an installation vessel, separated one from the other by a distance roughly equal to the water depth. The payload is hung to the transportation vessel by a cable which the other extremity is xed to the installation vessel. Then the payload is detached from the transportation vessel and lowers following a pendulous motion [START_REF] Wang | Research on inuence of horizontal osets of underwater equipment based on pendulous installation method[END_REF].

Another possibility is based on the pencil buoy method [START_REF] Mork | A cost-eective and safe method and for transportation and installation of subsea and structures -the pencil and buoy method[END_REF]. The payload is immersed inshore, then towed to site suspended from a pencil shaped buoy and nally lowered oshore using a winch. 4 State of the art of the numerical simulation of lowering or lifting operations

To schedule the operations, establish the operational conditions (weather windows and uptime levels), design the equipments or predict the body motions to reduce risks, collisions or any injury to the workers, numerical simulation has become an essential tool.

The numerical simulation of lowering or lifting operations at sea has been extensively studied in the literature. These works can be gathered in three domains:

Structural analysis;

Hydrodynamic analysis; Motion control analysis.

Structural analysis

Oshore lifting or lowering operations require to model an articulated multibody system. The rst structural studies were based on simplied approaches. For instance, by assumption, the crane tip follows a prescribed motion. This mechanical model uncouples the dynamics of the payload from the dynamics of the oating crane. It assumes that the payload does not inuence the crane and the supporting vessel. This hypothesis is justied in case of small ratio of the payload mass over the vessel mass [START_REF] Hannan | Numerical simulation of submerged payload coupled with crane barge in waves[END_REF]. Elling and McClinton [START_REF] Elling | Dynamic loading of shipboard cranes[END_REF] and Chin et al. [START_REF] Chin | Nonlinear dynamics of a boom crane[END_REF] computed the dynamics of a single concentrated mass, such as a pendulum, linked to a crane tip by a rigid cable with a prescribed harmonic displacement of the boom. This approach allows studying the motion of the lifted object subject to a parametric excitation. For instance, Elling and McClinton [START_REF] Elling | Dynamic loading of shipboard cranes[END_REF] highlighted a resonance phenomenon when the frequency of the harmonic imposed crane tip motion matches the pendulum natural frequency. Nevertheless mechanical coupling eects can be important [START_REF] Nojiri | Motion characteristics of crane vessels in lifting operation[END_REF].

It is also possible to linearize the motion equation at the system's equilibrium position. The purpose of a linearized model is to perform a frequency-domain analysis and compute the natural frequencies of the system. Comparisons between nonlinear and linearized models along with experimental measurements showed a good agreement for small amplitude motions but not near the resonance [START_REF] Schellin | Crane ship response to wave groups[END_REF][START_REF] Schellin | Response analysis and operating limits of crane ships[END_REF].

Another simplication is to consider a two-dimensional problem. Ellermann et al. [START_REF] Ellermann | Nonlinear dynamics of oating cranes[END_REF][START_REF] Ellermann | Nonlinear dynamics in the motion of oating cranes[END_REF] simulated a oating crane linked to a suspended lumped mass by assuming a motion in a vertical plane. Then, this model was simplied in order to study the inuence of dierent parameters using a multiple scales method and a path-following technique to perform a bifurcation analysis.

Uncoupled or linearized or two-dimensional models suer from a lack of generality.

Payload dynamics may inuence the motion of the oating crane in case of heavy lifting operations, nonlinearities are present in the physics (mechanics, mooring, hoisting cable dynamics, etc.) and a two-dimensional model restricts the simulation to planar motions.

Thus, more complex coupled models were necessary. Lots of studies (Schellin et al. [START_REF] Schellin | Response analysis and operating limits of crane ships[END_REF], Van Den Boom et al. [START_REF] Van Den | Computer analysis of heavy lift operations[END_REF], Witz [START_REF] Witz | Parametric excitation of crane loads in moderate sea states[END_REF], Malenica et al. [START_REF] Malenica | Some aspects of seakeeping of the oating body with attached pendulum[END_REF], etc.) were devoted to the simulation of a oating crane with a payload hanging in the air. The problem is in three dimensions and the cable is either rigid or elastic. Only few studies consider the payload in the water. The dynamics of a submerged payload with a constrained motion using a rigid cable was studied by Bai et al. [START_REF] Bai | Numerical simulation of fully nonlinear wave interaction with submerged structures: Fixed or subjected to constrained motion[END_REF]. Hannan and Bai [START_REF] Hannan | Analysis of nonlinear dynamics of fully submerged payload hanging from oshore crane vessel[END_REF] performed a dynamical analysis of the same constrained payload near a oating barge. The barge is motionless, which simplies the mechanical problem. They used phase trajectories, Poincaré maps and bifurcation diagrams to analyse the payload motion in waves. Bashir et al. [START_REF] Bashir | Simulations of dynamic interaction between a blu body and installation vessel during launch and recovery in rough seas[END_REF] and Nam et al. [START_REF] Nam | Experimental and numerical study on coupled motion responses of a oating crane vessel and a lifted subsea manifold in deep water[END_REF] did a numerical study with both a oating crane and a payload in water.

Since Cha et al. [START_REF] Cha | Dynamic response simulation of heavy cargo suspended by a oating crane based on multibody system dynamics[END_REF], mechanical models have started being based on fully coupled multibody dynamics. These authors used constraint equations to include the internal degrees of freedom of their multibody system. Using this approach, they could simulate two rigid bodies with six degrees of freedom each and linked by an elastic cable (a oating crane with a lifted heavy cargo). This model was also used by Ku and Roh [START_REF] Ku | Dynamic response simulation of an oshore wind turbine suspended by a oating crane[END_REF] in order to simulate the dynamic response of an oshore wind turbine suspended in the air by a oating crane.

So far, motion equations were derived manually. In case of a change in the multibody system, they had to be derived again. Hence, it became convenient to use multibody dynamic algorithms to write automatically the motion equation for most or any multibody systems. Ku and Ha [START_REF] Ku | Dynamic response analysis of heavy load lifting operation in shipyard using multi-cranes[END_REF] used multibody dynamic algorithms based on the Newton-Euler equation to model a heavy load lifting operation with multi-cranes. These algorithms are based on a recursive formulation and can be applied to all multibody systems. Ham et al. [START_REF] Ham | Multibody dynamic analysis of a heavy load suspended by a oating crane with constraint-based wire rope[END_REF] preferred the Euler-Lagrange equation. They formulated it with constraint equations. An application was done with a dynamic analysis of a oating crane with two booms and a heavy load in the air.

Hitherto, only rigid bodies were considered. According to Ren et al. [START_REF] Ren | Dynamic response analysis of a moored crane-ship with a exible boom[END_REF], rigid crane boom can be used in case of short crane booms and small payload-to-ship ratios, otherwise the crane boom exibility should be taken into account. Some studies considers a exible crane boom. Ren et al. [START_REF] Ren | Dynamic response analysis of a moored crane-ship with a exible boom[END_REF] simulated a exible boom oating crane with a point mass payload linked by a rigid rope by assuming a two-dimensional motion.

The authors showed the exibility involves high-frequency vibrations in the surge vessel motion along with a reduction of the payload-swing angle. Park et al. [START_REF] Park | Dynamic factor analysis considering elastic boom eects in heavy lifting operations[END_REF] modeled a exible boom oating crane in three dimensions using a nite-element method and two six-degree-of-freedom rigid bodies for the crane and the payload. Surge and pitch motions were increased in case of the exibility. The high-frequency vibrations found in [START_REF] Ren | Dynamic response analysis of a moored crane-ship with a exible boom[END_REF] are not reported.

Thus, the mechanical model of the lifting and lowering operations becomes more complex with time, from uncoupled bi-dimensional approaches to fully coupled nonlinear dynamics handling mechanical interactions. The same applies for cable models, from a single rigid cable to possibly several elastic cables (slings).

Hydrodynamic and aerodynamic analysis

The main external loads acting on a oating crane and the payload are:

Weight;

Wind loads;

Mooring loads; Hydrodynamic loads.

Wind loads are not always considered in the numerical simulations of lowering or lifting operations. They may induce large motions in case of blade installations [START_REF] Gao | A summary of the recent work at ntnu on marine operations related to installation of oshore wind turbines[END_REF] or light payloads [START_REF]Modelling and analysis of marine operations[END_REF]. When they are computed, the wind loads are mainly expressed following a drag equation [START_REF] Van Den | Computer analysis of heavy lift operations[END_REF][START_REF] Ku | Dynamic response analysis of heavy load lifting operation in shipyard using multi-cranes[END_REF][START_REF] Cha | Development of a simulation framework and applications to new production processes in shipyards[END_REF]:

F W ind = 1 2 C W ind D ρ air AV 2 W ind (1)
with:

F W ind the wind loads;

C W ind D the dimensionless drag coecient;

ρ air the air density;

A the projected area of the bodies which contact with wind;

V W ind the wind velocity.

More complex models using a time-varying wind velocity [START_REF] Ham | Development and validation of a simulation-based safety evaluation program for a mega oating crane[END_REF] and taking into account wind turbulence [START_REF] Ren | Development and application of a simulator for oshore wind turbine blades installation[END_REF] are also developed. No wind loads will be considered in this PhD work.

Regarding mooring loads, dierent theories are used: quasi-static [START_REF] Ham | Development and validation of a simulation-based safety evaluation program for a mega oating crane[END_REF], linear [START_REF] Witz | Parametric excitation of crane loads in moderate sea states[END_REF] or nonlinear [START_REF] Schellin | Response analysis and operating limits of crane ships[END_REF]. These models are presented in chapter II.

Hydrodynamic loads are split into several components: Hydrostatic loads;

Hydrodynamic loads which include loads due to:

• Wave-structure interaction eects;

• Current;

• Viscous eects.

When the current and viscous loads are computed, they follow a drag equation [START_REF] Schellin | Response analysis and operating limits of crane ships[END_REF][START_REF] Cha | Development of a simulation framework and applications to new production processes in shipyards[END_REF][START_REF] Ham | Development and validation of a simulation-based safety evaluation program for a mega oating crane[END_REF]. Hydrostatic loads may be either linear or nonlinear. A linear approach is used when the hydrostatic loads are computed based on the mean wetted surface of the bodies. They are expressed with a linear hydrostatic matrix [START_REF] Nam | Experimental and numerical study on coupled motion responses of a oating crane vessel and a lifted subsea manifold in deep water[END_REF][START_REF] Schellin | Response analysis and operating limits of crane ships[END_REF][START_REF] Van Den | Computer analysis of heavy lift operations[END_REF]:

F Hydrostatic = F B -K Hydrostatic X (2)
where F Hydrostatic , F B , K Hydrostatic and X represent the hydrostatic loads, the buoyancy force, the hydrostatic stiness matrix and the position vector of the body, respectively.

When hydrostatic loads are nonlinear, the pressure is integrated over the exact wetted body surface [START_REF] Ku | Dynamic response simulation of an oshore wind turbine suspended by a oating crane[END_REF][START_REF] Cha | Dynamic response simulation of heavy cargo suspended by a oating crane based on multibody system dynamics[END_REF]. Lee et al. [START_REF] Lee | Dynamic response of a oating crane in waves by considering the nonlinear eect of hydrostatic force[END_REF] compared the use of linear and nonlinear hydrostatics in case of a lowering operation and showed that nonlinearities in the hydrostatic loads increase the dynamic motion.

Concerning the wave-structure interaction, most of the studies are based on the linear frequency-domain potential ow theory [START_REF] Ku | Dynamic response simulation of an oshore wind turbine suspended by a oating crane[END_REF][START_REF] Schellin | Response analysis and operating limits of crane ships[END_REF][START_REF] Van Den | Computer analysis of heavy lift operations[END_REF][START_REF] Malenica | Some aspects of seakeeping of the oating body with attached pendulum[END_REF][START_REF] Cha | Dynamic response simulation of heavy cargo suspended by a oating crane based on multibody system dynamics[END_REF]. A linear time-domain potential ow theory is used by Ku and Ha [START_REF] Ku | Dynamic response analysis of heavy load lifting operation in shipyard using multi-cranes[END_REF]. Linear theory (both in frequency and time domains) assumes a small amplitude motion of the bodies and a small steepness of the waves. It is valid when the payload is in the air as it is the case in the majority of the works. When the payload is in the water and is lowered or lifted, the linear potential ow theory is not applicable with consistency. The presence of two bodies close to each other leads to hydrodynamic interactions and the lowering or the lifting of the payload involves a large relative amplitude motion. That is why, Hannan [START_REF] Hannan | Numerical simulation of submerged payload coupled with crane barge in waves[END_REF] used a fully nonlinear potential ow theory to simulate such marine operations in his PhD work. In this case, the assumptions of the linear theory do not have to be satised.

One part of his work focused on the motion of a fully immersed cylinder in waves and subject to a constrained pendulum motion with a lowering velocity [START_REF] Bai | Numerical simulation of fully nonlinear wave interaction with submerged structures: Fixed or subjected to constrained motion[END_REF]. He also studied the hydrodynamic interactions between a xed oating barge and the same cylindrical payload [START_REF] Hannan | Analysis of nonlinear dynamics of fully submerged payload hanging from oshore crane vessel[END_REF]. The presence of the barge close to the payload modied the motion of this latter due to the hydrodynamic interactions.

More details about the dierent hydrodynamic approaches are given in chapter III.

Thus, as for the mechanical models, the wave-structure interaction models become more complex with time. Most of the studies are based on the linear frequency-domain potential ow theory which involves two strong assumptions on the body motion and the wave steepness. A whole lowering operation cannot be consistently simulated with this theory. The use of an unsteady potential ow based solver to simulate the oating crane and the payload in waves gains in importance.

Motion control analysis

In order to attenuate the heave motion of the lifted object and reduce the tension variation in the hoisting cable, heave compensation systems are used. Their nal goal is to decouple the payload motion from the wave-induced motion of the vessel, especially during a deepwater lowering or lifting operation. Three families of heave compensation exist [START_REF] Nam | Experimental and numerical study on coupled motion responses of a oating crane vessel and a lifted subsea manifold in deep water[END_REF][START_REF] Woodacre | A review of vertical motion heave compensation systems[END_REF]:

Passive heave compensation: a spring-damper system is placed along the hoisting line to shift the heave natural frequency of the lifting equipment. These systems do not require any energy to operate.

Active heave compensation: sensors and controlled actuators (winches, pistons) are used to oppose the heave motion. These systems require energy to operate and are more complex, expensive and ecient than passive heave compensators.

Hybrid active-passive heave compensation: combination of features of passive and active heave compensation systems.

Other motion compensation systems are possible such as an active tugger line force control for single blade installation [START_REF] Zhengru Ren | Active tugger line force control for single blade installation[END_REF]. Motion compensation systems will not be considered in this PhD work.

Norms

Det Norske Veritas (DNV ) provides guidelines about the modelling and the analysis of marine operations [START_REF]Modelling and analysis of marine operations[END_REF]. Three models are proposed for estimating the loads acting on a lowered payload through the wave zone:

A simplied method;

A regular design wave approach;

A time-domain approach.

The simplied method assumes:

The horizontal extent of the payload is small compared to the wave length;

The vertical motion of the payload follows the crane tip motion;

The vertical acting loads are dominant.

This method gives simple conservative estimates of the loads acting on the lifted object. Only characteristic values are included in the computation. For example, the characteristic wave amplitude η a is:

η a = 0.9H S (3)
with H S the signicant wave height.

The characteristic hydrodynamic force may be evaluated by:

F Hyd = (F D + F Slam ) 2 + (F M -F ρ ) 2 (4)
with F D the characteristic hydrodynamic drag force;

F Slam the characteristic slamming impact force;

F M the characteristic hydrodynamic mass force;

F ρ the characteristic varying buoyancy force.

In the regular design wave approach, a single regular wave is applied to the payload which is subject to a prescribed harmonic motion. The design wave amplitude η is: η = 0.9H S for operations performed within 30 min H S for operations performed in more than 30 min

(5)

The time-domain approach of an object lowered into the wave zone is based on the motion equation for a vertical motion only. No horizontal displacement is considered.

The payload is subject to the following forces: hoisting cable force, weight, buoyancy force, steady force due to current, inertia force, wave damping force, drag force, wave excitation force and slamming or water exit force.

Regarding the simulation of both the oating crane and the payload, DNV guidelines split the problem into two categories:

Light lifts, where the mass of the payload is less than 2 % of the oating crane displacement;

Heavy lifts, where the mass of the payload is more than 2 % of the oating crane displacement.

In the rst case, it is assumed that the vessel motion is not aected by the payload motion and the mechanical model is uncoupled. In the second case, the coupled dynamics is required.

The coupled approach of DNV is a nine degree-of-freedom model (six for the crane and three for the payload). The two bodies are linked by a spring. The hydrodynamic loads are evaluated for the crane and the payload using a hydrodynamic database provided by a linear potential ow based solver. Hydrostatic loads are linear too. The mechanical coupling is ensured by a coupling stiness matrix. Thus, this model takes into account both hydrodynamic and mechanical interactions.

In case of a deepwater lowering, DNV norms provide a more accurate approach which models the control of the vertical motion of the lifted object and the horizontal oset due to the current. Additional recommendations are given in case of landing on seabed or retrieval.

DNV norms have been compared to commercial numerical tools (cf. subsection 4.5).

Jacobsen and Leira [4] simulated a submerged towing operation. The simplied method and the numerical results of SIMO were compared to experiments. It was shown that DNV guidelines did not provide an accurate response estimation and should only be used for feasibility studies at an early design stage. Kimiaei et al. [START_REF] Kimiaei | Comparing the results of a simplied numerical model with DNV guidelines for installation of subsea platforms[END_REF] used DNV guidelines released in 1996 and 2008 and compared them to OrcaFlex on the installation of a subsea frame through the wave zone. They concluded that the two versions of the norms overestimates the numerical results of OrcaFlex, the earliest guidelines being less conservative.

Software packages

Several commercial time-domain numerical tools are available for performing simulations of marine operations:

SIMO 15 (SImulation of Marine Operations) developed by SINTEF Ocean and DNV [START_REF][END_REF];

aNySIM 16 developed by MARIN [START_REF]aNySIM Time Domain Analysis of Multi Body Dynamics for Oshore Operations[END_REF];

OrcaFlex 17 developed by Orcina [START_REF][END_REF];

DeepLines 18 developed by Principia and IFP Energies Nouvelles [START_REF]DeepLines 4.5 Theory Manual[END_REF];

ProteusDS 19 developed by DSA [START_REF]Dynamic Systems Analysis Ltd[END_REF];

ARIANE 20 developed by Bureau Veritas [START_REF]Ariane8 theoretical manual[END_REF];

ANSYS Aqwa Mechanical interactions between bodies are taken into account;

Wave loads are computed using a frequency-domain potential ow theory;

Hydrostatic loads are based on a linear approach;

Wave-structure interaction between oating bodies are taken into account;

Additional loads (wind, current, mooring, second-order wave loads, etc.) may be specied.

Nevertheless, there are dierences in the details. For example, OrcaFlex and ProteusDS Hydrodynamic interactions between the oating crane and the payload are rarely studied and when it is done, it is at the price of a too simple mechanical model;

The use of a linear frequency-domain potential ow based solver is not consistent when the payload is lowered or lifted due to the violation of the small amplitude motion hypothesis;

Similarly, the steadiness of the wave-structure interaction due to the linear potential ow theory is not consistent when oating and immersed bodies have a large relative motion;

Commercial software packages are based on the same approach: a multibody dynamical solver coupled with a linear potential ow based solver and suer of the aforementioned conclusions when lowering and lifting operations are simulated;

Norms present guidelines based models which can be too simple.

That is why a new approach is required for performing numerical simulations of lowering and lifting operations. It is necessary to couple both a dynamical solver which would be able to handle multibody numerical computations involving several bodies, articulations, cables and a winch and a wave-structure interaction solver which would be consistent enough to evaluate the unsteady hydrodynamic loads and the hydrodynamic interactions between bodies with a large relative amplitude motion. This analysis is the motivation for this PhD project.

Context of the project

This PhD work was supported by a CIFRE 24 scholarship through a partnership between the LHEEA25 laboratory of Ecole Centrale de Nantes and INNOSEA.

The LHEEA laboratory of Ecole Centrale de Nantes

The LHEEA laboratory of Ecole Centrale de Nantes is an internationally well-known laboratory specialized in numerical simulations and experiments at model scale and in situ in four themes: Free surface hydrodynamics; Fluid-structure interaction; Dynamics of the atmosphere; Systems approach for ground and marine propulsion systems.

One research topic, dedicated to the Ocean Energy and Ocean Waves Group of the lab., is the study of the marine renewable energy devices (oshore wind turbines, wave energy converters, tidal turbines) by numerical simulation of the resource (wave, wind), the motion of the systems and their performance. This PhD is the continuation of several works of the LHEEA. For example Rongère and Clément [START_REF] Rongère | Systematic dynamic modeling and simulation of multibody oshore structures: application to wave energy converters[END_REF] presented a methodology to model and simulate multibody oshore structures. To do so, a dynamic algorithm was coupled with the linear potential ow theory in case of a single oating body (a wave energy converter in the paper).

The laboratory has also developed a potential ow based solver using the weakscatterer hypothesis [START_REF] Letournel | Développement d'un outil de simulation numérique basé sur l'approche weak-scatterer pour l'étude des systèmes houlomoteurs en grands mouvements[END_REF][START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF], named WS_CN. This numerical tool computes the unsteady hydrodynamic loads without doing any assumption on the amplitude of the body motion nor on the wave steepness. So far, the application of this tool has been the simulation of single wave energy converters to evaluate more accurately their performance. This solver will be used in our work.

Thus, the interests of the LHEEA for this PhD are the pursuit of research about the numerical simulation of multibody oshore structures, the continuation of the development of an unsteady potential ow based solver, the quantication of its interest and its application in a new eld: marine operations.

INNOSEA

INNOSEA is an engineering company specialized in marine renewable energy projects and is a spin-o company from Ecole Centrale de Nantes. INNOSEA's main areas of expertise include hydrodynamic, aeroelastic, structural and metocean analyses. The company is also an editor of software tools for the numerical simulation of marine renewable energy devices. This PhD is part of the research and development strategy of INNOSEA to create a numerical tool able to simulate multibody oshore structures. To do so, the software InWave has been developed by INNOSEA and the LHEEA to simulate articulated marine energy devices [START_REF] Combourieu | InWave: a new exible design tool dedicated to wave energy converters[END_REF]. This tool is coupled with the linear potential ow based solver Nemoh [START_REF] Babarit | Theoretical and numerical aspects of the open source BEM solver NEMOH[END_REF].

The interests of INNOSEA for this PhD are the extension of InWave to handle the simulation of marine operations and the coupling with an unsteady potential ow based solver when the use of Nemoh could not be consistent and accurate enough.

Objectives

In order to ll the knowledge gaps exposed in subsection 4.6 and bring a contribution to the numerical simulation of lowering and lifting operations, the main objectives of the PhD thesis are:

The implementation of cable and winch modelling capabilities in the existing multibody dynamic numerical tool (InWave );

The extension of the existing hydrodynamic numerical tool (WS_CN ) to perform simulations involving several immersed or surface-piercing bodies;

The coupling between these two solvers to simulate lowering and lifting operations;

Layout

The comparison of the classical approach to model lowering operations using the linear potential ow theory and the developed numerical tool to quantify the eects of the wave-structure interaction theory used in WS_CN.

Layout

The rst chapter of the thesis introduces the multibody theory used in InWave.

After a state of the art of the multibody system dynamics, the modelling approach based on relative coordinates along with the dynamic algorithm, named The third chapter is dedicated to the hydrodynamic theory. A state of the art of the dierent potential ow approach is presented. The theory of the model used in this PhD, based on the weak-scatterer hypothesis, is presented. Its numerical implementation through the solver WS_CN is introduced for the solving of the hydrodynamic problem as well as for the mesh generator. This numerical tool has been extended to multibody simulations and validated from comparisons with measurements in case of forced motions and with a linear potential-ow based solver for free motions. Other developments such as the free surface remeshing, the parallelization of the computation of the inuence coecients or the use of a sparse linear system solver are also presented.

The fourth chapter introduces the uid-structure coupling between the multibody solver InWave and the hydrodynamic solver WS_CN. A state of the art of the uidstructure interaction is provided. Then, the theory of the tight coupling is explained.

The language binding between the two existing numerical tools is detailed as InWave is implemented in C++ and WS_CN in Fortran. The tight coupling is checked from comparisons with the hydrodynamic solver alone. Other coupling strategies are also studied. The coupling between InWave and CableDyn is presented.

The fth chapter proposes a new mesh strategy to overcome the lack of robustness of the initial mesh generator in WS_CN originally. The explanation of this choice along with a state of the art of the mesh generation are provided. The development of each step of this new mesh generator and its comparison with the initial one are given.

The last chapter is dedicated to the experiments of an upending operation of a spar conducted in the wave basin of Ecole Centrale de Nantes. The simulation of marine operations requires solving the dynamics of a multibody system, constituted, at least, of a oating crane, a cable and a payload in the case of a lowering operation. This chapter introduces a state of the art of the multibody system dynamics which leads to the presentation of the dierent methods to model a multibody system. Several dynamic algorithms are exposed. The Composite-Rigid-Body Algorithm, which is used through this PhD thesis, is presented in details.

I.1.2 Dynamic approaches to multibody simulation

Multibody dynamics is composed of:

Forward or direct dynamics: the acceleration is evaluated from the knowledge of the loads acting on the multibody system;

Inverse dynamics: the loads produced by a given acceleration are computed.

Direct dynamics is used for the simulation of the free body motion. Inverse dynamics is required for the motion control or the trajectory planning. Both approaches are based on the motion equation of the multibody system but used dierently. The mathematical model of the forward dynamics (FD) and the inverse dynamics (ID) can be written as follows [START_REF] Featherstone | Rigid body dynamics algorithms[END_REF]:

q = FD(model, q, q, Γ)

(I.1)
and Γ = ID(model, q, q, q)

(I.2)
with: q the position vector; q the velocity vector;

q the acceleration vector;

Γ the vector of the internal loads;

model the input characteristics of the multibody system (mass, inertia, external loads, etc.).

Only forward dynamics will be used in this PhD thesis.

I.1.3 Direct dynamics algorithms

Direct dynamics algorithms can mainly be formulated according to four dierent approaches [START_REF] Ham | Development and validation of a simulation-based safety evaluation program for a mega oating crane[END_REF][START_REF] Featherstone | Robot dynamics: Equations and algorithms[END_REF]:

The Articulated-Body Algorithm (ABA);

The Composite-Rigid-Body Algorithm (CRBA);

The Augmented Formulation;

The Discrete Euler-Lagrange Equation. propagates the constraints from one body to the next one and calculates the acceleration of one body at a time. The Composite-Rigid-Body Algorithm is more general, because it consists in writing the motion equation of the whole system to nd the accelerations of all bodies at one time [START_REF] Featherstone | Rigid body dynamics algorithms[END_REF].

The ABA has been applied by Rongère and Clément [START_REF] Rongère | Systematic dynamic modeling and simulation of multibody oshore structures: application to wave energy converters[END_REF] to simulate the motion of a single wave energy converter. To evaluate the hydrodynamic loads, the Articulated-Body Algorithm was coupled with the linear potential ow-based theory. Within this theory, one part of the hydrodynamic loads depends on the acceleration of the body:

the added-mass loads. The ABA enables the simulation of a single body subject to hydrodynamic loads. But the simulation of several oating or immerged bodies in hydrodynamic interactions is more challenging and requires the use of the Composite-Rigid-Body Algorithm. Indeed, in this case, the added-mass loads of one body depend not only on the acceleration of the body itself, but also on the acceleration of all other bodies subject to hydrodynamic loads. As the ABA computes the acceleration of one body at a time, it fails to simulate hydrodynamic interactions, hence the need to use the CRBA. Combourieu et al. [START_REF] Combourieu | InWave: a new exible design tool dedicated to wave energy converters[END_REF] coupled the CRBA with a linear potential owbased theory to simulate a complex wave energy converter with several moving bodies in hydrodynamic interactions.

Closed-loop systems can be modeled by the ABA and the CRBA but with additional developments compared to kinematic trees. Closed-loops involve joint variables which are not independent anymore. The main strategy to deal with such multibody systems is to make cuts in the closed-loop system in order to nd a kinematic tree, named a spanning tree. At the cuts, a constraint equation is used based on Lagrangian multipliers [START_REF] Featherstone | Rigid body dynamics algorithms[END_REF].

I.1.3.2 The Augmented Formulation

Lagrangian multipliers form the key point of the Augmented Formulation. Within this approach, the redundant Cartesian coordinates are used to locate each body of the multibody system. The motion equation of the whole multibody system is established assuming each body has six degrees of freedom (dof ). As the redundancy of the chosen coordinate system is due to the geometrical constraints in the multibody system, it is necessary to add constraint equations to take into account these kinematic constraints.

For the sake of clarity, let us consider two six-dof bodies linked by a revolute joint.

There are seventeen unknowns in total: the six degrees of freedom of each body and ve components of the internal loads in the joint. But only twelve scalar motion equations are available. Nevertheless, an algebraic equation may be written to ensure the contact between the two bodies despite their relative rotational motion. This constraint equation allows to express the internal loads in the joint and then to close the system of equations.

Constraint equations and inequations (represented by the function φ) are sorted in several categories (Figure I.2). Equality kinematic constraints represent permanent physical contacts between bodies, whereas inequality constraints arise when bodies can make contact or impact and be separated [START_REF] Siciliano | Handbook of robotics[END_REF].

Constraint equations mainly depend on time and positions only (holonomic constraints) or are velocity-dependent (nonholonomic constraints). Nonholonomic constraints appear for rolling contact whereas holonomic constraints occur for sliding con- x the vertical position of the mass;

y the horizontal position of the mass;

l the rod length.

Figure I.3 Simple pendulum

If this example was modelled using a reduced coordinate set, the unique variable in two dimensions would be θ, the angular position of the mass. With the Augmented Formulation, the Newton-Euler motion equation of the whole multibody system is:

M Ẍ = F c + F ext (I.4)
with:

M the whole system mass matrix;

Ẍ the vector of the Cartesian accelerations of all the bodies; F c the internal loads in the whole system;

F ext the external loads acting on the whole system;

Both F c and Ẍ are unknown in a direct dynamics algorithm. Constraint equations are added to close the system of equations:

φ(X, t) = 0

(I.5)
Internal loads are linked to the constraint equations by the use of Lagrange multipliers λ:

F c = ∂φ ∂X T λ (I.6)
Two time-dierentiations of (I.5) allow to couple the constraint equations to the whole system motion equation (I.4):

   M - ∂φ ∂X T ∂φ ∂X 0    Ẍ λ =   F ext ∂ 2 φ ∂t∂X Ẋ  (I.7)
Equations (I.4) and (I.5) form a dierential-algebraic system of equations (DAE ) of index 3. The index is given by the number of time-dierentiations required to transform a DAE into a rst-order system of ordinary dierential equations (ODE ). Two timedierentiations were necessary to write (I.7). In order to obtain the time-derivation of the Lagrange multipliers, a third time-dierentiation is applied and leads to a DAE of index 3. The index also indicates the numerical diculty to solve the DAE [START_REF] Negrut | On the implicit integration of dierential-algebraic equations of multibody dynamics[END_REF].

Several techniques exist to solve index-3 DAE. The direct integration of index-3 DAE (Equations (I.4) and (I.5)) being subject to numerical instabilities, other strategies are preferred [START_REF] Uchida | Real-time dynamic simulation of constrained multibody systems using symbolic computation[END_REF]. One approach is to use a stabilization method. Equation I.7 is integrated in time. Doing so, the validity of (I.5) is not ensured and a drift motion appears because of the time integration. This is why a stabilization method is used to take into account the violation of (I.5). It is based on an alteration of the internal loads to compensate the errors of the direct integration [START_REF] Negrut | On the implicit integration of dierential-algebraic equations of multibody dynamics[END_REF][START_REF] Baumgarte | Stabilization of constraints and integrals of motion in dynamical systems[END_REF]. Another possible method is based on the reduction of the DAE-index to 1 or 2 [START_REF] Uchida | Real-time dynamic simulation of constrained multibody systems using symbolic computation[END_REF]. Additional Lagrange multipliers (reduction to index 2) or new variables which match the same time-derivative of the Lagrange multipliers (reduction to index 1) are added to the system of equations.

The use of a relative coordinate set with the ABA or the CRBA to model kinematic trees always leads to an ODE system which is greatly easier to solve numerically.

I.1.3.3 The Discrete Euler-Lagrange equation

The last main direct dynamics algorithm formulation is the use of the Euler-Lagrange equation instead of the Newton-Euler equation. Ham et al. [START_REF] Ham | Multibody dynamic analysis of a heavy load suspended by a oating crane with constraint-based wire rope[END_REF] discretized the Euler-Lagrange equation in presence of kinematic constraints, dierent kinds of joints and closed-loops. This method also leads to an automatic writing of the multibody motion equations.

I.1.3.4 Discussion

Each formulation of the direct dynamics algorithms enables the simulation of all multibody systems but with dierent numerical diculties. Both the Articulated-Body-Algorithm and the Composite-Rigid-Body-Algorithm are ecient and straightforward for computing the dynamics of kinematic trees but need much more attention on how the input data are carried out due to the use of a reduced coordinate set. The extension to closed-loop systems can be challenging too. Regarding the Augmented Formulation, no distinction is done between kinematic trees and closed-loop systems as constraint equations are always required. The solving of the kinematic constraints coupled with the motion equations is a dicult numerical task.

I.1.4 InWave

The multibody dynamics solver used through this PhD thesis is InWave. It has been the result of a joint development between the LHEEA laboratory of Ecole Centrale de Nantes and INNOSEA since 2014. The direct dynamics algorithm of InWave is the Composite-Rigid-Body Algorithm for kinematic trees. It is implemented in C++. This algorithm has been coupled with the linear potential ow-based solver NEMOH [START_REF] Babarit | Theoretical and numerical aspects of the open source BEM solver NEMOH[END_REF]. In its early developments, InWave was a numerical tool purely dedicated to the modeling of Wave Energy Converters (WEC ) [START_REF] Combourieu | InWave: a new exible design tool dedicated to wave energy converters[END_REF]. Since 2015, three PhD projects have been launched to extend the capabilities of InWave.

The work of Vincent Leroy, PhD student at the LHEEA laboratory and INNOSEA, is about the unsteady behaviour of oshore oating wind turbines. InWave has been coupled to a free-vortex wake theory-based unsteady aerodynamic solver [START_REF] Leroy | Development of a simulation tool coupling hydrodynamics and unsteady aerodynamics to study oating wind turbines[END_REF]. This coupling was rstly used to investigate the behaviour of oating horizontal-axis wind turbines and was compared to state-of-the-art steady aerodynamic models. Eventually, a steady double multiple streamtube theory-based solver has been implemented and coupled to InWave to study oating vertical-axis wind turbines. The impact of the aerodynamic solver on the behaviour at sea of such systems has been investigated with a comparison to the unsteady aerodynamic coupling results [START_REF] Leroy | Impact of the aerodynamic model on the modelling of the behaviour of a oating vertical axis wind turbine[END_REF].

The work of David Ogden, EngD student at University of Edinburgh (UK) and IN-NOSEA focuses on the numerical simulation of complex wave energy converters with kinematic loops, exible bodies, etc. To do so, InWave has been coupled to the multibody numerical tool HotInt. This has opened new features: complex joints between bodies, closed-loops or ropes. The developed coupling is being validated with a series of WECs [START_REF] Ogden | New mechanical and features for time-domain and WEC modelling and in InWave[END_REF].

The last axis of development of InWave is the aim of this PhD thesis: the simulation of marine operations.

I.2 Modeling of multibody systems I.2.1 Numbering of the bodies

A multibody system is composed of n + 1 bodies. As InWave only considers kinematic trees, the number of joints is n. If the number of joints is higher than n, there is at least one kinematic loop. In A number is given to each body. The body numbered 0 is the base of the multibody system. It is the unique body for which Cartesian coordinates with respect to the global earth-xed inertial frame are used . The base can be either xed or oating (moving).

Other bodies are numbered with increasing order from the base. Their position is based on the modied Denavit-Hartenberg parameters (mDH ). The numbering of the bodies denes an antecedence relationship through the multibody system. Mathematically, if the i-th body is the ancestor of the j-th body, then i < j and the antecedence relationship is written:

i = a j (I.8)
By denition, the base is the single body without ancestor and the terminal bodies, considered as the leaves of the kinematic tree, are the bodies without successor. Joints are numbered such as the joint j connects the body a j to the body j .

I. The advantages of these coordinates are that they allow to describe any kind of multibody systems with a minimum of parameters without ambiguities.

Only one degree-of-freedom joints are considered, either revolute or prismatic. A revolute joint, respectively a prismatic joint, grants for a rotation, respectively a translation, between two bodies. To produce more complex joints, virtual massless bodies are added to ensure the necessary degrees of freedom. For instance, in case of a ball joint linking two physical bodies, two virtual massless bodies are required to create three concurrent-axis and orthogonal to each other revolute joints. The articular variable dening the rotation around the revolute axis joint or the translation along the prismatic axis joint is q j . The set of the articular variables, written q, is the articular position vector of the multibody system.

Each body has a local frame, Σ j , of origin O j and orthonormal basis (x j , y j , z j ).

An extra unit vector, u j is also built to be able to dene the mDH parameters in every situation. The local base frame, Σ 0 is dened with respect to the global inertial earth xed reference frame Σ e , of origin O e and orthonormal basis (x e , y e , z e ). O e , x e and y e lie on the mean water free surface and z e points upwards.

The step-by-step construction of these local frames is based on the following rules [START_REF] Rongère | Systematic dynamic modeling and simulation of multibody oshore structures: application to wave energy converters[END_REF][START_REF] Rongère | Simulation dynamique des systèmes Bateau-Avirons-Rameur(s)[END_REF]:

z j is along the axis of the joint j;

x j is along the mutual perpendicular to z j and one of the succeeding joint axis z k such as a k = j. This denition leads to three possible cases:

• If x a j is not perpendicular to z j , u j is dened along the mutual perpendicular to z a j and z j . • If x a j is perpendicular to z j , then u j is equal to x a j ;

• If the body j is a terminal body, x j can be dened arbitrarily. y j completes the basis such as (x j , y j , z j ) forms an orthonormal basis.

Once the local frames are established, the six modied Denavit-Hartenberg parameters are dened such as: i = a j ; γ j the angle between x i and u j around z i ; b j the distance between x i and u j along z i ; α j the angle between z i and z j around u j ; d j the distance between z i and z j along u j ; θ j the angle between u j and x j around z j ; r j the distance between u j and x j along z j .

If u j = x a j , then γ j = 0 and b j = 0, so only four parameters are necessary. Otherwise, the six parameters are useful. For a given multibody system, the set of mDH parameters is not unique. It depends on the numbering of the bodies, the order of the joints and the orientation of the local frame unit-vectors. The six modied Denavit-Hartenberg parameters (γ j , b j , α j , d j , θ j , r j ) form the reduced coordinate set used in InWave. If the joint j is revolute, then q j = θ j , otherwise q j = r j for a prismatic joint. The general denition of the articular variable becomes: q j = σj θ j + σ j r j (I.9) with:

σ j = 0 if the joint j is revolute; σ j = 1 if the joint j is prismatic; σj = 1 -σ j .
Other mDH parameters stay constant during the simulation. From the mDH parameters, it is possible to dene the rotation matrix from Σ j to the ancestor local frame Σ a j [START_REF] Rongère | Systematic dynamic modeling and simulation of multibody oshore structures: application to wave energy converters[END_REF]: The composition of rotation matrices ensure:

a j R j =   C γ j C θ j -S γ j C α j S θ j -C γ j S θ j -S γ j C α j C θ j S γ j S α j S γ j C θ j + C γ j C α j S θ j -S γ j S θ j + C γ j C α j C θ j -C γ j S α j -S α j S θ j S α j C θ j C α j
j R i = j R k k R i (I.11)
Regarding their inverse, the well-known formula is:

j R -1 i = j R T i (I.12)
The position of the origin of Σ j from the origin of Σ a j and expressed in Σ a j is:

a j P j =   d j C γ j + r j S γ j C α j d j S γ j -r j C γ j S α j r j C α j + b j   (I.13)
In other words, if e O a j O j represents the position of the frame Σ j with respect to the frame Σ a j and expressed in Σ e , it comes: Rotation from Σ e to Σ ψ 0 around z e through an angle ψ 0 . Σ ψ 0 is dened by the orthonormal basis (x ψ 0 , y ψ 0 , z ψ 0 ) where z ψ 0 = z e . The associated rotation matrix is:

a j P j = a j R
e R ψ 0 =   C ψ 0 -S ψ 0 0 S ψ 0 C ψ 0 0 0 0 1   (I.15)
Rotation from Σ ψ 0 to Σ θ 0 around y ψ 0 through an angle θ 0 . Σ θ 0 is dened by the orthonormal basis (x θ 0 , y θ 0 , z θ 0 ) where y θ 0 = y ψ 0 . The associated rotation matrix is:

ψ 0 R θ 0 =   C θ 0 0 S θ 0 0 1 0 -S θ 0 0 C θ 0   (I.16)
Rotation from Σ θ 0 to Σ ϕ 0 = Σ 0 around x θ 0 through an angle ϕ 0 . Σ ϕ 0 is dened by the orthonormal basis (x ϕ 0 , y ϕ 0 , z ϕ 0 ) where x ϕ 0 = x θ 0 . The associated rotation matrix is:

θ 0 R ϕ 0 =   1 0 0 0 C ϕ 0 -S ϕ 0 0 S ϕ 0 C ϕ 0   (I.17)
The six-component generalized position vector to locate Σ 0 with respect to Σ e in term of position and orientation is: x 0 , y 0 , z 0 , ϕ 0 , θ 0 and ψ 0 represent the surge, sway, heave, roll, pitch and yaw of the base body, respectively. If the base is xed, then Σ 0 = Σ e and η0 = 0 6×1 .

η 0 = η trans
The rotation matrix from Σ 0 to Σ e becomes [START_REF] Rongère | Systematic dynamic modeling and simulation of multibody oshore structures: application to wave energy converters[END_REF]:

e R 0 = e R ψ 0 ψ 0 R θ 0 θ 0 R ϕ 0 (I.19) =   C ψ 0 C θ 0 -S ψ 0 C ϕ 0 + C ψ 0 S θ 0 S ϕ 0 S ψ 0 S ϕ 0 + C ψ 0 C ϕ 0 S θ 0 S ψ 0 C θ 0 C ψ 0 C ϕ 0 + S ϕ 0 S θ 0 S ψ 0 -C ψ 0 S ϕ 0 + S ψ 0 C ϕ 0 S θ 0 -S θ 0 C θ 0 S ϕ 0 C θ 0 C ϕ 0   (I.20)
The results of this section allows to locate every point of each body in all local frames Σ j and in the global inertial frame Σ e . To perform automatic dynamic computations, it is necessary to gure out the kinematic recursive relationships for the velocities and the accelerations of each body. This is the goal of the next section.

I.3 Kinematic recursive equations

The six-component generalized velocity vector of body j with respect to Σ e and expressed in Σ j is: j V j = j v j j ω j (I.21) with: j v j the linear velocity of body j with respect to Σ e at the point O j and expressed in Σ j .

j ω j the angular velocity of body j with respect to Σ e and expressed in Σ j .

To establish the kinematic recursive equation for the velocities, let us introduce two extra notations to manipulate more easily both linear and angular velocities:

e v(O j , Σ j /Σ e ) the velocity of the point O j of Σ j with respect to Σ e and expressed in Σ e . The denition of j v j involes:

j v j = j R e e v(O j , Σ j /Σ e ) (I.22)
e Ω(Σ j /Σ e ) the angular velocity of Σ j with respect to Σ e and expressed in Σ e . The denition of j ω j involes: j ω j = j R e e Ω(Σ j /Σ e )

(I .23)
In this section, it is assumed that i = a j .

The transport of the linear velocity from O j of Σ j to O i of Σ i provides: e v(O j , Σ j /Σ e ) = e v(O j , Σ j /Σ i ) + e v(O i , Σ i /Σ e ) + e Ω(Σ i /Σ e ) × e O i O j (I. [START_REF] Bashir | Simulations of dynamic interaction between a blu body and installation vessel during launch and recovery in rough seas[END_REF] with:

e v(O j , Σ j /Σ i ) the relative linear velocity between the frames Σ j and Σ i . It only depends on the nature of the joint j: e v(O j , Σ j /Σ i ) = σ j qj e z j (I.25) e z j the joint axis of joint j in Σ e . Its value in the local frame of the body j is:

j z j = j R e e z j = 0 0 1 T (I.26)
The transport of the angular velocity from Σ j to Σ i gives:

e Ω(Σ j /Σ e ) = e Ω(Σ j /Σ i ) + e Ω(Σ i /Σ e )

(I.27)
with e Ω(Σ j /Σ i ) the relative angular velocity between the frames Σ j and Σ i . It only depends on the nature of the joint j:

e Ω(Σ j /Σ i ) = σj qj e z j

j v j = j R i i v i -j R i S( i P j ) i ω i + σ j qj j z j (I.29) j ω j = j R i i ω i + σj qj j z j (I.30)
with:

S the skew-symmetric matrix of the cross product associated with the vector u such as:

u × v = S(u)v (I.31)
The expression of S for a vector u = u 1 u 2 u 3 T is:

S(u) =   0 -u 3 u 2 u 3 0 -u 1 -u 2 u 1 0   (I.32)
The adoption of a matrix format for (I.29) and (I.30) allows to write the nal kinematic recursive equation for the velocities:

j V j = j T i i V i + qj j a j (I.33)
with:

j T i the transformation matrix between the six-component generalized velocity vectors:

j T i = j R i -j R i S( i P j ) 0 3×3 j R i (I.34)
These transformation matrices follow the same rule for their composition as the rotation matrices:

p T q = p T k k T q (I.35)
j a j the six-component generalized joint axis:

j a j = σ j j z T j σj j z T j T (I.36)
As the base has no ancestor, (I.33) is not valid for j = 0. 0 V 0 is directly expressed from the time-derivative of η 0 . By denition:

ηtrans 0 = e R 0 0 v 0 (I.37)
The angular velocity of the base depends on the time-dierentiation of the Cardan angles:

ω 0 = ψ0 z e + θ0 y ψ 0 + φ0 x θ 0 (I.38)
The three terms of ω 0 are written in three dierent frames, consequently it is necessary to express this vector is the same unique frame. Two choices are possible:

In Σ e [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF]:

e ω 0 =   0 0 ψ0   + e R ψ 0   0 θ0 0   + e R ψ 0 ψ 0 R θ 0   φ0 0 0   (I.39) = S 0 ηrot 0 (I.40)
S 0 denotes the transformation matrix between the intermediate Cardan frames and Σ e :

S 0 =   C θ 0 C ψ 0 -S ψ 0 0 C θ 0 S ψ 0 C ψ 0 0 -S θ 0 0 1   (I.41)
This quantity is not used in the Composite-Rigid-Body Algorithm but is used in the weakly nonlinear potential ow-based theory which is presented in chapter III.

In Σ 0 [START_REF] Rongère | Systematic dynamic modeling and simulation of multibody oshore structures: application to wave energy converters[END_REF]:

0 ω 0 = ϕ 0 R θ 0 θ 0 R ψ 0   0 0 ψ0   + ϕ 0 R θ 0   0 θ0 0   +   φ0 0 0   (I.42) = e Ω -1 0 ηrot 0 (I.43)
e Ω 0 noties the transformation matrix between Σ 0 and the intermediate Cardan frames:

e Ω 0 =     1 S ϕ 0 T θ 0 C ϕ 0 T θ 0 0 C ϕ 0 -S ϕ 0 0 S ϕ 0 C θ 0 C ϕ 0 C θ 0     (I.44)
with T * = tan( * ).

e Ω 0 and S 0 are linked to e R 0 by the following formula:

S 0 e Ω 0 = e R 0 (I.45)

Finally the relationship between 0 V 0 and η0 becomes:

η0 = e J 0 0 V 0 (I.46)
e J 0 denotes the Jacobian matrix between Σ 0 and Σ e :

e J 0 = e R 0 0 3×3 0 3×3 e Ω 0 (I.47)
The kinematic recursive equation for the accelerations is derived using the same method as for the velocities (cf. Appendix A):

j Vj = j T i i
Vi + j γ j + qj j a j (I. [START_REF] Featherstone | Rigid body dynamics algorithms[END_REF] with: j Vj the projection in Σ j of the generalized acceleration vector of body j with respect to Σ e : j V j = j vj j ωj = j R e e vj j R e e ωj (I.49) j γ j the Coriolis and relative angular and linear accelerations:

j γ j = j R i S( i ω i )S( i ω i ) i P j + 2σ j qj S( j R i i ω i ) j z j σj qj S( j R i i ω i ) j z j (I.50)
Regarding the linear acceleration, j vj does not match the time-dierentiation of j v j = j R e e v j . A correction is required. Two results about the time-derivation of rotation matrices needs to be reminded: The time-dierentiation of j v j leads to:

d e R j dt
d j v j dt / Σe = d j R e e v
d j v j dt / Σe = -j R e S( e ω j ) e R j j v j + j vj (I.58)
Making e R j appear in the cross product matrix S and using (I.53), it comes:

d j v j dt / Σe = j vj -S( j ω j ) j v j (I.59)
Regarding the angular acceleration:

d j ω j dt / Σe = j ωj (I.60)
Let us dene ν j as equal to j V j . The application of (I.59) and (I.60) to the base body, by using the six-component generalized accelerations, gives:

ν0 = 0 V0 - S( 0 ω 0 ) 0 v 0 0 3×1 (I.61)
This section established the kinematic recursive equations for the velocities (I. [START_REF] Woodacre | A review of vertical motion heave compensation systems[END_REF] and the accelerations (I.48). These relationships are at the base of the Composite-Rigid-Body Algorithm, presented in details in the next section.

I.4 The Composite-Rigid-Body Algorithm

The Newton-Euler motion equation of the whole multibody system is: V0 nor on the articular articulation q.

H(q) 0 V0 q = 0 6×1 Γ -C(q, q) (I.
Γ the (6 + n) vector of the internal loads along or around the joint axes; 0 V0 the acceleration of the base with respect to the global inertial frame and expressed in the base local frame;

q the vector of the articular accelerations.

The solving of (I.62) requires the knowledge of H, C and Γ. This latter is dened by the user of the multibody solver. For instance, for a wave energy converter, Γ represents the power take-o force (PTO ) of the device. Spring-damper systems are often used [START_REF] Leroy | Benchmarking of the new design tool InWave on a selection of wave energy converters from NumWEC project[END_REF]. In that case, the internal load in a joint takes the following form: [START_REF] Spraul | Parameter calibration in dynamic simulations of power cables in shallow water to improve fatigure damage estimation[END_REF] with: q j the articular variable; k P T O the power take-o stiness; B P T O the power take-o damping.

Γ j = -k P T O q j -B P T O qj (I.
The Composite-Rigid-Body Algorithm aims at computing H and C. To do so, the preliminary step is the writing of the motion equation for each body. The motion equation of body j at O j expressed in its local frame Σ j is: [START_REF] Buckham | Dynamics Modelling of Low-Tension Tethers for Submerged Remotely Operated Vehicles[END_REF] with: j Φ j the time-dierentiation of the linear and angular momentum; j F T j the sum of external and internal loads acting on body j.

j Φ j = j F T j (I.
The time-dierentiation of the linear and angular momenta at O j projected in Σ j is (cf. Appendix A): j Φ j = j M j j Vj + m j S( j ω j )S( j ω j ) j S j S( j ω j ) j I Oj j ω j (I. [START_REF] Davidson | Mathematical modelling of mooring systems for wave energy converters -A review[END_REF] with: j M j the (6 × 6) mass matrix of body j expressed in Σ j :

j M j = m j I 3 -m j S( j S j ) m j S( j S j ) j I Oj (I.66)
m j the mass of body j; j I Oj the (3 × 3) inertia matrix of body j reduced at O j with respect to Σ j ; j S j the position of the center of gravity of body j with respect to Σ j . In other words, if j O j G j denotes the position of the center of gravity of body j with respect to the origin of Σ j and expressed in Σ j , then:

j S j = j O j G j (I.67) I 3 the (3 × 3) identity matrix.
The loads applied to the body j are:

The external loads at O j expressed in Σ j such as weight, hydrodynamic loads, aerodynamic loads, mooring loads, etc.: j F ej ;

The internal loads due to the unique ancestor, at O j and expressed in Σ j : j F j ;

The internal loads j F k due to the successor bodies k such as a k = j, at O j and expressed in Σ j : -

k/a k =j j F k .
After the transport of j F k to O k and in Σ k , it yields:

j F k = k T T j k F k (I.68)
Finally:

j F T j = j F ej + j F j - k/a k =j k T T j k F k (I.69)
Hence, (I.64) becomes:

j M j j Vj + m j S( j ω j )S( j ω j ) j S j S( j ω j ) j I Oj j ω j = j F ej + j F j - k/a k =j k T T j k F k (I.70)
The weight is an example of external loads applied to each body. Its expression in the local frame Σ j is: j F W eight ej = -m j g j R e z e m j gS( j S j ) j R e z e (I. [START_REF] Masciola | Extending the capabilities of the mooring analysis program: A survey of dynamic mooring line theories for integration into FAST[END_REF] where g is the gravity constant.

The Coriolis and relative accelerations along with the external loads are grouped together to give the generalized loads acting on body j: j β j = -j F ej + m j S( j ω j )S( j ω j ) j S j S( j ω j ) j I Oj j ω j (I.72)

Finally, Equation I.70 is simplied using (I.72):

j M j j Vj = j F j - k/a k =j k T T j k F k -j β j (I.73)
The Composite-Rigid-Body Algorithm unfolds in three stages:

A forward loop through the bodies ∀j ∈ 1 ; n ;

A backward loop through the bodies ∀j ∈ n ; 0 ;

The construction of the matrix H and the vector C.

The state vector of this algorithm is:

Y CRBA =     η 0 ν 0 q q     (I .74)

I.4.1 The forward loop

The aim of this loop is the computation of all quantities which depend neither on the articular accelerations nor on the base acceleration nor on the internal loads. That is to say, ∀j ∈ 1 ; n : i = a j ; i R j from (I.10); i P j from (I.13); j T i from (I.34); j V j from (I.33); j γ j from (I.50); j β j from (I.72).

The initialization of this loop is achieved using the state vector (I.74) to calculate:

e R 0 from (I.20); η0 from (I.46).

I.4.2 The backward loop

The aim of this loop is the computation of the (6 × 6) mass matrix and the generalized loads of the composite bodies. The composite body j is composed of the body j and all its successors until the terminal bodies of the branch, whose the branch-root body is body j. For instance, in Figure I.1b, the composite body 1 is made of the bodies 1, 3 and 4. By denition, a composite body has no successor. The composite body of a terminal body is the terminal body itself. The composite body of the base is the whole multibody system. The use of composite bodies justies the name of this direct dynamics algorithm.

The motion equation of the composite body j, using (I.73), is: [START_REF] Huang | Dynamic analysis of three dimensional marine cables[END_REF] with: j M c j the (6 × 6) mass matrix of the composite body j expressed in Σ j ; j β c j the generalized loads (external and inertial) acting on the composite body j and expressed in Σ j ; j F j the internal loads due to the unique ancestor of the composite body j, at O j and expressed in Σ j . Applying (I.73) to the ancestor of body j (the body i such as i = a j ) and by introducing (I.75) and then (I.48), it comes:

j M c j j Vj = j F j -j β c j (I.
i M i i Vi = i F i - k/a k =i k T T i k M c k k T i i Vi + k γ k + qk k a k + k β c k -i β i (I.76)
After the identication of the terms of (I.76) with (I.75), the following recursive equations arise:

j M c j = j M j + k/a k =j k T T j k M c k k T j (I.77) j β c j = j β j + k/a k =j k T T j k M c k k γ k + qk k a k + k β c k (I.78)

I.4.3 Construction of H and C

The rst row of (I.62) is:

H 11 0 V0 + H 12 q + C 1 = 0 6×1 (I.79)
By denition, the base has no ancestor therefore 0 F 0 = 0 6×1 . Thus, (I.75) for j = 0 becomes:

0 M c 0 0 V0 = -0 β c 0 (I.80)
0 β c 0 is obtained by developing recursively j β c j from n to 0 using (I.78):

0 β c 0 = 0 β 0 + n k=1 k T T 0 k M c k k a k qk + n k=1 k T T 0 k β k + k M c k k γ k (I.81)
The identication of the terms of (I.80) and (I.81) with (I.79) gives [START_REF] Rongère | Modèle dynamique des système multicorps ottants avec prise en compte des interactions hydrodynamiques entre les corps[END_REF]:

H 11 = 0 M c 0 (I.82)
col q (H 12 ) = q T T 0 q M c q q a q ∀q ∈ 1 ; n (I.83)

C 1 = 0 β 0 + n k=1 k T T 0 k β k + k M c k k γ k (I.84)
col q (H 12 ) denotes the q-th column of the matrix H 12 .

The second row of (I.62) is:

H 21 0 V0 + H 22 q = Γ -C 2 (I.85)
Γ j denotes the j-th component of Γ and by denition of Γ:

Γ j = j a T j j F j (I.86)
The left-multiplication of (I.75) by j a j allows to write:

j a T j j M c j j Vj = Γ j -j a T j j β c j (I.87)
Let β(j) be the set of the bodies on the direct branch between the base (not included) and the body j (included). The iterative introduction of (I.48) in (I.87) leads to:

j a T j j M c j j T 0 0 V0 + j a T j j M c j k∈β(j) j T k k a k qk = Γ j -j a T j j M c j k∈β(j) j T k k γ k -j a T j j β c j (I.88)
Equation I.85 involves that if 0 V0 = 0 6×1 and q = 0 n×1 then Γ = C 2 . By denoting

j βc j = j β c j 0 V0 =0 6×1 q=0 n×1
, (I.78) becomes:

j βc j = j β j + k/a k =j k T T j k M c k k γ k + k βc k (I.89)
The identication of the terms of (I.88) and (I.89) with (I.85) gives [START_REF] Rongère | Modèle dynamique des système multicorps ottants avec prise en compte des interactions hydrodynamiques entre les corps[END_REF]:

row p (col q (H 22 )) = p a T p p M c p p T q q a q ∀(p, q) ∈ 1 ; n 2 such as p q (I.90)
The upper triangle of H is lled by symmetry.

H 21 = H T 12
(I.91) row p (C 2 ) = p a T p   p M c p k∈β(p) p T k k γ k + p βc p   (I.92)
row p (H 22 ) denotes the p-th row of the matrix H 22 .

Eventually, the backward loop may be summed up in the following way: ∀j ∈ n ; 0 : i = a j ;

i βc i is computed from (I.89); i M c i is computed from (I.77).
The computation of H is achieved using: Once the motion equation of the whole multibody system is formed, it remains to solve this linear system of size (6 + n):

(I.
0 V0 q = H -1 0 6×1 Γ -C (I.93)
A LU decomposition method is used.

Degrees of freedom can be blocked by removing columns and rows from H, C and Γ in order to delete the interactions between the blocked and active dof. Hence, the corresponding components of 0 V0 and/or q are zeroed. Much attention is necessary in case of base-dof blocking because 0 V0 = η0 . For example, if the fourth dof of the base is blocked then 0 ω0 (1) = 0 which is not always equivalent to φ0 = 0.

I.5 Time integration

The nal set of governing equations leads to an ordinary dierential equation system such as Ẏ = f (Y, t). The time-dierentiation of the state vector (I.74) is obtained using:

(I.46) for η0 ;

(I.93) and (I.61) for ν0 ;

(I.93) for q

In InWave, the state vector is time-stepped using either a fourth-order explicit Runge-Kutta (RK4 ) scheme or an adaptive Adams-Moulton scheme. Only the RK4 scheme will be used in this work. We remind the classical time-step equation of the RK4 scheme:

Y(t + dt) = Y(t) + dt 6 (k 1 + 2k 2 + 2k 3 + k 4 ) (I.94) with:
dt the xed time step of the RK4 scheme;

k 1 = f (Y, t); k 2 = f (Y + dt 2 k 1 , t + dt 2 ); k 3 = f (Y + dt 2 k 2 , t + dt 2 ); k 4 = f (Y + dtk 3 , t + dt).

I.6 Conclusion

This chapter presented a review of the general concepts used for multibody dynamics. Several multibody modelings and direct dynamics algorithms were exhibited. In this PhD thesis, bodies are modeled using the modied Denavit-Hartenberg parameters.

The details of the Composite-Rigid-Body Algorithm were also given and demonstrated. The simulation of marine operations involves the use of cables and thus solving the cable dynamics. This chapter introduces a state of the art of cable dynamics. Two models are presented in details: the low-order lumped mass model and the multibody model. This latter is based on the Composite-Rigid-Body Algorithm presented in the last chapter. A validation of these theories is exposed for each model.

This chapter also presents a state of the art of the winch modeling to wind or unwind a cable. The chosen winch approach is detailed and an example of its use is given.

II.1 State of the art

Cables are extensively used in oshore engineering. Mooring lines maintain the oating bodies on station by preventing their drift motion. Mooring cables are also the key point of the stability of tension-leg platforms for instance. Power cables convey the energy production of a marine renewable energy device to the xed power-grid connections resting on the seabed [START_REF] Spraul | Parameter calibration in dynamic simulations of power cables in shallow water to improve fatigure damage estimation[END_REF]. A wide range of marine operations require the use of cables. For example, towing operations need towlines and lowering / lifting operations use hoisting lines. This is why it is necessary to compute the cable dynamics to achieve a marine operation simulation. If the cable is not considered as rigid, the stretching is of rst importance. Bending and torsion eects are signicant in case of low-tension cables where internal loads dominate the cable dynamics [START_REF] Buckham | Dynamics Modelling of Low-Tension Tethers for Submerged Remotely Operated Vehicles[END_REF]. But, the modeling of the bending and torsion loads increase markedly the complexity of the numerical models. A physical eect must be avoided in any cable: snap loads. Snap or snatch load is a spike of very short duration in the cable tension caused when a line becomes slack and then suddenly taut [START_REF] Davidson | Mathematical modelling of mooring systems for wave energy converters -A review[END_REF]. The amplitude of these loads is much higher than the maximum of the dynamic loads. Snapping phenomenon produces shocks in the line material and reduces the fatigue life [START_REF] Hsu | Snap loads on mooring lines of a oating oshore wind turbine structure[END_REF]. DNV denes a criterion, in case of a lowering operation, to avoid snap loads using a 10 % margin for safety [START_REF]Modelling and analysis of marine operations[END_REF]:

F hyd 0.9F static (II.1)
with:

F hyd the characteristic hydrodynamic force, from (4);

F static the static weight of the payload (dierence between the weight and the buoyancy force).

II.1.1 Mathematical modellings

Six dierent mathematical models exist to evaluate cable loads:

Quasi-static approach;

Force-displacement-velocity model;

Lumped mass models;

Finite-element method;

Finite-dierence method;

Multibody approach.

II.1.1.1 Quasi-static approach

The quasi-static theory computes the line position and the tension from the catenary equation for a continuous slack cable. Cable dynamics, internal axial damping loads and hydrodynamic loads are omitted. This model assumes each cable element is only subject to the internal axial tension, the weight and the buoyancy. The line is permanently in static equilibrium during the simulation and the tension only depends on the position of its two end points (the fairlead for the extremity attached to the vessel and the anchor for the other extremity, connected to the seabed). This approach is widely used to describe a mooring system due to its quickness and its ease of implementation [START_REF] Masciola | Implementation of a multisegmented, quasi-static cable model[END_REF]. But the important assumptions lead to large possible discrepancies [START_REF] Davidson | Mathematical modelling of mooring systems for wave energy converters -A review[END_REF]. Nevertheless this method can also be used to initialize the position of a slack cable in order to perform dynamic computations [START_REF] Davidson | Mathematical modelling of mooring systems for wave energy converters -A review[END_REF]. The initialization of the position of a taut cable is gured out by assuming a straight line between the extremities of the cable.

The motion equation for a cable element in quasi-static equilibrium, in the plane dened by these two end points (Figure II.2), gives:

Along the normal axis of the cable element:

T dφ = (W -B) cos(φ) + D n (II.2)
with:

• T the tension;

• W the weight;

• B the buoyancy;

• D n the normal drag force;

• φ the angle between the cable element and the horizontal axis;

• dφ a small variation of φ.

Along the tangential axis of the cable element:

dT = (W -B) sin(φ) -D t (II.3)
with:

• D t the tangential drag force;

• dT a small variation of the tension T .

Figure II.2 Loads on a cable element in quasi-static equilibrium

The quasi-static hypothesis assumes gravity forces (weight and buoyancy) are predominant on drag loads, leading to:

T dφ = (W -B) cos(φ) (II.4) dT = (W -B) sin(φ) (II.5)
These equations implies that the horizontal tension is constant throughout the line and the vertical tension follows the formula [START_REF] Masciola | Implementation of a multisegmented, quasi-static cable model[END_REF]:

V = µs + V A (II.6)
with:

V the vertical tension;

µ the linear weight such as µ = (ρ -ρ water )Ag; ρ the cable density;

ρ water the water density;

A the cross sectional area of the cable;

V A the vertical tension at the anchor;

s the curvilinear coordinate along the cable such as s = 0 at the anchor and s = L u at the fairlead;

L u the unstretched length of the cable.

The tension at any point of the cable is given by:

T (s) = H 2 + (µs + V A ) 2 (II.7)
where H denotes the (constant) horizontal tension in the cable.

The cable position is described by the catenary equation [START_REF] Masciola | Implementation of a multisegmented, quasi-static cable model[END_REF]:

           x(s) = H µ sinh -1 µs + V A H -sinh -1 V A H + Hs EA z(s) = H µ   1 + µs + V A H 2 -1 + V A H 2   + 1 EA µs 2 2 + V A s (II.8)
E is the cable Young's modulus.

Equation II.8 only depends on the horizontal and vertical tensions at the anchor (or at the fairlead). Their computation is achieved by using a Newton-Raphson algorithm to zero the vector F, dened by:

F(H, V A ) = x(L u ) -x input (L u ) z(L u ) -z input (L u ) (II.9)
x input (L u ) and z input (L u ) are the horizontal and vertical displacements between the two end points of the cable.

II.1.1.2 Force-displacement-velocity model

Another simple approach is the use of a force-displacement-velocity model based on cable stiness and damping matrices [START_REF] Borg | Oshore oating vertical axis wind turbines, dynamics modelling state of the art. part II: Mooring line and structural dynamics[END_REF]:

F cable = F 0 -K(x)x -C(x) ẋ (II.10)
with:

F cable the cable loads;

F 0 a constant vertical force representing the mooring system weight, the cable pretension, etc.;

K the mooring stiness matrix;

C the mooring damping matrix

x the body position.

If K and C are constant, the model is linear, otherwise it is nonlinear.

The cable loads computed in this way only represent the global mooring system.

There is no representation of the individual lines. But this approach is easy to implement and leads to fast computations as soon as the two matrices K and C are known.

Slack cables cannot adequately be simulated like this, except in case of small amplitude motions. An example of the use of this forced-displacement-velocity model can be found in [START_REF] Gueydon | Update on the comparison of second-order loads on a tension leg platform for wind turbines[END_REF].

The four other approaches are dynamic models. Inertia eects and hydrodynamic loads are taken into account. Thus, a dynamic model is more accurate for predicting the loads in a cable.

II.1.1.3 Lumped mass models

In the lumped mass theory, the cable is discretized into N Elements cable elements and N Elements + 1 nodes where the mass is concentrated (or lumped) (Figure II.3).

Nodes are connected by spring-damper systems and located with respect to a global inertial frame. Then, the motion equation is written at the position of the nodes [START_REF] Hall | Validation of a lumped-mass mooring line model with DeepCwind semisubmersible model test data[END_REF]:

∀i ∈ 0 ; N (M i + A i )r i = W i + B i + T i + C i + D i (II.11)
with, at the node i: The nodes are the black points and the cable elements the black lines.

The main advantage of the lumped mass theory lies in the strictly diagonal total mass matrix (including mass and added-mass matrices). No matrix inversion is required in this approach. By doing that, the coupling terms of the total mass matrix are neglected. Cases where lines are connected between them, such as nets, may lead to a loss of accuracy [START_REF] Masciola | Extending the capabilities of the mooring analysis program: A survey of dynamic mooring line theories for integration into FAST[END_REF]. Equation II.11 does not include either bending or torsion eects, deriving a low-order lumped mass formulation. Buckham [START_REF] Buckham | Dynamics Modelling of Low-Tension Tethers for Submerged Remotely Operated Vehicles[END_REF] developed a highorder lumped mass formulation including bending and torsion eects. Bending and torsion loads are computed based on the continuous cable equation (II.12) and a niteelement model which approximates the cable by a cubic spline. The total mass matrix is maintained strictly diagonal.

Mr = F + h + w J ω = N + r × F + n (II.12)
with:

M the total mass matrix per cable unit length; r the position vector;

F the internal forces per cable unit length;

h the hydrodynamic loads per cable unit length;

w the weight and buoyancy per cable unit length;

J the inertia matrix per cable unit length; ω the angular velocity;

N the internal moments per cable unit length;

n the external moments per cable unit length;

( ) the spatial dierentiation.

The lumped mass theory is widespread and several examples of its use can be found in the literature [START_REF] Hall | Validation of a lumped-mass mooring line model with DeepCwind semisubmersible model test data[END_REF][START_REF] Driscoll | Development and validation of a lumped-mass dynamics model of a deep-sea ROV system[END_REF][START_REF] Vissio | Expanding ISWEC modelling with a lumped-mass mooring line model[END_REF]. The low-order lumped mass theory is detailed in section II.2.

II.1.1.4 Finite-element method

The nite-element method uses the same cable discretization as the lumped mass theory but using the integral form of (II.12). The motion equation is derived from the virtual work principle [START_REF] Garrett | Dynamic analysis of slender rods[END_REF]. The mass distribution is considered as continuous compared to the lumped mass approach where the mass is discretized at the position of the nodes.

It results in a total mass matrix with o-diagonal coecients expressing the inertial coupling terms. The nite-element method is known to require less cable elements compared to the lumped mass approach but requires a complex mathematical formulation and larger computational time [START_REF] Davidson | Mathematical modelling of mooring systems for wave energy converters -A review[END_REF].

II.1.1.5 Finite-dierence method

The lumped mass and nite-element formulation only discretize the cable in space.

The time-stepping is achieved by an external integrator scheme. Time derivatives are considered as continuous. The nite-dierence method discretizes (II.12) in both space and time. The time-step dt becomes inherent to the discretization as the cable discretization in nodes was inherent in the previous mathematical modellings. Thus, a time-dependent variable x may be discretized, for instance, using a forward, backward or centred scheme [START_REF] Davidson | Mathematical modelling of mooring systems for wave energy converters -A review[END_REF]:

∂x ∂t ≈ x i+1 -x i dt ≈ x i -x i-1 dt ≈ x i+1 -x i-1 2dt (II.13)
This approach is easier to implement than a nite-element theory but is prone to numerical diculties [START_REF] Masciola | Extending the capabilities of the mooring analysis program: A survey of dynamic mooring line theories for integration into FAST[END_REF]. Details on the discretization of the motion equations can be found in [START_REF] Huang | Dynamic analysis of three dimensional marine cables[END_REF][START_REF] Gobat | The Dynamics of Geometrically Compliant Mooring Systems[END_REF].

II.1.1.6 Multibody approach

A multibody approach, based on the Augmented Formulation presented in section I.

φ =   p i+1 -p i θ i Ω i   = 0 5×1 (II.14)
Where p i , θ i and Ω i denote the position of the body i, its bend and twist angles, respectively. The stretching (elasticity), bending and torsion coecients are introduced in the system through constraint regularization [START_REF] Servin | Rigid body cable for virtual environments[END_REF][START_REF] Servin | Massless cable and for real-time and simulation[END_REF]. This approach was also used by Ham et al. [START_REF] Ham | Multibody dynamic analysis of a heavy load suspended by a oating crane with constraint-based wire rope[END_REF] to model a oating crane with a heavy load suspended with a constraintbased wire rope.

II.1.2 Time integration and numerical damping

Regarding the numerical integration of cable dynamics, several schemes are possible. The fourth-order Runge-Kutta scheme, presented in section I.5, remains a popular technique. This time-stepper is robust but requires a small time step due to the stiness of the system of equations [START_REF] Buckham | Dynamics Modelling of Low-Tension Tethers for Submerged Remotely Operated Vehicles[END_REF]. That is why another approach is often chosen: the use of an implicit time integrator which includes numerical damping to attenuate highfrequency waves in cables and more generally in structural dynamics. Most of them are based on the Newmark's scheme [START_REF] Newmark | A method of computation for structural dynamics[END_REF]. It integrates numerically second-order ordinary dierential equations whereas a RK4 scheme deals with rst-order ordinary dierential equations. Let us consider the displacement u n , the velocity un and the acceleration ün at the time t n , the second-order ODE to solve is:

Mü n = F(t n , u n , un ) (II.15)
Integration algorithms are compared in terms of accuracy, stability and numerical dissipation. Accuracy depends on the local truncation error to compute the timedependent variables. An algorithm is accurate to n-order if the error accumulated is O(dt n ). The stability is the property of an algorithm, under any initial conditions, to limit the numerical error during a simulation. An algorithm is said unconditionally stable if the convergence of the solution is independent of the time step dt, otherwise it is conditionally stable for a range of values of dt [START_REF] Kontoe | Development of time integration schemes and advanced boundary conditions for dynamic geotechnical analysis[END_REF]. As all explicit methods, the RK4 scheme is conditionally stable. The stability relies on the spectral radius of the numerical amplication matrix, A, dened by:

  u n+1 un+1 ün+1   = A   u n un ün   (II.16)
The spectral radius of A, notied ρ(A), is the largest absolute value of its eigenvalues.

To ensure the stability of an algorithm, it must satisfy [START_REF] Kontoe | Development of time integration schemes and advanced boundary conditions for dynamic geotechnical analysis[END_REF]:

ρ(A) 1 (II.17)
The numerical dissipation is the capacity of an algorithm to eliminate spurious highfrequency oscillations. The spectral radius of the amplication matrix also measures the dissipation. If ρ(A) = 1 then no dissipation is present but if ρ(A) is lower, the numerical damping increases. The Newmark's scheme time-steps the displacement and the velocity by the following equations [START_REF] Newmark | A method of computation for structural dynamics[END_REF]:

   u n+1 = u n + dt un + 1 2 -β dt 2 ün + βdt 2 ün+1 un+1 = un + (1 -γ)dtü n + γü n+1 (II.18)
Where β and γ are the Newmark's parameters. This scheme is unconditionally stable

if 2β γ 1 2 . Numerical damping is introduced if γ > 1 2 and β 1 4 γ + 1 2
but the accuracy is only of rst order [START_REF] Fung | Numerical dissipation in time-step integration algorithms for structural dynamic analysis[END_REF].

This scheme has been improved to reach a second-order accuracy. Hilber et al. [START_REF] Hilber | Improved numerical dissipation for time integration algorithms in structural dynamics[END_REF] proposed to weight the internal loads between two time steps by adding a third parameter, this is the HHT-α scheme, whereas Wood et al. [START_REF] Wood | An alpha modication of Newmark's method[END_REF] weighted the inertial loads, obtaining the WBZ scheme. Finally, Chung and Hulbert [START_REF] Chung | A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-alpha method[END_REF] decided to weight both internal and inertial loads, leading to the Generalized-α method [START_REF] Mahjoubi | Méthode générale de couplage de schéma d'intégration multi-échelles en temps en dynamique des structures[END_REF]. The equations of the Newmark's scheme (II.18) are used along with a variant of the motion equation solved at each time step:

     Mü n+1-α M = F(t n+1-α F , u n+1-α F , un+1-α F ) ( * ) n+1-α M = (1 -α M )( * ) n+1 + α M ( * ) n ( * ) n+1-α F = (1 -α F )( * ) n+1 + α F ( * ) n (II.19)
The Generalized-α method includes the HHT-α scheme (α M = 0), the WBZ scheme (α F = 0) and the Newmark's scheme (α M = α F = 0). This method is second-order

accurate if γ = 1 2 -α M + α F and unconditionally stable if α M α F 1 2 and β 1 4 + 1 2 (α F -α M ).
The optimal high-frequency dissipation with minimal low-frequency impact is achieved by using [START_REF] Chung | A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-alpha method[END_REF]:

β = 1 4 (1 -α M + α F ) 2 (II.20) α M = 2ρ ∞ -1 ρ ∞ + 1 (II.21) α F = ρ ∞ ρ ∞ + 1 (II.22)
where ρ ∞ is the desirable value of spectral radius at innite frequency, leading to a controllable numerical dissipation.

The Generalized-α method is used, for example, by Buckham [START_REF] Buckham | Dynamics Modelling of Low-Tension Tethers for Submerged Remotely Operated Vehicles[END_REF] with a high-order lumped mass theory and by Gobat [START_REF] Gobat | The Dynamics of Geometrically Compliant Mooring Systems[END_REF] with a nite-dierence approach.

II.1.3 Cable software packages

Several commercial or o-the-shelf numerical tools dedicated to cable dynamics exist. Some of them, considered as the most common, are listed in Table II.1.

Software package

Status Cable theory Time-stepper MAP++ [START_REF] Masciola | Implementation of a multisegmented, quasi-static cable model[END_REF] Open-source Quasi-static ø

MoorDyn [START_REF] Hall | Validation of a lumped-mass mooring line model with DeepCwind semisubmersible model test data[END_REF] Open-source Low-order lumped mass RK21 OrcaFlex [START_REF][END_REF] Commercial High-order lumped mass Generalized-α2 DeepLines [START_REF]DeepLines 4.5 Theory Manual[END_REF] Commercial

Finite elements Newmark

ProteusDS [START_REF]Dynamic Systems Analysis Ltd[END_REF] Commercial High-order lumped mass RK4 Another open-source software package is MoorDyn5 [START_REF] Hall | MoorDyn User's Guide[END_REF]. It uses the low-order lumped mass theory and handles line interconnections, clump weights, oats and seabed friction [START_REF] Hall | Ecient modelling of seabed friction and multi-oater mooring systems in MoorDyn[END_REF]. It is released under a GPLv3 license for its C++-based version and under an Apache License for its Fortran-based version.

Regarding the commercial numerical tools, OrcaFlex and ProteusDS use the highorder lumped mass theory whereas DeepLines relies on a nite-element approach.

II.2 The low-order lumped mass theory

II.2.1 The need of an in-house cable solver

The simulation of lowering or lifting operations involves the computation of cable dynamics. Based on the review of the previous section, it appears that a quasi-static or force-displacement-velocity approaches are not sucient for lowering or lifting operations as neither inertial loads nor hydrodynamic loads are evaluated. Bending and torsion eects may be signicant for slack cables whereas hoisting cables are taut. They also lead to higher computational cost [START_REF] Hall | Validation of a lumped-mass mooring line model with DeepCwind semisubmersible model test data[END_REF]. Consequently, a dynamic model with only the eects of the stretching seems enough in our case. Finite-element and nite-dierence approaches require both complex developments and a coupling between these theories and the Composite-Rigid-Body Algorithm. This leads to two potential solutions for the problem of interest in this PhD: the lumped mass theory and the multibody approach.

The former leads to simple developments if bending and torsion are omitted but the coupling between the lumped mass theory and the CRBA remains. The latter leads to the simulation of cables within the multibody system theory. At the beginning of this PhD work, InWave could not simulate cable dynamics. Therefore, the decision was taken to use both theories.

As seen in subsection II. 1.3, MoorDyn is an open-source low-order lumped mass theory-based numerical tool which matches our specications. This software package is available with a C++-based version and a Fortran-based version. As InWave is implemented in C++, the Fortran-based version involves overcoming the non-interoperability of the two programming languages. Regarding the C++-based version of MoorDyn, it is released under the GPLv3 license. Therefore, if this numerical tool is included in InWave, the total source code must be released. This is not possible for INNOSEA as InWave is their in-house numerical tool.

That is why, it was decided to develop an in-house low-order lumped mass-based cable solver. This solution allows to reach multiple goals:

The comparison between this cable theory and the CRBA;

In case of failure of the simulation of cables by the CRBA, this solver could be used as a third-party module;

There is no problem of license; It can also be used as a dynamic mooring solver.

II.2.2 Theoretical developments

As explained in subsubsection II.1.1.3, the cable is discretized into N Elements elements and N Elements + 1 nodes. In case of a mooring cable, the node 0 represents the fairlead whereas the node N Elements is the anchor. The motion equation at the position of the nodes and in the global inertial frame is:

∀i ∈ 0 ; N Elements (M i + A i )r i = W i + B i + T i + C i + D i (II.23)
The expressions of each load component can be found in [START_REF] Hall | Validation of a lumped-mass mooring line model with DeepCwind semisubmersible model test data[END_REF]. For a better understanding of the physics of the low-order lumped mass theory, they are reminded below.

Figure II.4 Loads at the node i

Each force at the node i is due to the two adjacent cable elements i and i + 1. Thus, the weight W i is the result of the weight of both the half element i and the half element i + 1:

W i = W Element i + W Element i+1 (II.24)
with:

W Element i = 1 2 AL u ρ c g (II.25)
where L u , ρ c and g represent, respectively, the unstretched length of the cable element, the cable density and the gravity acceleration vector. L u is assumed to be constant through the cable.

The expression of the buoyancy force is given by:

B i = B Element i + B Element i+1 (II.26)
with:

B Element i = - 1 2 AL u ρg (II.27)
where ρ is the water density.

The axial tension due to the cable element i is:

T Element i =    - EA L u ( r i -r i-1 -L u ) r i -r i-1 r i -r i-1 if r i -r i-1 L u 0 3×1 otherwise (II.28)
The second condition means there is no compression of the cable.

The axial tension at the node i is expressed by:

T i = T Element i -T Element i+1 (II.29)
The friction between the strands of the cable dissipates energy and is modeled by a linear viscous damper [START_REF] Buckham | Dynamics Modelling of Low-Tension Tethers for Submerged Remotely Operated Vehicles[END_REF]:

C Element i = - CA L u ∂ ∂t ( r i -r i-1 -L u ) r i -r i-1 r i -r i-1 (II.30) with: ∂ ∂t ( r i -r i-1 -L u ) = 1 r i -r i-1 3 j=1 (r j i -r j i-1 )( ṙj i -ṙj i-1 ) (II.31)
r j i is the j th component of the position vector r i . C is the internal viscous damping coecient of the cable. Finally, the axial damping force at the node i is:

C i = C Element i -C Element i+1 (II.32)
The drag loads are computed from Morison's equation [START_REF] Morison | The force exerted by surface waves on piles[END_REF] as the cable can be seen as a slender cylinder. By neglecting the wave kinematics, the relative velocity at the node i is -ṙ i . Its tangential component is (-ṙ i • q i )q i . q i is the tangent unit-vector at the node i, dened by:

q i = r i+1 -r i-1 r i+1 -r i-1 (II.33)
The normal and tangential drag loads, D n i and D t i , are:

D n i = 1 2 ρC dn dL u (ṙ i • q i )q i -ṙi [(ṙ i • q i )q i -ṙi ] (II.34) D t i = 1 2 ρC dt dL u (-ṙ i • q i )q i (-ṙ i • q i )q i (II.35)
where C dn , C dt and d are the normal and tangential drag coecients and the cable diameter. Hence:

D i = D n i + D t i (II.36)
With the same reasoning, the normal and tangential added-mass loads, A n i ri and A t i ri , are:

A n i ri = ρC an AL u [(r i • q i )q i -ri ]
(II.37)

A t i ri = ρC at AL u [(-r i • q i )q i ] (II.38)
where C an and C at are the normal and tangential added-mass coecients. The tangential added-mass force may be used, for example, in case of chains [START_REF] Hall | Validation of a lumped-mass mooring line model with DeepCwind semisubmersible model test data[END_REF]. Finally, it comes:

A i = A n i + A t i (II.39) = ρAL u C an (I 3×3 -q i q T i ) + C at (q i q T i ) (II.40)
Thus, once all the loads are computed, the acceleration of every node may be evaluated:

∀i ∈ 0 ; N Elements ri = (M i + A i ) -1 (W i + B i + T i + C i + D i ) (II.41)

II.2.3 State vector and time integration

The state vector of the low-order lumped mass theory (LM ) is:

Y LM = r ṙ (II.42)
where r and ṙ gather the positions and the velocities of all nodes in following the num-

bering of Figure II.3.
The time-derivative of the state vector is obtained from (II.41). The time-stepping is achieved using a fourth-order explicit Runge-Kutta scheme with a xed time step (I.94).

These theoretical developments have been implemented in a Python numerical tool named CableDyn.

II.2.4 Validation

Buckham [START_REF] Buckham | Dynamics Modelling of Low-Tension Tethers for Submerged Remotely Operated Vehicles[END_REF] presented a validation of his low-order lumped mass-based numerical tool using two test cases considering a single towline. A kinematic boundary condition is applied at the towpoint, located at the node 0 of the cable and at the mean sea level (r 2 0 = r 3 0 = 0). The rst maneuver consists in a horizontal acceleration of this point while the second one is a horizontal deceleration. Comparisons are made with the numerical results of Buckham and experimental data of Vaz and Patel [START_REF] Vaz | Transient behaviour of towed marine cables in two dimensions[END_REF]. The characteristics of the 6-element cable are listed in Table II.2. This cable represents an armoured optical bre cable.

Parameter Value

Equivalent diameter (m)

0.0332

Cable density (kg/m 3 )

3121

Young's modulus (GPa)

77.5

Damping coecient (N s/m 2 ) 11 551

Total unstretched length (m) 300

Normal drag coecient

1.649

Tangential drag coecient 0

Normal added-mass coecient 1

Tangential added-mass coecient 0

Table II.2 Cable mechanical and hydrodynamic properties

The water density, ρ, is 1025 kg /m 3 and the time step is 0.1 s.

In the acceleration test case, the cable is towed at a velocity of 0.566 m/s for a 1000 s period so that it reaches a steady state, then the towpoint is accelerated at a constant acceleration of 0.011 15 m/s 2 over a 60 s interval and achieves a constant speed of 1.235 m/s. The kinematic boundary condition at the towpoint is: Regarding the deceleration test case, the cable is towed at a velocity of 1.286 m/s for the same period of 1000 s. Then the towpoint is decelerated at a constant acceleration of -0.0129 m/s 2 over a 60 s interval and achieves a constant speed of 0.514 m/s. The kinematic boundary condition at the towpoint in this test case is: The comparisons between the experimental data provided by Vaz and Patel [START_REF] Vaz | Transient behaviour of towed marine cables in two dimensions[END_REF],

r 1 0 =      0.566t if t
r 1 0 =      1.286t if t 1000 s 1286 + 1.286(t -1000) -0.00643(t -1000)
the numerical results of Buckham [START_REF] Buckham | Dynamics Modelling of Low-Tension Tethers for Submerged Remotely Operated Vehicles[END_REF] and CableDyn are shown in Figure II.6 Cable proles during the deceleration test case at dierent times: in red at t =1000 s, in blue at t =1120 s, in green at t =1300 s and in black at t =1720 s.

II.3 Cable dynamics using the CRBA

II.3.1 Theoretical developments

In the low-order lumped mass theory, each node has three degrees of freedom.

Within the multibody system modeling, each node needs three joints to ensure two rotations at the position of the node and one translation to take into account the stretching along the cable element linked to the node. As the theory of the Composite-Rigid-Body Algorithm, presented in section I.4, only used single-degree-of-freedom joints, three bodies are necessary, including two massless bodies and one physical body. To sum-up, every cable element j is made of:

A massless body 3j -2;

A revolute joint 3j -2 around y 3(j-1) = z 3j-2 ;

A massless body 3j -1;

A revolute joint 3j -1 around y 3j-2 = z 3j-1 ;

A physical body 3j where the mass is lumped;

A prismatic joint 3j along y 3j-1 = z 3j .

The numbering of the bodies, given in this list for indication, assumes the cable starts at the body 1 which, obviously, is not always the case. The local frames are dened in Figure II.7. For each body u j = x i with i = a j .

Figure II.7 Local frames for the cable element j in assuming q 3j-1 = 0. The red color is dedicated to the prismatic joints (3(j -1) and 3j) while the blue and green colors are used for the revolute joins 3j -1 and 3j -2 respectively.

The modied Denavit-Hartenberg parameters, dened in subsection I.2.2, for these three bodies are given in Table II.3. Based on the construction of the local frames, q 3j is negative. In case of a straight vertical cable, q 3j-2 = q 3j-1 = -π 2 rad.

σ j γ j (rad) b j (m) α j (rad) d j (m) θ j (rad) r j (m) Body 3j -2 0 0 0 - π 2 0 q 3j-2 0 Body 3j -1 0 0 0 - π 2 0 q 3j-1 0 Body 3j 1 0 0 - π 2 0 - π 2 q 3j
Table II.

Modied Denavit-Hartenberg parameters for the cable element j

To ensure the internal loads in the cable, articular torques and forces must be applied to match (II.29) and (II.32). They are dened by [START_REF] Wuillaume | Development and adaptation of the composite rigid body algorithm and the weakscatterer approach in view of the modeling of marine operations[END_REF]:

Γ 3j-2 = 0 (II.45) Γ 3j-1 = 0 (II.46) Γ 3j =      - EA L u (q 3j + L u ) - CA L u q3j if |q 3j | L u - CA L u q3j otherwise (II.47)
Equations (II.45) and (II.46) involve that no internal torque is added in the revolute joints as neither bending nor torsion eect are modeled. Equation II.47 ensures the stretching eects due to the elasticity and the viscous damping. As with the low-order lumped mass theory, no compression of the cable is modeled.

Regarding the hydrodynamic loads, they are only applied on the physical body 3j.

The expression of the buoyancy force is:

3j F Buoyancy e3j = ρAL u g 3j R e z e 0 3×1
(II.48) 3j v 3j is the velocity of the physical body 3j in its own local frame. Its two rst components, 3j v 3j (1) and 3j v 3j (2), represent the normal velocity while the third component, 3j v 3j (3), is the tangential velocity of the cable element. So the drag force at the body 3j is expressed by:

3j F Drag e3j = -     1 2 ρdL u   C dn   3j v 3j (1) 3j v 3j (2) 0     3j v 3j (1) 3j v 3j (2) 0   + πC dt | 3j v 3j (3)|   0 0 3j v 3j (3)     0 3×1     (II.49)
By the same reasoning, the added-mass matrix arises:

3j M Added-mass 3j = 3j A 3j 0 3×1 0 3×1 0 3×1 (II.50) 3j A 3j = ρAL u   C an 0 0 C an 0 0 0 C at   (II.51)
These developments have been implemented in InWave. Furthermore, as nothing in this section is specic to the Composite-Rigid-Body Algorithm, this model can also be applied to the Articulated-Body Algorithm.

II.3.2 Validation

The multibody approach for computing cable dynamics, based on the Composite-Rigid-Body Algorithm, is now compared to the low-order lumped mass theory. The two test cases presented in subsection II.2.4 require the control of the towpoint position.

The cable modeling described in the previous section cannot be used immediately for this purpose. Extra joints should be added to enable the translational motion of the towpoint. For the sake of understanding, a more suitable test case is dened. A 3element cable is considered with the same mechanical properties as dened in Table II.2, except for:

The Young's modulus: 77.5 MPa;

The damping coecient:

10 5 A N s/m 2 ;
The total unstretched length: 10 m.

The time step is 0.001 s and the total duration of the simulation is 10 s.

The node 0 is kept at the position r 0 = 0 3×1 during the simulation while the starting position of the node 3 is r 3 = (-2, 0, -10) T . The two internal nodes are along the straight line between the nodes 0 and 3. In the multibody modeling, 10 bodies are necessary: 9 due to the three cable elements and one more for the base. The cable motion is in the the plane (O e , x e , z e ) so q 3j-1 = -π 2 rad ∀j ∈ 1 ; 3 . As the cable is straight, q 4 = q 7 = -π 2 rad at the starting time. Finally, the initial angle is imposed by xing q 1 =-1.3734 rad. very good agreement is observed. Some dierences are observed, due to the dierent formulation of the hydrodynamic loads. For example, with the lumped mass approach, to dene the drag loads, the tangent direction is given by Equation II.33 and is the approximated direction of a line passing between the two adjacent node points, whereas with the CRBA, q j = e z j , the direction of the cable element j. Each cable element results in three bodies in the multibody system and two of them are massless. This leads to useless computations. In order to speed up the CRBA, instead of using single-degree-of-freedom, cable joints are created. They are formed of the same three bodies presented in the last section.

The forward loop of the Composite-Rigid-Body Algorithm stays identical, because all elementary physical quantities must still be computed. Regarding the backward loop of the CRBA, (I.77) and (I.89) become [START_REF] Wuillaume | Development and adaptation of the composite rigid body algorithm and the weakscatterer approach in view of the modeling of marine operations[END_REF]:

3(j-1) M c 3(j-1) = 3(j-1) M 3(j-1) + 3j T T 3(j-1) 3j M c 3j 3j T 3(j-1)
(II.52)

3(j-1) βc 3(j-1) = 3(j-1) β 3(j-1) + 3j T T 3(j-1) 3j M c 3j 2 k=0 3j T 3j-k 3j-k γ 3j-k + 3j βc 3j (II.53)
Regarding the construction of the matrix H and the vector C, (I.83), (I.90) and (I.92) turn to:

∀q ∈ 1 ; N Elements : col [3q-2,3q-1,3q] (H 12 ) = 3q T T 0 3q M c 3q 3q P q (II.54) ∀(p, q) ∈ 1 ; N Elements 2 such as p q: row [3p-2,3p-1,3p] (col [3q-2,3q-1,3q] (H 22 )) = 3p P T p 3p M c 3p 3p T 3q 3q P q (II.55) ∀p ∈ 1 ; N Elements : row [3p-2,3p-1,3p] (C 2 ) = 3p P T p   3p M c 3p i∈β(3p) 3p T 3i 2 k=0 3i T 3i-k 3i-k γ 3i-k + 3p βc 3p   (II.56) col [3q-2,3q-1,3q] (A), respectively row [3q-2,3q-1,3q] (A)
, denotes the (3q -2)-th, (3q -1)-th and 3q-th columns, respectively rows, of A. N Elements is the number of cable elements. 3j P j represents the (6×3) generalized projection matrix of the cable joint j expressed in the local frame of the body 3j, equivalent of j a j for the single-dof joints, and dened by:

3j P j = 3j T 3j-2 3j-2 a 3j-2 3j T 3j-1 3j-1 a 3j-1 3j a 3j (II.57)

II.4.2 Validation

The test case presented in subsection II. As expected, the ABA is less time-consuming than the CRBA when both of them are based on single-dof joints. Indeed, the Composite-Body-Algorithm needs the inversion of H whereas the Articulated-Body Algorithm does not. It is interesting to notice that the CRBA with a cable joint formulation is less cumbersome than the ABA with single-dof joints.

The time gap between the multibody approaches and the lumped-mass theory originates from the mechanical interactions which are not taken into account by the latter theory. Only one loop is achieved over the nodes of the cable per Runge-Kutta step whereas several loops are made through the multibody system in both the CRBA and the ABA.

II.5 Winch modeling

In a lowering or lifting operation, the hoisting cable is unwound or wound using a winch. A winch is also used in case of a towed body [START_REF] Kamman | Modeling of variable length towed and tethered cable systems[END_REF] or a tethered underwater vehicle system [START_REF] Prabhakar | Dynamics modeling and control of a variable length remotely operated vehicle tether[END_REF]. Therefore, a winch model has to be added to our numerical tool to perform marine operation simulations.

II.5.1 State of the art

Although lot of works exist about the simulation of the dynamics of xed length cables, only a few deal with variable length cables [START_REF] Prabhakar | Dynamics modeling and control of a variable length remotely operated vehicle tether[END_REF]. Banerjee and Do [START_REF] Banerjee | Deployment control of a cable connecting a ship to an underwater vehicle[END_REF] developed an underwater cable dynamics model using a cable controller to achieve the cable deployment or retrieval at the ship. The cable deployment control law was:

L = K V (L C -L) (II.58)
with:

L is the cable scope;

L C denotes the controlled cable length; K V represents the gain.

Wang et al. [START_REF] Wang | Finite element analysis of a threedimensional underwater cable with time-dependent length[END_REF] used a nite-element method to model a time-varying length cable.

The cable length L(t) followed the formula:

L = L 0 + v d t (II.59)
where L 0 is the initial cable length, v d the deployment cable velocity and t the time.

Kamman and Huston [START_REF] Kamman | Modeling of variable length towed and tethered cable systems[END_REF] presented a model of variable length towed cable based on a lumped mass theory. They assumed the number of cable elements to be held constant during the simulation. Thus, when a cable was paid out, respectively reel-in, cable elements were expanded, respectively shortened, one after the other. The thrust force created by the mass ux between the vessel and the cable through the winch was not accounted for. The mass ux is created by the increase or the decrease of the cable mass due its variable length. Prabhakar and Buckham [START_REF] Prabhakar | Dynamics modeling and control of a variable length remotely operated vehicle tether[END_REF] developed a model in which the topmost cable element length became a time varying quantity while the rest of the cables remained constituted of constant length elements. The force induced by the winch was added to the motion equation of the topmost element.

When the element connected to the winch is too long, it is split into a xed length element and a new variable length element. On the contrary, when the topmost element is too small, it is merged with the subsequent downstream element. This model was used by Zand et al. [START_REF] Zand | Ship and winch regulation for remotely operated vehicle waypoint navigation[END_REF] too.

In the software OrcaFlex, the unstretched length of the cable element linked to the winch is controlled and time-dependent [START_REF][END_REF].

A winch model based on the work of Zand et al. [START_REF] Zand | Ship and winch regulation for remotely operated vehicle waypoint navigation[END_REF] and the theory of OrcaFlex has been developed for both the low-order lumped mass theory and the multibody theory.

II.5.2 Thrust force and cable loads

The deployment or the retrieval of a cable by a winch involves a variation of the mass of the cable hanging from the winch. The force due to the momentum ux at the winch boundary, which matches the position of the node 0 is [START_REF] Zand | Ship and winch regulation for remotely operated vehicle waypoint navigation[END_REF]:

F W inch 0 = v d r 1 -r 0 r 1 -r 0 dM 0 dt (II.60)
where v d is the deployment velocity of the winch, positive for a payout, negative for a retrieval.

The unstretched length of the rst cable element is given by:

L u (t) = L u (0) + ˆt 0 v d (τ ) dτ (II.61)
Thus, the mass ux follows the expression:

dM 0 dt = Aρ c v d (II.62)
Equation II.23 for the rst node becomes:

(M 0 + A 0 )r 0 = W 0 + B 0 + T 0 + C 0 + D 0 + F W inch 0 (II.63)
The axial tension and damping due to the top cable element are updated with the time-varying unstretched length.

T Element

1 =    - EA L u (t) ( r 1 -r 0 -L u (t)) r 1 -r 0 r 1 -r 0 if r 1 -r 0 L u (t) 0 3×1 otherwise (II.64) C Element 1 = - CA L u (t) ∂ ∂t ( r 1 -r 0 -L u (t)) r 1 -r 0 r 1 -r 0 (II.65) with: ∂ ∂t ( r 1 -r 0 -L u (t)) = 1 r 1 -r 0 3 j=1 (r j 1 -r j 0 )( ṙj 1 -ṙj 0 ) -v d (t) (II.66)
Regarding the multibody theory, the cable loads are expressed by:

Γ 3 =      - EA L u (t) (q 3 + L u (t)) - CA L u (t) ( q3 + v d (t)) + v d (t) dM 0 dt if |q 3 | L u (t) - CA L u (t) ( q3 + v d (t)) + v d (t) dM 0 dt otherwise (II.67)

II.5.3 Cable deployment

During a pay-out, the unstretched length of the cable element connecting to the winch is increased using (II.61). Once the cable length is too long, the rst element is divided into two elements. The one linked to the winch is a variable-length element while the other ones are constant-length elements of unstretched length L u (0). The splitting criterion is:

L u (t) = αL u (0) (II.68) with α > 1.
If α is too small, the new variable-length element is too small and numerical errors may appear. If α is too big, large elements arise which could also lead to numerical errors. A good compromise is to create a new element whose its length is the half of the initial cable element size, so:

r N ew = r 0 + 1 3 (r 1 -r 0 ) (II.70) ṙNew = ṙ0 + 1 3 (ṙ 1 -ṙ0 ) (II.71)
The mass distribution is updated, which gives for the new node:

M N ew = 1 2 Aρ c (L u (t) + L u ) (II.72)
where L u (t) represents the length of the new rst cable element.

II.5.3.2 Multibody model

Figure II.13 displays the scheme of the cable splitting using the multibody approach.

A cable element, made of three bodies and joints as explained in subsection II.

q 6 = q 3 -q3 (II.79) q6 = q3 -q3 (II.80)
The mass distribution distribution is updated as for the low-order lumped mass model.

II.5.3.3 Comparison

These two models, for the lumped mass theory and the multibody theory, are compared using the test case presented in subsection II.

Cable retrieval

During a pay-in, the unstretched length of the cable element connected to the winch is decreased using (II.61). Once the cable length is too small, the two rst elements are merged. The resulting element is a variable-length element. The merging criterion is:

L u (t) = βL u (0) (II.81) with β < 1.
For the same reasons as given in subsection II.5.3, a good compromise is to merge the elements when: The mass distribution is updated, which gives for the node 1:

M 1 = 1 2 Aρ c (L u (t) + L u ) (II.83)
where L u (t) represents the length of the new rst cable element.

II.5.4.2 Multibody model

Figure II.18 shows the scheme of the cable merging using the multibody approach. A cable node is deleted, which involves the deletion of two cable elements and the creation of a new one. The impact of this transformation also concerns the initial third cable element.

Figure II.18 Cable merging with the multibody theory

The initialization of the articular positions and velocities requires the use of the Cartesian positions and velocities. The Cartesian positions are computed from the homogeneous transformation matrices j h i , dened for the body j with i = a j , by:

j h i = j R i i P j 0 1×3 1 (II.84)
And for the base, by:

0 h e = 0 R e 0 R e η 0 0 1×3 1 (II.85)
The recursive equation for the homogeneous matrices is:

j h e = j h i i h e (II.86)
Finally, the Cartesian position of the body j, e P j , is evaluated by:

e h j = j h -1 e = e R j e P j 0 1×3 1 (II.87)
Regarding the Cartesian velocities, they are computed from:

e V j = e R j j V j (II.88)
Finally, the articular positions and velocities are [START_REF] Buckham | Dynamics Modelling of Low-Tension Tethers for Submerged Remotely Operated Vehicles[END_REF]:

q1 = - π 2 + arctan r 1 2 -r 1 0 r 3 2 -r 3 0 (II.89) q1 = ( ṙ1 2 -ṙ1 0 )(r 3 2 -r 3 0 ) -( ṙ3 2 -ṙ3 0 )(r 1 2 -r 1 0 ) (r 1 2 -r 1 0 ) 2 + (r 3 2 -r 3 0 ) 2 (II.90) q2 =                - π 2 + arctan   -(r 2 2 -r 2 0 )cos q1 + π 2 (r 3 2 -r 3 0 )   if cos q1 + π 2 sin q1 + π 2 ; - π 2 + arctan   -(r 2 2 -r 2 0 )sin q1 + π 2 (r 1 2 -r 1 0 )   otherwise. (II.91) q2 = -( ṙ2 2 -ṙ2 0 )cos q1 + π 2 -q1 sin q1 + π 2 (r 2 2 -r 2 0 ) (r 3 2 -r 3 0 ) -( ṙ3 2 -ṙ3 0 )(r 2 2 -r )cos q1 + π 2 (r 3 2 -r 3 0 ) 2   1 +   -(r 2 2 -r 2 0 )cos q1 + π 2 (r 3 2 -r 3 0 )   2    (II.92) q3 = -r 2 -r 0 (II.93) q3 = 1 r 2 -r 0 3 j=1 (r j 2 -r j 0 )( ṙj 2 -ṙj 0 ) (II.94) q4 = - π 2 + arctan r 1 3 -r 1 2 r 3 3 -r 3 2 -q1 (II.95) q4 = ( ṙ1 3 -ṙ1 2 )(r 3 3 -r 3 2 ) -( ṙ3 3 -ṙ3 2 )(r 1 3 -r 1 2 ) (r 1 3 -r 1 2 ) 2 + (r 3 3 -r 3 2 ) 2 -q1 (II.96) q5 =                -π + arctan   -(r 2 3 -r 2 2 )cos q4 + π 2 (r 3 3 -r 3 2 )   -q2 if cos q4 + π 2 sin q4 + π 2 ; -π + arctan   -(r 2 3 -r 2 2 )sin q4 + π 2 (r 1 3 -r 1 2 )   -q2 otherwise. (II.97) q5 = -( ṙ2 3 -ṙ2 2 )cos q4 + π 2 -q4 sin q4 + π 2 (r 2 3 -r 2 2 ) (r 3 3 -r 3 2 ) -( ṙ3 3 -ṙ3 2 )(r 2 3 -r )cos q4 + π 2 (r 3 3 -r 3 2 ) 2   1 +   -(r 2 3 -r 2 2 )cos q4 + π 2 (r 3 3 -r 3 2 )   2    (II.98) q6 = q 9 (II.99) q6 = q9 (II.100)
where r j i is the j th component of the position vector r i of the node i.

The mass distribution distribution is updated as for the low-order lumped mass model.

II.5.4.3 Comparison

These two models of reel-in are compared from a modied version of the test case presented in subsection II.3.2. The initial angle of -78.69 • is kept but the cable is three times longer and split into six elements instead of three. The node 0 stays at r 0 = 0 3×1 but the other extremity is now at r 6 = (-6, 0, -30) T . The lifting velocity is xed to 0.5 m/s between 8 s and 44 s. No ramp is used on the lifting velocity.

II.6 Conclusion

This chapter presented a review of the dierent theories to model cables. Two models were detailed and compared: the low-order lumped mass theory implementing in CableDyn and a multibody approach using the CRBA of InWave. These two theories model the stretching of the cable but neither the bending nor the torsion. The mass is discretized at the nodes of the cable. First, the low-order lumped mass theory has been validated with two test cases using numerical results and experimental data of the literature. A good agreement is observed. Then, a comparison was done between the two theories of cable dynamics and showed a perfect agreement. A cable joint has also been developed in the CRBA to speed-up the numerical simulations.

A state of the art of the winch models was exposed in this chapter. The cable element connecting to the winch becomes variable-unstretched length. The spatial discretization of the cable changes to ensure the elements are not too long or too short. This model may be used with both the low-order lumped mass theory and the multibody theory.

Its implementation in the low-order lumped mass theory and the multibody theory gave the same results. The simulation of marine operations may involve the presence of two bodies subject to hydrodynamic loads which have a large relative amplitude motion. Then, the classical frequency-domain linear potential ow approach is not applicable anymore. After presenting a review of the hydrodynamic theories, this chapter details the potential ow theory based on the weak-scatterer hypothesis, used in the numerical tool WS_CN. The developments done during this PhD are exposed: the extension to multibody simulation, the free surface remeshing and the parallelization of the computation of the inuence coecients.

III.1 State of the art III.1.1 Potential ow theory

Three hypotheses are done:

The ow is irrotational (H1 );

The ow is incompressible (H2 );

The uid is inviscid (H3 ).

The ows based on these assumptions are named the potential ows. Viscosity and turbulence are neglected. This approximation is valid in case of wave propagation for instance. When a body is present, viscous eects may be important, depending on the body shape. Viscous loads are less important in case of slender body shapes.

Nevertheless, the potential ow theory is widely used, shows some good agreements in case of seakeeping, manoeuvring and wave propagation between the theory and the experiments and presents a good compromise between the accuracy and the CPU-time.

This theory is applied here in hydrodynamics but it may be also used in aerodynamics [START_REF] Borg | Oshore oating vertical axis wind turbines, dynamics modelling state of the art. part I: Aerodynamics[END_REF]. III.1.1.1 Laplace's equation (H1 ) involves:

∇ × v = 0 3×1 (III.1)
where v is the uid velocity.

Equation III.1 leads to:

v = ∇φ (III.2)
where φ is the velocity potential. Thus, only one scalar function is necessary to compute the three dimensional uid velocity, reducing the number of unknowns.

The incompressibility of the ow (H2 ) allows to write:

∇ • v = 0 (III.3)
From (III.3), the Laplace's equation in the uid domain D arises:

∆φ = 0 (III.4)
Consequently, the velocity potential is a harmonic function.

III.1.1.2 Bernoulli's equation

The Bernoulli's equation is derived from the momentum conservation using the three hypotheses of the potential ow theory. Its expression is:

p = -ρ ∂φ ∂t + 1 2 ∇φ • ∇φ + gz (III.5)
where p denotes the total pressure, ρ the uid density, g the gravity constant and z the vertical position.

∂ * ∂t

represents the Eulerian (or partial) derivative.

The last term of the right-hand side of (III.5) is the hydrostatic pressure:

p Static = -ρgz (III.6) III.1.1.

Boundary conditions

Two types of boundaries have to be considered:

Free surface;

Solid surfaces (immersed or oating bodies, surfaces of the numerical tank walls and the sea bottom).

Free surface boundary conditions

The free surface is dened by the function η named the free surface elevation or wave elevation. It is assumed that η is a single-valued function. In this condition, the wave breaking cannot be modeled. The equation of the free surface is:

z = η(x, y, t) (III.7)
The kinematic free surface boundary condition reects that a particle on the free surface always remains part of the free surface:

∂η ∂t = -∇φ • ∇η + ∂φ ∂z at z = η(x, y, t) (III.8)
The dynamic free surface boundary condition ensures the continuity of the pressure at the interface. The atmospheric pressure being taken equal to 0, it yields from the Bernoulli's equation:

∂φ ∂t = - 1 2 ∇φ • ∇φ -gη at z = η(x, y, t) (III.9)

Solid surface boundary condition

The inviscidness of the uid (H3 ) involves a slip condition on solid boundaries of

normal n: v • n = ∂φ ∂n = v Solid • n (III.10)
This body condition ensures the impermeability of the surface. The equality of the tangential components would lead to the creation of a boundary layer which is not consistent with the hypothesis of inviscidness.

III.1.1.4 Hydrodynamic loads

By integration of the total pressure over the instantaneous wetted surface of the body j, S B j (t), of normal vector n, pointing outwards the uid domain, the hydrodynamic loads at the center of gravity G j arise:

e F Hydro j =      ¨SB j (t)
pn dS ¨SB j (t) p( e G j M) × n dS

     (III.11)
The hydrostatic loads are obtained by the integration of the hydrostatic pressure (III.6):

e F Static j =      ¨SB j (t) p Static n dS ¨SB j (t) p Static ( e G j M) × n dS      (III.12)

III.1.2 Numerical simulation of potential ows

Even if the potential ow approximation is the result of many assumptions, the problem stays complex and several nonlinearities are present:

The free surface nonlinearities: the free surface boundary equations needs to be fullled on an unknown surface z = η(x, y, t);

The slip condition is applied on the instantaneous position of the bodies;

The hydrodynamic loads are nonlinear and coupled with the unknown motion of the bodies.

Thus, several classes of potential ow models exist and their dierences lie in the treatment of these nonlinearities:

Fully nonlinear models;

Weakly nonlinear models based on the weak-scatterer hypothesis;

Body-exact models;

Linear models.

III.1.2.1 Fully nonlinear models

When the potential ow theory presented above is directly applied, it leads to fully nonlinear models. The bodies and the free surface are meshed at their real position and the pressure is integrated over the instantaneous wetted surface as shown in Figure III.1.

It is the most accurate method based on the potential ow approximation. It is widely used to simulate wave propagation [START_REF] Ducrozet | 3-D HOS simulations of extreme waves in open seas[END_REF].

But, the uid-structure simulation makes the diculties of the method appear.

Indeed, the boundary conditions on the bodies and the free surface are expressed at the exact positions. Thus, the regridding of the mesh is a necessity: rstly, to update the position of the boundaries, and secondly, to keep a good quality mesh which is deformed by the waves and the bodies. In case of surface-piercing bodies, sawtooth instabilities can appear and smoothing techniques are required. The computation of the intersection curves between the free surface and the bodies for any translational and rotational motion is a dicult task. To avoid the reection of the perturbed waves at the numerical boundaries, articial damping coecients are used on the free surface.

Wave breaking may occur in such a description of the free surface and causes the stop of the simulation. Finally, the space discretization of the mesh must be small enough to simulate all the perturbed waves generated by the presence of the bodies.

All these characteristics make the fully nonlinear models cumbersome and numerically challenging, despite their accuracy.

Nevertheless, fully nonlinear models are subject of an important number of publications [START_REF] Ferrant | Non-linear time-domain models for irregular wave diraction about oshore structures[END_REF][START_REF] Bai | Fully nonlinear simulation of wave interaction with xed and oating ared structures[END_REF]. To reduce the complexity of the fully nonlinear approach, other models have been developed. In most of the other models, both the velocity potential and the wave elevation are decomposed into an incident component and a perturbed component. This latter may also be decomposed into several elementary quantities as it will be presented further.

For the moment:

φ = φ I + φ P η = η I + η P (III.13)
with:

( * ) I the incident component;

( * ) P the perturbed component.

The incident component is assumed to be known while the perturbed part is unknown.

This decomposition allows to apply everywhere in the domain any kind of incident waves (regular or irregular). The incident wave does not have to be propagated, hence no wave maker model is necessary.

The weak-scatterer hypothesis assumes the perturbed quantities have to be small compared to the incident quantity:

φ P φ I η P η I (III.14)
Doing so, the free surface boundary equations are linearized around the incident free surface elevation z = η I (x, y, t). The free surface nonlinearities are simplied. Contrary to the fully nonlinear approach, this surface is known. Thus, it is not necessary to mesh the perturbed waves, only the incident waves needs to be meshed. The pressure is integrated over the instantaneous wetted body surface delimited by the incident wave elevation as presented in Figure III.2. The weak-scatterer hypothesis involves the possibility to use a larger spatial discretization and therefore reduces the CPU-time. But the fullment of this hypothesis is required, which could limit the application of this theory.

No assumption has been done about the body nonlinearities. As it will be presented in subsection III.1.3, this method is particularly adapted to slender surface-piercing bodies with or without forward speed and immersed bodies.

This model is the hydrodynamic theory used in this PhD work. More details and explanations about this approach are given in the rest of this chapter.

Body-exact models

A further simplication may be applied to the free surface conditions. These conditions can be linearized around the mean free surface elevation z = 0. Thus, the free surface mesh remains planar (Figure III.3), which enables a faster mesh convergence and the reduction of computing time. In that case, the pressure is integrated over the wetted body surface delimited by the mean wave elevation. This method is only consistent if small steepness waves are present. The wave steepness, , for a regular wave of amplitude A and wave number k, is dened by: = kA

(III.15)
This method is named the body-exact approximation. An example of such an approach is given in [START_REF] Watai | A time-domain boundary elements method for the seakeeping analysis of oshore systems[END_REF].

φ P = φ P (0) + φ P (1) + 2 φ P (2) + O( 3 ) η P = η P (0) + η P (1) + 2 η P (2) + O( 3 ) (III.16)
with ( * ) P (j) the j th order perturbed component.

This decomposition is introduced in each equation of the potential ow theory.

Terms of the same order of magnitude are collected, dening several problems of different order. Each one depends on the problems of smaller order of magnitude. The zeroth-order matches the hydrostatics. The rst-order problem corresponds to the fully linear problem while the second-order problem bring quadratic terms. In these models, the body boundary conditions are linearized on the mean position of the bodies and the pressure is integrated over this xed mean wetted body surface. Hence, the mesh of the bodies is xed. The linearization of the free surface boundary conditions is also applied, leading to a totally still mesh. The CPU-time is much lower than for the aforementioned models. An example of such an approach for the rst-order problem is also given in [START_REF] Watai | A time-domain boundary elements method for the seakeeping analysis of oshore systems[END_REF]. Another decomposition is classically applied. Based on the superposition principle, the velocity potential is written:

φ P = φ D + φ R (III.17)
φ D represents the velocity potential of diraction, when the body is xed and the incoming wave is present. φ R denotes the total radiation velocity potential, when the body is moving with a forced motion without incoming wave. This latter velocity potential is decomposed into elementary quantities:

φ R = 6 j=1 φ R j (III.18)
where φ R j is the velocity of radiation for the j th degree of freedom. The elementary problems for these velocity potentials are solved and the global solution is obtained by superposition.

In order to add some nonlinearities, some works integrate the incident and hydrostatic pressure over the instantaneous wetted body surface delimited by the incident free surface elevation while the dynamic pressure is computed over the mean body surface. This is the so-called nonlinear Froude-Krylov approximation. The Froude-Krylov loads are dened by:

e F F K j =      ¨SB j (t) p I n dS ¨SB j (t) p I ( e G j M) × n dS      (III.19)
where p I represents the pressure due to the incident velocity potential φ I :

p I = -ρ ∂φ I ∂t + 1 2 ∇φ I • ∇φ I (III.20)
Although this method can show good results, it suers from an inconsistency as the hydrodynamic loads are not assessed over the same surfaces. An example of application of this approach is given by Gilloteaux [START_REF] Gilloteaux | Mouvements de grande amplitude d'un corps ottant en uide parfait[END_REF].

The linear models can also be solved in frequency domain. In this case, the body motion is assumed steady and all time-dependent quantities f (x, y, z, t) are written as follows:

f (x, y, z, t) = Re f (x, y, z)e -iωt (III.21)
where i is the imaginary unit, ω the wave frequency and f the complex amplitude of f . The radiation problems are considered for an harmonic motion and for each of the wave frequencies and directions of interest. Once these hydrodynamic problems are solved, a hydrodynamic database is created with the added-mass and damping coecients and the exciting forces [START_REF] Babarit | Theoretical and numerical aspects of the open source BEM solver NEMOH[END_REF]. These frequency-dependent quantities serve to compute the impulse responses and then to achieve time-domain simulations using Cummin's equation [START_REF] Cummins | The impulse response function and ship motions[END_REF] (cf. Appendix B).

Based on the Stokes' series expansion (III.16), a second-order problem may also be dened and leads to a new set of frequency-dependent coecients, named the quadratic transfer functions or QTF [START_REF] Pinkster | Low Frequency Second Order Wave Exciting Forces on Floating Structures[END_REF]. Even if some extra physical phenomenon can be taken into account by this way (mean drift, resonance due to the low-frequency and highfrequency second-order wave loads), the assumptions of the theory always have to be fullled: small steepness waves and small amplitude body motion. The rst-order and second-order frequency-domain models are among the most used hydrodynamic theories both in the industry and in the laboratories, for ship resistance, manoeuvrability, design optimization as well as seakeeping of marine renewable energy devices. Examples of such solvers are: Nemoh1 [START_REF] Babarit | Theoretical and numerical aspects of the open source BEM solver NEMOH[END_REF], WAMIT2 , Diodore3 , HydroSTAR4 , ANSYS Aqwa5 , etc.

The small amplitude body motion condition may often be very restrictive especially in multibody simulations. This is why some researchers working with a frequencydomain model update their hydrodynamic database during their simulation when the relative motion between the bodies becomes too important. Nevertheless, such an approach is inconsistent due to the retardation functions, taking into account the ow memory, which are not computed at the same position anymore [START_REF] Bunnik | A simulation approach for large relative motions of multi-body oshore operations in waves[END_REF].

III.1.2.5 Selected model

Table III.1 summarizes the main dierences between the aforementioned potential ow theories.

As highlighted in subsection 4.2, so far, only three approaches have been used to compute the hydrodynamic loads in marine operation simulations: a frequency-domain model [START_REF] Ku | Dynamic response simulation of an oshore wind turbine suspended by a oating crane[END_REF][START_REF] Schellin | Response analysis and operating limits of crane ships[END_REF][START_REF] Cha | Dynamic response simulation of heavy cargo suspended by a oating crane based on multibody system dynamics[END_REF], a linear time-domain model [START_REF] Ku | Dynamic response analysis of heavy load lifting operation in shipyard using multi-cranes[END_REF] and a fully nonlinear potential ow model [START_REF] Hannan | Nonlinear hydrodynamic responses of submerged moving payload in vicinity of a crane barge in waves[END_REF]. The hydrodynamic theory used in this PhD is a weakly nonlinear model based on the weak-scatterer hypothesis. This approach is consistent with the simulation of marine operations since they often involve several bodies with, possibly, a large relative motion. Therefore, the full linearization of the body condition made in a linear potential-ow based solver is likely too restrictive for this application. Moreover, a frequency-domain model assumes a steady state motion. Using a weakly nonlinear theory, no hypothesis is done about the body motion amplitude, the unsteadiness of the ow and the wave steepness. Compared to a fully nonlinear approach, it is expected to lead to more stable simulations. But some nonlinear eects (wave drift, hydrodynamic impact)

may
+ φ P φ = φ I + φ P φ = φ I + φ P φ = φ I + φ D + φ R φ = φ I + φ P or φ = φ I + φ D + φ R φ = φ I + φ D + φ R Assumptions ø φ P φ I 1 1 A m 1 1 A m 1      1 A m 1 Steady state Meshed free surface z = η(x, y, t) z = η I (x, y, t) z = 0 z = η I (x,
O(1 h) O(1 h) O(1 h)

III.1.3 State of the art of the potential ow theory based on the weak-scatterer hypothesis

The weak-scatterer hypothesis was introduced by Pawlowski and Bass [START_REF] Pawlowski | A theoretical and numerical model of ship motions in heavy seas[END_REF] in 1991 to perform time-domain computations of large ship motions in heavy seas. They described their assumption by the words:

The disturbance induced by the moving ship in the wave ow is considered to be of smaller magnitude than the wave ow quantities which are proportional to the wave height, but at least of the same magnitude as the wave ow quantities proportional to the square of the wave height. This assumption, explained here in simple terms, is called the weak scatterer hypothesis.

Thus, the scattered wave components are assumed much smaller than the incident wave components. The scope of application of this hypothesis is [START_REF] Pawlowski | A theoretical and numerical model of ship motions in heavy seas[END_REF]:

In the theoretical model described [. . . ], the oncoming wave which interacts with the ship, is assumed to be high and steep, so that the quantities proportional to the square of its height cannot be neglected. The motions of the ship, induced by the wave, are also considered to be large, of a magnitude proportional to the wave height.

This theory, according to Pawlowski and Bass [START_REF] Pawlowski | A theoretical and numerical model of ship motions in heavy seas[END_REF], should be applied in case of: SWAN-4 : a time-domain code based on the weak-scatterer hypothesis. SWAN-4 was used with a Series 60 and a Snowdrift moving in head seas and compared to SWAN-2. Both the linear theory and the weak-scatterer approach were in good agreement with the experimental data. The weak-scatterer simulations only brought a small correction [START_REF] Kring | Nonlinear ship motions and wave-induced loads by a Rankine method[END_REF]. The extension of SWAN to the weak-scatterer hypothesis was the PhD work of Huang [START_REF] Huang | Nonlinear ship motions by a Rankine panel method[END_REF]. In his thesis, other comparisons were made between the linear and weak-scatterer theories with a containership in head seas. A signicant improvement was noticed using SWAN-4 compared to SWAN-2 especially at the resonance. Moreover, for small wave slopes, the weak-scatterer method converged to the linear theory results. But, the steeper the incoming waves were, the larger the dierences between the weak-scatterer method and the linear theory were.

SWAN-1, SWAN-2 and SWAN-4 were compared by Grigoropoulos et al. [START_REF] Grigoropoulos | Experimental verication of the linear and non-linear versions of a panel code[END_REF] to experimental results for several kinds of ships: a Series 60, a reefer and a ROPAX. Both SWAN-1 and SWAN-2 gave robust numerical results for every test cases, the use of nonlinear Froude-Krylov loads added accuracy to the predictions. But, SWAN-4 gave some unreasonable results which could be very dierent from the experiments. No explanations were given of these surprising results.

A third numerical tool, named WISH (Wave-Induced loads and Ship motion), of the Seoul National University, uses the weak-scatterer hypothesis. As for LAMP and SWAN, WISH exists with several hydrodynamic theories [START_REF] Kim | Time-domain analysis of nonlinear motion responses and structural loads on ships and oshore structures: development of WISH programs[END_REF]:

WISH-1 : a linear time-domain code; WISH-2 : a linear time-domain code with nonlinear Froude-Krylov loads; WISH-3 : a time-domain code based on the weak-scatterer hypothesis.

Comparisons between SWAN-4, WISH-3 and experimental data were achieved in [START_REF] Kim | Time-domain analysis of nonlinear ship motion responses based on weak-scatterer hypothesis[END_REF] for a Series 60 and a containership in head seas and showed a good agreement albeit the formulations were slightly dierent. Other comparisons to experimental data and the three versions of WISH in case of a containership in head seas showed an overall good agreement of every method. Best agreements were obtained using the weak-scatterer formulation, especially in rough seas. WISH has been coupled to a exible-body mechanical solver, giving WISH-FLEX.

The weak-scatterer theory has only been applied to a non-ship like body by Bretl [START_REF] Bretl | A time domain model for wave induced motions coupled to energy extraction[END_REF]. His PhD work focused on the study of a wave energy converter made of a oating rigid hemisphere coupled with a planar pendulum.

To sum up, it stems from this state of the art that the weak-scatterer approach has been mainly applied to the computation of the motions and the hydrodynamic loads of a moving slender ship in steep head regular waves. Large deformations of the mesh, for instance due to a lateral motion, have not been encountered. Irregular trains of waves have not been tested either. Since 2012, a potential ow based solver following the weak-scatterer hypothesis has been under development at LHEEA laboratory of Ecole Centrale de Nantes. It is named WS_CN.

Letournel [START_REF] Letournel | Développement d'un outil de simulation numérique basé sur l'approche weak-scatterer pour l'étude des systèmes houlomoteurs en grands mouvements[END_REF] developed a rst version of WS_CN dealing with a single submerged body with translational motions. Several validation cases were presented, all of them consider a submerged sphere. First, the weak-scatterer theory was compared to the linear approach in the case of the diraction of regular waves by a xed sphere. Small steepness regular waves involved the matching of the hydrodynamic loads from the two theories for every frequency, whereas high steepness regular waves caused important dierences. Then, radiation cases due to a heave harmonic motion of the sphere were treated. The weak-scatterer approach was compared to several other theories: linear, body-exact and fully nonlinear. A good agreement between all theories was observed for the harmonic coecients in case of a small amplitude motion. Afterwards, three decay tests were displayed with dierent initial positions. The motion of the sphere from the linear and weak-scatterer approaches matched in every scenario. These results constituted the rst validation of the weak-scatterer approach compared to other uid potential theory for a non-ship like body. An application was performed by Letournel with CETO-type wave energy converter with a spring-damper power-take-o. Numerical comparisons were achieved using NEMOH and WS_CN. Once again, the two models showed a good agreement for small amplitude motions and small steepness waves, but some dierences appeared otherwise. The mean absorbed power by the WEC was modied in this case. Another submerged WEC has been simulated using the weak-scatterer approach: an oscillating wave surge converter or wave roller-type WEC [START_REF] Letournel | Weakly nonlinear modeling of submerged wave energy converters[END_REF]. The same conclusions were obtained as with the CETO-type WEC. Letournel also showed the weak-scatterer hypothesis could be not veried punctually when the heaving sphere got close to the free surface.

Chauvigné [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF] extended WS_CN to a single surface-piercing body with an arbitrary motion. An advance front method was used to generate the total mesh or regenerate the body mesh. Two mesh morphing methods were also adopted to avoid the remeshing in case of small deformations: a linear spring analogy on the body surface and an interpolation scheme based on a radial basis function on the free surface. Several validation cases were presented by Chauvigné. The diraction of a regular wave by either a bottom-xed or truncated surface-piercing vertical cylinder was studied. Comparisons were made with respect to linear and fully nonlinear theories and experimental data for the maximum run-up around the cylinder. This case was highly nonlinear, such as a linear approach was not accurate enough. In the two congurations (xed-bottom or truncated cylinder), the weak-scatterer model showed a good agreement with the fully nonlinear theory and experimental data on the bow side of the cylinder but overestimated the maximum run-up on the lee side, probably due to important free surface nonlinearities. Then the radiation of a truncated cylinder in surge or pitch motion was evaluated and compared to a nonlinear approach. In both cases, a very good agreement between the two approaches was observed in the time series of the maximum run-up.

In details, the relative error on the rst harmonic was very low but larger discrepancies were observed for higher harmonics. A free motion validation test with a truncated cylinder linked to its equilibrium position by a spring was presented. As with Letournel [START_REF] Letournel | Développement d'un outil de simulation numérique basé sur l'approche weak-scatterer pour l'étude des systèmes houlomoteurs en grands mouvements[END_REF], numerical results from WS_CN matched the linear theory in case of small amplitude regular wave but dierences appear with a higher amplitude wave. Finally, an application with the WaveStar wave energy converter was featured. Comparisons were done with the oshore numerical tool DeepLines and the experimental data. The numerical results with the weak-scatterer method for a diraction case were conclusive and matched DeepLines and the experimental measures. But, the free motion of the WaveStar device was not accurately simulated by the two numerical tools due to an improper modeling of the power take o. Nevertheless, the linear and weak-scatterer results matched again. As explained by Chauvigné [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF], the weak-scatterer hypothesis was not always validated in this latter application. The disturbed wave elevation could be roughly of the same order of magnitude as the incident component.

These two PhDs have brought the weak-scatterer theory in a new direction: the simulation of a single submerged or oating wave energy converter. Intensive comparisons with the linear and nonlinear potential ow theories were achieved and demonstrated the capacity of the method to perform seakeeping computations. As said by Huang [START_REF] Huang | Nonlinear ship motions by a Rankine panel method[END_REF], non-slender body shapes may induce important wave perturbations and this phenomenon was observed in [START_REF] Letournel | Développement d'un outil de simulation numérique basé sur l'approche weak-scatterer pour l'étude des systèmes houlomoteurs en grands mouvements[END_REF][START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF], leading to an unvalidated weak-scatterer hypothesis.

Regarding the computation time, Letournel et al. [START_REF] Letournel | Weakly nonlinear modeling of submerged wave energy converters[END_REF] found the weak-scatterer approach is one order of magnitude faster than a fully nonlinear approach and two orders of magnitude greater than the real time (in sequential calculations). It appears from these works that the weak-scatterer theory has been compared to a body-exact approach only once. Yet, this latter method could be ecient if the steepness of the waves is not too large. In the case of still water, the two theories match.

Regarding the CPU-time reduction, several types of symmetries were implemented by Letournel et al. [START_REF] Letournel | Weakly nonlinear modeling of submerged wave energy converters[END_REF] in WS_CN : vertical along the (xOz) plane and horizontal on the at sea bottom. Simulations in open domain are also possible when the lateral numerical tank surfaces are far enough. Chauvigné [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF] have allowed a partial calculation of the inuence coecients when sub-domains are motionless.

At the beginning of this PhD work, the simulations using WS_CN dealt with a single submerged or oating body in sequential computations. Surface-piercing body simulations have been performed by Chauvigné [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF] but only with a small amplitude motion of the body and so with only small deformations of the free surface mesh. Therefore, simulations with a large deformation of the free surface mesh were not possible, as well as multibody simulations.

Note

In the literature, the term weakly nonlinear covers a variety of situations. For example, in [START_REF] Kim | Time-domain analysis of nonlinear motion responses and structural loads on ships and oshore structures: development of WISH programs[END_REF][START_REF] Kim | Time-domain analysis of nonlinear ship motion responses based on weak-scatterer hypothesis[END_REF] and [START_REF] Song | Numerical analysis and validation of weakly nonlinear ship motions and structural loads on a moden containership[END_REF], the so-called weakly nonlinear approach matched the nonlinear Froude-Krylov approach. Greco and Lugni [START_REF] Greco | 3-D seakeeping analysis with water on deck and slamming. Part 1: Numerical solver[END_REF] dened their potential ow based solver relied on the weak-scatterer hypothesis as weakly nonlinear but their hydrodynamic loads were estimated as linear in frequency domain while the Froude-Krylov and hydrostatic loads were nonlinear. Ruggeri et al. [START_REF] Ruggeri | A higher order time domain rankine panel method for linear and weakly non-linear computation[END_REF] presented some weakly nonlinear computations but their time-domain solver was fully linear using a high-order panel method and computing second-order mean drift forces. Thus, the expression weakly nonlinear includes dierent types of approximations and is denitely unclear. In this PhD thesis, following the denominations of Letournel et al. [START_REF] Letournel | Weakly nonlinear modeling of submerged wave energy converters[END_REF] and Chauvigné [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF],

there is an equivalence between the terms weak-scatterer and weakly nonlinear.

III.2 Theory of the potential ow based on the weak-scatterer hypothesis III.2.1 Assumptions and governing equations

The hydrodynamic theory used in this PhD assumes:

A tri-dimensional, unsteady and potential ow;

A at sea bottom;

A single-valued free surface elevation;

The unawareness of the surface tension;

The knowledge of the incoming wave eld;

The nullity of the pressure above the free surface;

The rigidity of the oating and immersed bodies;

The validity of the weak-scatterer hypothesis.

The uid domain, D, has a boundary, S, constituted of the free surface F S, the wetted body surfaces S B and the numerical tank surfaces (including the sea bottom) T . The reference frame is Σ e , as dened in subsection I.2.2. The number of hydrodynamic bodies is N W SC .

From the assumptions, the potential ow theory is applied. The governing equations have already been demonstrated:

(III.4) in the ow;

(III.8) and (III.9) for the free surface;

(III.10) for the solid boundaries (bodies and sea bottom);

A permeability condition on the numerical tank walls:

∂φ ∂n = ∂φ I ∂n ⇒ ∂φ P ∂n = 0 (III.22)
A radiation condition, the wave eld far from the bodies matches the incident wave eld:

   φ ----→ r→+∞ φ I η ----→ r→+∞ η I (III.23)
The use of the decomposition (III.13) gives:

                                                   ∆φ P = -∆φ I in
φ P ----→ r→+∞ 0 η P ----→ r→+∞ 0 (III.24)
The incoming wave eld is computed following a potential ow approximation, consequently:

∆φ I = 0 in the uid domain D

(III.25)

The weak-scatterer approximation enables the linearization of the free surface boundary equations at the incident wave elevation. By application of the Taylor's theorem for a function f at the point z = η I (x, y, t), it yields:

f (x, y, z, t) z=η(x,y,t) = f (x, y, z, t) z=η I (x,y,t) +(η-η I ) ∂f (x, y, z, t) ∂z z=η I (x,y,t) +O(φ P 2 )+O(η P 2 )

(III.26)
Equation III.26 is used with the free surface boundary equations of (III.24) and, after applying the weak-scatterer hypothesis (III.14):

                         ∂η P ∂t = - ∂η I ∂t -∇φ I • ∇η I -∇φ I • ∇η P -∇φ P • ∇η I + ∂φ I ∂z + ∂φ P ∂z +η P ∂ 2 φ I ∂z 2 -∇η I • ∂∇φ I ∂z at z = η I (x, y, t) ∂φ P ∂t = - ∂φ I ∂t - 1 2 ∇φ I • ∇φ I -∇φ I • ∇φ P -g(η I + η P ) -η P ∂ 2 φ I ∂z∂t + ∇φ I • ∂∇φ I ∂z at z = η I (x, y, t) (III.27)

III.2.2 Arbitrary Lagrangian-Eulerian description

The solving of the uid governing equations involves the use of a computational mesh. There are three possible descriptions of the motion in a continuum media:

The Lagrangian description: the nodes of the mesh follow the material points, so that the boundaries can be easily followed in case of small deformations. If large deformations occur, the quality of the panels geometry is not guaranteed and the mesh has to be regenerated. This approach is mainly used in structural mechanics or in the Smooth Particle Hydrodynamics method [START_REF] Fourey | An ecient FSI coupling strategy between smoothed particle hydrodynamics and nite element methods[END_REF] for instance.

The Eulerian description: the nodes of the mesh are xed. This approach is easy to implement, robust and large deformations of the mesh may be applied. The mesh quality is preserved, but tracking the boundaries of the domain requires a ne mesh.

The arbitrary Lagrangian-Eulerian (ALE ) description combines the better characteristics of the two previous approaches. The boundary equations are solved at the material nodes taking into account the arbitrary motion of the mesh and ensuring its good quality. This approach is preferred in case of free surface ows.

In order to track the physical quantities at the nodes of the mesh under an arbitrary motion, the ALE description is preferred. Thus, the free surface boundary equations need to be updated at the nodes of the mesh moving with an arbitrary motion.

Let Ω a (t) be an arbitrary control volume moving at the velocity v a . Then the derivative of a quantity f of Ω a (t) is:

δf δt = ∂f ∂t + (v a • ∇)f (III.28)
where δ * δt represents the derivative with respect to an ALE description.

In our case, v a is the velocity of the nodes of the mesh, denoted v mesh . The free surface boundary equations become:

                         δη P δt = - ∂η I ∂t -∇φ I • ∇η I -(∇φ I -v mesh ) • ∇η P -∇φ P • ∇η I + ∂φ I ∂z + ∂φ P ∂z +η P ∂ 2 φ I ∂z 2 -∇η I • ∂∇φ I ∂z at z = η I (x, y, t) δφ P δt = - ∂φ I ∂t - 1 2 ∇φ I • ∇φ I -(∇φ I -v mesh ) • ∇φ P -g(η I + η P ) -η P ∂ 2 φ I ∂z∂t + ∇φ I • ∂∇φ I ∂z at z = η I (x, y, t) (III.29)

III.2.3 The boundary value problem

Eventually, the boundary value problem (BVP ) to solve is:

                                                               ∆φ P = 0 in the uid domain D δη P δt = - ∂η I ∂t -∇φ I • ∇η I -(∇φ I -v mesh ) • ∇η P -∇φ P • ∇η I + ∂φ I ∂z + ∂φ P ∂z +η P ∂ 2 φ I ∂z 2 -∇η I • ∂∇φ I ∂z ) at z = η I (x, y, t) δφ P δt = - ∂φ I ∂t - 1 2 ∇φ I • ∇φ I -(∇φ I -v mesh ) • ∇φ P -g(η I + η P ) -η P ∂ 2 φ I ∂z∂t + ∇φ I • ∂∇φ I ∂z at z = η I (x, y, t) ∂φ P ∂n = - ∂φ I ∂n + v Solid • n on S B (t)
and the seabed ∂φ P ∂n = 0 on the numerical tank walls

φ P ----→ r→+∞ 0 η P ----→ r→+∞ 0 (III.30)
Dierent methods exist to solve this problem, for instance:

The nite-dierence method;

The nite-element method;

The boundary element method (BEM ).

Apart from their dierent formulations, these methods mainly dier on the mesh requirement and the sparsity of the linear systems they involve. The nite-dierence method and the nite-element method require volume meshes whereas the boundary element methods need surface meshes and so less unknowns and smaller linear systems.

But, these linear systems are dense in case of BEM and sparse otherwise. Sparse matrices require less memory to store their coecients and ecient algorithms exist to solve large sparse linear systems. Nevertheless, the easier mesh management oered by the BEM makes its method widely used. WS_CN is based on the boundary element method.

III.2.4 The boundary element method III.2.4.1 The boundary integral equation Note

So far the normal unit vector n pointed outward the uid, from now on, the normal unit vector points inward the uid.

The boundary element method is based on the Green's second identity: Theorem 1. Let Ω(t) be a volume of R 3 , ∂Ω(t) its boundary and n the inward unit normal to this boundary. Let dV and dS be an elementary volume and surface of Ω(t) and ∂Ω(t), respectively. Let f and g be both twice continuously dierentiable scalar functions. Then:

˚Ω(t) (f ∆g -g∆f ) dV = - ¨∂Ω(t
∂f ∂n = ∇f • n = f n (III.32)
The functions f and g are chosen such as:

f = φ P g = G (III.33)
where G represents the Green's function, solution of the equation:

∆G(x i , x j ) = δ(x i -x j ) (III.34)
δ denotes the Dirac delta function, x i and x j are two points of R 3 .

Basically, it exists two ways to nd the Green's function. Either G only satises (III.34) and becomes a Rankine source Green's function or G also satises the free surface boundary equations and the radiation condition and becomes a Kelvin source Green's function or free surface Green's function. In WS_CN, a Rankine source Green's function is used, that is to say:

G(x i , x j ) = - 1 4π x i -x j = - 1 4π r ij (III.35)
In WS_CN, with no loss of generality, the Green's function is simply dened by:

G(x i , x j ) = 1 r ij (III.36)
Its normal derivative is:

∂G ∂n j (x i , x j ) = - r ij • n j r ij 3
(III.37)
∂ * ∂n j is the normal derivative with respect to n j .

Applying (III.31) over the whole uid domain D bounded by the surface S at the point x i gives:

˚D φ P (x j )∆G(x i , x j ) -G(x i , x j )∆φ P (x j ) dV = - ¨S φ P (x j ) ∂G ∂n j (x i , x j ) -G(x i , x j ) ∂φ P ∂n (x j ) dS (III.38)
Here x j is the variable of integration.

The integral of the Laplacian of the Green's function may be expressed by:

˚D ∆G(x i , x j ) dV = ˚D ∆ 1 r dV (III.39) = ˚D ∇ • - r r 3 dV (III.40) = ¨S r • n r 3 dS (III.41) = - ¨S ∂G ∂n j (x i , x j ) dS (III.42) = Ω(x i ) (III.43)
Ω(x i ) may be seen as the solid angle at the point x i . The generalization of this latter expression yields:

˚D φ P (x j )∆G(x i , x j ) dV = -φ P (x i ) ¨S ∂G ∂n j (x i , x j ) dS = φ P (x i )Ω(x i ) (III.44)
The rst term of the left hand side of (III.38) is zeroed:

∆φ P (x j ) = 0 in D ⇒ ˚D G(x i , x j )∆φ P (x j ) dV = 0 (III.45)
Finally (III.38) becomes:

-φ P (x i )Ω(x i ) -¨S φ P (x j ) ∂G ∂n j (x i , x j ) dS + ¨S G(x i , x j ) ∂φ P ∂n (x j ) dS = 0 (III.46)
Equation III.46 is the boundary integral equation (BIE ) in the uid. Only two surface integrals are present so the computational domain is the boundary S of the uid domain D: the free surface F S, the wetted body surfaces S B and the numerical tank surfaces (including the sea bottom) T :

S = F S ∪ S B ∪ T (III.47)
Where S B is composed of the wetted surface S B j of each body j:

S B = N W SC j=1 S B j (III.48)
Two terms of (III.46) need to be claried:

¨S φ P (x j ) ∂G ∂n j (x i , x j ) dS and ¨S G(x i , x j ) ∂φ P ∂n (x j ) dS (III.49)

III.2.4.2 The inuence coecients

The boundary integral equation is solved using the collocation method. The boundary surface of the uid domain is discretized into N p panels of surface S p and N nodes:

S = Np p=1 S p (III.50)
The integral equation is written at each node i of the mesh:

-φ P (x i )Ω(x i ) - Np p=1 ¨Sp φ P (x j ) ∂G ∂n j (x i , x j ) dS + Np p=1 ¨Sp G(x i , x j ) ∂φ P ∂n (x j ) dS = 0 (III.51)
In WS_CN, panels are triangular. This ensures their atness and it is also a common mesh format. Each triangle p is dened by its three vertices (x p 1 , x p 2 , x p 3 ). Its centre of gravity is at x p G . Its contour is denoted C p . The unknowns are at the nodes. The computation of the surface integrals is achieved analytically, assuming a linear approximation of the physical quantities over the panels. For instance, for a quantity f over the panel p:

f (x) = f (x p G ) + ∇ S (f ) • (x -x p G ) (III.52)
∇ S (f ) is the surface gradient, dened by:

∇ S (f ) = Σ   f (x p 1 ) f (x p 2 ) f (x p 3 )   (III.53)
with:

Σ = 1 ∆ -(A + B) A B (III.54) ∆ = x p 2 -x p 1 2 x p 3 -x p 1 2 -[(x p 2 -x p 1 ) • (x p 3 -x p 1 )] 2 (III.55) A = x p 3 -x p 1 2 (x p 2 -x p 1 ) -[(x p 2 -x p 1 ) • (x p 3 -x p 1 )](x p 3 -x p 1 ) (III.56) B = -[(x p 2 -x p 1 ) • (x p 3 -x p 1 )](x p 2 -x p 1 ) + x p 2 -x p 1 2 (x p 3 -x p 1 )
(III.57)

Letournel [START_REF] Letournel | Développement d'un outil de simulation numérique basé sur l'approche weak-scatterer pour l'étude des systèmes houlomoteurs en grands mouvements[END_REF] gives the expressions of the two surface integrals:

¨Sp φ P (x j ) ∂G ∂n j (x i , x j ) dS =   ∆ ¨Sp ∂G ∂n j (x i , x j ) dS - ˛Cp G(x i , x j )(x j -x i ) × dl T Σ     φ P (x p 1 ) φ P (x p 2 ) φ P (x p 3 )   (III.58) = C i p   φ P (x p 1 ) φ P (x p 2 ) φ P (x p 3 )   (III.59) ¨Sp G(x i , x j ) ∂φ P ∂n (x j ) dS =   ∆ ¨Sp G(x i , x j ) dS - ˛Cp n p × dl G(x i , x j ) T Σ          ∂φ P ∂n (x p 1 ) ∂φ P ∂n (x p 2 ) ∂φ P ∂n (x p 3 )        (III.60) = D i p        ∂φ P ∂n (x p 1 ) ∂φ P ∂n (x p 2 ) ∂φ P ∂n (x p 3 )        (III.61)
with:

∆ = 1 3 1 1 1 + (x i -x p G ) T Σ (III.62)
n p represents the normal of the panel p.

Equation III.51 expresses the boundary integral equation with respect to the panels of the mesh, but the use of the collocation method involves writing this equation with respect to the nodes of the mesh:

-φ P (x i )Ω(x i ) - N j=1 G ij φ P (x j ) + N j=1 H ij ∂φ P ∂n (x j ) = 0 (III.63)
where G and H are the matrices of the inuence coecients or inuence matrices. They represent the inuence of every node on every node. Their elements are:

G ij = p∈P(j) C i p (δ j p ) (III.64) H ij = p∈P(j) D i p (δ j p ) (III.65)
δ j p = q ∈ {1, 2, 3} such as x j = x p q (III.66) P(j) represents the set of the adjacent panels of which the node j is a vertex. C i p (k) and D i p (k) denote the k th component of the row vectors C i p and D i p .

Regarding the solid angle, Equations (III.63) and (III.44) give:

φ P (x i )Ω(x i ) = -φ P (x i ) ¨S ∂G ∂n j (x i , x j ) dS (III.67) = -φ P (x i ) N j=1 G ij (III.68)
leading to:

N j=1 G ij φ P (x j ) = N j=1 H ij ∂φ P ∂n (x j ) (III.69)
with:

G ii = - N j=1 j =i G ij (III.70)
Finally the boundary integral equation can be written with a matrix format:

Gφ P = Hφ P n (III.71)

φ P and φ P n are the velocity potential and normal velocity vectors of size N . The i th component of φ P , respectively φ P n , is φ(x i ), respectively φ n (x i ).

Note

As the unknowns are at the nodes, there is a singularity at the interfaces free surface / bodies and free surface / numerical tank because both the free surface boundary equations and the slip condition are applied. Moreover, at the sharp edges in the body meshes, it arises a discontinuity of the normal vectors so the slip condition is not well dened. Consequently multiple node technique is used at the interfaces and at the sharp edges. Doing so, several boundary conditions are satised at the same location. This method increases the number of nodes in the mesh but ensures that all the boundary equations are treated properly. The continuity of the velocity potential at the multiple nodes is also checked during the simulation.

III.2.4.3 The linear system

In our problem, the velocity potential is known on the free surface and its normal derivative is known on the solid surfaces (bodies, sea bottom, numerical tank walls).

The goal of the BEM is to compute the missing quantity on each surface. Thus, the velocity potential will be evaluated on the solid surface and its normal derivative will be found on the free surface.

Mathematically, it means the boundary integral equation (III.71) is decomposed into block matrices and vectors to separate the known quantities from the unknown ones.

Consequently, the inuence matrices are written:

G = G(F S) G(T ) • • • G(B j ) • • • (III.72) H = H(F S) H(T ) • • • H(B j ) • • • (III.73)
F S, T and B j represents respectively the free surface, the numerical tank and the body j. Each block is a rectangular matrix of size the number of nodes in the whole mesh times the number of the nodes in the considered part of the mesh. For instance, G(B j ) is of size N × N (B j ) with N the number of nodes in the mesh and N (B j ) the number of nodes in the mesh of the body j.

Equation III.71 becomes a linear system AX = B of size N such as:

A = H(F S) -G(T ) • • • -G(B j ) • • • (III.74) X =         φ P n (F S) φ P (T ) . . . φ P (B j ) . . .         (III.75) B = -H(T ) • • • -H(B j ) • • • G(F S)         φ P n (T ) . . . φ P n (B j ) . . . φ P n (F S)         (III.76)
The matrix A is dense, as explained in subsection III.2.3. The solving of this linear system is done using a generalised minimal residual iterative scheme (GMRES ) with a diagonal preconditioner.

III.2.5 Gradients

Once the velocity potential is known at every node of the mesh, it remains to compute the gradient of both the velocity potential and the free surface elevation to apply the free surface boundary equation (III.29). Regarding wave elevation η(x, y), its gradient is given by:

∇η P =       ∂η P ∂x ∂η P ∂y 0       (III.77)
As the velocity potential also depends on z, its gradient is split into a surface component ∇ S and a normal component ∂ ∂n such as:

∇φ P = ∇ S φ P + ∂φ P ∂n n (III.78)
The normal component on the bodies comes from the slip condition (III.30) and on the free surface from the boundary element method (III.76). The horizontal derivatives of the wave elevation and the surface gradient of the velocity potential are computed using a polyharmonic spline approximation [START_REF] Letournel | Développement d'un outil de simulation numérique basé sur l'approche weak-scatterer pour l'étude des systèmes houlomoteurs en grands mouvements[END_REF]. The main results are reminded hereinafter. The approximation of the function s at the node x i is:

s(x i ) = N i v j=0 α j ψ( x i -x j ) + p(x i ) (III .79) with:
ψ the radial basis function;

p the unknown polynomial;

N i v the number of rst-order and second-order neighboring nodes of the node i;

x j the neighboring nodes;

α j the unknown coecients.

Note

The rst-order neighboring nodes are dened as the immediate neighboring nodes while the second-order neighboring nodes are dened as the neighboring nodes of the rst-order neighboring nodes, as shown in Figure III.5.

Figure III.5 Example of the neighborhood (from [44])

In W S_CN , a third-order polynomial spline is used so the kernel ψ and the polynomial p are:

ψ(x) = x 3 (III.80) p(x) = α N i v +1 + α N i v +2 x + α N i v +3 y + α N i v +4 xy + α N i v +5 x 2 + α N i v +6 y 2 (III.81)
α j and p are determined by the interpolation conditions on the nodes x j , neighbors of the node x i : s(x j ) = s j (III.82)

N i v j=1 α j q(x j ) = 0 (III.83)
where s j is the known velocity potential or wave elevation of the node j and q denotes all polynomials with a degree less or equal than two.

Finally, these equations form a linear system of size N i v + 7: [START_REF] Chung | A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-alpha method[END_REF] with:

M P P T 0 3×3 α β = s 0 3×1 (III.
M the matrix containing the evaluation of the radial basis function:

M ij = ψ( x i -x j ) (III.85)
P the matrix with row j is given by the vector 1 x j y j x j y j x 2 j y 2 j ;

α the vector of the coecients α j , including the coecients of p.

The spatial derivatives of the wave elevation arise:

               ∂η P ∂x (x i ) = N i v j=0 α j ∂ψ ∂x ( x i -x j ) + ∂p ∂x (x i ) ∂η P ∂y (x i ) = N i v j=0 α j ∂ψ ∂y ( x i -x j ) + ∂p ∂y (x i ) (III.86)
Concerning the gradient of the velocity potential, a change of variables is necessary to take into account the z-dependency [START_REF] Letournel | Développement d'un outil de simulation numérique basé sur l'approche weak-scatterer pour l'étude des systèmes houlomoteurs en grands mouvements[END_REF].

To ensure the continuity of the gradient of the velocity potential at the multiple nodes in case of intersection curve or sharp edges, the following system is solved (here for a double node):

             ∇φ P • n D1 = ∂φ P ∂n 1 ∇φ P • n D2 = ∂φ P ∂n 2 ∇φ P • (n D1 × n D2 ) = ∇φ P • (n D1 × n D2 ) (III.87)
where n D1 , n D2 , ∂φ P ∂n 1 and ∂φ P ∂n 2 represent the normals and the normal derivative of the velocity potential of the two intersecting surfaces.

If n D1 ≈ n D2 , the two rst equations of the linear system are identical, a new system has to be dened:

         ∇φ P • n D1 = ∂φ P ∂n 1 ∇φ P • (n D1 × n D2 ) = ∇φ P • (n D1 × n D2 ) ∇φ P • [n D2 × (n D1 × n D2 )] = ∇φ P • [n D2 × (n D1 × n D2 )] (III.88)

III.2.6 Fluid-structure interaction

If the oating and immersed bodies are still or following a prescribed motion, the hydrodynamic problem is completely described by the previous sections. But, in case of free body motions, the motion equation of each body needs to be solved. In W S_CN , bodies are assumed not to be in mechanical interactions but only in hydrodynamic interactions, so that the motion equation for the body j at its centre of gravity G j takes the following form: η j the generalized position vector of the body j:

η j =         x j y j z j φ j θ j ψ j         (III.92)
where these coordinates represent, respectively, the surge, sway, heave, roll, pitch and yaw of the body j.

Note

In [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF][START_REF] Letournel | Weakly nonlinear modeling of submerged wave energy converters[END_REF], the expression of the inertial loads is dierent:

e F Inertia j = 0 3×3
e I G j Ṡj + e İG j S j ηj (III. [START_REF] Banerjee | Deployment control of a cable connecting a ship to an underwater vehicle[END_REF] This latter expression matches (III.91).

The hydrodynamic loads are obtained by the integration of the pressure from Bernoulli's equation. To be consistent with the hydrodynamic theory developed here, the decomposition (III.13) and the weak-scatterer hypothesis (III.14) are applied in (III.5), leading to the following expression of the pressure:

p W SC = -ρ ∂φ I ∂t + ∂φ P ∂t + 1 2 ∇φ I • ∇φ I + ∇φ I • ∇φ P + gz (III.94)
Then, the expression of the hydrodynamic loads of the body j at its center of gravity is:

e F W SC j =      - ¨SB j (t) p W SC n dS - ¨SB j (t) p W SC (x -e S j ) × n dS      (III.95)
In the expression of the pressure:

z is known from the node positions;

φ I , ∇φ I and ∂φ I ∂t are known from the incoming wave eld which is assumed to be known;

φ P is known from the boundary value problem subsection III.2.3 and the solving of the boundary integral equation (III.2.4.3);

∇φ P is known using the B-spline approximation subsection III.2.5;

∂φ P ∂t is unknown.

Thereby, (III.89) involves two unknowns: the acceleration ηj and the time-dierentiation of the velocity potential ∂φ P ∂t

. The uid-structure interaction appears here.

III.2.6.1 State of the art of the computation of the time-dierentiation of the velocity potential

Five methods exist to solve this uid-structure interaction in case of an unsteady potential ow theory [START_REF] Koo | Freely oating-body simulation by a 2D fully nonlinear numerical wave tank[END_REF][START_REF] Bandyk | The acceleration potential in uid-body interaction problems[END_REF][START_REF] Guerber | Modélisation numérique des interactions non-linéaires entre vagues et structures immergées, appliquée à la simulatio de systèmes houlomoteurs[END_REF]:

The backward nite dierence method;

The iterative method;

The mode-decomposition method;

The indirect method;

The implicit boundary method.

The backward nite dierence method is the most straightforward approach due to its simplicity. The time-dierentiation of the velocity potential is obtained from the velocity potential at the current and previous time steps:

Dφ Dt = φ(t) -φ(t -dt) dt (III.96)
where

D * Dt

denotes the Lagrangian derivative. This method is known to be numerically unstable [START_REF] Koo | Freely oating-body simulation by a 2D fully nonlinear numerical wave tank[END_REF].

The iterative method adds a retroactive loop along with a predictor-corrector to converge on the values of the body accelerations and the time-derivative of the velocity potential. This method is used by Cao et al. [START_REF] Cao | Nonlinear computation of wave loads and motions of oating bodies in incident waves[END_REF] for example.

The mode-decomposition method splits the time-dierentiation of the velocity potential into seven elementary functions (or modes) which correspond to an unit acceleration for each degree of freedom (ψ j ) and the acceleration due to the velocity eld (ψ 7 ):

∂φ ∂t = 6 j=1 a j ψ j + ψ 7 (III.97)
Each mode is solved using a boundary integral equation. Once these modes are found, the body acceleration can be determined. This method was applied by Koo and Kim [START_REF] Koo | Freely oating-body simulation by a 2D fully nonlinear numerical wave tank[END_REF] for instance.

The indirect method does not compute directly ∂φ ∂t but only its integration over the body surface. To do so, auxiliary functions are introduced and the use of the Green's second identity leads to an expression of the hydrodynamic forces from these auxiliary functions. Therefore the boundary integral equation of the time-derivative of the velocity potential is not solved. Nevertheless, the body-surface pressure cannot be obtained with this approach. This method has been proposed by Wu and Eatock Taylor [START_REF] Wu | Transient motion of a oating body in steep water waves[END_REF] and has been widely used so far [START_REF] Hannan | Numerical simulation of submerged payload coupled with crane barge in waves[END_REF][START_REF] Wu | The coupled nite element and boundary and element analysis of nonlinear interactions between waves and bodies[END_REF][START_REF] Li | Fully nonlinear numerical simulations of wave interactions with multiple structures at resonance[END_REF].

The implicit boundary method couples the motion equation to the boundary integral equation satised by ∂φ ∂t

. Then, this latter quantity is integrated over the body surface to evaluate the hydrodynamic loads. This approach requires to develop the timederivative of the body boundary condition, which is a dicult task. Two expressions of this body condition were derived:

One developed by Cointe et al. [START_REF] Cointe | Nonlinear and linear motions of a rectangular barge in a perfect uid[END_REF] in 2D and by Van Daalen [START_REF] Van Daalen | Numerical and theoretical studies of water waves and oating bodies[END_REF] in 3D using local normal and tangential components of body and uid velocities;

The second proposed by Tanizawa [START_REF] Tanizawa | A nonlinear simulation method of 3-D body motions in waves (1st report)[END_REF] in 2D and extended in 3D by Berkvens [START_REF] Berkvens | Floating bodies interacting with water waves: devlopment of a time-domain panel method[END_REF] based on the acceleration of a uid particle sliding on the body surface.

The proof of the equivalence of these two expressions is demonstrated by Letournel et al. [START_REF] Letournel | Proof of the equivalence of Tanizawa-Berkvens' and Cointe-van Daalen's formulations for the time derivative of the velocity potential for non-linear potential ow solvers[END_REF] and a unied expression is given.

In WS_CN, the implicit boundary method was preferred for its eciency in terms of stability and time-consuming to the other methods and follows the formalism developed by Letournel [START_REF] Letournel | Développement d'un outil de simulation numérique basé sur l'approche weak-scatterer pour l'étude des systèmes houlomoteurs en grands mouvements[END_REF].

III.2.6.2 The implicit boundary method

The method consists in solving a new boundary value problem. The Laplacian of the ∂φ P ∂t is found using (III.30):

∆ ∂φ P ∂t = ∂∆φ P ∂t = 0 in the uid domain D (III.98)

Note

The time-derivative of the velocity potential, ∂φ ∂t

, is named the acceleration potential sometimes but it is not equal to the Lagrangian time-dierentiation of the velocity:

Dv Dt = D∇φ Dt = ∂∇φ ∂t + (∇φ) • ∇φ = ∇ ∂φ ∂t + 1 2 ∇φ • ∇φ (III.99)
Because for all vectors a:

1 2 ∇(a • a) = (a • ∇)a + a × (∇ × a) (III.100) ∂φ ∂t + 1 2
∇φ • ∇φ does not satisfy Laplace's equation because of the velocity-squared term [START_REF] Bandyk | The acceleration potential in uid-body interaction problems[END_REF]. The terms time-derivative or time-dierentiation of the velocity potential are preferred.

The free surface boundary equation is given by (III.28):

∂φ P ∂t = δφ P δt -+(v mesh • ∇)φ P at z = η I (x, y, t) (III.101)
The slip condition needs to use the normal derivative of the time-dierentiation of the velocity potential, dened by:

∂ 2 φ P ∂t∂n = ∂ 2 φ P ∂n∂t = ∇ ∂φ P ∂t • n (III.102)
On the seabed and the numerical tank walls, this quantity is:

∂ 2 φ P ∂t∂n = - ∂ 2 φ I ∂t∂n on the seabed (III.103) ∂ 2 φ P ∂t∂n = 0
on the numerical tank walls (III.104)

On the wetted body surfaces for the body j, the slip condition is given by Letournel et al. [START_REF] Letournel | Proof of the equivalence of Tanizawa-Berkvens' and Cointe-van Daalen's formulations for the time derivative of the velocity potential for non-linear potential ow solvers[END_REF] in case of translational motions and extended to rotational motions by Chauvigné [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF], here at the node x i :

∂ 2 φ P ∂t∂n = - ∂ 2 φ I ∂t∂n + n [(x i -e S j ) × n] T S j • ηj + q j (III.105)
q j is given by:

q j = e ω j • s 1 ∂φ P ∂s 2 -2( ηtrans j • s 2 ) -e ω j • s 2 ∂φ P ∂s 1 -2( ηtrans j • s 1 ) + ( ηtrans j • s 1 ) R 1 ∂φ P ∂s 1 -(η trans j • s 1 ) + ( ηtrans j • s 2 ) R 2 ∂φ P ∂s 2 -(η trans j • s 2 ) + ( ηtrans j • n) ∂ 2 φ P ∂s 2 1 + ∂ 2 φ P ∂s 2 2 + 1 R 1 + 1 R 2 ∂φ P ∂n + ( Ṡj ηrot j ) × (x i -e S j ) + e ω j × ( e ω j × (x i -e S j )) • n (III.106)
where s 1 and s 2 are the two local tangent vectors and R 1 and R 2 denote the local curvature associated to these vectors.

The boundary integral equation used with the velocity potential (III.46) is also valid with its time-derivative, here written at the point x i :

- ∂φ P ∂t (x i )Ω(x i ) - ¨S ∂φ P ∂t (x j ) ∂G ∂n j (x i , x j ) dS + ¨S G(x i , x j ) ∂ 2 φ P ∂n∂t (x j ) dS = 0 (III.107)

III.2.6.3 The discretization Note

The time-dierentiation is written either ∂ * ∂t or ( * ).

As the boundary integral equation is the same as in the rst boundary value problem, except that the velocity potential is substituted for its time-dierentiation, then the inuence matrices G and H stay identical. Therefore, (III.71) becomes:

G φ = H φn (III.108)
Regarding the slip condition of the body j (III.105), its discretization gives:

φP n (B j ) = -φI n (B j ) + CK j ηj + Q j (III.109)
CK j is a rectangular matrix of size N (B j ) × 6 and Q j is a vector of size N (B j ).

The hydrodynamic loads of the body j (III.95) becomes:

e F W SC j = e CT j φP (B j ) + e T W SC j (III.110)
where e CT j is a 6 × N (B j ) matrix and e T W SC j is a vector of size 6 including the known components of the hydrodynamic loads.

Thus, the discretized motion equation arises:

e M Hydro

III.2.6.4 The linear system

The set of ordinary dierential equations to solve is:

       G φ = H φn e M Hydro j ηj -e CT j φP (B j ) = e T W SC j + e F Inertia j + e F Other j ∀j ∈ 1 ; N W SC φP n (B j ) = -φI n (B j ) + CK j ηj + Q j ∀j ∈ 1 ; N W SC (III.112)
Equation III.112 forms a linear system AX = B of size N +

N W SC j=1 N (B j ) + 6N W SC .
The matrix A is dense so the solving of this linear system is achieved using a GMRES method with a diagonal preconditioner. The vector X is:

X =                   φP n (F S) φP (T ) . . . φP (B j ) . . . φP n (B j ) . . . ηj . . .                   (III.113)

III.3 Mesh generation

The solving of the two boundary value problems requires the use of a surface mesh on the whole domain: free surface, wetted body surfaces and numerical tank walls. To avoid numerical discrepancies, a good quality triangulated mesh is necessary, in terms of size and shape. Using an unsteady hydrodynamic solver, the mesh changes with time.

Consequently, not only a mesh has to be created at the initial stage, but also at any time during the simulation, independently of the wave eld and the body motions.

In his PhD work, Letournel [START_REF] Letournel | Développement d'un outil de simulation numérique basé sur l'approche weak-scatterer pour l'étude des systèmes houlomoteurs en grands mouvements[END_REF] only performed simulations with a single submerged body so the surface mesh of the domain was easy to generate. Indeed, in this case, the body mesh and the free surface do not intersect each other. Then, Chauvigné [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF] has extended the code to the simulation of a single surface-piercing body. This involves the tracking of the intersection curve and the continuity of the mesh at this interface. The (free-)surface-to-(body-)surface intersection is seeked using a marching method which starts from a known starting point on the intersection curve and then steps along it in a direction prescribed by the curve local geometry. Regarding the mesh generation, an advance front method has been developed by Chauvigné [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF] and applied for both the body and the free surface meshes.

III.3.1 Intersection curve tracking III.3.1.1 Initial point

Each body is formed of a set of parametric surfaces S(u, v). For example, a cylinder is made of two discs and a cylindrical body. Its parametric equation is:

S Cylinder (u, v) =   R cos(u) R sin(u) v   (III.114)
A cube is formed of six planes. The parametric equation of a plane is:

S P lane (u, v) =   u v 0   (III.115)
The initial point on the intersection curve is searched along an isoparametric curve

III.3.1.2 Marching method

Once the initial point P 0 = S(u 0 , v 0 ) of the intersection line is found, the rest of the curve is tracked using a marching method. The intersection curve C(s) = S(u C (s), v C (s)) is dened by the equation:

T(s) • n F S = 0 (III.116)
where:

T is the local tangent vector of the surface S(u, v):

T(s) = S u ∂u ∂s + S v ∂v ∂s (III.117) with:      S u = ∂S ∂u S v = ∂S ∂v (III.118)
n F S is the free surface normal vector;

s is the curvilinear coordinate.

The rates of change of (u C , v C ) along the intersection are solutions of the following system of dierential equation:

         du C ds = S v • n F S E(S v • n F S ) 2 -2F (S v • n F S )(S u • n F S ) + G(S u • n F S ) 2 dv C ds = S u • n F S E(S v • n F S ) 2 -2F (S v • n F S )(S u • n F S ) + G(S u • n F S ) 2 (III.119)
where:

     E = S u • S u F = S u • S v G = S v • S v (III.120)
As the initial point on the intersection curve is known, the other points are found from the integration of (III.119):

P = S(u 0 + δ u , v 0 + δ v ) (III.121) with:                  δ u = s 0 +ds ŝ0 du δ v = s 0 +ds ŝ0 dv (III.122)
A fourth-order explicit Runge-Kutta scheme with a xed spatial step ds is used (I.94).

III.3.2 Grid generation

As soon as the intersection curve is known ( The grid generation is based on the advance front method. The mesh front is initialized by the meshed edges of a surface. Then, new triangular panels are created successively, which updates the front of the mesh, until the whole surface is meshed. A new panel is generated from three existing nodes of the front or, if it is not possible, from two existing nodes of the same edge of the front and a new node. The position of this new node P new satises the following equation:

MP new = δ 2 1 - δ ref 2 2 n F ront (III.123)
with:

n F ront the normal vector oriented from the mesh part of the surface to the unmeshed part;

M the midpoint of the edge of the front;

δ ref the reference panel size;

δ 1 a distance dened by:

δ 1 =      δ ref if 0.5 < δ ref and > δ ref 0.55δ ref if 0.55 > δ ref 2δ ref if 2 > δ ref (III.

III.3.3 Reference panel size

The reference panel size δ ref is often constant over a surface but not on the free surface. Indeed, to avoid the creation of meshes with too many panels, the free surface mesh is ner close to the surface-piercing bodies than far from the bodies. The evolution of δ ref over the free surface is gured out by solving the following problem [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF]: The boundaries of the free surface are the intersection curves with the surface-piercing bodies and the intersection between the free surface and the numerical tank walls. δ input is the input panel size xed by the user.

∆δ ref =
These equations are discretized over a rst free surface mesh matching the plane z = 0. The mesh is structured with a constant spatial discretization h ref in both x and y directions. To make the dierence with the nal unstructured mesh, the structured mesh is named the Cartesian grid. The solution is searched at the center of every cell of this grid. A second-order central nite-dierence scheme is used to solve the Laplace's equation. And after discretization: system is considered as dense. Nevertheless, the linear system is not dense: (III.126) involves a large sparse linear system. In his PhD work, Chauvigné [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF] did not use the sparsity of the linear system to solve it and it was not possible to compute δ ref in case of ne meshes. The quality of the mesh could not be maintained during the simulation if δ ref was not small enough. Thus, a new linear system solver has been coupled with WS_CN in order to improve the memory eciency: PARDISO6 (Parallel Sparse Direct And Multi-Recursive Iterative Linear Solvers). A LU decomposition is always performed but the method is optimized for sparse linear systems. The non-zero coecients of the matrix A are stored using a Compressed Sparse Row format (or CSR), based on three arrays:

δ i+1,j ref -2δ i,j ref + δ i-1,j ref h 2 ref + δ i,j+1 ref -2δ i,j ref + δ i,j
aa, containing the non-zero coecients;

ja, storing the column indices of the non-zero coecients in increasing order;

ia, listing the rst column indices of the previous array for every row.

For example, if the matrix A is7 :

A =       4 * * 7 * * * 6 * * -3 * * -2 9 * 1 * 3 * * * 8 * 6       (III.127)
Then, the CSR format of A is:

         aa = 4

III.4 Mesh morphing

Once the mesh is generated, it has to be updated at each time step of the simulation to take into account the motion of both the bodies and the free surface. Two choices are possible. Either the generation of a new mesh or the deformation of the existing mesh.

The rst method ensures a good mesh quality through the time-domain simulation but involves the interpolation of the physical quantities on the free surface (cf. section III.8)

and is time-consuming. Nevertheless this rst method can handle large deformations.

The second approach is possible when small deformations occur and requires a smaller CPU time.

To avoid too many remeshings, two mesh morphing algorithms are implemented in W S_CN [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF]:

A spring analogy method for the mesh of the bodies;

A mesh deformation method using radial basis functions for the free surface mesh.

When the deformations are too large, a remeshing process is applied.

III.4.1 Body mesh morphing

The spring analogy method consists of a physical analogy which replaces every edge of the mesh by a ctitious spring connecting two vertices. The method was introduced by Batina [START_REF] Batina | Unsteady Euler airfoil solutions using unstructured dynamic meshes[END_REF] for unstructured meshes in case of an unsteady aerodynamic analysis of oscillating airfoils. But it is also used in hydrodynamics, for example by Leroyer [START_REF] Leroyer | Etude du couplage écoulement / mouvement pour des corps solides ou à déformations imposée par résolution des équations de Navier-Stokes[END_REF] and Jacquin [START_REF] Jacquin | Navire autopropulsé en manoeuvres : simulation numérique et optimisation des performances hydrodynamiques[END_REF] with a viscous ow solver. The spring analogy method is applied in case of [START_REF] Blom | Considerations on the spring analogy[END_REF]:

Moving boundary problems;

Unstructured mesh smoothing.

In this latter application, the nodes are moved inside the mesh to obtain better shape panels and a higher quality mesh. This method will be used in section V.1. An example of such a mesh optimization may be found in [START_REF] Schmidt | Dynamic mesh optimization based on the spring analogy[END_REF]. The spring analogy method used in the present hydrodynamic solver enables to deform the mesh subject to moving boundaries.

The force exerted on the node i by its neighboring nodes follows Hooke's law [START_REF] Blom | Considerations on the spring analogy[END_REF]:

F i = N i v j=1 k ij (x j -x i -d ij ) (III.129)
where:

x i is the position of the node i;

d ij is the equilibrium vector between the node i and j; k ij denotes the spring stiness;

N i v represents the number of rst-order neighboring nodes of the node i.

Two dierent linear spring analogy exist, depending on the expression of the equilibrium vector:

The segment spring analogy when the equilibrium lengths are equal to the initial lengths of the edges: d ij = x old j -x old i . The expression of the spring loads becomes:

F i = N i v j=1 k ij (δ j -δ i ) (III.130)
with δ i = x i -x old i , the displacement of the node i. The mesh is only deformed if its boundaries are deformed themselves. Consequently this approach is chosen to solve moving boundary problems.

The vertex spring analogy when the equilibrium lengths are zeroed:

d ij = 0 3×1 .
Doing so, the mesh can be deformed even if its boundaries are still, which is the case for a mesh smoothing.

Here, the mesh deformation algorithm is used to update the mesh after a modication of the position of the bodies, thus the segment spring analogy is chosen.

In order to prevent the colliding of nodes and the interpenetration of neighboring panels, the spring stiness is inversely proportional to the square of the edge length.

Thus, if two vertices get closer, the spring becomes stier and conversely.

k ij = 1 x j -x i 2 (III.131)
The static equilibrium of the mesh leads to:

F i = 0 3×1 (III.132)
This equation may be solved iteratively for every node i or the whole linear system can be written and solved for all the nodes at a time.

Torsional springs also exist to handle larger deformations [START_REF] Farhat | Torsional springs for twodimensional dynamic unstructured uid meshes[END_REF][START_REF] Degand | A three-dimensional torsional spring analogy method for unstructured dynamic meshes[END_REF]. Other modications of the linear spring analogy are available too [START_REF] Selim | Mesh deformation approaches a survey[END_REF].

To ensure the spring loads are along the edge axis, Chauvigné [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF] preferred the following expression:

F i = N i v j=1 k ij (δ j • n ij -δ i • n ij ) (III.133)
with n ij the tangential unit vector along the edge formed by the nodes i and j.

The displacement is written in the local direct orthonormal base (u i , v i , n i ) of the node i:

δ i = d i u u i + d i v v i + d i w n i (III.134)
where:

n i is the inward normal vector of the body surfaces;

u i and v i are the two tangential vectors.

The slip condition on the body j at the node i involves:

d i w = [ e v j + e ω j × (x i -e S j )] • n i dt (III.135)
On the intersection curve or at the sharp edges of the body meshes, a node on this intersection has to remain on it. Thus, n i is normal to one of the surfaces, u i is along the intersection line and v i is chosen such as the local basis is orthonormal and direct.

In that case, d v is dened by:

d i v =    ∂φ I ∂n F S n F S • v i dt on the intersection curve [ e v j + e ω j × (x i -e S j )] • v i dt at the sharp edges (III.136)
n F S is the normal vector of the double node linked to the free surface on the intersection curve.

If three surfaces or more intersect each other at the same node i, then the condition is imposed on d i u :

d i u = [ e v j + e ω j × (x i -e S j )] • u i dt (III.137)
The use of the slip condition at every node makes the problem bi-dimensional, so the static equilibrium of this segment spring analogy method is for every node i:

F i • u i = 0 F i • v i = 0 (III.138)
After mathematical developments, the system of equations becomes for a node i on a smooth surface:

                               N i v j=1 k ij (u j • n ij )(u i • n ij )d j u - N i v j=1 k ij (u i • n ij ) 2 d i u + N i v j=1 k ij (v j • n ij )(u i • n ij )d j v - N i v j=1 k ij (u i • n ij )(v i • n ij ) d i v = N i v j=1 k ij (n j • n ij )(u i • n ij )d j w - N i v j=1 k ij (n i • n ij )(u i • n ij ) d i w N i v j=1 k ij (u j • n ij )(v i • n ij )d j u - N i v j=1 k ij (u i • n ij )(v i • n ij ) d i u + N i v j=1 k ij (v j • n ij )(v i • n ij )d j v - N i v j=1 k ij (v i • n ij ) 2 d i v = N i v j=1 k ij (n j • n ij )(v i • n ij )d j w - N i v j=1 k ij (n i • n ij )(v i • n ij ) d i w (III.139)
Equation III.139 leads to a sparse linear system of size twice the number of nodes in the body mesh AX = B with:

X =       . . . d i u d i v . . .       (III.140)
The solution is achieved using a LU factorization. This algorithm enables the denition of the node velocity in the body meshes:

v mesh = d i u dt u i + d i v dt v i + d i w dt n i (III.141)

III.4.2 Free surface mesh morphing

Regarding the free surface mesh morphing, the spring analogy method is not performed. It would lead to important CPU times, consequently another approach is chosen: an interpolation scheme of the free surface node displacement based on radial basis functions (RBF ) [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF][START_REF] De Boer | Mesh deformation based on radial basis function interpolation[END_REF].

The interpolation function s is expressed by:

s(x) = Nc j=1 α j ψ( x -x c j ) + p(x) (III.142)
with:

ψ the radial basis function dened by:

ψ(x) = x 2 log(x) (III.143)
p an unknown rst degree polynomial;

x c j the control nodes where the displacements are known;

N c the number of control nodes;

α j the unknown coecients.

α j and p are determined by the interpolation conditions:

s(x c j ) = d c j (III.144) Nc j=1 α j q(x c j ) = 0 (III.145)
where d c j is the known displacement of the node j and q denotes all polynomials with a degree less or equal than that of p.

Finally, these equations form a linear system of size N c + 3:

M P P T 0 3×3 α β = d c 0 3×1 (III.146)
with M the matrix containing the evaluation of the radial basis function such as M ij = ψ( x c i -x c j ) and P is the matrix with row j is given by the vector 1, x c j , y c j .

α is the vector of the coecients α j and β the vector of the coecients of p.

The coecients α and β are computed twice: a rst time for the displacements along the x-axis and a second time along the y-axis. Two interpolation functions are obtained: s x and s y . Then, the node velocity in the free surface mesh is gured out by solving the following system of equations:

               v mesh • n F S = ∂φ ∂n F S v mesh • e x e = s x (x i ) dt v mesh • e y e = s y (x i ) dt (III.147)

III.5 Incident wave models

In the decomposition (III.13), the incident components are assumed to be known.

Two models of incoming wave elds are used in WS_CN :

The Airy wave theory:

φ I (x, t) = Nwaves j=1 A j g ω j cosh[k(z + D)] cosh(kD) sin(k j • x -ω j t + φ j ) (III.148)
where A j , ω j , k j and φ j represent the wave amplitude, the wave frequency, the wave vector and the phase of the j th Airy wave. N waves is the total number of waves.

III.6. Absorbing numerical beach

The stream function theory of Rienecker and Fenton [START_REF] Rienecker | A Fourier approximation method for steady water waves[END_REF]:

φ I (x, t) = N RF j=1 B j cosh[jk(z + D)] cosh(jkD) sin[j(k RF • x -ω RF t + φ RF )] (III.149)
where B j and N RF are the coecients and the order of the Fourier series. ω RF , k RF and φ RF represent the wave frequency, the wave vector and the phase of the Rienecker and Fenton's wave.

The Airy wave model is linear whereas the Rienecker and Fenton's wave theory is nonlinear and therefore more consistent with the weakly nonlinear hydrodynamic solver used in this PhD work.

The dispersion relation for a wave of frequency ω and wave number is:

ω 2 = gk tanh(kh) (III.150)
where h is the water depth and g the gravity constant.

The incident wave eld is present from the starting time of the simulation. To avoid the generation of non-physical and spurious waves due to the abrupt appearance of the bodies in the ow, a ramp function is applied on the body conditions (III.30) and

(III.105):

f (t) =          0 when t T 1 1 when t T 2 1 2 1 -cos π t -T 1 T 2 -T 1 otherwise (III.151)
with T 1 and T 2 the starting and nal time of application of the ramp function.

III.6 Absorbing numerical beach

To ensure the permeability condition on the numerical tank walls (III.22) and to avoid the wave reection, a so-called absorbing numerical beach is applied. Its length is often equal to the greatest wave length of the incident wave eld. A damping coecient ν is added in the free surface boundary conditions:

                         δη P δt = - ∂η I ∂t -∇φ I • ∇η I -(∇φ I -v mesh ) • ∇η P -∇φ P • ∇η I + ∂φ I ∂z + ∂φ P ∂z +η P ∂ 2 φ I ∂z 2 -∇η I • ∂∇φ I ∂z ) -νη P at z = η I (x, y, t) δφ P δt = - ∂φ I ∂t - 1 2 ∇φ I • ∇φ I -(∇φ I -v mesh ) • ∇φ P -g(η I + η P ) -η P ∂ 2 φ I ∂z∂t + ∇φ I • ∂∇φ I ∂z -νφ P at z = η I (x, y, t) (III.152)
The expression of the damping coecient is:

ν(r) =          0 when r r 0 1 when r R 1 2 1 -cos π r -r 0 R -

III.7 State vector and time integration

The state vector of the weakly nonlinear potential ow theory based on the weakscatterer hypothesis is:

Y W SC =                φ P (F S) η P P mesh . . . η j . . . ηj . . .                (III.154)
P mesh represents the position of all the nodes of the mesh. Its time-derivative is v mesh of

The time-dierentiation of Y W SC is obtained using:

(III.29) for φP (F S) and ηP ;

(III.141) and (III.147) for v mesh ;

(III.113) for ηj .

The time-stepping is achieved using a fourth-order explicit Runge-Kutta scheme with a xed time step (I.94).

At the initial time t = 0s, the ow is assumed to be at rest so the scattered quantities are zeroed:

φ P (F S) = 0 N F S ×1 (III.155) φP (F S) = 0 N F S ×1
(III.156)

η P (F S) = 0 N F S ×1 (III.157) ηP (F S) = 0 N F S ×1 (III.158)
where N F S is the number of nodes of the free surface mesh.

III.8 Free surface remeshing

The simulations of oating bodies with large amplitude motions involve important deformations of the mesh and the appearance of degenerated panels even if mesh morphing methods are available (cf. Figure III.13). This leads to numerical errors. Therefore, it becomes necessary to regenerate the mesh of the domain when the panel shape quality decreases. The regeneration of the wetted body surface meshes only requires the call to the mesh generator as presented in section III.3. The hydrodynamic solver is not aected by this change. Chauvigné [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF] developed the remeshing of the bodies in his PhD work. In contrast, the remeshing of the free surface mesh has an impact on the hydrodynamic simulation. Indeed, two quantities of the state vector (III.154) are located at the free surface nodes: the perturbed velocity potential φ P (F S) and the scattered wave elevation η P . These quantities are known for the current free surface mesh but unknown for the new free surface mesh. An interpolation scheme based on third-order polynomial splines used for the computation of the surface gradients (cf.

subsection III.2.5) is applied to evaluate φ P (F S) and η P on the new free surface mesh from the current one. The free surface remeshing process unfolds in three steps:

A new free surface mesh (x N ew i

) i∈ 1 ; N N ew F S
is created using the advance front method, the former (old ) mesh

(x Old i ) i∈ 1 ; N Old F S is saved (Figure III.14a);
For each node of this new mesh, the nearest node of the former mesh is searched

(Figure III.14b);
A third-order polynomial spline interpolation is applied to evaluate the perturbed velocity potential and the scattered wave elevation at the location of the new nodes, based on the nearest node and its neighboring nodes in the former mesh

(Figure III.14c).
For example, for the wave elevation, the spline approximation is:

η P (x N ew i ) = N i,Old v +1 j=0 α j ψ x N ew i -x Old j + p(x N ew i ) (III.159) N i,Old v
refers to the number of rst-order and second-order neighboring nodes of the nearest point in the former mesh of x N ew i . +1 signies that the nearest point of x N ew i in the former mesh is included. α j and p are determined by the interpolation conditions on the nodes x Old j , neighbors of the nearest node in the former mesh of the node x N ew i in the new mesh:

s(x Old j ) = η P (x Old j )

(III.160)

N i,Old v +1 j=1 α j q(x Old j ) = 0 (III.161)
The same notations are used here as in subsection III.2.5. The case of a cylinder in waves with a heaving motion is used for the validation of the free surface remeshing. Its mass is 12.88 kg, its radius 0.2 m, its length 0.2 m and its draft 0.1 m. An incoming regular wave of amplitude 0.005 m and wave frequency 8 rad/s is used. A wave probe is located at the position (0.4, 0, 0). The mesh and time step convergences were checked and lead to a mesh of 10 000 panels and a time step of 0.01 s. Two sets of numerical results are compared based on when the free surface remeshing process is called:

When it is necessary: the size or the shape of a panel on the free surface is too poor (20 remeshings for 20 s);

At every time step: the remeshing is forced (2000 remeshings in total).

III.9 Gaussian lter

The free surface remeshing is not able to remove all the numerical errors on the free surface mesh. It is necessary to use a lter to remove the high-frequency nonphysical waves or saw-tooth waves. Letournel [START_REF] Letournel | Quantication de l'eet du lissage Gaussien ajouté dans le code WS[END_REF] implemented a Gaussian lter on the perturbed wave elevation in WS_CN. The lter is classically used every ve time steps. Its formula is:

η P (x i ) = 1 α ˆS η P (x)G f ilter (x i , x) dS (III.162) with α = ˆS G f ilter (x i , x) dS (III.163) G f ilter (x i , x) = 1 √ 2πσ exp - x -x i 2 2σ 2 (III.164)
The standard deviation σ is xed to 1 2

√ A i (1 + ν)
where A i is the area associated with the node i and ν the damping of the absorbing numerical beach (III.153).

After discretization, (III.162) becomes:

η P (x i ) = 1 α N i v j=1 η P (x j )G f ilter (x i , x j )A j (III.165)
where N i v is the number of rst-order and second-order neighboring nodes of the node i.

III.10 Parallelization

Several techniques are used to decrease the CPU time of WS_CN. In the PhD work of Letournel [START_REF] Letournel | Développement d'un outil de simulation numérique basé sur l'approche weak-scatterer pour l'étude des systèmes houlomoteurs en grands mouvements[END_REF]:

The (xOz) plane is used as symmetry plane to divide the number of panels by two if the physical problem is symmetric;

The atness of the sea bottom allows not to mesh this surface because it is used as a symmetry plane;

A far-eld approximation of the inuence coecients enables to speed-up their computation;

Chauvigné [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF] developed a partial computation of the inuence coecients. The parts of the mesh which only have a small relative motion between each other keep the same inuence coecients. For example the panels of the numerical tank wall are almost still during a simulation, so their inuence on themselves is constant and does not need to be evaluated again.

Three new developments were added:

The use of the sparse linear system solver PARDISO in the body mesh morphing algorithm (cf. subsection III.4.1), as it was done in subsection III.3.3;

The respect of the column-major order because WS_CN is implemented in Fortran;

The parallelization using OpenMP8 (Open Multi-Processing) of the computation of the inuence coecients.

III.11 Extension to multibody simulations

Letournel [START_REF] Letournel | Développement d'un outil de simulation numérique basé sur l'approche weak-scatterer pour l'étude des systèmes houlomoteurs en grands mouvements[END_REF] and Chauvigné [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF] only run single body simulations with WS_CN.

The extension to multibody simulations was necessary to perform marine operation simulations involving at least two bodies subject to hydrodynamic loads. It unfolds in two steps:

The extension of the mesh generator;

The extension of the hydrodynamic solver.

These developments were directly presented by taking into account a multibody format for writing the equations in the previous sections. It stays to validate them both in forced and free motions.

III.11.1 Forced motion

The test cases presented by Watai et al. [START_REF] Watai | A new time domain Rankine panel method for simulations involving multiple bodies with large relative displacements[END_REF] are used for the validation of the capacity of WS_CN to deal with multiple interacting oating bodies.

III.11.1.1 Presentation of the test cases

Watai et al. [START_REF] Watai | A new time domain Rankine panel method for simulations involving multiple bodies with large relative displacements[END_REF] conducted experimental tests at the University of Sao Paulo (Brazil) with two cylinders in regular waves:

One xed cylinder, named Body 1 ;

One moving cylinder, named Body 2, with a large prescribed harmonic motion along the x-axis.

The characteristics of these cylinders are listed in Table III Table III.

Wave probe locations

The incident wave eld coming from the negative y, so WP1 is upstream and WP3 Watai et al. [START_REF] Watai | A new time domain Rankine panel method for simulations involving multiple bodies with large relative displacements[END_REF] considered four regular waves, listed in Table III.5.

For each regular wave, three wave frequencies, ω pm , of the harmonic prescribed motion of Body 2 were selected. The motion amplitude, A pm is xed at 0.37 m. Overestimations are also present in the work of Watai et al. [START_REF] Watai | A new time domain Rankine panel method for simulations involving multiple bodies with large relative displacements[END_REF] and are likely due to the main hypotheses of the uid solver (inviscid uid, irrotational ow). Nevertheless Watai et al. [START_REF] Watai | A new time domain Rankine panel method for simulations involving multiple bodies with large relative displacements[END_REF] needed much less panels (4010 panels in total for Case 1) than the present uid method (around 20 000 on the free surface mesh for Case 1). These authors used a body-exact approximation (using a linearized free surface at z = 0) whereas a weak-scatterer hypothesis is applied here, which could explain the need of many more panels and so a slower mesh convergence. The use of many remeshing processes, because of the large relative motion, leads to interpolations of the physical quantity on the free surface and could deteriorate the accuracy. In Case 1, the total mesh is regenerated around 200 times during the simulations (5 % of the number of times steps). Figure III.33 shows the eect of a forced remeshing process at every time step. It is only noticeable along the x-axis. The remeshing processes cannot be deleted otherwise large numerical error arise. Therefore, the eect of the required 200 remeshing processes cannot be quantied. Another possible explanation of the dierences between the numerical results and the experimental data is the use of the Gaussian ltering.

In our simulations, the ltering is applied every ve time steps. If the ltering is not used, sawtooth waves appear (cf. Finally, the weak-scatterer method seems to be slightly more accurate than the body-exact method. This conrms that the weak-scatterer method leads to a bodyexact approximation when the incident wave stiness and the free surface nonlinearities are small as required by the body-exact approximation.

If the wave steepness (III.15) is increased, and so the free surface nonlinearities too, then the dierences between the two methods increase. Numerical results are showed in Figures III. [START_REF]Ariane8 theoretical manual[END_REF] WS_CN may also be switched to a fully linear approximation. In this case, the free surface is linearized at its mean position (z = 0) and the body conditions are linearized at the equilibrium position of the bodies (initial position here). The mesh convergence is shown in Figures III. [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF] and III.46. The mesh with 9000 panels is chosen. The time step is 0.005 s. Less panels are required compared to the weak-scatterer and body-exact approaches. Figures III. [START_REF] Babarit | Theoretical and numerical aspects of the open source BEM solver NEMOH[END_REF] and III.48 present the comparison between the experimental data, the weak-scatterer, the body-exact and the fully linear approximations. The amplitude and the frequency of F y 1 and F z 1 are good but the modulation of the amplitude is not captured. The hydrodynamic load along the x-axis, F x 1 , does not match the ex- perimental data and the other numerical methods at all. Regarding the wave elevations, the modulation is not captured either. Larger dierences are observed for η 1 and η 2 because the hydrodynamic interactions are more important than for η 3 . Indeed, the wave probe 3 is located in a sheltered area provided by Body 1 [START_REF] Watai | A new time domain Rankine panel method for simulations involving multiple bodies with large relative displacements[END_REF], which explains the better agreement of the linear model with the weak-scatterer and body-exact models.

Finally, the linear model is not accurate enough to simulate the hydrodynamic interactions in the present case. The free surface nonlinearities are small, which enables the use of the body-exact approximation but the body nonlinearities are too important to use a linear approximation. This conrms that an unsteady potential ow based solver is required in case of large relative motions. In this section, the extension of WS_CN to multibody simulations is veried for several bodies in free motion. A comparison is done between WS_CN and the diractionradiation code Nemoh [START_REF] Babarit | Theoretical and numerical aspects of the open source BEM solver NEMOH[END_REF] coupled with InWave.

Two identical vertical cylinders (named Cylinder 1 and Cylinder 2 ) of radius 0.2 m, length 0.2 m and draft 0.1 m are considered. The initial positions of Cylinder 1 and Cylinder 2 are (0, -0.4, 0) and (0, 0.4, 0).

Their mass is 12.88 kg. An incoming regular wave of amplitude 0.001 m and wave frequency 8 rad/s is used. A wave probe is present at the position (0.4, 0, 0). The cylinders only move in heave. The multibody system used in InWave-Nemoh is made of two bodies linked by a prismatic joint. Cylinder 1 is the base while Cylinder 2 is the body 1. The set of modied Denavit-Hartenberg coecients is given in Table III

.7. j σ j γ j (rad) b j (m) α j (rad) d j (m) θ j (rad) r j (m) 1 1 π 2 0 0 0.8 - π 2

III.12 Conclusion

This chapter presented the hydrodynamic theory used in this PhD work in WS_CN :

the potential ow approach based on the weak-scatterer hypothesis. A review of the dierent approximations of the potential ow theory was provided along with a state of the art of the use of the weak-scatterer approximation. In order to simulate marine operations involving several bodies subject to hydrodynamic loads, WS_CN has been extended to multibody simulations. A validation in forced motion was presented using experimental data. The comparison between the experimental data and the weakscatterer model gave good results for both the hydrodynamic loads and the wave elevations. Hydrodynamic interactions between the oating bodies were well captured.

An analysis of the weak-scatterer hypothesis was done through the comparison with a body-exact approximation and a fully linear approximation. As the wave steepness was small, the body-exact model also gave good results compared to the experiments but with fewer panels in the mesh. Nevertheless, when the wave steepness and the free surface nonlinearities were increased, the weak-scatterer and body-exact approaches started being dierent. The linear approximation gave insuciently accurate results compared to the experimental data.

These results have proved the interest of an unsteady potential ow theory in case of large relative motions. Furthermore, the weak-scatterer approximation is fully relevant when the free surface nonlinearities are important, otherwise a body-exact could be enough. If a body-exact approximation is used, in order to reduce the size of the mesh and therefore the CPU time, a free surface Green's function (or Kelvin source Green's function) could be preferred to the Rankine source Green's function used in WS_CN.

The free surface would not be meshed.

The extension to multibody simulations in free motion was validated by the comparison with the frequency-domain potential ow-based solver Nemoh coupled with InWave.

The stability of WS_CN has been improved by the implementation of the free surface remeshing process. The CPU time has been decreased by the parallelization of the computation of the inuence coecients using OpenMP. The memory footprint has been reduced by the use of the sparse linear system solver PARDISO.

The coupling of the potential ow theory based on the weak-scatterer hypothesis of WS_CN with the Composite-Rigid-Body Algorithm of InWave is the goal of the next chapter.

The previous chapters presented the main technical requirements to simulate marine operations: multibody dynamics, cable dynamics and hydrodynamics. The goal of this chapter is to introduce the coupling between these models. First, a state of the art of the multiphysics couplings is presented. Then, a coupling strategy, between the potential ow theory based on the weak-scatterer hypothesis of WS_CN and the Composite-Rigid-Body Algorithm of InWave, is proposed and detailed. A validation of these developments along with a comparison with other coupling strategies are presented.

IV.1 State of the art IV.1.1 Multiphysics problems

Multiphysics problems are met in many elds [START_REF] Felippa | Staggered transient analysis procedures for coupled mechanical systems: Formulation[END_REF][START_REF] Sigrist | Méthodes numériques de calculs couplés uide/structure -cas du uide stagnant : introduction[END_REF]:

Civil engineering: wind-structure interaction, soil dynamics, etc.;

Ocean engineering: uid-structure interaction, cavitation, acoustics, etc.; Aerospace, nuclear or automotive engineering: uid-structure interaction, thermalstructure interaction, etc.; Biological engineering: uid-structure interaction, etc.;

Other.

These multiphysics problems can be represented by partial dierential equations in space and time coupled at their physical boundaries. Software modules are generally available to solve one domain of the physics (named a eld). For coupled problems, it is necessary to couple several software packages. Park et al. [START_REF] Park | Stabilization of staggered solution procedures for uide-structure interaction analysis[END_REF] published one of the rst papers on uid-structure interaction (nite-element methods for the structure, boundary integral technique for the uid). Since the early works, three schemes to simulate multiphysics problems have arisen [START_REF] Felippa | Staggered transient analysis procedures for coupled mechanical systems: Formulation[END_REF][START_REF] Felippa | Partitioned analysis of coupled mechanical systems[END_REF]:

IV.1.1.1 Field elimination

The eld elimination method eliminates one or more elds in the coupled equations.

The other elds are time-stepped simultaneously. For instance, in a uid-structure problem, this method implies to introduce the uid equations into the motion equation, so that this latter is the only equation to solve. The advantage of this method is that it reduces the number of variables to time-step. But it leads to the increase in the complexity of the multiphysics problem to solve (high-order time-derivatives can appear) and the loss of mathematical properties [START_REF] Felippa | Staggered transient analysis procedures for coupled mechanical systems: Formulation[END_REF]. This approach is rarely used.

IV.1.1.2 Monolithic treatment

With a monolithic treatment, the governing dierential equations are time-stepped simultaneously. This system of dierential equations is called the coupling equation.

The advantages of a monolithic treatment are the robustness and the accuracy as the governing equations of the system are solved synchronously. But this approach involves a complex assembling and solving of the set of equations. Furthermore, the temporal discretization needs to be the same (although several time integrators may be used [START_REF] Li | Développement d'une méthode de simulation de couplage uide-structure à l'aide de la méthode SPH[END_REF]). Fixed time-step through the eld solvers are inconvenient if one of them requires a much smaller time step compared to the other ones. An example of monolithic treatment is given by Blom [START_REF] Blom | A monolithical uid-structure interaction algorithm applied to the piston problem[END_REF] with a monolithic uid-structure coupling using Eu- al. [START_REF] Park | Partitioned transient analysis procedures for coupled-eld problems: Stability analysis[END_REF][START_REF] Park | Partitioned transient analysis procedures for coupled-eld problems: Accuracy analysis[END_REF]. In case of sequential execution of the eld solvers, partitioned approaches are named staggered solutions. This case was studied in detail by Felippa and Park [START_REF] Felippa | Staggered transient analysis procedures for coupled mechanical systems: Formulation[END_REF]. Dierent algorithms were developed for the staggered approach [START_REF] Piperno | Partitioned procedures for the transient solution of coupled aeroelastic problems -Part 1: Model problem, theory and two-dimensional application[END_REF][START_REF] Farhat | Two ecient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems[END_REF][START_REF] Matthies | Algorithms for strong coupling procedures[END_REF].

The positive aspects of a partitioned treatment are the inherent modularity along with the possibility to t the time step, the time integrator and more generally the physical model for every eld solver. Thereby, each eld solver is independent from the others.

For complex coupled problems, this approach is more software-wise and less challenging than the monolithic treatment. The drawbacks are the numerical errors and the stability problems.

As the solvers are called asynchronously, it always exists a time-lag between the time integrations of the elds. For instance, the exact position of a uid-structure interface is not necessarily the same in every solver because the spatial discretizations are dierent.

Therefore, the slip condition for an inviscid uid or the no-slip condition for a viscous uid is not exactly enforced. The result of this is an articial increase or decrease of the energy in the system which leads to a loss of accuracy. The conservation of the energy at the uid-structure interface allows to improve the robustness and accuracy of such partitioned approaches [START_REF] Piperno | Partitioned procedures for the transient solution of coupled aeroelastic problems -Part 2: energy transfer analysis and three-dimensional applications[END_REF]. The energy conservation has also to be ensured with a monolithic treatment when several spatial discretizations (meshes) are used [START_REF] Mahjoubi | A monolithic energy conserving method to couple heterogeneous time integrators with incompatible time steps in structural dynamics[END_REF]. The use of a predictor can reduce the eects of the time-lag [START_REF] Piperno | Partitioned procedures for the transient solution of coupled aeroelastic problems -Part 2: energy transfer analysis and three-dimensional applications[END_REF].

Amongst the staggered methods or loose couplings, there are two classes: the purely sequential methods or explicit couplings and the iterative methods or implicit couplings [START_REF] Matthies | Algorithms for strong coupling procedures[END_REF][START_REF] Gasmi | Numerical stability and accuracy of temporally coupled multi-physics modules in wind-turbine CAE tools[END_REF]. Each method can also be divided into two versions: serial or parallel, leading to four classical staggered coupling algorithms [START_REF] Yvin | Interaction uide-structure pour des congurations multi-corps. Applications aux liaisons complexes, lois de commande d'actionneur et systèmes souples dans le domaine maritime[END_REF]

(Figure IV.2):
The explicit couplings:

• The conventional serial staggered method (CSS ) (Figure IV.3);

• The conventional parallel staggered method (CPS ).

The implicit couplings:

• The block Gauss-Seidel method (Figure IV.4);

• The block Jacobi method.

The only advantage of the parallel version of a coupling method is the possibility of parallelization. The numerical properties are better for the serial version [START_REF] Fourey | An ecient FSI coupling strategy between smoothed particle hydrodynamics and nite element methods[END_REF]. By using a retroactive loop, an implicit coupling gives the same results as a monolithic treatment, other things being equal [START_REF] Matthies | Algorithms for strong coupling procedures[END_REF]. But, the solvers are called more than once per time step. Consequently implicit couplings lead to an increase of the CPU time compared to explicit couplings. Other algorithms exist. For instance, subcyles can be added if a eld needs much smaller time step, couplings can be asynchronous, the exit condition in a iterative loop may be enhanced by using a Newton or Newton-like method in order to accelerate the implicit coupling [START_REF] Yvin | Interaction uide-structure pour des congurations multi-corps. Applications aux liaisons complexes, lois de commande d'actionneur et systèmes souples dans le domaine maritime[END_REF], etc.

In 1995, Belanger et al. [START_REF] Belanger | Time-marching analysis of uidcoupled systems with large added mass[END_REF] encountered numerical instabilities in their staggered uid-structure coupling when the uid added-mass became larger than the structural mass. They proposed a method to avoid these numerical discrepancies by adding a virtual added-mass in the motion equation to stabilize their simulation [START_REF] Belanger | Time-marching analysis of uidcoupled systems with large added mass[END_REF]. An explanation was given using a theoretical model [START_REF] Causin | Added-mass eect in the design of partitioned algorithms for uidstructure problems[END_REF]. It showed that the ratio of the uid density over the structural density is critical for the stability of the staggered coupling.

Moreover, when decreasing the time step, instabilities become more important. Förster et al. [START_REF] Förster | Articial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous ows[END_REF] showed that this problem does not depend on the time integration scheme.

The added-mass eect was also studied for compressible ows by van Brummelen [START_REF] Van Brummelen | Added mass eects of compressible and incompressible ows in uid-structure interaction[END_REF].

Yvin [START_REF] Yvin | Interaction uide-structure pour des congurations multi-corps. Applications aux liaisons complexes, lois de commande d'actionneur et systèmes souples dans le domaine maritime[END_REF] used a relaxation method to reduce the numerical instabilities without modications of the uid or structure solvers. As this added-mass eect depends on the uid density, it does not appear when the uid is the air, because its density is too small. This numerical problem is mostly encountered in partitioned water-structure coupling.

IV.1.2 Computer programming

In the previous section, coupling schemes were presented from a physical and numerical point of view. The software architecture of multiphysics numerical tools is an important consideration too. Two options are possible to solve a coupled problem:

either a new numerical tool is created and dedicated to the multiphysics problem or dierent existing numerical tools are coupled.

Partitioned approaches are modular by their nature. The use of legacy codes is possible. The independence of the implementation of the dierent solvers is guaranteed.

For the monolithic treatment of a coupled problem, it is not straightforward. This is why it is necessary to make a distinction between a monolithic formulation, where a new numerical tool is created ex nihilo, including all the solvers, and a tight coupling in which dierent independent subsystems solving a unique eld of the physics are linked through a coupling equation and time-stepped simultaneously. This latter coupling is considered as a partitioned approach [START_REF] Gasmi | Numerical stability and accuracy of temporally coupled multi-physics modules in wind-turbine CAE tools[END_REF]. It gathers the numerical advantages of a monolithic treatment (robustness, accuracy) and the modularity and the software-wise approach of a partitioned treatment. Loose couplings are distinct of tight couplings by the separate time-stepping of the solvers. For example, Li [START_REF] Li | Développement d'une méthode de simulation de couplage uide-structure à l'aide de la méthode SPH[END_REF] used a tight coupling approach with a Smooth-Particle-Hydrodynamics model and a nite-element method.

Finally, the dierent coupling strategies are summarized in

Language binding

In partitioned approaches, the implementation of the solvers is independent from the others. Consequently, dierent programming languages may be used, for instance to take advantage of the speed of low-level languages and the ease of implementation of high-level languages. Doing so input-output communication between solver may become challenging. Dierent methods exist to overcome this diculty.

The simplest approach relies on temporary text les. Each matrix, vector or scalar required by a solver is written in a temporary le. In case of an important bulk of data, the time of writing and reading makes this method prohibitive.

The use of a glue code is another possibility [START_REF] Jonkman | The new modularization framework for the FAST wind turbine CAE tool[END_REF]. All modules are interfaced with one high-level program which manages the input-output relationships for the whole code via DLLs (Dynamic Link Library) or the direct coupling of the source codes. This latter option needs the use of a specic programming language dedicated to the interfacing with other existing applications. Python

1 is an example of such a language. Dierent tools exist to compile low-level languages with Python. For example, Cython2 allows linking scripts written in C or C++ with Python [START_REF] Behnel | Cython: The best of both worlds[END_REF], f2py3 does the same between Fortran and Python [START_REF] Peterson | F2PY: a tool for connecting fortran and python programs[END_REF], etc. The availability of the source codes restrains clearly this method (for instance when commercial tools are used).

Finally the use of message passing between solvers is another possibility. For instance, MPI communication (Message Passing Interface) enabled the coupling between the uid solver SPH-ow4 and the structural solver Code_Aster5 [START_REF] Fourey | An ecient FSI coupling strategy between smoothed particle hydrodynamics and nite element methods[END_REF]. Yvin [START_REF] Yvin | Interaction uide-structure pour des congurations multi-corps. Applications aux liaisons complexes, lois de commande d'actionneur et systèmes souples dans le domaine maritime[END_REF] preferred employing network sockets to link the multibody dynamics software package MBDyn6 and the hydrodynamic solver ISIS-CFD7 .

IV.2 Coupling between the CRBA and the weakly nonlinear potential ow theory

IV.2.1 The original coupling in WS_CN

The terms to dene the dierent methods to solve the uid-structure interaction, presented in subsubsection III.2.6.1, do not match those presented in section IV.1. Each eld (elds of the potential ow theory and multiphysics coupling theory) has dened its own terms. Nevertheless, an equivalence is possible and is presented in Table IV . i represents the number of iterations to reach the convergence at each time step.

The mode decomposition and indirect methods are two monolithic couplings because the dierential equations are time-stepped simultaneously. But they are specic to the potential ow modelling because of the mode decomposition and the use of the Green's second identity. The implicit boundary method is strictly a monolithic coupling, as the dierential equations of the potential ow theory and the motion equations are solved at the same time.

IV.2.2 The choice of a coupling strategy

As presented in the previous sections, several coupling strategies are possible. In-Wave and WS_CN already existed before this PhD work, therefore a monolithic coupling, which leads to a new implementation of both solvers was not a meaningful option.

An implicit loose coupling would lead to a higher CPU time cost than a tight coupling.

An explicit loose coupling would only give a small reduction of the computation time, as the second Boundary Value Problem always needs to be gured out, for the price of a decrease of the accuracy and the stability of the simulation.

An advantage of a loose coupling approach is the possibility to t the time step for each solver. The mechanical solver needs to simulate cables and leads to a sti system of dierential equations, so small time steps are expected. The suitable order of magnitude of the uid solver time step is not predictable with accuracy. But the subcycling of the mechanical solver in a tight coupling could x this problem. This is why, in order to maintain the robustness, the stability and the accuracy, without increasing signicantly the CPU time while creating a modular architecture, a tight coupling between the Composite-Rigid-Body Algorithm of InWave and the weakly nonlinear potential ow theory of WS_CN is preferred.

IV.2.3 Theoretical developments

The tight coupling between the Composite-Rigid-Body Algorithm and the weakly nonlinear potential ow theory involves solving both the mechanical and uid governing equations in the same time. It requires the introduction of the hydrodynamic loads in the multibody motion equation (I.62) and the expression of the slip condition (III. [START_REF] Hannan | Nonlinear hydrodynamic responses of submerged moving payload in vicinity of a crane barge in waves[END_REF] with respect to the articular accelerations.

IV.2.3.1 Multibody motion equation

Two methods are possible for taking into account the hydrodynamic loads in the multibody motion equation. The rst method involves to include the hydrodynamic loads in the external load vector j F ej (Method 1 ). For example, the weight is added in the multibody motion equation following this method (cf. section I.4). The second method requires the projection of the hydrodynamic loads in the articular space using the articular variables (Method 2 ). For instance, the vector of the internal loads along the joint axes Γ (I.62) follows this approach.

In the case of the weight, Method 1 is straight forward. But, Method 2 could also be applied. Regarding the hydrodynamic loads, Method 1 is not as feasible. As seen in subsection III.2.6, the hydrodynamic loads bring a new unknown: the time-dierentiation of the velocity potential. This unknown must be accessible and isolated to form the linear system of the uid-structure coupling. Method 1 does not enable that. That is why, Method 2 is preferred.

The application of Method 2 to include the hydrodynamics loads in (I.62) gives:

H 0 V0 q = 0 6×1 Γ -C + 0 F W SC Γ W SC (IV.1)
0 F W SC is the sum of all hydrodynamic loads acting on the whole multibody system at O 0 and projected in the base frame:

0 F W SC = j∈F W SC 0 N j e e G j e F W SC j (IV.2)
with:

F W SC the set of the bodies subject to the hydrodynamic loads. Each hydrodynamic body has a number in the multibody system and a number in F W SC .

There is:

card(F W SC ) = N W SC (IV.3)
card( * ) represents the number of elements of a set.

e F W SC j the hydrodynamic loads acting on the body j at G j in Σ e given by (III.95); e G j the (6 × 6) matrix to change the point of computation of the hydrodynamic loads from G j to O j : e G j = I 3 0 3×3 S( e R j j S j ) I 3 (IV.4) e the (6 × 6) matrix to project the hydrodynamic loads into the base frame from O j to O 0 :

0 N j e = 0 R e 0 3×3 S( 0 P j ) 0 R e 0 R e (IV.5)
Γ W SC is the sum of the hydrodynamic articular loads:

Γ W SC = j∈F W SC \{0} Γ W SC j (IV.6)
where Γ W SC j is the projection of the hydrodynamic loads acting on the body j into the articular space [START_REF] Featherstone | Rigid body dynamics algorithms[END_REF]:

Γ W SC j = j J T j j ρ e e G j e F W SC j (IV.7)
with:

j ρ e the (6 × 6) projection matrix of the hydrodynamic loads from Σ e to Σ j :

j ρ e = j R e 0 3×3 0 3×3 j R e (IV.8)
j J j the (6 × n) Jacobian matrix of the body j, dened by:

j V j = j T 0 0 V 0 + j J j q (IV .9)
The Jacobian matrix represents the mapping between the operating space (using Cartesian coordinates) and the articular space (using articular variables). If the base is xed, the k th column of j J j represents the contribution of the k th articular velocity to the Cartesian velocity j V j . Following this denition, the Jacobian matrix does not exist for the base (j = 0).

The hydrodynamic loads are expressed by (III.110). Equation IV.1 becomes:

H 0 V0 q -    j∈F W SC 0 N j e e G j e CT j φP (B j ) j∈F W SC \{0} j J T j j ρ e e G j e CT j φP (B j )    = 0 6×1 Γ -C +    j∈F W SC 0 N j e e G j e T W SC j j∈F W SC \{0} j J T j j ρ e e G j e T W SC j    (IV.10)

IV.2.3.2 Slip conditions

The slip conditions (III.109) need to be expressed with respect to the articular accelerations instead of the Cartesian accelerations. The rst step is to write the Cartesian velocity with respect to the velocity of the base and the articular velocities. From (I.40), one has:

e ω j = S j   φj θj ψj   (IV.11)
with:

S j =   C θ j C ψ j -S ψ j 0 C θ j S ψ j C ψ j 0 -S θ j 0 1   (IV.12)
e V j is the Cartesian velocity of body j at O j with respect to Σ e . ηj is dened by (I.18) for the body 0 but its denition is extended for every body j. This velocity is expressed at G j and using the Cardan angles. e V j and ηj are related by: ηj = τ -1 j e V j (IV.13) with:

τ -1 j = I 3 -S( e R j j S j ) 0 3×3 S -1 j (IV.14)
The (6 ×(n+6)) generalized Jacobian matrix e Λ j allows to relate the (6 ×1) velocity vector of body j in Σ e to the base and articular velocities:

e V j = e Λ j 0 V 0 q (IV.15)
The k th column of e Λ j represents the contribution of the k th component of the base velocity 0 V 0 , if k 6, or the k th , if k > 6, articular velocity to the Cartesian velocity e V j .

After time-dierentiation of (IV.15) and (IV.13) and the introduction of (I.61), the Cartesian acceleration, ηj , can be expressed as function of the base and articular accelerations:

ηj = τ -1 j e Λ j 0 V 0 q + τ -1 j e Λj 0 V 0 q + τ -1 j e Λ j   0 V0 q -   S( 0 ω 0 ) 0 v 0 0 3×1 0 n×1     (IV.16)
The time-derivation of τ -1 j is expressed by:

τ -1 j = 0 3×3 -e R j S( j ω j )S( j S j ) j R e 0 3×3 -S -1 j Ṡj S -1 j (IV.17)
Finally, the introduction of (IV.16) in (III.109) yields:

φP n (B j ) = -φI n (B j ) + CK j τ -1 j e Λ j 0 V0 q + CK j   ( τ -1 j e Λ j + τ -1 j e Λj ) 0 V 0 q -τ -1 j e Λ j   S( 0 ω 0 ) 0 v 0 0 3×1 0 n×1     + Q j (IV.18)
It remains to clarify the expressions of the Jacobian matrix j J j , the generalized Jacobian matrix e Λ j and its time-dierentiation e Λj .

IV.2.3.3 Jacobian matrices

From the denition of the Jacobian matrices given in IV.2.3.1, let us consider the k th articulation of a multibody system and a body j such as j > k. The base is assumed motionless. The articular velocity qk leads to a velocity vector k V k j for the body j:

k V k j = σ k k z k + σk ( k z k × k P j ) qk σk k z k qk (IV.19)
By summing the contributions of all the joints of the direct branch β(j) (as dened in subsection I. 4.3), one can have:

V j = k∈β(j) k V k j (IV.20) =    k∈β(j) σ k k z k + σk ( k z k × k P j ) qk k∈β(j) σk k z k qk    (IV.21)
It is necessary to dene a common frame of projection, for instance Σ j :

j V j =    k∈β(j) σ k j z k + σk S( j z k ) j R k k P j ) qk k∈β(j) σk j z k qk    (IV.22)
Thus, following the matrix format of (IV.9), the expression of j J j is:

col k ( j J j ) =      σ k j z k + σk S( j z k ) j R k k P j ) qk σk j z k qk if k ∈ β(j) 0 6×1 if k / ∈ β(j) (IV.23)
The change of frames is achieved using the projection matrix (IV.8), for instance in the base frame:

0 J j = 0 ρ j j J j (IV.24) IV.2.3.4 Expression of e Λ j
The projection in Σ e of (IV.9) leads to: e V j = e ρ j j V j (IV.25)

= e ρ j j T 0 0 V 0 + e ρ j j J j q (IV.26)

= e ρ 0 0 L j 0 V 0 + e ρ 0 0 J j q (IV.27) = e ρ 0 0 L j 0 V 0 + 0 J j q (IV.28)
where:

0 L j changes the point of computation of 0 V 0 from O 0 to O j : 0 L j = I 3 -S( 0 P j ) 0 3×3 I 3 (IV.29)
the expression of 0 J j is given by (IV.24).

Based on (IV.15), the expression of the generalized Jacobian matrix is:

e Λ j =    e ρ 0 0 L j 0 J j if j = 0 e ρ 0 0 L j 0 6×n if j = 0 (IV.30)

IV.2.3.5 Expression of e Λj

The time-dierentiation of (IV.30) is challenging because of the Jacobian matrix 0 J j . Two methods exist to compute the time-derivation of such a Jacobian matrix. The rst one is a direct calculation as presented by Hourtash [START_REF] Hourtash | The kinematic hessian and higher derivatives[END_REF]. The drawback of this approach is its complexity. The second method is based on a recursive algorithm which computes e Λj 0 V 0 q instead of e Λj directly [169].

The time-derivative of (IV.15) is:

e Vj = e Λj 0 V 0 q + e Λ j   0 V0 q -   S( 0 ω 0 ) 0 v 0 0 3×1 0 n×1     (IV.31)
Assuming 0 V0 = 0 6×1 and q = 0 n×1 and applying (IV.30):

e V * j = e Λj 0 V 0 q -e ρ 0 S( 0 ω 0 ) 0 v 0 0 3×1 (IV.32)
By denoting:

e V * j = e Vj 0 V0 =0 6×1 q=0 n×1 (IV.33)
e V * j can also be expressed by:

e V * j = e ρ j j V * j (IV.34)
The introduction of (IV.34) in (IV. [START_REF] Lee | Dynamic response of a oating crane in waves by considering the nonlinear eect of hydrostatic force[END_REF] gives:

e Λj 0 V 0 q = e ρ j j V * j + e ρ 0 S( 0 ω 0 ) 0 v 0 0 3×1 (IV.35)
Using the assumption q = 0 n×1 in (I.48) allows writing for i = a j :

j V * j = j T i i V * i + j γ j (IV.36)
While the hypothesis 0 V0 = 0 6×1 initializes this recursive equation:

0 V * 0 = 0 6×1 (IV.37)
Using (IV.36) and (IV.37), j V * j can be computed for each body during the Composite- Rigid-Body Algorithm. Then, the quantity e Λj 0 V 0 q can be calculated from (IV.35).

Thereby:

e Λj 0 V 0 q =                                         e ρ j j V * j + e ρ 0 S( 0 ω 0 ) 0 v 0 0 3×1 with: j V * j = j T i i V * i + j γ j for i = a j 0 V * 0 = 0 6×1                if j = 0 e ρ 0 S( 0 ω 0 ) 0 v 0 0 3×1 if j = 0 (IV.38)

IV.2.3.6 Linear system and time integration

The tight uid-structure coupling between the Composite-Rigid-Body Algorithm and the weakly nonlinear potential ow theory is formed by (III.108), (IV.10) and (IV.18):

                                                   G φ = H φn H 0 V0 q -    j∈F W SC 0 N j e e G j e CT j φP (B j ) j∈F W SC \{0} j J T j j ρ e e G j e CT j φP (B j )    = 0 6×1 Γ -C +    j∈F W SC 0 N j e e G j e T W SC j j∈F W SC \{0} j J T j j ρ e e G j e T W SC j    φP n (B j ) -CK j τ -1 j e Λ j 0 V0 q = -φI n (B j ) + CK j   ( τ -1 j e Λ j + τ -1 j e Λj ) 0 V 0 q -τ -1 j e Λ j    S( 0 ω 0 ) 0 v 0 0 3×1 0 n×1       + Q j ∀j ∈ F W SC (IV.39)
Equation IV.39 may be re-written as a (N + j∈F W SC N (B j ) + n + 6) linear system: AX = B. N is the total number of nodes in the mesh, N (B j ) is the number of nodes in the mesh of the body j and n is the number of degrees of freedom in the multibody system. It is solved by using a GMRES method with a diagonal preconditioner. Doing so, the vector X is known:

X =                   φP n (F S) φP (T ) . . . φP (B j ) . . . φP n (B j ) . . . 0 V0 q                   (IV.40)
The state vector of this coupling (CRBA -WSC ) is:

Y CRBA-WSC =                      φ P (F S) η P P mesh . . . η j . . . ηj . . . ν 0 q q                      (IV.41)
Its time-dierentiation is obtained using:

(III.29) for φP (F S) and ηP ;

(III.141) and (III.147) for v mesh ;

(IV.16) for ηj ;

(I.93) and (I.61) for ν0 ;

(I.93) for q.

The time-stepping is achieved using a fourth-order explicit Runge-Kutta scheme with a xed time step (I.94).

IV.2.4 Language binding

The Composite-Rigid-Body Algorithm is implemented in C++ in the numerical tool

InWave while the weakly nonlinear potential ow theory is implemented in Fortran 90 in the solver WS_CN. Thus, a language binding is mandatory because of the two dierent programming languages. As explained in subsection IV.1.3, dierent possibilities of language binding exist. The source code of both InWave and WS_CN are available so the use of a glue code to guarantee the modularity of the numerical uid-structure coupling is favoured. The co-simulation is driven in a Python environment. Cython makes the language binding between the C++ source code and the Python glue code whereas f2py does the same for the Fortran 90 source code. These tools compile the source code and create a Python library (.pyd) for each solver which can be called in a Python environment.

The creation and the solving of the tight uid-structure coupling along with the temporal loop of the co-simulation are achieved by a Python numerical tool named InWaveS_CN.

IV.2.5 Implementation

The implementation of the tight coupling between InWave and WS_CN unfolds in seven steps:

InWave performs the three stages of the Composite-Rigid-Body Algorithm while WS_CN computes the inuence coecients and solves the rst boundary value problem;

The quantities, required for solving the uid-structure interaction, are sent to InWaveS_CN ;

The linear system (IV.39) is formed;

The linear operator, the right-hand side along with the initialization of the solution are sent to WS_CN ;

The iterative method GMRES is applied to solve the linear system;

The solution of the linear system is sent to InWaveS_CN ;

The solution is assigned to the uid and mechanical solvers, the multibody acceleration is expressed into the Cartesian space. The state vector of each solver is time-stepped.

IV.3 Validation

The uid-structure coupling presented in this chapter is now compared to an academic test case which can also be run by the weakly nonlinear potential ow-based solver alone.

Two identical surface-piercing oating circular cylinders are considered. Both oating bodies (named Cylinder 1 and Cylinder 2 ) have a radius of 0.2 m, a height of 1 m and a draft of 0.5 m. The position of the centers of gravity along with inertia characteristics are given in Table IV.2. At the starting time, the two cylinders are aligned along the y-axis and separated by 1 m. The incident wave is regular, with an amplitude of 0.01 m, a frequency of 8 rad/s and propagating towards the positive x. A cylindrical domain of radius 2λ is used with an absorbing beach length of λ. The initial mesh, with 10 650 panels, is showed in Figure IV.8. The time step is 0.01 s.

Each body has three degrees of freedom; surge, heave and pitch. Regarding the multibody modelling, Cylinder 1 is considered as the base of the multibody system. Three joints are necessary to ensure the three dof of Cylinder 2. The two rst joints are prismatic while the last one is revolute. The modied Denavit-Hartenberg parameters of the multibody system are given in Table IV Table IV.2 Inertia characteristics and initial position of the centres of gravity of Cylinder 1 and 2

j σ j γ j (rad) b j (m) α j (rad) d j (m) θ j (rad) r j (m) 1 1 π 2 0 π 2 1 π 0 2 1 0 0 π 2 0 - π 2 0 3 0 0 0 π 2 0 π 2 0
Table IV.3 mDH parameters associated with the multibody system or vice versa. In case of an iterative loose coupling (or implicit loose coupling) a test condition is necessary within the internal loop. It is given by (IV.42). The triplet (p, n, i) denotes the p th component of the vector, the n th time step and the i th internal step.

       0 V0 (p, n, i + 1) -0 V0 (p, n, i) ∀p ∈ 1 ; 6 |q(p, n, i + 1) -q(p, n, i)| ∀p ∈ 1 ; n max φP (B j , n, i + 1) -φP (B j , n, i) ∀j ∈ 1 ; N (IV.42)
represents the tolerance of the GMRES method. For the sake of speed, a maximum of 30 internal steps is xed.

Three following couplings are considered: Tight;

Explicit loose with a uid formulation;

Implicit loose with a uid formulation.

They are compared using the test case presented in section IV.3. based on a tight, loose explicit and loose implicit couplings. Loose couplings use a uid formulation. As expected, the tight and implicit loose couplings match perfectly while the explicit loose coupling shows some dierences. This is due to the time lag between the computations of the hydrodynamic loads and the accelerations. Regarding the implicit loose coupling, this approach needs between 15 and 25 internal iterations per computation. The addition of a predictor and/or corrector within the internal loop could reduce the number of iterations. Table IV.4 presents the CPU time for each coupling strategy without using the parallelization of the inuence coecients. The explicit loose coupling is quicker than the tight coupling but it is less accurate. The implicit loose coupling has an opposite behaviour. Finally, the tight coupling is a good compromise between the CPU cost, the robustness and the accuracy. The increase of the CPU time when the tight coupling is used is a consequence of the communication time between the dierent modules and the dierent languages.

The level of dierences between a tight and a loose explicit coupling depends on the case (incident wave, hydrodynamic interactions, etc.). For instance, if the centerto-center distance the two cylinders is 0.6 m instead of 1.0 m (d 1 = 0.6), the dierences in heave motion increase because of the more important hydrodynamic interactions between the bodies (Figure IV.13).

Moreover, the tight and implicit loose couplings of InWaveS_CN along with the weakly nonlinear potential ow-based solver WS_CN lead to the same results whereas three dierent sets of dierential equations need to be solved. This fact ensures the validation of the dierent uid-structure couplings presented in this chapter. The coupling between the InWave and WS_CN may be used to performed a lowering operation including a vessel, a hoisting cable and a payload. But, as soon as there is a cable with one extremity connected to the multibody system and the other extremity xed in the Cartesian frame, the CRBA cannot be used. This is the case, for example, for the mooring cables where the endpoints are xed to the vessel and the seabed or the hoisting cable used in chapter VI where the position of the winch and the rotation axis of the buoy are xed in the global frame. In the direct dynamics algorithms of InWave, it is only possible to prescribe the position of the base in Σ e , the other bodies being located using relative coordinates. The body positions may be prescribed when inverse dynamics algorithms are used. Consequently, two choices are possible to gure out this problem. Either developing some inverse dynamics algorithms or simulating cables using the Cartesian coordinates. This latter approach is preferred because, as explained in section II.2, a low-order lumped mass theory based solver, named CableDyn, has been developed and can be easily coupled with InWaveS_CN.

IV.5.1 Coupling strategy

The low-order lumped mass theory and the Composite-Rigid-Body algorithm are coupled using an explicit loose coupling for the sake of ease of development (cf. subsubsection IV.1.1.3). In other words, at each time step, the spring loads are evaluated from the position of the multibody system at the previous time step. Then, these loads are added to the multibody motion equation (IV.1). A tight coupling would have been more

complex to implement as it would require to solve both the multibody motion equation IV.5. Coupling between the CRBA and the low-order lumped mass theory (IV.1) and the motion equation for each cable node (II.23) simultaneously. Furthermore, the low-order lumped mass theory is based on Cartesian coordinates whereas the multibody theory uses relative coordinates, therefore a change of coordinate system for one of the two solvers would be mandatory. A loose coupling between a oating body and cables (for example a mooring system) provides good results compared to a tight coupling as shown by Jacob et al. [START_REF] Jacob | Parallel implementations of coupled formulations for the analysis of oating production systems, part I: Coupling formulations[END_REF].

One advantage of the loose coupling with the cable solver is the possibility to t the time step, by the use of the subcycling technique. Indeed, the time step to solve the cable dynamics is usually much smaller than for the uid-structure dynamics. It is also possible to use another time integration scheme than the fourth-order Runge-Kutta scheme of InWave to add some numerical damping in the cable solver (cf. subsection II.1.2). If necessary, it is easy to transform the present explicit loose coupling in an implicit loose coupling with the addition of an retroactive loop.

Finally, the uid-structure coupling is tight while the cable-structure coupling is loose and explicit.

IV.5.2 Theoretical developments

Let us consider a cable of which one extremity, the anchor, is position-controlled and the other extremity, the fairlead, is xed to the j th body of the multibody system at the point A j . The notations dened in chapter I are used in the rest of this section.

Following the method used in subsubsection IV.2.3.1 to apply the hydrodynamic loads, the cable loads are directly added to the multibody motion equation (IV.1):

H 0 V0 q = 0 6×1 Γ -C + 0 F W SC Γ W SC + 0 F Cable Γ Cable (IV.43)
0 F Cable is the sum of all cable loads acting on the whole multibody system at O 0 and projected in the base frame: F Cable the set of the bodies subject to cable loads;

0 F Cable = j∈F Cable 0 N j e e G Cable
e F Cable j the cable loads acting on the body j at A j in Σ e and expressed by:

e F Cable j = T 0 + C 0 (IV.45)
The subscript 0 refers to the node 0 of the cable (cf. subsection II. Γ Cable is the sum of the cable articular loads:

Γ Cable = j∈F Cable \{0} Γ Cable j (IV.47)
where Γ Cable j is the projection of the cable loads acting on the body j into the articular space: where:

Γ Cable j = j J T j j ρ e e G Cable
j ρ e is obtained from (IV.8); j J j results of (IV.23).

IV.5.3 Time integration

The time integration of the whole coupling is achieved using a fourth-order explicit Runge-Kutta scheme. The positions and the velocities of the fairleads are updated from (IV.49) and (IV.50).

e OA j = e P j + e R j j O j A j (IV.49)

e v(O j , Σ j /Σ e ) = e R j j v j + S( j ω j ) j O j A j (IV.50)

IV.5.4 Language binding

CableDyn is implemented in Python, so it becomes a new module of the existing coupling between InWave and WS_CN.

IV.6 Conclusion

This chapter presented the theory, the implementation and the language binding of a tight coupling between the CRBA of InWave and the potential ow theory based on the weak-scatterer of WS_CN. The Python numerical tool dedicated to this coupling has been named InWaveS_CN. A validation of its development was achieved by comparison with WS_CN alone and showed a perfect agreement. The tight coupling was also compared to dierent loose coupling strategies. The implicit loose coupling gave the same results as the tight coupling but required a larger CPU time while the explicit loose coupling presented some dierences which could be more important if the case was changed. The tight coupling represents a robust compromise between accuracy and CPU time.

CableDyn, the cable dynamics solver based on the low-order lumped mass theory has been included into InWaveS_CN as a new module to perform cable simulations.

The theory of this coupling was presented.

The next chapter presents a new strategy for generating meshes in WS_CN, which will be used in the last chapter of this PhD work to validate InWaveS_CN with experimental data.

The mesh generator of WS_CN, which was implemented before the start of this PhD, presents a lack of robustness when the surface-piercing body is not analytic and, so far, has only been used with academic geometries (sphere, vertical cylinder, cube, Wigley hull). Consequently a new mesh generator is required. This is the aim of this chapter. A state of the art of the mesh generation and the intersection curve tracking is presented. Then, all the steps of the new mesh strategy are detailed. Two validation test cases and a study of the CPU time are exposed.

V.1 The need of a new mesh generator

The mesh generator presented in section III.3 is able to deal with academic surfacepiercing bodies (vertical cylinder, sphere, cube, Wigley hull) or immersed bodies. However, it suers of a lack of robustness for more complex shapes. For example, the simulation of the experiments, detailed in chapter VI, requires to deal with a horizontal oating cylinder and the mesh generator fails to properly mesh it. The problem comes from the intersection curve tracking (cf. subsection III.3.1) which is not robust enough to nd the interface in this case. As pointed out by Ko et al. [START_REF] Ko | Development of panel generation system for seakeeping analysis[END_REF], the marching method, as used in WS_CN, becomes problematic when the shape of the body is complicated and the conguration of the intersection with the free surface is complex.

With a horizontal cylinder, sharp edges are present on the intersection curve and two types of surfaces intersect the free surface: the cylindrical surface and the discs.

Following the observation of the lack of robustness of the present mesh generator, two choices are possible. Either trying to enhance it or launching the development of a new mesh generator which could be more robust, universal and usable for industrial applications. We chose the second option for four reasons:

The implementation of the initial mesh generator makes the task of enhancement hard and the success uncertain;

The improvement of the initial mesh generator involves the development of a robust geometric modeller for any oating body, for instance based on NURBS [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF]. This requires lots of work and means the development of a computer-aided design tool from scratch, whereas some tools already exist and some of them are free and open-source.

The initial mesh generator suers of important memory leaks which deeply aect the performance of the code when long simulations are run;

The development of a new mesh generator gives us the opportunity to test a new strategy of grid generation.

V.2 Functional specication

The development of a new approach could solve some problems of the initial mesh generator. Its functional specication is: Quality: the new mesh generator has to create good quality meshes able to be used in an unsteady potential ow based solver; Non-regression: the new mesh generator has to give the same or better results in terms of stability and accuracy as the initial mesh generator; the wave elevation such as the nonlinear steady free surface boundary conditions were satised. Between 5 and 25 iterations are usually necessary in such problems according to Raven [START_REF] Raven | A solution method for the nonlinear ship wave resistance problem[END_REF]. 25 computations represent less than 7 time steps when a fourth-order Runge-Kutta integration scheme is used (with four solvings of the hydrodynamic problem per time step). For comparison, a time-domain method needs several hundreds or even thousands of time steps per numerical simulation. Consequently, the mesh quality required for a time-domain simulation is much higher than for an iterative method.

Moreover, Choi et al. [START_REF] Choi | Panel cutting method: new approach to generate panels on a hull in Rankine source potential approximation[END_REF] used quadrangular panels which were approximately rectilinear with the intersection curve, therefore the panel cutting method did not generate lots of tiny panels which could lead to numerical errors. The panels smaller than 10 % of the smallest panel of the initial mesh were eliminated. Nevertheless, in another conguration, with triangular panels, the number of tiny panels could be much more important and would lead to poor quality meshes. Finally, the panel cutting method has only been used by Choi et al. [START_REF] Choi | Panel cutting method: new approach to generate panels on a hull in Rankine source potential approximation[END_REF] in case of ships, that is to say with smooth surfaces without sharp edges.

Note

In CFD solvers 9 , the cutting method is also used and is called the cut-cell method (because volume meshes, made of cells, are required instead of surface meshes, made of panels) [START_REF] Bigay | Développement d'un solveur faiblement compressible sur maillage cartésien pour les écoulements hydrodynamiques autour de corps[END_REF]. The cells are cut to t the body geometry. When poorly formed cells are generated, a cell merging method is applied to delete them [START_REF] Ingram | Developments in cartesian cut cell methods[END_REF]. As it will be described in subsection V.4.2, this problem is also present with surface meshes.

V.4 Coupling between the panel cutting method and the advance front method

The originality of the proposed method is the generation of meshes adapted for a time-domain unsteady potential ow approach requiring high quality connected nodes surface meshes using triangular panels with possibly sharp edges for thousands of time steps. This method assumes the initial mesh is of best quality. It unfolds in ve steps:

An initial mesh is obtained from an external mesh generator (Figure V.1a);

The mesh is cut at the incident wave eld (Figure V.1b);

The intersection curve is tracked from the cut mesh (Figure V.1c);

The free surface and the numerical tank walls are meshed using the same advance front method as in subsection III. The cut mesh is connected with the free surface mesh, leading to the nal mesh

(Figure V.1e).
9 Computational Fluid Dynamics, for the simulation of viscous and turbulent ows. The panel cutting method comes from the numerical tool: Meshmagick 10 . This software package has been developed at Ecole Centrale de Nantes by François Rongère.

It enables the management of surface meshes encountered in the potential ow theory:

The conversion between major le formats for hydrodynamic computations tools (Nemoh, WAMIT, HydroSTAR or Diodore); Elementary transformations such as translation, rotation, scaling, symmetry, normals orientation verication, etc.; Hydrostatics computations: stiness matrix, position of the center of buoyancy, displacement, etc.;

The clipping of the mesh by a plane. x

• n P lane = c (V.1)
where n P lane and c are the panel normal and constant. For instance for the plane z = 0, n P lane = e z e and c = 0.

The intersection point x between the edge (p 0 p 1 ) and the plane equation (V.1) satises:

x = tp 0 + (1 -t)p 1 (V.2)
with:

t = c -p 1 • n P lane p 0 • n P lane -p 1 • n P lane (V.3)
For an arbitrary single-valued incident wave eld, a mathematical model is not necessarily available. For the sake of simplicity and robustness, a numerical approach is preferred. A bisection method has been implemented in Meshmagick to do so. Thus, if an edge has one node above the free surface and one below, the intersection node is The density of nodes and panels is much higher at the interface than in the rest of the mesh;

The panel shape is poor.

As a consequence, the mesh cannot be used in a hydrodynamic solver. An extra step of mesh enhancement is mandatory. To preserve the geometry of the mesh, these two algorithms are applied conditionally.

Two geometric criteria are dened per algorithm: the rst one about the panel area, the second one about a characteristic distance. These criteria are detailed in Table V.1.

Criterion Vertical Horizontal

Area A is the panel area;

A α V A mean A α H A mean Distance p 1 p 2 β V mean z -η I β H mean
A mean denotes the mean of the panel areas in the initial mesh;

mean represents the mean of the edge lengths in the initial mesh; The values of α V , α H , β V and β H are chosen empirically. An example of a set of coecients is given below:

           α V = 0.4 α H = 0.1 β V = 0.3 β H = 0.1 (V.4)
A special attention is paid to the nodes along the sharp edges of the mesh. They are not moved. The sharp edges are tracked using the discontinuity of the panel normals. Equation V.12 leads to a sparse linear system of size twice the number of nodes in the body mesh such as AX = B with:

X =       . . . x i u x i v . . .       (V.13)
The solution is obtained using the same sparse linear system solver PARDISO [184] as in subsection III.4.1. The use of the spring analogy method faintly deforms the geometry of the body. To avoid this numerical error, the algorithm is not applied to the full body mesh but only to a part of the mesh close to the intersection curve. The nodes below this part are considered as xed and the three displacement conditions (V.8), (V.9) and (V.10) are assumed satised. The geometry is preserved and the number of non-zero coecients in the linear system is reduced, involving a CPU time reduction.

V.4.4 Multiple node tracking

In the previous section, it was assumed multiple nodes were known. Therefore, the boundary conditions (V.9) and (V.10) are directly applied at the proper nodes. But, when an initial mesh is used, it only includes a set of nodes and a table of connectivities.

Consequently, multiple nodes need to be automatically identied.

The multiple node tracking algorithm unfolds in three steps:

The detection of multiple nodes is achieved from the discontinuity of the outward panel normals in the body mesh (at this stage, the free surface mesh is not generated yet). A node is multiple if and only if it exists at least two neighboring panels of normal vectors u and v such as:

u • v < (V.

V.4.5 Mesh qualities

To evaluate the mesh qualities, algebraic metrics f are used. They have the following properties [START_REF] Knupp | Algebraic mesh quality metrics for unstructured initial meshes[END_REF][START_REF] Stimpson | The verdict geometric quality library[END_REF]:

f is dimensionless;
f is referenced to an ideal element, here an equilateral triangle of edge length ;

f ∈ 0 ; 1 with f = 1 if and only if the panel matches an equilateral triangle of edge length and f = 0 if and only if the triangle is degenerated;

f is invariant by translation;

f does not depend on the orientation of the panel (except for the orientation metrics which are not studied here);

f is a function of the node positions.

Three metrics are used:

The relative size metric f size which detects triangles which are unusually large or small relative to an equilateral triangle of reference. Its denition is:

f size = min τ, 1 τ (V.15)
with τ the ratio of the panel area A to the area of an equilateral triangle of edge length :

τ = A √ 3 4 2 (V.16)
f size = 1 if and only if the triangle has the same area as an equilateral triangle of edge length and f size = 0 if the triangle is degenerate.

The shape metric f shape which detects distortions in the shape of a triangle, independently of its size, compared to an equilateral triangle. Its denition is:

f shape = 2 √ 3A L 1 • L 1 + L 1 • L 2 + L 2 • L 2 (V.17)
with L 1 = p 0 -p 2 and L 2 = p 1 -p 0 , by keeping the notations of Figure V.3. f shape = 1 if and only if the triangle is equilateral and f shape = 0 if the triangle is degenerate.

The size-shape metric f size-shape that measures both size and shape simultaneously:

f size-shape = f size f shape (V.18)
f shape = 1 if and only if the triangle is equilateral with an area equal to √ 3 4

2 and f shape = 0 if the triangle is degenerate.

The area of a panel is given by:

A = 1 2 L 1 × L 2 (V.19)
Figures V.12, V.13 and V.14 show the evolution of f size , f shape and f size-shape during the mesh generation. In this cases, = 0.02 m. The use of both the panel merging and the vertex spring analogy method improves signicantly the quality of the mesh, especially on the intersection curve. If the mesh optimization algorithm is not applied, the mesh stays of poor quality. Once the nal body mesh is obtained, the intersection curve is tracked from the nodes which satises z = η I . But, doing so, the list of nodes does not form an oriented curve which is mandatory to generate the free surface mesh. Consequently, the nodes on the intersection curve have to be sorted. The process unfolds in three steps.

An initial node x init is randomly chosen on the intersection curve. This point has exactly two neighbors, x a and x b . The curve orientation demands to dene a predecessor, x pred , and a successor, x suc . For instance, let us assume x pred = x a .

x b becomes the successor of x init , so its ancestor is known.

For each node, the ancestor is known, so there is only one possibility for the successor. Thus, the curve is built step by step along the succeeding nodes.

When the successor matches the initial point, the curve is dened and oriented.

When the intersection curve is found, the free surface mesh is generated using the advance front method presented in subsection III.3.2. Then, the body mesh and the free surface mesh are connected. An example of an intersection curve and a nal mesh are displayed in Figures V. [START_REF] Schellin | Crane ship response to wave groups[END_REF]

Note

If the sparsity of the linear system obtained in the spring analogy method is not used (i.e. PARDISO is not used) and if each node is considered with three degrees of freedom instead of two (tri-dimensional problem instead of bi-dimensional), then it involves an increase of the mesh optimization part of 1270 %, or 8.513 s instead of 0.621 s. The method becomes too long to be applied eciently. The proling is updated in V.25. The cylinder mass is 12.88 kg and its inertia in pitch 0.5 kg m2. The center of gravity of the cylinder is located at its geometrical center. The same incoming regular wave of amplitude 0.005 m and wave frequency 8 rad/s is used in the two test cases. A wave probe is present at the position (0.4, 0, 0).

V.7.1 Heave test case

The mesh convergence for the two strategies is displayed in A very good agreement is obtained for all these results. Some slight dierences are observed, probably due to the permanent remeshing process in the new mesh generator whereas in the initial mesh strategy the mesh is only deformed if the regeneration is not necessary. The remeshing involves an interpolation between the old and the new mesh as explained in section III.8 and so numerical errors occur.

The sum of the incident wave amplitude and the amplitude of the heave body motion represents 9 % of the draft, which involves a signicant deformation of the body mesh during the simulation. A good agreement is obtained for all these results, except the appearance of a slow phase shift in the pitch motion. As in subsection V.7.1, it is probably the consequence of the permanent remeshing with the new mesh strategy which leads to numerical errors.

Moreover, the mesh quality of the new mesh generator is less good compared to the initial one.

To prove the eect of the permanent remeshing in the dierences observed in The functional specication of the new mesh generator, listed in section V.2, is now checked:

The quality of the new meshes is good and sucient for performing a time-domain simulation. Nevertheless, at the time of writing, the free surface mesh close to the intersection curve is not as good as in the initial mesh. For example, Figure V.36

shows a close-up of the mesh around the intersection curve in the case of a surfacepiercing vertical cylinder. The distribution of the area of the panels should be axisymmetric with the two mesh strategies. It is the case with the initial mesh generator but not with the new mesh generator where some regions with smaller panels arise. This is due to the non-equidistance of the nodes on the intersection line in the new mesh generator, despite the use of the spring analogy method. This equidistance is guaranteed in the initial mesh generator because the intersection curve is meshed rst, the body and free surface meshes being generated from it.

Possible enhancements would be the improvement of the panel merging method and the projection of the nodes on the intersection curve before clipping the mesh to avoid the creation of tiny panels. The use of an another open-source library such as GTS12 which deals with surface meshes is also a possibility.

The non-regression of the new mesh generator is proved by the two test cases presented in section V.7. The same simulations could be performed, involving a good accuracy in the numerical results.

V.8. Compliance with the functional specication and conclusions

The robustness is demonstrated by the creation of meshes which could not be taken into account by the initial mesh generator (horizontal cylinder, SEAREV, TLP, FPSO). The possibilities oered by the new mesh generator mainly depend on the quality of the initial mesh. So far, only one body in one piece and involving a single intersection curve (no moon pool) is considered. The new mesh strategy is feasible with any wave model by the use of a bisection method to nd the intersection curve as presented in subsection V.4.1.

The modularity is checked as displayed on the scheme of the coupling in Fig-

ure V.23;

The user-friendlyness of the new mesh strategy is ensured by the use of an initial mesh. The rest of the process is totally automatic for any body. No extra-coding is required.

The memory allocation is good, without memory leak in the scripts of the new mesh generator which were implemented for this purpose. Consequently, only the memory leaks coming from the use of the advance front method remain. But those created for the body mesh generation and the intersection curve tracking in the initial mesh generator disappear.

The process time is slightly more important with the new mesh generator than with the initial one due to the spring analogy method and some parts in Python (cf. section V.6). The translation of the Python code into Fortran and the use of another smoothing method would reduce the CPU time.

Hence, a new mesh strategy has been developed in this chapter, based on the idea:

the body mesh for the user, the free surface mesh for the solver. In order to validate the numerical tool created during this PhD work, some experiments were conducted in a wave basin of Ecole Centrale de Nantes. The numerical simulations of these experiments involve the use of multibody dynamics (InWave), cable dynamics and a winch (CableDyn), hydrodynamics for a body with a large amplitude motion (WS_CN) and with a complex geometry (Meshmagick). All the modules constituting InWaveS_CN are required. First, the experiments are detailed. The numerical results and the comparison with the measurements close this nal chapter.

VI.1 Presentation of the experiments

In order to validate the coupling between the Composite-Rigid-Body algorithm and the unsteady potential ow theory based on the weak-scatterer hypothesis, experiments were conducted in the shallow water tank of Ecole Centrale de Nantes. These experiments represent the upending of a spar in waves. A spar is moved from a horizontal position to its equilibrium vertical position using a cable on a winch. These experiments follow three steps: At the beginning of the experiments, the spar is kept horizontal with a cable An upending operation was preferred to a lowering operation because the experimental set-up was more straightforward. Only one body, available in the laboratory, was required and the set-up was easier and quicker to install: one body linked with a cable and a winch to the xed footbridge instead of two bodies in free motion linked to each other through a cable and winch.

The interests of these experiments are:

The presence of an articulated multibody system (the spar and its revolute joint); A cable is unwound using a winch;

The wetted surface of the spar is subject to a large deformation;

The spar has a free motion so a uid-structure coupling arises.

Test matrix

The test matrix depends on four parameters:

The wave amplitude;

The wave period;

The orientation of the moving buoy with respect to the xed wave direction;

The lowering velocity of the cable.

Only

VI.1.3 Output data

Five output data are measured:

The position of the buoy;

The position of the cable;

The tension in the cable;

The lowering velocity;

The wave elevation.

The positions of the buoy, the wave elevation and the cable prole are tracked using spherical markers. The motion capture software Qualisys follows the markers with two infrared cameras. Three markers are necessary to track the three-dimensional motion of the buoy. The projection of (VI.1) along the vertical axis and (VI.2) along the axis perpendicular to the plane including the buoy and the cable give:

-mg + ρV g + T = 0

(VI.3) -mgKG + ρV gKB + T KT = 0 (VI.4)
where m, g and V are the mass of the buoy, the gravity constant and the immersed volume.

Equation VI.3 leads to:

V = 1 ρ m - T g (VI.5)
Equation VI.4 involves:

T = mg KB -KG KB -KT (VI.6)
The cable is xed at the keel of the buoy so KT = 0 and at the horizontal initial position KB = L 2 with L the length of the buoy, consequently:

T = mg 1 - 2 L KG (VI.7)
Thus, the tension in the initial conguration is 120.65 N. This does not match perfectly with the experimental value observed in Figure VI.13 because the buoy is not totally horizontal in its initial position. The dynamic tension clearly depends on the wave period. But, when the buoy reaches its nal position, the tension is independent of the waves, because the cable is slack. show the lowering of the cable and its horizontal displacement due to the buoy motion.

When the cable is slack, the marker trajectories become arbitrary. Once the cable markers are in the water, they could not be tracked anymore by the infrared cameras.

The markers on the buoy could be tracked because there were never immersed. It is made of 8862 panels.

The multibody system is composed of two bodies:

The base (body 0) which represents the footbridge of the basin and is motionless:

η 0 = 0 6×1 (VI.8)
The buoy (body 1) linked to the base by a revolute joint to model the rotation axis. The initial modied Denavit-Hartenberg parameters are given in Table VI.4.

j σ j γ j (rad) b j (m) α j (rad) d j (m) θ j (rad) r j (m) 1 0 π 2 0.0183 π 2 0 - π 2 0

Table VI.4 mDH parameters for Case 15

The cable is made of one element. The fairlead position in the body frame is:

1 O 1 A 1 = (-0.1, -0.908, 0) T (VI.9)
The hydrodynamic loads on the cable are neglected. The cable used in the experiments is sti so the value of the numerical cable stiness has to be high enough not to modify the numerical results. If the stiness increases, the necessary time step to be converged decreases. Consequently, the inuence of the cable stiness is studied at the same time as the time step convergence. Three cases are considered:

An axial stiness of 6.7 × 10 5 N and an axial damping coecient of 3 × 10 7 N s/m 2 , which form the reference;

A stiness which is twice as high as the reference with the same damping coecient;

A stiness which is ve times as high as the reference with the same damping coecient.

The time step convergence for these three cases are presented in Figures VI.27 data and the numerical results. As with Cases 8 and 15, a dierence is noticed in the initial tension. The eects of the waves is well captured for both the tension and the rotational motion. The tension decreases due to the winch and oscillates at the wave frequency. The angular motion is not aected by the waves. Once more, the simulation stops when one extremity of the buoy is close to be immersed.

j γ j (rad) b j (m) α j (rad) d j (m) θ j (rad) r j (m) 1 π 2 0.024 π 2 0 - π 2

VI.3 Conclusion

This chapter presented the experiments of an upending operation conducted in the shallow water basin of Ecole Centrale de Nantes and the comparisons with the numerical results of InWaveS_CN. All the modules constituting this numerical tool were required. Five cases were investigated, two cases without waves and with dierent lowering velocities and ramps, three cases with an incident regular wave with dierent wave parameters. Each comparison gave a good agreement between the experimental data and the numerical results, in term of cable tension and angular position of the buoy.

The eects of the winch (velocity and linear ramp) and the waves were well captured.

The dierence in the initial horizontal position of the cylinder at the starting time could be responsible of the dierences observed. In every case, the numerical simulation stopped when one extremity of the buoy was close to be immersed, leading to a body mesh tangent to the free surface mesh and therefore numerical errors. Nevertheless, these comparisons to experimental data has given a rst and promising validation of InWaveS_CN.

boundary conditions were linearized around the incident free surface elevation. This approach is consistent in case of large (relative) amplitude motions and steep waves. The velocity potential was computed using a boundary element method. The uid-structure interaction was formulated with a monolithic coupling using a second boundary element method to evaluate both the time-derivative of the velocity potential and the body acceleration. The implementation of this hydrodynamic theory, in the numerical tool WS_CN, was already done before the start of our work in case of a single immersed or surface-piercing body.

The main contribution to WS_CN has been the extension to multibody simulations with large motions. Both the mesh generator and the hydrodynamic solver have been modied accordingly. A rst validation has been achieved in forced motion by comparison with the experimental data of Watai et al. [START_REF] Watai | A new time domain Rankine panel method for simulations involving multiple bodies with large relative displacements[END_REF]. Two cylinders were considered.

One body was still while the second body was subject to a prescribed large amplitude motion. Comparisons were done based on the hydrodynamic loads of the xed cylinder and the wave elevations at three wave probes. The weak-scatterer approach showed a very good agreement with the experiments, being able to simulate the hydrodynamic interactions between the oating bodies and the modulation of the hydrodynamic loads.

A large number of panels were required to reach the converge of the numerical results, which led to important CPU times and computer memory demands for every simulation. Super-computing facilities were used. WS_CN has also been switched to a body-exact formulation (linearization of the free surface boundary equations around the mean position) and a fully linear formulation (linearization of both the free surface and the body boundary equations around the mean position). The body-exact approach provided a good agreement with the experimental data too with a lower CPU time because of coarser meshes. The agreement was good due to the small steepness of the waves. When the wave steepness was increased, more dierences were observed between the weak-scatterer and the body-exact models. The fully linear approach did not give accurate results, the body nonlinearities being important. Theses results have proved the interest of an unsteady weakly nonlinear potential ow theory, compared to a fully linear approach, in case of large relative motions between bodies.

A second validation of the multibody extension was proposed, based on a comparison between the weak-scatterer approach and a classical linear frequency-domain approach in free motion. This case presented a good agreement between the numerical results of the two theories. Other developments have been achieved in WS_CN : the implementation of a free surface remeshing process to ensure more robust simulations, the parallelization of the computation of the inuence coecients to speed up the code and the reduction of the memory footprint with the use of the sparse linear system solver PARDISO. Thus, our contribution to WS_CN has enabled to reach multibody simulations which are more robust, stable and faster. From the comparisons between the dierent approximations included in the code (weak-scatterer, body-exact and linear), it seems relevant to us that WS_CN is seen as a multi-level potential ow based numerical tool able to t the hydrodynamic theory for every case. The weak-scatterer method should be used in case of large amplitude motion and steep waves. If the wave steepness is low then a body-exact approximation would be sucient and a linear approximation would only be required when both the motion amplitude and the wave steepness are small.

The potential ow theory based on the weak-scatterer hypothesis and the Composite-Rigid-Body Algorithm have been coupled to solve the uid-structure interaction and enable the simulation of multibody systems in waves with large relative motions, as expected in marine operations. For the sake of robustness and CPU time, a tight coupling has been selected. The coupling equation, which gathered the multibody motion equation provided by the CRBA and the second boundary value problem of the potential ow theory, was derived and demonstrated. These developments have led to the creation of a new numerical tool, named InWaveS_CN. A signicant emphasis was placed on its modularity through the language binding using Python as glue code language.

A validation of this coupling has been done by the comparison between InWaveS_CN and WS_CN. A perfect agreement was noticed, giving the proof of the good derivation of the coupling equation. The tight coupling was tested by comparison with two loose couplings (explicit and implicit). As expected, the tight coupling matched the implicit loose coupling results and the explicit loose coupling presented some dierences because of the time lag between the solvers.

The numerical tool CableDyn has been added as a module of InWaveS_CN, using an explicit loose coupling, to simulate cables when kinematic loops appeared, as they could not be handled by the CRBA.

The mesh generator of WS_CN, implemented prior to this PhD, was based on a semi-analytical approach to track the intersection curve between a surface-piercing body and the incident free surface. This approach was a marching method which required the knowledge of a parametric equation of the body surface. Then, an advance front method was used to generate the mesh of the bodies, the free surface and the numerical tank walls. This mesh strategy suered of a lack of robustness and was only used with academic bodies. Furthermore, it would involve the development of a computer-aided design tool inside WS_CN, which is dedicated to the simulation and the analysis of wave-structure interaction, without using the capacity of a professional mesh generator.

Consequently, it was decided to develop a new mesh generator based on a simple idea:

the body mesh for the user, the free surface mesh for the solver. The body mesh was assumed to be obtained using a third-party mesh generator, then WS_CN had only to generate the free surface mesh. The inclusion of the body mesh in the free surface mesh was done with a panel cutting method, using the Python numerical tool Meshmagick, coupled with the advance front method of WS_CN through the language binding developed previously. A special care was taken to generate good quality meshes by the use of a panel merging algorithm and a spring analogy method to avoid tiny panels and optimize the position of the nodes. Meshmagick has become a new module of In-WaveS_CN. Two validation test cases were presented based on comparisons between the initial mesh generator using WS_CN and the new mesh generator of InWaveS_CN.

A very good agreement was noticed. Some dierences were observed, probably due to the remeshing process used at every time step.

Finally, the experiments of an upending operation in waves were conducted in the shallow water basin of Ecole Centrale de Nantes and were presented in details. A horizontal cylinder in waves was kept horizontal using a cable, then the cable was unwound and the cylinder reached its vertical equilibrium position. InWaveS_CN was used to simulate the experiments. All the modules constituting this numerical tool were required: InWave for the multibody dynamics, CableDyn for the cable dynamics and the winch, WS_CN for the hydrodynamics with a body subject to a large amplitude motion and Meshmagick for the complex geometry. Five cases were presented: two cases without waves and with dierent lowering velocities and ramps, three cases with an incident regular wave with dierent wave parameters. A good agreement was observed in term of buoy position and cable tension. The eects of the winch velocity, its linear ramp and the waves were well captured. But, when one extremity of the cylinder crossed the free surface, large numerical errors occurred. Nevertheless, these numerical simulations constituted the rst and promising validation of InWaveS_CN with experiments of a marine operation.

This PhD work made eective progress in the eld of the simulation of marine operations:

The potential ow theory based on the weak-scatterer hypothesis has been extended to multibody simulations;.

The interest of the weak-scatterer theory has been compared to other approaches:

body-exact and linear, leading to a better understanding of its use;

A multibody solver, including a cable model and a winch model, has been coupled to an unsteady potential ow based solver through a tight coupling;

A new strategy of mesh generation based on a panel cutting method coupled with an advance front method has been developed and applied with an unsteady potential ow based solver;

A consistent frame has been developed to simulate operations at sea with large relative motions and compared to experimental data.

From this PhD work and its conclusions, future works may be considered. The Composite-Rigid-Body Algorithm used in InWave cannot handle kinematic loops. As seen in the simulation of the experiments, it may be problematic. A more general mechanical solver could be useful for considering kinematic loops, contact-force models, exible bodies, etc. This is the goal of the PhD of David Ogden which has coupled the frequency-domain potential ow-based solver Nemoh with the multibody solver HotInt [START_REF] Ogden | New mechanical and features for time-domain and WEC modelling and in InWave[END_REF]. A coupling of this multibody solver with WS_CN could lead to the simulation of more complex articulated systems with large relative motions.

The simulation of cables requires small time steps because of the presence of springdamper systems. In InWaveS_CN, the time integrator is the fourth-order explicit Runge-Kutta scheme (RK4 ) with a constant time step and is shared by all modules.

Consequently, the time step is xed by the cable solver CableDyn or the multibody solver InWave whereas most of the CPU time is due to the hydrodynamic solver WS_CN.

Simulations are too time-consuming uselessly when cables are present. This time step could be reduced by tting the time integrator: for instance by using a Newmark's scheme to add numerical damping (cf. subsection II.1.2) in the mechanical and cable solvers only and by keeping the RK4 scheme in WS_CN by simplicity or by changing the time integrator globally. Another possibility would be to add subcyles in the temporal loop to use a lower time step with the cable and mechanical solvers.

Bending and torsion loads could be added to the cable model in case of low-tension tethers, for example with a marine operation involving a Remotely Operated Vehicles (ROV ) [START_REF] Buckham | Dynamics Modelling of Low-Tension Tethers for Submerged Remotely Operated Vehicles[END_REF] or with a cable laying operation.

Motion compensation systems have not been considered in this PhD whereas they are present for some particular operations in the lowering and lifting operations. Therefore, their inclusion in the model is necessary for more realistic simulations.

Vincent Leroy has coupled InWave with two aerodynamic solvers to compute the aerodynamic loads aecting oating wind turbines: a free-vortex wake theory-based unsteady aerodynamic solver and a steady double multiple streamtube theory-based solver [START_REF] Leroy | Development of a simulation tool coupling hydrodynamics and unsteady aerodynamics to study oating wind turbines[END_REF]. It would be interesting to gather the couplings presented in our PhD work with those developed by Vincent Leroy in order to simulate oating wind turbines subject to large amplitude motions in waves (because of the drift and/or the inclination of the turbine due to the wind for instance).

Regarding WS_CN, several works could be planned. The code suers of memory leaks in the marching method to track intersection curves and in the advance front method. They reduce the performance and the duration of the simulations. If these memory leaks cannot be xed in the present implementation, a new implementation of the algorithms would be necessary. The CPU times required by WS_CN are still large, even after the parallelization of the computation of the inuence coecients. This is why, the PhD work of Yohan Poirier has started in 2017 accordingly. He has achieved a GPU parallelization of the code and he now is developing a Parareal algorithm in WS_CN. The Parareal algorithm is a parallel-in-time integration method [START_REF] Lions | Résolution d'EDP par un schéma en temps "pararéel[END_REF]. The goal is to reach real-time computations with a single body. If the body-exact approximation is used, the implementation of the free surface Green's function could reduce the size of the mesh and so the CPU time. The expression of the free surface Green's function in the case of a weak-scatterer approximation does not exist yet. So far, WS_CN has only handled incident regular waves, consequently, it would be interesting to extend the incident wave models to irregular waves, by using a High-Order Spectral (HOS ) model [START_REF] Ducrozet | 3-D HOS simulations of extreme waves in open seas[END_REF] for instance. A fully nonlinear potential ow model could be implemented in WS_CN in the cases where the weak-scatterer hypothesis is not satised. WS_CN could also be extended to a second order of magnitude of the weak-scatterer hypothesis (on the same principle as the Stokes' series expansion used in the linear models) to add more free surface nonlinearities. Lots of application cases of the code have not

been tested yet: ships with forward speed (including hydrofoils for a larger motion), side-by-side operations, slamming, etc.

The enhancement of the new mesh strategy, developed in this PhD, would require the improvement of the panel merging method to reach better quality meshes, especially at the intersection curve. The projection of the nodes on the intersection curve before clipping the mesh could also be a solution to study. A body mesh morphing algorithm should be developed to reduce the numerical errors due to the permanent remeshing and update the intersection curve. The CPU time would be smaller too. An extension of the new mesh generator to multibody simulations could be done easily.

InWaveS_CN could be more intensively used in case of lowering and lifting operations in order to quantify with accuracy the interest of the weak-scatterer method. The eects of the distance between the payload and the vessel, the free surface or the seabed and of the lowering and lifting velocities could be studied for example. The extension to simulations with a payload in the air which is then immersed should be considered through the modeling of the slamming loads in WS_CN. Finally, aerodynamic loads, neglected in our work whereas winds may be strong far oshore [START_REF] Ren | Development and application of a simulator for oshore wind turbine blades installation[END_REF], could be included in the model to improve accuracy of the analyses performed for marine operations.

To the happy few Les logiciels commerciaux de simulation numérique oshore (OrcaFlex [START_REF][END_REF], DeepLines [START_REF]DeepLines 4.5 Theory Manual[END_REF], SIMO [START_REF][END_REF], etc.) sont capables de modéliser les interactions mécaniques mais sont Γ le vecteur des eorts internes projectés selon l'axe des liaisons; 0 V0 l'accélération de la base par rapport au repère inertiel et exprimée dans le repère de la base;

q le vecteur des accélérations articulaires. ( * ) I la composante incidente;

( * ) P la composante perturbée.

L'hypothèse de faible perturbation implique que la composante perturbée est faible devant la composante incidente. This PhD work focuses on the development of a numerical tool to simulate marine operations with consistency, in particular lowering and lifting operations. The Composite-Rigid-Body Algorithm, implemented in the numerical tool InWave, is used to model multibody systems. A cable model and a winch model are developed following this multibody approach and compared to the classical low-order lumped mass theory.

Hydrodynamic loads and hydrodynamic interactions are simulated using an unsteady potential flow theory based on the weakscatterer hypothesis, implemented in the numerical tool WS_CN. This approach is extended to multibody simulations and validated with comparisons to experimental data.

InWave and WS_CN are coupled to solve wavestructure interaction for articulated multibody systems with large relative motions in waves. A tight coupling is selected for its robustness. The coupling equation is derived and validated from comparisons with WS_CN. This leads to the creation of a new numerical tool, InWaveS_CN, using Python as glue code language.

A new mesh strategy, based on the coupling between a panel cutting method and an advance front method, is developed in WS_CN.

Experiments of an upending operation were conducted at Ecole Centrale de Nantes. The comparison between the numerical simulations and the experimental data leads to a first and promising validation of InWaveS_CN.

8 h 9 h 4 Block

 894 vertical cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.ref = 0.3 my = 0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . III.ref = 0.3 my = 0.15 m . . . . . . . . . . . . . . . . . . . . . . . . . III.10 h ref = 0.6 my = 0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . III.11 h ref = 0.01 my = 0.15 m . . . . . . . . . . . . . . . . . . . . . . . . . III.12 Example of a damping numerical beach with L abs = 1 m and R = 3 m . III.13 Mesh distortion due to the translation of a cylinder along the y-axis (left). The second cylinder is xed (right). . . . . . . . . . . . . . . . . III.14 The three steps of the interpolation scheme on the free surface mesh . . III.15 Comparison of time series of the heave motion (top) and the wave elevation (bottom) from numerical results with and without forced free surface remeshing processes . . . . . . . . . . . . . . . . . . . . . . . . . III.16 Eect of the ltering of the scattered wave elevation on a simulation with a translation cylinder along the y-axis (left) and a xed cylinder (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.17 Inuence of the parallelization on the CPU time reduction for a partial and total calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi List of Figures III.18 Proling of W S_CN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 III.19 Comparison of time series of the heave motion (top) and the wave elevation (bottom) from numerical results using a sequential (nth = 1) and parallel (nth = 5) computation . . . . . . . . . . . . . . . . . . . . . . . 133 III.20 Sketch of the top view of the experimental set-up . . . . . . . . . . . . 134 III.21 Comparison of time series of the hydrodynamic loads on Body 1 from numerical results for dierent meshes for Case 1 . . . . . . . . . . . . . 136 III.22 Comparison of time series of the wave elevations at WP1, WP2 and WP3 from numerical results for dierent meshes for Case 1 . . . . . . . 136 III.23 Comparison of time series of the hydrodynamic loads on Body 1 from numerical results for dierent time steps for Case 1 . . . . . . . . . . . 137 III.24 Comparison of time series of the wave elevations at WP1, WP2 and WP3 from numerical results for dierent time steps for Case 1 . . . . . 137 III.25 Top view of the mesh for Case 1 . . . . . . . . . . . . . . . . . . . . . . 138 III.26 Bottom view of the mesh for Case 1 . . . . . . . . . . . . . . . . . . . . 138 III.27 Prescribed harmonic motion of Body 2 for Case 1 . . . . . . . . . . . . 138 III.28 Comparison of time series of the hydrodynamic loads on Body 1 from numerical and experimental results for Case 1 . . . . . . . . . . . . . . 139 III.29 Comparison of time series of the wave elevations at WP1, WP2 and WP3 from numerical and experimental results for Case 1 . . . . . . . . 140 III.30 Time series of the hydrodynamic loads onBody 2 from numerical results for Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 III.31 Comparison of time series of the hydrodynamic loads on Body 1 from numerical and experimental results for Case 5 . . . . . . . . . . . . . . 141 III.32 Comparison of time series of the hydrodynamic loads on Body 1 from numerical and experimental results for Case 10 . . . . . . . . . . . . . . 141 III.33 Comparison of time series of the hydrodynamic loads on Body 1 from numerical and experimental results for Case 1 with and without forced remeshing processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 III.34 Comparison of time series of the hydrodynamic loads on Body 1 from numerical and experimental results for Case 1 for dierent frequencies of call of the Gaussian ltering (nliss ) . . . . . . . . . . . . . . . . . . . 143 III.35 Perturbed component of the wave pattern (η P ) at t = 24.595 s . . . . . 144 III.36 Comparison of time series of the incident and scattered components of the wave elevation at WP1, WP2 and WP3 from numerical results for Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 III.37 Comparison of time series of the hydrodynamic loads on Body 1 from numerical results for Case 1 with a body-exact approximation and several meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 III.38 Comparison of time series of the wave elevations at WP1, WP2 and WP3 from numerical results for Case 1 with a body-exact approximation and several meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 III.39 Comparison of time series of the hydrodynamic loads on Body 1 from numerical and experimental results for Case 1 with two hydrodynamic approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 III.40 Comparison of time series of the wave elevations at WP1, WP2 and WP3 from numerical and experimental results for Case 1 with two hydrodynamic approximations . . . . . . . . . . . . . . . . . . . . . . . . 147 xvii III.41 Comparison of time series of the hydrodynamic loads on Body 1 from numerical results for = 0.096 . . . . . . . . . . . . . . . . . . . . . . . III.42 Comparison of time series of the wave elevations at WP1, WP2 and WP3 from numerical results for = 0.096 . . . . . . . . . . . . . . . . . III.43 Comparison of time series of the hydrodynamic loads on Body 1 from numerical results for = 0.144 . . . . . . . . . . . . . . . . . . . . . . . III.44 Comparison of time series of the wave elevations at WP1, WP2 and WP3 from numerical results for = 0.144 . . . . . . . . . . . . . . . . . III.45 Comparison of time series of the hydrodynamic loads on Body 1 from numerical results for Case 1 with a linear approximation and several meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.46 Comparison of time series of the wave elevations at WP1, WP2 and WP3 from numerical results for Case 1 with a linear approximation and several meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.47 Comparison of time series of the hydrodynamic loads on Body 1 from numerical and experimental results for Case 1 with three hydrodynamic approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.48 Comparison of time series of the wave elevations at WP1, WP2 and WP3 from numerical and experimental results for Case 1 with three hydrodynamic approximations . . . . . . . . . . . . . . . . . . . . . . . III.49 Mesh convergence using WS_CN for the heave motion of Cylinder 1 (top) and Cylinder 2 (bottom) . . . . . . . . . . . . . . . . . . . . . . . III.50 Time step convergence using WS_CN for the heave motion of Cylinder 1 (top) and Cylinder 2 (bottom) . . . . . . . . . . . . . . . . . . . . . . III.51 Mesh convergence of the hydrodynamic coecients using Nemoh. The number of panels is given for one cylinder. . . . . . . . . . . . . . . . . III.52 Final meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.53 Time step convergence using InWave for the heave motion of Cylinder 1 III.54 Comparison of time series of the heave motion of Cylinder 1 from numerical results of InWave-Nemoh and WS_CN . . . . . . . . . . . . . III.55 Comparison of time series of the heave motion of Cylinder 1 and Cylinder 2 from numerical results of WS_CN . . . . . . . . . . . . . . . . . III.56 Comparison of time series of the relative heave motion between Cylinder 1 and Cylinder 2 from numerical results of Nemoh . . . . . . . . . . . IV.1 Monolithic approach in case of a uid-structure coupling.X and F represent the motion of the structure and the hydrodynamic loads. . . . IV.2 Main loose coupling algorithms . . . . . . . . . . . . . . . . . . . . . . . IV.3 CSS algorithms. X and F represent the motion of the structure and the hydrodynamic loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV.Gauss-Seidel algorithm in case of a uid-structure coupling. X and F represent the motion of the structure and the hydrodynamic loads.163 IV.5 Main coupling strategies . . . . . . . . . . . . . . . . . . . . . . . . . . IV.6 Language binding between InWave and WS_CN using Python as glue code language. The red, green and blue colors represent the C++, Python and Fortran languages. . . . . . . . . . . . . . . . . . . . . . . . xviii IV.7

Figure 2 Figure 3

 23 Figure 2 Towing of the oating wind turbine WindFloat 6

I. 1

 1 Figure I.1 Sketches of the dierent multibody system families. Black circle: body, black point: joint and red arrow: kinematic loop.

I. 1 . 3 . 1

 131 The ABA and the CRBA Both the Articulated-Body Algorithm and the Composite-Rigid-Body Algorithm are part of the embedding technique as dened by Shabana[START_REF] Shabana | Computational dynamics Second Edition[END_REF]. Instead of using the classical Cartesian coordinates (also named the redundant coordinate set[START_REF] Masarati | Comprehensive multibody aeroservoelastic analysis of integrated rotorcraft active controls[END_REF]), these algorithms are based on a joint coordinate set (also named a reduced or relative coordinate set). With the Cartesian coordinates, each body is located from a unique global frame whereas by using joint coordinates, each body is located from one of its ancestors. The Articulated-Body Algorithm and the Composite-Rigid-Body Algorithm are directly applicable to kinematic trees (which include open-chain systems). Both of them are based on the Newton-Euler equation, written in a recursive way. But they dier in the way they compute the accelerations. The Articulated-Body Algorithm

  tact. Holonomic constraint group is split into two categories according to the time dependency. Rheonomic constraints are time-dependent and used to add prescribed motions (excitation) in joints. Scleronomic constraints are time-independent and form the most common category of constraints.

Figure I. 2 (

 2 Figure I.2 Classication of constraints (from [48])

  Figure I.1b, n = 5 and is equal to the number of joints, whereas in Figure I.1c, n = 5 but the number of joints is 6 and one closed-loop is present.

  Figure I.4 presents a sketch of the local frames and the modied Denavit-Hartenberg parameters in case of a set of three bodies of a kinematic tree.

Figure I. 4

 4 Figure I.4 Local frames and modied Denavit-Hartenberg parameters in case of a kinematic tree (from [43])

C

  * = cos( * ); S * = sin( * ).

=rot 0 = ϕ 0 θ 0 ψ 0 T

 0 x 0 y 0 z 0 T = e P 0 the Cartesian position of O 0 from O e ; η the angular position using the Cardan angles.

(I. 28 )

 28 Equations (I.24) and (I.27) are now projected in the local frame of body j using (I.14), (I.22) and (I.23):

/

  Σe = S( e ω j ) e R j (I.51) d e R -1 j dt / Σe = -e R -1 j d e R j dt / Σe e R -1 j (I.52) Furthermore, for any rotation matrix M and any 3-component vectors a and b, the cross product follows the formula: (Ma) × (Mb) = M(a × b) (I.53) Or using the cross product matrix S: S(Ma)Mb = MS(a)b (I.54)

  Cables are subject to three types of deformation (Figure II.1): An axial deformation: the stretching; A curvature deformation: the bending; A twisting deformation: the torsion.

Figure II. 1

 1 Figure II.1 Deformations of a cable element: stretching of length dx , bending with a radius of curvature R and torsion by an angle θ

M i the 3 × 3

 33 mass matrix; A i the 3 × 3 added-mass matrix; W i the weight; B i the buoyancy force;T i the axial tension due to the adjacent nodes; C i the axial damping loads due to the adjacent nodes; D i the hydrodynamic loads; r i the position vector.

Figure II. 3

 3 Figure II.3 Spatial discretization of a cable. The position vector of the nodes are in red.The nodes are the black points and the cable elements the black lines.

  Figure II.5 for the acceleration test case and in Figure II.6 for the deceleration test case. A fair agreement is observed in both cases. Nevertheless, numerical results obtained from CableDyn and from Buckham do not exactly match. This may be explained by the dierent formulation of the hydrodynamic loads (drag and added-mass loads) in the two low-order lumped mass theories compared here. For example, Buckham [64] used a loading function in the expression of the drag loads for accounting for the non-linear breakup between the normal and tangential components.

Figure II. 5

 5 Figure II.5 Cable proles during the acceleration test case at dierent times: in red att =120 s, in blue at t =180 s, in green at t =240 s and in black at t =540 s.

  Dierent cable proles are presented in Figure II.8.

Figure II. 8

 8 Figure II.8 Proles of an oscillating 3-element cable. The triangles represent the position of the nodes. The black lines denote the cable elements.

Figure II. 9

 9 Figure II.9 Comparison of q 9 from numerical results of the CRBA and the lumped mass theory (LM )

  3.2 now serves to compare the cable modeling with the CRBA using the single-dof joint formulation and the cable joint formulation. Numerical results are displayed in Figure II.11. The two formulations match perfectly.

Figure II. 11

 11 Figure II.11 Comparison of q 9 from numerical results of the CRBA with both the single-dof formulation and the cable joint formulation

α = 3 2 (

 2 Figure II.12 shows the scheme of the cable splitting using the lumped mass approach.It consists of adding a new node between the two top nodes.

Figure II. 12

 12 Figure II.12 Cable splitting with the low-order lumped mass theory

Figure II. 13

 13 Figure II.13 Cable slitting with the multibody theory

3 . 2 .

 32 The lowering velocity is xed to 0.5 m/s between 8 s and 44 s. No ramp is used on the lowering velocity.

Figure II. 14

 14 Figure II.14 shows the length of the topmost cable element or the articular variable of the rst prismatic joint. The discontinuity of the length indicates the addition of a new cable element while the straight lines indicate the increase of the unstretched length. The cut-o length for adding a new element, equal to

Figure II. 14

 14 Figure II.14 Comparison of the length of the topmost cable element from numerical results of the CRBA and the lumped mass theory (LM )

β = 1 2 (

 2 Figure II.17 displays the scheme of the cable merging using the lumped mass approach. The second node is simply deleted. Then, the rst node is directly connected to the third one.

Figure II. 17

 17 Figure II.17 Cable merging with the low-order lumped mass theory

Figure II. 19

 19 Figure II.19 shows the length of the topmost cable element or the articular variable of the rst prismatic joint. The discontinuity of the length outlines the deletion of a new cable element while the straight lines indicate the decrease of the unstretched length. The two curves match perfectly, demonstrating both Cartesian and multibody approaches allow to model the winding of a cable. Figure II.20 presents the angle of the top cable element with respect to the vertical axis. Contrary to the reel-in test case, some discontinuities were expected. Indeed, the suppression of a node involves an angular error close to the winch as it can be seen in Figure II.18. In the present test case, the cable remains almost vertical so this error is very small. Figure II.21 displays the cable prole at dierent time steps.

Figure II. 19

 19 Figure II.19 Comparison of the length of the topmost cable element from numerical results of the CRBA and the lumped mass theory (LM )

Figure III. 1

 1 Figure III.1 Fully nonlinear model of a oating body (yellow) in waves at their exact elevation (solid blue line) and its wetted surface (hatched area)

Figure III. 2

 2 Figure III.2 Weakly nonlinear model based on the weak-scatterer hypothesis of a oating body (yellow) in waves at their incident elevation (solid blue line) and its wetted surface (hatched area)

Figure III. 3

 3 Figure III.3 Body exact model of a oating body (yellow) in waves at their mean elevation (solid blue line) and its wetted surface (hatched area)

Figure III. 4

 4 Figure III.4 Linear model of a oating body (yellow) in waves at their mean elevation (solid blue line) and its wetted surface (hatched area)

  be underestimated. No comparison has been made between the weakly nonlinear model based on the weak-scatterer hypothesis and a classical frequency-domain model in the case of a marine operation. Thus, it is of interest to compare them and quantify their dierences.

  used in the hydrodynamic theory: e M Hydro j = m j I 3 0 3×3 0 3×3 e I G j S j (III.90) This matrix is dierent from e M j used in section I.4 in expression and in point of computation (G j for e M Hydro j , the center of the joint O j for e M j ). S j comes from (I.41) and e I G j is the (3 × 3) inertia matrix of body j reduced at G j with respect to Σ e . e F Inertia j the inertial loads: e F Inertia j = 0 3×3 -e I G j Ṡj S -1 j e ω j -S( e ω j ) e I G j e ω j (III.91) e F W SC j the hydrodynamic loads based on the weak-scatterer hypothesis; e F Other j the other external loads (weight, aerodynamic loads, mooring loads, power take-o loads, etc.);

j

  ηje CT j φP (B j ) = e T W SC j + e F Inertia j + e F Other j (III.111)

Figure III. 6

 6 Figure III.6 and found using a dichotomy method.

Figure III. 6

 6 Figure III.6 Example of an isoparametric line for a cylinder (from [45])

  Figure III.7b) from the parametric surfaces of the bodies (Figure III.7a), the mesh generation process may start. It unfolds in ve steps: The extremities of the edges of the geometries are tracked (for instance, the vertices of a cube); The intersection curves are meshed (Figure III.7c); The edges of all geometries are meshed, the multiple nodes are dened (Figures III.7c and III.7d); The wetted body surfaces are meshed (Figure III.7e); The free surface and the numerical tank walls are meshed and connected to the wetted body meshes (Figure III.7f).

  Figure III.7 The steps of the mesh generation of the whole domain with a oating vertical cylinder

  0 on the free surface δ ref = δ input at the boundaries of the free surface (III.125)

  ref represents the value of δ ref in the cell (i, j) of the Cartesian grid. Equation III.126 leads to a linear system AX = B of size N ref , the number of cells in the grid. A LU factorization is used. The value of δ ref at the position of the created point is found from a bilinear interpolation.

Figure

  Figure III.8a shows the values of δ ref over the free surface for a domain of radius 3 m with δ u = 0.2 m with a surface-piercing oating cylinder of radius 0.2 m and δ u = 0.022 m. Figure III.8b displays the nal mesh with high quality panels and a smooth transition between the nest area to the coarsest area.

  Figure III.8 h ref = 0.3 my = 0 m

  Figure III.11 h ref = 0.01 my = 0.15 m

  distance of a node with respect to the center of the domain, R the radius of the cylindrical domain and r 0 = R -L abs with L abs the length of the absorbing numerical beach.

Figure III. 12

 12 Figure III.12 Example of a damping numerical beach with L abs = 1 m and R = 3 m

Figure III. 13

 13 Figure III.13 Mesh distortion due to the translation of a cylinder along the y-axis (left).The second cylinder is xed (right).

  Figure III.14 The three steps of the interpolation scheme on the free surface mesh

Figure III. 15

 15 Figure III.15 presents the comparison of the time series of the cylinder motion and the wave elevation at the wave probe. The numerical results are similar even if few dierences are observed. They are caused by the interpolations during the free surface remeshing algorithm which involves numerical errors.

Figure III. 15

 15 Figure III.15 Comparison of time series of the heave motion (top) and the wave elevation (bottom) from numerical results with and without forced free surface remeshing processes

  Figure III.16 Eect of the ltering of the scattered wave elevation on a simulation with a translation cylinder along the y-axis (left) and a xed cylinder (right).

Figure III. 17

 17 Figure III.17 presents the results of the parallelization for a simulation with two surface-piercing cylinders and a mesh of 8300 panels and 4550 nodes and without free surface remeshing. The top gure shows that the time saving is more important if the inuence coecients are completely evaluated. The interest of the partial computation of the inuence coecients depends on the relative velocity between the mesh parts and so of the dynamics of the problem. The bottom gure shows that the same nal time is reached with the two methods once the main eects of the parallelization are obtained.From 5 threads, the interest of the partial computation of the inuence coecients is negligible. The partial computation is interesting for a sequential computation as it lowers the simulation time of 50 %.

  Figure III.18 Proling of W S_CN

Figure III. 19

 19 Figure III.19 Comparison of time series of the heave motion (top) and the wave elevation (bottom) from numerical results using a sequential (nth = 1) and parallel (nth = 5) computation

  Initial position of Body 2 (m) (0, -0.3, 0)

  WP1 (m) (0, 0.7, 0) WP2 (m) (0, 0, 0) WP3 (m) (0, -0.7, 0)

  downstream. The positions of every device are highlighted in Figure III.20.

Figure III. 20

 20 Figure III.20 Sketch of the top view of the experimental set-up (from [145])

  and III.[START_REF] Bai | Numerical simulation of fully nonlinear wave interaction with submerged structures: Fixed or subjected to constrained motion[END_REF]. It is noticed that the mesh convergence is slow along the x axis of the hydrodynamic loads compared to the two other directions. A mesh of 20 000 panels is used. The time step convergence is presented in Figures III.[START_REF] Hannan | Analysis of nonlinear dynamics of fully submerged payload hanging from oshore crane vessel[END_REF] and III.24. It is also slow along the x-axis. A time step of 0.005 s is used hereinafter. Figures III.25 and III.26 show the mesh of the domain for Case 1.

Figure III. 21

 21 Figure III.21 Comparison of time series of the hydrodynamic loads on Body 1 from numerical results for dierent meshes for Case 1

Figure III. 27

 27 Figure III.27 Prescribed harmonic motion of Body 2 for Case 1

Figure III. 28 1 Figure

 281 Figure III.28 Comparison of time series of the hydrodynamic loads on Body 1 from numerical and experimental results for Case 1

Figure III. 16 )

 16 Figure III.34. Only few dierences are present. For the sake of illustration, the perturbed wave pattern for Case 1 at t = 24.595 s is displayed in Figure III.35.

Figure III. 33

 33 Figure III.33 Comparison of time series of the hydrodynamic loads on Body 1 from numerical and experimental results for Case 1 with and without forced remeshing processes

Figure III. 36

 36 Figure III.36 Comparison of time series of the incident and scattered components of the wave elevation at WP1, WP2 and WP3 from numerical results for Case 1

  Figures III.[START_REF] Rongère | Systematic dynamic modeling and simulation of multibody oshore structures: application to wave energy converters[END_REF] and III.44 for a wave amplitude three times bigger than Case 1. The wave steepness for Case 1 is = 0.048. The wave length is kept constant, only the wave amplitude is modied.

Figure III. 37

 37 Figure III.37 Comparison of time series of the hydrodynamic loads on Body 1 from numerical results for Case 1 with a body-exact approximation and several meshes

Figure III. 45

 45 Figure III.45 Comparison of time series of the hydrodynamic loads on Body 1 from numerical results for Case 1 with a linear approximation and several meshes

Figure III. 49

 49 Figure III.49 Mesh convergence using WS_CN for the heave motion of Cylinder 1 (top) and Cylinder 2 (bottom)

  Figure III.52 Final meshes

Figure III. 53

 53 Figure III.53 Time step convergence using InWave for the heave motion of Cylinder 1

Figure III. 54

 54 Figure III.54 Comparison of time series of the heave motion of Cylinder 1 from numerical results of InWave-Nemoh and WS_CN

Figure III. 55

 55 Figure III.55 Comparison of time series of the heave motion of Cylinder 1 and Cylinder 2 from numerical results of WS_CN

  ler's equation for the uid and the motion equation for a rigid body. A scheme of a monolithic coupling is shown in Figure IV.1.

Figure IV. 1

 1 Figure IV.1 Monolithic approach in case of a uid-structure coupling. X and F represent the motion of the structure and the hydrodynamic loads.

Figure IV. 2

 2 Figure IV.2 Main loose coupling algorithms

Figure IV. 4

 4 Figure IV.4 Block Gauss-Seidel algorithm in case of a uid-structure coupling. X and F represent the motion of the structure and the hydrodynamic loads.

Figure IV. 5 .

 5

Figure IV. 5

 5 Figure IV.5 Main coupling strategies

  Figure IV.6 presents the relationships between InWave, WS_CN and InWaveS_CN.

Figure IV. 6

 6 Figure IV.6 Language binding between InWave and WS_CN using Python as glue code language. The red, green and blue colors represent the C++, Python and Fortran languages.

Figure IV. 7

 7 Figure IV.7 presents the sketch of the seven dierent steps. Only one stage of the fourth-order Runge-Kutta method is represented.

Figure IV. 7

 7 Figure IV.7 The seven steps of the tight coupling between InWave and WS_CN. The red, green and blue colors represent the C++, Python and Fortran languages. The arrows denote the communication between the modules. The color of the arrows depends on the programming language which sends the data. The superscript n denotes the n th time step.

  .3. d 1 represents the relative distance between the two cylinders. The time series of the motions of both Cylinder 1 and Cylinder 2 are presented in Figures IV.9 and IV.10. Numerical results coming from WS_CN are based on (III.112) while those obtained with InWaveS_CN use (IV.39). They show a perfect agreement between the two formulations which validates the implementation of the present coupling.

Figure IV. 8

 8 Figure IV.8 Initial mesh of the comparison test case between WS_CN and InWaveS_CN

Figure IV. 9

 9 Figure IV.9 Comparison of time series of the motions of Cylinder 1 from numerical results of InWaveS_CN and WS_CN

Figure IV. 11 and

 11 Figure IV.12 show the surge, heave and pitch motions for the Cylinders 1 and 2

Figure IV. 11 Table IV. 4

 114 Figure IV.11 Comparison of time series of the motions of Cylinder 1 from numerical results of InWaveS_CN for a tight, loose explicit and implicit couplings

  Figure IV.14 shows the algorithm of the coupling between the three solvers: InWave, WS_CN and CableDyn.

Figure IV. 14

 14 Figure IV.14 Tight-loose coupling between InWave (structure solver), WS_CN (uid solver) and CableDyn (cable solver). X, F and T represent the motion of the structure, the hydrodynamic loads and the cable loads.

3 (

 3 2.2), which is the fairlead by denition. The axial tension T 0 and the axial damping force C 0 are evaluated from (II.29) and (II.32), for each cable. e G Cable j the (6 × 6) matrix to change the point of computation of the cable loads fromA j to O j : e G Cable j = I 3 0 3×3 S( e R j j O j A j ) I IV.46)j O j A j the position of the fairlead connected to the body j with respect to Σ j ; 0 N j e given by (IV.5).

  Figure IV.15 presents the updated scheme of the coupling.

Figure IV. 15

 15 Figure IV.15 Language binding between InWave, WS_CN and CableDyn using Python as glue code language. The red, green and blue colors represent the C++, Python and Fortran languages.

3 . 2 (

 32 Figure V.1d);

  Figure V.1 The steps of the mesh generation of the whole domain with a oating vertical cylinder using the panel cutting method

  10 https://github.com/LHEEA/meshmagick This tool is implemented in Python and released under the GPLv3 license.The mesh clipping starts with a partition of the mesh: for instance in Figure V.2 for a cylinder of radius 0.2 m and height 1 m. The panels which are strictly above the incident wave elds are not taking into account while those strictly below the incident wave elds are automatically kept. Regarding the panels where some vertices are above the incident free surface and some others below, they form the crown mesh. The panel cutting method is only applied to this part of the mesh.(a) Initial mesh of a cylinder (b) Partition of the mesh. Blue: lower mesh, green: upper mesh, red: crown mesh. The black line represents the mean sea level. The space between the parts is added for the visualization.

Figure V. 2

 2 Figure V.2 Partition of a mesh in Meshmagick

  searched iteratively. Examples of clipping are displayed in Figure V.4. The clipping of the present cylinder with a regular wave of amplitude 0.1 m and wave frequency 12 rad/s is used as example in the rest of this section.

Figure V. 4

 4 Figure V.4 Clippings of a vertical cylinder against a plane and a regular wave. The red line represents the analytical incident wave elevation.

Figure V. 5

 5 Figure V.5 Deformed panels close to the intersection curve. Zoom from Figure V.4b.

Figure V. 6

 6 Figure V.6 Examples of vertical (red solid line) and horizontal (red dashed line) tiny triangles with α H = β H = 0.3, α V = 0.4 and β V = 0.3

p 1 and p 2

 2 are dened in Figure V.7a z is the vertical coordinate of the node p 0 in Figure V.7b; η I means the incident wave elevation at the vertical of the node p 0 in Figure V.7b; α V , α H , β V and β H are the panel merging coecients. The subscript indicates if the coecient is used for the vertical (V ) or horizontal (H ) tiny panels. The higher these coecients are, the more important the number of deleted panels is, but, the more important the risk of interpenetration of neighboring panels is. An example of a panel overlapping is shown in Figure V.8. (a) α H = β H = 0.3, α V = 0.4 and β V = 0.3 (b) α H = β H = α V = β V = 0.5. The dark area indicates the panel overlapping.

Figure V. 8

 8 Figure V.8 Example of a panel overlapping. The wave height is 0.15 m.

Figure V. 9

 9 Figure V.9 shows the application of the panel merging algorithm. The density of nodes and panels is reduced on the intersection curve as wanted, but the panel shape is still not good enough. This problem is addressed in the next section.

Figure V. 10

 10 Figure V.10 presents the application of the spring analogy method once the mesh clipping and the panel merging are applied. The panel shape is improved on the intersection curve as expected. This mesh is now good enough to be used in the uid solver.

  Figure V.11 Examples of multiple node tracking. The panels including a simple, double or triple node are, respectively, in blue, green and red.

  Figure V.12 Evolution of f size during the mesh generation

  and V.[START_REF] Schellin | Response analysis and operating limits of crane ships[END_REF]. The same vertical cylinder is used as in the previous section. The amplitude of the regular wave is 0.02 m and its wave frequency 12 rad/s. Other nal meshes for two academic geometries are shown: a horizontal cylinder in FigureV.[START_REF] Ellermann | Nonlinear dynamics of oating cranes[END_REF] and a cube in Figure V.18. Three non-academic geometries are also displayed: The wave energy converter: SEAREV 11 ; A FPSO; A tension-leg platform (TLP ). These three geometries are presented in Figure V.19. Their incorporation in a free surface mesh is displayed in Figure V.20 for the SEAREV, in Figure V.21 for the FPSO and in Figure V.22 for the TLP.

Figure V. 15

 15 Figure V.15 Example of an intersection curve for a vertical cylinder. The red points are the nodes, the blue line is the intersection curve, the black arrow represents the orientation of the curve, the black cross denotes the gravity center of the water-plane area. 11 Système Électrique Autonome de Récupération de l'Energie des Vagues

  Figure V.23 displays the updated scheme of the coupling. This is the nal state of the numerical tool developed during this PhD work. In the rest of the PhD thesis, when Meshmagick is used, the mechanical solver is always InWave.

Figure V. 23

 23 Figure V.23 Language binding between InWave, WS_CN, CableDyn and Meshmagick using Python as glue code language. The red, green and blue colors represent the C++, Python and Fortran languages

  Figure V.24 Proling of the new mesh generator. The colors follow the rule dened in Figure V.23: green for Python, blue for Fortran.

Figure V. 25

 25 Figure V.25 Proling of the new mesh generator without using PARDISO. The colors follow the rule dened in Figure V.23: green for Python, blue for Fortran.

  Figure V.26. It shows that a mesh of 10000 panels for the initial mesh generator and of 9600 panels for the new one are sucient. The initial meshes for the heave test case are shown in Figure V.28. The time step convergence is presented in Figure V.27, showing that a time step of 0.01 s is enough. The comparison between the numerical results of the two mesh generators is shown in Figure V.29 for the heave motion and the wave elevation at the wave probe. The perturbed wave pattern at t = 5.15 s is displayed in Figure V.30.

  Figure V.26 Mesh convergence for the heave test case

  Figure V.31 Mesh convergence for the pitch test case

  Figure V.36 Area of the free surface panels close to the intersection curve with the two mesh strategies

(

  Figure VI.1a); The cable is unwound using a winch, the spar has a rotating motion around a horizontal xed axis (Figure VI.1b); The spar reaches its vertical equilibrium position (Figure VI.1c).

  These characteristics are at the heart of the numerical tool developed during our work (InWaveS_CN ) and match the work achieved in the previous ve chapters of the present PhD thesis.

  Figure VI.1 Three steps of the spar upending. The xed axis is indicated by the cross and the winch by the circle.

Figure VI.3 Buoy 1 Figure

 1 Figure VI.3 Buoy 1

Figure VI. 5

 5 Figure VI.5 Revolute joint system

Figure VI. 6

 6 Figure VI.6 Winch

  Figure VI.8 shows the markers of the buoy, the cable and the winch.

Figure VI. 9

 9 Figure VI.9 presents the results of Qualisys at the initial stage for Case 15.

Figure VI. 8

 8 Figure VI.8 Positions of the markers

Figure VI. 10

 10 Figure VI.10 Winch velocity for Cases 15, 16, 17 and 18. t = 0 s denotes the starting time of the winch.

Figure VI. 12

 12 Figure VI.12 presents the wave elevation for Case 17 for a regular wave of amplitude 0.01 m and a wave period of 1 s. The eect of the waves on the buoy rotation motion is insignicant (Figure VI.13). But it is clearly noticeable on the cable tension. The tension dierence between the tension in still water and in waves is shown in Figure VI.14.

Figure VI. 12

 12 Figure VI.12 Wave elevation for Case 17. t = 0 s denotes the starting time of the winch.

Figure VI. 13

 13 Figure VI.13 Cable tension and rotational motion for Cases 15, 16, 17 and 18. t = 0 s denotes the starting time of the winch.

Figure VI. 14

 14 Figure VI.14 Tension dierence between Case 15 (still water) and Cases 16, 17 and 18 (in waves). t = 0 s denotes the starting time of the winch.

  The in-plane horizontal and vertical motions of the markers trajectory are displayed in Figure VI.16 while the out-of-plane motion is shown in Figure VI.17. The cable is out-of-plane when it becomes slack.

Figure VI. 15

 15 Figure VI.15 Trajectory of the markers for Case 15. CoG denotes the position of the center of gravity of the buoy. C# represents the cable markers.

Figure VI. 16

 16 Figure VI.16 Horizontal (x) and vertical (z) motion of the markers for Case 15. CoG denotes the position of the center of gravity of the buoy. C# represents the cable markers. t = 0 s is the starting time of the winch.

Figure VI. 17

 17 Figure VI.17 Out-of-plane (y) motion of the markers for Case 15. CoG denotes the position of the center of gravity of the buoy. C# represents the cable markers. t = 0 s is the starting time of the winch.

Figure VI. 18

 18 Figure VI.18 Trajectory of the markers for Case 18. CoG denotes the position of the center of gravity of the buoy. C# represents the cable markers.

Figure VI. 19

 19 Figure VI.19 Horizontal (x) and vertical (z) motion of the markers for Case 18. CoG denotes the position of the center of gravity of the buoy. C# represents the cable markers. t = 0 s is the starting time of the winch.

Figure VI. 20 VI. 2 . 1

 2021 Figure VI.20 Out-of-plane (y) motion of the markers for Case 18. CoG denotes the position of the center of gravity of the buoy. C# represents the cable markers. t = 0 s is the starting time of the winch.

Figure VI. 22

 22 Figure VI.22 presents the comparison of the initial position of the winch, the cable, the buoy center of gravity and the rotation axis between the experiments and the numerical simulation.

Figure VI. 21

 21 Figure VI.21 Initial mesh of the buoy with 8862 panels

Figure VI. 23

 23 Figure VI.23 Comparison of time series of the lowering velocity at the winch from numerical and experimental results for Case 15. t = 0 s denotes the starting time of the winch.

Figure VI. 26

 26 Figure VI.26 Bi-dimensional view of the initial mesh for Case 15. The cable is in red.

  , VI.28 and VI.29. As expected, a very small time step is required when the cable stiness is higher. A comparison of the numerical results after convergence for the three values of the cable stiness is displayed in Figure VI.30. The tension and the rotational motion are independent of the cable stiness. The simulations do not last the same duration because of memory leak problems. Finally, an axial stiness of 6.7 × 10 5 N, an axial damping coecient of 3 × 10 7 N s/m 2 and a time step of 0.001 s are chosen.

Figure VI. 27

 27 Figure VI.27 Comparison of time series of the cable tension and the angular position from numerical results for Case 15 for a cable stiness of 6.7 × 10 5 N. t = 0 s denotes the starting time of the winch.

Figure VI. 28

 28 Figure VI.28 Comparison of time series of the cable tension and the angular position from numerical results for Case 15 for a cable stiness of 1.34 × 10 6 N. t = 0 s denotes the starting time of the winch.

Figure VI. 29

 29 Figure VI.29 Comparison of time series of the cable tension and the angular position from numerical results for Case 15 for a cable stiness of 3.35 × 10 6 N. t = 0 s denotes the starting time of the winch.

Figure VI. 30

 30 Figure VI.30 Comparison of time series of the cable tension and the angular position from numerical results for Case 15 for a cable stiness of 3.35 × 10 6 N. t = 0 s denotes the starting time of the winch.

Figure VI. 31

 31 Figure VI.31 presents the comparison of the angular position of the buoy and the cable tension between the experimental and numerical results. It can be seen the horizontality of the buoy at the initial time was not perfect during the experiments. Despite that, for both the motion and the tension, the agreement is very good. The tension decrease during the increase of the lowering velocity due to the ramp is well captured. The perturbed wave pattern at t = 4.5 s is shown in Figure VI.32. The simulation only lasts 5.8 s because at that moment, one extremity of the buoy (bottom) is fully immersed whereas it was piercing at the initial time. Consequently the body mesh is tangent to the free surface mesh which leads to important numerical errors. The mesh at the nal time step is displayed in Figure VI.33.

Figure VI. 31

 31 Figure VI.31 Comparison of time series of the cable tension and the angular position from numerical and experimental results for Case 15. t = 0 s denotes the starting time of the winch.

Figure VI. 32

 32 Figure VI.32 Perturbed component of the wave pattern (η P ) at t = 4.5 s. The cable is in red.

Figure VI. 33

 33 Figure VI.33 Bi-dimensional view of the nal mesh for Case 15. The cable is in red.

Figure VI. 35

 35 Figure VI.35 Comparison of time series of the cable tension and the angular position from numerical and experimental results for Case 8. t = 0 s denotes the starting time of the winch.

Figure VI. 36

 36 Figure VI.36 Comparison of time series of the lowering velocity at the winch from numerical and experimental results for Case 17. t = 0 s denotes the starting time of the winch.

Figure VI. 37

 37 Figure VI.37 Comparison of time series of the cable tension and the angular position from numerical and experimental results for Case 17. t = 0 s denotes the starting time of the winch.

Figure VI. 39

 39 Figure VI.39 Comparison of time series of the cable tension and the angular position from numerical and experimental results for Case 18. t = 0 s denotes the starting time of the winch.

Figure VI. 41

 41 Figure VI.41 Comparison of time series of the cable tension and the angular position from numerical and experimental results for Case 20. t = 0 s denotes the starting time of the winch.

Figure C. 3 C.2 Case 2 Figure C. 4

 324 Figure C.3 Cable tension and rotational motion for Case 1. t = 0 s denotes the starting time of the winch.

Figure C. 5

 5 Figure C.5 Wave elevation for Case 2. t = 0 s denotes the starting time of the winch.

Figure C. 6 C.3 Case 3 Figure C. 7

 637 Figure C.6 Cable tension and rotational motion for Case 2. t = 0 s denotes the starting time of the winch.

Figure C. 8

 8 Figure C.8 Wave elevation for Case 3. t = 0 s denotes the starting time of the winch.

Figure C. 9

 9 Figure C.9 Cable tension and rotational motion for Case 3. t = 0 s denotes the starting time of the winch.

Figure C. 11

 11 Figure C.11 Wave elevation for Case 4. t = 0 s denotes the starting time of the winch.

Figure C. 12

 12 Figure C.12 Cable tension and rotational motion for Case 4. t = 0 s denotes the starting time of the winch.

Figure C. 14

 14 Figure C.14 Wave elevation for Case 5. t = 0 s denotes the starting time of the winch.

Figure C. 15

 15 Figure C.15 Cable tension and rotational motion for Case 5. t = 0 s denotes the starting time of the winch.

Figure C. 17

 17 Figure C.17 Wave elevation for Case 6. t = 0 s denotes the starting time of the winch.

Figure C. 18

 18 Figure C.18 Cable tension and rotational motion for Case 6. t = 0 s denotes the starting time of the winch.

Figure C. 20

 20 Figure C.20 Wave elevation for Case 7. t = 0 s denotes the starting time of the winch.

Figure C. 21

 21 Figure C.21 Cable tension and rotational motion for Case 7. t = 0 s denotes the starting time of the winch.

Figure C. 23

 23 Figure C.23 Wave elevation for Case 8. t = 0 s denotes the starting time of the winch.

Figure C. 24

 24 Figure C.24 Cable tension and rotational motion for Case 8. t = 0 s denotes the starting time of the winch.

Figure C. 26

 26 Figure C.26 Wave elevation for Case 9. t = 0 s denotes the starting time of the winch.

Figure C. 27

 27 Figure C.27 Cable tension and rotational motion for Case 9. t = 0 s denotes the starting time of the winch.

Figure C. 29

 29 Figure C.29 Wave elevation for Case 10. t = 0 s denotes the starting time of the winch.

Figure C. 30

 30 Figure C.30 Cable tension and rotational motion for Case 10. t = 0 s denotes the starting time of the winch.

Figure C. 32

 32 Figure C.32 Wave elevation for Case 11. t = 0 s denotes the starting time of the winch.

Figure C. 33

 33 Figure C.33 Cable tension and rotational motion for Case 11. t = 0 s denotes the starting time of the winch.

Figure C. 35

 35 Figure C.35 Wave elevation for Case 12. t = 0 s denotes the starting time of the winch.

Figure C. 36

 36 Figure C.36 Cable tension and rotational motion for Case 12. t = 0 s denotes the starting time of the winch.

Figure C. 38

 38 Figure C.38 Wave elevation for Case 13. t = 0 s denotes the starting time of the winch.

Figure C. 39

 39 Figure C.39 Cable tension and rotational motion for Case 13. t = 0 s denotes the starting time of the winch.

Figure C. 41

 41 Figure C.41 Wave elevation for Case 14. t = 0 s denotes the starting time of the winch.

Figure C. 42

 42 Figure C.42 Cable tension and rotational motion for Case 14. t = 0 s denotes the starting time of the winch.

Figure C. 44

 44 Figure C.44 Wave elevation for Case 15. t = 0 s denotes the starting time of the winch.

Figure C. 46

 46 Figure C.46 Wave elevation for Case 16. t = 0 s denotes the starting time of the winch.

Figure C. 47

 47 Figure C.47 Cable tension and rotational motion for Case 16. t = 0 s denotes the starting time of the winch.

Figure C. 51

 51 Figure C.51 Wave elevation for Case 18. t = 0 s denotes the starting time of the winch.

Figure C. 52 Figure C. 53

 5253 Figure C.52 Cable tension and rotational motion for Case 18. t = 0 s denotes the starting time of the winch.

Figure C. 54

 54 Figure C.54 Wave elevation for Case 19. t = 0 s denotes the starting time of the winch.

Figure C. 55

 55 Figure C.55 Cable tension and rotational motion for Case 19. t = 0 s denotes the starting time of the winch.

H

  tous basés sur une approche d'écoulement potentiel linéarisé. C'est pourquoi il est nécessaire de développer un outil de simulation numérique prenant en compte les interactions mécaniques et étant cohérent d'un point de vue hydrodynamique pour modéliser le couplage uide-structure lors d'une opération de descente ou de remontée de colis. Les objectifs de la thèse sont : L'implémentation d'un modèle de câble et de treuil dans le logiciel de simulation mécanique utilisé (InWave ); L'extension du logiciel de simulation hydrodynamique (WS_CN ), basé la théorie des écoulements potentiels instationnaires satisfaisant l'hypothèse de faible perturbation, dite weak-scatterer , aux simulations multicorps; Le couplage entre InWave et WS_CN dans le but de simuler numériquement des opérations de descente et de remontée de colis; La comparaison entre une modélisation classique utilisant la théorie des écoulements potentiels linéarisés et l'outil développé pour quantier l'intérêt de la théorie instationnaire weak-scatterer . D.2 Dynamique des systèmes multicorps La théorie de mécanique multicorps utilisée dans cette thèse permet de simuler des systèmes multicorps articulés de type arbre cinématique, c'est-à-dire qu'ils ne disposent pas de boucles cinématiques. Concrétement, chaque corps du système multicorps ne posséde qu'un unique prédécesseur mais potentiellement plusieurs successeurs. Dans le cas où plusieurs successeurs seraient présents, des boucles cinématiques apparaîtraient. Les corps sont considérés comme rigides et séparés les uns des autres par une articulation autorisant un seul degré de liberté soit en translation (liaison glissière) soit en rotation (liaison pivot). Chaque corps est repéré via des coordonnées relatives par rapport à son prédécesseur en utilisant les paramètres modiés de Denavit-Hartenberg. Le corps racine (ou la base) du système multicorps est repéré avec des coordonnées Cartésiennes. Cette modélisation permet d'écrire les équations cinématiques récursives du système multicorps. L'algorithme utilisé se nomme l'Algorithme aux Corps Rigides Composites (ACRC ) et permet la résolution d'un problème de dynamique multicorps directe, c'est-à-dire où les eorts extérieurs et intérieurs sont connus et l'accélération des corps inconnue. L'équation du mouvement du système multicorps est : la matrice masse généralisée du système multicorps; C le vecteur des eorts extérieurs et des accélérations d'entrainement et de Coriolis;

L

  'ACRC permet le calcul de la matrice H et du vecteur C à partir des équations cinématiques récursives. Connaissant Γ, les accélérations peuvent être calculées. L'ACRC est implémenté dans le logiciel InWave. L'intégration en temps se fait grâce à un schéma de type Runge-Kutta d'ordre 4. D.3 Dynamique de câbles Les câbles sont simulés en utilisant un modèle dynamique, basé sur la discrétisation du câble en éléments et en noeuds, où la masse est concentrée. Chaque élément comprend un ressort et un amortisseur, an de modéliser l'élasticité du câble et les frottements internes. Lorsque cette approche utilise des coordonnées Cartésiennes pour repérer les noeuds, elle est appelée la méthode lumped mass . Un programme, nommé CableDyn a été développé en ce sens. Il est validé par comparaisons avec les résultats numériques et expérimentaux fournis par Buckham [64]. Un bon accord est observé. L'Algorithme aux Corps Rigides Composites est également utilisé pour simuler des câbles par cette même approche, via InWave. Chaque élément câble est modélisé par trois articulations : deux liaisons pivots et une liaison glissière. Les eorts internes à la liaison glissière permettent de modéliser l'élasticité et les frottements. Ils s'expriment comme suit : + L u ) -CA L u qj si |q j | L u ; C, A et L u représentent le module d'Young, le coecient d'amortissement, l'aire de la section du câble et la longueur à vide de l'élément câble. L'indice j indique le numéro du corps dans le système multicorps. Il n'y a pas d'eorts internes dans les liaisons pivots, la exion et la torsion du câble ne sont donc pas modélisées. La méthode lumped mass et l'ACRC sont comparés et prouvent, par leur accord, que les approches sont identiques. An d'accélérer les calculs de l'ACRC lorsque des câbles sont simulés, une réécriture de l'algorithme dynamique est eectuée. Une articulation câble est créee pour séparer deux corps par trois liaisons mécaniques, soit un élément câble. Une accélération de 49.2 % du temps de calcul est notée. Enn, un modèle de treuil est développé pour simuler le déroulement et l'enroulement d'un câble. Il se base sur l'ajout ou la suppression d'éléments câble pendant la simulation temporelle. L'élément câble connecté au treuil voit sa longueur à vide être modiée. Les méthodes lumped mass et multicorps (ACRC ) donnent les mêmes résultats pour ce modèle de treuil après comparaison. D.4 Hydrodynamique La théorie hydrodynamique utilisée dans cette thèse est la théorie des écoulements potentiels satisfaisant l'hypothèse de faible perturbation, dite weak-scatterer . Le potentiel de vitesse (φ) et l'élévation de surface libre (η) sont décomposés en une partie incidente connue et une partie perturbée inconnue. φ = φ I + φ P η = η I + η P (D.3) avec :

φ

  P = o(φ I ) η P = o(η I ) (D.4)Les équations limites de surface libre sont ensuite linéarisées au niveau de la surface libre incidente. Cette méthode est cohérente en cas de mouvements de grandes amplitudes et de houles cambrées, contrairement à la théorie des écoulements potentiels linéarisés.La composante perturbée du potentiel de vitesse est calculée à partir d'une méthode aux éléments frontières en utilisant une fonction de Green de type Rankine. Le couplage uide-structure est basé sur une approche monolithique et nécessite l'utilisation d'une deuxième méthode aux éléments frontières pour calculer la dérivée temporelle du potentiel de vitesse. L'implémentation de cette théorie hydrodynamique a été eectuée dans le code WS_CN avant le début de cette thèse, dans le cas d'un seul corps immergé ou ottant[START_REF] Letournel | Développement d'un outil de simulation numérique basé sur l'approche weak-scatterer pour l'étude des systèmes houlomoteurs en grands mouvements[END_REF][START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF].La contribution principale apportée à WS_CN est l'extension du solveur à des simulations multicorps en cas de mouvements de grandes amplitudes. Une première validation est entreprise par comparaison avec les résultats expérimentaux de Watai et al. [145]. Le dispositif expérimental comprend deux cylindres dont l'un est xe et l'autre a un mouvement harmonique de grande amplitude. Une houle régulière est présente de direction l'axe centre à centre des cylindres en conguration initiale. L'approche hydrodynamique utilisée dans notre travail permet d'obtenir de très bons résultats numériques par rapport aux données expérimentales, que ça soit en terme d'eorts hydrodynamiques ou d'élévations de surface libre. Les interactions hydrodynamiques entre les diérents corps sont bien captées. Une validation de l'extension aux simulations multicorps est également faite en mouvement libre, par comparaison avec InWave couplé au logiciel Nemoh. Ce dernier est basé sur la théorie des écoulements potentiels linéarisés. Les résultats numériques des deux théories sont en accord. D'autres développements dans WS_CN sont eectués : l'implémentation d'un algorithme de remaillage de surface libre, la parallélisation du calcul des coecients d'inuence et la réduction de la quantité de mémoire utilisée via l'emploi d'un solveur de systèmes linéaires creux. D.5 Interaction uide-structure La théorie des écoulements potentiels basés sur l'hypothèse de faible perturbation et l'Algorithme aux Corps Rigides Composites sont couplés pour calculer l'interaction uide-structure d'un système multicorps ayant un mouvement relatif de grande amplitude dans une houle cambrée . Un couplage fort est choisi pour sa robustesse. L'équation de couplage, qui rassemble l'équation du mouvement du système multicorps et le deuxième problème aux éléments frontières, est établie et démontrée. Ces développements engendrent la création d'un nouveau logiciel nommé InWaveS_CN. Comme InWave est implémenté en C++ et WS_CN en Frotran, il est nécessaire d'assurer les communications entre les deux solveurs. An de garantir la modularité du logiciel, le code d'intégration est en Python. Une validation est eectuée par comparaison entre In-WaveS_CN et WS_CN dans un cas multicorps sans interaction mécanique. Ceci permet de prouver la justesse de la démonstration et de l'implémentation de l'équation de couplage. Le couplage fort est comparé à d'autres formes de couplages (faibles, explicite et implicite) pour en démontrer la pertinence. Le code de simulation de câbles, CableDyn est aussi couplé à InWaveS_CN en utilisant un couplage faible explicite. D.6 Développement d'un nouveau mailleur Le mailleur de WS_CN, implémenté avant le début de ce travail de doctorat [45], est basé sur une approche semi-analytique pour calculer la position de la courbe d'intersection entre le corps ottant et la surface libre incidente. Les maillages des corps et de la surface libre sont générés via une méthode d'avance de front. Un manque de robustesse de cette méthode est constaté, notamment pour mailler des géometries complexes. Une nouvelle stratégie de maillage est développée en supposant que le maillage du corps entier a été généré au préalable en utilisant un logiciel de maillage externe. Le maillage du corps entier est alors coupé au niveau de la surface libre incidente par un algorithme de découpe de maillages. Cette méthode déforme signicativement le maillage au niveau de la courbe d'intersection avec la surface libre. Pour améliorer la qualité du maillage du corps au niveau de la découpe, des algorithmes de fusion de facettes et de déformation de maillages sont utilisés. Puis, le maillage du corps est connecté au maillage de surface libre, généré par la méthode d'avance de front comme utilisée initialement. L'outil de découpe de maillage se nomme Meshmagick et est incorporé à InWaveS_CN en tant que nouveau module. Deux cas tests sont eectués an de valider la méthode par des comparaisons entre le mailleur initial et la nouvelle stratégie de maillage. Un très bon accord est obtenu, validant la nouvelle approche de génération de maillages pour WS_CN. D.7 Expériences An de valider le couplage entre InWave et WS_CN, implémenté dans InWaveS_CN, des essais en bassin ont été menés à l'École Centrale de Nantes. Ces expériences représentent une opération de redressement dans la houle d'un otteur d'éolienne de type spar (cylindre). Initialement, le cylindre est maintenu horizontal au moyen d'un câble, puis ce câble est déroulé grâce à un treuil entrainant la rotation autour d'un axe xe du cylindre, qui, nalement, atteint sa position d'équilibre verticale. Ces expériences sont pertinentes au regard du modèle développé dans ce travail de thèse car il y a un système mécanique articulé (le cylindre et l'axe xe), un câble, un treuil et un corps soumis à des eorts hydrodynamiques avec un mouvement libre de grande amplitude. InWaveS_CN est utilisé pour simuler numériquement ces expériences. Tous les modules le consituant sont sollicités : InWave pour la dynamique multicorps, CableDyn pour la dynamique du cable et du treuil, WS_CN pour l'hydrodynamique, Meshmagick pour la génération de maillages complexes. Cinq cas sont présentés : deux sans houle et avec diérentes lois de vitesses de déroulement du câble et trois en présence d'une houle incidente régulière avec diérents paramètres de houle. Les eets de la vitesse de déroulement, de la rampe sur cette dernière et des vagues sont bien capturés. Mais, quand une extrémité du cylindre traverse la surface de l'eau, des erreurs numériques importantes apparaîssent. Néanmoins, la comparaison entre les simulations numériques et les données expérimentales ore une première et prometteuse validation d'InWaveS_CN. Titre : Simulation numérique des opérations d'installation pour les fermes d'éoliennes offshore Mots-clés : Opérations marines, Câbles, Ecoulement potentiel, Weak-scatterer, Interaction fluide-structure Résumé : L'éolien offshore est l'énergie marine la plus avancée et utilisée dans le monde. Afin d'accroître l'énergie extraite du vent, les dimensions des éoliennes deviennent plus importantes et les parcs éoliens sont installées de plus en plus loin des côtes, où les mers sont plus agitées et les vents plus forts. De fait, les opérations marines sont plus complexes et plus chères et les fenêtres météo sont écourtées et se raréfient. Dans le cadre de cette thèse, un logiciel de simulation numérique des opérations marines est développé, en particulier pour des applications de descentes et de remontées de colis lourds. L'Algorithme aux Corps Rigides Composites, implémenté dans le logiciel InWave, est utilisé pour modéliser le système multicorps. Un modèle de câble et de treuil est développé, suivant la théorie multicorps utilisée, et comparé à la théorie câble classique dite « lumped mass ». Les efforts hydrodynamiques ainsi que les interactions hydrodynamiques sont modélisés par une théorie potentiel instationnaire satisfaisant l'hypothèse de faible perturbation, dite « weak-scatterer ». L'approche « weak-scatterer » du logiciel WS_CN est étendue aux simulations multi-flotteurs et validée par comparaison avec des données expérimentales. InWave et WS_CN sont couplés afin de résoudre l'interaction houle-structure pour des systèmes multicorps articulés en mer. Un couplage fort est adopté pour sa robustesse. L'équation de couplage est établie et validée via des comparaisons avec WS_CN. Le logiciel ainsi crée se nomme InWaveS_CN et utilise un code d'intégration en Python. Une nouvelle stratégie de maillage, basée sur un algorithme de découpe de maillages et une méthode par avance de front, est développée dans WS_CN. Enfin, des essais en bassin d'une opération de redressement ont été menés à l'ECN. La comparaison entre les simulations numériques et les données expérimentales offre une première et prometteuse validation d'InWaveS_CN. Title : Numerical simulation of installation operations for offshore wind farms Keywords: Marine operations, Multibody dynamics, Cable dynamics, Potential flow, Weak-scatterer, FSI Abstract: Offshore wind represents the most advanced and used marine energy in the world. To increase the wind power extraction, turbines grow in size and wind farms are installed further offshore in presence of rough seas and strong winds. Marine operations become more challenging and expensive, weather windows are shorter and less frequent.

  The superscript n denotes the n th time step. . . . . . . 175 IV.8 Initial mesh of the comparison test case between WS_CN and In-WaveS_CN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 IV.9 Comparison of time series of the motions of Cylinder 1 from numerical results of InWaveS_CN and WS_CN . . . . . . . . . . . . . . . . . . . 177 IV.10 Comparison of time series of the motions of Cylinder 2 from numerical results of InWaveS_CN and WS_CN . . . . . . . . . . . . . . . . . . . 177 IV.11 Comparison of time series of the motions of Cylinder 1 from numerical results of InWaveS_CN for a tight, loose explicit and implicit couplings 179 IV.12 Comparison of time series of the motions of Cylinder 2 from numerical results of InWaveS_CN for a tight, loose explicit and implicit couplings 179 IV.13 Comparison of the heave motion of Cylinder 1 with d 1 = 0.6 from numerical results of WS_CN and InWaveS_CN with a loose explicit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 IV.14 Tight-loose coupling between InWave (structure solver), WS_CN (uid solver) and CableDyn (cable solver). X, F and T represent the motion 1 Marine renewable energies Marine renewable energies (MRE ) are the renewable energies that can be extracted from the ocean. Their main use is electricity generation. They are expected to con-

	tribute to the mitigation of climate change. Marine renewable energies encompass
	various kinds of energies:
	Wave energy;
	Tidal energy;
	Oshore wind energy;
	Ocean thermal energy;

  21 developed by ANSYS; FryDoM 22 (Flexible and Rigid body Dynamic modeling for Marine operations) developed by D-ICE ENGINEERING and Ecole Centrale de Nantes;

	MOSES 23 (Multi Operational Structural Engineering Simulator) developed by
	Bentley Systems [42].

These software packages have the same features overall: Dynamical computations are multibody; Bodies are rigid with six degrees of freedom;

  the Composite Rigid-Body Algorithm or CRBA, are presented in details. The motion equation of the multibody systems is established.The second chapter is devoted to cable dynamics. A state of the art of the dierent cable modelling approaches is exposed. Two approaches are used to simulate cables, one is based on the CRBA while the second method uses the low-order lumped mass theory.

The latter is developed into a solver named CableDyn, validated from comparisons with experiments. Comparisons are done between the two approaches. A winch model is also presented to wind or unwind a cable by modifying the number of cable elements during the numerical simulations.

  Classication of multibody systems . . . . . . . . . . . . . . I.1.2 Dynamic approaches to multibody simulation . . . . . . . . I.1.3 Direct dynamics algorithms . . . . . . . . . . . . . . . . . . The forward loop . . . . . . . . . . . . . . . . . . . . . . . . I.4.2 The backward loop . . . . . . . . . . . . . . . . . . . . . . . I.4.3 Construction of H and C . . . . . . . . . . . . . . . . . . .

	Chapter I
	Multibody System Dynamics
	Contents
	The experimental device is
	presented, together with the measurements obtained. These results are used for compar-
	ison with the numerical results coming from the coupling between InWave, CableDyn,
	WS_CN and the new mesh generator.

I.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.1.1 I.1.3.1 The ABA and the CRBA . . . . . . . . . . . . . . I.1.3.2 The Augmented Formulation . . . . . . . . . . . . I.1.3.3 The Discrete Euler-Lagrange equation . . . . . . I.1.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . I.1.4 InWave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2 Modeling of multibody systems . . . . . . . . . . . . . . . . . . . I.2.1 Numbering of the bodies . . . . . . . . . . . . . . . . . . . . I.2.2 The modied Denavit-Hartenberg parameters . . . . . . . . I.3 Kinematic recursive equations . . . . . . . . . . . . . . . . . . . . I.4 The Composite-Rigid-Body Algorithm . . . . . . . . . . . . . . . I.4.1 I.5 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . I.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  2.2 The modied Denavit-Hartenberg parameters

InWave uses a reduced set of coordinates to locate each body of the multibody system. It is dierent from the classical Cartesian coordinates. It follows the geometric conguration created by Denavit and Hartenberg for open chain systems and extended to any multibody system (with or without closed-loops) by Khalil and Kleinnger [59].

  So far, local frames have been built except for the base. The base frame is directly dened with respect to the earth-xed inertial frame Σ e . Σ 0 is obtained after a translation of vector O e O 0 and, then, three rotations based on the denition of the Cardan angles. Their construction follows the next steps: Translation of vector O e O 0 = e P 0 to move Σ e from O e to O 0 ;

	e	e O a j O j	(I.14)

  Comparison . . . . . . . . . . . . . . . . . . . . . 77 II.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

		II.5.4.3
	Chapter II
	Cable Dynamics
	Contents This algorithm is based on few hypotheses: multibody systems are limited to kinematic II.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . trees of rigid bodies linked by one-dof joints. The CRBA unfolds in three steps which II.1.1 Mathematical modellings . . . . . . . . . . . . . . . . . . . lead to the construction and the solving of the motion equation for the whole multibody II.1.1.1 Quasi-static approach . . . . . . . . . . . . . . . . system. The numerical tool InWave relies on this algorithm and is used in this PhD to perform time-domain simulations of marine operations. The next chapter focus on cable II.1.1.2 Force-displacement-velocity model . . . . . . . . .
	dynamics, presents the application of the CRBA to model cables and its comparison to II.1.1.3 Lumped mass models . . . . . . . . . . . . . . . .
	a classical cable dynamic theory. II.1.1.4 Finite-element method . . . . . . . . . . . . . . .
		II.1.1.5 Finite-dierence method . . . . . . . . . . . . . .
		II.1.1.6 Multibody approach . . . . . . . . . . . . . . . .
		II.1.2 Time integration and numerical damping . . . . . . . . . .
		II.1.3 Cable software packages . . . . . . . . . . . . . . . . . . . .
	II.2	The low-order lumped mass theory . . . . . . . . . . . . . . . . .
		II.2.1 The need of an in-house cable solver . . . . . . . . . . . . .
		II.2.2 Theoretical developments . . . . . . . . . . . . . . . . . . .
		II.2.3 State vector and time integration . . . . . . . . . . . . . . .
	II.3	II.2.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cable dynamics using the CRBA . . . . . . . . . . . . . . . . . .
		II.3.1 Theoretical developments . . . . . . . . . . . . . . . . . . .
	II.4	II.3.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cable joints in the CRBA . . . . . . . . . . . . . . . . . . . . . .
		II.4.1 Theoretical developments . . . . . . . . . . . . . . . . . . .
		II.4.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	II.5	Winch modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
		II.5.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . .
		II.5.2 Thrust force and cable loads . . . . . . . . . . . . . . . . . .
		II.5.3 Cable deployment . . . . . . . . . . . . . . . . . . . . . . . .
		II.5.3.1 Low-order lumped mass model . . . . . . . . . . .
		II.5.3.2 Multibody model . . . . . . . . . . . . . . . . . .
		II.5.3.3 Comparison . . . . . . . . . . . . . . . . . . . . .
		II.5.4 Cable retrieval . . . . . . . . . . . . . . . . . . . . . . . . .
		II.5.4.1 Low-order lumped mass model . . . . . . . . . . .
		II.5.4.2 Multibody model . . . . . . . . . . . . . . . . . .

Table II .

 II 1 Cable theory and time-stepper in dierent software packages MAP++ 3 models mooring lines based on a multi-segmented quasi-static approach. It handles seabed contact and friction. It is released as an open-source library under the Apache License by NREL 4 and can be run with Python, C, C++ or Fortran programs.

  1000 s 566 + 0.566(t -1000) + 0.005575(t -1000) 2 if t > 1000 s and t 1060 s 620.03 + 1.235(t -1060)

if t > 1060 s (II.

[START_REF] Rongère | Systematic dynamic modeling and simulation of multibody oshore structures: application to wave energy converters[END_REF]

  The mean CPU-time over 10 runs of the pendulum test case for each approach is presented in TableII.4. The ABA used here has a single-dof formulation. Obviously, the implementation of cable joints could also be achieved in this direct dynamics TableII.4 CPU-time of dierent cable dynamics theories applied to the pendulum test caseThe use of cable joints in the CRBA involves a time reduction of 49.2 % compared to the use of single-dof joints. The forward loop is not aected by this reduction because every elementary quantity (velocities, Coriolis accelerations, external loads, etc.) always has to be evaluated for each body. The backward loop, which now iterates the number of

	algorithm.
	ABA Single-dof CRBA Cable joint CRBA LM
	160.3 s	175.1 s	89.0 s	35.2 s

cable elements instead of the number of bodies, and the construction of the generalized mass matrix H and the load vector C, previously coecient-by-coecient and now by sets of 3 × 3 matrices for H and 3 × 1 vectors for C, are at the origin of the CPU-time reduction.

  III.1.2.3 Body-exact models . . . . . . . . . . . . . . . . . III.1.2.4 Linear models . . . . . . . . . . . . . . . . . . . . III.1.2.5 Selected model . . . . . . . . . . . . . . . . . . . III.1.3 State of the art of the potential ow theory based on the The boundary integral equation . . . . . . . . . . III.2.4.2 The inuence coecients . . . . . . . . . . . . . . III.2.4.3 The linear system . . . . . . . . . . . . . . . . . . III.2.5 Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.2.6 Fluid-structure interaction . . . . . . . . . . . . . . . . . . . III.2.6.1 State of the art of the computation of the timedierentiation of the velocity potential . . . . . . III.2.6.2 The implicit boundary method . . . . . . . . . . III.2.6.3 The discretization . . . . . . . . . . . . . . . . . . III.2.6.4 The linear system . . . . . . . . . . . . . . . . . . Initial point . . . . . . . . . . . . . . . . . . . . . III.3.1.2 Marching method . . . . . . . . . . . . . . . . . . III.3.2 Grid generation . . . . . . . . . . . . . . . . . . . . . . . . . III.3.3 Reference panel size . . . . . . . . . . . . . . . . . . . . . . III.4 Mesh morphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.4.1 Body mesh morphing . . . . . . . . . . . . . . . . . . . . . . III.4.2 Free surface mesh morphing . . . . . . . . . . . . . . . . . . III.5 Incident wave models . . . . . . . . . . . . . . . . . . . . . . . . . III.6 Absorbing numerical beach . . . . . . . . . . . . . . . . . . . . . . III.7 State vector and time integration . . . . . . . . . . . . . . . . . . III.8 Free surface remeshing . . . . . . . . . . . . . . . . . . . . . . . . III.9 Gaussian lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.10 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.11 Extension to multibody simulations . . . . . . . . . . . . . . . . . Inuence of the body nonlinearities . . . . . . . . III.11.2 Free motion . . . . . . . . . . . . . . . . . . . . . . . . . . . III.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	Chapter III
	Hydrodynamics
	Contents

The next chapter presents the hydrodynamic theory used in this PhD work. III.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.1.1 Potential ow theory . . . . . . . . . . . . . . . . . . . . . . III.1.1.1 Laplace's equation . . . . . . . . . . . . . . . . . III.1.1.2 Bernoulli's equation . . . . . . . . . . . . . . . . . III.1.1.3 Boundary conditions . . . . . . . . . . . . . . . . III.1.1.4 Hydrodynamic loads . . . . . . . . . . . . . . . . III.1.2 Numerical simulation of potential ows . . . . . . . . . . . . III.1.2.1 Fully nonlinear models . . . . . . . . . . . . . . . III.1.2.2 Weakly nonlinear models based on the weak-scatterer hypothesis . . . . . . . . . . . . . . . . . . . . . . weak-scatterer hypothesis . . . . . . . . . . . . . . . . . . . III.2 Theory of the potential ow based on the weak-scatterer hypothesis III.2.1 Assumptions and governing equations . . . . . . . . . . . . III.2.2 Arbitrary Lagrangian-Eulerian description . . . . . . . . . . III.2.3 The boundary value problem . . . . . . . . . . . . . . . . . III.2.4 The boundary element method . . . . . . . . . . . . . . . . III.2.4.1 III.3 Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 III.3.1 Intersection curve tracking . . . . . . . . . . . . . . . . . . . III.3.1.1 III.11.1 Forced motion . . . . . . . . . . . . . . . . . . . . . . . . . . III.11.1.1 Presentation of the test cases . . . . . . . . . . . III.11.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . III.11.1.3 Impact of the weak-scatterer hypothesis and inuence of the free surface nonlinearities . . . . . . . III.11.1.4

  y, t) for HS and FK;

	z = 0	S B (0)
	z = 0	S B (0)
	z = 0 otherwise.	S B (t) for HS and FK;	S B (0) otherwise.
		S B (t)	O(1 day)
		S B (t)	O(Days)
		S B (t)	O(1 week)
		Mesh body surface	Time

Table III

 III

.1 Main dierences between several potential ow theories. NL: Fully nonlinear, WSC: Weakly nonlinear based on the weak-scatterer hypothesis, BE: Body exact approximation, NFK: Nonlinear Froude-Krylov, LTD: Linear time-domain, FD: Frequency-domain. A m denotes the body motion amplitude. HS and FK represent the hydrostatic and Froude-Krylov loads.

  Series 60 and a trawler between numerical results and experimental data in small amplitude and steep regular waves. A good agreement was observed which conrmed the validity of the hypothesis. Only few research studies have been published so far. Lin et al.[START_REF] Lin | Large amplitude motions and wave loads for ship design[END_REF] developed a time-domain solver based on the weak-scatterer hypothesis, called LAMP-4. This numerical tool is part of the program LAMP (Large Amplitude Motions Program) for the computation of the motions and loads of a ship operating in extreme sea conditions. It exists four versions of the solver LAMP[START_REF] Lin | Recent hydrodynamic and tool development and validation for and motions and slam and loads on oceangoing and high-speed vessels[END_REF]: , short of Ship Wave ANanlysis, developed at the Massachusetts Institute of

	Technology. It includes:
	SWAN-1 : a linear frequency-domain code;
	SWAN-2 : a linear time-domain code with linear or nonlinear Froude-Krylov loads;
	LAMP-1 follows a linear approach;

Large body motions;

Steep waves.

If the weak-scatterer hypothesis is not fullled

[START_REF] Pawlowski | On the application of the weak-scatterer hypothesis to the prediction of ship motions in heavy seas[END_REF]

: [. . . ] this theory still gives a non-linear time domain formulation of the scattering problem, albeit not consistent in its non-linear part. In other words, the applicability of the weak scatterer hypothesis is necessary to construct a consistent non-linear scattering theory in which the linear free surface condition is used. This method was applied by Pawlowski in

[START_REF] Pawlowski | A theoretical and numerical model of ship motions in heavy seas[END_REF][START_REF] Pawlowski | On the application of the weak-scatterer hypothesis to the prediction of ship motions in heavy seas[END_REF][START_REF] Pawlowski | A nonlinear theory of ship motion in waves[END_REF]

. Comparisons were made for a LAMP-2 is based on a nonlinear Froude-Krylov model; LAMP-3 is similar to LAMP-2 but allows large lateral displacements of the ship; LAMP-4 applies the weak-scatterer method. SWAN

  the uid domain D • ∇η I -∇φ I • ∇η P -∇φ P • ∇η I -∇φ P • ∇η P Solid • n on S B (t) and the seabed

	∂η P ∂t -∇φ I + = -∂η I ∂t ∂φ I ∂z + ∂φ P ∂z at z = η(x, y, t)
	∂φ P ∂t	= -	∂φ I ∂t	-	1 2	∇φ I • ∇φ I -∇φ I • ∇φ P -	1 2	∇φ P • ∇φ P -g(η I + η P ) at z = η(x, y, t)
	∂φ P ∂n + v ∂φ P = -∂φ I ∂n ∂n = 0 on the numerical tank walls

Table III .

 III 2 presents the CPU-time for solving (III.125) in the present test case by considering the sparsity of the linear system (Sparse ) or not (Dense ). As it can be seen, not only the use of PARDISO allows to deal with larger linear systems but also it is quite faster. As a reminder, a mesh counts roughly between 5000 and 15 000 nodes, giving the size of the linear system to solve for the rst boundary value problem. Hence, the computation of the δ ref with a ne Cartesian grid can easily lead to larger linear systems to solve than the boundary element methods.

	h ref (m)	N ref	Density (%)	CPU time (s) Dense Sparse
	0.6	100	2.92	0.001	0.094
	0.3	400	9.45 × 10 -1 0.004	0.094
	0.1	3600	1.29 × 10 -1 0.525	0.125
	0.07	7225	6.59 × 10 -2 2.494	0.140
	0.06	10 000	4.80 × 10 -2 5.785	0.187
	0.05	14 400	3.36 × 10 -2	ø	0.234
	0.01	360 000	1.38 × 10 -3	ø	4.227
	0.005	1 440 000 3.46 × 10 -6	ø	ø

Table III

 III

.2 CPU-times for dierent sizes of linear system using or not its sparsity

The density of a linear system is the number of non-zero coecients in A divided by the total number of coecients, i.e. N 2 ref . ø indicates that the calculation was not possible because of the lack of memory on the local computer used during this PhD work.

  Table III.3 Characteristics of the two cylinders Three wave probes (WP1, WP2, WP3 ) are also used. Their position is given in Table III.4.

  Table III.6 presents the twelve test cases. Regular wave characteristics. λ and A represent the wave length and the wave amplitude.

	Reg 1	6.400	1.506	0.0115
	Reg 2	6.800	1.330	0.0100
	Reg 3	7.000	1.259	0.0095
	Reg 4	7.200	1.190	0.0090
	Table III.5 Case	Wave	ω pm (rad/s) A pm (m)
	1	Reg 1		0.427	0.37
	2	Reg 1		0.213	0.37
	3	Reg 1		0.107	0.37
	4	Reg 2		0.453	0.37
	5	Reg 2		0.227	0.37
	6	Reg 2		0.113	0.37
	7	Reg 3		0.467	0.37
	8	Reg 3		0.233	0.37
	9	Reg 3		0.117	0.37
	10	Reg 4		0.480	0.37
	11	Reg 4		0.240	0.37
	12	Reg 4		0.120	0.37
	Table III.6 Characteristics of the twelve test cases

III.11.1.2 Results

A cylindrical domain is used for the simulations, with a radius of two wave lengths (2λ). One wave length is used for the numerical absorbing beach. Bodies are meshed ID ω (rad/s) λ (m) A (m) at real scale. The ramp function (III.151) is applied for a wave period from the start of the simulation.

The mesh convergence for Case 1 is presented in Figures III.

[START_REF] Malenica | Some aspects of seakeeping of the oating body with attached pendulum[END_REF]

0

  Table III.7 mDH parameters for the free motion test case in InWave-Nemoh The mesh convergence using WS_CN is shown in Figure III.49 while the time step

convergence for a mesh of 15 000 panels is presented in Figure III.50. From these results, a mesh of 15 000 panels with a time step of 0.005 s are selected in WS_CN.

Table IV .

 IV .1.The number of boundary value problems to solve for computing the time-dierentiation of the velocity potential is given for each method. The boundary value problem for solving the velocity potential is not included. 1 Equivalence of the terms to dene the uid-structure couplings used in the potential ow theory and in the multiphysics coupling theory. N BIE denotes the number of boundary integral equations to solve at each time step to compute

	Potential ow theory	Multiphysics coupling theory	N BIE
	Backward nite dierence	Loose explicit	0
	Iterative	Loose implicit	1 + i
	Mode decomposition	Monolithic	7
	Indirect	Monolithic	6
	Implicit boundary	Monolithic	1
	∂φ
	∂t

Table V .

 V 1 Geometric criteria to apply the panel merging

where:

Table V .

 V 2 shows the distribution of each step of the new mesh generator amongst the modules and the languages.

	Module	Meshmagick InWaveS_CN WS_CN
	Language	Python	Python	Fortran
	Mesh clipping	X
	Panel merging	X
	Multiple node tracking			X
	Spring analogy method			X
	Intersection curve tracking		X	X
	Free surface mesh generation			X
	Final mesh			X

Table V .

 V 2 Distribution of the steps of the new mesh generator amongst the modules and the languagesV.6 Proling and CPU time comparisonThe CPU time used by the initial mesh generators is presented in TableV.3, while the results for the new mesh generator are displayed in TableV.4. The same test case is run to obtain these results and leads to a mesh of roughly 9500 panels with a single surface-piercing body. As expected, the time necessary to create the free surface and numerical walls meshes is almost constant with the two methods. The time to track the intersection curve is always small. The use of the new mesh generator involves an increase of 62 % of the CPU time. Nevertheless, it stays in the same order of magnitude and small compared to the time of a whole WS_CN simulation.Table V.3 CPU time for every main task of the initial mesh generatorThe proling of the two mesh generators is shown in Figure V.24. 40 % of CPU time to create a whole mesh based on the new mesh strategy comes from Meshmagick, that is to say the Python scripts (in blue on the pie chart). As Python is slower than Fortran, the time required for the mesh clipping and the panel merging could be signicantly reduced by a translation of the Python scripts into Fortran.

	V.6. Proling and CPU time comparison
	Task	CPU time (s)
	Mesh clipping		0.696
	Panel merging		0.558
	Mesh optimization		0.621
	Intersection curve tracking		0.009
	Free surface and numerical walls meshes	1.270
	Total		3.154
	Task	CPU time (s)
	Intersection curve tracking	0.003
	Body meshes	0.771
	Free surface and numerical walls meshes	1.170
	Total	1.944

Table V .

 V 4 CPU time for every main task of the new mesh generator

Table VI .

 VI regular waves were used as WS_CN cannot deal with irregular waves without adding extra developments. The test matrix is given by Table VI.3. Cases 1, 8 and 15 represent three tests without waves. Each test lasts less than one minute. 3 Test matrix of the experimental tests. The heading angle is the angle between the buoy vertical axis and the wave direction.

	Case	Wave amplitude (m)	Wave period (s)	Heading angle ( • )	Lowering velocity (m/s)
	1	0	ø	90	0.067
	2	0.01	0.7	90	0.067
	3	0.01	1	90	0.067
	4	0.01	1.5	90	0.067
	5	0.02	0.7	90	0.067
	6	0.02	1	90	0.067
	7	0.02	1.5	90	0.067
	8	0	ø	0	0.067
	9	0.01	0.7	0	0.067
	10	0.01	1	0	0.067
	11	0.01	1.5	0	0.067
	12	0.02	0.7	0	0.067
	13	0.02	1	0	0.067
	14	0.02	1.5	0	0.067
	15	0	ø	0	0.033
	16	0.01	0.7	0	0.033
	17	0.01	1	0	0.033
	18	0.01	1.5	0	0.033
	19	0.02	0.7	0	0.033
	20	0.02	1	0	0.033
	21	0.02	1.5	0	0.033

0

  Table VI.6 mDH parameters for Case 17

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Laboratoire de recherche en Hydrodynamique, Énergétique et Environnement Atmosphèrique

A constant-time-step second order Runge-Kutta integration scheme.

For the implicit scheme only, an explicit integration scheme based on a semi-implicit Euler method is also available.

https://map-plus-plus.readthedocs.io/en/latest

https://www.nrel.gov

http://www.matt-hall.ca/moordyn.html

L u is time-varying due to the elasticity of the cable, q

denoting the real length of the cable element and not its

https://lheea.ec-nantes.fr/logiciels-et-brevets/nemoh-presentation-192863.kjsp

https://www.wamit.com

http://www.principia-group.com/blog/product/diodore

https://www.veristar.com/portal/veristarinfo/detail/software/ SeakeepingandMooringAnalysis/HYDROSTAR/Hydros

https://www.ansys.com/fr-fr/products/structures/ansys-aqwa

https://www.pardiso-project.org

* denotes a zero.

https://www.openmp.org

https://www.python.org

http://cython.org

https://docs.scipy.org/doc/numpy/f2py

http://www.sph-flow.com

https://www.code-aster.org

https://www.mbdyn.org

https://lheea.ec-nantes.fr/software-and-patents/isis-cfd-193387.kjsp

http://gts.sourceforge.net

Compliance with the functional specication and conclusions . . . Robustness: the new mesh generator has to be able to deal with any oating body with a single intersection curve with the free surface and for any wave model;

Modularity: the new mesh generator has to retain the modularity of the coupling between InWave and WS_CN ;

User-friendliness: the new mesh generator does not have to require any extra coding when a new geometry is used. This also means the solver WS_CN may be run as a black box by a user who is not a developer.

Memory allocation: the new mesh generator does not have to suer of memory leaks;

Process time: the new mesh generator does not have to involve an important increase of the CPU-time compared to the initial mesh generator.

V.3 State of the art

A mesh generation algorithm needs to know the shape of the bodies to compute their intersection with waves. This is achieved using an interface tracking method. The body surfaces can be modeled using several formats [START_REF] Ko | A survey: application of geometric modeling techniques to ship modeling and design[END_REF]:

Analytical expressions of the parametric surfaces when the geometries are simple (cylinder, sphere, cube, axisymmetric geometry, etc.); B-splines [START_REF] Ko | Development of panel generation system for seakeeping analysis[END_REF][START_REF] Park | Hydrodynamic hull form design using an optimization technique[END_REF]; NURBS (Non-uniform rational basis splines), as it is commonly done in the computer-aided design tools [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF].

Several methods exist to compute the intersection curve between a body and the free surface, that is to say a surface-to-surface intersection [START_REF] Chauvigné | Tenue à la mer d'un otteur animé de grands mouvements pour les Energies Marines Renouvelables[END_REF][START_REF] Gilloteaux | Mouvements de grande amplitude d'un corps ottant en uide parfait[END_REF][START_REF] Ko | A survey: application of geometric modeling techniques to ship modeling and design[END_REF][START_REF] Foufou | Surface / surface intersections: a three states classication[END_REF]:

The analytical methods where the exact solution is seeked;

The lattice methods where the surface-to-surface intersection is transformed into a curve-to-surface intersection;

The subdivision methods where the problem is decomposed recursively into subproblems easier to solve;

The marching methods, which start from a known starting point on the intersection curve and then step along it in a direction prescribed by the curve local geometry.

Once the intersection curve is obtained, the mesh generation follows. Several algorithms are available, for instance:

The quadtree method where the domain is enclosed into a bounding box which is recursively subdivided into smaller panels by taking into account the presence of the bodies. The positioning of the nodes is not optimal but the method is simple and robust.

The advance front method where the boundaries (including the intersection curves) are meshed and form the initial front of the mesh. Then, this front is updated by creating new nodes and panels until all the surfaces are covered. The panels have a good shape but the convergence is not always ensured.

The Delaunay triangulation method creates panels from a given set of nodes such as no point is present in the circumcircle of any triangular panel. This method requires in a rst step the creation of the nodes. This approach may be time consuming.

In WS_CN, as explained in subsection III.3.1 and subsection III.3.2, a marching method and an advance front method are used, respectively, for the intersection curve tracking and the mesh generation.

The main drawback of this approach (Parametric curves -Intersection curves -Mesh generation) is the mandatory knowledge of a parametric representation of the bodies.

In most hydrodynamic solvers (StarCCM+ 1 , OpenFOAM 2 , FINE/Marine 3 , Nemoh, WAMIT, etc.), the mesh of the bodies is an input of the numerical tool, using an external mesh generator (BlockMesh 4 , HEXPRESS 5 , Gmsh 6 , Rhinoceros 3D 7 , SALOME 8 , etc.). Then, the mesh is created by the solver. So, another approach is possible if the mesh of the bodies is known initially. The initial mesh is cut at the known free surface and then the the free surface mesh is generated from the intersection curves. This approach is called the panel cutting method and has been implemented at the end of this PhD.

So far, the panel cutting method has been used in two cases:

The computation of the nonlinear hydrostatic and Froude-Krylov loads;

The solving of the steady nonlinear wave resistance problem.

Examples of the computation of the nonlinear hydrostatic and Froude-Krylov loads based on a panel cutting method may be found in [START_REF] Ham | Development and validation of a simulation-based safety evaluation program for a mega oating crane[END_REF][START_REF] Rodrigues | Exact pressure integration on submerged bodies in waves using a quadtree adaptative mesh algorithm[END_REF][START_REF] Rodrigues | Froude-Krylov forces from exact pressure integrations on adaptive panel meshes in a time domain partially nonlinear model for ship motions[END_REF]. The original mesh is recursively subdivided on the intersection curve using a quadtree process. Then the adjacent underwater subpanels are agglomerated to form bigger panels and coarse meshes. Horel et al. [START_REF] Horel | A method of immersed surface capture for broaching application[END_REF] and Sengupta et al. [START_REF] Sengupta | A simplied approach for computation of nonlinear ship loads and motions using a 3D time-domain panel method[END_REF] applied directly the panel cutting method by clipping the panels of the initial mesh to t the incoming waves. Lee and Lee [START_REF] Lee | Nonlinear hydrostatic analysis of exible oating structures[END_REF] used the panel cutting method in case of hydrostatic calculations with exible structures and non-matching meshes.

Regarding the solution of the steady nonlinear wave resistance problem, this was achieved by Choi et al. [START_REF] Choi | Panel cutting method: new approach to generate panels on a hull in Rankine source potential approximation[END_REF]. The panels were cut at the real wave elevation and then the mesh generated was used to perform a hydrodynamic computation and not only the hydrostatic and the Froude-Krylov calculations. Nevertheless, due to the steadiness of the problem, an iterative method was used to compute the velocity potential and The panel merging method reduces the density of nodes and panels close to the intersection curve but the node positions and the panel shapes are still not good enough. To improve them, an unstructured mesh smoothing algorithm based on the spring analogy is used. As explained in subsection III.4.1, it is based on a vertex spring analogy.

The expression of the spring loads is:

As we want the small panels to get larger and the large panels to get smaller, all the panels have to be able to be deformed. So, the stiness is taken constant. Its value has no inuence so it is chosen as unity:

The same notations as in subsection III.4.1 are used. The position is decomposed into three components:

To ensure the nodes stay on the body surfaces, the normal displacement is zeroed:

On the intersection curve or at the sharp edges in the body mesh, a node on this intersection has to remain on it. Thus, n i is normal to one of the surfaces, u i is along the intersection line and v i is chosen such as the local basis is orthonormal and direct.

In that case, the displacement along v i is zeroed:

If three surfaces intersect each other at the same node i, then the displacement along u i is also zeroed:

In case of a node on a smooth surface, the nal system of equations to solve yields:

And after some mathematical developments and by including the condition on the normal displacement, the system of three equations is reduced to a bi-dimensional problem:

is a constant, arbitrary dened as equal to cos(20 • ). Consequently, an angle of 20

• between two neighboring normal vectors leads to a multiple node.

Then, the order of multiplicity of the node needs to be gured out. It is necessary to know if the node is double, triple, quadruple, etc. The following rule is applied:

, then the node is at least triple, otherwise it is double;

• If it exists a neighboring normal vector x such as

, then the node is at least quadruple, otherwise it is triple;

• Etc.

Once a multiple node is found with its order of multiplicity, new nodes are created at the same position. Therefore, each multiple node is made of several elementary nodes: two for a double node, three for a triple node, four for a quadruple node, etc. The table of connectivities needs to be updated to associate each elementary node to a surface. Each neighboring panel of normal vector n is connected to the correct elementary node from the following rule (here for a quadruple node) using u, v and w previously dened:

, then the neighboring panel belongs to the same surface as u;

, then the neighboring panel belongs to the same surface as v;

, then the neighboring panel belongs to the same surface as w;

• Otherwise, n belongs to the fourth surface. The size of the basin makes it suitable for short waves. The basin is lled of freshwater so the water density is 1000 kg/m 3 .

Figure VI.2 Shallow water basin of Ecole Centrale de Nantes

The spar is a cylindrical buoy (illustrated in Figures VI. [START_REF] Gao | A summary of the recent work at ntnu on marine operations related to installation of oshore wind turbines[END_REF] Regarding the cable tension, the initial tension when the buoy is horizontal is 122 N.

When the oating body is lowered into the water, this value decreases due to the increase of the hydrostatic loads. Once the vertical equilibrium position of the cylinder is reached, the cable is slack and so the tension is zero. T Buoy the cable tension;

K the keel position;

G the position of the center of gravity;

B the buoyancy center position;

F the fairlead position.

Meshmagick input coecients are:

Conclusion and perspectives

The goal of our PhD work was the numerical simulation of marine operations, in particular of lowering and lifting operations. Such simulations require models of multibody dynamics, cable dynamics, hydrodynamics and uid-structure interaction. This analysis has driven our project.

A multibody theory has been handled to simulate mechanical interactions in multibody systems. No kinematic loops were considered. Bodies were rigid, with six degrees of freedom, separated one from each other by a single-degree-of-freedom joint and located using relative coordinates (modied Denavit-Hartenberg parameters), except the base of the multibody system which was tracked with Cartesian coordinates. The direct dynamic algorithm was the Composite-Rigid-Body Algorithm (CRBA). Its implementation, in the numerical tool InWave, was done prior to the beginning of this PhD work.

The CRBA has been used to simulate cable dynamics. Cable elements were modelled using three joints (two revolute joints and one prismatic joint). The internal loads in the prismatic joint was modeled using a spring-damper system for simulating the stretching of the cable. No bending nor torsion were considered. This multibody model has been validated by comparison with the classical low-order lumped mass theory. This latter theory is based on the discretization of a cable into lumped masses linked to each other by a spring-damper system. The lumped mass theory has been implemented in a numerical tool, CableDyn and validated with the numerical and experimental data given by Buckham [START_REF] Buckham | Dynamics Modelling of Low-Tension Tethers for Submerged Remotely Operated Vehicles[END_REF]. In order to speed-up the multibody solver when cables were simulated, a cable joint has been elaborated and the CRBA updated in that way. A reduction of 49.2 % of the CPU time was observed.

A winch model has been developed to simulate the lowering or the lifting of a payload. The approach was based on the adding or deletion of cable elements during the simulation. The cable element linked to the winch became variable-unstretched-length.

Both the multibody and the lumped mass approaches provided the same results after comparisons.

In the literature, marine operations are mainly modelled using the linear potential ow theory for the computation of hydrodynamic loads. This theory assumes small amplitude motions of the bodies around their mean position and small steepness waves.

In case of large (relative) amplitude motions and/or steep waves, this theory is not valid anymore. In contrast, the originality of this PhD thesis has been the use of an unsteady potential ow theory based on the weak-scatterer hypothesis to simulate marine operations. The velocity potential and the wave elevation were split into a known incident component and an unknown perturbed component. The perturbed quantities were assumed small in comparison to the incident ones. The free surface Appendix A

Multibody equations

This appendix presents the demonstration of two equations used in chapter I.

A.1 Kinematic recursive equation for accelerations

The goal of this section is to prove (I.48).

A.1.1 Linear acceleration

The time-dierentiation of (I.24) gives:

By using (I.25):

And also:

Finally:

+ 2σ j qj e Ω(Σ i /Σ e ) × e z j + σ j qj e z j (A.9)

And by projecting in the local frame of body j:

Angular acceleration

The time-dierentiation of (I.27) gives:

After using (I.28):

e Ω(Σ j /Σ e ) = e Ω(Σ i /Σ e ) + σj qj e Ω(Σ i /Σ e ) × e z j + σj qj e z j (A.14)

By projecting in the local frame of body j:

A.1.3 Final results

Equations (A.10) and (A.15) lead to:

A.2. Time-dierentiation of the linear and angular momenta with:

A.2 Time-dierentiation of the linear and angular momenta

The goal of this section is to prove (I.65). By denition, the time-dierentiation of the linear and angular momenta at O j and expressed in Σ e is:

where e δ(O j , Σ j /Σ e ) is the time-derivative of the angular momentum at O j .

A.2.1 Linear momentum

The transport of the linear momentum at O j gives:

But each body is rigid, consequently:

So:

Angular momentum

The transport of the angular momentum at G j gives:

e δ(O j , Σ j /Σ e ) = e δ(G j , Σ j /Σ e ) + m j e v(G j , Σ j /Σ e ) × e G j O j e Ω(Σ j /Σ e ) (A.29)

But each body is rigid, consequently:

And using (I.51):

e δ(G j , Σ j /Σ e ) = e I Gj e Ω(Σ j /Σ e ) + e Ω(Σ j /Σ e ) × e I Gj e Ω(Σ j /Σ e )

(A.31)

The transport of the inertia matrix from G j to O j , using the Huygens theorem gives:

e I Oj = e I Gj -m j S( e Ω(Σ j /Σ e ) )S( e Ω(Σ j /Σ e ) )

(A.32)

So:

e δ(O j , Σ j /Σ e ) = e I Oj e Ω(Σ j /Σ e ) + m j S( e O j G j ) e v(O j , Σ j /Σ e ) + S( e Ω(Σ j /Σ e ) ) e I Oj e Ω(Σ j /Σ e ) (A.33)

A.2.3 Final results

After projecting in the local frame of body j, Equations (A.25) and (A.33) lead to:

Vj + m j S( j ω j )S( j ω j ) j S j S( j ω j ) j I Oj j ω j (A. [START_REF] Zhengru Ren | Active tugger line force control for single blade installation[END_REF] with:

j M j = m j I 3 -m j S( j S j ) m j S( j S j ) j I Oj A and B are square matrices of size 6 × N b while f Exc is a complex vector of size 6 × N b , with N b the number of bodies.

From the hydrodynamic database, the time-domain hydrodynamic loads are computed:

The excitation force which represents the sum of the Froude-Krylov loads and the diraction loads:

where η I denotes the incident wave elevation and K Exc is the impulse response function vector of the complex excitation force vectors:

The radiation force:

where X is the Cartesian position of the bodies, A(∞) represents the added mass matrix at innity frequency and K R is the impulse response function matrix of the radiation force:

Finally the motion equation of a body or Cummin's equation arises [START_REF] Cummins | The impulse response function and ship motions[END_REF]:

with M is the mass matrix and C denotes the hydrostatic stiness matrix.

Other methods exist to compute the convolution integral of Cummins' equation, such as Prony's method [START_REF] Armesto | Comparative analysis of the methods to compute the radiation term in Cummins' equation[END_REF]. La simulation numérique des opérations de descente et de remontée de colis a été intensément étudiée dans la littérature. Ces travaux peuvent être classés en trois catégories : les analyses mécaniques, hydrodynamiques, et de contrôle des mouvements.

Une opération de descente nécessite de simuler un système multicorps articulé. Le navire, les câbles et le colis sont en interaction mécanique. Les premiers modèles mécaniques étaient basés sur des approches simpliées : découplage de la dynamique du colis de celle du navire [START_REF] Elling | Dynamic loading of shipboard cranes[END_REF], linéarisation des équations du mouvement [START_REF] Schellin | Crane ship response to wave groups[END_REF] ou problème bidimensionnel [START_REF] Ellermann | Nonlinear dynamics of oating cranes[END_REF]. Ensuite, des modèles tridimensionnels, non-linéaires et résolvant l'interaction mécanique entre les corps apparurent [START_REF] Schellin | Response analysis and operating limits of crane ships[END_REF][START_REF] Van Den | Computer analysis of heavy lift operations[END_REF][START_REF] Witz | Parametric excitation of crane loads in moderate sea states[END_REF]. Ces études avaient pour point commun de maintenir le colis en l'air, de sorte que seul le navire subissait des eorts hydrodynamiques. An d'écrire et de résoudre automatiquement les équations de mouvement, des algorithmes de mécanique multicorps furent utilisés, basées sur les équations de Newton-Euler [START_REF] Ku | Dynamic response analysis of heavy load lifting operation in shipyard using multi-cranes[END_REF] ou de Euler-Lagrange [START_REF] Ham | Multibody dynamic analysis of a heavy load suspended by a oating crane with constraint-based wire rope[END_REF].

Concernant le modèle hydrodynamique d'une opération de descente de colis, la plupart des travaux utilisèrent la théorie des écoulements potentiels linéarisés dans le domaine fréquentiel [START_REF] Ku | Dynamic response simulation of an oshore wind turbine suspended by a oating crane[END_REF][START_REF] Schellin | Response analysis and operating limits of crane ships[END_REF][START_REF] Van Den | Computer analysis of heavy lift operations[END_REF][START_REF] Cha | Dynamic response simulation of heavy cargo suspended by a oating crane based on multibody system dynamics[END_REF]. Cette théorie suppose que les corps ont un mouvement