Ocean Engineering 55 (2012) 206–218

Contents lists available at SciVerse ScienceDirect

Ocean Engineering

0029-80

http://d

n Corr

E-m

rab@lam

fabricio

bruno@

journal homepage: www.elsevier.com/locate/oceaneng

Parallel implementations of coupled formulations for the analysis of floatingproduction systems, part I: Coupling formulations

Breno Pinheiro Jacob n, Rodrigo de Almeida Bahiense, Fabrıcio Nogueira Correa,Bruno Martins Jacovazzo

LAMCSO—Laboratory of Computer Methods and Offshore Systems, COPPE/UFRJ—Post-Graduate Institute of the Federal University of Rio de Janeiro, Civil Engineering Department,

Centro de Tecnologia Bloco B sala B-101, Cidade Universitaria, Ilha do Fund~ao, Caixa Postal 68.506, 21945-970 Rio de Janeiro, RJ, Brazil

a r t i c l e i n f o

Available online 4 August 2012

Keywords:

Floating production systems

Mooring systems

Risers

Nonlinear dynamics

Coupled analysis

18/$ - see front matter & 2012 Published by

x.doi.org/10.1016/j.oceaneng.2012.06.019

esponding author. Tel./fax: þ55 21 2562 738

ail addresses: breno@coc.ufrj.br (B.P. Jacob),

cso.coppe.ufrj.br (R.d.A. Bahiense),

@lamcso.coppe.ufrj.br (F.N. Correa),

lamcso.coppe.ufrj.br (B.M. Jacovazzo).

a b s t r a c t

Considering the analysis of Floating Production Systems (FPS), this work describes two different

formulations for the coupling of the hydrodynamic behavior of the hull and the hydrodynamic/

structural behavior of the mooring lines and risers. These formulations are characterized by how the

equations of motion of the hull and the lines are associated and solved.

The first formulation is a ‘‘weak coupling’’ (WkC) scheme, focused on the equations of motion of the

hull: the coupling is performed by forces acting on the right-hand side of the hull equations. The second

formulation is a ‘‘strong coupling’’ (StC) scheme, where the Finite Element meshes of all mooring lines

and risers are assembled together, and the hull is considered as a ‘‘node’’ of this model.

Results of case studies employing the WkC and StC formulations are presented; based on these

results, the behavior of these formulations is compared, in terms of their accuracy, computational

performance, and applicability to different types of offshore systems. These results correspond to

sequential implementations of the coupling formulations; further studies are presented in a subsequent

paper, regarding different domain decomposition strategies associated to these coupled formulations,

implemented on computers with parallel architecture.

& 2012 Published by Elsevier Ltd.

1. Introduction

Along the last decades, offshore oil production activities havebeen advancing towards even deeper waters, reaching newfrontiers so far not conceivable. Considering for instance theBrazilian case, very large deep-water oil fields have been assessedin the Campos Basin, and more recently in the pre-salt layer in theSantos Basin. For such deep and ultra-deep water scenarios, thecurrent trend is the use of Floating Production Systems (FPS),based on moored ships or semisubmersible platforms, and con-nected to risers that convey the oil, gas and other fluids that resultfrom the exploitation process.

The traditional design practice of FPS employed numericaltools based on uncoupled formulations, where the hydrodynamicbehavior of the hull is not influenced by the nonlinear dynamicbehavior of the mooring lines and risers. In this ‘‘classic’’approach, the hydrodynamic analysis of the hull was performed

Elsevier Ltd.

5

with the mooring lines represented by simplified scalar models,and without consideration of the risers. This analysis leads to themotions of the hull, and to the design of the mooring linescomplying with specified limit values for the motions. In asubsequent design step, these motions are prescribed at the topof Finite Element (FE) models of the risers, on nonlinear dynamicanalyses to determine their structural behavior.

Nowadays, it is widely acknowledged that the design of FPS fordeep-water applications should employ coupled analysis tools,considering that the moored system and the risers comprise atruly integrated system: its overall behavior is dictated not onlyby the hydrodynamic behavior of the hull, but also by itsinteraction with the hydrodynamic-structural behavior of thelines. The pioneering works proposing coupled models for theanalysis of FPS were mostly related to Tension Leg Platforms(TLPs); see for instance (Paulling and Webster, 1986; Davies andMungall, 1991; Sircar et al., 1993; Phifer et al., 1994; Schott et al.,1994). The development of numerical tools considering couplingeffects for other types of FPS has become more widespreadtowards the end of the 90’s, see for instance (Astrup et al.,2001; Chaudhury, 2001; Correa et al., 2002; Correa, 2003; Finnet al., 2000; Garrett et al., 2002a; Heurtier et al., 2001; Hong andHong, 1996; Jacob and Masetti, 1998; Jacob, 2005; Kim et al.,

B.P. Jacob et al. / Ocean Engineering 55 (2012) 206–218 207

2001,2005; Ormberg et al., 1997; Ormberg and Larsen, 1998;Senra et al., 2002; Senra, 2004; Tahar and Kim, 2008; Wichers andDevlin, 2001). Innovative research on this line continues to thepresent days (Yang and Kim, 2010; Low, 2011).

Nonlinear time-domain coupled analysis tools have been usedin the context of different analysis methodologies (Ormberg et al.,1998; Correa et al., 2002). ‘‘Fully coupled’’ methodologies assumethat the lines are modeled by FE meshes refined enough tosimultaneously provide accurate results for the platform motionsand the structural response of the lines (i.e., the fatigue behaviorof the risers). On the other hand, ‘‘hybrid methodologies’’ combinethe use of coupled and uncoupled models; one example is the‘‘coupled motion analysis’’ (Ormberg et al., 1998) where a coupledmodel with the lines represented by coarser meshes providesaccurate results for the platform motions, which subsequently areinput to uncoupled models of the risers for fatigue analysis.

Currently, the use of coupled motion analysis is being estab-lished on the design practice of risers. However, the use of fullycoupled analysis methodologies for the simultaneous nonlineardynamic analysis and design of the moored system and its risersmay still be hindered by excessive computational costs. Extensiveuse of time-domain fully coupled analyses may not be feasible forthe current design practice, which requires, for instance, hun-dreds or thousands of analyses for the assessment of fatiguebehavior of the risers.

In this context, to obtain simpler and more efficient solutionprocedures involving reasonable approximations, previous workshave presented frequency-domain analysis methods. Garrett et al.(2002a) and Garrett (2005) presented both time and frequencydomain procedures using the same model; the model appropriateto the application is used, and both procedures were shown to beaccurate and efficient. Garrett et al. (2002b) and Low and Grime(2010) also presented coupled frequency-domain analysis proce-dures. Combined schemes where the low-frequency and wave-frequency components of the hull motions are solved separatelyin the time and frequency domain respectively, and the lines arerepresented by a lumped-mass approach, are presented in (Lowand Langley, 2006,2008; Low, 2008,2011).

Focusing on improved computational efficiency while main-taining higher levels of accuracy (attained by a full time-domainmethod and a rigorous representation of the lines by FiniteElement models), this work and the companion paper (Jacobet al., in press) describe coupled formulations for the analysis offloating production systems, and present parallel implementa-tions associated to these formulations.

This paper presents two different formulations that are con-sidered for the coupling of the equations of motion that repre-sents the hull and the lines. The first coupling formulation is a‘‘weak coupling’’ (WkC) scheme, which is focused on the equa-tions of motion of the hull: the coupling between the hydro-dynamic model of the hull and the hydrodynamic/structuralmodel of the lines is performed by forces acting on the right-hand side of the hull equations. The second formulation is a‘‘strong coupling’’ (StC) scheme, where the FE meshes of allmooring lines and risers are assembled together, and the hull isconsidered as a ‘‘node’’ of this model.

In the remainder of this paper, Section 2 presents the equa-tions of motion that represent the behavior of the hull and thelines. The different coupling formulations are characterized byhow these equations are associated and solved, as described inSection 3. Firstly, Section 3.1 describes the ‘‘weak’’ couplingscheme WkC; then, Section 3.2 presents the ‘‘strong’’ couplingscheme StC. Since the implementation of parallel techniques ishighly associated to these different coupling schemes, initialcomments regarding this issue will already be presented alongthe description of the coupling formulations. More details

regarding the parallel implementations will be presented inJacob et al. (in press).

Section 4 presents a brief description of the methods consid-ered for the calculation of the environmental forces, and thehydrodynamic models that represent the hull and the lines.Results of case studies comparing the WkC and StC formulationsare presented on Section 5. Based on these results, Section 6comments on the behavior of these formulations, in terms of theiraccuracy, computational performance, and applicability to differ-ent types of offshore systems.

2. Equations of motion

This section summarizes the formulation of the equations ofmotion that represent the behavior of the hull and the lines. Thedifferent coupling formulations considered in this work will becharacterized by how these equations are associated and solved,as described next in Section 3.

2.1. Hull: large amplitude, rigid body equations of motion

The rigid-body motions of the hull are represented by theexact large amplitude equations of motion (Meirovitch, 1970;Paulling, 1992). In order to assemble these equations, twocoordinate systems must be considered: oxyz and OXYZ, initiallycoincident and with their origin located at the center of gravity(c.g.) of the platform. The ‘‘global’’ system oxyz remains fixed inspace, and is taken as an inertial reference system, but the ‘‘local’’or ‘‘body’’ system OXYZ moves, following the rigid-body motion ofthe hull.

The position vector of a given point p of the body in localcoordinates is defined as Xp

¼{X,Y,Z}, and the correspondingvector in global coordinates is xp

¼{x,y,z}. The motion may beexpressed as the sum of a translation of the origin of the localsystem OXYZ, and a rotation about an axis passing through it:

a)

The translational motion xl¼{xl, yl, zl} is expressed by the timedependent coordinates of the origin of the local system OXYZ,measured in the global xyz directions; and

b)

The rotational relative position of the two coordinate systemsis expressed employing the Euler angles g, a and b as follows:The body first rotates about OZ through the yaw angle b, thenrotates about the resulting new position OY through the pitchangle a, and, finally, about this last position OX through theroll angle, g.

The expression that defines the coordinate transformationrelating the local ‘‘body’’ system to the global system is thengiven by

Xp¼Aðxp�xlÞ ð1Þ

where A is a 3�3 rotation matrix defined in terms of the angles g,a and b as follows (where sa¼sin a, ca¼cos a, and so on):

A¼

cbca sbca �sa�sbcgþcbsasg cbcgþsbsasg casgsbsgþcbsacg �cbsgþsbsacg cacg

0B@

1CA ð2Þ

Considering that A is an orthogonal matrix, the inversetransformation is expressed as

xp ¼ xlþAT Xpð3Þ

Newton’s second law may be written in terms of f and m (theexternal force and moment vectors) respectively for translational

B.P. Jacob et al. / Ocean Engineering 55 (2012) 206–218208

and rotational motions, as follows:

f ¼d

dtðMvÞ ð4Þ

m¼d

dtðIxÞ ð5Þ

where v and x are respectively the translational and angularvelocity vectors of the body; M is a 3�3 diagonal matrix with thedry mass of the hull; and I is a 3�3 matrix with moments andproducts of inertia.

Regarding the translational momentum Eq. (4), if the vectorsof translational velocity v and force f are expressed in the globalinertial system oxyz, the time derivative of the momentum (Mv)at the right hand side of Eq. (4) becomes:

f ¼Mdv

dtð6Þ

where

v¼dx

dtð7Þ

and x¼xl¼{xl, yl, zl} are the coordinates of the body c.g. in theglobal oxyz system (as can be seen in Eq. (3) with Xp

¼0).On the other hand, it is convenient to evaluate the angular

momentum Eq. (5) in the body coordinate system OXYZ, since theinertia matrix I is constant in this system. The time derivative ofangular momentum (Ix) is therefore evaluated in a coordinatesystem that is rotating, and Eq. (5) becomes:

m¼ Idxdtþx� ðIxÞ ð8Þ

It remains to establish the relation between x and the timederivatives of the rotation angles g, a and b, as follows:

x¼ Bdhdt

, B¼

1 0 �sa0 cg casg0 �sg cacg

0B@

1CA ð9Þ

Now, Eqs. (6)–(9) may be rearranged in the form of thefollowing set of twelve first order differential equations, wherethe unknown variables are v, x, x and h (respectively transla-tional and angular components of velocities and position of thebody as functions of time):

dv

dt¼M�1f,

dx

dt¼ v ð10aÞ

dxdt¼ I�1 m�x� ðIxÞ½ �,

dhdt¼ B�1x ð10bÞ

The external force and moment vectors f and m include theenvironmental loadings of wind, wave and current. The calcula-tion of these loads is associated to the hydrodynamic model of thehull, as will be described later in Section 4. In the weak couplingscheme, as will be shown in Section 3.1, these vectors also includethe resultant forces and moments of the lines at the connectionswith the hull.

Several types of nonlinear effects are involved in these equa-tions; for instance, vectors f and m include hydrodynamic forcesand moments that are nonlinear functions of body position andvelocities. Also, the vector product x� (Ix) and the B–1 transfor-mation contain nonlinear terms involving products and powers ofangular velocities and trigonometric functions of Euler angles. Atthis point, simplified linear formulations could assume smallamplitudes of motion and disregard higher other terms, contain-ing products or powers of quantities with lower order of magni-tude. However, this formulation keeps all nonlinear terms andtherefore is valid for large amplitudes of motion, since the

integration is performed in the time domain by the fourth-orderRunge–Kutta method, as will be described in Section 3.1.

2.2. Lines: semi-discrete equations of motion

The structural dynamic behavior of the lines is described byequations of motion that arise from the Finite Element (FE) spatialdiscretization. According to the type of line considered (mooringlines, flexible or steel catenary risers, umbilical cables, etc.), eithernonlinear truss or frame elements can be considered for thespatial discretization.

The FE spatial discretization yields a set of second-orderordinary differential equations (ODE) (Bathe, 1996; Belytschko,1983); ‘‘semi-discrete’’ because they are still continuous functionsof time. This ODE correspond to the equations of motion that canbe written as follows, where the unknowns are acceleration,velocity and displacement vectors €uðtÞ, _uðtÞ and u(t); eachcomponent of these vectors corresponds to a dof of the FE mesh:

M €uðtÞþC _uðtÞþRðuðtÞÞ ¼ Fðu,tÞ ð11Þ

In this expression, R(u) is a vector of internal elastic forces(nonlinear functions of the unknown displacements u). Inertiaand damping forces are represented by the terms affectedrespectively by M (the global mass matrix) and C (that may bedefined as a Rayleigh proportional damping matrix—a linearcombination of the global mass and stiffness matricesC¼amMþakK (Bathe, 1996)). The global matrices M and K areassembled from the contributions of the corresponding matricesfrom each element of the FE mesh.

Finally, F is the vector of external loads (that can also be anonlinear function of the unknown displacements u), includingthe environmental loadings of dead weight, wave and current viathe Morison formulation (Morison et al., 1950), as will becommented later in Section 4.

3. Coupling formulations

Having defined the formulation of the equations of motionthat represent the hull and the lines, we can now establish thedifferent formulations for the coupling between the hydrody-namic/motion model of the hull and the FE models of the lines,according to the way these equations are associated.

3.1. Weak coupling formulation

The weak coupling formulation (WkC) is centered on theequations of motion of the hull. In this formulation, the couplingis performed by forces on the right-hand side of Eq. (10). At agiven step of the Runge–Kutta (R–K) time-integration procedurethat solves the equations of motion of the hull (10), one (or more)steps of the time-integration procedure of the equations ofmotion (11) are performed for each line.

That is, at each step of the dynamic analysis of the hull, stepsof nonlinear dynamic analyses of the FE model of each line areindependently performed, with the components of hull motion(obtained from the previous R–K step) prescribed at the top of theline, and also under the action of all environmental loadings. Theresults of these FE analyses are the forces and moments at the topof all lines. These forces are then accumulated in the vectors f andm at the right-hand side of the hull equations (10), added to theenvironmental forces due to wind, wave and current acting onthe hull.

Fig. 1 summarizes the computational implementation for thedynamic analysis in this WkC scheme, and the following subsections

t = 0

Evaluate hull external forces;Store at f1 and m1, Eqs. (12)

Increment line counter i

Steps of integration ofEqs of Motion (11) of Line i

(See Table 1)

Assemble and Invert HullMass Matrix A−1, Eqs. (13)

Solve hull equations (13)by the R-K method

Add Line Forces fromprevious step

Hull Motions

Line Forces

t = t + Δt

i ≥ Num. of Lines ?

Initialize line counter i = 0

Prescribe Hull Motions

no

yes

t ≥ Total time ?no

yes

End

Fig. 1. Computational implementation of the WkC scheme.

B.P. Jacob et al. / Ocean Engineering 55 (2012) 206–218 209

describe the time-integration procedures involved, respectively forthe hull and the lines.

3.1.1. Integration of the equations of motion of the hull

As mentioned before, the solution of the equations of motionof the hull (10) is performed using the fourth-order Runge–Kuttatime integration method. Before applying the R–K method, somemanipulations in Eq. (10) should be performed to take intoaccount that the force and moment vectors (f and m) havecomponents proportional to the body accelerations (e.g. theinertia terms of the Morison formula). These components willgenerate the so-called ‘‘added mass’’ terms, which may present anonlinear variation in time that is not considered in standardfrequency-domain hydrodynamic models, but is taken intoaccount by the hybrid Morison/radiation-diffraction model thatwill be described in Section 4.2).

Details of such manipulations can be found in Jacob (2005) andPaulling (1992); briefly speaking, they consist in separating theterms of f and m that depend on the body accelerations, andtransferring them, affected by ‘‘added mass’’ values, to the left-hand side of the equations. As a result, Eq. (10) can be rewritten inthe following matrix form:

MþA B

C IþD

" #dvdt

dxdt

( )¼

f1

m1�x� ðIxÞ

( )ð12Þ

Therefore, f1 and m1 at the right-hand side of these expres-sions are the terms of forces and moments that depends on theposition, velocity and time, but do not depend on the acceleration.Moreover, the matrix in brackets at the left-hand side corre-sponds to a symmetric global mass matrix, which will be referredas A. Besides the submatrices M and I, it also includes the addedmass matrices A and D, that are updated at every time step, andthe coupling added mass terms B and C. Considering that thismatrix A is symmetric and, in general, nonsingular, Eq. (12) can berewritten as

dvdt

dxdt

( )¼A

�1 f1

m1�x� ðIxÞ

( )ð13Þ

or, partitioning the matrix A�1

dv

dt¼A11f1þA12 m1�x� ðIxÞ½ � ð14Þ

dxdt¼A21f1þA22 m1�x� ðIxÞ½ � ð15Þ

These equations are now in a form appropriate for theapplication of the R–K method, since the right-hand side doesnot contain terms with derivatives of the basic unknowns.

3.1.2. Integration of the equations of motion for each line

For each mooring line and riser, one set of the nonlinear semi-discrete equations of motion (11) is independently assembled and

B.P. Jacob et al. / Ocean Engineering 55 (2012) 206–218210

solved in the time domain using variants of an implicit Newmarktime-integration algorithm (Newmark, 1959), such as the aH-

Newmark or HHT method (Hilber et al., 1977) or the aB-Newmark

method (Wood et al., 1980) that present numerical dissipation toeliminate spurious high-frequency components of the response.Adaptive, automatic time-stepping variation schemes can also beincorporated (Jacob and Ebecken, 1992,1993,1994a).

The implementation of such implicit algorithms is associatedto an iterative technique for the solution of nonlinear systems ofequations—the Newton–Raphson (N–R) method and its variants(Bathe, 1996). This implementation consists in writing a time-discretized form of Eq. (11) at time t¼tnþ1, with €uðtnþ1Þ, _uðtnþ1Þ

and u(tnþ1) replaced by anþ1, vnþ1 and dnþ1, and assuming alinearization in the neighborhood of the displaced positioncorresponding to the instant tnþ1 (Geradin et al., 1983), byexpressing the nonlinear term R(dnþ1) as a truncated Taylorseries that includes KT, the tangent stiffness matrix. The N–Rmethod then operates on the following incremental-iterativeform for the discrete equations of motion:

MaðkÞnþ1þCvðkÞnþ1þKDdðkÞ ¼ Fnþ1�Rðdðk�1Þnþ1 Þ, dðkÞnþ1 ¼ dðk�1Þ

nþ1 þDdðkÞ

ð16Þ

Predictor and corrector expressions for the Newmark opera-tors can be written as follows (Jacob and Ebecken, 1994b), whereb and g are parameters that define a particular member of theNewmark family:

an ¼�1

bDtvn�

1

2b�1

� �an, vn ¼ 1�

gb

� �vnþ 1�

g2b

� �Dtan ð17Þ

aðkÞnþ1 ¼ anþ1

bDt2DdðkÞ, vðkÞnþ1 ¼ vnþ

gbDt

DdðkÞ ð18Þ

The application of these operators to the time-discretizedincremental-iterative Eq. (16) leads to a set of effective systemof linear algebraic equations (Bathe, 1996; Hughes, 1987), to besolved at each N–R iteration of each time step. A general form forthis effective system of equations can be expressed as

Aðk�1Þ

nþ1 DdðkÞ ¼ bðk�1Þ

nþ1 ð19Þ

The effective matrix A is defined as a combination of the mass,damping and stiffness matrices affected by scalar coefficients interms of the Newmark parameters, the time step value, and theRayleigh damping parameters. The vector of ‘‘effective residuals’’b is calculated in terms of the external loads and of elastic,

Table 1Procedures for each time step of the integration of the FE equations of moti

(a) Initial computations for time instant nþ1

1. Initialize total displacements, incremental displacements, and e

dð0Þnþ1 ¼ dn; Ddð0Þ ¼ 0; Rðdð0Þnþ1Þ ¼ R dnð Þ

2. Evaluate external loadings Fnþ1

3. Use element routines to update and assemble global tangent st

Calculate and triangularize effective matrix A of Eq. (19)

4. Estimate predictor values for accels an and vels vn, Eq. (17)

5. Calculate constant term of effective residuals bn

/ effective resid

(b) Iterative Newton–Raphson cycle: k¼1, Nitmax

1. Solve effective system Eq. (19) dðkÞ ¼ A�1¼ b

ðk�1Þ

Update total displacements dðkÞnþ1 ¼ dðk�1Þnþ1 þDdðkÞ .

2. Use element routines to calculate internal forces RðdðkÞnþ1Þ).

3. Calculate effective residuals for next iteration bðkÞ

(Jacob and Eb

4. Check appropriate equilibrium norms (displacements and inter

Terminate iterative cycle if less than specified tolerance values

(c) End of iterative cycle, final computations for time instant nþ1

1. Update accelerations and velocities, Eq. (18).

damping and inertia forces from the previous time step. Adetailed formulation of this implementation of implicit Newmarktime integration algorithms in association with the N–R techni-que, including the particular form of the effective matrix A andthe effective residual vector b may be found in Jacob and Ebecken(1994b).

Table 1 summarizes the procedures for each time step for theintegration of the FE equations of motion (11), based on thecombination of an implicit variant of the Newmark algorithmwith the Newton–Raphson method as described in this section,for each line in the WkC formulation.

3.1.3. Remarks regarding the computational implementation

The fact that the equations of motion of each mooring line andriser are solved individually (as illustrated by the loop on thenumber of lines in Fig. 1) leads to some interesting properties forthis scheme. Firstly, different time steps can be employed for theintegration of the equations of motion of each line, ranging typicallyfrom 0.005 s to 0.01 s. Moreover, the time step required to integratethe hull equations can be significantly larger than these values: thisis because the 6-dof natural periods of moored hulls typically rangefrom 20 s to 200 s (in the case of the horizontal motions), andtherefore the accuracy and stability of the integration procedure forthe equations of motion of the hull is maintained with time stepsranging typically between 0.5 s and 1 s.

This naturally leads to the use of the subcycling technique(Belytschko and Lu, 1993; Daniel, 1997; Hughes, 1987; Smolinski,1996), as described in Jacob et al. (in press) along the presentation ofthe parallel implementations. This technique can be useful even insequential implementations, but is more efficiently exploited whenassociated to parallel implementations. It will be seen that this WkCscheme is particularly adequate for the implementation on compu-ters with parallel architecture: there is no single global FE matrixassembled for all lines, and the global stiffness matrix correspondingto each individual line may be naturally assigned to one node of acluster of processors. Moreover, these individual global matricespresent small bandwidth, contributing to the computationalefficiency.

3.2. Strong coupling formulation

3.2.1. Motivation

Several studies have demonstrated that the WkC formulationpresents accurate results for typical applications of floating

on (Eq. 11) (WkC formulation: Each line/StC formulation: Full system).

lastic forces:

iffness matrix KT;

uals for first iteration bð0Þ

(Jacob and Ebecken, 1994b)

ecken, 1994b)

face/global residuals);

Evaluate hull external hydrodynamic forces:Store at positions of global vector F (Eq. (11))corresponding to the d.o.f.s of the CG node

t = 0

Generation of a single, global FE mesh for all lines, includingthe hull CG node and the connection elements

no

Assemble Hull Mass Matrix A ,Store at positions of global matrix M (Eq. (11))

corresponding to the d.o.f.s of the CG node

Step of Integration ofEqs of Motion (11) of Full System

(See Table 1)

t = t + Δt

t ≥ Total time ?

End

yes

Fig. 2. Computational implementation of the StC scheme.

B.P. Jacob et al. / Ocean Engineering 55 (2012) 206–218 211

production systems (Bahiense et al., 2008; Correa et al., 2008;Lima, 2006; Lima et al., 2006; Senra et al., 2010). However, doubtsregarding the ability of this formulation to provide good results toany class of moored floating system still persist.

As could be observed above in the description of the imple-mentation of the WkC scheme, it may be seen as a hybridexplicit–implicit integration procedure, where the hull is inte-grated by the explicit R–K method, and the lines are integrated byan implicit Newmark algorithm. It can be seen in Fig. 1 that thereis a lag between the calculation of the forces at the top of thelines, and their application at the hull: in a given time step, thehull equations consider values of line forces obtained from theintegration of the equations of motion of the lines, submitted to aprescribed motion in turn obtained at the previous time step. Dueto this lag, the dynamic equilibrium of the system is notrigorously assured; this can hinder the time-marching solutionand generate spurious high-frequency noise.

It can be argued that this issue would be critical only inproblems where the highest frequencies of the response couldeffectively influence the behavior of the hull. Fortunately, this isnot the case of most typical applications of floating productionsystems, and in fact the accuracy of the WkC formulation hasalready been validated by several studies published elsewhere(see for instance Bahiense et al., 2008; Correa et al., 2008; Lima,2006; Lima et al., 2006; Senra et al., 2010). However, this issuemay be important in problems involving the coupling of a floatingbody (integrated by an explicit algorithm) to a stiff structure(integrated implicitly), or even to more conventional offshoreapplications such as the installation of rigid pipelines by a floatingbarge in a S-Lay configuration (Masetti et al., 2004).

Anyway, these are important motivations for the development,implementation and study of a strong coupling scheme (referredas StC) (Bahiense, 2007; Bahiense et al., 2008). As will bedescribed next, this scheme requires modifications in the equa-tions of motion of the hull and the lines, resulting in a differenttime integration procedure.

3.2.2. Simultaneous integration of all equations of motion

While the weak coupling formulation (WkC) was centered onthe equations of motion of the hull (10), the strong couplingformulation (StC) is focused on the equations of motion of thelines (11). The StC scheme consists in incorporating, in a single setof equations of motion, stored in a single global matrix, the FEmatrices (mass, stiffness and damping) of all lines. The 6-dofequations of the hull are now associated to a node of this globalFE model, located at the spatial position corresponding to the c.g.;all calculated hydrodynamic forces acting on the hull are refer-enced to this node.

As a result, instead of one set of Eq. (10) and several sets ofEq. (11) (one for each line), the system is now represented by asingle set of equations of motion (11), totally coupled. This leadsto the assembly and solution of a single global effective system ofalgebraic equations similar to (19), where the effective globalmatrix A now incorporates the dof’s of all elements that representthe lines, plus the 6 dofs of the node that represents the c.g. of thehull. This way, both the structural response of the lines and themotions of the hull are determined simultaneously at the sametime step, avoiding the lag mentioned above that happens in theWkC formulation.

Fig. 2 summarizes the computational implementation for thedynamic analysis in the StC scheme, noting that, before the timeintegration loop, there is a stage where additional connectionelements are automatically generated (as will be described in thenext item). It can also be seen that the set of fully coupled Eq. (11)may be solved by employing the same procedure that was

considered for the solution of an isolated line in the WkCformulation (summarized in Table 1): the combination of animplicit variant of the Newmark time integration algorithm withthe Newton–Raphson method to deal with the nonlinearities.Another solution procedure that is being considered is the use ofextensions of the hybrid Time-Frequency domain solutionmethod based on a Green approach (HTF-GA) that was introducedby (Correa et al., 2010).

3.2.3. Remarks regarding the computational implementation

For the computational implementation of the StC scheme,where a single FE mesh represents all mooring lines, risers andthe hull, the topology of the data structure of the program isaltered to allow the gathering of the matrices of the finiteelements of all lines into a single global matrix, instead of intodifferent global matrices, one for each line.

The StC scheme also requires the transference of motions andforces between the nodes of the top line connections and the nodethat represent the c.g. of the hull. This can be made by differentprocedures. A more elegant solution would be to implementmaster–slave relationships between the dof of these nodes(Jelenic and Crisfield, 1996); however a more expedite andperhaps more effective solution is simply to automatically gen-erate 3-D frame elements connecting the c.g. node to theconnection nodes at the top of the lines. This way, one ‘‘connec-tion element’’ will be generated for each line of the model. Theseelements, illustrated in green on Fig. 3, are not intended torepresent the elastic, inertial of hydrodynamic behavior of thehull. Their role is simply to transfer motions and forces, and sincethe hull is assumed as a rigid body, they will be generated with

Fig. 3. Strong coupling model: connection elements.

B.P. Jacob et al. / Ocean Engineering 55 (2012) 206–218212

stiffness sufficiently high to assume a rigid behavior, and withnegligible mass.

Also, the Morison hydrodynamic coefficients associated tothese connection elements are taken as zero. This is because thehydrodynamic properties of the hull, involved in the calculationof the environmental loads (wave, wind and current), are alreadyrepresented by the same hydrodynamic models considered in theWkC formulation (mentioned in Section 4.2) that leads to thedetermination of the vectors f and m included in the right-handside of Eq. (12). The values of forces and moments that result fromthese loads are now accumulated on the single global vector ofequivalent nodal loads F of Eq. (11), on the positions thatrepresent the dofs of the node that represent the c.g. of the hull.

Finally, the inertial properties of the hull are incorporated inthe 6-dof matrix of coefficients indicated at the left-hand side ofEq. (12). As mentioned before, this matrix incorporates the 3�3diagonal matrix M with the dry mass of the hull, the 3�3 matrix Iwith moments and products of inertia, and also the added-massterms. The coefficients of this 6-dof matrix will be accumulated atthe corresponding positions of the global mass matrix of the FEmesh of Eq. (11).

Differently from the WkC formulation, where a parallel imple-mentation naturally arises from the fact that each line is solvedseparately and therefore the lines can be assigned to the proces-sors, now in the StC formulation parallel implementations are notso evident. Nevertheless, the StC formulation can benefit from theuse of different parallelization techniques, especially domaindecomposition methods such as those studied in Rodrigueset al. (2007). Details of such parallel implementations will bepresented in Jacob et al. (in press).

3.3. Static analysis

The standard procedure for performing dynamic analysis of FPSconsists in two analysis stages. The first comprises a static analysisapplying only the static components of the environmental loadings.The dynamic analysis itself, including with all loadings, is thenrestarted from the system already statically balanced, resulting in adecrease of the transient part of the dynamic response, and there-fore leading to considerable reductions of CPU time.

The static analysis associated to the StC formulation consistssimply in the classical implementation of the Newton–Raphsonmethod for the solution of the single, global nonlinear static equili-brium equations. The WkC formulation, however, requires someadditional considerations since there is no single global set ofequations. Different solution strategies can be considered, includinga preliminary ‘‘uncoupled’’ static analysis phase where the lines areindividually submitted to dead weight only, followed by a ‘‘coupled’’static analysis phase where variants of the N–R or bisection methodscan be independently applied for translational and rotational compo-nents, see for instance Rodrigues and Jacob (2003).

4. Calculation of environmental forces; hydrodynamic models

The developments presented here and in Jacob et al. (in press)are implemented in the in-house, non-commercial Prosimcoupled analysis program (Jacob and Masetti, 1998). This sectionsummarizes the formulation for the calculation of environmentalforces implemented in the program. Only a brief description willbe presented here; more details can be found elsewhere(Chakrabarti, 1987; Hooft, 1982; Jacob, 2005; Morison et al.,1950; Paulling, 1992).

4.1. Environmental loads

The environmental loads include wave and current acting onthe submerged portions of the hull and lines, and wind acting onthe exposed portions of the hull.

The wave description may be defined as a single regular waveof specified height, period, direction and phase; or by two or moresuperimposed wave components, each of different height, periodand direction. Irregular seastates can also be considered, withunidirectional or bi-directional Pierson–Moskowitz or Jonswapspectra; the seastate is represented in the time domain by anensemble of regular wave components generated from the wavespectrum, at equal intervals of wave period with randomlydistributed phases. The water velocities, accelerations and pres-sures in a point are obtained by the sum of the values calculatedfor each wave component. The Airy linear wave theory(Chakrabarti, 1987) is employed for the calculation of velocitiesand accelerations of fluid particles.

A current profile may be input, as a table of horizontal velocityversus depth. This profile is assumed to move with the wavesurface, i.e., during computation, the depth for interpolationwithin the current table is measured below the instantaneouswave surface. The interpolated value of the current velocity isadded vectorially to the wave velocity.

Wind forces are computed at each time step by interpolatingfrom tables of wind drag coefficients, functions of the wind angleof attack. These coefficients are usually determined from windtunnel tests. The wind velocity can be considered as a static,constant value; optionally a time varying term determined from arandom wind spectrum such as the API spectrum (API, 1993) canbe added.

4.2. Hydrodynamic models

After the velocities and accelerations of fluid particles arecalculated, the forces acting over the submerged portions of thehull and lines are calculated, using different hydrodynamicmodels briefly summarized in the text that follows.

4.2.1. Lines: classical Morison model

Forces due to wave and current loads acting over the sub-merged segments of the lines are calculated using extensionsof the classical Morison formulation for cylindrical members(Chakrabarti, 1987; Jacob, 2005; Morison et al., 1950).This formulation assumes that the incident waves are notdisturbed by presence of the structure, and therefore is appro-priate for structures with small diameters relative to the wavelengths: mooring lines, risers, and framed members of fixedplatforms such as jackets or compliant towers. The resultingforces on the lines are incorporated on the vector F at the right-hand side of Eq. (11).

Table 2Hull properties.

Length (m) 88.40 Coordinates of CG (m) x 0.4034

Breadth (m) 88.40 y 0.5575

Height (m) 40.50 z 24.00

Operational draft (m) 26.00 Radii of gyration (m) roll, pitch 35.00

Diameter of pontoons (m) 16.31 yaw 37.00

Diameter of columns (m) 20.74 Displacement (t) 57,818.00

Fig. 4. Visualization of the numerical model of the hull.

Table 3Properties of the mooring line segments.

Chain Polyester

Nominal diameter (m) 0.21 0.222

Axial stiffness EA (kN) 1100000.00 480712.00

Weight in air (kN/m) 2.8253 0.3169

Weight in water (kN/m) 2.4580 0.0833

B.P. Jacob et al. / Ocean Engineering 55 (2012) 206–218 213

4.2.2. Hull: classical radiation-diffraction model

For large floating bodies, the presence of the body causessignificant changes on the fluid flow, generating diffraction,interference and radiation effects. This is the case, for instance,of platforms based on ships such as FPSOs. For such cases, a fullradiation/diffraction model based on the Potential Theory isavailable (Chakrabarti, 1987; Jacob, 2005). On the WkC formula-tion, the resulting forces on the hull are incorporated on thevectors f1 and m1 at the right-hand side of the hull Eq. (12); onthe StC formulation, they are incorporated on the vector F at theright-hand side of Eq. (11) for the full system.

4.2.3. Hull: hybrid Morison/radiation-diffraction model

Finally, the program incorporates a hybrid Morison/radiation-diffraction model (Hooft, 1982; Jacob, 2005; Paulling, 1992) thatcombines positive characteristics of the Morison and the classicalradiation/diffraction models. This hybrid model allows the repre-sentation of diffraction and radiation effects that occur in large-diameter cylindrical members of Tension-Leg, Semi-Submersibleor Spar platforms (or framed members in general that can berepresented by equivalent cylindrical members), by combiningthe following hydrodynamic forces: (a) the 1st-order forces fromthe Morison formula, including drag and added-mass inertiaforces; (b) the 1st-order Froude–Krylov forces; and (c) forcesfrom the Potential Theory, including 2nd-order wave diffractioneffects (generating mean and slow drift forces) and also radiationeffects (generating frequency-dependent radiation damping),both calculated from coefficients previously determined by ahydrodynamic analysis program such as Wamit (Lee, 1998).

5. Case studies

5.1. Model description

The behavior of the different coupling formulations is nowassessed by comparing the results of their application to a typicalfloating production system. The FPS is comprised by a semisubmer-sible platform similar to those employed for deep waters in theCampos Basin, Southeastern Brazil, installed at a water depth of1800 m. The platform is moored by 16 lines in a taut-leg configura-tion, with four lines at each corner. The production system com-prises 47 flexible lines, including risers and umbilicals.

The main objective of the studies presented in this section is tocompare the relative computational performance of the couplingformulations, since, as mentioned in Section 3.2, the accuracy ofthe WkC formulation for deep-water moored floating productionsystems has already been validated by several studies publishedelsewhere (see for instance Bahiense et al., 2008; Correa et al.,2008; Lima, 2006; Lima et al., 2006; Senra et al., 2010).

5.1.1. Characteristics of the hull

The hull of the platform comprises four columns and twopontoons. Table 2 presents the main properties of the hull. In thistable, the coordinates of the CG are referred to the ‘‘structuralsystem’’ of the platform, where the horizontal xy plane is located the

base of the pontoons, the x axis is aligned with the East direction,and the origin is at the midsection of the hull. On the other hand, theradii of gyration are defined in another reference system, parallel tothat structural system, but with origin at the CG.

For this application, the hull is represented by the hybridhydrodynamic model described in Section 4.2. Fig. 4 presents thevisualization of the array of cylindrical members of this model,with equivalent properties devised to adequately represent thedisplaced volume and the hydrodynamic characteristics of theactual members. The values for these equivalent properties,including the hydrodynamic drag and inertia coefficients, arecalculated following the recommendations of the DNV code(DNV/POSMOOR, 1989). As mentioned before, the hybrid modelis complemented by slow drift and potential damping coeffi-cients, obtained by performing a hydrodynamic analysis with theWamit program.

5.1.2. Characteristics of the lines

The mooring lines are comprised by an intermediate polyestersegment with length of 2500 m, and by top and bottom chainsegments with lengths of 200 m each. The properties of thesesegments are presented in Table 3.

Depending on the type of results desired for the coupledanalysis (hull motion response and/or structural response of thelines), the risers could be modeled either by truss or frameelements; moreover, different levels of mesh refinements couldbe employed. In the context of the studies presented here, where

Fig. 5. Visualization of platform and full system of lines.

Fig. 6. Perspective and top view of the full system (63 lines).

Fig. 7. Perspective and top view of the reduced system with 30 lines.

Fig. 8. Perspective and top view of the reduced system with 16 lines.

Fig. 9. Perspective and top view of the reduced system with 8 lines.

0Time (s)

0

10

20

30

40

Dis

plac

emen

t (m

)

SurgeWkCStC

1200800400

Fig. 10. Surge motion.

B.P. Jacob et al. / Ocean Engineering 55 (2012) 206–218214

the main objective is to compare the relative computationalperformance of the different coupling formulations (regardlessof the type and refinement of the Finite Element mesh), all linesare modeled with truss elements, and with not very refinedmeshes. Each line employs a mesh gradation where the smallerelements are on the top and TDP segments (with element lengthsof about 5 m), and the intermediate, suspended segments havelarger elements (with lengths of about 50 m).

5.2. Coupled models

Since the StC formulation stores the FE matrices of all lines in asingle global matrix, it is reasonable to expect its computationalperformance to be more sensitive to the total number of lines ofthe system. Therefore, in order to assess the relative computa-tional performance of the coupling formulations, different modelsare analyzed: for a full system, including all 63 lines (47 risers andumbilicals, and 16 mooring lines), and three other ‘‘reduced’’models with only part of the lines: respectively 30, 16 and 8 lines.Fig. 5 presents a 3D view of the full system, with the hull barelyvisible due to the scale of the model; the mooring lines arerepresented in green, and the risers/umbilicals are represented inblue. Then follows Figs. 6–9 illustrating the models with 63, 30,16 and 8 lines.

5.3. Loading data, analysis parameters

Wave and current loadings, aligned with the structural x axisof the platform, are considered for the analyses. The wave loadsare represented by an irregular seastate defined by the Jonswapspectral model, with significant height Hs¼7.16 m and peakperiod Tp¼14.78 s. Current loads are represented by a triangularprofile of velocities, with the surface velocity equal to 1.23 m/s.Results for other loading cases are presented in Bahiense (2007).

The best strategy would be to firstly perform a static analysisunder the action of the steady loads only (dead weight andcurrent), and then to start the dynamic analysis from the static

equilibrium configuration obtained. However, since the mainobjective of the analyses presented here is to compare theformulations in terms of computational performance in thedynamic analyses, static analyses are not performed and thedynamic analyses are executed considering a ramp time of 700 suntil the full application of the loads. The total simulation time is1200 s, and the time step for the integration of the equations ofmotion is 0.01 s.

5.4. Results

5.4.1. Hull motions and line tensions

Figs. 10–13 compares the results of the WkC and StC formula-tions for the full system with 63 lines, in terms of time series ofsurge and heave motions, and tensions at the top of a ooring line.

600Time (s)

18

22

26

30

34

Dis

plac

emen

t (m

)

SurgeWkCStC

1000900800700

Fig. 11. Surge motion (detail).

600Time (s)

-2

-1

0

1

2

Dis

plac

emen

t (m

)

HeaveWkCStC

1000900800700

Fig. 12. Heave motion (detail).

600Time (s)

1200

1400

1600

1800

2000

2200

Tens

ion

(kN

)

Top TensionWkCStC

12001000800

Fig. 13. Tension at the top of a mooring line (detail).

Table 4CPU times.

CPU time (s) Ratio StC/WkC

No. of lines StC WkC

08 13.85 23.27 0.60x

16 55.20 45.58 1.21x

30 181.35 72.28 2.51x

63 1207.28 148.63 8.12x

B.P. Jacob et al. / Ocean Engineering 55 (2012) 206–218 215

Of course, on actual design activities, the results of time-domain analysis with irregular waves should not be presentedonly in terms of time series; they must be complemented bymotion and tension statistics. The statistical values could becalculated from a given realization, provided the total simulationtime is enough to achieve the desired confidence interval or errorestimate for the parameters of the response. According to Garrettet al. (2002a), a way to get a measure of variability fromsimulation to simulation in order to assess if the total time isenough is to run repeated simulations (or replicates), withdifferent realizations for the same loading condition, and examinethe distribution of response statistics over the replicates; five ormore replicates allow one to estimate the standard deviation ofstatistical measures from each of the samples, with error esti-mates calculated by Student’s t and the given number of repli-cates. Requirements for the simulation parameters (e.g. frequencyspacing to represent stochastic loadings, time of transient part ofthe dynamic response, total simulaton time and number ofsimulation replicates) should be determined as presented byGarrett et al. (2002a).

Here, it should be recalled that the analyses presented in thiscase study are not intended to provide values for design para-meters, to be compared with the limit values of correspondingdesign criteria. The goal is merely to compare the relativeperformance of the WkC and StC formulations, in terms of CPUtimes (as presented next), and of the time series that result forone given realization. In any case, this need of performing severalreplications for the same loading condition also serves as amotivation for the development of more efficient solution

procedures, including the parallel implementations associated tothese formulations that are described in Jacob et al. (in press).

A very good agreement can be observed between the results. Itis expected that the StC formulation would naturally lead toresults more accurate than those obtained by the WkC formula-tion, since the former is conceptually more rigorous. Therefore,this agreement can also be seen as a contribution to the validationof the WkC formulation for this type of application, confirmingprevious studies that also included comparison with experimen-tal results (Bahiense et al., 2008; Correa et al., 2008; Lima, 2006;Lima et al., 2006; Senra et al., 2010).

5.4.2. Computational performance

Now regarding the computational performance of the StC andWkC formulations, Table 4 presents the CPU times for theanalyses of the different coupled models described in Section5.2 (the full model with 63 lines, and the reduced models with 30,16 and 8 lines).

Recalling that both formulations employ the Newton–Raphsoniterative procedure for the solution of the nonlinear systems ofeffective equations similar to (16), in all cases the strong couplingStC formulation required fewer iterations for convergence withthe single set of totally coupled equations of motion for all lines,while the weak coupling WkC formulation leads to a largernumber of iterations to solve the sets of equations for each line.

Now considering the solution of the effective systems of linearequations similar to (19) that arises at each N–R iteration, itshould be recalled that the solution of such systems by a directtechnique such as the Gauss method involves two main steps:(1) factorization of the matrix, and (2) Solution of the resultingtriangular system by forward or backward substitution. Consider-ing ‘n’ as the number of equations, for full (not sparse) symmetricmatrices step (1) requires about n3/3 multiplications and a similarnumber of additions, and step (2) requires about n2 multiplica-tions (Heath, 2002). For large matrices as is usual on practicalapplications, the cost of the factorization phase is dominant, andtherefore it can be stated that the solution of the systems of

B.P. Jacob et al. / Ocean Engineering 55 (2012) 206–218216

equations requires O(n3) work; that is, the cost of the solutionincreases cubically with the number of equations.

On the other hand it is well known that the FE matrices areoften sparse, with entries concentrated near the main diagonal.Effective implementations of the Gauss method are thenemployed to deal with such sparse matrices; considering forinstance a banded matrix with bandwidth b, such implementa-tions require only O(b2n) work (Heath, 2002), which representssubstantial savings over full systems if b5n.

Considering the WkC formulation, an increase on the numberof lines would represent an increase on the number of indepen-dent systems of equations to be solved, and therefore if no otherfactors are considered the CPU time would increase only linearly.However, for the StC formulation an increase on the number oflines (all of them connected to the single node representing theplatform) would represent not only an increase on the number ofequations n of the single effective matrix, but also an increase onits bandwith b, and therefore the CPU costs would still increaseexponentially (although less than cubically) even with the use ofa sparse solver and a nodal reordering procedure. Therefore theStC formulation tends to require more CPU time to solve thelarger, single set of equations for all lines.

These considerations are reflected in Table 4 where it can beobserved that, for the smaller model (with only 8 lines), the StCformulation is more efficient (spending only 60% of the CPU timerequired by the WkC formulation). A similar behavior was alsoobserved in Bahiense (2007) for other floating offshore systemswith a relatively small number of lines. This is due to the fact thatin these cases with fewer lines (and/or represented by a relativelysmaller number of Finite Elements), the computational perfor-mance is dominated by the cost of the additional N–R iterations.

On the other hand, for the models with more lines (and moreequations), the cost of the solution of the effective Eq. (19) nowdominates. Therefore, the WkC formulation is progressively moreefficient for the models with 16 lines or more, since the CPU timerequired by the StC formulation increases exponentially with thenumber of lines. It should be observed that, while this overallbehavior may be representative of typical floating systems,probably this particular threshold of 16 lines and the particularStC/WkC ratio reported on Table 4 cannot be generalized to othersystems: there are other factors involved that depend on eachparticular model, mainly differences on mesh refinement andnumber of equations for each individual line.

6. Final remarks

As mentioned before, the objective of this paper is to present,assess and compare two different schemes to couple the equa-tions of motion of the hull and lines of floating productionsystems, in terms of accuracy and computational performance.

Regarding the accuracy, it can be observed that the StCformulation is conceptually more rigorous, and therefore it isnaturally expected to provide results more accurate than thoseobtained by the WkC formulation. This way, the fact that bothformulations presented similar results for the application pre-sented here helps to validate the WkC formulation, confirmingprevious studies that also included comparisons with experimen-tal results (Bahiense et al., 2008; Correa et al., 2008; Lima, 2006;Lima et al., 2006; Senra et al., 2010).

It is important to remark that this conclusion strictly appliesonly to the particular type of application considered here, that is,large deep-water moored FPS based on semisubmersible plat-forms or ships. Therefore, the fact that the WkC formulation hasled to accurate results for such applications does not necessarilyimply that it can be adequate for use with other offshore systems.

This issue has already been mentioned in Section 3.2 of this work,associated for instance to the simulation of the installation ofrigid pipelines by a floating barge in a S-Lay configuration(Masetti et al., 2004), or the installation of large subsea equip-ments by floating vessels. For such applications, further studieswill be performed to assess not only the accuracy but also thecomputational performance of both formulations.

Regarding computational performance, taking the results pre-sented for the four different platform models (with 63, 30, 16 and8 lines), it could be observed that the simultaneous integration ofall equations of motion (corresponding to the hull and all lines) bythe StC formulation contributes with the efficiency of the iterativenonlinear solution procedure by the Newton–Raphson method.Thus a fewer number of N–R iterations are required for conver-gence, leading to the solution of fewer effective systems of linearEq. (19) along the total simulation time, and contributing to theperformance of the StC formulation.

It could also be observed that the performance of the StCformulation is strongly dependent on the number of degrees offreedom of the Finite Element meshes that represent the lines.This defines the size of the systems of Eq. (19) to be solved at eachN–R iteration, at each time step of the time integration procedure.Therefore, models with a higher number of lines, and/or withlines discretized by more refined meshes, require the solution oflarger systems of equations. The computational costs involved inthe solution of such systems by a direct technique such as theGauss method increase exponentially with the number of equa-tions; therefore such costs tend to dominate the CPU time,and cannot be compensated by the reduction on the number ofN–R iterations, thus penalizing the performance of the StCformulation.

This way, the StC formulation may comprise a more efficientsolution procedure for FPS with a relatively smaller number oflines, as in the case of the system with only eight lines presentedhere, and other similar applications with shorter/fewer lines or inshallower waters such as the presented in Bahiense (2007) andBahiense et al. (2008). On the other hand, for systems with morelines (or discretized by more refined meshes) the WkC formula-tion has been shown to be more efficient, as in the cases of thesystems with 16 or more lines presented here.

It is important to recall that the results presented in this papercorrespond to a sequential implementation of the coupling for-mulations. As mentioned before during the presentation of theformulations, further studies regarding their association withdomain decomposition strategies and the implementation oncomputers with parallel architecture are presented in the com-panion paper (Jacob et al., in press).

Acknowledgments

The authors would like to acknowledge the active support ofPetrobras (the Brazilian state oil company). Petrobras have beenboosting research activities in this area, and encouraging the useof innovative numerical tools in real-life design situations.

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