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

Non-linear dynamic analysis of the response of moored oating structures José E. Gutiérrez-Romero 1 a , Julio García-Espinosa bd , Borja Serván-Camas b , Blas Zamora-Parra c (a) Depart. Naval Technology, Universidad Politécnica de Cartagena (UPCT), 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/ Edici 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/ 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 behaviour of oshore marine structures requires advanced simulations tools for the accurate assessment of the seakeeping behaviour of these devices. The aim of this work is to present a time-domain model for solving the dynamics of oating marine devices, subjected to non-linear environmental loads and paying special attention on the mooring dynamics. First, the formulation of the hydrodynamic approach for solving the wave-oater interaction is introduced. Second, the solver of the mooring dynamics, based on a non-linear Finite Element Method approach, is presented. Third, a procedure for coupling the hydrodynamic along with other external loads, with the oating structure and mooring dynamics is described. Fourth, some validation examples and comparisons among dierent mooring approaches are presented. Fifth, an analysis of the OC3 oating wind turbine concept is performed to study the inuence of dierent mooring models, the eects of non-linear waves on the platform, and the tension in 1 Corresponding author. Tel. no.: +34 868 071 261. Fax no.: +34 968 325 422 . E-mail address: [email protected] (José E. Gutiérrez-Romero) Preprint submitted to Marine Structures February 24, 2017 the mooring system. The dynamic mooring model along with the secondorder wave model produce realistic simulations of the oating wind turbine performance. Keywords: FEM, Coupled Analysis, Mooring Dynamics, Second-order waves, Time-domain 1. Introduction Research trends in Marine Renewable Energies (MRE) are mainly focused in oshore wind energy due to the high expectations raised by this technology. The technology for marine wind turbines is currently well-developed, but limited for xed installations in shallow-water areas. The next horizon is focused on deep-water technology [1], but dierent challenges for Floating Oshore Wind Turbines (FOWT) are not solved yet [2], such as the dynamic stability in the presence of non-linear ambient loads [3]. In fact, an accurate prediction of the dynamic response of a FOWT, considering the interaction among the hydrodynamics, mooring, and aerodynamics of the turbine, is identied as one of the key challenges for the simulation tools required to design the future FOWTs [4, 5, 6]. Standard design procedures and simulation tools for marine structures come from the existing technology and experience of the oshore oil and gas industry. For instance, the classic simulation approaches are based on uncoupled formulations, where the hydrodynamic response of the oater is linearised and can be decoupled from the mooring [7, 8]. Recently, coupled simulations have been adopted to solve the seakeeping of FOWT devices, since the dynamic of FOWT oers a high complexity due to the variety of loads and non-linear eects. Anyhow, the interaction among dierent components, such as the wind turbine structure, the rotor dynamics, the mooring arrangements and the oating structure must be taken into account in a more accurate way [9, 10]. The analysis of FOWT should be carried out with simulation codes capable to include the physics governing the dynamic response of these devices. With regards to marine structures, Low and Langley [11] showed that the dynamic response of a oating production system in a random sea can be split in two timescales: low frequency and wave frequency responses. Moreover the seakeeping of the oating device and the dynamics of the mooring lines are coupled. Then, the analysis should take into account the interac- 2 tion between them, and the time-domain analysis seems to be the right way to simulate this sort of coupled problems. In fact, the American Bureau of Shipping [32] considers that the global seakeeping analysis of a FOWT should take into account: the unsteady wind loads, wind turbine control systems, wind turbine-platform interaction, wave actions over the platform, currents, mooring loads, and any other types of external actions. It will be presented later on that the time-domain approach allows to handle these actions in a natural manner. The rst works proposing coupled formulations for oating production systems were developed for TLP-type platforms [12, 13, 14], where it was found that the sum-frequency eects are important. Also other authors investigated and developed models to couple dierent eects and loads in oating devices [15, 16, 17, 18]. More recently, Cordle and Jonkman [5] made a review of the state of art of the simulation tools for FOWT. Bae and Kim [9] recently presented a research on the eects of second-order wave excitation in a mono-column-TLP type FOWT comparing with the uncoupled simulations in time-domain. Karimirad and Moan [19] developed a simpli- ed method for the coupled dynamics of a oshore wind turbine aiming at saving computational time. They also analysed a spar-type FOWT, and compared dierent hydro-elasticity codes [2]. In [3], the dynamic response of the spar-type platform subject to wave induced excitation was investigated numerically, including catenary mooring cables. Matha et al. [6] investigated a platform-tower coupled with a TLP-type FOWT. Jonkman and Sclavounos [20] presented a tool integrating an aeroelastic model for onshore wind turbines, and a hydrodynamic load model, together with a Quasi-Static (QS) catenary model for mooring lines. More sophisticated mooring models, which are usually based on Finite Element Methods (FEM) have been investigated; for instance, Kim et al. [21] presented a comparison between a linear spring and a non-linear FEM mooring for the analysis of the dynamic response when they are coupled with a oating structure. Several coupled seakeeping analyses with FEM mooring line models can also be found in the literature [22, 23, 24, 25, 26, 27]. Frequency-domain and time-domain approaches are used for seakeeping analyses of marine structures. However, the frequency-domain approach has diculties to accurately handle non-linearities such as those arising from the mooring lines, and the low frequency components of the wave-body interaction, as appointed by Low [29], while the time-domain analysis can straightforwardly include any non-linearity within each time step in a natu- 3 ral manner. However, the main drawback of the time-domain is an increase of the computational time required for the simulations. To overcome this limitation, Low and Langley [11] and Low [29] presented a hybrid time-frequency domain model to simulate low and wave frequency response of coupled problems. In this work, an extension of the time-domain method proposed by Serván- Camas and García-Espinosa [30, 31] is used. This method obtains a relevant reduction of the computational time thanks to the use of deation techniques and High Performance Computation based on Graphical Processor Units (GPU). The tool is based on the Finite Element Method (FEM) and is able to solve multi-body seakeeping problems on unstructured meshes including any type of external load. Other examples of FEM solvers for seakeeping analysis can be found in the literature. For instance, Hong and Nam [33], who presented a second-order analysis of wave forces using FEM in the timedomain, and investigated the interactions with multi-body devices. This work presents the development of a coupled FEM seakeeping and mooring models for the analysis of oating oshore structures. The model is able to take into account rst and the second-order irregular waves, wind loads, and mooring loads. A novel iterative scheme for coupling the wave diraction-radiation solver developed by Serván-Camas and García-Espinosa [28, 30, 31] with non-linear aerodynamics and mooring loads is also described. The paper is organized as follows. First, the governing equations of the body dynamics are introduced. Second, the second-order diractionradiation wave problem is described. Third, the non-linear FEM model for mooring lines is introduced in detailed. Fourth, the coupling between the dynamics of the oater with non-linear loads, such as mooring lines forces, is explained. Fifth, some validations with experimental results are shown, as well as a comparison of the non-linear FEM model with the Quasi-Static catenary model. Sixth, we analyse an operational case of a spar buoy FOWT based on the OC3 concept and NREL 5 MW turbine. This case study aims at evaluating the inuence of the type of mooring model, as well as the eects of non-linear wave on the dynamics of the platform. Finally, some relevant conclusions from the obtained results are presented. 4 2. Problem statement 2.1. General dynamics of oater The motion of a oater subject to ambient loads can be modelled using the rigid body dynamics. Let OXYZ be a xed global frame of reference, and let Gxyz be a local frame of reference located at the center of gravity, and whose axis are parallel to the axis of the global frame (see Fig. 1). Assuming small rotations, for each time step, the body accelerations can be obtained respect to the local frame from the rigid-body dynamic equations: M b·¨ x = f (t), (1) ¯ ¯ I b·¨ r b= m (t), (2) where M b is the mass matrix, ¨ x is the linear acceleration vector , ¯ ¯ I b is the instantaneous inertia tensor, f (t) and m (t) are the external loads vector (hydrostatics, wave loading, mooring, wind, etc.) and the external moments respectively; nally ¨ r b is the angular acceleration vector. xo x(t) G࢞ ࢟ ࢠ X Y Z O m (t) f (t) time: t time: t + ∆t Figure 1: Global and local frame of reference used in computation of oater dynamics. 5 The calculation of the external forces f (t) over the body constitutes an essential part of the dynamics, and will be described in more detailed later on. 2.2. Flow equation and boundary value problem Assuming incompressible and irrotational ow in a domain Ω , being Γb the wetted surface of the oating device, the wave problem governing equations are [28]: ∆ϕ= 0 on Ω, (3) ∂ξ ∂t +∂ϕ ∂x ∂ξ ∂x +∂ϕ ∂y ∂ξ ∂y −∂ξ ∂z = 0 on z=ζ, (4) ∂ϕ ∂t +1 2∇2ϕ+gξ + P fs ρ= 0 on z=ζ, (5) v ϕ· n + v b· n = 0 on Γb, (6) P b=−ρ∂ϕ ∂t −1 2ρ∇2ϕ−ρgzb on Γb, (7) where ϕ is the velocity potential, ξ is the free surface elevation, g is the gravity acceleration, P fs is the pressure on free surface, ρ is the uid density, ξ is the wave elevation, P b is a pressure at a point over body, v ϕ is the uid velocity, v b is the local velocity at a point on the wetted body surface, n is the vector normal to the body wetted surface (pointing upward this surface), and zb is the vertical coordinate of any point of the body. 2.3. Second-order diraction-radiation governing equations The governing equations for the second-order diraction-radiation wave problem are obtained applying Taylor expansion on the boundary surfaces of a time-independent domain. This approach allows to approximate the free surface on z=ζ and the mean body surface Γ0 b at time t . Then, a perturbed solutions based on Stokes expansion procedure is applied to the velocity potential, free surface elevation, and body motion. More details can be found in [28]. The solution can be decomposed as ϕ=ψ+θ, (8) ξ=η+ζ, (9) 6 where ψ is the incident wave velocity potential, θ is the diraction-radiation wave velocity potential, η is the incident wave elevation, and ζ is the diractionradiation wave elevation. Then, the wave diraction-radiation governing equations up to second-order are [28]: ∆θ1+2 = 0 in Ω, (10) ∂η1+2 ∂t −∂θ1+2 ∂z =−∂θ1 ∂x ∂η1 ∂x −∂θ1 ∂y ∂η1 ∂y −∂θ1 ∂x ∂ζ1 ∂x −∂θ1 ∂y ∂ζ1 ∂y −∂ψ1 ∂x ∂η1 ∂x −∂ψ1 ∂y ∂η1 ∂y on z= 0 (11) ∂θ1+2 ∂t + P fs ρ+gη1+2 =−η1∂ ∂z ∂θ1 ∂t −ζ1∂ ∂z ∂θ1 ∂t  −η1∂ ∂z ∂ψ1 ∂t −1 2∇θ1· ∇θ1− ∇ψ1· ∇θ1 on z= 0, (12) v 1+2 θ· n 1=− v 1+2 b+ v 1+2 ψ+ r 1 b·∇ v 1 θ+∇ v 1 ψ· n 1 on Γb, (13) P 1+2 b ρ=−gzb−g r 1+2 b−∂θ1+2 ∂t − r 1 b· ∇ ∂θ1 ∂t  −1 2∇θ1· ∇θ1− ∇ψ1· ∇θ1 on Γb, (14) where superscripts 1 and 1 + 2 denote the components at the rst-order and up to second-order solution, and rb is the displacement vector at a point over body. 2.4. Second-order wave loads With regards to the ambient loads applied over wetted body surface, hydrodynamic wave forces and moments can be obtained directly from pressure integration over the wetted body surface. Thus, it can be written that F 1+2 = F 0 H+ F 1+2 H+ F 1+2 D, (15) M 1+2 = M 0 H+ M 1+2 H+ M 1+2 D, (16) where subscript H denotes the hydrostatic loads and D denotes the dynamic loads. F 0 and M 0 are the initial hydrostatic forces and moments over the body, F 1+2 H= F 1 H+ F 2 H and M 1+2 H= M 1 H+ M 2 H are the up to second-order 7 hydrostatic loads and moments, and F 1+2 D and M 1+2 D are the up to secondorder dynamic loads and moments. Details of each component can be found in [28]. 3. Non-linear FEM model for mooring dynamics 3.1. Equations for cable dynamics The dynamic equations for a mooring cable with length L with negligible bending and torsional stiness can be formulated as [34, 35] (ρwCmA0+ρ0)∂2 r l ∂t2=∂ ∂l EA0 e e+ 1 ∂ r l ∂l + f (t) (1 + e), (17) where ρw is the water density, Cm is the added mass coecient, ρ0 is the mass per unit length of the unstretched cable, r l is the position vector, E is the Young's modulus, A0 is the cross-sectional area of the cable, e is the strain, f are the external loads applied on the cable, and l is the length along the unstretched cable. The boundary conditions are given by ∂2 r l ∂t2= 0 l= 0 ( at the anchor connection point ), (18) ∂2 r l ∂t2=¨ r fl l=L( at the fairlead connection point ), (19) where ¨ r fl is the second derivative of the position vector at the fairlead connection point. The external loads acting on the cable, considered in this work, are the self-weight of the cable, hydrostatic loads, drag forces [36], and seabed interaction. 3.2. Seabed interaction When the cable undergoes an excitation, the part which lies between the anchor and the touchdown point interacts with the seabed. This interaction is a complex non-linear eect. In this work, the seabed-mooring interaction is modelled as a spring-damping system [37, 38], and is implemented as follows: 1. The vertical forces due to the stiness of the seabed in each time step are calculated as 8 clw ll if GsAczr−zt+∆t> clwll, (20) GsAczt−zr+ ∆t˙zt+ ∆t21 2−β¨zt otherwise , (21) where cl>1 is a coecient limiting the seabed reactions, ll is the portion of the cable resting on the seabed, w is the weight per unit of length of the line, w is the apparent weight of the cable, β is the term related to the numerical integration of the dynamic FEM model, z is the vertical coordinate of the cable, Gs is the seabed stiness, Ac is the contact area of the line, and ∆t is the time step. The term zr is dened as zr=z+wll GsD, (22) where D is the diameter of line. The term zr can be dened as a limit of the line subsidence. 2. The vertical forces due to the damping system are modelled as dlzt+ (1 −γ) ∆t¨zt if zt+∆t< zr, (23) 0 otherwise (24) being γ a term related to the numerical scheme adopted. In addition dl is expressed as dl= 2ςlsGswll gDll, (25) where ςl is the seabed damping [39]. 3. Finally, the horizontal forces due to the seabed interaction are modelled as dlxt+ [1 −γ]∆t¨xt,(x− direction ), (26) dlyt+ [1 −γ]∆t¨yt,(y− direction ). (27) The algorithm developed to solve the mooring dynamics is described next. 9 EA = 50 N, the weight per unit length w= 0.4 N/m, and 14.1421 m of span length. The cable is discretized into twenty-two bar elements, and the time step adopted is 0.014 s. Figure 5 shows dierent positions of the cable until it gets the expected U form. Note that a damping increment leads to a faster convergence of the stationary solution. The reactions at end points are 3.4681 , and the maximum deection (0.715 m) is reached at X= 7.07105 m. Figure 5: Validation of the non-linear FEM mooring model. Case 1: dierent stages of time evolution of cable under its self weight. 16 Figure 6: Validation of the non-linear FEM mooring model. Case 2: dierent time step positions of free vibration cable obtained from computed results. Excel table with values o case 1 Case 2: Free vibration cable . The second case is based on that proposed by [48]. The experiment consists of a cable initially in a horizontal position, and letting one end free and leaving the cable evolve under gravitational loads. The computed results are compared with the experimental results taken from [48], as well as with the data obtained from simulation with the FEM structural solver RamSeries ( www.compassis.com ). The properties of the cable are: length L= 1.0 m, stiness EA = 50 N, weight per unit of length w= 0.98 N/m, and 0.881 m distance between the cable end points. The cable was divided into forty four bar elements, and the time step adopted was 0.001 s. This time step must be low enough for obtaining an accurate description of the cable motion. The obtained numerical results show a good agreement between our results, and those experimentally and numerically obtained by other authors. The cable position at dierent times obtained from computed results can be observed in Fig. 6. 17 Figure 7: Validation of the non-linear FEM mooring model. Case 2: comparison between computed results of the path of free end of the cable obtained and experimental results of [48] and numerical computed with RamSeries. Slight dierences can be appreciated at the lower end in Fig. 7. This fact can be explained by the dierent numerical parameters chosen to carry out the simulation. 18 Motor with load cell Chain 0,0 m 3,0 m 3,3 m 32,554 m Figure 8: Validation of the non-linear FEM mooring model. Case 3: the geometrical set-up of the experimental tests by [38]. Case 3: Cable attached to a circular rotating plate . Now, results obtained by the developed FEM solver are compared for the model test proposed by Lindhal and Sjoberg [38, 39]. The experimental set up is shown in Fig. 8. The lower end was attached to the concrete oor and the upper end was attached to a circular plate with a xed rotation speed. The radius of the circular motion was 0.2 m. Two cases with dierent rotational periods of Tr = 1.25 s and Tr= 3.5 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. 19 Figure 9: Validation of the non-linear FEM mooring model. Case 3: comparison between the experimental and numerical cable top end reaction forces. Rotational period Tr = 3.5 s. The properties of the cable are: length L= 33 m, stiness EA = 104 N, weight per unit length w= 0.18 N/m, and diameter D= 10−3 m. In this validation case, the cable is discretized into 200 bar elements, and the time step adopted is 0.001 s. The motion of the top end is dened by the following expressions (including an initialization period to build up the spinning of the plate): x(t) = 0.2· tanh t 2 cos −2π Tr t+δ− cos δ, (38) y(t) = 0.0, (39) z(t) = 0.2· tanh t 2 sin −2π Tr t+δ− sin δ. (40) The results are compared with the experimental results taken from [38]. A good agreement between the computed reaction forces and the experimental results [38] can be observed in Fig. 9 (for Tr= 3.5 s), and 10 (for Tr= 1.25 s) for both, the maximum values and the time evolution. Only slight dierences are appreciated due to the uncertainty of the experimental data. The entire cable loses stiness at some instants in time, and the numerical oscillations after the slack are larger in the case of the larger excitation frequency, as it was observed by [39]. 20 Figure 10: Validation of the nonl-inear FEM mooring model. Case 3: comparison between the experimental and numerical cable top end reaction forces. Rotational period Tr = 1.25 s. 6. Demonstration of the coupled seakeeping-mooring solver A fully coupled analysis of the OC3 spar buoy oshore wind turbine, called Hywind ( www.ieawind.org/task23/ ) is presented next. Main particulars of the spar buoy FOWT are presented in [52, 53]. A general view of the buoy concept can be observed in Fig. 11. In this section, several analysis are made: 21 Figure 11: General view of the spar buoy wind turbine concept (OC3-Hywind concept). • The Response Amplitude Operator (RAO) are compared for a rigid wind turbine with no wind loads, with those obtained by other authors [49]. • The analysis of a spar wind turbine subjected to dierent monochromatic waves with periods close to the pitch resonance. The eects of the linear, the quasi-static, and the dynamic mooring models are compared. • A simulation in real operational conditions of the spar FOWT OC3 is performed. First and second-order irregular waves, real wind ow, and mooring loads (with three dierent types of mooring models) are taken into account. The inuence of the mooring type model, the eects of second-order wave, and mooring tensions are studied. 22 6.1. RAO analysis and comparison First, a RAO analysis is performed in the absence of wind. The frequency results of the seakeeping solver are obtained after applying a Fourier transform to the time history generated. Figure 12 shows an inter-code comparison. Note that a good agreement among the dierent solvers is found [49]. 23 Figure 12: OC3-Hywind concept. Comparison between the computed result by SeaFEM with those taken from other authors [49] for a rigid wind turbine with no wind. 24 6.2. Mooring analysis around pitch resonance Below, the inuence of three dierent mooring models on the OC3 spar buoy FOWT is analysed. These models are: the linear model cable (that behaves as springs and is represented by a linearised mooring matrix [53]); the Quasi-Static model, similar to the one presented in [54]; and the dynamic model developed in this work. Six rst-order monochromatic waves are used in the analysis. Key parameters of the mooring layout and cable properties can be found in Table 1. Fig. 13 compares the pitch motion for each mooring model. Table 1: Key parameters of pitch resonance study [53]. Item Value Unit Wave Amplitude 1.0 m Wave Period analysed 10; 25; 20; 35; 40; 55 s Number of Mooring Lines 3 Angle Between Adjacent Lines 120 deg Depth to Anchors Below SWL 320 m Depth to Fairleads Below SWL 70 m Radius to Fairleads from Platform Centerline 853.9 m Unstretched Mooring Line Length 902.2 m Mooring Line Diameter 0.9 m Mooring Line Mass Density 77.71 kg/m Mooring Line Extensional Stiness 3.84 × 10 8 N It can be observed that there are big dierences in the results in those cases close to the pitch resonance (about Tw= 30 s), while the results are quite similar for the cases with Tw= 10 s and Tw= 55 s. These results suggest that the use of simpler mooring models can lead to big errors near the resonance frequency, and as a consequence to magnify safety factors, contributing to increase the cost of the FOWT. 6.3. Simulation in operational conditions Finally, four analyses of the OC3 spar buoy in operational conditions are presented. The dierent analyses are carried out in similar environmental conditions, but using rst and second-order irregular waves. Furthermore, additional studies including Quasi-Static and dynamic mooring models are 25 Table 4 compares the maximum, minimum, average, and RMS tension values at the fairlead points, obtaining similar values for both mooring models. Furthermore, based on Figure 17, the second-order simulation provided larger tension values than the rst-order. This result suggests that a rstorder approximation can underestimate the fatigue loads and might lead to a wrong mooring design. Table 4: Comparison between maximum, minimum, mean, and RMS values of fairlead tension for Quasi-static and Dynamic mooring models. Line Case Max. (N) Min. (N) Mean (N) RMS (N) Line 1 1 1.435 × 10 6 1.073 × 10 6 1.250 × 10 6 1.252 × 10 6 Line 2 1 1.446 × 10 6 1.072 × 10 6 1.240 × 10 6 1.242 × 10 6 Line 3 1 7.411 × 10 5 5.209 × 10 5 5.991 × 10 5 6.003 × 10 5 Line 1 2 1.457 × 10 6 1.052 × 10 6 1.246 × 10 5 1.248 × 10 6 Line 2 2 1.471 × 10 6 1.040 × 10 6 1.236 × 10 6 1.238 × 10 6 Line 3 2 7.421 × 10 5 4.510 × 10 5 5.923 × 10 5 5.936 × 10 5 It is emphasized that the NFEM mooring model allows to consider nonlinear dynamic eects which cannot be taken into account by QS models. However, in deep water, the dynamics of catenary mooring lines are negligible as reported in [11]. 7. Conclusions A FEM coupled seakeeping and mooring model for the analysis of oating wind turbines is presented. Based on the results obtained in this work, the following conclusions are made: • A second-order time-domain seakeeping model based on FEM and capable of using unstructured meshes [28, 30, 31] is successfully applied to simulate oating wind turbines. • A dynamic cable model based on Non-linear FEM is presented. A Lagrangian formulation is used to describe the dynamics of the mooring line. Several non-linear eects are considered, such as seabed interaction and drag forces. In order to accelerate the body dynamics convergence, the restoring mooring forces on the oating structure are linearised for each iteration within each time step. Therefore, a stiness matrix has to be evaluated for each iteration. 32 • An algorithm is developed to couple the non-linear cable dynamics with the body dynamics solver within the time-domain seakeeping solver [28]. • The dynamic cable model presented in this work is validated against several experimental results, obtaining a good agreement between the numerical and experimental ones. • Dierent analyses of a FOWT based on the Hywind concept [52, 53] are performed. RAO curves are compared with those obtained by other authors, showing a good agreement. Moreover, a comparison among the linear, the QS, and the NFEM models are carried out for frequencies around pitch resonance, obtaining large dierences among mooring models. • The Hywind FOWT is analysed under a realistic operational condition. Second-order coupled simulations are carried out considering a real wind prole, and with two types of mooring models; the QS and the NFEM. Results from both cases are compared. 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