Development and verification of a time-domain approach for determining forces and moments in structural components of floaters with an application to floating wind turbines

Structural design of the floater is an important aspect in developing cost efficient and reliable floating wind turbines. It is difficult to well account for the effect of strong nonlinear dynamic characteristics and transient loading events, e.g. wind turbine faults, of floating wind turbines in a frequency-domain finite element analysis. The time-domain approach which implements the Morison's formula cannot accurately account for the hydrodynamic loads on the hull of floating wind turbines. While, the conventional hybrid frequency-time domain approach (based on the potential flow theory) fails to capture structural responses of the hulls since a rigid-body global model rather than a finite element model of the hull is employed. The present paper deals with the development and verification of a time-domain approach that can be easily implemented in various state-ofthe-art computer codes for wind turbine analysis, e.g. Simo/Riflex/Aerodyn, OrcaFlex and FAST þ CHARM3D, to extend their capabilities to analyze global forces and moments in structural components of a generic floater subject environmental loads from e.g. wind and waves. The global forces and moments in the structural components might be used as inputs of design formulas for structural strength design checks and/or used as boundary conditions in a sub-model finite element analysis to determine structural responses such as stresses. The proposed approach focuses on modeling of the inertia and external loads on the hull and mapping of the loads in the finite element model of the hull. In the proposed approach, floating wind turbines are considered as a system of several structural components, e.g. blades, rotational shaft, nacelle, tower, mooring lines, columns, pontoons and braces, rather than one rigid-body, while a finite element model for the hull is developed to represent the global stiffness of the structural components. The external and inertial loads on the hull are modeled as distributed loads rather than the integrated forces and moments. The conventional hybrid frequency-time domain approach, which is available in the state-of-the-art computer codes, is implemented to model the hydrodynamic loads on each structural component with essential modifications with respect to the corresponding hydrodynamic coefficients, e.g. added mass and potential damping coefficients and wave excitation forces. Approaches for modeling the hydrostatic pressure forces, gravity loads, drag forces and inertial loads on each structural component are also illustrated. Second order and higher order terms of the hydrostatic and hydrodynamic loads and the hydroelasticity effects are not accounted for in the present paper but can be ), zhen.gao@ntnu.no (Z. Gao), torgeir.moan@ntnu.no (T. Moan). r Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4. C. Luan et al. / Marine Structures 51 (2017) 87e109 88 further included. So far, the proposed approach has been implemented in the computer code Simo/Riflex/Aerodyn to analyze global forces and moments in the hull of a semisubmersible wind turbine. Good agreement between the reference values and the simulation results has been observed and indicates that the developed time-domain numerical models are reliable. The simulation results show that the low-frequency aerodynamic loads and fluctuations of hydrostatic pressure forces on and gravity of the floating wind turbine are important contributions to the structural responses, in particular, in the lowfrequency range. © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


a b s t r a c t
Structural design of the floater is an important aspect in developing cost efficient and reliable floating wind turbines. It is difficult to well account for the effect of strong nonlinear dynamic characteristics and transient loading events, e.g. wind turbine faults, of floating wind turbines in a frequency-domain finite element analysis. The time-domain approach which implements the Morison's formula cannot accurately account for the hydrodynamic loads on the hull of floating wind turbines. While, the conventional hybrid frequency-time domain approach (based on the potential flow theory) fails to capture structural responses of the hulls since a rigid-body global model rather than a finite element model of the hull is employed. The present paper deals with the development and verification of a time-domain approach that can be easily implemented in various state-ofthe-art computer codes for wind turbine analysis, e.g. Simo/Riflex/Aerodyn, OrcaFlex and FAST þ CHARM3D, to extend their capabilities to analyze global forces and moments in structural components of a generic floater subject environmental loads from e.g. wind and waves. The global forces and moments in the structural components might be used as inputs of design formulas for structural strength design checks and/or used as boundary conditions in a sub-model finite element analysis to determine structural responses such as stresses. The proposed approach focuses on modeling of the inertia and external loads on the hull and mapping of the loads in the finite element model of the hull. In the proposed approach, floating wind turbines are considered as a system of several structural components, e.g. blades, rotational shaft, nacelle, tower, mooring lines, columns, pontoons and braces, rather than one rigid-body, while a finite element model for the hull is developed to represent the global stiffness of the structural components. The external and inertial loads on the hull are modeled as distributed loads rather than the integrated forces and moments. The conventional hybrid frequency-time domain approach, which is available in the state-of-the-art computer codes, is implemented to model the hydrodynamic loads on each structural component with essential modifications with respect to the corresponding hydrodynamic coefficients, e.g. added mass and potential damping coefficients and wave excitation forces. Approaches for modeling the hydrostatic pressure forces, gravity loads, drag forces and inertial loads on each structural component are also illustrated. Second order and higher order terms of the hydrostatic and hydrodynamic loads and the hydroelasticity effects are not accounted for in the present paper but can be implemented to convert the frequency dependent hydrodynamic pressure loads due to wave diffraction and radiation to the integrated forces and moments corresponding to these 6 d.o.f.s. While, wind loads on blades and tower are typically considered as distributed loads. This approach has been implemented in some computer codes [22] to analyze structural responses of the RNA, tower and mooring lines and rigid-body motions of the hull of floating wind turbines [21,40e45].
The flow chart of a time-domain numerical model of a generic horizontal axis floating wind turbine which implements the hybrid frequency-time domain approach and is developed in the computer code Simo/Riflex/Aerodyn [29e32] is given in Fig. 1 as an example. The hull of the floating wind turbine is considered as a rigid-body with 6 d.o.f.s, while the tower base and fairleads of the mooring lines rigidly follow the motions of the hull. Six motion equations that are composed of the resultant external loads, e.g. viscous loads, gravity loads and hydro loads, on and inertial loads of the hull are generated in Simo [27] with necessary input, e.g. mass properties of the hull, drag coefficients, hydrodynamic coefficients, i.e. the added-mass coefficient matrices (A(u)), potential damping coefficient matrices (B(u)) and first order wave excitation load transfer function (H fw (u)), and specified forces, moments and restoring stiffness matrix. A finite element model, for which the mooring lines, tower and RNA are modeled as bar and beam elements and coupled to the motion equations of the hull are generated and solved in Riflex [28] with necessary input, e.g. relevant mass and structural properties and drag and added mass coefficients. Aerodyn [32] and a Java controller [31] are coupled to Riflex through a dll file [31] to account for the aerodynamic loads on the RNA and tower, effect of pitch control on aerodynamic loads on the three blades and effect of the generator inside the nacelle on the power production and generator torque. More details are available in the later part of this paper and [47].
The conventional hybrid frequency-time domain approach is considered as the state-of-the-art approach and has been used by researchers, e.g. Refs. [21,43,46,47], to analyze responses of floating wind turbines. However, to calculate structural responses of the hull, we must develop a finite element model of the hull rather than a rigid-body formulation; while, to map the loads in the finite element model, we must develop accurate and effective time-domain approach to model the loads on the hull as distributed loads rather than three integrated forces and moments.
This paper addresses a time-domain approach to deal with the challenges mentioned above. The focus is on the modeling of the inertia and external loads on the floating wind turbine hull and the mapping of the loads to the finite element model of the hull. The proposed approach can be easily implemented in various state-of-the-art computer codes, e.g. Simo/Riflex/ Aerodyn, OrcaFlex [22] and FAST þ CHARM3D [22], to extend their capabilities to analyze global forces and moments in structural components of a generic floater subject to linear and non-linear environmental loads, e.g. wind and waves. Details of the proposed approach and verification are available in the later part of this paper. An application of the proposed approach for ULS design check for the structural design of the hull of a semi-submersible wind turbine is available in Ref. [15].

The proposed approach
In general, numerical models in finite element codes are developed in an earth fixed coordinate system, such as the global coordinate system (O g -x g -y g -z g ) shown in Fig. 2.
In the proposed approach, the hull of floating structures is considered as an assemble of d structural components. d is specified by designer. A beam element finite element model, which includes d nodes (red colored in Fig. 2), can be developed in the global coordinate system to represent the global stiffness of the structural components. Each node has 6 d.o.f.s and corresponds to a structural component. The external loads on and inertia loads of each structural component are calculated, integrated and transferred to the node that corresponds to the structural component in the finite element model. In particular, the hydrostatic and the hydrodynamic loads on each structural component are obtained by integrating the hydrostatic and hydrodynamic pressure loads on the wet surface of the structural component. The pressure loads are normally calculated based on a frequency-domain hydrodynamic code using a panel method. Global forces and moments in structural components of the hull can be obtained by carrying out a finite element analysis using a time-domain code. Accurate global forces and moments are given at the cross-sections corresponding to the red dashed lines, see Fig. 2. The number of the structural components and quality of the finite element model of the hull affect the accuracy of the global forces and moments. The beam element finite element model of the hull should accurately represent the global stiffness of the hull, in particular for statically indeterminate structures.
The flow chart of a time-domain numerical model of a generic horizontal axis floating wind turbine which implements the proposed approach and is developed in the computer code Simo/Riflex/Aerodyn is given in Fig. 3. Comparing to Fig. 1 (the hybrid frequency-time domain approach), the proposed approach models the hull as a beam element finite element model while the approaches for modeling the external on and inertia loads of each structural component are illustrated in the following part of this section together with the limitations of the approaches.
The approaches are developed by extending the conventional approaches used in the state-of-the-art computer codes. Therefore, the proposed approach can be easily implemented in various state-of-the-art computer codes to extend their capabilities.
Generating and solving time-domain motion equations for a rigid-body oscillating with respect to its mean position in waves is a fundamental feature in some of the state-of-the-art computer codes used in the offshore wind and offshore oil and gas industry. In these computer codes, usually, three coordinate systems, i.e. Fig. 4, the O f -x f -y f -z f coordinate system is an earth fixed coordinate system located at the mean position of the geometrical center of the water plane area of the floater. The O b -x b -y b -z b coordinate system is a bodyfixed coordinate system. The position of O b and the orientation of the coordinate system rigidly follow rigid-body motions of the floater. The O r -x r -y r -z r coordinate system is a body-related coordinate system. O r rigidly follows horizontal movements of O b (the hull) but the orientation of the body-related coordinate system and vertical position of the O r are fixed (as the same as the body-related coordinate system when the hull is located at its initial position in time-domain simulation). When the floater is located at its mean position, the and O r -x r -y r -z r coordinate systems are coincident. In the motion equations, the hybrid frequency-time domain approach is implemented to account for the wave excitation load effects  and radiation load effects on the rigid-body motions. Hydrodynamic coefficients, i.e. the added-mass coefficient matrices (A(u)), potential damping coefficient matrices (B(u)) and first order wave excitation load transfer function (H fw (u)) must be calculated by 1) solving the potential-flow boundary value problem with the assumption that the hull of the floater is a rigidbody in the O f -x f -y f -z f coordinate system, 2) calculating pressure forces on the mean wet surface of the hull based on the Bernoulli's equation and corresponding velocity potential, 3) integrating the pressure on the wet surface of the hull using the coordinate system O f -x f -y f -z f to obtain the integrated forces and moments acting on O f , and 4) derive the hydrodynamic coefficients based on the corresponding resultant forces and moments on the O f in the O f -x f -y f -z f coordinate system.
In the proposed approach to calculate external and inertial loads on each structural component, we assumes that 1) the atmospheric pressure inside the hull is constantly equal to the atmosphere pressure at the still water plane and 2) the ballast fluid inside the hull is considered as ballast mass which introduce inertia loads on the hull rather than hydro pressure forces on the corresponding inner surface of the hull; while, in the boundary value problem for solving the hydrodynamic loads on each structural component, the hull is considered as a rigid-body. The second order and higher order terms of the hydro loads on the hull, except for the drag forces induced by viscous effect, and hydroelasticity effects are not included in the approaches discussed in the present paper, but they can be further included. The two assumptions are used to simplify the numerical models for the loads on the structural components. The second assumption can be implemented since the focus of the proposed approach is on capturing the global forces and moments in the structural components.
Details of the approaches for modeling inertial and external loads on each structural component are illustrated as follows.
For each structural component, a body-related coordinate system and a body-fixed coordinate system are established. We denote the origins of the body-related and body-fixed coordinate systems for the structural component i as O r,i and O b,i respectively. When the floating wind turbine is located at its mean position, the body-related and body-fixed coordinate systems for each structural component and the O f -x f -y f -z f coordinate system are coincident. The body-fixed coordinate system of the structural component i rigidly follows the motion of the corresponding node of the structural component i in the finite element model in the O g -x g -y g -z g coordinate system. In the O f -x f -y f -z f coordinate system, the motion of the O b,i and the orientation of the body-fixed coordinate system are described by h i ðtÞ ¼ ½h in the L t external;i , R potential_wave_i (t) is the resultants of the wave excitation loads on the structural component i obtained by applying inverse Fourier transform on R wave_i (u). R wave_i (u) is frequency dependent first order wave excitation vector for the structural component i. We have R wave_i (u) ¼ H fw_i (u)*Amp u $Amp u is determined by a spectrum of incident waves. For a sinusoidal wave with a given frequency u, Amp u is the amplitude of the sinusoidal wave. H fw_i (u) is first order wave excitation load transfer function for the structural component i. k i (t) is known as retardation or memory function for the structural component i and determined by corresponding to the high-frequency limit. A i (u) and B i (u) are frequency dependent added mass coefficient matrix and potential damping coefficient matrix for the structural component i.
is a 6 Â 1 vector, acting on O r,i and described in the body-related coordinate system.
Neglecting the second order and higher order terms, the expression of the R r r i ðtÞ is derived as: F ci is a 6Â1 vector. C i is a 6 Â 6 matrix with real coefficients. The expressions of F ci and C i h i (t) are available in Eqs. (3)e(8).
The coefficients in the C i and F ci are expressed by parameters that are defined in the O f -x f -y f -z f coordinate system with respect to the mean wet surface whereas R r r i ðtÞ represents forces and moments acting on O r,i in the body-related coordinate system. We assume that the mean outer wet surface of the structural component i and the corresponding inner surface are identical and are denoted as S 0 wet;i in the O f -x f -y f -z f coordinate system. s 0 is a point on the wet surface of the hull. The normal vector and position vector of s 0 at the mean position are denoted as n 0 ¼ [n 1 ,n 2 ,n 3 ] T and v 0 ¼ [v 1 ,v 2 ,v 3 ] T . n 0 is pointing away from the fluid field. Hydrostatic pressure on the s 0 (P 0 s0;hydro sta ) is given by applying Bernoulli's equation. P 0 represents the atmosphere pressure at the still water plane. Z 0 ¼ 0 since the O f -x f -y f -z f coordinate system is located at the still water plane. r f is density of sea water, taken as 1.025 tonnes/m 3 . g is gravity acceleration, 9.81 m/s 2 . We denote the atmospheric pressure inside the hull as P inner, atm . We assume P inner, atm is constantly equal to P 0 . P 0 s0;net denotes the net pressure on the s 0 at the mean position. P 0 s0;net is the difference between the hydrostatic pressure on the outer surface of the s 0 and the atmospheric pressure on the inner surface of the s 0 .
Flexibility effects of the hull on the hydro loads are not accounted for. The hydrodynamic loads on each structural component are derived from the velocity potential that are obtained by solving the boundary value problems with the assumption that the hull is a rigid-body. The kinematics of different structural components is constrained by the rigid-body assumption. Consequently, the hydrodynamic interaction effects between the structural components are included in the hydrodynamic loads on each structural component. However, the proposed approach may not be able to well model the hydroelasticity effect. Therefore, it is not recommended to be applied on floating structures with relatively large flexibility, for which hydroelasticity effect can be important.
The expressions of L t external;i and R r r i ðtÞ can be further modified. For example, additional terms can be included to account for second order and/or higher order hydro loads on each structural component, while the load effects of the ballast fluid can be modeled as pressure forces on the inner surface of each structural component.

Verification of the proposed approach
The proposed approach is implemented in Simo/Riflex/Aerodyn [29e32] to calculate global forces and moments of the 5-MW-CSC [15]. The layout of the semi-submersible floater is given in Figs. 5 and 6. The O g -x g -y g -z g and O f -x f -y f -z f coordinate systems are established at the mean position of the geometrical centre of the water plane area.
Five numerical models have been developed. Three comparisons, i.e. Comparison A, Comparison B and Comparison C, have been carried out to verify the proposed approach step by step. These numerical models and the comparisons are briefly explained below and illustrated in Figs. 7 and 8, respectively. Detailed descriptions for the numerical models are available in the later part of this paper. A summary of the features of the time-domain models is available in Table 3.   FDM is a frequency-domain model of the 5-MW-CSC developed in WADAM [26] to calculate wave induced global forces and moments by implementing a standard procedure used in the offshore oil and gas industry [26].
TDM-2B-L is a time-domain model of the 5-MW-CSC developed in Simo/Riflex [27,28]. The proposed approach is implemented to calculate wave induced global forces and moments in the cross-section as shown by the dashed line in Fig. 5. Aerodynamic loads are not available in the TDM-2B-L.
TDM-1BC is a time-domain model developed in Simo/Riflex/Aerodyn. Aerodynamic loads on the RNA and tower are appropriately accounted for in the TDM-1BC, while the conventional hybrid frequency-time domain approach is implemented to model the hydro loads on the semi-submersible hull. The modeling approach implemented in the TDM-1BC is considered as the state-of-the-art approach that has been used by researchers, e.g. Refs. [21,43,46,47], to analyze responses of floating wind turbines in wind and waves except for the global forces and moments in the hull since the approach models the hull as one rigid-body with 6 d.o.f.s.
TDM-2BN is an extension of the TDM-1BC. The proposed approach is implemented to calculate global forces and moments in the hull (in the cross-section as shown by the dashed line in Fig. 5). The TDM-1BC and TDM-2BN are identical except for the finite element model of the hull and method for modeling the external and inertial loads on the hull.
TDM-29BN is an extension of the TDM-2BN. The TDM-2BN includes two rigid-bodies for the hull, while, the TDM-29BN includes twenty-nine rigid-bodies for the hull. The TDM-29BN model is developed and compared to the TDM-2BN model in order to show that the proposed method can be generalized to a model consisting of any number of structural components. From the practical use point of review, it is convenient to use the TDM-29BN model to obtain the dynamic responses at any critical position of the hull by just one time-domain model. While, using the TDM-2BN approach, many different numerical models need to be built and analyzed.
It is expected that the time-domain model TDM-2BN can calculate the global forces and moments in the cross-section as shown by the dashed line in Fig. 5 while the time-domain model TDM-29BN can calculate the global forces and moments in the same cross-section and the other twenty-seven cross-sections.
As far as the authors know, there is no published experimental data for the global forces and moments in the hull of floating wind turbines in wind and waves. In addition, the state-of-the-art time-domain computer codes cannot accurately calculate the global forces and moments in the hull [22].
The accuracy of the calculated responses is related to two modeling issues: 1) whether or not the computer codes can accurately calculate the wind and waves induced external and inertial loads on the floating wind turbines and map the loads to the generated finite element models of the floating wind turbines; and 2) whether or not the finite element models generated in the computer codes can accurately represent the global stiffness of the floating wind turbines and calculate the structural responses for given loads. These two features are coupled. The hull of the 5-MW-CSC is a statically determinate structure. For a statically determinate structure, in general, the accuracy of the responses in the structure is purely determined by the accuracy of the external loads acting on the structural components, like aerodynamic loads on the blades and the tower and hydrodynamic loads on the floater and mooring lines and the inertial loads.
In linear theory, hydrodynamic and hydrostatic loads on structural components of the hull are determined by two issues. We denote these issues as ISA and ISB respectively.
ISA: The configuration and shape of the mean wet surfaces of the structural components. ISB: The motion responses of the structural components. The accuracy of the global forces and moments in the hull calculated by the TDM-2BN is indicated by the results of Comparison A and Comparison B. Comparison A focuses on verifying that the influence of the ISA on the hydro loads on the structural components can be accurately modeled in finite element codes which implement the proposed approach. Comparison B focuses on verifying that finite element codes which implement the proposed approach can accurately predict the motion responses of the structural components and responses of the RNA, tower and mooring lines of the reference semisubmersible wind turbine in wind and waves. Comparison C is carried out, to some extent, to address that the proposed approach can be applied to generic floating wind turbines, for which the hull may need to be modeled by any number of structural components. The proposed approach is not necessarily limited to two rigid-bodies for the hull. Due to the limitation of the proposed approach, we do not account for the hydroelasticity effect in the comparisons discussed in the present paper.
In Comparison A, the FDM is used as a reference model for the transfer functions of wave to global forces and moments in the hull. While the transfer functions can also be derived from structural responses of the hull calculated by carrying out regular and/or irregular wave analysis in time-domain numerical models that implement the proposed approach, e.g. TDM-2B-L, TDM-2BN and TDM-29BN. The FDM is a linear system and does not account for non-linear effects on the global forces and moments, while TDM-2B-L is developed to be, as much as possible, a linear system and equivalent to the FDM. The agreement in the transfer functions calculated in the FDM and TDM-2B-L is expected to be good if the proposed approach accurately models the hydro pressure forces on the structural components of the hull and maps the forces on the finite element model of the hull.
TDM-2B-L models the 5-MW-CSC as two structural components connected by three artificial beam elements. Aerodynamic loads are not accounted for in the TDM-2B-L. In contrast, aerodynamic loads are accounted for in the TDM-2BN while the TDM-2BN models the hull of the 5-MW-CSC as two structural components connected by three artificial beam elements and the mooring lines, tower and blades as beam elements. The mean wet surfaces of the two structural components of the TDM-2B-L are identical to the mean wet surfaces of the two structural components of the TDM-2BN correspondingly and respectively. Consequently, if the influence of the ISA on the hydro loads on the structural components can be accurately modeled in TDM-2B-L, the influence can be accurately modeled in the TDM-2BN.
The motions of the hull and responses of the RNA, tower and mooring lines predicted by TDM-2BN and TDM-1BC are compared in Comparison B. TDM-1BC is used as a reference model except for the global forces and moments in the hull. TDM-1BC and TDM-2BN are identical except for the finite element model of the hull and method for modeling the external and inertial loads on the hull. Therefore, agreement in the compared responses is expected to be good.
Since the hull of the 5-MW-CSC is a statically determinate structure, it is expected that the global structural stiffness of the hull does not affect the global forces and moments in the structural components of the hull except for the inertia loads and hydro loads induced by the flexible modes of the hull. The global structural stiffness of the hull is determined by properties of the equivalent cross-sections of the pontoons and columns and material properties, e.g. Young's modulus and modulus of rigidity. In Comparison C, artificial material properties are implemented to make the global structural stiffness of the hull of the TDM-29BN to be of the same magnitude as the one of the TDM-2BN. Consequently, the global forces and moments calculated by the TDM-2BN and TDM-29BN are expected to be identical. Research on the importance of the influence of the inertia loads and hydro loads induced by the flexible modes of the hull is interesting and will be investigated in future.

FDM
The FDM is a frequency-domain model developed in WADAM [26] to calculate wave induced global forces and moments in a cross-section as shown by the dashed line in Fig. 5. Only wave loads are considered here. A realistic cross-section of the pontoon of the hull may be composed of stiffened plates, stiffeners and girders. As shown in Fig. 6, in global analysis, the realistic cross-section can be simplified as a box-shape cross-section with equivalent thickness. The mean position of the geometrical center of the box-shape cross-section in the O f -x f -y f -z f coordinate system is (31.5, 0, À27). The cross-section divides the semi-submersible wind turbine into two parts: "Part A" and "Part B". The wind turbine is included in the "Part B". The global forces and moments in the cross-section are derived based on the fact that the global forces and moments in the cross-section and inertial and external loads on the "Part A" (as well as on Part B) must be in equilibrium. The standard procedure used in the offshore oil and gas industry [26] is implemented in WADAM to calculate the inertial and external loads on the "Part A".
The inertial loads on the "Part A" are determined by the mass and acceleration of the "Part A". Motion equations are generated and solved in frequency domain to derive the acceleration. In the motion equations, the RNA, tower and hull are modeled as a single rigid-body with 6 d.o.f.s. Structural flexibility of the floating wind turbine is neglected. The mooring lines are not included in the rigid-body formulation. Instead, a 6 Â 6 restoring stiffness matrix (C mooring ) is introduced in the motion equations to represent the restoring effect of the mooring lines on the motions. Hydrodynamic coefficients used in the motion equations are calculated by solving the potential-flow boundary value problem with the rigid-body assumption. The motion equations do not include viscous effect. The external loads on the "Part A" are composed of the first order hydrodynamic loads and the fluctuations of hydrostatic pressure forces and gravity loads.  Table 1. Each end node rigidly follows the motions of its corresponding rigid-body (h PA (t) or h PB (t)). The artificial beam elements are massless. There are no external loads on the artificial beam elements. Each artificial beam element only has axial and torsional stiffness. Artificial Young's modulus and modulus of rigidity are specified to make the artificial beams be stiff. For each beam element, the product of the Young's modulus and cross-section area is specified as 10 9 kN, while, the product of the torsional rigidity and modulus of rigidity are specified as 10 9 kNm 2 /rad. The proposed approach is implemented to calculate the first order hydrodynamic loads, gravity and hydrostatic pressure forces on the "Part A" and "Part B" and map the loads on the end nodes of the artificial beam elements. Viscous loads on the hull are not included. The mooring lines induced forces and moments on the floating wind turbine are accounted for by (À1)*C mooring *h PB (t), which are acting on O r,PB and described in the body-related coordinate system of the rigid-body that corresponds to the "Part B".

TDM-1BC
The TDM-1BC is a time-domain model developed in Simo/Riflex/Aerodyn. The time-domain model implements the conventional hybrid frequency-time domain approach. The hull is modeled as a rigid-body with 6 d.o.f.s in Simo. The conventional hybrid frequency-time domain approach is used to account for the first order wave excitation and radiation loads on the hull. The blades, shaft of the drive train inside the nacelle, tower and mooring lines are modeled as beam elements in Riflex. The motions of the lower end node of the tower and the upper nodes of the mooring lines rigidly follow the motions of the hull. The hub and nacelle are modeled as rigid mass points attached on the shaft and top of the tower. Aerodynamic loads on the blades and tower are calculated in Aerodyn [32]. A dll file [31] is used to account for the effect of pitch control on aerodynamic loads on the three blades and the effect of the generator inside the nacelle on the power production and generator torque. The torque of the generator is calculated by the dll file based on the rotational speed of the shaft. The shaft is rotational about its longitudinal axis. The rotational d.o.f. is achieved by applying a flex joint [28] on the beam element of the shaft. The blades are connected to the tower through the shaft. Loads on the blades, hub, shaft and generator torque are transferred through the flex joint to the beam element of the tower. Hydrodynamic loads on the mooring lines are accounted for by the Morison formula. The drag term of the Morison formula is used to account for the viscous loads on the hull. The non-dimensional drag coefficients (C d ) are specified in DNV [33]. C d for the width and height of the pontoons of the 5-MW-CSC is 1.95. C d for the central column is 0.8. C d for the side columns is 0.64. The work-flow chart is available in Fig. 1. Table 1 Positions of end nodes of three artificial beams in the body-fixed coordinate system (Units in meter).

TDM-2BN
The TDM-1BC and TDM-2BN are identical except for the finite element model of the hull and method for modeling the external and inertial loads on the hull. The cross-section, as shown by the dashed line in Fig. 5, discretizes the hull into two components. The two components of the hull are modeled as two rigid-bodies: "Body1" and "Body2". The wind turbine is mounted on the "Body 2". Each rigid-body has 6 d.o.f.s. The origins of the body-fixed and body-related coordinate systems for the "Body1" and "Body2" are denoted as O b,B1 and O r,B1 and O b,B2 and O r,B2 respectively. When the floating wind turbine is located at its mean position, the body-fixed and body-related coordinate systems are coincident to the O f -x f -y f -z f coordinate system. The positions and orientations of the O b,B1 and O b,B2 are described by h B1 (t) and h B2 (t) in the O f -x f -y f -z f coordinate system. The two rigid-bodies are connected by the artificial beam elements used in the TDM-2B-L. Each end node of the artificial beam element rigidly follows the motions of its corresponding rigid-body (h B1 (t) or h B2 (t)). The proposed approach is implemented to calculate the first order hydrodynamic loads, gravity and hydrostatic pressure forces on the "Body1" and "Body2" and map the loads on the end nodes of the artificial beam elements. The drag term of the Morison formula is used to account for the viscous loads on the hull. The non-dimensional drag coefficients used in TDM-2B-L and TDM-2BN are identical. The work-flow chart is available in Fig. 3.

TDM-29BN
The TDM-29BN is an extension of the TDM-2BN. Numerical models for the RNA, tower and mooring lines of the TDM-2BN and TDM-29BN are identical. In the TDM-29BN, we consider that the hull is composed of twenty-nine structural components. For example, the blue colored parts in Fig. 9 are the structural components named "ICP_S1" and "SP3_2" respectively. Each structural component corresponds to a reference node (the brown colored circle). Each reference point represents 6 d.o.f.s of the corresponding structural component. The reference points are connected by beam elements that represent the flexibility of the hull. The beam elements are massless and there are no external loads on the beam elements. The viscous drag is accounted for by the drag term of the Morison formula and being integrated and transferred to the corresponding reference nodes. The end nodes of beam elements rigidly follow the motions of the corresponding reference node. In the 5-MW-CSC, there are four interfaces between the columns and pontoons. The ICP_S1 represents an interface between the side column 1 and the pontoon 1. For the TDM-29BN, the stiffness of the interfaces is not modeled since the interfaces are modeled as rigidbodies. The stiffness of the beam elements are determined by properties of the equivalent cross-sections of the pontoons and columns and material properties, e.g. Young's modulus and modulus of rigidity. In the Comparison C, artificial material properties are implemented to make the global structural stiffness of the hull of the TDM-29BN be in the same level as the one of the TDM-2BN. The specified stiffness properties of the beam elements are tabulated in Table 2. EA g represents the product of the Young's modulus and cross-section area. EI g represents the product of the Young's modulus and the second moment of the area of the cross-section. GJ g represents the product of the torsional rigidity and modulus of rigidity. The work-flow chart is available in Fig. 3.   Figs. 5 and 6. The global forces and moments are acting on the origin of the O inp -x inp -y inp -z inp coordinate system and described in the O inp -x inp -y inp -z inp coordinate system. The O inp -x inp -y inp -z inp coordinate system is a body-fixed coordinate system. When the 5-MW-CSC is located at its mean position, the body-fixed coordinate system is coincident to the O f -x f -y f -z f coordinate system except that the origin of the body-fixed coordinate system is located at the geometrical center of the crosssection which is (31.5, 0, -27) For a given point on the cross-section, axial stress (s x ) and shear stress (t) are calculated by Eqs. (10) and (11).
A is the area of the cross-section. Alternatively, transfer function moduli can be derived from the spectral densities of the incident waves and global forces and moments. For each wave direction, a 10-hour realization of wave elevation is generated from a white noise spectrum. The frequency range of the spectrum is from 0.3 rad/s to 2.2 rad/s. Significant wave height of the spectrum is specified as 1.233 m (H s ¼ 4 ffiffiffiffiffiffi ffi m 0 p , m 0 denotes the variance-area under the spectral density function). The wave induced motions and global forces and moments are calculated by the TDM-2B-L. The spectral densities are obtained by applying inverse Fourier transform, with a fixed smoothing parameter, of the autocorrelation function of the incident wave and global forces and moments.
The relative difference (R b ) in the obtained transfer function is employed to show the difference in two groups of data. R b is used to address the relative difference with respect to the corresponding maximum value in the entire wave frequency range (from 0.35 rad/s to 2 rad/s). For example,e(u) ¼ {e(u 1 ), e(u 2 ),…e(u i ),…} and f(u) ¼ {f(u 1 ), f(u 2 ),…f(u i ),…} represent transfer function moduli for wave induced axial force (jH Fx j). The transfer function moduli are calculated by carrying out irregular wave analysis and regular wave analysis respectively. u i is a given frequency.
Transfer function moduli for wave induced global forces and moments and axial and shear stresses obtained by carrying out regular and irregular wave analysis in the TDM-2B-L are compared. The agreement is good. The largest R b for the transfer function moduli for axial and shear stresses is less than 6%. For most of the transfer function moduli, R b is less than 3%. Therefore, for the TDM-2B-L, in the following, the transfer function moduli are obtained by carrying out irregular wave analysis.
The transfer function moduli given by the FDM and TDM-2B-L are compared. Some selected results are shown in the present paper. Note that the trend of the transfer function moduli for the sectional forces and moments are not necessary to be the same as that of the total integrated wave excitation loads or motions. The sectional forces and moments are resultants of the difference between the effects of these two on the structure.
The main observations are discussed as follows: The agreement in transfer function moduli for the global forces and moments and stresses given by the FDM and TDM-2B-L is very good. The relative differences (R b ) vary with respect to the wave frequency and wave direction. In general, peak values of R b may appear at frequencies nearby troughs of the transfer function modulus curves. However, the effect of the peak values of R b on the accuracy of the global forces and moments calculated by the TDM-2B-L is very limited. This is because, in the frequency range from 0.35 rad/s to 2 rad/s, for most of the transfer function moduli, R b is less than 2.5%. The maximum value of R b for the transfer function moduli for stresses and for global forces and moments is no more than 8% and 5.9% respectively. Two examples are available in Figs. 10 and 11. Position of the points on the cross-section is shown in Fig. 6.
The difference between the FDM and TDM-2B-L is induced by: 1) inherent difference between frequency-domain and time-domain models; 2) accuracy limitation for the numerical solver and other numerical issues.
Detailed discussions are as follows: C Transfer function moduli for the global forces and moments and stresses subjected to the white noise irregular wave analysis with H s ¼ 1.233 m and H s ¼ 12.33 m are compared. For wave directions, where yaw motion is very small due to the shape of the wet surface of the hull, i.e. 0-degree-wave, 60-degree-wave and 120-degree-wave, R b for the transfer function moduli is close to zero and indicates that the difference is negligible. In the rest wave directions, the difference is significant. In the frequency range from 0.85 rad/s to 2 rad/s, R b for the transfer function moduli for the stresses can be up to 28% (in the area around 1.85 rad/s). In the frequency range from 0.35 rad/s to 0.85 rad/s, R b for the transfer function moduli for the stresses is less than 5%. An example is shown in Fig. 12.  C In the TDM-2B-L, the first order wave excitation loads on the hull are generated from the corresponding spectral densities of the wave excitation loads. "WETFF59" and "WETFF199" represent wave to wave excitation load transfer functions that correspond to a set of 59 selected frequencies and a set of 199 selected frequencies respectively. If the TDM-2B-L use the "WETFF59" and is subjected to 80-degree-wave, 90-degree-wave or 100-degree-wave, the TDM-2B-L will give strange results in the transfer functions for the global forces and moments. The strange results will disappear if the "WETFF59" is replaced by "WETFF199". Fig. 13 shows the transfer function modulus curves for the global lateral shear force (F y ) given by the FDM and TDM-2B-L, which use the "WETFF59" and is subjected to 90-degree-wave, as an example. Compare to the FDM, the TDM-2B-L gives strange transfer function moduli ("an impulse") in the frequency range from 0.79 rad/s to 0.89 rad/s. We can observe similar strange results in the spectral densities for the global forces and moments given by the TDM-2B-L that implements the "WETFF59" rather than the "WETFF199". Fig. 14 shows the spectral densities for the global lateral shear force (F y ) given by the TDM-2B-L that implements the "WETFF59" and "WETFF199" respectively. The circles and squares on the curves represent the set of the 59 selected frequencies and set of the 199 selected frequencies respectively. The spectral densities are generated based on the same smoothing factor for the inverse Fourier transform. The realizations of the global forces and moments are calculated based on the same realization of the wave elevation. In the frequency range from 0.79 rad/s to 0.89 rad/s, the set of the 59 selected frequencies for the "WETFF59" has two frequencies (0.797 rad/s and 0.877 rad/s). These two frequencies are nearby the boundary of the range and sufficient to represent the wave excitation transfer functions in the range. The set of the 199 selected frequencies for the "WETFF199" has 12 frequencies uniformly distributed in the range. The "WETFF59" agree with the "WETFF199", while, the "WETFF199" are smoother since the "WETFF199" include more frequencies. Fig. 15 shows the transfer function curves for the lateral wave excitation force on the "PartA" in 90-degree-wave, as an example. We do not observe similar strange results in the "WETFF59" and "WETFF199". We can conclude that the strange results are related to the selected frequencies for the wave excitation transfer functions. However, the reason is not clear yet. In this paper, the numerical models, which implement the proposed method, utilize the "WETFF199". In general, a refined frequency resolution should be considered when using irregular wave analysis to obtain the transfer function.

Comparison B and comparison C
Ten combined wind and wave conditions are selected from a site in northern North Sea [34] and tabulated in Table 4. The combined wind and wave conditions are composed of five different mean wind speeds covering the below rated, at rated, above rated and parked wind speed and two wave directions. In addition, we also looked at wave only conditions by removing the winds from the combined conditions. For each condition, one 1-hour time-domain simulation is carried out in the TDM-1BC, TDM-2BN and TDM-29BN respectively. Identical random seeds are used to eliminate stochastic uncertainties. Responses, i.e. the pitch angle of each blade, azimuth angle and rotational speed of the rotor, aerodynamic forces and moments on the rotor, torque on the rotational shaft of the drive train, generator torque, generated power, global forces and moments in a given cross section of the tower, global rigid-body motions of the hull and mooring line tension at the top end (fairlead) of each mooring line, are calculated and compared. Definition of the directions of wind and waves is available in Fig. 16. Results and discussions with respect to the Comparison B are given as follows: The responses of the TDM-1BC and TDM-2BN subjected to the wave only conditions are firstly compared. We find that the responses are identical to each other (the difference is negligible). Part of the realization of the fore-aft bending moment at the tower base, in EC02000 (wave only), is given as an example, see Fig. 17.
Then, the responses of the TDM-1BC and TDM-2BN subjected to the combined wind and wave conditions are compared. When the relative wind speed at the hub is below the rated speed, the generator torque and rotational speed of the rotor will be adjusted by the controller to optimize the power generation. A slightly numerical difference in the TDM-1BC and TDM-2BN can result in slightly differences (phase shift) in the azimuth angle and rotational speed of the rotor. The differences will be accumulated with development of simulation time and result in developing differences in the aerodynamic loads on the rotor, global forces and moments at the tower base and mooring line tensions at the fairleads. As shown in Figs. 18 and 19, the realizations of the azimuth angle of the TDM-1BC and TDM-2BN are identical at the beginning but the phase shift is accumulated with the development of the simulation time. The differences induced by the phase shift have very limited effects on the realizations of the rigid-body motions of the hull and spectra of the global forces and moments at the tower base, mooring line tensions and rigid-body motions. Spectral densities of the fore-aft bending moment at the tower base of the TDM-1BC and TDM-2BN are given in Fig. 20 as an example.
For EC04000 and EC04090, the relative wind speed at nacelle is always above the rate speed. Therefore, the generator torque is constant and azimuth angles of TDM-1BC and TDM-2BN are in phase. Very slightly difference exists in the responses, see Fig. 21 as an example. The difference is induced by very slightly numerical difference in pitch actuator control. Identical responses can be obtained if the control model is removed from the TDM-1BC and TDM-2BN. In EC05000 and EC05090, where the wind turbine is parked, identical responses are observed.
Results and discussions with respect to the Comparison C are given as follows: We compare the responses of the TDM-2BN and TDM-29BN in wave only and in combined wind and wave conditions. Observations of the comparisons of the responses of the RNA, tower and mooring lines of the TDM-2BN and TDM-29BN are similar to the observations of the comparison of the TDM-1BC and TDM-2BN which have been illustrated in above.
Therefore, we focus on discussions with respect to the global forces and moments in the cross-section shown in Fig. 5. As mentioned in above, the global forces and moments in the cross-section (the dashed line) are described in the O inp -x inp -y inpz inp coordinate system and are denoted as F x , F y , F z , M x , M y , and M z . For each condition, time realizations of the global forces and moments of the TDM-2BN and TDM-29BN are in phase and almost identical. An example is given in Fig. 22.
When the floating wind turbine is located in calm water without wind loads, the global forces and moments in the hull are static and are determined by the gravity and hydrostatic pressures forces on the floating wind turbine. We find the difference in the static loads calculated by the TDM-2BN and TDM-29 and the analytical solution is less than 1%.
Figs. 23 and 24 show the spectral densities of the M y (bending moment) of the TDM-2BN and TDM-29BN in EC01000, EC02000, EC03000, EC04000 and EC05000. We do not observe any high frequency responses (in the frequency range 2 rad/s to 4 rad/s). The peaks of the density curves in the frequency range 0.5 rad/s to 2 rad/s correspond to the T p of the wave spectrum, while, in the operational conditions, considerable low-frequency (from 0 rad/s to 0.3 rad/s) components can be observed. The standard deviation of the M y , for example, in EC03000 with combined wind and waves is 21.4 MN*m, while the standard deviation of the M y in EC03000 with waves only is 15.2 MN*m. The relative difference is 41%.
In the low frequency range, the global forces and moments in the structural components of the hull are sensitive to fluctuations of the hydrostatic pressure forces on the structural components. This is because: 1) the aerodynamic loads on the RNA and tower can excite significant rotational motions (e.g. roll and pitch) in particular in the low frequency range comparing to the motions excited by the wave excitation loads on the hull, while the first order terms of the fluctuations of the hydrostatic pressure forces are proportional to the motions; 2) wave excitation loads are expected to be small since wave energy is expected to be very limited (except for swell) in the low-frequency range; and 3) inertia and radiation loads are proportional to the first order derivative (velocity) and second order derivative (acceleration) of the motions and are expected to be small in the low-frequency range.

Conclusions
The present paper deals with the development and verification of a time-domain approach that can be easily implemented in various state-of-the-art computer codes to extend their capabilities to analyze global forces and moments in structural components of a generic floater subject to linear and non-linear environmental loads from wind and waves. The global forces and moments in the structural components might be used as inputs of design formulas for structural strength design checks and/or used as boundary conditions in a sub-model finite element analysis to determine structural responses such as stresses, etc.
The proposed approach focuses on the modeling of the inertia and external loads on the hull and the mapping of the loads in the finite element model of the hull. In the proposed approach, floating wind turbines are considered as an assemblage of several structural components. The conventional hybrid frequency-time domain approach is extended to model the external loads on and inertia loads of each structural component. Hydrodynamic loads on each structural component are obtained by integrating the pressure loads that are obtained by solving the linear hydrodynamic problem with the assumption that the hull is a rigid-body. The kinematics of different structural components is constrained by the rigid-body assumption. The proposed approach does not account for hydroelasticity effects. The expressions of the hydro loads on each structural component can be further modified to account for, for example, second order and/or higher order hydro loads on each structural component. Beam elements are used to represent the global stiffness of the structural components but the proposed approach can be further extended to use other finite element models to represent the stiffness of the structural components.
So far, the proposed approach has been implemented in the computer code Simo/Riflex/Aerodyn to analyze global forces and moments in the hull of a semi-submersible wind turbine. Responses calculated by the numerical models that implement the proposed approach and obtained by the reference models are compared step by step. The agreements are very good. Accuracy of the proposed approach needs to be further checked by sensitivity studies and by comparison to the model test data.
In the operational conditions, considerable low-frequency components can be observed in the spectra of the obtained global forces and moments in the pontoons of the reference floating wind turbine. The results indicate that the low-frequency aerodynamic loads, the fluctuations of the hydrostatic pressure forces and the fluctuations of the gravity loads of floating wind turbines are important contributions to the structural responses, in particular, in the low-frequency range. This feature needs to be further investigated by experimental studies.

Mass and structural models
The hull The floating wind turbine is divided into two parts: "Part A" and "Part B". The two parts are modeled as two rigid-bodies. Each rigid-body has 6 d.o.f.s. The two rigid-bodies are connected by three artificial beam elements. Integrated mass (corresponding to each rigid-body).
The hull is modeled as one rigid-body with 6 d.o.f.s. Integrated mass.
The hull is discretized as two rigid-bodies: "Body1" and "Body2"/twenty-nine rigid-bodies. Each rigidbody has 6 d.o.f.s. The two rigid-bodies are connected by three artificial beam elements/The twenty-nine rigid-bodies are connected by beam elements. Integrated mass (corresponding to each rigid-body) Nacelle Rigid-bodies with integrated mass (Included in the rigid-body for the "Part B").