Time and frequency domain coupled analysis of deepwater floating production systems

Transcription

1 Applied Ocean Research 28 (2006) Time and frequency domain coupled analysis of deepwater floating production systems Y.M. Low a,, R.S. Langley b a Nanyang Technological University, School of Civil and Environmental Engineering, Block N1, Nanyang Avenue, Singapore , Singapore b Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK Received 11 September 2006; received in revised form 24 April 2007; accepted 5 May 2007 Available online 18 June 2007 Abstract The dynamic analysis of a deepwater floating structure is complicated by the fact that there can be significant coupling between the dynamics of the floating vessel and the attached risers and mooring lines. Furthermore, there are significant nonlinear effects, such as geometric nonlinearities, drag forces, and second order (slow drift) forces on the vessel, and for this reason the governing equations of motion are normally solved in the time domain. This approach is computationally intensive, and the aim of the present work is to develop and validate a more efficient linearized frequency domain approach. To this end, both time and frequency domain models of a coupled vessel/riser/mooring system are developed, which each incorporate both first and second order motions. It is shown that the frequency domain approach yields very good predictions of the system response when benchmarked against the time domain analysis, and the reasons for this are discussed. It is found that the linearization scheme employed for the drag forces on the risers and mooring lines yields a very good estimate of the resulting contribution to slow drift damping. c 2007 Elsevier Ltd. All rights reserved. Keywords: Deepwater; Floating structures; Coupled analysis; Time domain; Frequency domain; Lumped mass 1. Introduction As shallow water hydrocarbon reserves continue to reduce in contrast to ever increasing global demand, recent years have seen an increasing use of floating production systems to develop deepwater sites, with water depths in the region of m being of interest. Floating production systems normally have three main components: the floating vessel, the mooring lines, and the marine risers, all of which are subjected to environmental forces. Dynamic response is therefore a key consideration in the design of such systems, and various aspects of the physics of deepwater systems make dynamic analysis a particularly challenging computational task. Firstly, although the main purpose of the mooring system is to provide restoring forces to the vessel, the action of the mooring system cannot be approximated by simple nonlinear quasi-static springs, since the inertia and damping forces arising from the moorings may be comparable to those acting directly on the floating vessel. Corresponding author. Tel.: ; fax: address: ymlow@ntu.edu.sg (Y.M. Low). The dynamic analysis must therefore consider the coupled dynamics of all three components. Secondly, it is well known that floating vessels are subjected to both first and second order wave forces, with the second order difference frequency forces exciting the low frequency resonances in surge, sway, and yaw. The dynamic response in a random sea is therefore at two timescales: the wave frequency response (WF) at the wave frequency (0.2 2 rad/s) and the low frequency (LF) response at the in-plane resonances (around 0.02 rad/s). These two types of motion are coupled via the geometric nonlinearity of the mooring lines and risers (collectively referred to as lines in what follows) and the nonlinear drag forces acting on the lines. Any dynamic analysis must therefore account simultaneously for both types of motion. Given the nonlinearities and other complexities in the problem, the most common dynamic analysis approach is time domain analysis, and there are several commercial packages available which employ this method, such as DNV s DeepC 1]. In view of the high computational cost of a fully coupled time domain analysis of the vessel and lines, many approximate methods have been developed with a view to balancing /$ - see front matter c 2007 Elsevier Ltd. All rights reserved. doi: /j.apor

2 372 Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) accuracy and efficiency. Several of these methods (e.g. Ormberg and Larsen 2]; Senra et al. 3]) consider a time domain analysis of the vessel with quasi-static lines, but enhance this model by adding the dynamic influence of the lines as equivalent linear damping and/or inertia coefficients acting on the vessel. As pointed out by Garrett, Gordon and Chappell 4], these methods are not rigorous, since assumptions regarding the fairlead motions are made to estimate the damping from the lines; in reality an iterative approach is strictly necessary as the line damping and vessel motions are coupled. In the search for alternative approaches, especially during the early design stage, the highly efficient frequency domain method is appealing. The crux to its viability lies in the judicious treatment of the nonlinearities inherent in the system. The quadratic drag force on the lines can be linearized by known methods, while geometric nonlinearities of the lines are traditionally linearized by assuming small oscillations about a mean position. Due to the approximations made, the linearized frequency domain approach cannot be expected to match the nonlinear time domain method exactly, and the expected degree of accuracy is not as well established due to the limited literature on the topic. Garrett 5] noted that most of the current analysis tools undertake coupled analysis only in the time domain, but one exception with both time and frequency domain capabilities is Stress Engineering Services RAMS. In the time and frequency domain comparisons presented by Garrett 5] for a semi-submersible in 1800 m water depth using RAMS, there was excellent agreement for the most damaging seastate for fatigue (significant waveheight H S = 3 m), whereas stresses close to the seabed did not match as well for the 100-year hurricane case (H S = 12.2 m). It was suggested that a proper linearization of the nonlinear seabed interaction would bring the results closer. Ran, Kim and Zheng 6] also compared the coupled analysis of a spar in the time and frequency domains using proprietary programs. For the case without current, the difference in the estimation of slow drift surge motions was about 30%, and the top tension differed by a factor of two. The discrepancies were found to increase slightly in the presence of current. This level of agreement is clearly much worse than that reported by Garrett 5], leaving some doubt as to the suitability of frequency domain analysis. The objective of this paper is to make comparisons of time domain and frequency domain coupled analyses for a typical deepwater floating system, in order to quantify the range of accuracy of the frequency domain approach. An important issue is the ability to capture the coupling behavior precisely. Both methods must be founded on an identical framework to permit a consistent comparison, and in-house programs have been developed specifically for this purpose. In the choice of the discretization approach for the lines, the majority of previous studies have employed a finite element model, such as the one introduced by Garrett 7]. However, a formulation based on the lumped mass approach is preferred for the present work. The numerical efficiency of this approach is an obvious advantage for time domain simulations, while its simplicity and transparency are well suited to approximate frequency domain analysis. Moreover, the approach has been demonstrated by several authors (as detailed below) to be fully viable and accurate for structures that are not dominated by flexural rigidity. The lumped mass formulation adopted herein is based on a global coordinate system, thus removing the need for transformations between coordinate systems. The axial stiffness of the line is modeled by simple linear spring elements between nodes, while bending is modeled by rotational springs using a different approach to those previously reported in the literature. The previous methods are: (i) the method of Ghadimi 8], who derives the bending moment from the change of slope between adjacent elements, and subsequently calculates the equivalent shear forces; (ii) the method employed in the software Orcaflex 9] where a node is modeled as a short rod, and rotational spring dampers are applied on either side of the node; (iii) the method of Raman-Nair and Baddour 10] who derived the generalized active forces of an equivalent rotational spring in a local coordinate system. In this paper, the forces from an equivalent rotational spring are derived directly from the potential energy expression in global coordinates. Aside from simplicity in the resulting equations, the method may be used directly to obtain the tangent bending stiffness matrix. The time domain version of the present analysis employs the Wilson theta implicit integration scheme, which is inherently more stable than many other methods when relatively large time steps are employed 11]. Although the scheme is iterative, the computational cost of each iteration is relatively small as matrix factorization is performed only at the start of each time step. An additional benefit is that unrealistic high frequency axial responses of the line are automatically filtered out by using a large time step. As such, axial structural damping need not be considered, although it can be readily included if required. The analysis tool developed herein consists of several modules that can be used to perform coupled and/or uncoupled analysis of the lines and vessel in the time and frequency domains. The time domain line dynamics code is validated against the commercial software Orcaflex, which employs the lumped mass approach and an explicit integration scheme. The test case employed for this purpose consists of a hanging riser under regular wave excitation loads. Frequency domain analyses are also carried out for comparisons, and selected results are presented. Having validated the line dynamics code, the developed methods are used to analyze a coupled vessel/line system in both the time domain and frequency domain. Apart from the advantage of computational cost, it is shown that the frequency domain approach can also be used to provide useful post-processing information, such as the degree of damping supplied to the vessel by the lines, and the effective inertia of the lines. It is found that the frequency domain approach yields surprisingly accurate results for both the first and second order response of the system, despite the importance of nonlinear drag forces on the lines. One reason for this is that the drag linearization scheme employed is found to be simultaneously optimum for both the second and first order motions.

3 Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) Vessel model 2.1. Structural model The present model of the floating system comprises three distinct components: the vessel, the lines and a set of connecting springs. The displacement vector x of the coupled system is represented by s x =, (1) y] where s and y are the displacement vectors of the vessel and the lines respectively. Models of the lines and connecting springs will be discussed in Section 3. The vessel is described as a rigid body with six degrees of freedom at the CG, with s representing surge, sway, heave, roll, pitch and yaw in that order. The translational degrees of freedom are measured in the same coordinate system as the lines; hence, there is no need for transformations to local reference frames Wave forces on the vessel The wave forces acting on a floating vessel are well documented in the literature, and thus only a brief outline of the key results is given here. A linear diffraction analysis provides the vector of first order transfer functions T (1), defined so that in a regular wave of frequency ω F (1) (ω) = T (1) (ω)η(ω), (2) where F (1) is the vector of first order forces and η is the wave amplitude. For time domain simulations of a random seastate, the first order forces can be expressed as a sum over constituent seastate components, and this sum can be evaluated efficiently by using the Fast Fourier Transform technique to yield a time history of the forces 12]. In addition to the first order forces, the vessel is subjected to second order forces arising from nonlinear hydrodynamic effects. These forces are determined from a second order diffraction analysis, and only the slowly varying forces caused by difference frequencies in the surge, sway and yaw are pertinent to the present study. The vector of slow drift quadratic transfer functions (QTFs) T (2) is defined from the interaction between a pair of waves with frequencies ω n and ω m and is defined so that F (2) (ω m, ω n ) = T (2) (ω m, ω n )η(ω m )η(ω n ), (3) where F (2) is the vector of second order forces and η(ω n ) and η(ω m ) are two wave amplitudes. For time domain analysis the time history of the second order wave forces in a random seastate can be expressed as a double summation over the first order wave components that comprise the seastate; this expression may be evaluated using the efficient method described by Langley 13]. For frequency domain analysis, the cross-spectra matrix of the second order forces S (2) F F is needed, Fig. 1. Schematic diagram of elements j and k. and this is given by 14] ] S (2) H F F (ω) = 8 T (2) (µ, ω + µ) T (2) (µ, ω + µ) 0 S ηη (µ)s ηη (ω + µ)dµ (4) where S ηη is the wave spectrum and the subscript H denotes the Hermitian transpose. Each diagonal component of S (2) F F is real and is consistent with the form of the second order force spectrum given by Pinkster 15]. The added mass and radiation (potential) damping matrices of the vessel can be obtained from a radiation analysis 16]. These are in general frequency dependent; in the time domain they can either be Fourier transformed to give the retardation functions 16] of the vessel, or treated more approximately by using fixed values chosen at a representative frequency. 3. Line model 3.1. Structural model A mooring line/riser is modeled as a series of lumped masses (nodes) that are connected by linear springs (elements), with additional rotational springs employed to model the bending stiffness of the system. Large deflections and small strains are assumed. Fig. 1 shows three nodes arbitrarily numbered 1 to 3 connected to elements j and k, where the three-dimensional nodal displacements are denoted by y 1, y 2 and y 3. Each element is modeled as an extensional spring (bending stiffness will be discussed later in this section) and the mass of the element is lumped in equal halves onto the adjoining nodes. Consider initially only element j. The strain energy due to extension is given by V j = 1 2 E A L j ( y 2 y 1 L j ) 2, (5) where L j is the original length and EA is the axial rigidity. The elastic force T 1q acting on node 1 in the q direction is given by T 1q = V j = E A ( ) y2q y 1q y 1q y 2 y 1 E A ( ) y2q y 1q, (6) L j where q may be 1, 2, 3, corresponding to global Cartesian coordinates x, y, z respectively.

4 374 Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) The 3 3 extensional tangent stiffness matrix for node 1 is denoted as K1 A, and the entries are given by ( ) K1qr A = 2 V j E A = δ qr E A y 1q y 1r L j y 2 y 1 + E A ( y 2q y 1q ) (y2r y 1r ) y 2 y 1 3 where δ qr is the Kronecker delta that is 1 when q = r and 0 when q r. It can be readily shown that the stiffness matrix for element j is given by K A (elm j) = K A 1 K A 1 K A 1 K A 1 (7) ]. (8) Eqs. (6) and (8) can be developed for each element in the model to fully describe the elastic forces arising from axial extension. In addition, rotational springs are used to model bending stiffness, and a rotational spring, which couples together the two elements and three nodes in Fig. 1. Assuming that the profile of the riser/mooring line is initially straight, the potential energy U stored in the rotational spring is expressed as U = 1 2 E I ( ) ( ) 1 2 L j + L k (9) R where EI is the flexural rigidity of the system and R is the radius of curvature. By estimating R as the radius formed by fitting a circle through nodes 1, 2 and 3, the relation between R and the angle θ between the elements can be derived from geometry assuming θ is small, in which case U = 1 2 k bθ 2, (10) where k b = 2E I L j + L k. (11) The bending moment at node 2 can be found by differentiating Eq. (10) with respect to θ; the equivalent forces acting on the nodal degrees of freedom are derived in what follows. For frequency domain analysis the tangent stiffness matrix associated with the bending moment is required, and to this end the angle θ can be expressed as a function of the 1 9 displacement vector of the neighboring elements, and this vector, y = y T 1 y T 2 y T 3 ] T is separated into mean and dynamic parts, denoted respectively by ȳ and ỹ. For small values of ỹ (as assumed in linear frequency domain analysis), θ can be written as θ = θ + α T ỹ, (12) ( 2 θ 2 = θ θα T ỹ + α ỹ) T, (13) where θ is the mean value of θ and α T = α T 1 α T 2 α T 3 ] is a vector of influence coefficients, which is given by α 1 = 1 L j sin θ tk ( t k t j ) t j ], (14a) α 3 = 1 t j ( ) ] t j t k tk, L k sin θ (14b) α 2 = α 1 α 3, (14c) where t j and t k are the unit tangent vectors for elements j and k, defined as t j = y 2 y 1, t y 2 y 1 k = y 3 y 2. (15) y 3 y 2 It follows that U ỹ = k b θα + αα ỹ] T (16) 2 U ỹ ỹ = k bαα T = K B. (17) The nodal force vector arising from the rotational spring is denoted as Q = Q T 1 Q T 2 Q T ] T, 3 and can be found by putting ỹ = 0 in Eq. (16); for nonlinear time domain analysis, θ is interpreted as the instantaneous value of θ in this equation. Eq. (17) gives the 9 9 tangent bending stiffness matrix K B. Together, Eqs. (6), (7), (16) and (17) fully describe the elastic forces in the line and the associated tangent stiffness matrix entities. The fully assembled tangent stiffness matrix is given by the sum of the axial and bending contribution, and it is denoted here by K L. The mass matrix of the lines M L can be regarded as the sum of the structural mass matrix M S L and the added mass matrix ML A. The matrix MS L is diagonal, being formed by lumping the structural mass onto the various nodes; ML A will be discussed in Section Wave forces on the lines Linear wave theory can be used to calculate the fluid particle velocities u and accelerations u at any point on the lines, and the three-dimensional wave forces are calculated by using Morison s equation. The inertia and drag forces are usually computed separately for directions normal and tangential to the line, since the hydrodynamic coefficients in the two directions are different in general. For example, for the system shown in Fig. 1, the drag forces from half of elements j and k are lumped onto node 2 as F D = 1 4 ρ D { L ξ C n D Vr ( ) ] V r t ξ tξ where ξ= j,k + CD t ( ) } Vr t ξ tξ, (18) V r = r + V c, (19) r = u ẏ. (20)

5 Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) Here, V c is the current velocity, CD n and Ct D are the normal and tangential drag coefficients, and D is the outer diameter of the line. The inertia forces are likewise expressed as F I = 1 {( ) ) ] ρπ D2 L ξ 1 + C n 8 A u ( u tξ tξ ξ= j,k + ( 1 + C t ) ) } A ( u tξ tξ = 1 {( ) ) } ρπ D2 L ξ 1 + C n 8 A u C n A ( u tξ tξ ξ= j,k if C t A = 0, (21) where C n A and Ct A are the normal and tangential added mass coefficients. If it is assumed that C t A = 0, which is usually a valid approximation, then the equation is simplified as indicated above. Under this assumption, the added mass matrix for node 2 in Fig. 1 can be shown to be M2 A = 1 8 ρπ )} D2 C n A L ξ {(I t T ξ t ξ. (22) ξ= j,k where I is the identity matrix. The assembled added mass matrix for the lines, denoted as ML A, is non-diagonal, with coupling between the three degrees-of-freedom at any given node; however, there is no coupling between different nodes Connection of the lines to the vessel The connection of the lines to the vessel is effected by coupling each line to the vessel via a set of very stiff springs. If a point P has coordinates ˆP = ˆP x ˆP y ˆP z ] T relative to the CG, and its position in global coordinates is P = Px P y P z ] T, then a linear compatibility relationship exists for small rigid body rotations such that P = As + ˆP, (23) where ˆP z ˆP y A = ˆP z 0 ˆP x. (24) ˆP y ˆP x 0 Now, if a force f p is applied on point P, then the generalized force F B acting in the vessel degrees of freedom is F B (6 1) = A T f p. (25) (3 1) Now, if the top node of a line is attached to the point P via a set of three very stiff linear springs, then the restoring spring force f p acting at P is given by f p = K S ( yt P ), (26) where K S = K S I; (27) here, y T is the position of the top node of the line, K S is a spring constant and I is the 3 3 identity matrix. The spring force acting on the line node is equal and opposite to f p. The spring forces can be introduced into the model by adding a stiffness matrix that couples the vessel degrees of freedom to the degrees of freedom of the line node. It follows from Eqs. (25) and (26) that this matrix has the form A T K S A K C = (6 6) K S A (3 6) A T K S (6 3) K S (3 3). (28) The role of the connecting springs is only to ensure that the separation of the line node from the point P on the vessel is infinitesimal. For this, K S should be large, but evidently not to the extent that would lead to ill conditioning of the equations. To simulate different hull to line connections, the properties of the upper elements of the line can be adjusted accordingly; further, although pin-joints are assumed in the present analysis, rotational springs could be introduced to the model to account for different conditions Seabed interaction For line nodes resting on the seabed, the upward contact forces are included in the model. Friction effects are considered to be less significant for the system analyzed in this paper and are neglected, since only a relatively small portion of the lines is resting on the seabed. A modified bilinear spring model with a gradual transition is proposed to avoid numerical stability, and the vertical contact force F sb on a node near the seabed is of the form F sb = 1 } 2 a 1 { z + 1a2 ln cosh(a 2 z + a 3 )] + a 4, (29) where a 1, a 2, a 3, and a 4 are suitably chosen constants. In particular, a 4 should be the value such that F sb 0 when z is a suitable distance away from the seabed. For frequency domain analysis, in order to maintain the symmetry of the stiffness matrix, the nodes near the seabed are assumed to be attached to grounded vertical springs, whose tangent stiffness can be derived from Eq. (29). 4. Solution of the coupled equations 4.1. Static analysis The static problem must be solved before undertaking a dynamic analysis. As nonlinearity and large displacements are involved, the static solution must be solved iteratively. One option is to employ a time domain analysis (described in the next section) and allow the system to settle to its final position after the transients have been damped out. However, it is more efficient to perform a fully static analysis using the Newton Raphson iteration scheme. The Jacobian needed for this method is similar to Eq. (44) of the following section, except that the mass and damping terms are neglected. To ease convergence, under-relaxation is employed by adding λi to the Jacobian, where I is the identity matrix, and λ is a positive real constant.

6 376 Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) Time domain analysis The equation of motion for the coupled system in the time domain is written as Mẍ(t) + ˆBẋ(t) + ˆKx(t) = F(t), (30) where x is the displacement vector, M, ˆB and ˆK are respectively the mass, damping and stiffness matrices of the coupled system, and F is the external force vector. The symbol is to indicate that there are no contributions to the damping and stiffness matrices from the lines and connecting springs, since these effects are included as external forces on the RHS of Eq. (30) for the purpose of time integration. It is convenient to partition the system matrices and vectors into blocks corresponding to the vessel and lines so that ] MV 0 M =, (31) 0 M L ] BV 0 ˆB =, (32) 0 0 ] KV 0 ˆK =, (33) 0 0 ] FV (t) F(t) =, (34) F L (t) where the subscripts V and L relate to the vessel and the lines respectively. The vessel forces F V have been discussed earlier. F V are the forces on the lines, and can be written as F L = T Q + W + F D + F I + F S, (35) where T and Q are the axial and rotational restoring forces, W is the effective weight, F D and F I are given by Eqs. (18) and (21), and F S represents forces from the vessel connecting springs. The matrix M V consists of the structural mass of the vessel and the added mass, while B V contains the damping on the vessel from viscous skin drag, wave drift damping and radiation damping. K V is the linear hydrostatic stiffness matrix of the vessel. Time integration is carried out using the Wilson theta implicit scheme, which is an extension of the Newmark Beta algorithm. Linear acceleration is assumed over an extended time step τ = Θh, where h is the normal time step and Θ 1.4. The kinematics at time t + τ are expressed as 11] x t+τ = x t + x, (36) ẋ t+τ = 3 τ x 2ẋ + τ ẍ, 2 (37) ẍ t+τ = 6 τ 2 x 6 τ ẋt 2ẍ t, (38) where represents increments in an extended time step. Equilibrium is enforced at time t + τ so that M (x t+τ ) ẍ t+τ + ˆBẋ t+τ + ˆKx t+τ = F t+τ. (39) Substituting Eq. (36) to (38) into Eq. (39) yields ( 6 f ( x) = F (x t + x) M (x t+τ ) τ 2 x 6 ) τ x t 2ẍ t ( 3 ˆB τ x 2ẋ + τ ) 2 ẍ = 0. (40) The Newton Raphson iterative technique can be used to solve this equation for x; the method is based on the result f ( x + ε) = f ( x) + J ( x) ε = 0, (41) ε = J ( x)] 1 f ( x) (42) where ε is the correction to the subsequent iteration and J is the Jacobian defined as J = f x. (43) The Jacobian need not be exact as its role is only to provide convergence of the algorithm, and it may be estimated to be J ( x) K (x t ) + 6 τ 2 M (x t) + 3 τ ˆB, (44) where K must include the contribution from the lines and springs, i.e. ] KV 0 K = + K 0 K C, (45) L where K C is the combined stiffness matrices from the connecting springs. Denoting the RHS of Eq. (44) as the matrix A, the following matrix equation must be solved to implement Eq. (42) Aε = f ( x). (46) The matrix A is positive definite and can be expressed as U T U using the Cholesky decomposition at the start of each time step, where U is an upper triangular matrix. The computational effort of each iteration is relatively small as it is limited only to forward and back substitution and the evaluation of the residual forces at time t + τ for the solution of ε. After a converged result is obtained, the incremental acceleration for the normal time step dẍ is found from linear interpolation using dẍ = ẍ/θ, after which the incremental velocity and displacement are solved accordingly 11] Frequency domain analysis A frequency domain analysis is inherently linear, and in order to apply the approach to a nonlinear problem such as the present one, all nonlinearities must be linearized. There are two sources of nonlinearity in the present model, these being the geometric nonlinearity arising from large deflections of the lines, and the nonlinear fluid drag forces which appear in Morison s equation (it can be noted that other fluid nonlinearities give rise to the second order wave forces acting on the vessel, but these forces appear only on the right hand side of the equations of motion and do not pose any difficulties for frequency domain analysis). The geometric nonlinearities

7 Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) are dealt with here by calculating the tangent stiffness of the lines at the equilibrium position, which allows for large static deflections but assumes that the dynamic deflections around the static position are small. The drag force nonlinearity is dealt with by either harmonic or statistical linearization, depending on whether the seastate is comprised respectively of regular or random waves. In order to facilitate the linearization, the full vectorial form of the drag force is replaced with an approximate version in which the drag force is computed independently in two orthogonal directions which are each perpendicular to the line. This approach is not strictly frame invariant, as pointed out by Hamilton 17] and Langley 18], but given the approximate nature of both Morison s equation and linearization techniques in general, the method is considered to be sufficiently accurate for present purposes. The method is detailed later in this section. In principle, the equations of motion can be solved efficiently in the frequency domain by firstly transforming the degrees of freedom to modal coordinates. The mass and stiffness matrices are diagonalized by this transformation, although the resulting damping matrix will be non-diagonal. For this approach to be effective, one or both of the following approximations must be made: (1) the off-diagonal terms of the damping matrix are neglected, in which case the equations of motion are uncoupled and can be solved very easily; (2) the modal coordinates are truncated to give a significant reduction in the number of degrees of freedom. In the present problem, the drag forces on the lines lead to a highly coupled damping matrix in modal coordinates, and given the banded nature of the equations in physical coordinates there is little to be gained by a modal transformation; thus physical coordinates are retained in the following analysis. A modal analysis can however provide a useful diagnostic technique, and this is discussed in Section 4.4. In order to linearize the drag forces, consider an arbitrary node 2, which is attached to elements j and k as shown in Fig. 1. The drag force is approximated by computing the force independently along two vectors which are orthogonal to each other and normal to the line. Using element j as an example, the choice of the unit normal vectors n j and j is arbitrary, so long as they satisfy n j j = 0 and j = ±n j t j. It is found that a good choice of unit normal vectors is such that one lies in the plane of the line. The relative velocity r in global coordinates at the node is transformed into coordinates of the normal directions using n r ξ = r n ξ ξ = j, k, (47a) Π r ξ = r ξ ξ = j,k, (47b) where the superscripts and subscripts refer respectively to the normal direction and the element. The nonlinear drag force in each direction is replaced by a linearized version of the form (V c + r) V c + r βr + γ, (48) where r represents the relative velocity in the considered direction, and the linearization coefficients β and γ are functions of max(r) for regular waves 19], and the standard deviation of r, σ r for random waves 20], Clearly, an iterative solution procedure is required, as detailed later in this section, since the linearization coefficients depend on the system response. For the case where the current velocity V c is zero, then γ = 0 and β reduces to 8/3π max(r) for regular waves and 8/πσ r for random waves. Using the foregoing procedure, the drag force in global coordinates as given by Eq. (18) is linearized as F D = B L (u ẏ) + F D, (49) where B L = 1 4 ρcn D D ξ= j,k L ξ ( n β ξ n T ξ n ξ + Π β ξ T ξ ξ ), (50) and F D is the mean drag force which has a similar form to Eq. (50), but with β replaced with γ. The damping matrix B L has the same coupling characteristics between the various degrees of freedom as the added mass matrix. In the case of random seas, it is convenient to compute the response to the first and second order wave forces (WF and LF) separately (but concurrently). The variance of the relative velocity can be regarded as the sum of the first and second order contributions, i.e. σ 2 r = σ 2 r (1) + σ 2 r (2). (51) Having linearized the system, the equation of motion can be written in the frequency domain as ( ) ω 2 M + iωb + K x(ω) = F(ω) (52) where M and K have been previously defined, and B includes the damping from the lines so that ] BV 0 B =. (53) 0 B L The matrix B L depends on the linearization coefficients, which in turn depend on the response of the system, and thus an iterative solution to Eq. (52) is required. It can further be noted that the static problem is coupled to both the WF and the LF response since the mean forces F are also functions of the linearization coefficients. At the end of a typical iteration loop, the matrices M, B and K (which are all position dependent) are reassembled based on the updated result for x. In order to compute the WF response, the transfer functions between the system response and the surface elevation must be found. The first order forces F (1) acting on the vessel and the lines can be written as F (1) T (1) ] (ω) (ω) = G (1) η, (54) (ω) where G (1) contains the transfer functions for the drag and inertia forces on the lines, which also depend on the linearization coefficients. It follows from Eq. (52) that the transfer functions for the system response can now be written as R (1) T (1) ] (ω) (ω) = H(ω) G (1), (55) (ω)

8 378 Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) where H(ω) = ( ω 2 M + iωb + K) 1. (56) The transfer function for the first order relative velocity at any point on the lines r (1) can then be found and, for random seas, the spectrum and variance follow from standard random vibration theory (e.g. Newland 12]). For the analysis of the second order LF response, it is necessary to work directly from the force cross-spectra matrix. To solve for σ 2 r (2), the spectrum of the LF relative displacement in the corresponding normal direction must first be found. The LF relative normal displacement X (2) can be written as a linear combination of x (2) such that X (2) = L T x (2). (57) The spectral density of X (2) can then be expressed as 11] S (2) X X (ω) = LT H(ω)S (2) F F (ω)h(ω)h L, (58) where S (2) F F is the cross-spectral matrix of F(2) and the superscript H denotes the Hermitian transpose. Due to the large number of zeros in the matrix S (2) F F, Eq. (58) can be evaluated more efficiently by converting it to a 3 3 matrix covering surge, sway and yaw, and adjusting the sizes of the other matrices accordingly. Finally, it can be noted that putting L = I in Eq. (58) yields the cross-spectral matrix of the second order (LF) response of the system Modal analysis and inertia/damping coefficients Following a linearized frequency domain analysis, a modal analysis of the system can be performed to yield important physical information such as the natural frequencies of the system, the system damping, the amount of damping provided by the lines relative to the vessel etc. To this end the undamped natural frequencies and mode shapes of the coupled system can be found by solving the standard eigenvalue problem. Let represent the modal matrix, i.e. the matrix whose columns are the eigenvectors of the system. The system degrees of freedom can be written in terms of a set of modal coordinates q via the relation x = q. (59) The equations of motion can now be transformed to modal coordinates to yield T M q + T B q + T K q = T F. (60) The total damping ratio ζ S in degree of freedom S can now be estimated as ( ) 2ζ S ω S = T B, (61) SS where ω S is the associated natural frequency and B is defined in Eq. (53). The damping ratio ζ S is the sum of contributions from the vessel and the lines. The former can be found from Eq. (61), but putting B L = 0 in Eq. (53); and the latter by putting Fig. 2. Static configuration of hanging riser. B V = 0. The inertia contribution from the lines may also be significant for deepwater floating structures. By definition, the modes are normalized such that T M = I. (62) By using Eqs. (31) and (62), the contribution to the normalized generalized mass from the lines can be found by putting M V = 0. The results yielded by Eqs. (61) and (62) allow a physical insight into the nature of the system response, and the degree of coupling that exists between the vessel and line dynamics. The linear coefficients for inertia and damping from the lines estimated in this way could, if required, be applied to a traditional uncoupled analysis of the vessel. As the linearized damping matrix is calculated iteratively from a fully coupled frequency domain analysis, it is specific to the problem and takes account of the seastate and the actual motions of the vessel and lines. 5. Validation of the line dynamics code The time domain line dynamics software developed as part of the present work has been validated against the commercial software Orcaflex, 9] and the time domain results have then been used to benchmark the frequency domain analysis software. Due to space constraints, results from only two load cases are presented here; a more comprehensive report on the validation study can be found in Reference 21] Description of the test case The test case is a hanging riser pinned at both ends, with the static configuration depicted in Fig. 2, and the input parameters summarized in Table 1. Two load cases are considered. For Case 1, the bottom end is pinned while the top end is prescribed a horizontal simple harmonic motion in still water (i.e. no wave kinematics). The amplitude of the prescribed motion is 10 m, and the period is 27 s, close to the first in-plane mode. For Case 2, both ends are pinned, and an in-plane regular wave acts on the riser. The wave has a waveheight of 10 m and a period

9 Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) Table 1 Input data for hanging riser Total unstretched length 170 m Outer diameter, D m Dry mass 165 kg m 1 EA 500 MN EI kn m 2 Density of water, ρ 1000 kg m 3 Gravitational acceleration, g m s 1 Drag coefficients C n D = 1, Ct D = 0 Added mass coefficients C n A = 1, Ct A = 0 Number of elements 68 Element size (constant) 2.5 m Table 2 Comparisons of static tension and reactions for hanging riser Inhouse Orcaflex Theory Top tension (kn) Bottom tension (kn) Vert reaction, top (kn) Horiz reaction, top (kn) Vert reaction, bottom (kn) Horiz reaction, bottom (kn) of 10 s. The top node is positioned at a water depth of 5 m which implies that all elements are fully submerged for each of the considered load cases. However, the line model described herein can be readily extended to accommodate partially or intermittently submerged elements by adjusting the effective weight and fluid loads on these elements accordingly Results Results for the static tension and the support reactions at the top and bottom end of the riser are presented in Table 2. The results labeled theory are derived from the classical catenary equations, neglecting elasticity and bending stiffness. It is found that the static results from the present code and Orcaflex are virtually indistinguishable, and are also close to the theoretical result, indicating that the adopted level of discretization is sufficient. The time history of the top tension for the two test cases is plotted in Fig. 3(a) and (b) respectively. The transient buildup stage during which loads are ramped up from zero is not included so that the results represent steady-state motion. In both cases, the curves produced by the present code and Orcaflex are nearly coincident despite the use of different numerical integration schemes and differing methods of analyzing the bending moments in the system. It is interesting to observe the high degree of nonlinearity exhibited in the time history for Case 1, where a horizontal motion of 10 m distorts the catenary shape appreciably. In contrast, the nonlinearity is not pronounced in Case 2, as the top and bottom ends are restrained. In order to compare the time and frequency domain predictions for the two test cases, the amplitudes of the dynamic tension are plotted along the riser in Fig. 4. In these figures, the amplitude of the dynamic tension for the time domain results Fig. 3. Time history of top tension; (a) Case 1 (prescribed top motion), (b) Case 2 (regular wave). is taken to be half the difference between the maximum and minimum values. The accuracy of the linearized frequency domain analysis is evidently consistent with the extent of the nonlinear behavior displayed in Fig. 3. For Case 1, the tension predicted by the frequency domain analysis differs from the time domain analysis by an average of 25%, while in Case 2, the mean discrepancy is 15%. It is found that the disparity drops for both cases when the loading amplitude is reduced 21]. 6. Case study of coupled analysis 6.1. Description of the floating system and the loading The foregoing theory has been implemented on a spread moored FPSO installed in a water depth of 2000 m, and the essential characteristics of the vessel are summarized in Table 3. The centre of gravity is located midway with respect to the length and breadth, and 5 m above the still water level. It is common for mooring lines to consist of a combination of chains and wires, and for different types of risers to be used for various functions, such as oil and gas production/export and water injection. For the purpose of the present study, only four catenary lines with uniform properties are considered. The plan of the configuration of the lines is illustrated in Fig. 5(a), while

10 380 Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) Table 4 Line data Water depth 2000 m Length of line 3300 m Pre-tension 4370 kn External diameter, D 0.15 m Dry mass 150 kg m 1 Scaling factor 6 EA 1000 MN EI 50 kn m 2 Drag coefficient 1 (normal) 0 (tangential) Added mass coefficient 1 (normal) 0 (tangential) No. of elements per line 50 Element size (constant) 66 m z coordinate of top node 10 m Table 5 Simulation parameters Fig. 4. Dynamic tension comparisons; (a) Case 1, (b) Case 2. Time domain Wave freq Low freq Width of strip dω (rad/s) No. of strips Buildup duration (min) 20 Main simulation duration (min) 83.8 Time step (s) 0.04 Frequency domain Wave freq Low freq Width of strip dω (rad/s) No. of strips Table 3 Vessel data Fig. 5. Illustration of vessel and lines; (a) plan view, (b) 3D view. Length Width Draft Displacement 240 m 46 m 10 m tons a 3D view is depicted in Fig. 5(b). Each line in the model represents a group of mooring lines and risers. This is easily achieved by scaling the mass, stiffness and external forces (e.g. the hydrodynamic coefficients) of the lines by the number of lines in the group. This scaling scheme does not modify the static position of the lines, assuming that the draft of the vessel is unchanged by also scaling the buoyancy. Thus, parametric studies on the number of lines can be performed without the need to change the model. In this paper, the scaling factor is 6, representing a total of 24 lines, and the properties of the lines (prior to scaling) are given in Table 4. The line tensions obtained from the analyses are then reduced by the same factor so that results are characteristic for a single line. In this example, mean forces from various sources are not considered for simplicity as they do not affect the main features of the dynamic analysis. Unless otherwise stated, the random wave environment is described by a three-parameter Jonswap spectrum with H S = 15.7 m, T z = 13.5 s and γ = 2. The random waves are unidirectional and approach the vessel at an angle of 22.5 measured from the bow, as illustrated in Fig. 5(a). The intent is to exercise all six degrees of freedom of the vessel, as opposed to head or beam waves. Simulation parameters for time and frequency domain analysis are given in Table 5. Wave frequencies are defined from 0.2 to 1.2 rad/s, and low frequency below 0.2 rad/s. The wave spectrum is non-zero in the WF range, and is divided into a number of strips of equal width dω. It is necessary to use a consistent dω in the time domain. However, for frequency domain simulation, it is possible to select a smaller dω for the second order analysis to capture the narrow-banded response. For the time domain analysis, there is setting period during which the loads are ramped up gradually to allow the transients

11 Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) Table 6 Comparison of standard deviations Fig. 6. Time history of surge excursion. to dissipate. Statistical estimates are obtained from the main analysis period only, and the variance is averaged over ten runs. This is to ensure that sufficient cycles are considered for the non-gaussian second order response, since each run has a simulation duration of 83.4 min and includes only approximately 20 LF cycles. Spectral densities are recovered from the time histories to allow direct comparison with the frequency domain. The clear delineation of wave and low frequencies makes it straightforward to assign responses to first or second order forces Results Selected results from the numerical simulations are presented in this section. An excerpt of the time history for the surge excursion taken from one of the time domain runs is shown in Fig. 6. Spectral densities have been recovered from the time histories generated by the time domain analysis and are shown in Fig. 7(a) (g). It is interesting to observe that there is significant LF roll motion, even though low frequency forces for roll are absent, and the natural frequency for roll is about 21 s (which can be deduced from the peak of the roll RAO). The reason is that for the particular model used in this study, roll is strongly coupled to the sway and yaw motions through the added mass matrix and the line motions. This effect can be captured in the frequency domain analysis by including roll in the second order analysis, but not defining any LF roll wave excitation forces. Fig. 7(a) (g) show the spectra generated by both the time and frequency domain analyses for the six vessel responses and the top tension of one of the lines. Table 6 gives the standard deviations, which are calculated from the spectra and separated into WF and LF components. The standard deviation of the tension (dominated by WF) along the line from the bottom end is plotted in Fig. 8. The natural periods obtained from a modal analysis of the coupled system are presented in Table 7. The equivalent linear damping and inertia coefficients from the lines for the relevant modes are also included in the table. The mode shapes are illustrated in Fig. 9, where dotted lines depict the deflected shapes. The first three modes primarily correspond to yaw, sway and surge in order of decreasing periods, and the fourth mode evidently involves mainly the line dynamics. Wave frequency Time Freq domain domain Low frequency Time Freq domain domain Surge (m) Sway (m) Heave (m) Negligible Roll (deg) Pitch (deg) Negligible Yaw (deg) Top tension (kn) Bottom tension (kn) Table 7 Natural periods and damping/inertia coefficients Mode Nat period (s) Type Yaw Sway Surge Lines Damping ratio, vessel (%) NA Damping ratio, lines (%) NA Mass from lines (%) NA It is interesting to explore the effect of various factors on the damping provided by the lines. Parametric studies are carried out by varying the drag coefficient C D, and calculating the damping ratio from repeated frequency domain simulations. The damping ratio for surge and sway for a range of C D values are plotted in Fig. 10(a). The procedure has been repeated by varying the significant waveheight H S, assuming that the wave drift damping remains unchanged, and the results are displayed in Fig. 10(b). The time domain simulations require around 90 h of CPU time for the ten runs on a Pentium GHz laptop. Comparatively, the frequency domain analysis require only approximately 3 min Discussion A severe seastate corresponding to a 100-year storm has been selected for the examples. This maximizes the opportunity for exercising the nonlinearities in the problem, and constitutes the critical test of the accuracy of the coupled frequency domain analysis. WF vessel motions predicted by time domain and frequency domain simulations are in excellent agreement, which can be expected since apart from resonant roll motions, the other vessel modes are either stiffness dominated (pitch and heave) or inertia dominated (surge, sway and yaw) and therefore insensitive to damping. For the LF motions, the frequency domain method performs well for surge and sway (deviating about 2% from the time domain). The roll and yaw results may appear to be comparatively less ideal (within 7%), but they are still satisfactory within engineering expectations. In addition to the motions, excellent agreement is also obtained for dynamic tensions along the line, with the frequency domain approach under predicting by a marginal 1%.

12 382 Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) Fig. 7. Spectral density plots; (a) surge, (b) sway, (c) heave, (d) roll, (e) pitch, (f) yaw, (g) top tension.

13 Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) Fig. 8. Comparison of standard deviation of tension along the line. It can be noted that in the LF vessel response, there is significant scatter between the results from the ten individual time domain runs. For example, the mean-square for LF surge over ten runs has a sample standard deviation of 3.55 m (Langley 13] elaborates on this subject). Taking the average from the runs improves the predictions of LF motions. Damping ratios for surge, sway and yaw are estimated from the damping matrix computed iteratively from the coupled frequency domain analysis. The variance of the relative velocity needed to define the damping matrix is treated as the sum of WF and LF contributions. However, the LF composition is in fact minimal. For instance, the damping ratio for surge calculated without LF velocities is 30.4%, compared to 31.1% for the combined WF and LF velocities. This implies that using the present approach, the linearized damping forces critical for LF motions are primarily governed by WF velocities. Since stochastic linearization is based on minimizing the mean-squared error across the entire spectrum, the dominance of WF velocities means that the linearization is likely to be accurate for WF drag, but not necessarily for the LF drag forces. However the simulations results demonstrate that the linearization approach captures the LF drag forces accurately, although the underlying reason is not immediately obvious. The topic is not addressed in the literature owing to the scarcity of coupled frequency domain analysis. As the assumption is central to the methodology, it warrants further investigation. Hence, a theoretical justification that supports the assumption is supplied in the Appendix. The derivation therein is valid when LF velocities are assumed to be much smaller than WF velocities. The other type of nonlinearity commonly found in floating systems is the geometric nonlinearity of the lines, resulting in nonlinear restoring forces provided to the vessel. However, this form of nonlinearity is especially weak for deepwater systems, as typical vessel motions are small in comparison to the dimensions of the lines, and the catenary shape is not distorted appreciably. In a previous work, a simplified two degree-of-freedom system is studied to glean insight into the coupling mechanisms arising from the nonlinearities in drag and the restoring force 23]. It was discovered that the latter is an influential source of coupling. The LF vessel responses are sensitive to Fig. 9. Mode shapes of coupled system; (a) mode 1 yaw, (b) mode 2 sway, (c) mode 3 surge, (d) mode 4 lines. the line drag dominated by WF velocities, whereas the WF line dynamics also depends on the instantaneous position of the vessel, and thus the magnitude of LF motions. The present case where geometric nonlinearity is negligible constitutes a considerable advantage in coupled analysis, as the WF line dynamics are no longer seriously influenced by LF vessel motions. It is apparent that the frequency domain method also owes its favorable performance to this factor. The effect of C D on the damping ratio is studied (see Fig. 10(a)). As expected, damping ratio increases with increasing C D. Nonlinearity of the drag forces is exemplified from the decreasing slope of the graph. This is because an increase in drag forces is accompanied by reduced motions, and subsequently reduced linearization coefficients. Fig. 10(b) shows that the damping ratio is also highly dependent on