The challenge of service life prediction related corrosion modelling

==Non-linear dynamic analysis of the response of moored floating structures==

'''José E. Gutiérrez-Romero<span id="fnc-1"></span>[[#fn-1|<sup>1</sup>]]<math>^{a}</math>, Julio García-Espinosa<math>^{b}</math><math>^{d}</math>, Borja Serván-Camas<math>^{b}</math>, Blas Zamora-Parra<math>^{c}</math>'''

{|
|-
|(a) Depart. Naval Technology, Universidad Politécnica de Cartagena (UPCT), [http://www.upct.es/ http://www.upct.es/] 
|-
| Paseo Alfonso XIII 48, 30203 Cartagena, Spain 
|-
| (b) Centre Internacional de Metodes Numerics en Enginyeria (CIMNE), [http://www.cimne.upc.edu/ http://www.cimne.upc.edu/] 
|-
| Edifici C1, Campus Norte, UPC, Gran Capitán s/n, 08034 Barcelona, Spain 
|-
| (c) Depart. Thermal and Fluids Engineering, Universidad Politécnica de Cartagena (UPCT), [http://www.upct.es/ http://www.upct.es/] 
|-
| Doctor Fleming s/n, 30202 Cartagena, Spain 
|-
| (d) Universidad Politécnica de Cataluña (UPC) 
|-
| C. Gran Capitán s/n, Campus Nord, 08034 Barcelona, Spain
|}

==Abstract==

The complexity of the dynamic response 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 new time-domain model for solving the dynamics of moored floating marine devices, specifically offshore wind turbines, subjected to non-linear environmental loads. The paper first introduces the formulation of the second-order wave radiation-diffraction solver, designed for calculating the wave-floater interaction. Then, the solver of the mooring dynamics, based on a non-linear Finite Element Method (FEM) approach, is presented. Next, the procedure developed for coupling the floater dynamics model with the mooring model is described. Some validation examples of the developed models, and comparisons among different mooring approaches are presented. Finally, an study of the OC3 floating wind turbine concept is performed to analyze the influence of the mooring model in the dynamics of the platform and the tension in the mooring lines. The work comes to the conclusion that the coupling of a dynamic mooring model along with a second-order wave radiation-diffraction solver can offer realistic predictions of the floating wind turbine performance.

'''keywords'''

FEM, Coupled Analysis, Mooring Dynamics, Second-order waves, Time-domain

<span id="fn-1"></span>
<span style="text-align: center; font-size: 75%;">([[#fnc-1|<sup>1</sup>]]) Corresponding author. Tel. no.: +34 868 071 261. Fax no.: +34 968 325 422 . E-mail address: jose.gutierrez@upct.es (José E. Gutiérrez-Romero)</span>

==1 Introduction==

Research trends in marine renewable energies are mainly focused on offshore wind energy due to the high expectations raised in this field. Currently, the technology for marine wind turbines is well-developed, but limited for fixed installations in shallow-water areas. The next horizon is focused on developing Floating Offshore Wind Turbines (FOWT) technology to enable the access to deep waters <span id='citeF-1'></span>[[#cite-1|[1]]]. But, different challenges still have to be overcome to enable the development of this industry <span id='citeF-2'></span>[[#cite-2|[2]]]. In particular, an accurate prediction of the dynamic response is identified as one of the key challenges for the analysis tools required to design the future FOWTs <span id='citeF-3'></span><span id='citeF-4'></span><span id='citeF-5'></span><span id='citeF-6'></span>[[#cite-3|[3,4,5,6]]].

Standard design procedures and simulation tools for marine structures come from the existing experience in the offshore oil and gas industry. For instance, the classic simulation approach used to assess the mooring systems is based on uncoupled formulations, where the hydrodynamic loads and dynamic response of the floater is evaluated, based on frequency-domain solvers <span id='citeF-7'></span><span id='citeF-8'></span>[[#cite-7|[7,8]]]. This approach has difficulties to accurately handle non-linear effects, such as those arising from the mooring lines and the low frequency components of the wave-body interaction, as appointed by Low <span id='citeF-9'></span>[[#cite-9|[9]]]. Although, different works has presented models aiming to overcome this limitation <span id='citeF-10'></span><span id='citeF-9'></span>[[#cite-10|[10,9]]], the dynamics of FOWT presents a higher complexity due to the variety of loads and non-linear effects, and more robust and accurate models, able to handle the interaction among the different components, have to be developed <span id='citeF-11'></span><span id='citeF-12'></span>[[#cite-11|[11,12]]]. In fact, the American Bureau of Shipping considers that the global seakeeping analysis of a FOWT should take into account <span id='citeF-13'></span>[[#cite-13|[13]]]: unsteady wind loads, wind turbine control systems, wind turbine-platform interaction, wave actions over the platform, currents, mooring loads, and any other type of external actions. These loads has different characteristic time-scales <span id='citeF-10'></span>[[#cite-10|[10]]] what makes ineffective the use of classic analysis procedures.

In this regard, several works can be found in the literature that deal with the problem of handling different forces acting on marine structures. One of the first works in this field, was presented by Pauling and Webster <span id='citeF-15'></span>[[#cite-15|[15]]], which showed a procedure to couple the action of waves, wind, and the mooring system, acting on a floating structure. Other pioneer works can be found in <span id='citeF-16'></span><span id='citeF-17'></span><span id='citeF-18'></span><span id='citeF-19'></span><span id='citeF-20'></span>[[#cite-16|[16,17,18,19,20]]], where several procedures were developed in order to solve the floater dynamics in frequency-domain, including various mooring models. Recent research works <span id='citeF-21'></span><span id='citeF-22'></span>[[#cite-21|[21,22]]] have presented a procedure to reduce the computational cost of the time-domain dynamics simulation of moored FOWTs, and have carried out an extensive fatigue study of the mooring system. Other relevant work is <span id='citeF-23'></span>[[#cite-23|[23]]], which presented a frequency-domain hydro-aeroelastic solver coupled with a quasi-static mooring model. Finally, some works should be cited that have presented sophisticated mooring models, including elastic deformation <span id='citeF-11'></span><span id='citeF-24'></span><span id='citeF-25'></span>[[#cite-11|[11,24,25]]]. It has to be highlighted that still most of the above mentioned works combine the Boundary Element Method (BEM) to solve the wave-floater interactions with mooring models in the frequency-domain. However, it can be found some works using different methodologies, such as the Finite Element Method (FEM), to solve the problem in the time-domain <span id='citeF-26'></span>[[#cite-26|[26]]]. The time-domain approach allows to handle different types of actions in a natural manner.

But, despite the advances presented in the above referred works, several challenges related to the accuracy of the analysis of FOWT dynamics can still be identified <span id='citeF-6'></span><span id='citeF-14'></span>[[#cite-6|[6,14]]]. This fact has motivated this work that presents the development of a non-linear coupled FEM seakeeping model for the analysis of moored floating offshore structures. The starting point is the time-domain method proposed by Serván-Camas and García-Espinosa <span id='citeF-27'></span><span id='citeF-28'></span>[[#cite-27|[27,28]]], which is extended in this work to take into account second-order irregular waves, wind loads, and mooring effects. The paper is organized as follows. First, the governing equations of the body dynamics are introduced. Second, the second-order diffraction-radiation seakeeping solver is described. Third, the non-linear FEM model for mooring lines analysis is introduced. Fourth, the algorithm designed to solve the body dynamics equations, including the different non-linear loads, is explained. Fifth, some validations of the presented mooring model against experimental results are shown. Sixth, an operational case of a spar buoy FOWT based on the OC3 concept and NREL 5 MW turbine is analysed. 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 of the work are presented.

==2 Problem statement==

===2.1 General dynamics of the floater===

The motion of a floater subject to ambient loads can be described using the rigid body dynamics equations. Let '''OXYZ''' be a fixed global frame of reference, and let '''Gxyz''' be a local frame of reference located at the center of gravity of the floater, and whose axis are parallel to those of the global frame (see Fig. [[#img-1|1]]). In most of the cases, under operating conditions of FOWTs, small rotations simplification can be assumed. Hence, the floater accelerations respect to the local frame can be obtained from the following set of equations,

<span id="eq-1"></span>
<span id="eq-2"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{M}_{b} \; \ddot{\mathbf{x}} = \mathbf{f}\left(t\right) </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
|-
| style="text-align: center;" | <math> \bar{\bar{\mathbf{I}}}_{b} \; \ddot{\mathbf{r}}_{b} = \mathbf{m}\left(t\right) </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
|}
|}

where <math display="inline">\mathbf{M}_{b}</math> is the mass matrix, <math display="inline">\ddot{\mathbf{x}}</math> is the linear acceleration vector, <math display="inline">\bar{\bar{\mathbf{I}}}_{b}</math> is the instantaneous inertia tensor, <math display="inline">\mathbf{f}\left(t\right)</math> and <math display="inline">\mathbf{m}\left(t\right)</math> are the external forces and moments, respectively, and finally <math display="inline">\ddot{\mathbf{r}}_{b}</math> is the angular acceleration vector.

The calculation of the forces acting on the floater constitutes an essential part of the problem, and will be described in detail later on.

===2.2 Fluid flow governing equations===

Assuming incompressible and irrotational flow, the seakeeping problem of a floating body is governed by the following equations:

<span id="eq-3"></span>
<span id="eq-8"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\Delta \varphi=0    \mbox{ on}\;\Omega  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
|-
| style="text-align: center;" | <math> \frac{\partial \xi }{\partial{t}} + \frac{\partial{\varphi }}{\partial{x}}\frac{\partial{\xi }}{\partial{x}} + \frac{\partial{\varphi }}{\partial{y}}\frac{\partial{\xi }}{\partial{y}} - \frac{\partial{\varphi }}{\partial{z}}  = 0    \mbox{ on}\; z=\xi </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
|-
| style="text-align: center;" | <math> \frac{\partial{\varphi }}{\partial{t}}+ \frac{1}{2} \left|\nabla ^{2}\varphi \right|+ g\xi  = 0    \mbox{ on}\; z=\xi </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
|-
| style="text-align: center;" | <math> \mathbf{\mbox{v}}_{\varphi }\cdot \mathbf{n}+\mathbf{\mbox{v}}_{b}\cdot \mathbf{n} = 0    \mbox{ on}\;\Gamma _{b} </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
|-
| style="text-align: center;" | <math> \frac{\partial{\varphi }}{\partial{z}}=0   \mbox{ on}\; z=-H </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
|-
| style="text-align: center;" | <math> \mbox{ P}_{b}=-\rho \frac{\partial{\varphi }}{\partial{t}}-\frac{1}{2}\rho \left|\nabla ^{2}\varphi \right|-\rho g z_{b}    \mbox{ on}\;\Gamma _{b}   </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
|}
|}

where <math display="inline">\varphi </math> is the velocity potential, <math display="inline">\Omega </math> is the fluid domain, <math display="inline">\Gamma _{b}</math> is the wetted surface of the floater, <math display="inline">\xi </math> is the free surface elevation, <math display="inline">g</math> is the gravity acceleration, <math display="inline">\rho </math> is the fluid density, <math display="inline">\xi </math> is the wave elevation, <math display="inline">\mbox{P}_{b}</math>, <math display="inline">\mathbf{v}_{b}</math>, <math display="inline">z_{b}</math> are the pressure, velocity and vertical coordinate of any point on the floater wetted surface, respectively, <math display="inline">\mathbf{v}_{\varphi }</math> is the fluid velocity, <math display="inline">\mathbf{n}</math> is the vector normal to the floater wetted surface (pointing upward this surface), and <math display="inline">H</math> is the water depth.

===2.3 Second-order diffraction-radiation governing equations===

The governing equations for the second-order wave problem are obtained from Eqs. ([[#eq-3|3]]-[[#eq-8|8]]) by applying Taylor expansion on the boundary surfaces of a time-independent domain. This approach allows to approximate the free surface on <math display="inline">z=\zeta </math> and the mean floater surface <math display="inline">\Gamma ^0_{b}</math> at time <math display="inline">t</math>. Then, a perturbed solution based on the Stokes expansion procedure is applied to the velocity potential, free surface elevation, and floater motion. Retaining terms up to second order, the resulting equations are

<span style="text-align: center; font-size: 75%;">

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\begin{array}{l} \Delta \varphi ^{1+2}=0 && \mbox{ on}\,\Omega \\ \frac{\partial \xi ^{1+2}}{\partial{t}} - \frac{\partial{\varphi ^{1+2}}}{\partial{z}} = \xi ^1\frac{\partial{}}{\partial{z}}\left(\frac{\partial{\varphi ^1}}{\partial{z}}\right)- \frac{\partial{\xi ^1}}{\partial{x}}\frac{\partial{\varphi ^1}}{\partial{x}} - \frac{\partial{\xi ^1}}{\partial{y}}\frac{\partial{\varphi ^1}}{\partial{y}} && \mbox{ on }z=0 \\ \frac{\partial{\varphi ^{1+2}}}{\partial{t}} + g\xi ^{1+2} = - \xi ^1\frac{\partial{}}{\partial{z}}\left(\frac{\partial{\varphi ^1}}{\partial{t}}\right)- \frac{1}{2} \nabla \varphi ^1 \cdot \nabla \varphi ^1 && \mbox{ on }z=0 \\ \left(\bold{v}^{1}_{\varphi } + \bold{v}^{1}_{p} \right)\cdot \bold{n}^{1}  + \left(\bold{v}^{2}_{\varphi } + \bold{v}^{2}_{p}  \right)\cdot \bold{n}^{0} = -\left(\left(\bold{r}_{b}^{1} \cdot \nabla \right)\bold{v}^{1}_{\varphi } \right)\cdot \bold{n}^{0} && \mbox{ on}\,\Gamma _{b} \\ \frac{\partial{\varphi ^{1+2}}}{\partial{z}}=0 & & \mbox{ on}\; z=-H \end{array}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
|}

</span> and the total pressure at any point <math display="inline">p</math> on the wetted surface of the body is

<span style="text-align: center; font-size: 75%;">

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math> \frac{\mbox{P}_{b}^{1+2}}{\rho } = - g z_{b} - g r_{zb}^{1+2} - \frac{\partial{\varphi }^{1+2}}{\partial{t}} - \mathbf{r}_{b}^{1}\cdot \nabla \left(\frac{\partial{\varphi ^{1}}}{\partial{t}} \right)-\frac{1}{2} \nabla \varphi ^1 \cdot \nabla \varphi ^1  \mbox{ on}\,\Gamma _{b}\,\,  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
|}

</span>

where superscript 0 denotes the initial position of the body, 1 and <math display="inline">1+2</math> denote the components at the first-order and up to the second-order solution and <math display="inline">\mathbf{r}_{b}</math> is the displacement vector at any point over the body. The following relationships are considered <math display="inline">\varphi ^{1+2} = \varphi ^{1} + \varphi ^{2}</math>, <math display="inline">\xi ^{1+2}= \xi ^{1} + \xi ^{2}</math>, <math display="inline">\mbox{P}_{b}^{1+2} = \mbox{P}_{b}^{0} + \mbox{P}_{b}^{1} + \mbox{P}_{b}^{2}</math>, <math display="inline">\mathbf{r}_{b}^{1+2} = \mathbf{r}_{b}^{1} + \mathbf{r}_{b}^{2}</math>, <math display="inline">\mathbf{\mbox{v}}^{1+2}_{b} = \mathbf{\mbox{v}}^{1}_{b} + \mathbf{\mbox{v}}^{2}_{b}</math> and <math display="inline">\mathbf{\mbox{v}}^{1+2}_{\varphi } = \mathbf{\mbox{v}}^{1}_{\varphi } + \mathbf{\mbox{v}}^{2}_{\varphi }</math>.

The solution of the governing equations can be decomposed as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\varphi = \psi + \theta , </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
|-
| style="text-align: center;" | <math> \xi = \eta + \varsigma , </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (16)
|}
|}

where <math display="inline">\psi </math> is the incident wave velocity potential, <math display="inline">\theta </math> is the diffraction-radiation wave velocity potential, <math display="inline">\eta </math> is the incident wave elevation, and <math display="inline">\varsigma </math> is the diffraction-radiation wave elevation. Using above decomposition, the resulting wave diffraction-radiation governing equations up to second-order can be written as:

<span id="eq-17"></span>
<span id="eq-18"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\Delta \theta ^{1+2} = 0   \mbox{ on}\,\Omega , </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (17)
|-
| style="text-align: center;" | <math> \frac{\partial \eta ^{1+2}}{\partial{t}} - \frac{\partial{\theta ^{1+2}}}{\partial{z}} = -\frac{\partial{\theta ^1}}{\partial{x}} \frac{\partial{\eta ^1}}{\partial{x}} - \frac{\partial{\theta ^1}}{\partial{y}} \frac{\partial{\eta ^1}}{\partial{y}}   </math>
|-
| style="text-align: center;" | <math>  - \frac{\partial{\theta ^1}}{\partial{x}} \frac{\partial{\varsigma ^1}}{\partial{x}} - \frac{\partial{\theta ^1}}{\partial{y}} \frac{\partial{\varsigma ^1}}{\partial{y}} - \frac{\partial{\psi ^1}}{\partial{x}} \frac{\partial{\eta ^1}}{\partial{x}} - \frac{\partial{\psi ^1}}{\partial{y}} \frac{\partial{\eta ^1}}{\partial{y}}   \mbox{ on } z=0  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (18)
|-
| style="text-align: center;" | <math> \frac{\partial{\theta ^{1+2}}}{\partial{t}} + g\eta ^{1+2} = - \eta ^1\frac{\partial{}}{\partial{z}}\left(\frac{\partial{\theta ^1}}{\partial{t}}\right)- \varsigma ^1\frac{\partial{}}{\partial{z}}\left(\frac{\partial{\theta ^1}}{\partial{t}}\right)  </math>
|-
| style="text-align: center;" | <math> - \eta ^1\frac{\partial{}}{\partial{z}}\left(\frac{\partial{\psi ^1}}{\partial{t}}\right)- \frac{1}{2} \nabla \theta ^1 \cdot \nabla \theta ^1 - \nabla \psi ^1 \cdot \nabla \theta ^1   \mbox{ on }z=0 </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (19)
|-
| style="text-align: center;" | <math> \bold{v}^{1}_{\theta } \cdot \bold{n}^{1} + \bold{v}^{2}_{\theta } \cdot \bold{n}^{0}= -\left(\bold{v}^{1}_{b} + \bold{v}^{1}_{\psi }  \right)\cdot \bold{n}^{1}   </math>
|-
| style="text-align: center;" | <math> -\left(\bold{v}^{2}_{b} + \bold{v}^{2}_{\psi } \right)\cdot \bold{n}^{0} - \left(\left(\bold{r}_{b}^{1} \cdot \nabla \right)\left(\bold{v}^{1}_{\theta } + \bold{v}^{1}_{\psi } \right)\right) \cdot \bold{n}^{0}   \mbox{ on}\,\Gamma _{b} </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (20)
|-
| style="text-align: center;" | <math> \frac{\partial{\theta ^{1+2}}}{\partial{z}}=-\frac{\partial{\psi }^{1+2}}{\partial{z}}   \mbox{ on}\; z=-H </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (21)
|}
|}

and the total pressure at any point <math display="inline">p</math> on the wetted surface of the body is

<span style="text-align: center; font-size: 75%;">

<span id="eq-22"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\begin{array}{l} \frac{\mbox{P}_{b}^{1+2}}{\rho } = - g z_{b} - g r_{zb}^{1+2} - \frac{\partial{\theta }^{1+2}}{\partial{t}} - \frac{\partial{\psi }^{1+2}}{\partial{t}} - \mathbf{r}_{b}^{1}\cdot \nabla \left(\frac{\partial{\psi ^{1}}}{\partial{t}} \right)\\ - \mathbf{r}_{b}^{1}\cdot \nabla \left(\frac{\partial{\theta ^{1}}}{\partial{t}} \right)-\frac{1}{2} \left(\nabla \theta ^1 \cdot \nabla \theta ^1 -\nabla \psi ^1 \cdot \nabla \psi ^1 \right)-\nabla \psi ^1 \cdot \nabla \theta ^1 && \mbox{ on}\,\Gamma _{b}\,\,  \end{array}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (22)
|}

</span>

where superscripts 1 and <math display="inline">1+2</math> denote the components at the first-order and up to second-order solution.

In this work, the wave diffraction-radiation equations are solved using the FEM, while a fourth-order time-marching compact Padé scheme is used for integrating the combined kinematic-dynamic free surface boundary condition. Applying the standard Galerkin method <span id='citeF-30'></span>[[#cite-30|[30]]] to Eq. ([[#eq-17|17]]), the resulting global system of algebraic equations can be written in the associated matrix form as <span id='citeF-27'></span>[[#cite-27|[27]]]:

<span id="eq-23"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bar{\bar{\bold{L}}}\bold{\theta } = \bold{b}^{B} + \bold{b}^{R} + \bold{b}^{Z_{0}} + \bold{b}^{Z_{-H}},  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (23)
|}

where <math display="inline">\bar{\bar{\bold{L}}}</math> is the standard finite element Laplacian matrix, and <math display="inline">\bold{b}^{B}</math>, <math display="inline">\bold{b}^{R}</math>, <math display="inline"> \bold{b}^{Z_{0}}</math>, and <math display="inline">\bold{b}^{Z_{-H}}</math> are the vectors resulting of integrating the corresponding boundary condition terms. Regarding the bottom boundary for the diffracted and radiated potential, it is imposed naturally by taken <math display="inline">\bold{b}^{Z_{-H}} = 0</math>.

Once the velocity potential is solved, free surface elevation and dynamic pressure on the surface of the floater can be calculated using Eqs. [[#eq-22|22]] and [[#eq-18|18]], respectively. Using that information, hydrodynamic loads acting on the floater can be evaluated <span id='citeF-29'></span>[[#cite-29|[29]]].

The aforementioned solution scheme has been implemented in the seakeeping analysis solution SeaFEM <span id='citeF-29'></span><span id='citeF-31'></span>[[#cite-29|[29,31]]]. Further details regarding the mathematical models, numerical schemes, and hydrodynamic loads evaluation can be found in <span id='citeF-27'></span><span id='citeF-28'></span><span id='citeF-29'></span>[[#cite-27|[27,28,29]]].

==3 Non-linear FEM model for mooring dynamics==

===3.1 Equations for cable dynamics===

Most mooring cables used in marine applications are very slender structures, with characteristic length/diameter ratio of several thousands. Hence, the bending stresses become several orders of magnitude smaller than axial stresses, and therefore can be neglected for practical purposes <span id='citeF-7'></span><span id='citeF-32'></span>[[#cite-7|[7,32]]]. The equation governing the dynamics of a cable with negligible bending and torsional stiffness can be formulated as <span id='citeF-33'></span>[[#cite-33|[33]]]

<span id="eq-24"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\left(\rho C_{m}A_{0}+\rho _{0}\right)\frac{\partial ^{2}\mathbf{r}_{l}}{\partial t^{2}}=\frac{\partial }{\partial l}\left(EA_{0}\frac{s}{s+1}\frac{\partial \mathbf{r}_{l}}{\partial l}\right)+\mathbf{f}\left(t\right)\left(1+s\right)  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (24)
|}

where <math display="inline">\rho </math> is the water density, <math display="inline">C_{m}</math> the added mass coefficient, <math display="inline">\rho _{0}</math> the mass per unit length of the unstretched cable, <math display="inline">\mathbf{r}_{l}</math> the position vector, <math display="inline">E</math> the Young's modulus, <math display="inline">A_{0}</math> the cross-sectional area of the cable, <math display="inline">s</math> the axial strain, <math display="inline">\mathbf{f}</math> the vector of external forces acting on the cable, and <math display="inline">l</math> the length of the unstretched cable.

Eq. [[#eq-24|24]] is completed with the corresponding boundary conditions, given by

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{\partial ^{2}\mathbf{r}_{l}}{\partial t^{2}}=0    l=0\;\;\;\; \left(\mbox{ at the anchor connection point}\right)</math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (25)
|-
| style="text-align: center;" | <math> \frac{\partial ^{2}\mathbf{r}_{l}}{\partial t^{2}}=  \ddot{\mathbf{r}}_{fl}    l=L\;\;\;\; \left(\mbox{ at the fairlead connection point}\right) </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (26)
|}
|}

where <math display="inline">\ddot{\mathbf{r}}_{fl}</math> is the second derivative of the position vector at the fairlead connection point.

===3.2 External forces acting on the cable===

The external forces acting on the cable, to be considered in this work, comprises the viscous forces due to the non-linear drag <span id='citeF-35'></span>[[#cite-35|[35]]], self-weight, friction and hydrostatic forces and seabed interaction. The force components due to non-linear drag are calculated as follows <span id='citeF-29'></span>[[#cite-29|[29]]]:

<span id="eq-27"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{f}_{bf}\left(t\right)=\frac{1}{2} C_{d}\rho D_{c}\left|\mathbf{l}\times \mathbf{u}_{rel} \times \mathbf{l}\right|\left(\mathbf{l}\times \mathbf{u}_{rel} \times \mathbf{l}\right)+ \frac{1}{2} C_{f}\rho \pi D_{c}\left|\mathbf{l}\times \mathbf{u}_{rel}\right|\left(\mathbf{l}\times \mathbf{u}_{rel}\right)\times \mathbf{l}  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (27)
|}

where <math display="inline">C_{d}</math> is the non-linear drag coefficient, <math display="inline">D_{c}</math> the diameter of the cable, <math display="inline">\mathbf{l}</math> the unit vector defining the local orientation of a considered section of the cable, <math display="inline">C_{f}</math> the friction coefficient, and <math display="inline">\mathbf{u}_{rel}</math> the relative fluid velocity of the cable respect to the velocity induced by incidents waves.

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, which, in this work, is modelled as a spring-damping system <span id='citeF-36'></span><span id='citeF-37'></span>[[#cite-36|[36,37]]], and implemented as follows:

<ol>

<li>The vertical forces due to the stiffness of the seabed in each time step are calculated as </li>

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\begin{array}{l} c_{l}w\,l_{l} & & \mbox{ if}\; G_{s}A_{c}\left(z_{r}-z\right)>c_{l}wl_{l} \\ G_{s}A_{c}\left(z-z_{r}\right)&& \mbox{ otherwise} \end{array}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (29)
|}

where <math display="inline">c_{l}>1</math> is a coefficient limiting the seabed reactions, <math display="inline">l_{l}</math> the length of the cable element resting on the seabed, <math display="inline">w</math> the weight per unit of length of the line, <math display="inline">w</math> the apparent weight of the cable, <math display="inline">z</math> the vertical coordinate of the cable node, <math display="inline">G_{s}</math> the seabed stiffness, and <math display="inline">A_{c}</math> the contact area of the line element. The term <math display="inline">z_{r}</math> is defined as

<span id="eq-30"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>

z_{r}=z+\frac{wl_{l}}{G_{s}D_c}  

</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (30)
|}

The term <math display="inline">z_{r}</math> can be regarded as a limit of the line subsidence.

<li>The vertical forces due to the damping are modelled as </li>

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>

d_{l} \dot{z}    \mbox{ if}\; z<z_{r} </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (31)
|-
| style="text-align: center;" | <math> 0    \mbox{ otherwise} 

</math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (32)
|}
|}

being <math display="inline">\gamma </math> a term related to the numerical scheme adopted. In addition, <math display="inline">d_{l}</math> is expressed as

<span id="eq-33"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>

d_{l}=2\varsigma _{l}\sqrt{G_{s}\left(\frac{w l_{l}}{g}\right)D l_{l}}  

</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (33)
|}

where <math display="inline">\varsigma _{l}</math> is the seabed damping Palm.

<li>Finally, the horizontal forces due to the seabed interaction are modelled as </li>

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>

d_{l}\dot{x} \;\;\;\; (x-\mbox{direction}) 

</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (34)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>

d_{l}\dot{y} \;\;\;\; (y-\mbox{direction}) 

</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (35)
|}

</ol>

===3.3 Non-linear FEM algorithm solution===

In this work, a FEM approach combined with a co-rotational updated Lagrangian formulation is used for describing the dynamics of the mooring cable. Bar elements, with just three translational degrees of freedom per node, are used for modelling the mooring elements. Figure [[#img-2|2]] shows a scheme of the discretization adopted in this work.

Applying the standard FEM formulation for non-linear elastodynamics problems <span id='citeF-39'></span>[[#cite-39|[39]]], the equations for the dynamic equilibrium of the forces acting on a cable, can be written as:

<span id="eq-36"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bold{f} = \bold{M}\;\ddot{\mathbf{x}} + \bold{C}\;\dot{\mathbf{x}} + \bold{K}\;\mathbf{x}  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (36)
|}

where <math display="inline">\bold{f}</math> is the vector of external loads, <math display="inline">\bold{M}</math> is the inertia matrix of the line, including the added mass effect, <math display="inline">\bold{C}</math> is the matrix corresponding to the structural damping, <math display="inline">\bold{C}</math> is the stiffness matrix, and '''x''' is the displacement vector.

The consistent mass matrix of the line can be obtained as:

<span id="eq-37"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bold{M} = \mathbb{A}\; \left(m^{e} + am^{e}\right)  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (37)
|}

being <math display="inline">\mathbb{A}</math> an suitable finite element assembly operator, the superscrit <math display="inline">e</math> denotes an individual cable element, and <math display="inline">m^{e}</math>, <math display="inline">am^{e}</math> are the elemental inertia and added mass matrixes.

The elemental inertia mass matrices can be obtained as,

<span id="eq-38"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>m^{e} = \rho \int _{V^{e}}{\bold{N}^{T} \bold{N}\,\mbox{d}V}  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (38)
|}

where <math display="inline">\bold{N}</math> is the matrix of shape functions, and <math display="inline">V</math> the volume of an individual element. Similar considerations can be used to calculate the elemental added mass matrices.

Considering the linearization of the current configuration, Eq. ([[#eq-36|36]]) can be expressed in terms of the incremental displacement of the cable nodes, <math display="inline">\delta \mathbf{x}</math>, as

<span id="eq-39"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bold{f} = \bold{M}\;\ddot{\mathbf{x}} + \bold{C}\;\dot{\mathbf{x}} + \bold{K}\;\delta \mathbf{x}+ \bold{P}^0 + \bold{R}^{t}  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (39)
|}

where <math display="inline">\bold{P}^0</math> is the tension vector corresponding to the initial configuration, and <math display="inline">\bold{R}^{t}</math> is the vector of internal forces of the cable in the previous configuration.  The tangent stiffness matrix of the line, <math display="inline">\bold{K}</math>, can be obtained as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bold{K} = \mathbb{A}\; k^{e} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (40)
|}

where <math display="inline">k^{e}</math> is the elemental tangent stiffness matrix, which can be derived by differentiating the internal force, <math display="inline">\bold{u}</math>, with respect to the node displacements,

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>k^{e} = \frac{\partial{\bold{u}}}{\partial{x}} = \frac{\partial{}}{\partial{x}}\left(V^{e}s \frac{\partial{s}}{\partial{x}}\right)= V^{e}E \frac{\partial{s}}{\partial{x}} \otimes \frac{\partial{s}}{\partial{x}} + V^{e}S \frac{\partial ^2{s}}{\partial{x^2}} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (41)
|}

where <math display="inline">V^{e}</math> and <math display="inline">S</math> are the volume and the axial stress of the bar, respectively.

The FEM allows calculating those matrixes as follows:

<span id="eq-42"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>k^{e} = {\bold{B}^{T} V^{e} \mbox{E} \bold{B}} + {V^{e} s \bold{H}}   </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (42)
|}

where <math display="inline">\bold{B}</math> and <math display="inline">\bold{H}</math> are constant-over-element matrixes, whose derivation is detailed in <span id='citeF-39'></span><span id='citeF-40'></span>[[#cite-39|[39,40]]].

The damping matrix is evaluated using a linear Rayleigh's damping model by:

<span id="eq-43"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bold{C} = \mathbb{A}\; \left(c_{1}k^{e} + c_{2}m^{e}\right)  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (43)
|}

where <math display="inline">c_{1}</math> and <math display="inline">c_{2}</math> are coefficients calculated taking into account the characteristic frequencies of the problem.

Further details on the application of the non-linear FEM approach to the cable dynamics equation can be found in <span id='citeF-39'></span><span id='citeF-41'></span>[[#cite-39|[39,41]]].

In this work, an implicit Bossak-Newmark scheme <span id='citeF-42'></span>[[#cite-42|[42]]] is used to integrate Eq. ([[#eq-39|39]]) in time. This offers the following set of algebraic equations, which can be solved iteratively,

<span id="eq-45"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\left[\left(1-\alpha \right)\bold{M}^{t+\Delta t, i} + \Delta t\gamma \,\bold{C}^{t+\Delta t, i} + \Delta t^2 \beta \,\bold{K}^{t+\Delta t, i}\right]\;\ddot{\mathbf{x}}^{t+\Delta t, i+1}  = </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (44)
|-
| style="text-align: center;" | <math> \Delta t^2 \beta \bold{K}^{t+\Delta t, i}\;\ddot{\mathbf{x}}^{t+\Delta t, i} + \bold{f}^{t+\Delta t, i} - \bold{P}^0 - \bold{R}^{t+ \Delta t,i}</math>
|-
| style="text-align: center;" | <math> -\bold{C}^{t+\Delta t, i}\left[\dot{\mathbf{x}}^{t} - \Delta t \left(\gamma + 1\right)\ddot{\mathbf{x}}^t \right]- \alpha \,\bold{M}^{t+\Delta t, i} \;\ddot{\mathbf{x}}^t  </math>
|}
|}

where <math display="inline">\Delta t</math> is the time step, <math display="inline">i</math> denotes the <math display="inline">i</math>-th iteration and <math display="inline">\alpha </math>, <math display="inline">\beta </math> and <math display="inline">\gamma </math> are numerical parameters of the Bossak-Newmark method <span id='citeF-42'></span>[[#cite-42|[42]]].

Eq. ([[#eq-45|45]]) is solved iteratively, and a criterion based on the maximum difference between the position reached by the nodes in two consecutive iterations is applied to evaluate the convergence. Furthermore, an Aitken's <math display="inline">\Delta ^2</math> method is used to accelerate the iterative process <span id='citeF-43'></span>[[#cite-43|[43]]].

Once Eq. ([[#eq-45|45]]) is solved, the updated position and velocity of each node of the cable discretization can be evaluated as (see Fig. [[#img-3|3]])

<span id="eq-46"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{x}^{t+\Delta t} = \mathbf{x}^t+\Delta t \dot{\mathbf{x}}^t+\frac{\Delta t^2}{2}\left[\left(1+2\beta \right)\ddot{\mathbf{x}}^t+2\beta \,\ddot{\mathbf{x}}^{t+\Delta t}\right]</math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (45)
|-
| style="text-align: center;" | <math> \dot{\mathbf{x}}^{t +\Delta t} = \dot{\mathbf{x}}^t + \Delta t\left[\left(1-\gamma \right)\ddot{\mathbf{x}}^t + \gamma \,\ddot{\mathbf{x}}^{t + \Delta t}  \right]  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (46)
|}
|}

The iterative procedure for solving the cable dynamics can be summarized as follows:

<ol>

<li>Setting the problem statement and boundary conditions, as well as the initial equilibrium configuration, of the mooring line. For instance, initial position based on a static catenary with touchdown can be chosen. The tension corresponding to the initial configuration is used to define <math display="inline">\bold{P}^0</math>. </li>
<li>Evaluation of mass, damping, and tangent stiffness matrixes is carried out by assembling the different elemental matrixes for the current configuration. </li>
<li>Calculation of external forces for each cable element for the current configuration. </li>
<li>Solving cable dynamics Eq. ([[#eq-45|45]]) and update position and velocity of each node of the cable using Eq. ([[#eq-46|46]]). </li>
<li>Iteration from the step 2 through 4 until convergence is reached for each element of the line. </li>

</ol>

==4 Algorithm to solve the floater dynamics==

Most of the models found in the literature to solve the moored-body dynamics use a frequency-domain analysis of the floater response together with a convolution integral <span id='citeF-44'></span>[[#cite-44|[44]]] for solving the motions of the coupled system <span id='citeF-11'></span><span id='citeF-21'></span><span id='citeF-23'></span>[[#cite-11|[11,21,23]]]. This work uses a different strategy, taking advantage of the characteristics of the presented time-domain FEM seakeeping model to straightforwardly couple the floater dynamics with the developed non-linear mooring model.

Fig. [[#img-3|3]] shows the algorithm used to solve the dynamics of the coupled system. In order to accelerate the solution of the non-linear solver, it includes two nested loops; the outer loop iterates on the non-linear wave diffraction-radiation problem, while the inner loop takes into account the remaining external forces vector acting on the floater, including the mooring reaction forces. In this figure, <math display="inline">\bold{F}^0</math>, are the external forces different from the hydrodynamic and the mooring forces, and <math display="inline">\bold{K}_{M}</math> is the linear stiffness matrix of the mooring system.

Cable dynamics solver is formulated in term of the acceleration, and this is also valid for the boundary condition of the fairlead connection point. Taking into account the time integration scheme used, the displacement of the end node, <math display="inline">\mathbf{x}^{*}</math>, must fulfill the following compatibility relationship:

<span id="eq-47"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{x}^* = \mathbf{x}^t + {\Delta t} \dot{\mathbf{x}}^t + \frac{{\Delta t}^2}{2}  \left[\left(1 + 2\beta \right)\ddot{\mathbf{x}}^t + 2\beta \ddot{\mathbf{x}}^* \right],  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (47)
|}

where <math display="inline">\ddot{\mathbf{x}}^*2</math> is the acceleration of the fairlead connection point, that can be evaluated in each time step, as follows

<span id="eq-48"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\ddot{\mathbf{x}}^* = \frac{1}{\beta {\Delta t}^2} \left[\mathbf{x}^* - \mathbf{x}^t - {\Delta t}\dot{\mathbf{x}}^t - \frac{{\Delta t}^2}{2} \left(1 + 2\beta \right)\ddot{\mathbf{x}}^t \right]  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (48)
|}

In order to reduce the computing time, mooring reaction forces within the inner loop are evaluated using a linear approach. For this purpose, the stiffness matrix of the mooring system, <math display="inline">\bold{K}_{M}</math>, is estimated every first iteration of the loop, by calculating the Jacobian of the mooring reaction forces using numerical differentiation,

<span id="eq-49"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\bold{K}_{M}=\left[\begin{array}{ccc}\partial R_x/\partial x & \partial R_x/\partial y & \partial R_x/\partial z \\ \partial R_y/\partial x & \partial R_y/\partial y & \partial R_y/\partial z \\ \partial R_z/\partial x & \partial R_z/\partial y & \partial R_z/\partial z \end{array}\right],  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (49)
|}

where <math display="inline">R</math> 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

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{f}_x = R_x + k_{11}\Delta x + k_{12}\Delta y + k_{13}\Delta z, </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (50)
|-
| style="text-align: center;" | <math> \mathbf{f}_y = R_y + k_{21}\Delta x + k_{22}\Delta y + k_{23}\Delta z, </math>
|-
| style="text-align: center;" | <math> \mathbf{f}_z = R_z + k_{31}\Delta x + k_{32}\Delta y + k_{33}\Delta z, </math>
|}
|}

where the terms <math display="inline">k_{ij}</math> are the elements of the mooring stiffness matrix, and <math display="inline">\Delta x</math>, <math display="inline">\Delta y</math> and <math display="inline">\Delta z</math> are the displacements of the fairlead point from the position at the beginning of the current time step.

Finally, the solution procedure can be summarised as follows:

<ol>

<li>For each iteration <math display="inline">k</math> of the solver loop, the wave diffraction-radiation problem given by Eq. ([[#eq-17|17]]), is solved using the floater velocities obtained in the previous converged solution of the floater dynamics loop (see Fig. [[#img-3|3]]). </li>
<li>Once the velocity potential <math display="inline">\varphi </math> is known, hydrodynamic loads acting on the floater are computed.  </li>
<li>Then, the stiffness matrix of the mooring lines is obtained using numerical differentiation techniques. These matrices are used to estimate the mooring forces within the inner loop of the dynamic solver.  </li>
<li>For each iteration <math display="inline">l</math> of the inner loop, the floater dynamic Eqs. ([[#eq-1|1]])-([[#eq-2|2]]) are solved. </li>
<li>Once convergence of the internal loop is reached, new floater position and velocity are used to update the boundary conditions for the wave diffraction-radiation problem. </li>
<li>Iterations in the outer loop are done until convergence is reached. </li>
<li>Once the outer loop converges, it is proceeded to the calculation of the next time step. </li>

</ol>

==5 Validation of the non-linear FEM mooring model==

In this section, some validation examples of the presented non-linear FEM model are presented.

==Case 1: Free vibration cable==

The first validation case is based on that proposed by <span id='citeF-45'></span>[[#cite-45|[45]]]. The experiment consists of a cable initially in a horizontal position with one end free, which is let to evolve under gravitational loads. The computed results are compared with the experimental results taken from <span id='citeF-45'></span>[[#cite-45|[45]]], as well as with the data obtained from a simulation with the FEM structural solver RamSeries <span id='citeF-46'></span>[[#cite-46|[46]]]. The properties of the cable are: length <math display="inline">L =</math> <math display="inline">1.0</math> m, stiffness <math display="inline">EA =</math> 50 N, weight per unit of length <math display="inline">w =</math> <math display="inline">0.98</math> N/m. The distance between the cable end points is <math display="inline">0.881</math> m. The cable was divided into forty four bar elements, and the time step used was <math display="inline">0.001</math> s. The cable position at different times, obtained from the computed results are shown in Fig. [[#img-4|4]]. The obtained numerical results show a good agreement between our results, and those experimentally and numerically obtained by other authors. However, slight discrepancies can be appreciated at the lower end in Fig. [[#img-5|5]]. This fact can be explained by the different numerical parameters chosen to carry out the simulation.

Furthermore, a mesh sensitivity study has been carried out using different discretizations of the cable. As it can be shown in Fig. [[#img-6|6]], the increase on the number of elements leads to a better approximation, showing a rapid convergence of the results towards the experimental data.

==Case 2: Cable attached to a circular rotating plate==

Now, results obtained by the developed FEM cable solver are compared for the model test proposed by Lindhal and Sjoberg <span id='citeF-37'></span><span id='citeF-38'></span>[[#cite-37|[37,38]]]. The experimental set up is shown in Fig. [[#img-7|7]]. The lower end was attached to the concrete floor and the upper end to a circular plate rotating with a fixed speed. The radius of the circular motion was <math display="inline">0.2</math> m. Two cases with different rotational periods of <math display="inline">T_{r}</math> = 1.25 s and <math display="inline">T_{r}=3.5</math> 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.

The properties of the cable are: length <math display="inline">L =</math> <math display="inline">33</math> m, stiffness <math display="inline">EA =</math> <math display="inline">10^4</math> N, weight per unit length <math display="inline">w =</math> <math display="inline">0.18</math> N/m, and diameter <math display="inline">D_{c} =10^{-3}</math> m. In this validation case, the cable is discretized into 200 bar elements, and the time step is set to <math display="inline">0.001</math> 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):

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>x\left(t\right) =  0.2\,\mbox{ tanh} \left(\frac{t}{2}\right)\left(\mbox{ cos}\left[-\frac{2\pi }{T_{r}}t + \delta \right]-\mbox{ cos}\delta \right)\; </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (51)
|-
| style="text-align: center;" | <math> y\left(t\right) =  0.0\; </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (52)
|-
| style="text-align: center;" | <math> z\left(t\right) =  0.2\,\mbox{ tanh} \left(\frac{t}{2}\right)\left(\mbox{ sin}\left[-\frac{2\pi }{T_{r}}t + \delta \right]-\mbox{ sin}\delta \right)\; </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (53)
|}
|}

The results are compared with the experimental results taken from <span id='citeF-37'></span>[[#cite-37|[37]]]. A good agreement between the computed reaction forces and the experimental results <span id='citeF-37'></span>[[#cite-37|[37]]] can be observed in Figs. [[#img-8|8]] (for <math display="inline">T_{r}=3.5</math> s) and [[#img-9|9]] (for <math display="inline">T_{r}=1.25</math> s) for both the maximum values and the time evolution. Only slight differences are appreciated, and probably lying within the experimental uncertainty range. The entire cable loses stiffness at some instants, and the numerical oscillations after the slack are larger in the case of <math display="inline">T_{r}=3.5</math> s, as it was observed in <span id='citeF-38'></span>[[#cite-38|[38]]].

==6 Demonstration of the coupled seakeeping-mooring solver==

A fully coupled analysis of the OC3 spar buoy offshore wind turbine, called OC3&#8211;''Hywind'' (www.ieawind.org/task23/) is presented in this section. Main particulars of the spar buoy FOWT are introduced in <span id='citeF-47'></span><span id='citeF-48'></span>[[#cite-47|[47,48]]]. A general view of the buoy concept can be observed in Fig. [[#img-10|10]]. In this section, several analysis are made:

* The linear Response Amplitude Operators (RAOs) due to waves action are computed and compared with those obtained by other authors <span id='citeF-49'></span>[[#cite-49|[49]]].
* The dynamics of the OC3 spar wind turbine subject to different monochromatic waves, with periods close to the pitch resonance, is calculated. The influence in the results of different mooring models (i.e. linear,  quasi-static catenary, and FEM cable) is studied.
* A simulation in real operational conditions of the spar OC3 is finally performed. First and second-order irregular waves, real wind flow, and mooring loads (with quasi-static catenary and FEM cable models) are taken into account. The influence of the mooring model, the effects of second-order wave, and mooring tensions are studied.

===6.1 RAO analysis and comparison===

First, a RAO analysis is performed in the absence of wind. The response of the OC3 spar in the frequency-domain are obtained after applying a Fourier transform to the time history computed by the seakeeping solver. For this purpose, a white spectrum with 50 monochromatic waves, wave amplitude 1 meter, and wave periods ranging between 20 and 200 seconds, is used. The sampling time required for performing a Fast Fourier Transfer analysis was 1089 seconds. Figure [[#img-11|11]] shows an inter-code comparison. Note that a good agreement among the different solvers is found Ramanchandran.

===6.2 Mooring analysis around pitch resonance===

In this section, the influence of three different mooring models on the OC3 spar buoy FOWT is analysed. The models are:

* A lineal model that behaves as a spring and is represented by a constant stiffness matrix. This model ignores inertia and damping terms. The loads acting on the FOWT from mooring lines are calculated as,

<span id="eq-54"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>

\bold{f}_{mooring} = \bold{f}_{mooring}^{0} - \bold{C} \mbox{x},  

</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (54)
|}

where <math display="inline">\bold{f}_{mooring}^{0}</math> are the loads acting on the FOWT in the undisplaced position, <math display="inline">\bold{C}</math> is the restoring matrix obtained by linearization <span id='citeF-48'></span>[[#cite-48|[48]]], and <math display="inline">\mbox{x}</math> is the displacements vector of the FOWT.

* A quasi-static catenary model similar to the one presented in <span id='citeF-50'></span>[[#cite-50|[50]]]. In this model, the mooring lines are analyzed in a local frame of reference. When FOWT moves, the fairlead position of each mooring line change. In this case, the mooring model transforms the fairlead point position from the global to local reference of frame. Then, the non-linear system, defined in terms of the unknown horizontal and vertical distances of the end points of the mooring line, is solved <span id='citeF-29'></span>[[#cite-29|[29]]].

* Finally, the dynamic FEM cable model presented in this work.

Every mooring model has been run in six different cases, defined by first-order monochromatic waves around the pitch resonance frequency. Key parameters of the mooring layout and cable properties of the model can be found in Table [[#table-1|1]]. Fig. [[#img-12|12]] compares the pitch motion for each mooring model. It can be observed that there are big differences in the results offered by the linear model close to the pitch resonance (about <math display="inline">T_{w} =</math> 30 s), while the results are quite similar for the cases farther from that point.

The results obtained suggest that the use of simple mooring models, which cannot account for some important effects such as geometrical stiffness (linear model) or damping effects (spring), can lead to big errors near the resonance frequency of the device. As can be shown, the linear mooring model can diminish the amplitude of floater dynamic response close to the resonance frequency, and it could lead to underestimate the safety factors on the FOWT design.

===6.3 Simulation in operational conditions===

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. Similar investigation can be found on <span id='citeF-51'></span>[[#cite-51|[51]]], where the OC3 spar buoy was subjected to different operational conditions. 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 <span id='citeF-52'></span><span id='citeF-53'></span>[[#cite-52|[52,53]]] has been used to linearise with FAST <span id='citeF-52'></span>[[#cite-52|[52]]] the behaviour of the wind turbine around the operating wind speed. It should be remarked that the rotational and periodicity effects are considered in the calculation of the steady state characteristics of the rotor.

The wind loads are estimated considering a non-uniform wind flow, with an average wind speed of 11.4 m/s. The wind flow profile is obtained using Turbsim <span id='citeF-54'></span>[[#cite-54|[54]]], and the wind loads on the wind turbine are obtained from FAST/AeroDyn <span id='citeF-52'></span>[[#cite-52|[52]]]. The aerodynamic model used to compute the loads on the FOWT is the Generalized Dynamic Wake, available on FAST. Dynamic stall and pitching moments are also considered.

A JONSWAP spectrum with a mean wave period <math display="inline">T_m</math>= 12.0 s, and significant wave height <math display="inline">H_s</math>= 6.0 m, is simulated. The key parameters of the different studied cases are presented in Table [[#table-2|2]]. As stated above, two different types of mooring models are analysed; one based on the quasi-static catenary model <span id='citeF-50'></span>[[#cite-50|[50]]], and the other based on the dynamic FEM cable model presented in this work. For the dynamic cable analysis, each mooring line is divided into 200 bar elements.

Figures [[#img-13|13]], [[#img-14|14]] and [[#img-15|15]] 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 quasi-static and cable FEM mooring models offer quite similar results.

Figures [[#img-16|16]], [[#img-17|17]] and [[#img-18|18]] present the Power Spectral Density of the heave, roll, and pitch movements. It can be observed noticeable differences between the results using the first and the second-order wave models, and the quasi-static and FEM cable mooring models. The higher value of 5.846<math display="inline">%</math> of difference between the two mooring models can be found in the case of second-order waves for heave motion. It also can be found large differences between first and second-order waves for both models, in the range between 0.1 and 0.3 rad/s for heave motions. However, regarding rotations (roll and pitch motions) the largest discrepancies are found for higher frequency ranges (from 0.1 to 0.9 rad/s, and from 0.18 to 1.2 rad/s, respectively). The higher differences between first and second-order values (approximately 15 <math display="inline">%</math>) are found for the pitch motion.

Table [[#table-3|3]] shows the mean and RMS values, as well as the motion amplitude for the first and the second-order movements. When comparing the quasi-static and the FEM cable models, only slight differences are observed. In particular, the second-order pitch motion is higher when using the FEM cable model, while the other values remain with similar values for both models.

Figure [[#img-19|19]] shows the  tension for each mooring line at the fairlead point. The FEM cable model recorded larger amplitude oscillations (in the range of 5 s to 10 s period) compared with quasi-static model. The differences observed in Fig. [[#img-19|19]] in the tension amplitude reach up to 24 <math display="inline">%</math>. This fact suggests that using a quasi-static model for fatigue assessment of the mooring lines could overestimate their fatigue life. Furthermore, based on Figure [[#img-19|19]], the second-order simulation provided larger tension values than the first-order simulations. This result suggests that a first-order approximation can underestimate the fatigue loads and might lead to a wrong mooring design.

Table [[#table-4|4]] compares the maximum, minimum, average, and RMS tension values at the fairlead points, obtaining similar values for both mooring models.

Figure [[#img-20|20]] presents the Power Spectral Density of the tension of the mooring lines, showing significant differences between the two mooring models for the higher frequencies.

Finally, a summary of the computational effort required in every case (CPU-seconds) is shown in Table [[#table-5|5]]. Three items regarding with time consumption are evaluated: the calculation of the Jacobian of the mooring system; the iterative algorithm, which includes the Laplacian solver; and the rest of calculations. It is noticed that the quasi-static mooring model spends little time (0.01%) in each time step compared with dynamic FEM cable mooring model (52.45% As can be deducted from Table [[#table-5|5]], the computational cost of using the dynamics models increases the computational effort up to 3 times.

==7 Conclusions==

A non-linear FEM solver for the analysis of the response of moored floating structures, in particular floating wind turbines, has been presented. Based on the results obtained in this work, the following conclusions can be remarked:

* A second-order time-domain seakeeping model based on the FEM has been successfully applied to simulate floating wind turbines.
* A non-linear dynamic FEM cable model, based on a co-rotational updated Lagrangian formulation, has been presented to describe the dynamics of the mooring system. Moreover, the model includes the effect of some non-linear external forces, such as seabed interaction and drag forces.
* An algorithm has been developed to couple the non-linear FEM cable dynamics with the time-domain seakeeping solver.
* In order to accelerate the convergence of the iterative loop of the floater dynamics, the restoring mooring forces are linearised, evaluating the stiffness matrix of the mooring system. Furthermore, the Aitken's method is used to reduce the number of the required iterations.
* The dynamic FEM cable model presented in this work has been validated against several experimental results, obtaining a good agreement between the numerical and the experimental data.
* Different analyses of a FOWT based on the OC3&#8211;''Hywind'' concept has been performed. RAO curves has been compared with those obtained by other authors, showing a good agreement. Moreover, a comparison among a linear, a quasi-static, and the FEM cable models has been carried out for frequencies around pitch resonance, obtaining noticeable differences among the mooring models.
* The OC3&#8211;''Hywind'' FOWT has been analysed under operational conditions. Second-order waves simulations are carried out including a realistic wind profile, and with two different types of mooring models. The FEM cable mooring model combined with second-order waves provides a realistic simulation of the performance of the floating wind turbine. The results suggests that using a quasi-static model for fatigue assessment of the mooring lines could overestimate their fatigue life. Furthermore, first-order waves approximation could underestimate tension values on the mooring systems.

==8 Acknowledgement==

This work was partially supported by the X-SHEAKS project (ENE2014-59194-C2-1-R) of the Ministerio de Economía y Competitividad (Spain).

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==Captions for Tables==

Table 1: Key parameters of pitch resonance study <span id='citeF-48'></span>[[#cite-48|[48]]].

Table 2: Key parameters of fully coupled simulations for Hywind floating offshore wind turbine.

Table 3: Comparison between mean, amplitude and RMS values of FOWT motions for first and second-order wave environment.

Table 4: Comparison between maximum, minimum, mean, and RMS values of fairlead tension for Quasi-static and Dynamic mooring models.

Table 5: Comparison of computational effort in seconds required in coupled simulations.

==Captions for Figures==

Figure 1: Global and local frame of reference used in computation of floater dynamics.

Figure 2: Scheme showing the general approach adopted for the spatial discretization of the cable mooring.

Figure 3: General approach to algorithm for solving the floater dynamics.

Figure 4: Validation of the non-linear FEM mooring model. Case 1: different time step positions of free vibration cable obtained from computed results.

Figure 5: Validation of the non-linear FEM mooring model. Case 1: comparison between computed results of the path of free end of the cable obtained and experimental results of <span id='citeF-45'></span>[[#cite-45|[45]]] and numerical computed with RamSeries.

Figure 6: Validation of the non-linear FEM mooring model. Case 1: comparison between computed results with different spatial discretizations.

Figure 7: Validation of the non-linear FEM mooring model. Case 2: the geometrical set-up of the experimental tests by <span id='citeF-37'></span>[[#cite-37|[37]]].

Figure 8: Validation of the non-linear FEM mooring model. Case 2: comparison between the experimental and numerical cable top end reaction forces. Rotational period <math display="inline">T_{r}</math> = 3.5 s.

Figure 9: Validation of the non-linear FEM mooring model. Case 2: comparison between the experimental and numerical cable top end reaction forces. Rotational period <math display="inline">T_{r}</math> = 1.25 s.

Figure 10: General view of spar buoy wind turbine concept (OC3-Hywind concept).

Figure 11: OC3-Hywind concept. Comparison between the computed result by SeaFEM with those taken from other authors Ramanchandran for a rigid wind turbine with no wind.

Figure 12: Results obtained for mooring analysis around pitch resonance  of OC3-Hywind.

Figure 13: Comparison between heave motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).

Figure 14: Comparison between roll motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).

Figure 15: Comparison between pitch motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).

Figure 16: Comparison between Power Spectral Density of heave motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).

Figure 17:Comparison between Power Spectral Density  of roll motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).

Figure 18: Comparison between Power Spectral Density  of pitch motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).

Figure 19: Comparison of fairlead tension of each mooring line for Cases 1, and 2.

Figure 20: Comparison of Power Spectral Density of lines tension for Cases 1-4.

==Tables==


{|  class="floating_tableSCP wikitable" style="text-align: left; margin: 1em auto;"
|+ <span id='table-1'></span>Table. 1 Key parameters of pitch resonance study <span id='citeF-48'></span>[[#cite-48|[48]]].
|- style="border-top: 2px solid;"
|  Item                                          
| style="text-align: right;" | Value                  
| Unit 
|- style="border-top: 2px solid;"
|    Wave Amplitude                                
| style="text-align: right;" | 1.0                    
| m     
|-
|  Wave Period analysed                          
| style="text-align: right;" | 10; 25; 20; 35; 40; 55 
| s     
|-
|  Number of Mooring Lines                       
| style="text-align: right;" | 3                      
| 
|-
|  Angle Between Adjacent Lines                  
| style="text-align: right;" | 120                    
| deg   
|-
|  Depth to Anchors Below SWL                    
| style="text-align: right;" | 320                    
| m     
|-
|  Depth to Fairleads Below SWL                  
| style="text-align: right;" | 70                     
| m     
|-
|  Radius to Fairleads from Platform Centerline  
| style="text-align: right;" | 853.9                  
| m     
|-
|  Unstretched Mooring Line Length               
| style="text-align: right;" | 902.2                  
| m     
|-
|  Mooring Line Diameter                         
| style="text-align: right;" | 0.9                    
| m     
|-
|  Mooring Line Mass Density                     
| style="text-align: right;" | 77.71                  
| kg/m  
|- style="border-bottom: 2px solid;"
|  Mooring Line Extensional Stiffness            
| style="text-align: right;" | 3.84<math display="inline">\times </math>10<math display="inline">^8</math>     
| N     

|}


{|  class="floating_tableSCP wikitable" style="text-align: left; margin: 1em auto;"
|+ <span id='table-2'></span>Table. 2 Key parameters of fully coupled simulations for OC3-Hywind floating offshore wind turbine.
|- style="border-top: 2px solid;"
| colspan='4' style="text-align: center;" | Quasi-Common key parameter
|- style="border-top: 2px solid;"
|   Average Wind velocity   
| <math>W_{rel}</math>
| (m/s) 
| style="text-align: center;" | 11.4  
|-
| Wind direction          
| <math>\theta _{rel}</math>
| (deg) 
| style="text-align: center;" | 0.0   
|-
| Significant wave height 
| <math>H_{s}</math>
| (m)   
| style="text-align: center;" | 6.0   
|-
| Peak period             
| <math>T_{p}</math>
| (s)   
| style="text-align: center;" | 12.0  
|-
| Mean wave direction     
| <math>\theta _{w}</math>
| (deg) 
| style="text-align: center;" | 0     
|-
| Number of mooring lines 
| 
| (-)   
| style="text-align: center;" | 3     
|-
| Mooring lines elements  
| 
| (-)   
| style="text-align: center;" | 200   
|- style="border-top: 2px solid;"
|                          
| Mooring Model  
| Wave Spectrum    
| style="text-align: center;" | 
|- style="border-top: 2px solid;"
|  Case 1                  
| Quasi-static   
| JONSWAP 1<math display="inline">^{st}</math> 
| style="text-align: center;" | 
|-
| Case 2                  
| Non-linear     
| JONSWAP 1<math display="inline">^{st}</math> 
| style="text-align: center;" | 
|-
| Case 3                  
| Quasi-static   
| JONSWAP 2<math display="inline">^{nd}</math> 
| style="text-align: center;" | 
|- style="border-bottom: 2px solid;"
| Case 4                  
| Non-linear     
| JONSWAP 2<math display="inline">^{nd}</math> 
| style="text-align: center;" | 

|}


{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;"
|+ <span id='table-3'></span>Table. 3 Comparison between mean, amplitude and RMS values of FOWT motions for first and second-order wave environment.
|- style="border-top: 2px solid;"
| colspan='7' | Quasi-static
|- style="border-top: 2px solid;"
| style="text-align: left;" |                      
| Surge (m) 
| Sway (m) 
| Heave (m) 
| Roll (deg) 
| Pitch (deg) 
| Yaw (deg)
|- style="border-top: 2px solid;"
| style="text-align: left;" |  Mean 1<math display="inline">^{st}</math>      
| 0.04      
| 0.00     
| 0.00     
| 0.29      
| 0.41        
| 0.21  
|-
| style="text-align: left;" | Mean 2<math display="inline">^{nd}</math>      
| 0.04      
| 0.00     
| 0.00     
| 0.29      
| 0.41        
| 0.21  
|-
| style="text-align: left;" | Amplitude 1<math display="inline">^{st}</math> 
| 14.96     
| 1.34     
| 2.75      
| 1.91       
| 6.12        
| 10.38 
|-
| style="text-align: left;" | Amplitude 2<math display="inline">^{nd}</math> 
| 14.96    
| 1.34     
| 2.75      
| 1.91       
| 6.12        
| 10.38 
|-
| style="text-align: left;" | RMS 1<math display="inline">^{st}</math>       
| 2.76      
| 0.21     
| 0.53      
| 0.52       
| 1.09        
| 1.92  
|-
| style="text-align: left;" | RMS 2<math display="inline">^{nd}</math>       
| 2.73      
| 0.24     
| 0.45      
| 0.48       
| 1.22        
| 1.91  
|- style="border-top: 2px solid;"
| colspan='7' | Non-linear FEM
|- style="border-top: 2px solid;"
| style="text-align: left;" |                      
| Surge (m) 
| Sway (m) 
| Heave (m) 
| Roll (deg) 
| Pitch (deg) 
| Yaw (deg)
|- style="border-top: 2px solid;"
| style="text-align: left;" |  Mean 1<math display="inline">^{st}</math>      
| 0.00      
| 0.00     
| -0.01     
| 0.29       
| 0.40        
| 0.21  
|-
| style="text-align: left;" | Mean 2<math display="inline">^{nd}</math>      
| 0.10      
| 0.00     
| -0.03     
| 0.30       
| 0.42        
| 0.28  
|-
| style="text-align: left;" | Amplitude 1<math display="inline">^{st}</math> 
| 13.82     
| 1.12     
| 2.61      
| 1.92       
| 6.03        
| 10.31 
|-
| style="text-align: left;" | Amplitude 2<math display="inline">^{nd}</math> 
| 15.19     
| 1.55     
| 2.59      
| 2.02       
| 8.00        
| 10.83 
|-
| style="text-align: left;" | RMS 1<math display="inline">^{st}</math>       
| 2.54      
| 0.18     
| 0.50      
| 0.52       
| 1.09        
| 1.92  
|- style="border-bottom: 2px solid;"
| style="text-align: left;" | RMS 2<math display="inline">^{nd}</math>       
| 2.73      
| 0.24     
| 0.45      
| 0.48       
| 1.22        
| 1.91  

|}


{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;"
|+ <span id='table-4'></span>Table. 4 Comparison between maximum, minimum, mean, and RMS values of fairlead tension for Quasi-static and Dynamic mooring models.
|- style="border-top: 2px solid;"
| style="text-align: left;" |  Line    
| Case 
| Max. (N)            
| Min. (N)            
| Mean (N)            
| RMS (N)   
|- style="border-top: 2px solid;"
| style="text-align: left;" |   Line 1  
| 1    
| 1.435<math display="inline">\times </math>10<math display="inline">^6</math> 
| 1.073<math display="inline">\times </math>10<math display="inline">^6</math> 
| 1.250<math display="inline">\times </math>10<math display="inline">^6</math> 
| 1.252<math display="inline">\times </math>10<math display="inline">^6</math> 
|-
| style="text-align: left;" | Line 2  
| 1    
| 1.446<math display="inline">\times </math>10<math display="inline">^6</math> 
| 1.072<math display="inline">\times </math>10<math display="inline">^6</math> 
| 1.240<math display="inline">\times </math>10<math display="inline">^6</math> 
| 1.242<math display="inline">\times </math>10<math display="inline">^6</math> 
|-
| style="text-align: left;" | Line 3  
| 1    
| 7.411<math display="inline">\times </math>10<math display="inline">^5</math> 
| 5.209<math display="inline">\times </math>10<math display="inline">^5</math> 
| 5.991<math display="inline">\times </math>10<math display="inline">^5</math> 
| 6.003<math display="inline">\times </math>10<math display="inline">^5</math> 
|- style="border-top: 2px solid;"
| style="text-align: left;" |  Line 1  
| 2    
| 1.457<math display="inline">\times </math>10<math display="inline">^6</math> 
| 1.052<math display="inline">\times </math>10<math display="inline">^6</math> 
| 1.246<math display="inline">\times </math>10<math display="inline">^5</math> 
| 1.248<math display="inline">\times </math>10<math display="inline">^6</math> 
|-
| style="text-align: left;" | Line 2  
| 2    
| 1.471<math display="inline">\times </math>10<math display="inline">^6</math> 
| 1.040<math display="inline">\times </math>10<math display="inline">^6</math> 
| 1.236<math display="inline">\times </math>10<math display="inline">^6</math> 
| 1.238<math display="inline">\times </math>10<math display="inline">^6</math> 
|- style="border-bottom: 2px solid;"
| style="text-align: left;" | Line 3  
| 2    
| 7.421<math display="inline">\times </math>10<math display="inline">^5</math> 
| 4.510<math display="inline">\times </math>10<math display="inline">^5</math> 
| 5.923<math display="inline">\times </math>10<math display="inline">^5</math> 
| 5.936<math display="inline">\times </math>10<math display="inline">^5</math> 

|}


{|  class="floating_tableSCP wikitable" style="text-align: right; margin: 1em auto;"
|+ <span id='table-5'></span>Table. 5 Comparison of computational effort in seconds required in coupled simulations
|- style="border-top: 2px solid;"
| style="text-align: left;" |                   
| style="text-align: left;" | 
| style="text-align: left;" | 
| colspan='2' style="text-align: center;" | Quasi-static mooring
| colspan='2' style="text-align: center;" | Dynamic mooring
|- style="border-top: 2px solid;"
| style="text-align: left;" |   Total time       
| style="text-align: left;" | 
| style="text-align: left;" | 
| '''13882.25''' 
| 100.00% 
| '''42965.03''' 
| 100.00% 
|- style="border-top: 2px solid;"
| style="text-align: left;" |  
| style="text-align: left;" | Jacobian Mooring 
| style="text-align: left;" | 
| '''0.02''' 
| 0.00% 
| '''4317.56''' 
| 10.05% 
|- style="border-top: 2px solid;"
| style="text-align: left;" |  
| style="text-align: left;" | Iterative solver 
| style="text-align: left;" | 
| '''8503.18''' 
| 61.25% 
| '''26046.76''' 
| 60.62% 
|- style="border-top: 2px solid;"
| style="text-align: left;" |  
| style="text-align: left;" | 
| style="text-align: left;" | Mooring solver     
| 1.20 
| 0.01% 
| 22534.77 
| 52.45% 
|-
| style="text-align: left;" | 
| style="text-align: left;" | 
| style="text-align: left;" | Laplacian solver   
| 8142.74 
| 58.66% 
| 3112.52 
| 7.24% 
|-
| style="text-align: left;" | 
| style="text-align: left;" | 
| style="text-align: left;" | Other calculations 
| 359.24 
| 2.59% 
| 399.47 
| 0.93% 
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="text-align: left;" |  
| style="text-align: left;" | Others           
| style="text-align: left;" | 
| '''5379.05''' 
| 38.75% 
| '''12600.71''' 
| 29.33% 

|}

==FIGURES==

<div id='img-1'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_01.png|842px|Global and local frame of reference used in computation of floater dynamics.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 1:''' Global and local frame of reference used in computation of floater dynamics.
|}

<div id='img-2'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_02.png|842px|Scheme showing the general approach adopted for the spatial discretization of the cable mooring.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 2:''' Scheme showing the general approach adopted for the spatial discretization of the cable mooring.
|}

<div id='img-3'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_03.png|842px|General approach to algorithm for solving the floater dynamics.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 3:''' General approach to algorithm for solving the floater dynamics.
|}

<div id='img-4'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_04.png|600px|Validation of the non-linear FEM mooring model. Case 1: different stages of time evolution of cable under its self weight.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 4:''' Validation of the non-linear FEM mooring model. Case 1: different stages of time evolution of cable under its self weight.
|}

<div id='img-5'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_05.png|600px|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 <span id='citeF-45'></span>[[#cite-45|[45]]] and numerical computed with RamSeries.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 5:''' 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 <span id='citeF-45'></span>[[#cite-45|[45]]] and numerical computed with RamSeries.
|}

<div id='img-6'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_06.png|600px|Validation of the non-linear FEM mooring model. Case 1: comparison between computed results with different spatial discretization.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 6:'''  Validation of the non-linear FEM mooring model. Case 1: comparison between computed results with different spatial discretization.
|}

<div id='img-7'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_07.png|842px|Validation of the non-linear FEM mooring model. Case 2: the geometrical set-up of the experimental tests by <span id='citeF-37'></span>[[#cite-37|[37]]].]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 7:''' Validation of the non-linear FEM mooring model. Case 2: the geometrical set-up of the experimental tests by <span id='citeF-37'></span>[[#cite-37|[37]]].
|}

<div id='img-8'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_08.png|600px|Validation of the non-linear FEM mooring model. Case 2: comparison between the experimental and numerical cable top end reaction forces. Rotational period T<sub>r</sub> = 3.5 s.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 8:''' Validation of the non-linear FEM mooring model. Case 2: comparison between the experimental and numerical cable top end reaction forces. Rotational period <math>T_{r}</math> = 3.5 s.
|}

<div id='img-9'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_09.png|600px|Validation of the non-linear FEM mooring model. Case 2: comparison between the experimental and numerical cable top end reaction forces. Rotational period T<sub>r</sub> = 1.25 s.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 9:''' Validation of the non-linear FEM mooring model. Case 2: comparison between the experimental and numerical cable top end reaction forces. Rotational period <math>T_{r}</math> = 1.25 s.
|}

<div id='img-10'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_10.png|842px|General view of the spar buoy wind turbine concept (OC3-Hywind concept).]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 10:''' General view of the spar buoy wind turbine concept (OC3-Hywind concept).
|}

<div id='img-11'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_11.png|600px|OC3-Hywind concept. Comparison between the computed result by SeaFEM with those taken from other authors <span id='citeF-49'></span>[[#cite-49|[49]]] for a rigid wind turbine with no wind.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 11:''' OC3-Hywind concept. Comparison between the computed result by SeaFEM with those taken from other authors <span id='citeF-49'></span>[[#cite-49|[49]]] for a rigid wind turbine with no wind.
|}

<div id='img-12'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_12_10s.png|600px|]]
|[[Image:draft_tresdddst_455990605-Fig_12_25s.png|400px|]]
|-
|[[Image:draft_tresdddst_455990605-Fig_12_30s.png|600px|]]
|[[Image:draft_tresdddst_455990605-Fig_12_35s.png|400px|]]
|-
|[[Image:draft_tresdddst_455990605-Fig_12_40s.png|600px|]]
|[[Image:draft_tresdddst_455990605-Fig_12_55s.png|400px|Results obtained for mooring analysis around pitch resonance  of OC3-Hywind.]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 12:''' Results obtained for mooring analysis around pitch resonance  of OC3-Hywind.
|}

<div id='img-13'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_13.png|600px|Comparison between heave motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 13:''' Comparison between heave motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).
|}

<div id='img-14'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_14.png|600px|Comparison between roll motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 14:''' Comparison between roll motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).
|}

<div id='img-15'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_15.png|600px|Comparison between pitch motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 15:''' Comparison between pitch motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).
|}

<div id='img-16'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_16.png|600px|Comparison between Power Spectral Density of heave motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 16:''' Comparison between Power Spectral Density of heave motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).
|}

<div id='img-17'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_17.png|600px|Comparison between Power Spectral Density  of roll motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 17:''' Comparison between Power Spectral Density  of roll motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).
|}

<div id='img-18'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_18.png|600px|Comparison between Power Spectral Density of pitch motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 18:''' Comparison between Power Spectral Density of pitch motion for first and second-order wave environment for Cases 1-4 (described in Table [[#table-2|2]]).
|}

<div id='img-19'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_19.png|600px|Comparison of fairlead tension of each mooring line for Case 3, and 4.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 19:''' Comparison of fairlead tension of each mooring line for Case 3, and 4.
|}

<div id='img-20'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:draft_tresdddst_455990605-Fig_20.png|600px|Comparison of Power Spectral Density of lines tension for Cases 1-4.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 20:''' Comparison of Power Spectral Density of lines tension for Cases 1-4.
|}