time and frequency domain coupled analysis of deepwater

Applied Ocean Research 28 (2006) 371–385www.elsevier.com/locate/apor

Time and frequency domain coupled analysis of deepwater floatingproduction systems

Y.M. Lowa,∗, R.S. Langleyb

a Nanyang Technological University, School of Civil and Environmental Engineering, Block N1, Nanyang Avenue, Singapore 639798, Singaporeb 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 2007Available 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 ofthe 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 thetime domain. This approach is computationally intensive, and the aim of the present work is to develop and validate a more efficient linearizedfrequency domain approach. To this end, both time and frequency domain models of a coupled vessel/riser/mooring system are developed, whicheach incorporate both first and second order motions. It is shown that the frequency domain approach yields very good predictions of the systemresponse when benchmarked against the time domain analysis, and the reasons for this are discussed. It is found that the linearization schemeemployed 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 reducein contrast to ever increasing global demand, recent yearshave seen an increasing use of floating production systems todevelop deepwater sites, with water depths in the region of1000–3000 m being of interest. Floating production systemsnormally have three main components: the floating vessel, themooring lines, and the marine risers, all of which are subjectedto environmental forces. Dynamic response is therefore a keyconsideration in the design of such systems, and various aspectsof the physics of deepwater systems make dynamic analysisa particularly challenging computational task. Firstly, althoughthe main purpose of the mooring system is to provide restoringforces to the vessel, the action of the mooring system cannotbe approximated by simple nonlinear quasi-static springs, sincethe inertia and damping forces arising from the moorings maybe comparable to those acting directly on the floating vessel.

∗ Corresponding author. Tel.: +65 67905265; fax: +65 67910676.E-mail address: [email protected] (Y.M. Low).

0141-1187/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.apor.2007.05.002

The dynamic analysis must therefore consider the coupleddynamics of all three components. Secondly, it is well knownthat floating vessels are subjected to both first and second orderwave forces, with the second order difference frequency forcesexciting the low frequency resonances in surge, sway, and yaw.The dynamic response in a random sea is therefore at twotimescales: the wave frequency response (WF) at the wavefrequency (0.2–2 rad/s) and the low frequency (LF) responseat the in-plane resonances (around 0.02 rad/s). These two typesof motion are coupled via the geometric nonlinearity of themooring lines and risers (collectively referred to as lines inwhat follows) and the nonlinear drag forces acting on the lines.Any dynamic analysis must therefore account simultaneouslyfor both types of motion. Given the nonlinearities and othercomplexities in the problem, the most common dynamicanalysis approach is time domain analysis, and there are severalcommercial packages available which employ this method,such as DNV’s DeepC [1].

In view of the high computational cost of a fully coupledtime domain analysis of the vessel and lines, many approximatemethods have been developed with a view to balancing

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372 Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) 371–385

accuracy and efficiency. Several of these methods (e.g. Ormbergand Larsen [2]; Senra et al. [3]) consider a time domain analysisof the vessel with quasi-static lines, but enhance this modelby adding the dynamic influence of the lines as equivalentlinear damping and/or inertia coefficients acting on the vessel.As pointed out by Garrett, Gordon and Chappell [4], thesemethods are not rigorous, since assumptions regarding thefairlead motions are made to estimate the damping from thelines; in reality an iterative approach is strictly necessary as theline damping and vessel motions are coupled.

In the search for alternative approaches, especially duringthe early design stage, the highly efficient frequency domainmethod is appealing. The crux to its viability lies in thejudicious treatment of the nonlinearities inherent in the system.The quadratic drag force on the lines can be linearized byknown methods, while geometric nonlinearities of the lines aretraditionally linearized by assuming small oscillations about amean position. Due to the approximations made, the linearizedfrequency domain approach cannot be expected to match thenonlinear time domain method exactly, and the expected degreeof accuracy is not as well established due to the limitedliterature on the topic. Garrett [5] noted that most of thecurrent analysis tools undertake coupled analysis only in thetime domain, but one exception with both time and frequencydomain capabilities is Stress Engineering Services’ RAMS.In the time and frequency domain comparisons presented byGarrett [5] for a semi-submersible in 1800 m water depth usingRAMS, there was excellent agreement for the most damagingseastate for fatigue (significant waveheight HS = 3 m), whereasstresses close to the seabed did not match as well for the100-year hurricane case (HS = 12.2 m). It was suggestedthat a proper linearization of the nonlinear seabed interactionwould bring the results closer. Ran, Kim and Zheng [6] alsocompared the coupled analysis of a spar in the time andfrequency domains using proprietary programs. For the casewithout current, the difference in the estimation of slow driftsurge motions was about 30%, and the top tension differed by afactor of two. The discrepancies were found to increase slightlyin the presence of current. This level of agreement is clearlymuch worse than that reported by Garrett [5], leaving somedoubt as to the suitability of frequency domain analysis.

The objective of this paper is to make comparisons of timedomain and frequency domain coupled analyses for a typicaldeepwater floating system, in order to quantify the range ofaccuracy of the frequency domain approach. An important issueis the ability to capture the coupling behavior precisely. Bothmethods must be founded on an identical framework to permita consistent comparison, and in-house programs have beendeveloped specifically for this purpose.

In the choice of the discretization approach for the lines, themajority of previous studies have employed a finite elementmodel, such as the one introduced by Garrett [7]. However, aformulation based on the lumped mass approach is preferredfor the present work. The numerical efficiency of this approachis an obvious advantage for time domain simulations, whileits simplicity and transparency are well suited to approximatefrequency domain analysis. Moreover, the approach has been

demonstrated by several authors (as detailed below) to be fullyviable and accurate for structures that are not dominated byflexural rigidity.

The lumped mass formulation adopted herein is basedon a global coordinate system, thus removing the need fortransformations between coordinate systems. The axial stiffnessof the line is modeled by simple linear spring elements betweennodes, while bending is modeled by rotational springs using adifferent 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 slopebetween adjacent elements, and subsequently calculates theequivalent shear forces; (ii) the method employed in thesoftware Orcaflex [9] where a node is modeled as a short rod,and rotational spring dampers are applied on either side ofthe node; (iii) the method of Raman-Nair and Baddour [10]who derived the generalized active forces of an equivalentrotational spring in a local coordinate system. In this paper, theforces from an equivalent rotational spring are derived directlyfrom the potential energy expression in global coordinates.Aside from simplicity in the resulting equations, the methodmay be used directly to obtain the tangent bending stiffnessmatrix.

The time domain version of the present analysis employs theWilson–theta implicit integration scheme, which is inherentlymore stable than many other methods when relatively large timesteps are employed [11]. Although the scheme is iterative, thecomputational cost of each iteration is relatively small as matrixfactorization is performed only at the start of each time step.An additional benefit is that unrealistic high frequency axialresponses of the line are automatically filtered out by using alarge time step. As such, axial structural damping need not beconsidered, although it can be readily included if required.

The analysis tool developed herein consists of severalmodules that can be used to perform coupled and/or uncoupledanalysis of the lines and vessel in the time and frequencydomains. The time domain line dynamics code is validatedagainst the commercial software Orcaflex, which employs thelumped mass approach and an explicit integration scheme.The test case employed for this purpose consists of a hangingriser under regular wave excitation loads. Frequency domainanalyses are also carried out for comparisons, and selectedresults are presented. Having validated the line dynamics code,the developed methods are used to analyze a coupled vessel/linesystem in both the time domain and frequency domain. Apartfrom the advantage of computational cost, it is shown thatthe frequency domain approach can also be used to provideuseful post-processing information, such as the degree ofdamping supplied to the vessel by the lines, and the effectiveinertia of the lines. It is found that the frequency domainapproach yields surprisingly accurate results for both the firstand second order response of the system, despite the importanceof nonlinear drag forces on the lines. One reason for this isthat the drag linearization scheme employed is found to besimultaneously optimum for both the second and first ordermotions.

Y.M. Low, R.S. Langley / Applied Ocean Research 28 (2006) 371–385 373

2. Vessel model

2.1. Structural model

The present model of the floating system comprises threedistinct components: the vessel, the lines and a set of connectingsprings. The displacement vector x of the coupled system isrepresented by

x =

[sy

], (1)

where s and y are the displacement vectors of the vessel and thelines respectively. Models of the lines and connecting springswill be discussed in Section 3.

The vessel is described as a rigid body with six degrees offreedom at the CG, with s representing surge, sway, heave,roll, pitch and yaw in that order. The translational degreesof freedom are measured in the same coordinate system asthe lines; hence, there is no need for transformations to localreference frames.

2.2. Wave forces on the vessel

The wave forces acting on a floating vessel are welldocumented in the literature, and thus only a brief outline of thekey results is given here. A linear diffraction analysis providesthe vector of first order transfer functions T(1), defined so thatin 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 waveamplitude. For time domain simulations of a random seastate,the first order forces can be expressed as a sum over constituentseastate components, and this sum can be evaluated efficientlyby using the Fast Fourier Transform technique to yield a timehistory of the forces [12].

In addition to the first order forces, the vessel is subjectedto second order forces arising from nonlinear hydrodynamiceffects. These forces are determined from a second orderdiffraction analysis, and only the slowly varying forces causedby difference frequencies in the surge, sway and yaw arepertinent to the present study. The vector of slow drift quadratictransfer functions (QTFs) T(2) is defined from the interactionbetween a pair of waves with frequencies ωn and ωm and isdefined so that

F(2)(ωm, ωn) = T(2)(ωm, ωn)η(ωm)η(ωn), (3)

where F(2) is the vector of second order forces and η(ωn) andη(ωm) are two wave amplitudes. For time domain analysisthe time history of the second order wave forces in arandom seastate can be expressed as a double summation overthe first order wave components that comprise the seastate;this expression may be evaluated using the efficient methoddescribed by Langley [13]. For frequency domain analysis, thecross-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)F F (ω) = 8

∫∞

0T(2)(µ, ω + µ)

[T(2)(µ, ω + µ)

]H

× Sηη(µ)Sηη(ω + µ)dµ (4)

where Sηη is the wave spectrum and the subscript H denotesthe Hermitian transpose. Each diagonal component of S(2)

F F isreal and is consistent with the form of the second order forcespectrum given by Pinkster [15].

The added mass and radiation (potential) damping matricesof the vessel can be obtained from a radiation analysis [16].These are in general frequency dependent; in the time domainthey can either be Fourier transformed to give the retardationfunctions [16] of the vessel, or treated more approximately byusing 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), withadditional rotational springs employed to model the bendingstiffness of the system. Large deflections and small strains areassumed. Fig. 1 shows three nodes arbitrarily numbered 1 to3 connected to elements j and k, where the three-dimensionalnodal displacements are denoted by y1, y2 and y3. Each elementis modeled as an extensional spring (bending stiffness will bediscussed later in this section) and the mass of the element islumped in equal halves onto the adjoining nodes.

Consider initially only element j . The strain energy due toextension is given by

V j =12

E AL j

(∣∣y2 − y1∣∣− L j

)2, (5)

where L j is the original length and EA is the axial rigidity. Theelastic force T1q acting on node 1 in the q direction is given by

T1q =∂V j

∂y1q=

E A(y2q − y1q

)∣∣y2 − y1∣∣ −

E A(y2q − y1q

)L j

, (6)

where q may be 1, 2, 3, corresponding to global Cartesiancoordinates x , y, z respectively.


The 3 × 3 extensional tangent stiffness matrix for node 1 isdenoted as KA

1 , and the entries are given by

K A1qr =

∂2V j

∂y1q∂y1r= δqr

(E AL j

−E A∣∣y2 − y1

∣∣)

+E A

(y2q − y1q

)(y2r − y1r )∣∣y2 − y1∣∣3 (7)

where δqr is the Kronecker delta that is 1 when q = r and 0when q 6= r . It can be readily shown that the stiffness matrixfor element j is given by

KA(elm j)

=

[KA

1 −KA1

−KA1 KA

1

]. (8)

Eqs. (6) and (8) can be developed for each element in the modelto fully describe the elastic forces arising from axial extension.In addition, rotational springs are used to model bendingstiffness, and a rotational spring, which couples together thetwo elements and three nodes in Fig. 1. Assuming that theprofile of the riser/mooring line is initially straight, the potentialenergy U stored in the rotational spring is expressed as

U =12

E I(L j + Lk

) ( 1R

)2

(9)

where EI is the flexural rigidity of the system and R is the radiusof curvature. By estimating R as the radius formed by fitting acircle through nodes 1, 2 and 3, the relation between R and theangle θ between the elements can be derived from geometryassuming θ is small, in which case

U =12

kbθ2, (10)

where

kb =2E I

L j + Lk. (11)

The bending moment at node 2 can be found bydifferentiating Eq. (10) with respect to θ ; the equivalent forcesacting on the nodal degrees of freedom are derived in whatfollows.

For frequency domain analysis the tangent stiffness matrixassociated with the bending moment is required, and to thisend the angle θ can be expressed as a function of the 1× 9 displacement vector of the neighboring elements, andthis vector, y =

[yT

1 yT2 yT

3]T is separated into mean and

dynamic parts, denoted respectively by y and y. For smallvalues of y (as assumed in linear frequency domain analysis), θ

can be written as

θ = θ + αTy, (12)

∴ θ2= θ2

+ 2θαTy +

(αTy

)2, (13)

where θ is the mean value of θ and αT=[αT

1 αT2 αT

3]

is avector of influence coefficients, which is given by

α1 =1

L j sin θ

[tk −

(tk · t j

)t j], (14a)

α3 =−1

Lk sin θ

[t j −

(t j · tk

)tk], (14b)

α2 = −α1 − α3, (14c)

where t j and tk are the unit tangent vectors for elements j andk, defined as

t j =y2 − y1∣∣y2 − y1

∣∣ , tk =y3 − y2∣∣y3 − y2

∣∣ . (15)

It follows that

∂U∂ y

= kb

[θα + ααTy

](16)

∂2U∂ y∂ y

= kbααT= KB . (17)

The nodal force vector arising from the rotational spring isdenoted as Q =

[QT

1 QT2 QT

3]T, and can be found by putting

y = 0 in Eq. (16); for nonlinear time domain analysis, θ isinterpreted as the instantaneous value of θ in this equation.Eq. (17) gives the 9 × 9 tangent bending stiffness matrix KB .Together, Eqs. (6), (7), (16) and (17) fully describe the elasticforces in the line and the associated tangent stiffness matrixentities. The fully assembled tangent stiffness matrix is given bythe sum of the axial and bending contribution, and it is denotedhere by KL .

The mass matrix of the lines ML can be regarded as the sumof the structural mass matrix MS

L and the added mass matrixMA

L . The matrix MSL is diagonal, being formed by lumping the

structural mass onto the various nodes; MAL will be discussed in

Section 2.2.

3.2. Wave forces on the lines

Linear wave theory can be used to calculate the fluid particlevelocities u and accelerations u at any point on the lines,and the three-dimensional wave forces are calculated by usingMorison’s equation. The inertia and drag forces are usuallycomputed separately for directions normal and tangential to theline, since the hydrodynamic coefficients in the two directionsare different in general. For example, for the system shown inFig. 1, the drag forces from half of elements j and k are lumpedonto node 2 as

FD=

14ρD

∑ξ= j,k

Lξ

{Cn

D[Vr −

(Vr · tξ

)tξ]

+ C tD(Vr · tξ

)tξ}, (18)

where

Vr = r + Vc, (19)r = u − y. (20)


Here, Vc is the current velocity, CnD and C t

D are the normal andtangential drag coefficients, and D is the outer diameter of theline.

The inertia forces are likewise expressed as

FI=

18ρπ D2

∑ξ= j,k

Lξ

{(1 + Cn

A) [

u −(u · tξ

)tξ]

+(1 + C t

A) (

u · tξ)

tξ}

=18ρπ D2

∑ξ= j,k

Lξ

{(1 + Cn

A)

u − CnA(u · tξ

)tξ}

if C tA = 0, (21)

where CnA and C t

A are the normal and tangential added masscoefficients. If it is assumed that C t

A= 0, which is usually avalid approximation, then the equation is simplified as indicatedabove. Under this assumption, the added mass matrix for node2 in Fig. 1 can be shown to be

MA2 =

18ρπ D2Cn

A

∑ξ= j,k

Lξ

{(I − tT

ξ tξ)}

. (22)

where I is the identity matrix. The assembled added massmatrix for the lines, denoted as MA

L , is non-diagonal, withcoupling between the three degrees-of-freedom at any givennode; however, there is no coupling between different nodes.

3.3. Connection of the lines to the vessel

The connection of the lines to the vessel is effected bycoupling each line to the vessel via a set of very stiff springs.If a point P has coordinates P =

[Px Py Pz

]Trelative

to the CG, and its position in global coordinates is P =[Px Py Pz

]T, then a linear compatibility relationship existsfor small rigid body rotations such that

P = As + P, (23)

where

A =

1 0 0 0 Pz −Py

0 1 0 −Pz 0 Px

0 0 1 Py −Px 0

. (24)

Now, if a force fp is applied on point P , then the generalizedforce FB acting in the vessel degrees of freedom is

FB(6×1)

= ATfp(3×1)

. (25)

Now, if the top node of a line is attached to the point P viaa set of three very stiff linear springs, then the restoring springforce fp acting at P is given by

fp = KS(yT − P

), (26)

where

KS = KSI; (27)

here, yT is the position of the top node of the line, KS is a springconstant and I is the 3 × 3 identity matrix. The spring force

acting on the line node is equal and opposite to fp. The springforces can be introduced into the model by adding a stiffnessmatrix that couples the vessel degrees of freedom to the degreesof freedom of the line node. It follows from Eqs. (25) and (26)that this matrix has the form

KC =

ATKSA(6×6)

−ATKS(6×3)

−KSA(3×6)

KS(3×3)

. (28)

The role of the connecting springs is only to ensure that theseparation of the line node from the point P on the vessel isinfinitesimal. For this, KS should be large, but evidently not tothe extent that would lead to ill conditioning of the equations.To simulate different hull to line connections, the propertiesof 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 accountfor different conditions.

3.4. Seabed interaction

For line nodes resting on the seabed, the upward contactforces are included in the model. Friction effects are consideredto be less significant for the system analyzed in this paper andare neglected, since only a relatively small portion of the linesis resting on the seabed. A modified bilinear spring model witha gradual transition is proposed to avoid numerical stability, andthe vertical contact force Fsb on a node near the seabed is of theform

Fsb =12

a1

{−z +

1a2

ln [cosh(a2z + a3)] + a4

}, (29)

where a1, a2, a3, and a4 are suitably chosen constants. Inparticular, a4 should be the value such that Fsb → 0 when zis a suitable distance away from the seabed.

For frequency domain analysis, in order to maintain thesymmetry of the stiffness matrix, the nodes near the seabedare assumed to be attached to grounded vertical springs, whosetangent 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 adynamic analysis. As nonlinearity and large displacements areinvolved, the static solution must be solved iteratively. Oneoption is to employ a time domain analysis (described inthe next section) and allow the system to settle to its finalposition after the transients have been damped out. However,it is more efficient to perform a fully static analysis using theNewton–Raphson iteration scheme. The Jacobian needed forthis method is similar to Eq. (44) of the following section,except that the mass and damping terms are neglected. To easeconvergence, under-relaxation is employed by adding λI to theJacobian, where I is the identity matrix, and λ is a positive realconstant.


4.2. Time domain analysis

The equation of motion for the coupled system in the timedomain is written as

Mx(t) + Bx(t) + Kx(t) = F(t), (30)

where x is the displacement vector, M, B and K are respectivelythe mass, damping and stiffness matrices of the coupled system,and F is the external force vector. The symbol “∧” is to indicatethat there are no contributions to the damping and stiffnessmatrices from the lines and connecting springs, since theseeffects are included as external forces on the RHS of Eq. (30)for the purpose of time integration. It is convenient to partitionthe system matrices and vectors into blocks corresponding tothe vessel and lines so that

M =

[MV 0

0 ML

], (31)

B =

[BV 00 0

], (32)

K =

[KV 00 0

], (33)

F(t) =

[FV (t)FL(t)

], (34)

where the subscripts V and L relate to the vessel and the linesrespectively. The vessel forces FV have been discussed earlier.FV are the forces on the lines, and can be written as

FL = −T − Q + W + FD+ FI

+ FS, (35)

where T and Q are the axial and rotational restoring forces,W is the effective weight, FD and FI are given by Eqs. (18)and (21), and FS represents forces from the vessel connectingsprings. The matrix MV consists of the structural mass of thevessel and the added mass, while BV contains the dampingon the vessel from viscous skin drag, wave drift damping andradiation damping. KV is the linear hydrostatic stiffness matrixof the vessel.

Time integration is carried out using the Wilson–thetaimplicit scheme, which is an extension of the Newmark–Betaalgorithm. Linear acceleration is assumed over an extendedtime step τ = Θh, where h is the normal time step andΘ ≈ 1.4. The kinematics at time t + τ are expressed as [11]

xt+τ = xt + ∆x, (36)

xt+τ =3τ∆x − 2x +

τ

2x, (37)

xt+τ =6τ 2 ∆x −

6τ

xt − 2xt , (38)

where ∆ represents increments in an extended time step.Equilibrium is enforced at time t + τ so that

M (xt+τ ) xt+τ + Bxt+τ + Kxt+τ = Ft+τ . (39)

Substituting Eq. (36) to (38) into Eq. (39) yields

f (∆x) = F (xt + ∆x) − M (xt+τ )

(6τ 2 ∆x −

6τ

xt − 2xt

)− B

(3τ∆x − 2x +

τ

2x)

= 0. (40)

The Newton–Raphson iterative technique can be used tosolve 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 theJacobian defined as

J =∂f

∂∆x. (43)

The Jacobian need not be exact as its role is only to provideconvergence of the algorithm, and it may be estimated to be

−J (∆x) ≈ K (xt ) +6τ 2 M (xt ) +

3τ

B, (44)

where K must include the contribution from the lines andsprings, i.e.

K =

[KV 00 KL

]+ KC , (45)

where KC is the combined stiffness matrices from theconnecting springs. Denoting the RHS of Eq. (44) as the matrixA, the following matrix equation must be solved to implementEq. (42)

Aε = f (∆x) . (46)

The matrix A is positive definite and can be expressed asUTU using the Cholesky decomposition at the start of each timestep, where U is an upper triangular matrix. The computationaleffort of each iteration is relatively small as it is limited only toforward and back substitution and the evaluation of the residualforces at time t + τ for the solution of ε. After a convergedresult is obtained, the incremental acceleration for the normaltime step dx is found from linear interpolation using dx =

∆x/Θ , after which the incremental velocity and displacementare solved accordingly [11].

4.3. Frequency domain analysis

A frequency domain analysis is inherently linear, and inorder to apply the approach to a nonlinear problem such asthe present one, all nonlinearities must be linearized. Thereare two sources of nonlinearity in the present model, thesebeing the geometric nonlinearity arising from large deflectionsof the lines, and the nonlinear fluid drag forces whichappear in Morison’s equation (it can be noted that other fluidnonlinearities give rise to the second order wave forces actingon the vessel, but these forces appear only on the right handside of the equations of motion and do not pose any difficultiesfor frequency domain analysis). The geometric nonlinearities


are dealt with here by calculating the tangent stiffness of thelines at the equilibrium position, which allows for large staticdeflections but assumes that the dynamic deflections around thestatic position are small. The drag force nonlinearity is dealtwith by either harmonic or statistical linearization, dependingon whether the seastate is comprised respectively of regular orrandom waves. In order to facilitate the linearization, the fullvectorial form of the drag force is replaced with an approximateversion in which the drag force is computed independently intwo orthogonal directions which are each perpendicular to theline. This approach is not strictly frame invariant, as pointed outby Hamilton [17] and Langley [18], but given the approximatenature of both Morison’s equation and linearization techniquesin general, the method is considered to be sufficiently accuratefor present purposes. The method is detailed later in thissection.

In principle, the equations of motion can be solvedefficiently in the frequency domain by firstly transformingthe degrees of freedom to modal coordinates. The mass andstiffness 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 followingapproximations must be made: (1) the off-diagonal terms of thedamping matrix are neglected, in which case the equations ofmotion are uncoupled and can be solved very easily; (2) themodal coordinates are truncated to give a significant reductionin the number of degrees of freedom. In the present problem,the drag forces on the lines lead to a highly coupled dampingmatrix in modal coordinates, and given the banded nature of theequations in physical coordinates there is little to be gained by amodal transformation; thus physical coordinates are retained inthe following analysis. A modal analysis can however provide auseful diagnostic technique, and this is discussed in Section 4.4.

In order to linearize the drag forces, consider an arbitrarynode 2, which is attached to elements j and k as shown inFig. 1. The drag force is approximated by computing the forceindependently along two vectors which are orthogonal to eachother and normal to the line. Using element j as an example,the choice of the unit normal vectors n j and 5 j is arbitrary,so long as they satisfy n j · 5 j = 0 and 5 j = ±n j × t j . It isfound that a good choice of unit normal vectors is such that onelies in the plane of the line. The relative velocity r in globalcoordinates at the node is transformed into coordinates of thenormal directions using

nr ξ = r · nξ ξ = j, k, (47a)

Π r ξ = r · 5ξ ξ = j ,k, (47b)

where the superscripts and subscripts refer respectively to thenormal direction and the element. The nonlinear drag force ineach direction is replaced by a linearized version of the form

(Vc + r) |Vc + r | ≈ βr + γ, (48)

where r represents the relative velocity in the considereddirection, and the linearization coefficients β and γ arefunctions of max(r) for regular waves [19], and the standarddeviation of r , σr for random waves [20], Clearly, an iterative

solution procedure is required, as detailed later in this section,since the linearization coefficients depend on the systemresponse. For the case where the current velocity Vc is zero,then γ = 0 and β reduces to 8/3π max(r) for regular wavesand

√8/πσr for random waves.

Using the foregoing procedure, the drag force in globalcoordinates as given by Eq. (18) is linearized as

FD= BL (u − y) + FD

, (49)

where

BL =14ρCn

D D∑

ξ= j,k

Lξ

(nβξ nT

ξ nξ +Π βξ 5T

ξ 5ξ

), (50)

and FD is the mean drag force which has a similar form toEq. (50), but with β replaced with γ . The damping matrixBL has the same coupling characteristics between the variousdegrees of freedom as the added mass matrix. In the case ofrandom seas, it is convenient to compute the response to thefirst and second order wave forces (WF and LF) separately(but concurrently). The variance of the relative velocity can beregarded as the sum of the first and second order contributions,i.e.

σ 2r = σ 2

r (1) + σ 2r (2) . (51)

Having linearized the system, the equation of motion can bewritten in the frequency domain as(

−ω2M + iωB + K)

x(ω) = F(ω) (52)

where M and K have been previously defined, and B includesthe damping from the lines so that

B =

[BV 00 BL

]. (53)

The matrix BL depends on the linearization coefficients,which in turn depend on the response of the system, and thusan iterative solution to Eq. (52) is required. It can further benoted that the static problem is coupled to both the WF and theLF response since the mean forces F are also functions of thelinearization coefficients. At the end of a typical iteration loop,the matrices M, B and K (which are all position dependent) arereassembled based on the updated result for x.

In order to compute the WF response, the transfer functionsbetween the system response and the surface elevation must befound. The first order forces F(1) acting on the vessel and thelines can be written as

F(1)(ω) =

[T(1)(ω)

G(1)(ω)

]η, (54)

where G(1) contains the transfer functions for the dragand inertia forces on the lines, which also depend on thelinearization coefficients. It follows from Eq. (52) that thetransfer functions for the system response can now be writtenas

R(1)(ω) = H(ω)

[T(1)(ω)

G(1)(ω)

], (55)


where

H(ω) =

(−ω2M + iωB + K

)−1. (56)

The transfer function for the first order relative velocity atany point on the lines r (1) can then be found and, for randomseas, the spectrum and variance follow from standard randomvibration theory (e.g. Newland [12]).

For the analysis of the second order LF response, it isnecessary to work directly from the force cross-spectra matrix.To solve for σ 2

r (2) , the spectrum of the LF relative displacementin the corresponding normal direction must first be found. TheLF relative normal displacement X (2) can be written as a linearcombination of x(2) such that

X (2)= LTx(2). (57)

The spectral density of X (2) can then be expressed as [11]

S(2)X X (ω) = LTH(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 thelarge number of zeros in the matrix S(2)

F F , Eq. (58) can beevaluated more efficiently by converting it to a 3 × 3 matrixcovering surge, sway and yaw, and adjusting the sizes of theother matrices accordingly. Finally, it can be noted that puttingL = I in Eq. (58) yields the cross-spectral matrix of the secondorder (LF) response of the system.

4.4. Modal analysis and inertia/damping coefficients

Following a linearized frequency domain analysis, a modalanalysis of the system can be performed to yield importantphysical information such as the natural frequencies of thesystem, the system damping, the amount of damping providedby the lines relative to the vessel etc. To this end the undampednatural frequencies and mode shapes of the coupled system canbe found by solving the standard eigenvalue problem. Let 8

represent the modal matrix, i.e. the matrix whose columns arethe eigenvectors of the system. The system degrees of freedomcan be written in terms of a set of modal coordinates q via therelation

x = 8q. (59)

The equations of motion can now be transformed to modalcoordinates to yield

8TM8q + 8TB8q + 8TK8q = 8TF. (60)

The total damping ratio ζS in degree of freedom S can nowbe estimated as

2ζSωS =

(8TB8

)SS

, (61)

where ωS is the associated natural frequency and B is definedin Eq. (53). The damping ratio ζS is the sum of contributionsfrom the vessel and the lines. The former can be found from Eq.(61), but putting BL = 0 in Eq. (53); and the latter by putting

Fig. 2. Static configuration of hanging riser.

BV = 0. The inertia contribution from the lines may also besignificant for deepwater floating structures. By definition, themodes are normalized such that

8TM8 = I. (62)

By using Eqs. (31) and (62), the contribution to thenormalized generalized mass from the lines can be found byputting MV = 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 andline dynamics.

The linear coefficients for inertia and damping from thelines estimated in this way could, if required, be applied to atraditional uncoupled analysis of the vessel. As the linearizeddamping matrix is calculated iteratively from a fully coupledfrequency domain analysis, it is specific to the problem andtakes account of the seastate and the actual motions of the vesseland lines.

5. Validation of the line dynamics code

The time domain line dynamics software developed as partof the present work has been validated against the commercialsoftware Orcaflex, [9] and the time domain results havethen been used to benchmark the frequency domain analysissoftware. Due to space constraints, results from only two loadcases are presented here; a more comprehensive report on thevalidation study can be found in Reference [21].

5.1. Description of the test case

The test case is a hanging riser pinned at both ends, with thestatic configuration depicted in Fig. 2, and the input parameterssummarized in Table 1. Two load cases are considered. ForCase 1, the bottom end is pinned while the top end is prescribeda horizontal simple harmonic motion in still water (i.e. no wavekinematics). The amplitude of the prescribed motion is 10 m,and the period is 27 s, close to the first in-plane mode. ForCase 2, both ends are pinned, and an in-plane regular wave actson the riser. The wave has a waveheight of 10 m and a period


Table 1Input data for hanging riser

Total unstretched length 170 mOuter diameter, D 0.396 mDry mass 165 kg m−1

EA 500 MNEI 120.8 kN m2

Density of water, ρ 1000 kg m−3

Gravitational acceleration, g 9.807 m s−1

Drag coefficients CnD = 1, C t

D = 0Added mass coefficients Cn

A = 1, C tA = 0

Number of elements 68Element size (constant) 2.5 m

Table 2Comparisons of static tension and reactions for hanging riser

Inhouse Orcaflex Theory

Top tension (kN) 47.11 47.11 47.14Bottom tension (kN) 26.60 26.60 26.63Vert reaction, top (kN) 45.71 45.71 45.72Horiz reaction, top (kN) 11.40 11.40 11.47Vert reaction, bottom (kN) 24.04 24.04 24.03Horiz reaction, bottom (kN) −11.40 −11.40 −11.47

of 10 s. The top node is positioned at a water depth of 5 mwhich implies that all elements are fully submerged for each ofthe considered load cases. However, the line model describedherein can be readily extended to accommodate partially orintermittently submerged elements by adjusting the effectiveweight and fluid loads on these elements accordingly.

5.2. Results

Results for the static tension and the support reactions atthe top and bottom end of the riser are presented in Table 2.The results labeled “theory” are derived from the classicalcatenary equations, neglecting elasticity and bending stiffness.It is found that the static results from the present code andOrcaflex are virtually indistinguishable, and are also closeto the theoretical result, indicating that the adopted level ofdiscretization is sufficient.

The time history of the top tension for the two test casesis plotted in Fig. 3(a) and (b) respectively. The transientbuildup stage during which loads are ramped up from zerois not included so that the results represent steady-statemotion. In both cases, the curves produced by the presentcode and Orcaflex are nearly coincident despite the use ofdifferent numerical integration schemes and differing methodsof analyzing the bending moments in the system. It isinteresting to observe the high degree of nonlinearity exhibitedin the time history for Case 1, where a horizontal motion of10 m distorts the catenary shape appreciably. In contrast, thenonlinearity is not pronounced in Case 2, as the top and bottomends are restrained.

In order to compare the time and frequency domainpredictions for the two test cases, the amplitudes of the dynamictension are plotted along the riser in Fig. 4. In these figures, theamplitude of the dynamic tension for the time domain results

Fig. 3. Time history of top tension; (a) Case 1 (prescribed top motion), (b) Case2 (regular wave).

is taken to be half the difference between the maximum andminimum values. The accuracy of the linearized frequencydomain analysis is evidently consistent with the extent of thenonlinear behavior displayed in Fig. 3. For Case 1, the tensionpredicted by the frequency domain analysis differs from thetime domain analysis by an average of 25%, while in Case 2,the mean discrepancy is 15%. It is found that the disparity dropsfor 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 spreadmoored FPSO installed in a water depth of 2000 m, and theessential characteristics of the vessel are summarized in Table 3.The centre of gravity is located midway with respect to thelength and breadth, and 5 m above the still water level.

It is common for mooring lines to consist of a combinationof chains and wires, and for different types of risers to be usedfor various functions, such as oil and gas production/export andwater injection. For the purpose of the present study, only fourcatenary lines with uniform properties are considered. The planof the configuration of the lines is illustrated in Fig. 5(a), while


Fig. 4. Dynamic tension comparisons; (a) Case 1, (b) Case 2.

Fig. 5. Illustration of vessel and lines; (a) plan view, (b) 3D view.

Table 3Vessel data

Length 240 mWidth 46 mDraft 10 mDisplacement 110 000 tons

a 3D view is depicted in Fig. 5(b). Each line in the modelrepresents a group of mooring lines and risers. This is easilyachieved by scaling the mass, stiffness and external forces(e.g. the hydrodynamic coefficients) of the lines by the number

Table 4Line data

Water depth 2000 mLength of line 3300 mPre-tension 4370 kNExternal diameter, D 0.15 mDry mass 150 kg m−1

Scaling factor 6EA 1000 MNEI 50 kN m2

Drag coefficient 1 (normal) 0 (tangential)Added mass coefficient 1 (normal) 0 (tangential)No. of elements per line 50Element size (constant) 66 mz coordinate of top node −10 m

Table 5Simulation parameters

Time domainWave freq Low freq

Width of strip dω (rad/s) 0.00125 0.00125No. of strips 800 160Buildup duration (min) 20Main simulation duration (min) 83.8Time step (s) 0.04

Frequency domainWave freq Low freq

Width of strip dω (rad/s) 0.005 0.00125No. of strips 200 160

of lines in the group. This scaling scheme does not modify thestatic position of the lines, assuming that the draft of the vesselis unchanged by also scaling the buoyancy. Thus, parametricstudies on the number of lines can be performed without theneed to change the model. In this paper, the scaling factor is6, representing a total of 24 lines, and the properties of thelines (prior to scaling) are given in Table 4. The line tensionsobtained from the analyses are then reduced by the same factorso that results are characteristic for a single line.

In this example, mean forces from various sources are notconsidered for simplicity as they do not affect the main featuresof the dynamic analysis. Unless otherwise stated, the randomwave environment is described by a three-parameter Jonswapspectrum with HS = 15.7 m, Tz = 13.5 s and γ = 2. Therandom waves are unidirectional and approach the vessel atan angle of 22.5◦ measured from the bow, as illustrated inFig. 5(a). The intent is to exercise all six degrees of freedomof the vessel, as opposed to head or beam waves.

Simulation parameters for time and frequency domainanalysis are given in Table 5. Wave frequencies are definedfrom 0.2 to 1.2 rad/s, and low frequency below 0.2 rad/s. Thewave spectrum is non-zero in the WF range, and is divided intoa number of strips of equal width dω. It is necessary to usea consistent dω in the time domain. However, for frequencydomain simulation, it is possible to select a smaller dω for thesecond order analysis to capture the narrow-banded response.

For the time domain analysis, there is setting period duringwhich the loads are ramped up gradually to allow the transients


Fig. 6. Time history of surge excursion.

to dissipate. Statistical estimates are obtained from the mainanalysis period only, and the variance is averaged over tenruns. This is to ensure that sufficient cycles are consideredfor the non-Gaussian second order response, since each runhas a simulation duration of 83.4 min and includes onlyapproximately 20 LF cycles. Spectral densities are recoveredfrom the time histories to allow direct comparison with thefrequency domain. The clear delineation of wave and lowfrequencies makes it straightforward to assign responses to firstor second order forces.

6.2. Results

Selected results from the numerical simulations arepresented in this section. An excerpt of the time history forthe surge excursion taken from one of the time domain runsis shown in Fig. 6.

Spectral densities have been recovered from the timehistories generated by the time domain analysis and are shownin Fig. 7(a)–(g). It is interesting to observe that there issignificant LF roll motion, even though low frequency forcesfor roll are absent, and the natural frequency for roll is about21 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 throughthe added mass matrix and the line motions. This effect can becaptured in the frequency domain analysis by including roll inthe second order analysis, but not defining any LF roll waveexcitation forces.

Fig. 7(a)–(g) show the spectra generated by both the timeand frequency domain analyses for the six vessel responses andthe top tension of one of the lines. Table 6 gives the standarddeviations, which are calculated from the spectra and separatedinto WF and LF components. The standard deviation of thetension (dominated by WF) along the line from the bottom endis plotted in Fig. 8.

The natural periods obtained from a modal analysis of thecoupled system are presented in Table 7. The equivalent lineardamping and inertia coefficients from the lines for the relevantmodes are also included in the table. The mode shapes areillustrated in Fig. 9, where dotted lines depict the deflectedshapes. The first three modes primarily correspond to yaw, swayand surge in order of decreasing periods, and the fourth modeevidently involves mainly the line dynamics.

Table 6Comparison of standard deviations

Wave frequency Low frequencyTimedomain

Freqdomain

Timedomain

Freq domain

Surge (m) 1.785 1.783 4.270 4.163Sway (m) 1.192 1.191 4.320 4.268Heave (m) 2.344 2.341 NegligibleRoll (deg) 0.631 0.639 0.236 0.220Pitch (deg) 1.501 1.500 NegligibleYaw (deg) 1.336 1.334 6.869 7.331Top tension (kN) 186.3 180.6 24.79 24.47Bottom tension (kN) 156.4 153.1 25.01 24.17

Table 7Natural periods and damping/inertia coefficients

Mode 1 2 3 4

Nat period (s) 299.7 259.8 256.1 51.9Type Yaw Sway Surge LinesDamping ratio, vessel (%) 0.66 4.79 4.53 NADamping ratio, lines (%) 46.17 27.57 31.10 NAMass from lines (%) 6.31 4.32 4.46 NA

It is interesting to explore the effect of various factors on thedamping provided by the lines. Parametric studies are carriedout by varying the drag coefficient CD , and calculating thedamping ratio from repeated frequency domain simulations.The damping ratio for surge and sway for a range of CD valuesare plotted in Fig. 10(a). The procedure has been repeated byvarying the significant waveheight HS , assuming that the wavedrift damping remains unchanged, and the results are displayedin Fig. 10(b).

The time domain simulations require around 90 h of CPUtime for the ten runs on a Pentium 4 2.0 GHz laptop.Comparatively, the frequency domain analysis require onlyapproximately 3 min.

6.3. Discussion

A severe seastate corresponding to a 100-year storm hasbeen selected for the examples. This maximizes the opportunityfor exercising the nonlinearities in the problem, and constitutesthe critical test of the accuracy of the coupled frequency domainanalysis.

WF vessel motions predicted by time domain and frequencydomain simulations are in excellent agreement, which can beexpected since apart from resonant roll motions, the othervessel modes are either stiffness dominated (pitch and heave)or inertia dominated (surge, sway and yaw) and thereforeinsensitive to damping. For the LF motions, the frequencydomain method performs well for surge and sway (deviatingabout 2% from the time domain). The roll and yaw results mayappear to be comparatively less ideal (within 7%), but they arestill satisfactory within engineering expectations. In addition tothe motions, excellent agreement is also obtained for dynamictensions along the line, with the frequency domain approachunder predicting by a marginal 1%.


Fig. 7. Spectral density plots; (a) surge, (b) sway, (c) heave, (d) roll, (e) pitch, (f) yaw, (g) top tension.


Fig. 8. Comparison of standard deviation of tension along the line.

It can be noted that in the LF vessel response, there issignificant scatter between the results from the ten individualtime domain runs. For example, the mean-square for LF surgeover ten runs has a sample standard deviation of 3.55 m(Langley [13] elaborates on this subject). Taking the averagefrom the runs improves the predictions of LF motions.

Damping ratios for surge, sway and yaw are estimated fromthe damping matrix computed iteratively from the coupledfrequency domain analysis. The variance of the relative velocityneeded to define the damping matrix is treated as the sum of WFand LF contributions. However, the LF composition is in factminimal. For instance, the damping ratio for surge calculatedwithout LF velocities is 30.4%, compared to 31.1% for thecombined WF and LF velocities. This implies that using thepresent approach, the linearized damping forces critical for LFmotions are primarily governed by WF velocities.

Since stochastic linearization is based on minimizing themean-squared error across the entire spectrum, the dominanceof WF velocities means that the linearization is likely to beaccurate for WF drag, but not necessarily for the LF dragforces. However the simulations results demonstrate that thelinearization approach captures the LF drag forces accurately,although the underlying reason is not immediately obvious. Thetopic is not addressed in the literature owing to the scarcityof coupled frequency domain analysis. As the assumption iscentral to the methodology, it warrants further investigation.Hence, a theoretical justification that supports the assumptionis supplied in the Appendix. The derivation therein is validwhen LF velocities are assumed to be much smaller than WFvelocities.

The other type of nonlinearity commonly found in floatingsystems is the geometric nonlinearity of the lines, resultingin nonlinear restoring forces provided to the vessel. However,this form of nonlinearity is especially weak for deepwatersystems, as typical vessel motions are small in comparisonto the dimensions of the lines, and the catenary shape is notdistorted appreciably.

In a previous work, a simplified two degree-of-freedomsystem is studied to glean insight into the coupling mechanismsarising from the nonlinearities in drag and the restoringforce [23]. It was discovered that the latter is an influentialsource 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 WFline dynamics also depends on the instantaneous position ofthe vessel, and thus the magnitude of LF motions. The presentcase where geometric nonlinearity is negligible constitutes aconsiderable advantage in coupled analysis, as the WF linedynamics are no longer seriously influenced by LF vesselmotions. It is apparent that the frequency domain method alsoowes its favorable performance to this factor.

The effect of CD on the damping ratio is studied (seeFig. 10(a)). As expected, damping ratio increases withincreasing CD . Nonlinearity of the drag forces is exemplifiedfrom the decreasing slope of the graph. This is because anincrease 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


Fig. 10. Influence of parameters on damping ratio; (a) effect of CD , (b) effectof HS .

the design storm. The nonlinear behavior and dependence onseveral parameters make it difficult to estimate the dampingaccurately short of a coupled analysis.

7. Conclusions

A framework for the fully coupled analysis of a floatingstructure and the mooring/riser system is described. Thelines are modeled using a lumped mass approach, which issimple, efficient, and can be easily applied in the time domainor frequency domain. Numerical simulations show that aconsistently formulated frequency domain method can providegood predictions of vessel motions and line tensions, whencompared to time domain results. This makes frequency domaincoupled analysis an important tool, due to vastly superiorcomputational efficiency over the more commonly employedtime domain method. However, it is advisable to use timedomain coupled analysis to verify important cases at the finaldesign stage.

The very good agreement between the time domainand frequency domain approaches can be attributed totwo main aspects. Firstly, the drag linearization schemeprovides an optimal estimate of both the low and wavefrequency components of the drag force, as shown in theAppendix. Secondly, the geometric nonlinearity of the linesis insignificant for the deepwater floating system (2000 mwater depth) examined in this paper, as vessel motions aresmall in comparison to line dimensions. As such, it is worth

investigating shallower water systems (say 400 m), where bothcoupling and geometric nonlinearity may be crucial. In general,it is recommended that the frequency domain approach is likelyto be robust and accurate when geometric nonlinearity is notprevalent. The extent of geometric nonlinearity in a practicalsystem should be carefully assessed by an engineer.

A further feature of the frequency domain method isthe efficient estimation of inertia and damping contributionsfrom the lines in the form of linear coefficients. The resultsindicate that the effect of lines on vessel damping can be verylarge, especially for severe storms, thus emphasizing the needto capture the damping characteristics accurately through acoupled analysis.

Appendix

It is demonstrated in what follows that the standardequivalent linearization technique provides an optimal estimateof both the LF and WF components of the drag forces on thelines. The velocity-squared term which appears in the nonlineardrag force on an element of the line can be expanded as

(r1 + Vc + r2) |r1 + Vc + r2|

=

[(r1 + Vc)

2+ 2 (r1 + Vc) r2 + r2

2

]sgn(r1 + Vc)

= (r1 + Vc) |r1 + Vc| + 2 |r1 + Vc| r2 + r22 sgn(r1 + Vc)

(63)

where r1, r2, and Vc are respectively the WF, LF and steadycomponents of the relative velocity, and it has been assumedthat sgn (r1 + Vc) ≈ sgn(r1+r2+Vc) since r2 is small. The firsttwo terms on the right-hand side of Eq. (63) relate respectivelyto the WF + steady and LF force components respectively, andthe last term is taken to be negligibly small since it is secondorder in r2.

Firstly, consider the independent linearization of the LF dragforce. The LF term in Eq. (63) is replaced by a linear expressionof the form

2 |r1 + Vc| r2 = β2r2 + ε (64)

where ε is the error. Minimizing the expected mean-squarederror gives

∂

∂β2E[ε2]

= 0 (65)

∴ β2 = E [2 |r1 + Vc|] . (66)

Now consider the independent linearization of the WF +

steady contribution to Eq. (63), which is also replaced by alinear expression, written as

(r1 + Vc) |r1 + Vc| = β1r1 + γ1 + ε (67)

β1 can be found by minimization of the expected value of ε2,which can be shown to give

β1 =E [r1 (r1 + Vc) |r1 + Vc|]

E[r2

1] . (68)


Now, if r1 is assumed to be a Gaussian process, it is a standardresult that [22]

E [r1 f (r1)] = E[

f ′ (r1)]σ 2

r1(69)

for any function f . Using the above relationship, Eq. (68)simplifies to

β1 = E[

ddr1

{(r1 + Vc) |r1 + Vc|}

]= E [2 |r1 + Vc|] . (70)

The LF and WF components are simultaneously optimized bythe linearization procedure, since β1 = β2. Both are defined interms of σr1 , Vc and the associated error functions.

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