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An enhanced stiffness model for elastic lines and its application to the analysis of a moored floating offshore wind turbine
Ocean Engineering 109 (2015) 444–453
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
An enhanced stiffness model for elastic lines and its application to the analysis of a moored floating offshore wind turbine Zi Lin n, P. Sayer Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, UK
art ic l e i nf o
a b s t r a c t
Article history: Received 22 April 2015 Accepted 1 September 2015 Available online 8 October 2015
The performance of a polyester mooring line is non-linear and its elongation plays a significant role in the dynamic response of an offshore moored structure. However, unlike chain, the tension–elongation relationship and the overall behavior of elastic polyester ropes are complex. In this paper, by applying an enhanced stiffness model of the mooring line, the traditional elastic rod theory has been extended to allow for large elongations. One beneficial feature of the present method is that the tangent stiffness matrix is symmetric; in non-linear formulations the tangent stiffness matrix is often non-symmetric. The static problem was solved by Newton–Raphson iteration, whereas a direct integration method was used for the dynamic problem. The computed mooring line tension was validated against the proprietary OrcaFlex software. Results of mooring line top tension predicated by different elongations are compared and discussed. The present method was then used for a simulation of an offshore floating wind turbine moored with taut lines. From a comparison between linear and non-linear formulations, it is seen that a linear spring model under-estimates the mean position when the turbine is operating, but over-estimates the amplitude of the platform response at low frequencies when the turbine has shut down. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Large extension Elastic rod theory Finite element method Mooring system Motion response Dynamic response
1. Introduction The capture of offshore wind energy plays a key role across the maritime industry (EWEA, 2013). Offshore wind turbines are becoming larger and more powerful, and are being deployed in ever-deeper water. They can be mounted on a fixed or floating base, but the former starts to lose its economic advantage for water depths larger than 60 m (Goupee et al., 2014). Although the mooring system design for a floating offshore wind turbine (FOWT) has benefited from the experience of offshore oil and gas platforms, there are still several unknowns dependent on the type of floating bodies, e.g. size and environmental loading. From a report of EWEA (2013), it is recommended that more research must be done on mooring and anchoring systems for wind turbines. Numerical simulations of the dynamic response of mooring lines have been studied during the past few decades, for both elastic and inelastic lines. A massless spring (e.g. Kim et al., 2001) or the catenary equation (e.g. Agarwal and Jain, 2003) provide straightforward ways to model a mooring line, but it is difficult to account for the dynamic response and the interaction between the floating body and mooring line in an accurate manner. Multi-body n
Corresponding author. Tel.: þ 44 141 548 4911. E-mail address: [email protected] (Z. Lin).
http://dx.doi.org/10.1016/j.oceaneng.2015.09.002 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
system dynamics (e.g. Kreuzer and Wilke, 2003) divides the mooring line into several rigid bodies, but results in a large number of degrees of freedom even for a single line. Non-linear finite element methods (FEMs), accounting for geometric and material non-linearities, have been widely used for modelling mooring line response (e.g. Kim et al., 2013). Geometric nonlinearity is needed for large displacements of the mooring line, while material non-linearity can model the time-dependent properties of a polyester rope, e.g. Young's modulus. However, a major disadvantage of FEM is the transformation between local coordinate and global coordinate, which is often computationally intensive. The lumped mass and spring method can be categorized as a non-linear FEM method, for which the shape function becomes a single line (Low, 2006). Unlike traditional non-linear FEMs, the elastic rod theory is a global-coordinate-based method, which is considered to be more efficient (Kim et.al, 1994). The transformation between local and global coordinate is dealt within the element stiffness matrix. Following the elastic rod theory of Love (1944), Nordgren (1974) and Garrett (1982) developed this method and solved the governing non-linear equations by a finite difference method (FDM) and by FEM, respectively. Many researchers have further developed the elastic rod theory, including elongation of the line, seabed friction, non-linear material properties and the incorporation of buoys or clump weights in the mooring line model. Paulling and Webster (1986) considered the analysis of large amplitude
Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
motions of a TLP under the action of wind, wave and current, under the assumption of small line elongation. Ran (2000) proposed a finite element formulation for mooring lines and risers based on Garrett's rod theory, applicable to both frequency and time domains. Based on the traditional small extensible rod theory, the incorporation of large elongation has been presented by many researchers (e.g. Chen, 2002; Tahar, 2001; Kim et al., 2011). Based on the successful experience from offshore oil & gas platforms, the design and modelling of a FOWT has tended to use the same mathematical modelling and methods of solution as for offshore platforms, e.g. the hydrodynamic analysis of floating body, mooring design and the types of FOWTs (Spar, TLP and Semisubmersible, etc). The design of a station-keeping system for a FOWT uses the same methods and guidance as for a floating platform (e.g. ABS, 2014a). However, the geometry and operational water depth are different. Also, the turbine thrust force may have an effect on the motion response of the floating body and mooring line tension, and vice versa. These differences need to be examined for a FOWT. Polyester lines are made from visco-elastic materials and the stiffness characteristics rely on the loading history, the load duration and magnitude, etc (ABS, 2014b). The material nonlinearity of polyester lines is difficult to model and requires a longer simulation time. Thus some approximate models are used. For example, the dual stiffness method (Tahar et al., 2012) recommended by API RP 2SM (2001), and the visco-elastic model (Kim et al., 2011). In the present paper, a sensible balance has been sought between efficiency and accuracy. The traditional rod theory has been extended to allow for large stretch by applying an enhanced stiffness method. By using an approximation of the nonlinear tension-elongation relationship in a Taylor series expansion (Ćatipović et al., 2011), the mathematical and numerical formulation of large extensible mooring line are considered.
2. Mathematical formulation of a mooring line with large elongation
445
Mooring line normal component of acceleration r€ n , normal component of velocity r_ n , normal component of water particle _ n are given by velocity Vn and water practical acceleration V (Ćatipović et al., 2011) dr dr dr dr ; r€ n ¼ r€ r€ U ð3Þ r_ n ¼ r_ r_ U ds ds ds ds €n ¼V _ V _ U dr dr; V € V € U dr dr _n ¼V V ds ds ds ds
ð4Þ
In Eq. (1), ε is the elongation of the rod. Following Ćatipović et al., assuming equal principal stiffness, the relationship between the effective tension and elongation can be written as
ε¼
TE AE
ð5Þ
where AE is the axial stiffness The following elongation condition then has to be satisfied (Ćatipović et al., 2011) 1 dr dr U ¼1 ð1 þ εÞ2 ds ds
ð6Þ
In the static problem, the mass per unit length and diameter of the mooring line are related to the elongation ε. The crosssectional area and mass after elongation can be written as A=ð1 þ εÞ and m=ð1 þ εÞ, respectively (Ćatipović et al., 2011), where A and m are the cross-section area and mass of the mooring line without stretch. Applying the above relationship to the motion equation, we see that the term ð1 þ εÞ, multiplied by the applied force qE cancels out. For the hydrodynamic force calculated by Morison's equation, the mass per unit length and cross-sectional area for one element were assumed constant. Eqs. (1) and (6) show the rod motion equation and elongation condition, respectively: they are non-linear. In the following section, we will describe a numerical procedure for solving this nonlinear equation and the required order of approximation for the elongation condition. 2.2. Numerical Implementation
2.1. Equation of motion For polyester mooring lines bending and torsion stiffness can be neglected, but the elongation cannot be assumed to be small. The mooring line is discretized into a number of rods and the centreline of each rod is described by a space-time curve r ðs; tÞ. From Ćatipović et al. (2011), the equation of motion for a rod with large elongation can be written as: d T E dr þ 1 þ εÞqE ¼ ð1 þ εÞ ρr€ ð1Þ ds 1 þ ε ds ~ and A~ are distributed mass and cross-section area after where m extension.ρ and g are sea water density and gravitational acceleration, respectively.r€ represents the time derivation of the rod. T E is the effective tension of the rod. The relation between the real tension T R and the effective tension are (Ćatipović et al., 2011): T E ¼ T R þ pA, where p and A are hydrostatic pressure and crosssection area, respectively. qE is the load acting on the rod. For static ~ ~ ρAg, problem,qE ¼ mg while for dynamic problem ~ þ FH . ~ ρAg qE ¼ mg in which FH is the hydrodynamic loads on the mooring line (Paulling and Webster, 1986) calculated by Morison's equation (1950) as _ n þ 1C D ρDVn r_ n ðVn r_ n Þ FH ¼ C A ρAr€ n þ C M ρAV 2
ð2Þ
where n denotes the normal component. C A ,C M and C D are the added mass, inertial (Morison) and drag coefficients.
2.2.1. Static problem For the static problem, r is independent of time. Consequently the inertial term in Eq. (1) is deleted. We therefore have d T E dr þ qE ¼ 0 ð7Þ ds 1 þ ε ds Using the Taylor series expansion, the elongation relationship can be written as: 1 ð1 þ εÞ2
¼ 1 2ε þ 3ε2 þ oðε3 Þ
ð8Þ
However, it is not clear, a priori, whether the third-order term should be included explicitly. In the present paper, the order of expansion and subsequent results will be discussed. In the FEM, the variables r i and T E may be approximated (Garrett, 1982) as r i ðsÞ ¼
4 X
Ak ðsÞU ik
ð9Þ
k¼1
T E ðsÞ ¼
3 X
P m ðsÞ λm
ð10Þ
m¼1
where Ak and P m are shape functions. The definition of the shape functions can be found in the Appendix A. U ik and λm are unknown variables. The subscript i of U ik denotes the dimension of the element. For the 3-dimensional problem, i¼3. For k ¼1 and 3, U ik
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Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
represents the space position of the rod at two ends while U ik denotes the space derivative at both ends for k ¼2 and 4. λ is the Lagrange multiplier. The physical meaning of λ is mooring line tension at both ends and middle of the rod. The variable U ik and λm are defined as:
Re-arranging the terms and writing in the matrix form, we have 8 9 2 11 ðnÞ 3 ) ðnÞ ( < RðnÞ = K ijlk K 12 ΔU jk iln il 4 5 ð27Þ ¼ 21 ðnÞ ðnÞ : GðnÞ ; Δλn K mjk K 22 mn m
U i1 ¼ r i ð0Þ; U i2 ¼ r 0i ð0Þ U i3 ¼ r i ðLÞ; U i4 ¼ r 0i ðLÞ
ð11Þ
where
L ; λ3 ¼ T E ðLÞ λ1 ¼ T E ð0Þ; λ2 ¼ T E 2
ð12Þ
ðnÞ
Using Galerkin's method (Bathe, 1996) and integrating the motion equation from 0 to L over the length of the element, the final form of motion equation for static problem in notation form can be written as K^ nijlk λn U jk F il ¼ 0
ð13Þ
where K^ nijlk ¼ K 0nijlk þ λm K 1nmijlk þ λm λp K 2nmpijlk Z
K 0nijlk ¼
L
P n A0l A0k δij ds
0
Z K 1nmijlk
¼
Z F il ¼
1 P n P m A0l A0k δij ds EA
0
Z
K 2nmqijlk ¼
L
L
1 ðEAÞ2
0
Z
L
mg i ds
0
P n P m P q A0l A0k δij ds L
0
ðρAÞg i Al ds
ð14Þ ð15Þ
ð16Þ
ð17Þ
ð18Þ
ðnÞ ¼ K^ nijlk λn K 11 ijlk
ð28Þ
ðnÞ ¼ ðK 0nijlk þ2λm K 1nmijlk þ 3λm λp K 2nmpijlk ÞU ðnÞ K 12 iln jk
ð29Þ
ðnÞ ¼ B^ mkl U ðnÞ K 21 mjk jl
ð30Þ
ðnÞ K 22 ¼ ðB1mnkl þ 2λp B2pmnkl ÞU ðnÞ U ðnÞ mn jl jk
ð31Þ
ðnÞ ¼ K^ nijlk U ðnÞ F il RðnÞ il jk
ð32Þ
ðnÞ ¼ B^ mil U ki U kl C m Gm
ð33Þ
ðnÞ
The above formulation of the Newton–Raphson method can be written in matrix form K ðnÞ ðΔyÞ ¼ F ðnÞ
ð34Þ
where K and F are the same as the stiffness matrix and forcing vector in Eq. (27). Δy includes ΔU jk and Δλn . In the static problem, n represents the step of iteration. 2.2.2. Dynamic problem The inertial term in the equation of motion cannot be neglected in the dynamic problem. For the dynamic problem, the elongation condition can be approximated by
where δ is the Kronecker Delta function, L is the element length, and the standard double-suffix summation condition has been used. The elongation condition, incorporating Taylor series expansion to second order, can be written as
In the FEM, the variables r i and T E may be approximated (Garrett, 1982) as
B^ mkl U jl U jk C m ¼ 0
r i ðs; tÞ ¼
ð19Þ
1 ¼ 1 ε þ ε2 þ oðε3 Þ ð1 þ εÞ
B^ mkl ¼ B0mkl þ λn B1nmkl þ λn λp B2npmkl
ð36Þ
Z
L 0
P m A0k A0l ds
B1nmkl ¼
B2mnqkl ¼ Z
ð20Þ
T E ðs; tÞ ¼
3 X
P m ðsÞ λm ðtÞ
ð37Þ
m¼1
L 0
Z
L 0
ð21Þ
The variable U ik and λm are defined as: U i1 ¼ r i ð0; tÞ; U i2 ¼ r 0i ð0; tÞ
Z
Cm ¼
Ak ðsÞU ik ðtÞ
k¼1
where
B0mkl ¼
4 X
ð35Þ
2 P m P n A0k A0l ds EA 3
P m P n P q A0k A0l ds ðEAÞ2
L
P m ds 0
ð22Þ
λ1 ¼ T E ð0; tÞ; λ2 ¼ T E
ð24Þ
∂Ril ∂R ðΔU jk Þ þ il ðΔλn Þ ¼ 0 ∂U jk ∂ λn
ð25Þ
∂Gm ∂Gm ðΔU jk Þ þ ðΔλn Þ ¼ 0 ∂U jk ∂ λn
ð26Þ
ðn þ 1Þ ¼ GðnÞ Gm m þ
ð23Þ
Recalling Eq. (13) and the elongation condition (19), Newton– Raphson iteration was applied to the static problem (Ran, 2000). Omitting higher order components, we have þ 1Þ ¼ RilðnÞ þ Rðn il
U i3 ¼ r i ðL; tÞ; U i4 ¼ r 0i ðL; tÞ
L ; t ; λ3 ¼ T E ðL; tÞ 2
ð38Þ
ð39Þ
The definition of r i and T E are the same as in the static case. Integrating over the element generates the discretized form of the equation of motion. Incorporating the elongation condition, we have ^ ijlk Þ U€ jk ¼ λn K^ nijlk U jk þ F^ il ðM
ð40Þ
where ^ ijlk ¼ M ijlk þ M a M ijlk
ð41Þ M aijlk
where M ijlk and matrices,F^ il ¼ F il þ F H
are the standard mass and added mass
Gm ¼ B^ mkl U jl U jk C m C mn λn ¼ 0
ð42Þ
Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
For dynamic problem B^ mkl ¼ B0mkl þ λn B1nmkl þ λn λp B2npmkl B0mkl ¼
Z
L 0
P m A0k A0l ds Z
B1nmkl ¼
0
Z B2mnqkl ¼ L 0
ð44Þ
1 P m P n A0k A0l ds EA
L 0
Z C mn ¼
L
1 ðEAÞ2
ð43Þ
ð45Þ
P m P n P q A0k A0l ds
ð46Þ
1 P m P n ds EA
ð47Þ
To solve the second-order differential equation of motion (40), Ran (2000) introduced a new variable V:
447
Table1 Structural properties of the wind turbine. Items
Values
Draft (m) Elevation to platform top (Tower Base) above SWL (m) Depth to top of taper below SWL (m) Depth to bottom of taper below SWL (m) Platform diameter above taper (m) Platform diameter below taper (m) Platform mass, including ballast (Kg) CM location below SWL along platform centreline (m) Platform roll inertia about CM (kg m2) Platform pitch inertia about CM (kg m2) Platform yaw inertia about platform centreline (kg m2)
120 10 4 12 6.5 9.4 7.46E6 89.91 4229.2E6 4229.2E6 164.2E6
Table 2 Properties of mooring line (OrcaFlex, UK).
^ ^ V_ ¼ λn K^ M ijlk jk nijlk U jk þ F il
ð48Þ
Items
Values
U_ jk ¼ V jk
ð49Þ
Water depth(m) Number of mooring lines Angle between adjacent lines (degree) Depth to fairleads below SWL (m) Radius to fairleads from platform centerline (m) Upstretched mooring line length (m) Mooring line diameter (m) Mooring line mass (Kg/m) Mooring line elastic stiffness (EA) (N) Mooring line drag coefficient Mooring line added mass coefficient Mooring radius (m)
320 3 120 70 4.7 450 0.086 7.978 10.9E6 1.0 1.0 417.986
To solve these two equations, we need to integrate from t(n) to t(nþ 1), using the first-order Adam-Moulton method. Ran (2000) ^ ðn þ 0:5Þ during this time interval, assumed a constant value M ijlk
leading to the motion equation: 4 ^ n þ 0:5 n 0:5 ^ n ^ þ λ M K nijlk ΔU jk þ 2K nijlk U jk ðΔλn Þ n Δt 2 ijlk n n1 4 ^ n þ 0:5 n n 0:5 ^ V jk þ ð3F^ il F^ il Þ 2λn ¼ K nijlk U njk M Δt ijlk
ð50Þ
The elongation condition (42), using Taylor series expansion, þ 1Þ can be approximated as Gðn m þ 1Þ 0 ¼ 2Gðn 2GðnÞ m m þ2
∂GðnÞ ∂GðnÞ m ΔU jk þ 2 m Δλn ∂U jk ∂ λn
t1 ðnÞ ^ ¼ 2GðnÞ m þ 2K nijlk U il ðΔU jk Þ þC mn ðΔλn Þ
Re-writing Eqs. (50) and (51) in matrix form, we have 8 9 3 ) 011 ðnÞ ðnÞ ( < R0 ðnÞ = K 012 K ijlk Δ U jk iln il 4 5 ¼ 021 ðnÞ ðnÞ : G0mðnÞ ; Δλn K mjk K 022 mn
ð51Þ
2
ð52Þ
where 011 ðnÞ ¼ K ijlk
4 ^ ðn þ 0:5Þ ðn 0:5Þ ^ þ λn M ijlk K nijlk
Δt 2
ð53Þ
012 ðnÞ ¼ 2K^ nijlk U jk K iln
ð54Þ
021 ðnÞ ¼ 2K^ mijlk U il K mjk
ð55Þ
022 ðnÞ 22 ðnÞ ¼ 2ðK mn C mn Þ K mn
ð56Þ
R0ilðnÞ ¼
ðnÞ ðn 1Þ 4 ^ ðn þ 0:5Þ ðnÞ ðn 0:5Þ ^ V jk 2λn þ 3F^ il F^ il K nijlk U ðnÞ M ijlk jk
Δt
G0mðnÞ ¼ 2GðnÞ m
ð57Þ ð58Þ
The dynamic problem can be solved in a manner similar to the static case: K ðnÞ ðΔyÞ ¼ F ðnÞ
ð59Þ
Now, n denotes the time step (instead of the iteration step in the static analysis). The static model was first used to determine the mean position of the mooring line. The above numerical procedure was implemented by modifying FAST's (NWTC, 2014) FEAMooring. The original FAST program, based on the assumption of small elongations was extended to allow for large elongations,
Fig. 1. Comparison with Orcaflex and small extensible rod results in sea state 6 (Line1). (For interpretation of the references to color in this figure , the reader is referred to the web version of this article.)
and therefore suitable for polyester lines. An advantage of the present method is that the tangent stiffness matrix remains symmetric. 2.3. Validation of the enhanced model The present study considered a model of a spar-type floating platform, similar to that used for a wind turbine design of the National Renewable Energy Laboratory (NREL). The platform can be moored by slack or taut mooring lines (ABS, 2014a). In this paper, three equal taut mooring lines were selected for case studies. The parameters of the floating cylinder and the upper structure are the same as NREL's OC-3 Hywind Spar (Jonkman, 2010), except for the mooring system. The main properties of the wind turbine are shown in Table 1; those of the taut mooring line
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Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
Fig. 2. Comparison with Orcaflex and small extensible rod results in sea state 6 (Line2). (For interpretation of the references to color in this figure , the reader is referred to the web version of this article.)
Fig. 4. Comparison of mooring line tension(line length 420 m).
wind wave Line 2
Line 3
Line 1 Fig. 5. Plan view of the FOWT mooring configuration. Fig. 3. Comparison of line tension for a catenary chain under sea state 6.
in Table 2. For simplicity, the Radius to Fairleads from Platform Centreline in Table 1 was 4.7 m, instead of 5.2 m. Figs. 1 and 2 show the dynamic response of line tension in sea state 6 (H ¼5.5 m, T ¼11.3 s). The red line shows the results of Fastlink (FAST þOrcaFlex). In Fastlink, OrcaFlex solves the dynamic response of mooring line in the time domain and passes the mooring line tension to FAST for the coupled response of the mooring system. From this comparison we can see that the Taylor expansion to second order (present) shows little difference compared with the results from third order (extended stiffness). They both show very good agreement with the lumped mass and spring method. However, when assuming small elongation (equivalent to an expansion to first order, using the governing equation and elongation condition of the rod by Paulling & Webster,1986) the blue and green lines in Figs. 1 and 2 do not show good results for a polyester line. The present enhanced stiffness method is also appropriate for a slack mooring line (catenary chain, line length: 902.2 m; chain mass: 77.7 kg/m and elastic stiffness: 384.2*106 N). Fig. 3 compares the line tension results under sea state 6 for a catenary chain. Results from the approximation to second (reduced stiffness) and third order (extended stiffness) generate the same results as OrcaFlex and the small elongation assumption. From a comparison of Figs. 1–3 we can see that the present method can be used for modelling both traditional materials as well as high-performance fiber. For the extended stiffness condition (expansion to third order), the stiffness term and elongation are: K^ nijlk ¼ K 0nijlk þ λm K 1nmijlk þ λm λp K 2nmpijlk þ λq λm λp K 3nmpqijlk
ð60Þ
B^ mil ¼ B0mil þ λn B1nmil þ λn λp B2npmil þ λn λp λq B3npqmil Z K 3nmpqijlk ¼
L 0
1 ðEAÞ3
P n P m P p P q A0l A0k δij ds
ð61Þ
ð62Þ
For the static problem, Z B3npqmil ¼
L
4 ðEAÞ3
0
P n P p P q P m A0i A0l ds
ð63Þ
while for the dynamic problem, Z B3npqmil ¼
L 0
1 ðEAÞ3
P n P p P q P m A0i A0l ds
ð64Þ
2.4. Comparison of line tension for different approximations In order to check further the effect of differing elongation approximations, a reduced line length (420 m) was considered. Waves only were assumed, having the same height and frequency as above (sea state 6). The elongation of the mooring line is about 15%, but the difference between values of the mean and maximum line tension is around 1.3% (see Fig. 4, the mooring line layout is shown in Fig. 5). So, the approximation to second order is sufficient.
Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
3. Comparison between present method and linear spring method
449
Table 4 List of case studies, regular wave plus wind condition.
3.1. Methods applied The hydrodynamic coefficients and wave exciting forces for the OC-3 Spar-type wind turbine were pre-calculated by WAMIT (2013) and stored in FAST. Only first-order wave forces were included in the present study. The impulse response function method of Cummins (1962) was used in our time-domain study. Fourier transformations converted the frequency-dependent radiation damping terms for use in the time-domain model. The equation of motion of the floating platform is Z t Rðt τÞ X_ ðτÞdτ ¼ F e ð65Þ M þ M a ð1Þ X€ þ KX þ
Case studies
H (m)
Frequency (rad/s)
Mean wind speed (m/s)
1 2 3 4 5 6 7
2.56 2.56 2.56 2.56 2.56 2.56 2.56
0.157 0.185 0.201 0.242 0.299 0.483 0.897
11.4 11.4 11.4 11.4 11.4 11.4 11.4
1
where M is the mass and Ma the added mass at infinity, obtaining from the infinite-frequency limit.; K is the hydrostatic matrix; X is the motion response of the floating body. F e is the external force on the floating body, arising from waves and currents. The HydroDyn model of FAST calculates the retardation function and motion response of the platform; the former is given Z 2 t bðωÞ cos ðωtÞdω ð66Þ RðtÞ ¼
π
1
where b is the frequency-dependent radiation damping coefficients. The coupling between the floating body and mooring line applied a ‘loose coupling’ method, as introduced by Jonkman (2013).
Fig. 7.1. Surge RAO wave only.
3.2. Mooring system load-offset relationship Fig. 6 shows the surge restoring force against different initial horizontal position. From the graph we can see that the load-offset relationship is almost linear. For the linear spring method, the
Platform Restoring Force (N)
1E6
800E3
600E3
400E3
200E3
Fig. 7.2. Heave RAO wave only.
0 0
5
10
15
20
25
30
Offset (m), for direction 0.0 (deg)
Fig. 6. Platform surge load-offset graph.
Table 3 List of case studies, regular wave only condition. Case studies
H (m)
Frequency (rad/s)
1 2 3 4 5 6 7
2.56 2.56 2.56 2.56 2.56 2.56 2.56
0.157 0.185 0.201 0.242 0.299 0.483 0.897 Fig. 7.3. Pitch RAO wave only.
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Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
Fig. 7.4. Surge RAO wave plus wind.
Fig. 8.1. Surge time history, wave only condition T ¼ 26 s.
Fig. 7.5. Heave RAO wave plus wind.
Fig. 8.2. Heave time history, wave only condition T ¼ 26 s.
Fig. 7.6. Pitch RAO wave plus wind.
Fig. 8.3. Pitch time history, wave only condition T ¼ 26 s.
spring stiffness was derived from the same method- giving an initial offset and the spring stiffness was calculated with the following equation K¼
ΔF ΔS
ð67Þ
where K is the spring stiffness, F and S are the force and offset of the mooring system, respectively. The off-diagonal stiffness was ignored. In other words, in the linear stiffness method, the coupling effects (e.g. heave-pitch coupling) were not accounted for. The liner spring stiffness for surge, sway and heave are 30,680.5 N/m, 29,728.2 N/m and 23,178 N/m, respectively.
3.3. Results of case studies under wave-only and wave plus wind condition
Platform motion response The environmental conditions are shown in Tables 3 and 4. As potential theory fails to consider viscous effects, the additional linear damping was added. The additional damping for surge, sway and heave are 105 N/(m/s), 105 N/(m/s) and 1.3*105 N/(m/s), respectively. Figs. 7.1–7.6 show the motion RAOs for the Spar under wave-only and wave plus wind condition. Figs. 8–11 show a comparison of time history. Under the wave-only condition, the amplitude of heave
Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
Fig. 9.1. Surge time history, wave plus wind condition T ¼ 26 s.
451
Fig. 10.1. Surge time history, wave only condition T ¼7 s.
Fig. 9.2. Heave time history, wave plus wind condition T ¼ 26 s. Fig. 10.2. Heave time history, wave only condition T ¼ 7 s.
Fig. 9.3. Pitch time history, wave plus wind condition T ¼ 26 s.
response is not affected by the method of analysis, but the mean heave position has seen a large difference between the two methods, as can be seen from Figs. 8.2 and 9.2. Surge and pitch RAOs decrease under the wave plus wind condition, compared with wave-only condition. For the wave-only condition, there is little difference between the two methods for wave frequencies larger than 0.4 rad/s (e.g. Figs. 10.1 and 10.3), except for the mean position of the heave motion. These results indicate that for the primary design of substructure of the FOWT under some survival conditions (e.g. sea state 7 or sea state 8), the linear spring method can be applied, as it gives results as accurate as the FEM method but with less running time. However, for the wave plus wind condition, although the amplitude of motion response shows little difference between the linear spring
Fig. 10.3. Pitch time history, wave only condition T ¼ 7 s.
method and elastic rod theory, the mean position of surge and pitch were under-predicted by the linear spring method. Under the waveonly condition, the floating body oscillates about its mean position, but there is a very large mean offset (e.g. about 42 m in Fig. 11.1) when considering wave plus wind condition. Under the wave plus wind condition, the turbine thrust force is much larger than the wave forces.
Turbine thrust force Fig. 12 shows a comparison of mean rotor thrust force under linear spring method and present elastic rod theory. The mean
452
Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
Fig. 11.1. Surge time history, wave plus wind condition T ¼ 7 s.
Fig. 12. Comparison of turbine thrust force with different wave frequencies.
Fig. 11.2. Heave time history, wave plus wind condition T ¼7 s. Fig. 13.1. Comparison of mooring line tension under wave only and wave plus wind condition (T ¼26 s, Line1).
Fig. 11.3. Pitch time history, wave plus wind condition T ¼7 s.
thrust force is independent of wave frequency, as expected. However, linear spring method slightly underestimates the mean thrust force (around 1%).
Mooring line tension Figs. 13.1 and 13.2 show the mooring line tension for both waveonly and wave plus current condition. Under the wave-only condition, the mooring line tension does not vary much; the floating body oscillates around its initial mean position. The reason for this phenomenon is because current modelling only included first-order wave
Fig. 13.2. Comparison of mooring line tension under wave only and wave plus wind condition (T ¼ 26 s, Line2).
forces. The second-order effects are of little importance for the Spartype wind turbine (e.g. Roald et al., 2013). For the wave plus wind condition, the FOWT moves to a new equilibrium position and oscillates around this position, which results in one of the mooring lines becoming less taut. However, as discussed in the previous section, the proposed method is suitable for modelling both slack and taut mooring lines.
Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
453
4. Conclusions and discussion
References
The wind turbine simulation tool FAST has been modified to examine the response of a FOWT with polyester mooring lines. An enhanced stiffness model has been implemented to account for large elongations of the line. Its accuracy has been assessed numerically, and the results show that the proposed model is suitable for modelling both slack and taut mooring lines. The present approach has been applied to the simulation of a taut-moored FOWT. Comparison has been made against the linear spring method. Although the mooring system's static load-offset graph is linear, the linear spring method fails to consider the dynamic response of the mooring line. It under-predicts the motion of the floating body in the wave plus wind condition. This under prediction also affects the maximum mooring line tension, as its value is dependent on the instantaneous position of the floating platform. Due to the characteristics of the wind turbine thrust force, a coupled time-domain global performance analysis for a FOWT is much more time-consuming than the analysis of a floating platform. Therefore, this paper utilized an approximation focusing on the non-linear stress–strain relationship, but still is able to capture the nonlinear-stiffness characteristics-an important characteristics for polyester lines. A comparison between the current method and other approximations for modeling a polyester line, e.g. dual stiffness method (Tahar et al., 2012) and visco-elastic model (Kim et al., 2011), to establish the relative accuracy and efficiency would be a worthwhile future study.
ABS, 2014a. Global performance analysis for floating offshore wind turbine installations. ABS, 2014b. The application of fiber rope for offshore mooring. Agarwal, A.K., Jain, A.K., 2003. Dynamic behaviour of offshore spar platforms under regular sea waves. Ocean Eng. 30 (4), 487–516. API RP 2SM, 2001, Recommendation Practice for design, analysis, and testing of synthetic fiber ropes in offshore mooring applications. 1st ed. Bathe, K.J., 1996. Element Procedures. Prentice Hall. Ćatipović, I., Čorić, V., Radanović, J., 2011. An improved stiffness model for polyester mooring lines. Brodogradnja 62 (3), 235–248. Chen, X.H., 2002. Studies on Dynamic Iteration Between Deep-water Floating Structure and Their Mooring/tendon Systems. Texas A&M University, Dissertation. Cummins, W.E., 1962. The impulse response function and ship motions, DTMB Report 1661, Washington D.C. EWEA, 2013. Deep Water- The Next Step for Offshore Wind Energy. EWEA. Garrett, D.L., 1982. Dynamic analysis of slender rods. J. Energy Resour. Technol 104, 302–306. Goupee, A.J., Koo, B.J., Kimball, R.W., Lambrakos, K.F., Dagher, H.J., 2014. Experimental comparison of three floating wind turbine concepts. J. Offshore Mech. Arct. Eng. 136 (2). Jonkman, J., 2010. Definition of the floating System for phase IV of OC3. National Renewable Energy Laboratory. Golden, CO.NREL/TP-500-47535. Jonkman, J.M., 2013. The new modularization framework for the FAST wind turbine CAE tool, 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. Kim, B.W., Sung, H.G., Kim, J.H., Hong, S.Y., 2013. Comparison of linear spring and nonlinear FEM methods in dynamic coupled analysis of floating structure and mooring system. J. Fluids Struct. 42, 205–227. Kim, C.H., Kim, M.H., Liu, Y.H., Zhao, C.T., 1994. Time domain simulation of nonlinear response of a coupled TLP system. Int. J. Offshore Polar Eng. 4 (4), 284–291. Kim, J.W., Sablok, A., Kyoung, J.H. and Lambrakos, K., 2011. A nonlinear viscoelastic model for polyester mooring line analysis. In: Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering. OMAE 2011, pp. 797–803. Kim, M.H., Tahar, A., Kim, Y.B., 2001. Variability of spar motion analysis against design methodologies/parameters, In: Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering. OMAE 2001, pp. 153–161. Kreuzer, E., Wilke, U., 2003. Dynamics of mooring systems in ocean engineering. Arch. Appl. Mech. 73 (3-4), 270–281. Love, A.E.H., 1944. A Treatise on the mathematical theory of elasticity, 4th ed Dover Publications, New York. Low, Y.M., 2006. Efficient Methods for the Dynamic Analysis of Deepwater Offshore Production Systems. University of Cambridge, PhD Thesis. Morison, J.R., O'Brien, M.R., Johnson, J.W., Schaaf, S., 1950. The forces exerted by surface waves on piles. Pet. Trans. 2 (5), 149–154. Nordgren, R.P., 1974. On computation of the motion of elastic rods. J. Appl. Mech. Trans. ASME 41 (2), 777–780, Ser E. NWTC Computer-Aided Engineering Tools (FAST by Jason Jonkman, Ph.D.). 〈http://wind.nrel.gov/designcodes/simulators/fast/〉. Last modified 2-July-2014; (accessed 01.09.14). OrcaFlex user manual Version 9.7a. Orcina Ltd. Daltongate, Ulverston Cumbria. LA12 7AJ, UK. Paulling, J.R., Webster, W.C., 1986. A consistent large amplitude analysis of the coupled response of a TLP and tendon system. In: Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering 3, p. 126. Ran, Z., 2000. Coupled dynamic analysis of floating structures in waves and currents, ,. Texas A&M University, Dissertation. Roald, L., Jonkman, J., Robertson, A., Chokani, N., 2013. The effect of second-order hydrodynamics on floating offshore wind turbines. Energy Procedia, 253–264. Tahar, Arcandra, 2001. Hull/Mooring/Riser Coupled Dynamic Analysis of a Floating Platform with Polyester Mooring Lines. Texas A&M University, Dissertation. Tahar, A., Sidarta, D., Ran, A., 2012. Dual stiffness approach for polyester mooring line analysis in time domain. In: Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering - OMAE, pp. 513–521. WAMIT Inc. Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, Mass., USA, 2013.
Acknowledgment The authors gratefully acknowledge the assistance of Dr Jason Jonkman in his responses to questions about the simulation tool FAST. This research was supported by the Department of Naval Architecture, Ocean and Marine Engineering and the Faculty of Engineering at the University of Strathclyde and by the China Scholarship Council (CSC) (Grant no. 2011606021).
Appendix A Shape functions In the FEM, the shape functions Al and Pm are defined as follows (Garrett, 1982): A1 ¼ 1 3ξ þ 2ξ 2
3
A2 ¼ Lðξ 2ξ þ ξ Þ 2
A3 ¼ 3ξ 2ξ 2
3
3
A4 ¼ Lð ξ þ ξ Þ
ðA:1Þ
P 1 ¼ 1 3ξ þ 2ξ P 2 ¼ 4ξ ð1 ξÞ P 3 ¼ ξ ð2ξ 1Þ
ðA:2Þ
2
3
2
where ξ ¼ s=L
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
An enhanced stiffness model for elastic lines and its application to the analysis of a moored floating offshore wind turbine Zi Lin n, P. Sayer Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, UK
art ic l e i nf o
a b s t r a c t
Article history: Received 22 April 2015 Accepted 1 September 2015 Available online 8 October 2015
The performance of a polyester mooring line is non-linear and its elongation plays a significant role in the dynamic response of an offshore moored structure. However, unlike chain, the tension–elongation relationship and the overall behavior of elastic polyester ropes are complex. In this paper, by applying an enhanced stiffness model of the mooring line, the traditional elastic rod theory has been extended to allow for large elongations. One beneficial feature of the present method is that the tangent stiffness matrix is symmetric; in non-linear formulations the tangent stiffness matrix is often non-symmetric. The static problem was solved by Newton–Raphson iteration, whereas a direct integration method was used for the dynamic problem. The computed mooring line tension was validated against the proprietary OrcaFlex software. Results of mooring line top tension predicated by different elongations are compared and discussed. The present method was then used for a simulation of an offshore floating wind turbine moored with taut lines. From a comparison between linear and non-linear formulations, it is seen that a linear spring model under-estimates the mean position when the turbine is operating, but over-estimates the amplitude of the platform response at low frequencies when the turbine has shut down. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Large extension Elastic rod theory Finite element method Mooring system Motion response Dynamic response
1. Introduction The capture of offshore wind energy plays a key role across the maritime industry (EWEA, 2013). Offshore wind turbines are becoming larger and more powerful, and are being deployed in ever-deeper water. They can be mounted on a fixed or floating base, but the former starts to lose its economic advantage for water depths larger than 60 m (Goupee et al., 2014). Although the mooring system design for a floating offshore wind turbine (FOWT) has benefited from the experience of offshore oil and gas platforms, there are still several unknowns dependent on the type of floating bodies, e.g. size and environmental loading. From a report of EWEA (2013), it is recommended that more research must be done on mooring and anchoring systems for wind turbines. Numerical simulations of the dynamic response of mooring lines have been studied during the past few decades, for both elastic and inelastic lines. A massless spring (e.g. Kim et al., 2001) or the catenary equation (e.g. Agarwal and Jain, 2003) provide straightforward ways to model a mooring line, but it is difficult to account for the dynamic response and the interaction between the floating body and mooring line in an accurate manner. Multi-body n
Corresponding author. Tel.: þ 44 141 548 4911. E-mail address: [email protected] (Z. Lin).
http://dx.doi.org/10.1016/j.oceaneng.2015.09.002 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
system dynamics (e.g. Kreuzer and Wilke, 2003) divides the mooring line into several rigid bodies, but results in a large number of degrees of freedom even for a single line. Non-linear finite element methods (FEMs), accounting for geometric and material non-linearities, have been widely used for modelling mooring line response (e.g. Kim et al., 2013). Geometric nonlinearity is needed for large displacements of the mooring line, while material non-linearity can model the time-dependent properties of a polyester rope, e.g. Young's modulus. However, a major disadvantage of FEM is the transformation between local coordinate and global coordinate, which is often computationally intensive. The lumped mass and spring method can be categorized as a non-linear FEM method, for which the shape function becomes a single line (Low, 2006). Unlike traditional non-linear FEMs, the elastic rod theory is a global-coordinate-based method, which is considered to be more efficient (Kim et.al, 1994). The transformation between local and global coordinate is dealt within the element stiffness matrix. Following the elastic rod theory of Love (1944), Nordgren (1974) and Garrett (1982) developed this method and solved the governing non-linear equations by a finite difference method (FDM) and by FEM, respectively. Many researchers have further developed the elastic rod theory, including elongation of the line, seabed friction, non-linear material properties and the incorporation of buoys or clump weights in the mooring line model. Paulling and Webster (1986) considered the analysis of large amplitude
Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
motions of a TLP under the action of wind, wave and current, under the assumption of small line elongation. Ran (2000) proposed a finite element formulation for mooring lines and risers based on Garrett's rod theory, applicable to both frequency and time domains. Based on the traditional small extensible rod theory, the incorporation of large elongation has been presented by many researchers (e.g. Chen, 2002; Tahar, 2001; Kim et al., 2011). Based on the successful experience from offshore oil & gas platforms, the design and modelling of a FOWT has tended to use the same mathematical modelling and methods of solution as for offshore platforms, e.g. the hydrodynamic analysis of floating body, mooring design and the types of FOWTs (Spar, TLP and Semisubmersible, etc). The design of a station-keeping system for a FOWT uses the same methods and guidance as for a floating platform (e.g. ABS, 2014a). However, the geometry and operational water depth are different. Also, the turbine thrust force may have an effect on the motion response of the floating body and mooring line tension, and vice versa. These differences need to be examined for a FOWT. Polyester lines are made from visco-elastic materials and the stiffness characteristics rely on the loading history, the load duration and magnitude, etc (ABS, 2014b). The material nonlinearity of polyester lines is difficult to model and requires a longer simulation time. Thus some approximate models are used. For example, the dual stiffness method (Tahar et al., 2012) recommended by API RP 2SM (2001), and the visco-elastic model (Kim et al., 2011). In the present paper, a sensible balance has been sought between efficiency and accuracy. The traditional rod theory has been extended to allow for large stretch by applying an enhanced stiffness method. By using an approximation of the nonlinear tension-elongation relationship in a Taylor series expansion (Ćatipović et al., 2011), the mathematical and numerical formulation of large extensible mooring line are considered.
2. Mathematical formulation of a mooring line with large elongation
445
Mooring line normal component of acceleration r€ n , normal component of velocity r_ n , normal component of water particle _ n are given by velocity Vn and water practical acceleration V (Ćatipović et al., 2011) dr dr dr dr ; r€ n ¼ r€ r€ U ð3Þ r_ n ¼ r_ r_ U ds ds ds ds €n ¼V _ V _ U dr dr; V € V € U dr dr _n ¼V V ds ds ds ds
ð4Þ
In Eq. (1), ε is the elongation of the rod. Following Ćatipović et al., assuming equal principal stiffness, the relationship between the effective tension and elongation can be written as
ε¼
TE AE
ð5Þ
where AE is the axial stiffness The following elongation condition then has to be satisfied (Ćatipović et al., 2011) 1 dr dr U ¼1 ð1 þ εÞ2 ds ds
ð6Þ
In the static problem, the mass per unit length and diameter of the mooring line are related to the elongation ε. The crosssectional area and mass after elongation can be written as A=ð1 þ εÞ and m=ð1 þ εÞ, respectively (Ćatipović et al., 2011), where A and m are the cross-section area and mass of the mooring line without stretch. Applying the above relationship to the motion equation, we see that the term ð1 þ εÞ, multiplied by the applied force qE cancels out. For the hydrodynamic force calculated by Morison's equation, the mass per unit length and cross-sectional area for one element were assumed constant. Eqs. (1) and (6) show the rod motion equation and elongation condition, respectively: they are non-linear. In the following section, we will describe a numerical procedure for solving this nonlinear equation and the required order of approximation for the elongation condition. 2.2. Numerical Implementation
2.1. Equation of motion For polyester mooring lines bending and torsion stiffness can be neglected, but the elongation cannot be assumed to be small. The mooring line is discretized into a number of rods and the centreline of each rod is described by a space-time curve r ðs; tÞ. From Ćatipović et al. (2011), the equation of motion for a rod with large elongation can be written as: d T E dr þ 1 þ εÞqE ¼ ð1 þ εÞ ρr€ ð1Þ ds 1 þ ε ds ~ and A~ are distributed mass and cross-section area after where m extension.ρ and g are sea water density and gravitational acceleration, respectively.r€ represents the time derivation of the rod. T E is the effective tension of the rod. The relation between the real tension T R and the effective tension are (Ćatipović et al., 2011): T E ¼ T R þ pA, where p and A are hydrostatic pressure and crosssection area, respectively. qE is the load acting on the rod. For static ~ ~ ρAg, problem,qE ¼ mg while for dynamic problem ~ þ FH . ~ ρAg qE ¼ mg in which FH is the hydrodynamic loads on the mooring line (Paulling and Webster, 1986) calculated by Morison's equation (1950) as _ n þ 1C D ρDVn r_ n ðVn r_ n Þ FH ¼ C A ρAr€ n þ C M ρAV 2
ð2Þ
where n denotes the normal component. C A ,C M and C D are the added mass, inertial (Morison) and drag coefficients.
2.2.1. Static problem For the static problem, r is independent of time. Consequently the inertial term in Eq. (1) is deleted. We therefore have d T E dr þ qE ¼ 0 ð7Þ ds 1 þ ε ds Using the Taylor series expansion, the elongation relationship can be written as: 1 ð1 þ εÞ2
¼ 1 2ε þ 3ε2 þ oðε3 Þ
ð8Þ
However, it is not clear, a priori, whether the third-order term should be included explicitly. In the present paper, the order of expansion and subsequent results will be discussed. In the FEM, the variables r i and T E may be approximated (Garrett, 1982) as r i ðsÞ ¼
4 X
Ak ðsÞU ik
ð9Þ
k¼1
T E ðsÞ ¼
3 X
P m ðsÞ λm
ð10Þ
m¼1
where Ak and P m are shape functions. The definition of the shape functions can be found in the Appendix A. U ik and λm are unknown variables. The subscript i of U ik denotes the dimension of the element. For the 3-dimensional problem, i¼3. For k ¼1 and 3, U ik
446
Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
represents the space position of the rod at two ends while U ik denotes the space derivative at both ends for k ¼2 and 4. λ is the Lagrange multiplier. The physical meaning of λ is mooring line tension at both ends and middle of the rod. The variable U ik and λm are defined as:
Re-arranging the terms and writing in the matrix form, we have 8 9 2 11 ðnÞ 3 ) ðnÞ ( < RðnÞ = K ijlk K 12 ΔU jk iln il 4 5 ð27Þ ¼ 21 ðnÞ ðnÞ : GðnÞ ; Δλn K mjk K 22 mn m
U i1 ¼ r i ð0Þ; U i2 ¼ r 0i ð0Þ U i3 ¼ r i ðLÞ; U i4 ¼ r 0i ðLÞ
ð11Þ
where
L ; λ3 ¼ T E ðLÞ λ1 ¼ T E ð0Þ; λ2 ¼ T E 2
ð12Þ
ðnÞ
Using Galerkin's method (Bathe, 1996) and integrating the motion equation from 0 to L over the length of the element, the final form of motion equation for static problem in notation form can be written as K^ nijlk λn U jk F il ¼ 0
ð13Þ
where K^ nijlk ¼ K 0nijlk þ λm K 1nmijlk þ λm λp K 2nmpijlk Z
K 0nijlk ¼
L
P n A0l A0k δij ds
0
Z K 1nmijlk
¼
Z F il ¼
1 P n P m A0l A0k δij ds EA
0
Z
K 2nmqijlk ¼
L
L
1 ðEAÞ2
0
Z
L
mg i ds
0
P n P m P q A0l A0k δij ds L
0
ðρAÞg i Al ds
ð14Þ ð15Þ
ð16Þ
ð17Þ
ð18Þ
ðnÞ ¼ K^ nijlk λn K 11 ijlk
ð28Þ
ðnÞ ¼ ðK 0nijlk þ2λm K 1nmijlk þ 3λm λp K 2nmpijlk ÞU ðnÞ K 12 iln jk
ð29Þ
ðnÞ ¼ B^ mkl U ðnÞ K 21 mjk jl
ð30Þ
ðnÞ K 22 ¼ ðB1mnkl þ 2λp B2pmnkl ÞU ðnÞ U ðnÞ mn jl jk
ð31Þ
ðnÞ ¼ K^ nijlk U ðnÞ F il RðnÞ il jk
ð32Þ
ðnÞ ¼ B^ mil U ki U kl C m Gm
ð33Þ
ðnÞ
The above formulation of the Newton–Raphson method can be written in matrix form K ðnÞ ðΔyÞ ¼ F ðnÞ
ð34Þ
where K and F are the same as the stiffness matrix and forcing vector in Eq. (27). Δy includes ΔU jk and Δλn . In the static problem, n represents the step of iteration. 2.2.2. Dynamic problem The inertial term in the equation of motion cannot be neglected in the dynamic problem. For the dynamic problem, the elongation condition can be approximated by
where δ is the Kronecker Delta function, L is the element length, and the standard double-suffix summation condition has been used. The elongation condition, incorporating Taylor series expansion to second order, can be written as
In the FEM, the variables r i and T E may be approximated (Garrett, 1982) as
B^ mkl U jl U jk C m ¼ 0
r i ðs; tÞ ¼
ð19Þ
1 ¼ 1 ε þ ε2 þ oðε3 Þ ð1 þ εÞ
B^ mkl ¼ B0mkl þ λn B1nmkl þ λn λp B2npmkl
ð36Þ
Z
L 0
P m A0k A0l ds
B1nmkl ¼
B2mnqkl ¼ Z
ð20Þ
T E ðs; tÞ ¼
3 X
P m ðsÞ λm ðtÞ
ð37Þ
m¼1
L 0
Z
L 0
ð21Þ
The variable U ik and λm are defined as: U i1 ¼ r i ð0; tÞ; U i2 ¼ r 0i ð0; tÞ
Z
Cm ¼
Ak ðsÞU ik ðtÞ
k¼1
where
B0mkl ¼
4 X
ð35Þ
2 P m P n A0k A0l ds EA 3
P m P n P q A0k A0l ds ðEAÞ2
L
P m ds 0
ð22Þ
λ1 ¼ T E ð0; tÞ; λ2 ¼ T E
ð24Þ
∂Ril ∂R ðΔU jk Þ þ il ðΔλn Þ ¼ 0 ∂U jk ∂ λn
ð25Þ
∂Gm ∂Gm ðΔU jk Þ þ ðΔλn Þ ¼ 0 ∂U jk ∂ λn
ð26Þ
ðn þ 1Þ ¼ GðnÞ Gm m þ
ð23Þ
Recalling Eq. (13) and the elongation condition (19), Newton– Raphson iteration was applied to the static problem (Ran, 2000). Omitting higher order components, we have þ 1Þ ¼ RilðnÞ þ Rðn il
U i3 ¼ r i ðL; tÞ; U i4 ¼ r 0i ðL; tÞ
L ; t ; λ3 ¼ T E ðL; tÞ 2
ð38Þ
ð39Þ
The definition of r i and T E are the same as in the static case. Integrating over the element generates the discretized form of the equation of motion. Incorporating the elongation condition, we have ^ ijlk Þ U€ jk ¼ λn K^ nijlk U jk þ F^ il ðM
ð40Þ
where ^ ijlk ¼ M ijlk þ M a M ijlk
ð41Þ M aijlk
where M ijlk and matrices,F^ il ¼ F il þ F H
are the standard mass and added mass
Gm ¼ B^ mkl U jl U jk C m C mn λn ¼ 0
ð42Þ
Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
For dynamic problem B^ mkl ¼ B0mkl þ λn B1nmkl þ λn λp B2npmkl B0mkl ¼
Z
L 0
P m A0k A0l ds Z
B1nmkl ¼
0
Z B2mnqkl ¼ L 0
ð44Þ
1 P m P n A0k A0l ds EA
L 0
Z C mn ¼
L
1 ðEAÞ2
ð43Þ
ð45Þ
P m P n P q A0k A0l ds
ð46Þ
1 P m P n ds EA
ð47Þ
To solve the second-order differential equation of motion (40), Ran (2000) introduced a new variable V:
447
Table1 Structural properties of the wind turbine. Items
Values
Draft (m) Elevation to platform top (Tower Base) above SWL (m) Depth to top of taper below SWL (m) Depth to bottom of taper below SWL (m) Platform diameter above taper (m) Platform diameter below taper (m) Platform mass, including ballast (Kg) CM location below SWL along platform centreline (m) Platform roll inertia about CM (kg m2) Platform pitch inertia about CM (kg m2) Platform yaw inertia about platform centreline (kg m2)
120 10 4 12 6.5 9.4 7.46E6 89.91 4229.2E6 4229.2E6 164.2E6
Table 2 Properties of mooring line (OrcaFlex, UK).
^ ^ V_ ¼ λn K^ M ijlk jk nijlk U jk þ F il
ð48Þ
Items
Values
U_ jk ¼ V jk
ð49Þ
Water depth(m) Number of mooring lines Angle between adjacent lines (degree) Depth to fairleads below SWL (m) Radius to fairleads from platform centerline (m) Upstretched mooring line length (m) Mooring line diameter (m) Mooring line mass (Kg/m) Mooring line elastic stiffness (EA) (N) Mooring line drag coefficient Mooring line added mass coefficient Mooring radius (m)
320 3 120 70 4.7 450 0.086 7.978 10.9E6 1.0 1.0 417.986
To solve these two equations, we need to integrate from t(n) to t(nþ 1), using the first-order Adam-Moulton method. Ran (2000) ^ ðn þ 0:5Þ during this time interval, assumed a constant value M ijlk
leading to the motion equation: 4 ^ n þ 0:5 n 0:5 ^ n ^ þ λ M K nijlk ΔU jk þ 2K nijlk U jk ðΔλn Þ n Δt 2 ijlk n n1 4 ^ n þ 0:5 n n 0:5 ^ V jk þ ð3F^ il F^ il Þ 2λn ¼ K nijlk U njk M Δt ijlk
ð50Þ
The elongation condition (42), using Taylor series expansion, þ 1Þ can be approximated as Gðn m þ 1Þ 0 ¼ 2Gðn 2GðnÞ m m þ2
∂GðnÞ ∂GðnÞ m ΔU jk þ 2 m Δλn ∂U jk ∂ λn
t1 ðnÞ ^ ¼ 2GðnÞ m þ 2K nijlk U il ðΔU jk Þ þC mn ðΔλn Þ
Re-writing Eqs. (50) and (51) in matrix form, we have 8 9 3 ) 011 ðnÞ ðnÞ ( < R0 ðnÞ = K 012 K ijlk Δ U jk iln il 4 5 ¼ 021 ðnÞ ðnÞ : G0mðnÞ ; Δλn K mjk K 022 mn
ð51Þ
2
ð52Þ
where 011 ðnÞ ¼ K ijlk
4 ^ ðn þ 0:5Þ ðn 0:5Þ ^ þ λn M ijlk K nijlk
Δt 2
ð53Þ
012 ðnÞ ¼ 2K^ nijlk U jk K iln
ð54Þ
021 ðnÞ ¼ 2K^ mijlk U il K mjk
ð55Þ
022 ðnÞ 22 ðnÞ ¼ 2ðK mn C mn Þ K mn
ð56Þ
R0ilðnÞ ¼
ðnÞ ðn 1Þ 4 ^ ðn þ 0:5Þ ðnÞ ðn 0:5Þ ^ V jk 2λn þ 3F^ il F^ il K nijlk U ðnÞ M ijlk jk
Δt
G0mðnÞ ¼ 2GðnÞ m
ð57Þ ð58Þ
The dynamic problem can be solved in a manner similar to the static case: K ðnÞ ðΔyÞ ¼ F ðnÞ
ð59Þ
Now, n denotes the time step (instead of the iteration step in the static analysis). The static model was first used to determine the mean position of the mooring line. The above numerical procedure was implemented by modifying FAST's (NWTC, 2014) FEAMooring. The original FAST program, based on the assumption of small elongations was extended to allow for large elongations,
Fig. 1. Comparison with Orcaflex and small extensible rod results in sea state 6 (Line1). (For interpretation of the references to color in this figure , the reader is referred to the web version of this article.)
and therefore suitable for polyester lines. An advantage of the present method is that the tangent stiffness matrix remains symmetric. 2.3. Validation of the enhanced model The present study considered a model of a spar-type floating platform, similar to that used for a wind turbine design of the National Renewable Energy Laboratory (NREL). The platform can be moored by slack or taut mooring lines (ABS, 2014a). In this paper, three equal taut mooring lines were selected for case studies. The parameters of the floating cylinder and the upper structure are the same as NREL's OC-3 Hywind Spar (Jonkman, 2010), except for the mooring system. The main properties of the wind turbine are shown in Table 1; those of the taut mooring line
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Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
Fig. 2. Comparison with Orcaflex and small extensible rod results in sea state 6 (Line2). (For interpretation of the references to color in this figure , the reader is referred to the web version of this article.)
Fig. 4. Comparison of mooring line tension(line length 420 m).
wind wave Line 2
Line 3
Line 1 Fig. 5. Plan view of the FOWT mooring configuration. Fig. 3. Comparison of line tension for a catenary chain under sea state 6.
in Table 2. For simplicity, the Radius to Fairleads from Platform Centreline in Table 1 was 4.7 m, instead of 5.2 m. Figs. 1 and 2 show the dynamic response of line tension in sea state 6 (H ¼5.5 m, T ¼11.3 s). The red line shows the results of Fastlink (FAST þOrcaFlex). In Fastlink, OrcaFlex solves the dynamic response of mooring line in the time domain and passes the mooring line tension to FAST for the coupled response of the mooring system. From this comparison we can see that the Taylor expansion to second order (present) shows little difference compared with the results from third order (extended stiffness). They both show very good agreement with the lumped mass and spring method. However, when assuming small elongation (equivalent to an expansion to first order, using the governing equation and elongation condition of the rod by Paulling & Webster,1986) the blue and green lines in Figs. 1 and 2 do not show good results for a polyester line. The present enhanced stiffness method is also appropriate for a slack mooring line (catenary chain, line length: 902.2 m; chain mass: 77.7 kg/m and elastic stiffness: 384.2*106 N). Fig. 3 compares the line tension results under sea state 6 for a catenary chain. Results from the approximation to second (reduced stiffness) and third order (extended stiffness) generate the same results as OrcaFlex and the small elongation assumption. From a comparison of Figs. 1–3 we can see that the present method can be used for modelling both traditional materials as well as high-performance fiber. For the extended stiffness condition (expansion to third order), the stiffness term and elongation are: K^ nijlk ¼ K 0nijlk þ λm K 1nmijlk þ λm λp K 2nmpijlk þ λq λm λp K 3nmpqijlk
ð60Þ
B^ mil ¼ B0mil þ λn B1nmil þ λn λp B2npmil þ λn λp λq B3npqmil Z K 3nmpqijlk ¼
L 0
1 ðEAÞ3
P n P m P p P q A0l A0k δij ds
ð61Þ
ð62Þ
For the static problem, Z B3npqmil ¼
L
4 ðEAÞ3
0
P n P p P q P m A0i A0l ds
ð63Þ
while for the dynamic problem, Z B3npqmil ¼
L 0
1 ðEAÞ3
P n P p P q P m A0i A0l ds
ð64Þ
2.4. Comparison of line tension for different approximations In order to check further the effect of differing elongation approximations, a reduced line length (420 m) was considered. Waves only were assumed, having the same height and frequency as above (sea state 6). The elongation of the mooring line is about 15%, but the difference between values of the mean and maximum line tension is around 1.3% (see Fig. 4, the mooring line layout is shown in Fig. 5). So, the approximation to second order is sufficient.
Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
3. Comparison between present method and linear spring method
449
Table 4 List of case studies, regular wave plus wind condition.
3.1. Methods applied The hydrodynamic coefficients and wave exciting forces for the OC-3 Spar-type wind turbine were pre-calculated by WAMIT (2013) and stored in FAST. Only first-order wave forces were included in the present study. The impulse response function method of Cummins (1962) was used in our time-domain study. Fourier transformations converted the frequency-dependent radiation damping terms for use in the time-domain model. The equation of motion of the floating platform is Z t Rðt τÞ X_ ðτÞdτ ¼ F e ð65Þ M þ M a ð1Þ X€ þ KX þ
Case studies
H (m)
Frequency (rad/s)
Mean wind speed (m/s)
1 2 3 4 5 6 7
2.56 2.56 2.56 2.56 2.56 2.56 2.56
0.157 0.185 0.201 0.242 0.299 0.483 0.897
11.4 11.4 11.4 11.4 11.4 11.4 11.4
1
where M is the mass and Ma the added mass at infinity, obtaining from the infinite-frequency limit.; K is the hydrostatic matrix; X is the motion response of the floating body. F e is the external force on the floating body, arising from waves and currents. The HydroDyn model of FAST calculates the retardation function and motion response of the platform; the former is given Z 2 t bðωÞ cos ðωtÞdω ð66Þ RðtÞ ¼
π
1
where b is the frequency-dependent radiation damping coefficients. The coupling between the floating body and mooring line applied a ‘loose coupling’ method, as introduced by Jonkman (2013).
Fig. 7.1. Surge RAO wave only.
3.2. Mooring system load-offset relationship Fig. 6 shows the surge restoring force against different initial horizontal position. From the graph we can see that the load-offset relationship is almost linear. For the linear spring method, the
Platform Restoring Force (N)
1E6
800E3
600E3
400E3
200E3
Fig. 7.2. Heave RAO wave only.
0 0
5
10
15
20
25
30
Offset (m), for direction 0.0 (deg)
Fig. 6. Platform surge load-offset graph.
Table 3 List of case studies, regular wave only condition. Case studies
H (m)
Frequency (rad/s)
1 2 3 4 5 6 7
2.56 2.56 2.56 2.56 2.56 2.56 2.56
0.157 0.185 0.201 0.242 0.299 0.483 0.897 Fig. 7.3. Pitch RAO wave only.
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Fig. 7.4. Surge RAO wave plus wind.
Fig. 8.1. Surge time history, wave only condition T ¼ 26 s.
Fig. 7.5. Heave RAO wave plus wind.
Fig. 8.2. Heave time history, wave only condition T ¼ 26 s.
Fig. 7.6. Pitch RAO wave plus wind.
Fig. 8.3. Pitch time history, wave only condition T ¼ 26 s.
spring stiffness was derived from the same method- giving an initial offset and the spring stiffness was calculated with the following equation K¼
ΔF ΔS
ð67Þ
where K is the spring stiffness, F and S are the force and offset of the mooring system, respectively. The off-diagonal stiffness was ignored. In other words, in the linear stiffness method, the coupling effects (e.g. heave-pitch coupling) were not accounted for. The liner spring stiffness for surge, sway and heave are 30,680.5 N/m, 29,728.2 N/m and 23,178 N/m, respectively.
3.3. Results of case studies under wave-only and wave plus wind condition
Platform motion response The environmental conditions are shown in Tables 3 and 4. As potential theory fails to consider viscous effects, the additional linear damping was added. The additional damping for surge, sway and heave are 105 N/(m/s), 105 N/(m/s) and 1.3*105 N/(m/s), respectively. Figs. 7.1–7.6 show the motion RAOs for the Spar under wave-only and wave plus wind condition. Figs. 8–11 show a comparison of time history. Under the wave-only condition, the amplitude of heave
Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
Fig. 9.1. Surge time history, wave plus wind condition T ¼ 26 s.
451
Fig. 10.1. Surge time history, wave only condition T ¼7 s.
Fig. 9.2. Heave time history, wave plus wind condition T ¼ 26 s. Fig. 10.2. Heave time history, wave only condition T ¼ 7 s.
Fig. 9.3. Pitch time history, wave plus wind condition T ¼ 26 s.
response is not affected by the method of analysis, but the mean heave position has seen a large difference between the two methods, as can be seen from Figs. 8.2 and 9.2. Surge and pitch RAOs decrease under the wave plus wind condition, compared with wave-only condition. For the wave-only condition, there is little difference between the two methods for wave frequencies larger than 0.4 rad/s (e.g. Figs. 10.1 and 10.3), except for the mean position of the heave motion. These results indicate that for the primary design of substructure of the FOWT under some survival conditions (e.g. sea state 7 or sea state 8), the linear spring method can be applied, as it gives results as accurate as the FEM method but with less running time. However, for the wave plus wind condition, although the amplitude of motion response shows little difference between the linear spring
Fig. 10.3. Pitch time history, wave only condition T ¼ 7 s.
method and elastic rod theory, the mean position of surge and pitch were under-predicted by the linear spring method. Under the waveonly condition, the floating body oscillates about its mean position, but there is a very large mean offset (e.g. about 42 m in Fig. 11.1) when considering wave plus wind condition. Under the wave plus wind condition, the turbine thrust force is much larger than the wave forces.
Turbine thrust force Fig. 12 shows a comparison of mean rotor thrust force under linear spring method and present elastic rod theory. The mean
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Z. Lin, P. Sayer / Ocean Engineering 109 (2015) 444–453
Fig. 11.1. Surge time history, wave plus wind condition T ¼ 7 s.
Fig. 12. Comparison of turbine thrust force with different wave frequencies.
Fig. 11.2. Heave time history, wave plus wind condition T ¼7 s. Fig. 13.1. Comparison of mooring line tension under wave only and wave plus wind condition (T ¼26 s, Line1).
Fig. 11.3. Pitch time history, wave plus wind condition T ¼7 s.
thrust force is independent of wave frequency, as expected. However, linear spring method slightly underestimates the mean thrust force (around 1%).
Mooring line tension Figs. 13.1 and 13.2 show the mooring line tension for both waveonly and wave plus current condition. Under the wave-only condition, the mooring line tension does not vary much; the floating body oscillates around its initial mean position. The reason for this phenomenon is because current modelling only included first-order wave
Fig. 13.2. Comparison of mooring line tension under wave only and wave plus wind condition (T ¼ 26 s, Line2).
forces. The second-order effects are of little importance for the Spartype wind turbine (e.g. Roald et al., 2013). For the wave plus wind condition, the FOWT moves to a new equilibrium position and oscillates around this position, which results in one of the mooring lines becoming less taut. However, as discussed in the previous section, the proposed method is suitable for modelling both slack and taut mooring lines.
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4. Conclusions and discussion
References
The wind turbine simulation tool FAST has been modified to examine the response of a FOWT with polyester mooring lines. An enhanced stiffness model has been implemented to account for large elongations of the line. Its accuracy has been assessed numerically, and the results show that the proposed model is suitable for modelling both slack and taut mooring lines. The present approach has been applied to the simulation of a taut-moored FOWT. Comparison has been made against the linear spring method. Although the mooring system's static load-offset graph is linear, the linear spring method fails to consider the dynamic response of the mooring line. It under-predicts the motion of the floating body in the wave plus wind condition. This under prediction also affects the maximum mooring line tension, as its value is dependent on the instantaneous position of the floating platform. Due to the characteristics of the wind turbine thrust force, a coupled time-domain global performance analysis for a FOWT is much more time-consuming than the analysis of a floating platform. Therefore, this paper utilized an approximation focusing on the non-linear stress–strain relationship, but still is able to capture the nonlinear-stiffness characteristics-an important characteristics for polyester lines. A comparison between the current method and other approximations for modeling a polyester line, e.g. dual stiffness method (Tahar et al., 2012) and visco-elastic model (Kim et al., 2011), to establish the relative accuracy and efficiency would be a worthwhile future study.
ABS, 2014a. Global performance analysis for floating offshore wind turbine installations. ABS, 2014b. The application of fiber rope for offshore mooring. Agarwal, A.K., Jain, A.K., 2003. Dynamic behaviour of offshore spar platforms under regular sea waves. Ocean Eng. 30 (4), 487–516. API RP 2SM, 2001, Recommendation Practice for design, analysis, and testing of synthetic fiber ropes in offshore mooring applications. 1st ed. Bathe, K.J., 1996. Element Procedures. Prentice Hall. Ćatipović, I., Čorić, V., Radanović, J., 2011. An improved stiffness model for polyester mooring lines. Brodogradnja 62 (3), 235–248. Chen, X.H., 2002. Studies on Dynamic Iteration Between Deep-water Floating Structure and Their Mooring/tendon Systems. Texas A&M University, Dissertation. Cummins, W.E., 1962. The impulse response function and ship motions, DTMB Report 1661, Washington D.C. EWEA, 2013. Deep Water- The Next Step for Offshore Wind Energy. EWEA. Garrett, D.L., 1982. Dynamic analysis of slender rods. J. Energy Resour. Technol 104, 302–306. Goupee, A.J., Koo, B.J., Kimball, R.W., Lambrakos, K.F., Dagher, H.J., 2014. Experimental comparison of three floating wind turbine concepts. J. Offshore Mech. Arct. Eng. 136 (2). Jonkman, J., 2010. Definition of the floating System for phase IV of OC3. National Renewable Energy Laboratory. Golden, CO.NREL/TP-500-47535. Jonkman, J.M., 2013. The new modularization framework for the FAST wind turbine CAE tool, 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. Kim, B.W., Sung, H.G., Kim, J.H., Hong, S.Y., 2013. Comparison of linear spring and nonlinear FEM methods in dynamic coupled analysis of floating structure and mooring system. J. Fluids Struct. 42, 205–227. Kim, C.H., Kim, M.H., Liu, Y.H., Zhao, C.T., 1994. Time domain simulation of nonlinear response of a coupled TLP system. Int. J. Offshore Polar Eng. 4 (4), 284–291. Kim, J.W., Sablok, A., Kyoung, J.H. and Lambrakos, K., 2011. A nonlinear viscoelastic model for polyester mooring line analysis. In: Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering. OMAE 2011, pp. 797–803. Kim, M.H., Tahar, A., Kim, Y.B., 2001. Variability of spar motion analysis against design methodologies/parameters, In: Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering. OMAE 2001, pp. 153–161. Kreuzer, E., Wilke, U., 2003. Dynamics of mooring systems in ocean engineering. Arch. Appl. Mech. 73 (3-4), 270–281. Love, A.E.H., 1944. A Treatise on the mathematical theory of elasticity, 4th ed Dover Publications, New York. Low, Y.M., 2006. Efficient Methods for the Dynamic Analysis of Deepwater Offshore Production Systems. University of Cambridge, PhD Thesis. Morison, J.R., O'Brien, M.R., Johnson, J.W., Schaaf, S., 1950. The forces exerted by surface waves on piles. Pet. Trans. 2 (5), 149–154. Nordgren, R.P., 1974. On computation of the motion of elastic rods. J. Appl. Mech. Trans. ASME 41 (2), 777–780, Ser E. NWTC Computer-Aided Engineering Tools (FAST by Jason Jonkman, Ph.D.). 〈http://wind.nrel.gov/designcodes/simulators/fast/〉. Last modified 2-July-2014; (accessed 01.09.14). OrcaFlex user manual Version 9.7a. Orcina Ltd. Daltongate, Ulverston Cumbria. LA12 7AJ, UK. Paulling, J.R., Webster, W.C., 1986. A consistent large amplitude analysis of the coupled response of a TLP and tendon system. In: Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering 3, p. 126. Ran, Z., 2000. Coupled dynamic analysis of floating structures in waves and currents, ,. Texas A&M University, Dissertation. Roald, L., Jonkman, J., Robertson, A., Chokani, N., 2013. The effect of second-order hydrodynamics on floating offshore wind turbines. Energy Procedia, 253–264. Tahar, Arcandra, 2001. Hull/Mooring/Riser Coupled Dynamic Analysis of a Floating Platform with Polyester Mooring Lines. Texas A&M University, Dissertation. Tahar, A., Sidarta, D., Ran, A., 2012. Dual stiffness approach for polyester mooring line analysis in time domain. In: Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering - OMAE, pp. 513–521. WAMIT Inc. Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, Mass., USA, 2013.
Acknowledgment The authors gratefully acknowledge the assistance of Dr Jason Jonkman in his responses to questions about the simulation tool FAST. This research was supported by the Department of Naval Architecture, Ocean and Marine Engineering and the Faculty of Engineering at the University of Strathclyde and by the China Scholarship Council (CSC) (Grant no. 2011606021).
Appendix A Shape functions In the FEM, the shape functions Al and Pm are defined as follows (Garrett, 1982): A1 ¼ 1 3ξ þ 2ξ 2
3
A2 ¼ Lðξ 2ξ þ ξ Þ 2
A3 ¼ 3ξ 2ξ 2
3
3
A4 ¼ Lð ξ þ ξ Þ
ðA:1Þ
P 1 ¼ 1 3ξ þ 2ξ P 2 ¼ 4ξ ð1 ξÞ P 3 ¼ ξ ð2ξ 1Þ
ðA:2Þ
2
3
2
where ξ ¼ s=L