The complexity of the dynamic behaviour of offshore marine structures requires advanced simulations tools for the accurate assessment of the seakeeping behaviour of these devices. The aim of this work is to present a time-domain model for solving the dynamics of floating marine devices, subjected to non-linear environmental loads and paying special attention on the mooring dynamics. First, the formulation of the hydrodynamic approach for solving the wave-floater interaction is introduced. Second, the solver of the mooring dynamics, based on a non-linear Finite Element Method approach, is presented. Third, a procedure for coupling the hydrodynamic along with other external loads, with the floating structure and mooring dynamics is described. Fourth, some validation examples and comparisons among different mooring approaches are presented. Fifth, an analysis of the OC3 floating wind turbine concept is performed to study the influence of different mooring models, the effects of non-linear waves on the platform, and the tension in the mooring system. The dynamic mooring model along with the second-order wave model produce realistic simulations of the floating wind turbine performance.
Research trends in Marine Renewable Energies (MRE) are mainly focused in offshore wind energy due to the high expectations raised by this technology. The technology for marine wind turbines is currently well-developed, but limited for fixed installations in shallow-water areas. The next horizon is focused on deep-water technology , but different challenges for Floating Offshore Wind Turbines (FOWT) are not solved yet , such as the dynamic stability in the presence of non-linear ambient loads . In fact, an accurate prediction of the dynamic response of a FOWT, considering the interaction among the hydrodynamics, mooring, and aerodynamics of the turbine, is identified as one of the key challenges for the simulation tools required to design the future FOWTs [4,5,6].
Standard design procedures and simulation tools for marine structures come from the existing technology and experience of the offshore oil and gas industry. For instance, the classic simulation approaches are based on uncoupled formulations, where the hydrodynamic response of the floater is linearised and can be decoupled from the mooring [7,8]. Recently, coupled simulations have been adopted to solve the seakeeping of FOWT devices, since the dynamic of FOWT offers a high complexity due to the variety of loads and non-linear effects. Anyhow, the interaction among different components, such as the wind turbine structure, the rotor dynamics, the mooring arrangements and the floating structure must be taken into account in a more accurate way [9,10].
The analysis of FOWT should be carried out with simulation codes capable to include the physics governing the dynamic response of these devices. With regards to marine structures, Low and Langley  showed that the dynamic response of a floating production system in a random sea can be split in two timescales: low frequency and wave frequency responses. Moreover the seakeeping of the floating device and the dynamics of the mooring lines are coupled. Then, the analysis should take into account the interaction between them, and the time-domain analysis seems to be the right way to simulate this sort of coupled problems. In fact, the American Bureau of Shipping  considers that the global seakeeping analysis of a FOWT should take into account: the unsteady wind loads, wind turbine control systems, wind turbine-platform interaction, wave actions over the platform, currents, mooring loads, and any other types of external actions. It will be presented later on that the time-domain approach allows to handle these actions in a natural manner.
The first works proposing coupled formulations for floating production systems were developed for TLP-type platforms [12,13,14], where it was found that the sum-frequency effects are important. Also other authors investigated and developed models to couple different effects and loads in floating devices [15,16,17,18]. More recently, Cordle and Jonkman  made a review of the state of art of the simulation tools for FOWT. Bae and Kim  recently presented a research on the effects of second-order wave excitation in a mono-column-TLP type FOWT comparing with the uncoupled simulations in time-domain. Karimirad and Moan  developed a simplified method for the coupled dynamics of a offshore wind turbine aiming at saving computational time. They also analysed a spar-type FOWT, and compared different hydro-elasticity codes . In , the dynamic response of the spar-type platform subject to wave induced excitation was investigated numerically, including catenary mooring cables. Matha et al.  investigated a platform-tower coupled with a TLP-type FOWT. Jonkman and Sclavounos  presented a tool integrating an aeroelastic model for onshore wind turbines, and a hydrodynamic load model, together with a Quasi-Static (QS) catenary model for mooring lines. More sophisticated mooring models, which are usually based on Finite Element Methods (FEM) have been investigated; for instance, Kim et al.  presented a comparison between a linear spring and a non-linear FEM mooring for the analysis of the dynamic response when they are coupled with a floating structure. Several coupled seakeeping analyses with FEM mooring line models can also be found in the literature [22,23,24,25,26,27].
Frequency-domain and time-domain approaches are used for seakeeping analyses of marine structures. However, the frequency-domain approach has difficulties to accurately handle non-linearities such as those arising from the mooring lines, and the low frequency components of the wave-body interaction, as appointed by Low , while the time-domain analysis can straightforwardly include any non-linearity within each time step in a natural manner. However, the main drawback of the time-domain is an increase of the computational time required for the simulations. To overcome this limitation, Low and Langley  and Low  presented a hybrid time-frequency domain model to simulate low and wave frequency response of coupled problems.
In this work, an extension of the time-domain method proposed by Serván-Camas and García-Espinosa [30,31] is used. This method obtains a relevant reduction of the computational time thanks to the use of deflation techniques and High Performance Computation based on Graphical Processor Units (GPU). The tool is based on the Finite Element Method (FEM) and is able to solve multi-body seakeeping problems on unstructured meshes including any type of external load. Other examples of FEM solvers for seakeeping analysis can be found in the literature. For instance, Hong and Nam , who presented a second-order analysis of wave forces using FEM in the time-domain, and investigated the interactions with multi-body devices.
This work presents the development of a coupled FEM seakeeping and mooring models for the analysis of floating offshore structures. The model is able to take into account first and the second-order irregular waves, wind loads, and mooring loads. A novel iterative scheme for coupling the wave diffraction-radiation solver developed by Serván-Camas and García-Espinosa [28,30,31] with non-linear aerodynamics and mooring loads is also described. The paper is organized as follows. First, the governing equations of the body dynamics are introduced. Second, the second-order diffraction-radiation wave problem is described. Third, the non-linear FEM model for mooring lines is introduced in detailed. Fourth, the coupling between the dynamics of the floater with non-linear loads, such as mooring lines forces, is explained. Fifth, some validations with experimental results are shown, as well as a comparison of the non-linear FEM model with the Quasi-Static catenary model. Sixth, we analyse an operational case of a spar buoy FOWT based on the OC3 concept and NREL 5 MW turbine. This case study aims at evaluating the influence of the type of mooring model, as well as the effects of non-linear wave on the dynamics of the platform. Finally, some relevant conclusions from the obtained results are presented.
The motion of a floater subject to ambient loads can be modelled using the rigid body dynamics. Let OXYZ be a fixed global frame of reference, and let Gxyz be a local frame of reference located at the center of gravity, and whose axis are parallel to the axis of the global frame (see Fig. 1). Assuming small rotations, for each time step, the body accelerations can be obtained respect to the local frame from the rigid-body dynamic equations:
where is the mass matrix, is the linear acceleration vector , is the instantaneous inertia tensor, and are the external loads vector (hydrostatics, wave loading, mooring, wind, etc.) and the external moments respectively; finally is the angular acceleration vector.
|Figure 1: Global and local frame of reference used in computation of floater dynamics.|
The calculation of the external forces over the body constitutes an essential part of the dynamics, and will be described in more detailed later on.
Assuming incompressible and irrotational flow in a domain , being the wetted surface of the floating device, the wave problem governing equations are :
where is the velocity potential, is the free surface elevation, is the gravity acceleration, is the pressure on free surface, is the fluid density, is the wave elevation, is a pressure at a point over body, is the fluid velocity, is the local velocity at a point on the wetted body surface, is the vector normal to the body wetted surface (pointing upward this surface), and is the vertical coordinate of any point of the body.
The governing equations for the second-order diffraction-radiation wave problem are obtained applying Taylor expansion on the boundary surfaces of a time-independent domain. This approach allows to approximate the free surface on and the mean body surface at time . Then, a perturbed solutions based on Stokes expansion procedure is applied to the velocity potential, free surface elevation, and body motion. More details can be found in .
The solution can be decomposed as
where is the incident wave velocity potential, is the diffraction-radiation wave velocity potential, is the incident wave elevation, and is the diffraction-radiation wave elevation. Then, the wave diffraction-radiation governing equations up to second-order are :
where superscripts 1 and denote the components at the first-order and up to second-order solution, and is the displacement vector at a point over body.
With regards to the ambient loads applied over wetted body surface, hydrodynamic wave forces and moments can be obtained directly from pressure integration over the wetted body surface. Thus, it can be written that
where subscript denotes the hydrostatic loads and denotes the dynamic loads. and are the initial hydrostatic forces and moments over the body, and are the up to second-order hydrostatic loads and moments, and and are the up to second-order dynamic loads and moments. Details of each component can be found in .
The dynamic equations for a mooring cable with length with negligible bending and torsional stiffness can be formulated as [34,35]
where is the water density, is the added mass coefficient, is the mass per unit length of the unstretched cable, is the position vector, is the Young's modulus, is the cross-sectional area of the cable, is the strain, are the external loads applied on the cable, and is the length along the unstretched cable.
The boundary conditions are given by
where is the second derivative of the position vector at the fairlead connection point. The external loads acting on the cable, considered in this work, are the self-weight of the cable, hydrostatic loads, drag forces , and seabed interaction.
When the cable undergoes an excitation, the part which lies between the anchor and the touchdown point interacts with the seabed. This interaction is a complex non-linear effect. In this work, the seabed-mooring interaction is modelled as a spring-damping system [37,38], and is implemented as follows:
where is a coefficient limiting the seabed reactions, is the portion of the cable resting on the seabed, is the weight per unit of length of the line, is the apparent weight of the cable, is the term related to the numerical integration of the dynamic FEM model, is the vertical coordinate of the cable, is the seabed stiffness, is the contact area of the line, and is the time step. The term is defined as
where is the diameter of line.
The term can be defined as a limit of the line subsidence.
being a term related to the numerical scheme adopted. In addition is expressed as
where is the seabed damping Palm.
The algorithm developed to solve the mooring dynamics is described next.
Truss elements, with just three translational degrees of freedom per node, are used for modelling the mooring lines, since bending effects can be neglected in most of cases [7,40]. In this work, a FEM approach combined with an updated Lagrange formulation is used for describing the dynamics of the mooring cable. Some authors showed that the corotational formulation can get a better performance than the updated formulation. However some instabilities and lack of convergence can appear. Figure 2 shows a scheme of the discretization adopted in this work.
|Figure 2: Scheme showing the general approach adopted for the spatial discretization of the cable mooring.|
Applying the standard FEM formulation for the non-linear elastodynamics problems , the equations for the dynamic equilibrium of the forces on a cable for each time step can be written as
where are the external loads vector, is a consistent mass matrix of the line, considering inertia and added mass, is the pretension vector in the initial configuration, and is the internal forces vector of the cable. Damping effects of the mooring cable are introduced through a Rayleigh proportional damping matrix of . The application of the non-linear FEM approach to the cable equation can be found in [41,42].
Considering the linearization of the current configuration, Eq. (28) can be expressed as
where and are called the material stiffness matrix and the geometric stiffness matrix, respectively, in the FEM literature [41,42]. The sum of both matrices is the so-called tangent stiffness matrix. The geometric part of the stiffness matrix is considered as non-linear since no material properties appear, and it depends of the Second-Piola stress tensor of the current configuration. Incremental displacement of the cable nodes is .
where is the time step, denotes the ith iteration, is a parameter concerned with the Bossak-Newmark implicit method, and and are parameters related to the Newmark time integration scheme.
The new position and velocity of each node in each time step can be expressed as (see Fig. 3)
It can be highlighted that multi-segmented mooring systems can be solved with this approach, and the equilibrium between two or more mooring cables is determined using the Newton-Raphson method. A criterion based on the maximum difference between the position reached by the nodes in two consecutive iterations is applied to evaluate the convergence of the algorithm.
|Figure 3: General approach to algorithm for solving the floater dynamics.|
The iterative procedure for solving the cable dynamics can be summarized as
Most of the models found in the literature to solve the coupled mooring-body dynamics use a frequency domain analysis of the floater together with a convolution integral  for solving the motions of the coupled system [9,20,19]. The most common approach is based on a frequency-domain diffraction-radiation solver based on the Boundary Element Method (BEM) combined with a FEM model to solve the mooring dynamics. In this work, a different approach based on a time-domain FEM seakeeping model recently presented [30,31] is proposed. This approach allows to straightforwardly couple the seakeeping of the floater with non-linear loads.
The algorithm used in this work to solve the dynamics of the coupled system is presented in Fig. 3. In order to accelerate the solution of the non-linear solver, it includes two nested loops; the external loop iterates on the wave diffraction-radiation problem, while the internal loop takes into account the remaining external forces vector acting on the body. In this figure are the external forces different from the hydrodynamic and the mooring forces, and is the matrix of the linearised mooring. The procedure to carry out the coupled analysis is as follows:
where is the restoring force vector of the mooring cables.
As mentioned above, the cable dynamics is formulated in term of the acceleration, and this is also valid for the boundary condition of the fairlead connection point. A linear approximation to evaluate the end node acceleration based on the prescribed displacement is proposed in this work. Each time step is divided in sub-interval, that is to say, . So, in the -th time step, the acceleration of end node fulfills the following compatibility relationship
where is a coefficient to be determined in each time step. Applying an appropriate boundary condition, this coefficient can be formulated as
Above formulation for the boundary condition at the fairlead point has shown a better stability for the dynamics solver than higher order approximations.
As it was presented in the previous section, in the first iteration of the dynamic loop, the stiffness matrix of the cable is estimated by obtaining the Jacobian of the reaction forces using numerical differentiation,
where is the reaction of the mooring cable at the fairlead point. Thus, the cable response is linearised within each time step , by estimating the mooring restoring forces as
where the terms correspond to a linear stiffness matrix, and , and are the displacements of the fairlead point from the position at the beginning of the current time step.
In this section some validation examples of the non-linear FEM model are presented.
The first case is based on that presented in . It consists of an isolated cable, with fixed ends, subjected to its own weight (see Fig. 4). Initially the cable has a flat form. The expected deformation is a U form, and the reactions at the ends must be equal to the cable weight. The properties of the cable are: the stiffness 50 N, the weight per unit length N/m, and m of span length. The cable is discretized into twenty-two bar elements, and the time step adopted is s. Figure 5 shows different positions of the cable until it gets the expected U form. Note that a damping increment leads to a faster convergence of the stationary solution. The reactions at end points are , and the maximum deflection (0.715 m) is reached at m.
|Figure 4: Validation of the non-linear FEM mooring model. Case 1: cable under its self weight.|
|Figure 5: Validation of the non-linear FEM mooring model. Case 1: different stages of time evolution of cable under its self weight.|
The Ms Excel file below includes the results of the time evolution of the cable under its self weight (corresponding to Figure 5).
The second case is based on that proposed by . The experiment consists of a cable initially in a horizontal position, and letting one end free and leaving the cable evolve under gravitational loads. The computed results are compared with the experimental results taken from , as well as with the data obtained from simulation with the FEM structural solver RamSeries (www.compassis.com). The properties of the cable are: length m, stiffness 50 N, weight per unit of length N/m, and m distance between the cable end points. The cable was divided into forty four bar elements, and the time step adopted was s. This time step must be low enough for obtaining an accurate description of the cable motion. The obtained numerical results show a good agreement between our results, and those experimentally and numerically obtained by other authors. The cable position at different times obtained from computed results can be observed in Fig. 6.
|Figure 6: Validation of the non-linear FEM mooring model. Case 2: different time step positions of free vibration cable obtained from computed results.|
|Figure 7: Validation of the non-linear FEM mooring model. Case 2: comparison between computed results of the path of free end of the cable obtained and experimental results of  and numerical computed with RamSeries.|
Slight differences can be appreciated at the lower end in Fig. 7. This fact can be explained by the different numerical parameters chosen to carry out the simulation.
Now, results obtained by the developed FEM solver are compared for the model test proposed by Lindhal and Sjoberg [38,39]. The experimental set up is shown in Fig. 8. The lower end was attached to the concrete floor and the upper end was attached to a circular plate with a fixed rotation speed. The radius of the circular motion was m. Two cases with different rotational periods of = 1.25 s and s respectively, were investigated. The reaction forces at the top end of the cable are measured and compared with those computed by the proposed numerical model.
|Figure 8: Validation of the non-linear FEM mooring model. Case 3: the geometrical set-up of the experimental tests by .|
|Figure 9: Validation of the non-linear FEM mooring model. Case 3: comparison between the experimental and numerical cable top end reaction forces. Rotational period = 3.5 s.|
The properties of the cable are: length m, stiffness N, weight per unit length N/m, and diameter m. In this validation case, the cable is discretized into 200 bar elements, and the time step adopted is s. The motion of the top end is defined by the following expressions (including an initialization period to build up the spinning of the plate):
The results are compared with the experimental results taken from . A good agreement between the computed reaction forces and the experimental results  can be observed in Fig. 9 (for s), and 10 (for s) for both, the maximum values and the time evolution. Only slight differences are appreciated due to the uncertainty of the experimental data. The entire cable loses stiffness at some instants in time, and the numerical oscillations after the slack are larger in the case of the larger excitation frequency, as it was observed by .
|Figure 10: Validation of the nonl-inear FEM mooring model. Case 3: comparison between the experimental and numerical cable top end reaction forces. Rotational period = 1.25 s.|
A fully coupled analysis of the OC3 spar buoy offshore wind turbine, called Hywind (www.ieawind.org/task23/) is presented next. Main particulars of the spar buoy FOWT are presented in [52,53]. A general view of the buoy concept can be observed in Fig. 11. In this section, several analysis are made:
|Figure 11: General view of the spar buoy wind turbine concept (OC3-Hywind concept).|
First, a RAO analysis is performed in the absence of wind. The frequency results of the seakeeping solver are obtained after applying a Fourier transform to the time history generated. Figure 12 shows an inter-code comparison. Note that a good agreement among the different solvers is found .
|Figure 12: OC3-Hywind concept. Comparison between the computed result by SeaFEM with those taken from other authors  for a rigid wind turbine with no wind.|
The Ms Excel file below includes the RAO curves obtained in the present work and the data from HydroDyn and WAMIT used as reference (see Figure 12).
Below, the influence of three different mooring models on the OC3 spar buoy FOWT is analysed. These models are: the linear model cable (that behaves as springs and is represented by a linearised mooring matrix ); the Quasi-Static model, similar to the one presented in ; and the dynamic model developed in this work. Six first-order monochromatic waves are used in the analysis. Key parameters of the mooring layout and cable properties can be found in Table 1. Fig. 13 compares the pitch motion for each mooring model.
|Wave Period analysed||10; 25; 20; 35; 40; 55||s|
|Number of Mooring Lines||3|
|Angle Between Adjacent Lines||120||deg|
|Depth to Anchors Below SWL||320||m|
|Depth to Fairleads Below SWL||70||m|
|Radius to Fairleads from Platform Centerline||853.9||m|
|Unstretched Mooring Line Length||902.2||m|
|Mooring Line Diameter||0.9||m|
|Mooring Line Mass Density||77.71||kg/m|
|Mooring Line Extensional Stiffness||3.8410||N|
It can be observed that there are big differences in the results in those cases close to the pitch resonance (about 30 s), while the results are quite similar for the cases with 10 s and 55 s. These results suggest that the use of simpler mooring models can lead to big errors near the resonance frequency, and as a consequence to magnify safety factors, contributing to increase the cost of the FOWT.
Finally, four analyses of the OC3 spar buoy in operational conditions are presented. The different analyses are carried out in similar environmental conditions, but using first and second-order irregular waves. Furthermore, additional studies including Quasi-Static and dynamic mooring models are performed. The goal of these analyses is to evaluate the effects of the mooring model and the wave order on the dynamics of the system, as well as to estimate the tension in the mooring lines. On the one hand, the wind turbine system is assumed to be operating at an average wind speed of 11.4 m/s, which generates the maximum thrust and torque. FASTLognoter [50,51] has been used to linearise with FAST  the behaviour of the wind turbine around the operating wind speed. Restoration and damping matrices resulting from the linearisation of the wind turbine system are included into the global dynamics. It should be remarked that the rotational and periodicity effects are considered in the calculation of the steady state matrix. On the other hand, the wind loads are estimated considering non-uniform wind flow, with an average wind speed of 11.4 m/s. The wind flow profile is obtained using Turbsim , and the wind loads on the wind turbine are obtained from FAST/AeroDyn .
|Case 1||Case 2||Case 3||Case 4|
|Average Wind velocity||(m/s)||11.4||11.4||11.4||11.4|
|Wave Spectrum||(-)||JONSWAP 1||JONSWAP 1||JONSWAP 2||JONSWAP 2|
|Significant wave height||(m)||6.0||6.0||6.0||6.0|
|Mean wave direction||(deg)||0||0||0||0|
|Number of mooring lines||(-)||3||3||3||3|
|Mooring lines elements||(-)||200||200||50||200|
A JONSWAP spectrum with a mean wave period = 12.0 s, and significant wave height = 6.0 m, is simulated. The key parameters of the different case studies are presented in Table 2. As stated above, two different types of mooring models are analysed; one based on the Quasi-Static catenary model , and the other based on the dynamic (NFEM) cable model, presented in this work. For the dynamic cable analysis each mooring line is divided into 200 bar elements.
Figures 14, 15 and 16 show the computed heave, roll and pitch motions from 600 s to 900 s of simulation time. Noticeable differences are found between the first and the second-order movements, while the QS and NFEM mooring models offer quite similar results.
|Figure 13: Results obtained for mooring analysis around pitch resonance of OC3-Hywind.|
Table 3 shows the mean and RMS values, as well as the motion amplitude for the first and second-order movements. When comparing the QS and the NFEM models, only slight differences are observed. In particular, the second-order pitch motion is higher when using the NFEM model, compared to the QS model, while the other values remain with similar trends for both models.
|Surge (m)||Sway (m)||Heave (m)||Roll (deg)||Pitch (deg)||Yaw (deg)|
|Surge (m)||Sway (m)||Heave (m)||Roll (deg)||Pitch (deg)||Yaw (deg)|
Next, a video of the analysis case 1 is presented.
|Video 1. OC3 spar buoy wind turbine simulation (analysis case 1).|
Figure 17 shows the tension for each mooring line at the fairlead point. The NFEM model recorded larger amplitude oscillations (in the range of 5 s to 10 s period) compared with QS model. The differences observed in Fig. 17 in the tension amplitude reach up to 24 . This fact suggests that using a QS model for fatigue assessment of the mooring lines could overestimate their fatigue life.
|Figure 14: Comparison between heave motion for first and second-order wave environment for Case 1-4 (described in Table 2).|
|Figure 15: Comparison between roll motion for first and second-order wave environment for Case 1-4 (described in Table 2).|
|Figure 16: Comparison between pitch motion for first and second-order wave environment for Case 1-4 (described in Table 2).|
|Figure 17: Comparison of fairlead tension of each mooring line for Case 3, and 4.|
Table 4 compares the maximum, minimum, average, and RMS tension values at the fairlead points, obtaining similar values for both mooring models. Furthermore, based on Figure 17, the second-order simulation provided larger tension values than the first-order. This result suggests that a first-order approximation can underestimate the fatigue loads and might lead to a wrong mooring design.
|Line||Case||Max. (N)||Min. (N)||Mean (N)||RMS (N)|
It is emphasized that the NFEM mooring model allows to consider non-linear dynamic effects which cannot be taken into account by QS models. However, in deep water, the dynamics of catenary mooring lines are negligible as reported in Low.
A FEM coupled seakeeping and mooring model for the analysis of floating wind turbines is presented. Based on the results obtained in this work, the following conclusions are made:
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