coupled dynamic modeling of a moored floating platform

1

P

COUPLED DYNAMIC MODELING OF A MOORED

FLOATING PLATFORM WITH RISERS

Xiaoning Jing

Technip

Houston, Texas 77079, USA

Qi Xu

Technip


William C. Webster

Civil & Environmental Engineering

The University of California, Berkeley

California 94720-1710, USA

Kostas Lambrakos

Technip


Abstract

A time domain coupled analysis capability has been

developed to model the dynamic responses of an

integrated floating system incorporating the interactions

between vessel, moorings and risers in a marine

environment. Hydrodynamic responses of the vessel

allowing diffraction, radiation damping and wave drift

forces on panelized bodies in addition to loads on

Morison members, are modeled using the well-

established program, MLTSIM. RodDyn, a finite

element rod dynamics program, based on Garrett’s rod

theory, is an efficient program to model the nonlinear

dynamics of risers and moorings. The coupled MLTSIM-

RodDyn suite integrates the nonlinear motions and

structural analysis capabilities of RodDyn with the

extensive hydrodynamic simulation capabilities of

MLTSIM. With the fully-coupled dynamic analysis, the

integrated system can be analyzed consistently. That is,

the forces and moments applied by the rods to the

platform are concurrent with the motions imposed on the

rods by the platform at their multiple contact points. An

asynchronous coupling of these two programs has been

developed which allows for a fast simulation of this very

complex problem. A worked example showing the

nonlinear coupled analysis, is elaborated with

systematical comparison with uncoupled analysis.

Introduction

The focus of this paper is on techniques to improve the

computational efficiency of a consistent simulation of

offshore platform motions where the platform is attached

to a multiplicity of rods1 and where these rods are

modeled using a modern nonlinear finite-element

scheme. By consistent we mean that the motions of the

platform are coupled directly to the motions of the rods

and that the forces created by the platform’s excitation of

the rods and the excitation caused by waves and current

are coupled directly and concurrently into the platform

simulation. Computational efficiency is a significant

issue here since typical programs to compute platform

motions and typical programs to compute rod dynamics

are both computationally intensive. Coupling of these

two programs can easily lead to simulation times that

greatly exceed real time and this limits the utility of these

programs in the design of sophisticated new platforms.

Before discussing the proposed techniques for improving

the computational efficiency of a consistent coupled

platform/rod dynamics simulation, it is relevant to

discuss why such a complicated program is necessary.

Typical platform dynamics programs generally treat the

rods that connect to them in a simple way. Some

programs treat the rods as simple springs or spring

dampers; some treat them as static catenaries where the

forces exerted on the platform are assumed equivalent to

that imposed by a catenary whose top coordinates are the

same as the instantaneous values imposed by the

platform motions; still others may use tabular look-up

tables derived from simple static models. The

justification for these simplifications is that during

normal operations the forces exerted by these rods on the

platform are insignificant compared to the forces exerted

by the sea state. However, during extreme operations,

1 In mechanics a rod is a slender structural element that may have

bending rigidity. In this paper we will use the term rod as a collective descriptor of mooring lines, risers, tendons, etc.

Proceedings of the ASME 2011 30th International Conference on Ocean, Offshore and Arctic Engineering OMAE2011

June 19-24, 2011, Rotterdam, The Netherlands

OMAE2011-49553

Copyright © 2011 by ASME

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2

these rods do play a more important role. Slow drift

oscillations that can endanger the mooring system or any

SCRs, depend on the forces exerted by the rods on the

platform. Roll motions of the platform (especially on

drill ships) are much more responsive to forces from the

mooring lines than other motions, particularly in extreme

situations. It is in these extreme situations that the

simplified models of rod behavior are the most suspect.

In addition, as drilling depths get larger and larger, both

the mass of the collection of risers, mooring lines and the

like have become comparable to the platform mass and

the effects of their damping and inertia have become

more important. Much of the structural design of

platforms and their mooring systems depends on the

extreme situations rather than on operational situations.

As a result, it seems prudent to have computational tools

on hand to investigate and evaluate the extreme

situations, and having a consistent coupled platform/rod

dynamics program is one of these. For stochastic analysis

responding to current, wind and wave loads, the efficient

and accurate modeling is needed to produce the results

for design of floaters.

The computational efficiency of a coupled platform/rod

dynamics will depend on the individual efficiencies of

the component platform dynamics program and the rod

dynamics program, and will also depend on how these

two programs are coupled. We shall first discuss each of

these programs separately and then discuss how we

propose that they be coupled.

Rod Dynamics Simulation

A rod is a long, slender structure with perhaps bending

rigidity that is simply-connected (no branches). A rod is

not a rigid body and its motion in time may vary in a

complex way along the length of the rod and in time.

There are many finite-element programs for the

computation of the dynamics of a rod in use in the

offshore industry. They fall into two categories: general

finite element programs and rod-specific programs.

General programs that can be used for analysis of three-

dimensional, multi-connected structures are generally not

very efficient for the analysis of rods because these

programs do not take advantage of the narrow-banded

matrices produced by rod-specific programs. In our work

we use a rod-specific program, RodDyn, that is based on

the formulation developed by Garrett (1982). This

algorithm is formulated in global inertial coordinates and

as a result avoids the computation of a plethora of

trigonometric transformations involved with programs

where the coordinate system is formulated in the

element’s own coordinate system. As a result, Garrett’s

method is very efficient and is general enough to treat all

of the different types of rods used in offshore

engineering. In this formulation, each finite element has

15 time-varying coefficients. The matching that assures

the connectivity between neighboring elements reduces

this number to 8 new variables for each new element. A

typical representation for a rod connected to a platform

consists of about 30-50 finite elements corresponding to

about 250-400 variables which must be updated at each

simulation time step. RodDyn uses a one-step Adams-

Moulton integration scheme, but other similar programs

use Adams-Bashforth or higher-order schemes.

With rod-specific finite element programs, the computa-

tional complexity grows approximately linearly with the

number of elements and to a certain extent on the

number and type of constraints imposed on the rod by

the platform, on the sea floor or at an anchor or template.

A rod may contact the platform in many locations and

each contact point is represented by a separate constraint.

The rod imposes forces and moments on the platform at

each constraint. Typical constraints may include ball-

joint constraints (representing fixed connections like

fairleads or tie-offs), spring-dampers (representing ram

tensioners), or contact constraints (representing keel

guides), slider constraints (representing centralizers), etc.

Specification in inertial space of the instantaneous state

(location and orientation) of all of the constraints2 as a

function of time is sufficient to determine the interaction

forces between the rod and the platform.

If the rod is exposed to a sea state then the increase in

computational effort can be substantial. The wave-

induced forces affect the motions of the rod and the

forces transmitted by the rod to the platform. Typical

representations of a sea state may involve as many as

100 Airy wave components super-imposed to simulate a

random, directional sea state. Determination of the

particle velocities and accelerations at each of the 50 -

200 control points on a rod representation will then

involve thousands of evaluations of sines, cosines and

exponentials. These evaluations burden the overall

computational efficiency to the point where the

computation can be slowed by a factor of two or more.

An approach to mostly eliminate this burden is outlined

in a subsequent section.

Platform Motions Simulation

Through the years various computer programs have been

developed for the simulation of the motions of offshore

platforms. Platform motions programs are founded on

the Newton-Euler equations of motion of a rigid body

and the complexity arises from estimating the forces on

the platform as a result of its interaction with the sea

state around it. The underwater and near surface

configurations of typical offshore structures are

2 The state of the constraint is meant the position, orientation and

velocity of the reference point of the constraint in inertial coordinates.

For some constraints, for instance a ball joint constraint, this may correspond to the corresponding location and velocity of the rod at that

attachment; for other constraints, for instance a spring/damper, this may

correspond to the location and velocity where the force exerted on the rod is zero. The state of any constraint at any instant in time is simply a

Galilean transformation of the state of the platform at that instant.



3

composed of a large number of structural and non-

structural elements. The simulation of the forces acting

on these elements in response to the sea state requires

making several simplifying assumptions about the flow.

For instance, it is often assumed that the forces caused by

the motion of a slender tubular elements is equivalent to

that caused on the same element moving alone in an

infinite fluid. Further, it is often assumed that these

elements are “Morison elements” where the forces

created can be calculated using fixed added mass, cross-

flow drag and longitudinal drag coefficients. Larger

components, such as columns, hulls, etc. are often treated

using a diffraction model based on various panel-method

formulations and the assumption of inviscid, irrotational

flow. Each particular simulation program has its own

way of treating nonlinear effects and there is no general

agreement of which effects to include and which to

exclude. In any case, the major platform dynamics

programs are very sophisticated, have been refined and

optimized through the years and have a great deal of

computational complexity. The most used platform

simulation programs have been calibrated and adjusted

so that they reliably predict the motions measured during

model tests and some have been carefully compared with

full-scale platforms. In the discussion below, we will not

discuss the platform dynamics programs themselves.

Solution to the platform dynamics programs involves the

integration of the equations of motion subject to the

forces on the platform. The “state” of the platform will

be represented by a 12 component vector, ( )s t given by:

( ), ( ), ( ), ( ), ( ) , ( ),( )

( ), ( ), ( ), ( ), ( ), ( )

o o o o o o

o o o o o o

x t y t z t t t tt

u t v t w t p t q t r t

ϕ ψ θ =

s (1)

where ( ), ( ), ( )o o ox t y t z t are the three translational

motions along the x, y and z axes. ( ), ( ) , ( )o o ot t tϕ ψ θ are the three Euler angles of

the platform about the x, y and z axes. ( ), ( ), ( )o o ou t v t w t = ( ), ( ), ( )o o ox t y t z t& & & are the three

translational velocities of the platform. ( ), ( ), ( )o o op t q t r t = ( ), ( ) , ( )o o ot t tϕ ψ θ&& & are the

three rotational velocities of the platform.

It is typical to assume that the hydrodynamic forces due

to platform accelerations (either translator or angular) are

linearly proportional to the instantaneous acceleration3.

As a result, the platform accelerations multiplied by the

virtual mass matrix yields both the mechanical and

hydrodynamic acceleration forces.

3 This assumption can be avoided, but the integration of the

equations of motion will then involve convolution integrals and

even greater complexity.

With this assumption, the equations of motion can be

expressed as a set of 12 ordinary quasi-linear differential

equations in terms of the platform state:

1 2 3 4 5 6

( ), ( ), ( ), ( ), ( ), ( ),( )

( ), ( ), ( ), ( ), ( ), ( )

o o o o o ou t v t w t p t q t r tt

h t h t h t h t h t h t

=

s&

(2)

where

( )nh t are terms formed by the forces or moments

on the platform in the thn direction

(including those imposed by the attached

rods) multiplied by the inverse of the current

virtual mass matrix4. These terms are

usually highly nonlinear in the variables in

the state vector, but do not include any

acceleration terms. n =1,2,3 correspond to the surge, heave and

sway motions and n =4,5,6 correspond to

the roll, yaw and pitch motions.

Numerical solutions to equations like (2) above can be

solved with a variety of schemes. A popular scheme,

and one that will be used here, is the Runge-Kutta (RK)

fourth-order accurate scheme. In this scheme we

suppose that at time ot , the state (0) ( )ots is known along

with the time derivative of the state, (0) ( )ots& . We wish to

determine the state at some time ot t+∆ later.

According to the RK scheme, this is accomplished in 4

steps, as outlined in (3) below.

Step 1.

Estimate the state at time 2

o

tt

∆+ by

(1) (0) (0)( ) ( ) ( )2 2

o o o

t tt t t

∆ ∆+ = +s s s&

From (1) ( )

2o

tt

∆+s , determine the values of

(1) ( )2

n o

th t

∆+

and from (2), (1) ( )

2o

tt

∆+s&

Step 2.

Estimate the state at time 2

o

tt

∆+ by

(2) (0) (1)( ) ( ) ( )2 2

o o o

t tt t t

∆ ∆+ = +s s s&

From (2) ( )

2o

tt

∆+s , determine the values of

(2) ( )2

n o

th t

∆+

and from (2), (2) ( )

2o

tt

∆+s&

4 The virtual mass matrix can vary in time depending on the

immersion of various components of the platform. To correctly

account for this change, additional forces proportional to the

rate of change of added mass matrix times the velocity vectors

must be incorporated in the equations of motion. These will

not be discussed further here.



4

Step 3.

Estimate the state at time ot t+ ∆ by

(3) (0) (2)( ) ( ) ( )o o ot t t t t+ ∆ = + ∆s s s&

From (3) ( )ot t+ ∆s , determine the values of (3) ( )

n oh t t+ ∆

and from (2), (3) ( )ot t+ ∆s&

Step 4.

Make final estimate of the state at time ot t+ ∆ :

(0) (0)

(0) (1) (2) (3)

( ) ( )

1{ 2 2 }( )

6

o o

o

t t t

t t

+∆ =

+ ∆ + + +

s s

s s s s& & & &

(3)

From (0) ( )ot t+ ∆s , determine the values of (0) ( )

n oh t t+ ∆

and from (2) the final time derivative of the state vector, (0) ( )

ot t+ ∆s& .

With (0) ( )ot t+ ∆s and (0) ( )

ot t+ ∆s& as starting points for

the next time step, the procedure is repeated again and

again until the simulation is finished. We should note

that the implicit assumption is made here that the values

of ( )nh t vary slowly in the interval { , }o ot t t+ ∆ . It should

also be noted that these values need not be known at all

times, but rather need to be evaluated only at

, / 2,o o ot t t t t+∆ + ∆ . This fact will be used in our later

discussion of the rod dynamics. The complexity of this

process is clearly tied to the evaluation of the six

functions ( )nh t and this will depend on both the internal

algorithms used in the platform dynamics program and

the complexity of the platform itself. Following the RK

process, these evaluations must be made four times for

each time step.

Time scaling and coupling considerations

There are three time scales of importance in the coupled

platform/rod dynamics problem. The first is a time-scale

associated with the sea state. Typical sea states of

interest for the coupled problem have modal periods of

the order of 10-15 seconds. Different types of platforms

have very different natural periods. Heave periods can

vary from 10 seconds for drill ships to over 30 seconds

for semisubmersibles. Other restored motions such as

roll and pitch may have longer periods. For stability, an

integration scheme like the RK scheme mentioned above

must use a time step significantly smaller than any of

these typical periods. Since the periods of typical waves

and platforms are comparable, when used as stand-alone

platform dynamics program it is common to use time

steps in the order of 0.5 - 1.0 sec. for the integrations.

However, the situation for rods is quite different. Rods

have many possible oscillation modes. One is designated

as transverse waves, deformations normal to the axis of

the rod. The celerity of these waves depends on the mass

of the rod, its local tension and its local bending rigidity.

This velocity can be quite high. Longitudinal stress

waves which travel up and down the rod have a celerity

equal to the speed of sound in the rod material and this

speed can be very high. In this case, the period of the

first longitudinal mode of a steel TLP tendon of 3,000m

in length is about 1 sec. and the prominent higher modes

have correspondingly smaller periods. The situation with

the transverse waves is perhaps slightly less severe, but

because it is common for these to have higher modes,

they too have characteristic periods in the order of 1 sec

or less. In order for any numerical integration scheme to

be stable, it is necessary to use a time step that is much

smaller than any significant resonant period. As a result,

a typical choice for modern nonlinear finite-element rod

model when run as a stand-alone program is a time step

in the order of 0.01 -0.05 seconds, or 25-50 times smaller

than that for typical platform integration schemes.

The ideal computation time for the computation of a

coupled platform/rod dynamics would be the time

required to run the platform and the rods with their

corresponding stand-alone programs. The effort

involved in the per-time-step computation of the

platform dynamics for a typical complex platform is

perhaps 20-100 times that of a single rod with say 30 –

50 elements and with a sea state composed of few waves.

Of course, this rough comparison depends very much on

the exact details of each of these programs, details of the

platform and rod configurations. In other words, the

time required for the execution of one time step of a

platform dynamics program is approximately the same as

one time step of 20 – 50 rods, which is in fact the typical

number of rods attached to a platform. If a coupled

platform-rod dynamics model is constructed so that the

platform dynamics program and the rod dynamics run

synchronously (that is the time steps are chosen to be the

smallest required time step, i.e., that needed for the rod

dynamics) then the computation time will be

approximately double the ideal time. If there are fewer

rods attached to the platform, the ratio will be even

higher. What is proposed below is the development of

an asynchronous scheme where the coupled platform/rod

dynamics can achieve the ideal minimum computation

time.

Asynchronous coupling of platform dynamics

and rod dynamics

The form of the RK process for integrating the platform

equations of motions lends itself to the possibility of an

asynchronous coupling between the platform motions

program and the rod dynamics program. As shown in

Eqs. (2) and (3), what is needed for integration of the

rods into the platform integration are the forces at

, / 2,o o ot t t t t+∆ + ∆ only. However, to obtain these

forces, integration of the rod dynamics requires the



5

motion of the platform corresponding to all of the small

time steps between ot and / 2ot t+∆ and ot and ot t+∆ .

Consider first the motions of the platform in the x

direction. In the first step in Eq. (3), we know

( ), ( )o ox t x t& from the state, (0) ( )ots . We also know

( / 2)ox t t+∆ , ( / 2)ox t t+∆& from the RK estimate of the

state at / 2ot t+∆ given by (1) ( / 2)ot t+ ∆s . If the time

step for the platform motions is taken appropriately, that

is, small compared to the wave and platform periods,

then the motion between ot and / 2ot t+∆ can justifiably

assumed to be smooth and continuous. In this

circumstance, a cubic spline algorithm will be used to

interpolate the motions:

2 3

0 1 2 3( )x t a a t a t a tδ δ δ δ= + + + ,

0 / 2t tδ≤ < ∆ (4)

where

ot t tδ = −

0 ( )oa x t= (5)

1 ( )oa x t= &

2

2 3[ ( / 2) ( ) ] / ( / 2)

[2 ( ) ( / 2) ] / ( / 2)

o o

o o

a x t t x t t

x t x t t t

= +∆ − ∆

− + +∆ ∆& &

3

3

2

2[ ( / 2) ( ) ] / ( / 2)

[ ( ) ( / 2) ] / ( / 2)

o o

o o

a x t t x t t

x t x t t t

= − + ∆ − ∆

+ + + ∆ ∆& &

Eq. (4) represents a smooth curve that passes through the

points ( )ox t and ( / 2)ox t t+∆ and has velocities equal to

( )ox t& and ( / 2)ox t t+∆& at the ends of the intervals. It is

hard to conceive that the platform motion in the x

direction can vary much from this path within this

interval. A cartoon of the interpolation scheme is shown

in Figure 1 below.

timet t + ∆t/2

statevariable

x

x & xknownhere

.

x & xknown here

.

interpolated trajectory of x

between t and t+ ∆t/2

∆t for Adams-Moultonintegration of rod dynamics

~∆t for R-Kintegrationof platformmotions

Figure 1. Cubic spline interpolation of platform motion

If similar splines are constructed for the other 5 motions

of the platform, then these 6 splines provide an excellent

interpolation of the platform motions within the time

interval { , / 2}o ot t t+ ∆ without explicitly integrating the

platform equations of motion at each rod time step. This

known trajectory of the six state variables allows the

selection of a much smaller time step, t∆ % , appropriate

for integration of the rod dynamics within this interval.

A cartoon of the results of this integration for one force

exerted by the rod on the platform5 is shown in Figure 2

below.

time

t t + ∆t/2

rod forceon Platform

predicted by RodDyn

parabolic fit

rod loadingto be usedin R-K integrationof the platformmotions

Figure 2. History of rod force on platform

Due to the frequencies of the longitudinal and transverse

waves in the rod, the history of the rod force on the

platform is likely to have a character less smooth than

the motions of the platform, as shown in Figure 2. The

question then becomes, what force should be reported

back to the platform RK process as representative of the

rod force at the time / 2ot t+∆ . Several choices are

possible. One choice is simply to use the instantaneous

force calculated at the time / 2ot t+∆ (the block dot in

Figure 2). Another, perhaps somewhat more sophis-

ticated approach is to smooth the force time history using

a linear or higher order fit (a quadratic fit is shown in

Figure 2). In the current realization of our coupled

platform/rod dynamics program, the instantaneous value

of the force is used, although implementation of a

quadratic fit is neither difficult nor computationally

intensive. The reported contact forces and moments for

all rods at the end of this RK step are added to the other

estimated external loads on the platform to determine the

time derivative of the state, (1) ( / 2)ot t+∆s& by use of (2).

Similarly, at each step in the RK process the coordinate

and velocity of each platform motion is known at the

both ends of the corresponding time interval and similar

spline curves can be set up for interpolation within those

5 A single rod may contact the platform at a multiplicity of

points, each one of which can result in a force applied to the

platform. For simplicity in exposition we are considering only

one of these forces.



6

intervals. The rod dynamics program is run for each rod

much smaller time steps taken for this integration. The

actual time step may be different for different rods,

depending on their configuration.

The result of using this asynchronous time stepping of

platform dynamics and the rod dynamics is that the

computational effort is minimized and the run time for a

simulation is significantly shortened over that which

would be required if both the platform dynamics and rod

dynamics used the smaller rod dynamics time stepping.

Simplification of wave loads on the rod

As mentioned previously, the computation of the wave

loading on a rod when it is exposed to a random sea state

composed of a large number of components can

significantly slow down the simulation of rod dynamics.

First, the effects of surface wave motions extends only

within a layer near the water surface that is about one

wave-length in depth. That is, the vertical extent of these

effects is not the same for all the components in the sea

state . Thus, only rod elements whose current depth is

within one wave length of the longest wave need to be

considered in the computation of wave effects; elements

that are closer to the free surface than that have to only

consider the subset of waves whose lengths are longer

than the element depth. For many deep water rods, this

filtering eliminates much unneeded computation.

However, further reduction in the computational effort

associated with the waves can be achieved by making

use of the same discrepancy in time scales between the

wave periods and the rod time step mentioned in the

coupling discussion.

At any time 0t the particle velocity 0( )tu and pressure

gradient 0( )p t∇ at a given point on the rod,

0 0 0 0( ) ( ), ( ), ( )q q q qt x t y t z t = x can be found using the

standard formulas for Airy waves6. At this time the

velocity at any point on the rod is determined to be,

0 0 0 0( ) [ ( ), ( ), ( )]q q q qt x t y t z t=x& & & & from the rod dynamics

solution. We extrapolate the approximate position that

this point on the rod will occupy at a later time

0 wavet t t= + ∆ as

0 0 0( ) ( ) ( )q wave q wave qt t t t t+ ∆ ≈ + ∆x x x& (5)

where wavet∆ is large with respect to the RodDyn time

step and small with regard to any wave period.

We compute the wave particle velocity 0( )wavet t+ ∆u

and pressure gradient 0( )wavep t t∇ + ∆ at the forward

extrapolated point 0( )q wavet t+ ∆x using the same Airy

wave formulations. For times t , 0 0 wavet t t t≤ < + ∆ in

6 It is usual to modify these formulas using “Wheeler

stretching” to approximate the nonlinear effects near the free

surface.

the RodDyn computation, the water particle velocity and

pressure gradient for the point ( )qtx are estimated using

the following interpolation formulas

[ ]

0

00 0

( ) ( )

( )( ) ( )wave

wave

t t

t tt t t

t

≈

−+ + ∆ −

∆

u u

u u (6)

[ ]

0

00 0

( ) ( )

( )( ) ( )wave

wave

p t p t

t tp t t p t

t

∇ ≈ ∇

−+ ∇ + ∆ −∇

∆

When t exceeds wavet∆ , the values of the wave particle

velocity and pressure gradient both for the current time

and the new extrapolated time are re-computed and the

algorithm continues as before.

Typical values of wavet∆ are 0.2 – 0.4 sec and as a result,

this interpolation scheme reduces the computation of the

wave environment by a factor of 5 to 10 and this

virtually eliminates the penalty for including complex

sea state descriptions in the computation of the rod

dynamics. Numerical experiments have been carried out

that show that this approximation incurs negligible error

for typical offshore situations.

Cluster computing.

The asynchronous time-stepping developed above leads

to a possibility of further significant shortening of the

computation time for platforms with a large number of

rods that have a complexity that would demand larger

than normal numbers of finite elements to describe.

With the asynchronous approach, the connection

between the platform motions and rod forces is a “thin”

one. That is at step n of the RK algorithm, the rod

dynamics program needs only the projected state, ( )ns of

the platform at the corresponding time (12 numbers) and

the rod dynamics program returns only the resultant

forces and moment (6 numbers) on the platform from all

of the contacts between the rod and platform. As a

result, it is easily possible to use the approach developed

above to parse the computation out to a cluster of

computers, one “master” computer simulating the

platform motions dynamics and one or more “slave”

computers asynchronously simulating the rod dynamics

(receiving the platform state and outputting the platform

forces). With such a scheme it should be possible to

achieve simulation run times very close to that of the

stand-alone platform dynamics simulation.

Example

Application of the various simplifications discussed

above is illustrated by a sample simulation which

consists of one truss spar moored in 1500m water depth

in the Gulf of Mexico. Table 1 shows the characteristics

of the truss spar. This spar is moored with 12 mooring



7

lines which are grouped into a 3×4 pattern. Each group is

spaced by 120 deg and the mooring lines within one

group are divided by 5 deg. Each mooring line is

composed of three segments consisting of fairlead and

anchor chains at the both ends and polyester rope in the

middle. The connector between the chain and the

polyester rope is modeled as a point mass due to its short

length compared to other segments. The synthetic

material, such as polyester rope, exhibits creep behavior

which depends not only on the instantaneous value, but

also on the stress history. Thus, one viscoelastic material

model with two different modulii of elasticity is adopted

to simulate the complicated deformation of polyester

rope. The properties of mooring lines are provided in

Table 2. And the layout is shown in Figure 1.

Table 1. Spar Dimensions

Hull Diameter (m) 40

Hull Draft (m) 175

Hull Length (m) 194

Center of Gravity (m) -66

Roll/Pitch Radius of Gyration (m) 83

Yaw Radius of Gyration (m) 15

Displacement (tonnes) 103800

Table 2. Moor Line Properties

Segment Diameter

(mm)

Length

(m)

Dynamic

Stiffness

(KN)

Static

Stiffness

(KN)

Platform

Chain 172 91 3,071,190 -

Polyester

Rope 290 2,286 706,320 451,260

Anchor

Chain 172 122 3,071,190 -

Four top-tensioned risers (TTR) are arranged in a 9 meter

square pattern centered on the spar centerline and aligned

with the x and z axes. The top of the riser is supported by

the hydro-pneumatic tensioners and the bottom is

connected to the wellhead at the seafloor. The risers are

laterally restrained below the hard tank by guides

attached to the heave plates. At the keel of the platform,

the risers are strengthened with the tapered sections

which enhance the ability to withstand the higher

bending. The properties of TTRs are provided in Table 3.

Four steel catenary risers (SCR) consisting one oil export

riser, one gas export and two production risers, are hung

off at the outside of the soft tank. The properties of SCRs

are listed in Table 4.

Wave forces and hydrodynamic coefficients for the

submerged hull body are calculated from the

radiation/diffraction program WAMIT based on potential

theory.

Figure 3 shows the configuration of the platform and the

associated risers, SCRs and mooring lines as modeled in

the coupled simulation. The diameters of the various

riser, SCR and mooring line elements are exaggerated for

visibility.

Table 3. TTR Properties

Outside Diameter (mm) 406

Wall Thickness (mm) 25

Content Density (kg/m^3) 1025

Nominal Tension (KN) 5338

Table 4. SCR Properties

Oil

Export

Gas

Export Production

Outside Diameter (mm) 324 406 244

Wall Thickness (mm) 20 24 32

Product Density (kg/m^3) 1251 651 828

Table 5 shows the applied environment. Long-crested

waves with JONSWAP spectrum are used for the random

wave model and an NPD spectrum is used for the wind-

gust model. The time simulation for 1 hour global motion

is performed subjected to 100 years hurricane.

Table 5. 100-Year Hurricane Parameters

Wave

Spectrum JONSWAP

Hs (m) 15.2

Tp (sec) 15.6

Shape γ 2.4

Wind

Spectrum NPD

Vw (knot) 87.5

Current

Depth (m) Vel. (m/s)

0 1.9

40 1.4

75 0

1500 0

Results and Discussion



8

Figure 4 shows a comparison of two computations for

one hour real time with platform time steps of 0.5 sec.

and 1.0 seconds. The time steps for all of the rods in both

cases was 0.02 seconds. These graphs show that there is

no difference between these two computations and that

the computation is convergent.

Figure 5 shows a motion comparison between the

coupled analysis and an analysis modeling the response

of the TTRs, SCRs and mooring lines quasi-statically

which represents the uncoupled case. These figures show

small difference in the heave motions, but obvious

differences in the remainder of the motions.

Figure 6 shows the contact force at the keel joint for

TTR-1 and Figure 7 shows a comparison of the top

tensions at the fairlead for the mooring line 1 between

quasi-static mooring modeling in uncoupled analysis and

dynamic mooring modeling in coupled analysis. Figure

8 shows the von Mises stress distribution at the touch

down zone (TDZ) for the oil export SCR.

There are a large number of other important forces and

stresses that are output from this coupled simulation, but

space is too short to display them here. For instance

RodDyn reports the tensile stress and bending stress

everywhere in all of the risers, SCRs and mooring lines

and the contact forces at all of the contact points such as

guides and pull tubes.

Summary and Conclusions

In this paper we have outlined effective methods for

reducing the computation time required for consistent

coupled platform/rod dynamics simulations by exploiting

the time scale differences between the wave and platform

time scales and typical rod dynamics time scales. A

work example using these simplifications with the

coupling of MLTSIM and RodDyn shows the coupled

computation can be made on a PC with reasonable run

times, and also exhibits the consistence between coupled

and uncoupled simulations and some differences for the

example used.

ACKNOWLEDGMENTS

The authors would like to thank Technip for permitting

publication of this paper.

References

Garrett, D.L., 1982. Dynamic Analysis of Slender Rods.

Journal of Energy Resources Technology. Transactions

of ASME 104, 3002-307.

Garrett, D.L., 2005. Coupled analysis of floating

Production Systems. Ocean Engineering 32, 802-816.

Paulling, J. R. and Webster, W. C., 1986. Large-

Amplitude Analysis of the Coupled Response of a TLP

and Tendon System. In: Proceedings of the 5th

OMAE

Conference, Tokyo, Japan.

RodDyn Benchmark Report, 2008. Technip.

RodDyn Theory Manual, 2010. Technip.

Webster, W. C.,“Mooring-Induced Damping”, Ocean

Engineering, Vol. 22, No. 6, pp.571-591, 1995.

Figure 3 Truss Spar with 12 mooring lines, 4 TTRs and 4 SCRs



9

Figure 4. Effect of time step on predicted motions

with non-linear, fully coupled rods.

Figure 5. Comparison of non-linear, fully-coupled predicted

motions with quasi-static results.

25

30

35

40

45

50

2500 2600 2700 2800 2900 3000

Surg

e (m

)

Time (sec)

Dt=0.5 Sec Dt=1.0 sec

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

2500 2600 2700 2800 2900 3000

Heave (m

)

Time (sec)


-10

-8

-6

-4

-2

0

2

2500 2600 2700 2800 2900 3000

Pitch (deg)

Time (sec)


25

30

35

40

45

50

2500 2600 2700 2800 2900 3000

Surg

e (m

)

Time (sec)

Un-Coupled Coupled

6

8

10

12

14

16

2500 2600 2700 2800 2900 3000

Sw

ay (m

)

Time (sec)

Un-Coupled Coupled

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

2500 2600 2700 2800 2900 3000

Heave (m

)

Time (sec)

Un-Coupled Coupled



10

Figure 5 (continued). Comparison of non-linear, fully-coupled

predicted motions with quasi-static results.

Figure 6. Contact Force at Keel Joint (TTR#1)

Figure 7. Top Tension at Fairlead (Mooring #1)

Figure 8. von Mises Stress around TDZ (Oil Export SCR)

-2

-1

0

1

2

2500 2600 2700 2800 2900 3000

Roll (deg)

Time (sec)

Un-Coupled Coupled

-10

-8

-6

-4

-2

0

2

2500 2600 2700 2800 2900 3000

Pitch (deg)

Time (sec)

Un-Coupled Coupled

-2

-1

0

1

2

2500 2600 2700 2800 2900 3000

Yaw

(deg)

Time (sec)

Un-Coupled Coupled

0

100

200

300

400

500

600

2500 2600 2700 2800 2900 3000

Forc

e (kN)

Time (sec)

600

800

1000

1200

1400

1600

1800

2500 2600 2700 2800 2900 3000

Tensio

n (kN)

Time (sec)

Un-Coupled Coupled

0

30

60

90

120

150

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

Von M

ises S

tress (M

Pa)

ArcLength From Top (m)



coupled dynamic modeling of a moored floating platform

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