Krachten op een vertikale paal ten gevolge van de combinatie van stroom en golven. Deel 2: Mathematical simulation of the behaviour of moored and dynamically positioned floating structures

MaTS

MARIEN TECHNOLOGISCH SPEURWERK

MATHEMATICAL SUMULATION OF THE BEHAVIOUR OF MOORED AND DYNAMICALLY POSITIONED FLOATING STRUCTURES

VM-I-2

Industrie'le Raad voor de Oceanologie

Netherlands Industrial Council for Oceanology

Report No. Z42584-OR

MATHEMATICAL SIMULATION OF THE BEHAVIOUR OF MOORED AND DYNAMICALLY POSITIONED

FLOATING STRUCTURES

Report No. Z42584-OR

MATHEMATICAL SIMULATION OF THE BEHAVIOUR OF MOORED AND DYNAMICALLY POSITIONED FLOATING STRUCTURES

N.S.M.B. Order No. Z42584

Ordered by: Netherlands Industrial

Council for Oceanology (I.R.0.) P.O. Box 215

2600 AE DELFT THE NETHERLANDS

Reported by: H. van den Boom Approved by: G. van Oortmerssen

Approved by: Projektgroep Vloeistofmechanica 2 MARIEN TECHNOLOGISCH SPEURWERK:

Ir. J.B. van den Brug Prof.Dr.Ir. A.J. Hermans Dr.Ir. J.A. Pinkster

CONTENTS

page Summary

Nomenclature 1

Introduction 3

General description of the behaviour of moored

and dynamically positioned structures 10

Mathematical description of moored and

dynami-cally positioned structures 19

3.1 Hydrodynamic approaches 21

3.2 Excitation loads 28

3.2.1 Hydrodynamic loads 30

3.2.2 Aerodynamic loads 45

3.3 Hydromechanic reaction forces 49

3.4 Position keeping forces 54

3.4.1 General 54

3.4.2 Passive position keeping 56

3.4.3 Active position keeping 71

3.5 Special loads 81

A mathematical simulation model 83

Final remarks and recommendations 90

References 94

Appendix: Impulse response function applied

SUMMARY

Present-day offshore operations involve many types of floating structures which have to be kept on station by means of a mooring system or by means of thruster systems (dynamic positioning). Examples are drilling vessels, floating production systems, crane barges, floating storage barges and mooring terminals.

Sometimes, as for instance in pipe-laying or dredging, a certain track has to be followed. This can be

achieved by using thrusters or anchor winches.

When designing mooring or positioning systems, aspects of importance are:

- the motions of the floating structure in the onerating mode, with special reference to workability;

forces in the anchor system, with reference to holding

power of anchors, strength of anchor chains and other

structural components, both in the operational and survival condition;

power requirements and performance of thrusters;

control characteristics of a dynamic positioning system.

In a design phase, computer simulation models, which are sets of mathematical relations describing the process, can often be used to determine the behaviour of

moored and dynamically positioned floating structures. Usually a time domain approach is required due to the non-linear characteristics of the moored systems. The

mathe-matical model can be either purely theoretical or based

on a combination of theoretical and experimental data.

To determine the physical relationships, model tests may

be an indispensable tool.

The motion of a moored structure is governed by the exciting loads due to waves, wind and current, by fluid reactive

forces caused by the motions of the structure, and by the loads exerted by the mooring system, or controlled

ii

Typical aspects of the behaviour of moored floating

structures are elucidated by means of the following three typical examples:

a jetty mooring, exhibiting subharmonic motion

behaviour;

single point mooring systems, which are characterized

by large amplitude, low frequency coupled horizontal

motions;

dynamically positioned vessels, in which complex

control and propulsion systems are used to provide the

required restoring forces.

Apart from motions around a position of equilibrium

the analysis of transient motions of floating structures has become important.

In general mathematical methods as meant above are founded on several hydrodynamic approaches to the subject.

Classic approaches, which are based on the treatment of linear frequency domain equations of motions as instan-taneous relationships, cannot always be justified because of the frequency dependency of the fluid reactive forces and the non-linear effects in the restoring forces.

For several cases, however, the oscillations at wave

frequencies may be separated and computed by linear methods, such as strip theory and sink-source methods, while the

low frequency behaviour due to the slowly varying wind,

wave and current loads may be described by non-linear

equations of motions with constant hydrodynamic coefficients. The ultimate behaviour is then found from superposition of both components of motion.

The use of the impulse-response techniques to describe the

fluid reactive forces can be used when interactions between high and low frequency components of motion cannot be

ne-glected or in case of transient motions.

more rigorous attempts have to be made.

When considering the external loads on the floating

structure, three groups may be distinguished: the

exciting loads, the hydromechanic reaction loads and the

mooring loads. For each simulation these three groups

of loads have to be modelled. The exciting loads due to

the environmental flow fields may be separated into different parts depending on the characteristic properties of the

flow.

For large volume structures the wave loads can be deter-mined by existing computational methods while wind and

current forces may be predicted in a semi-empirical way.

Slender and small body structures experience non-linear fluid loading due to viscous effects. Therefore only use can be made of approximate methods to determine the environ-mental loading.

The determination of the fluid reactive forces, which

mainly defines the different simulation approaches, may

be carried out by known calculation programs and experimen-tal techniques. Special attention has to be given to the non-linear fluid reactive forces which are of prime

impor-tance in case of very slow motions or space frame

struc-tures.

In general the restoring forces due to mooring systems

can be determined while neglecting the dynamic behaviour of the mooring itself. In this way static and steady-state

solutions may be used in case of anchor line moorings, pile and leg-moorings and single point mooring systems.

In case of active position-keeping, the basic elements, viz.

the position measurement system, the automatic or manual

control system and the thruster or winch system have to be

simulated. Especially the modelling of controllable

pro-pulsion systems is quite elaborate. For these systems it is

iv

of prime importance to take into account the strong

interactions with the vessel's hull, the flow field

and appendages. Controllable cable mooring may easily

be incorporated when the control algorithm and the passive

cable restoring forces are known.

In addition to excitation, reaction and position-keeping forces special loads have to be incorporated sometimes. The loads may be due to the multi-body characteristics of

the structure resulting in mechanical and hydrodynamic interactions. Hydrodynamic interactions may also arise from neighbouring constructions and the tonography of sea bottom and shore.

The description of an existing mathematical model for

a D.P. vessel illustrates the structure and the use

of simulation models.

In order to improve existing simulation methods and to

develop new methods, some essential physical and

mathematical work has to be done. The following

recommen-dations are given:

Determination of wave-drift-forces under simultaneous

action of waves and current. Research on the low frequency fluid reactive forces in waves current. Both subjects, which seem to be linked, are of importance to model the

low

are indispensable for the simulation of any structure. Research on thruster-hull interaction especially in cases of wave and current action. This subject is strongly re-lated to the modelling of the restoring forces in case of dynamically positioned structures.

- The treatment of large amplitudes of motions with respect to the transformations of coordinates and the description of the environmental exciting loads. This study is indispen-sable for rational simulation of systems with a large degree of heading and position changes such as D.P.-vessels, and Single Point Moorings and in case of track following.

NOMENCLATURE

1.

A area, matrix

matrix of restoring force coefficients, coefficient

characteristic diameter, index; drag general force or moment

Fn Froude number

gust factor

water depth, height index; horizontal moment of inertia

Kc Keulegan-Carpenter number

characteristic length, index; lateral inertia matrix, number

number, moment around

x3-axis

in-phase part of second order transfer function quadrature part of second order transfer function impulse-response function, drag

Rn Reynolds number

spectral density function

Sn Strouhal number

period of oscillation, tension, draught, index; transverse

characteristic velocity

V velocity, vector, index; vertical

weight of cable element

X force in direction x1

force in direction

x2

a _{added mass coefficient, index; air}

damping coefficient

restoring force coefficient, coefficient, index; current

frequency, function gravity constant

index; mode of motion, component index; mode of motion, component

wave number, cable parameter, index; mode of motion

1 length

mass coefficient

number of revolutions, normal vector yaw velocity

distance along mooring cable time

surge velocity sway velocity

circular frequency

coordinate, displacement

x11x2,x3 direction fixed system of coordinates

x1,x2,x3 body fixed system of coordinates

coordinate

height, coordinate

a angle of incidence coefficient

coefficient, angle coefficient

rudder angle, impulse

phase angle, elasticity per unit length, error wave elevation

horizontal cable angle

X wave length, damping factor

1-1 angle of incidence

kinematic viscosity

ii half circumference of unit circle

density

time

(i) vertical cable angle, velocity potential

X velocity potential

velocity potential

1.

Physical and mathematical aspects involved in the theoretical approaches to the subject are illustrated by three typical mooring arrangements. Furthermore mathematical simulation is described in relation to

other methods. The purpose

of

the structure

of

1. INTRODUCTION

4.

Throughout the development of offshore engineering much attention has been given to the behaviour of moored floating structures. Many offshore structures are

self-floating, either permanently or during some stage

of their life. The evaluation of the operational

character-istics as well as the design of these structures (e.g. position keeping capability and design loads) require

accurate estimates of the behaviour due to the

environmental loading. For each design,specific

requirements have to be fulfilled in a number of well defined conditions. To indicate the wide scope of the subjects involved, examples of some typical designs are

given.

- Vessel moored to a jetty

To obtain design loads for the fixed structures and to establish the optimum characteristics of the mooring arrangement (fenders and mooring lines) the behaviour in the most severe wind, wave and current conditions has to be known. The mooring system is usually of a strong non-linear nature.

- Vessel moored to a single point mooring

Contrary to the previous case, the vessel is restrained at only one point (bow or stern) by means of a rigid yoke or a hawser allowing the vessel to rotate about the buoy to achieve an equilibrium between wave, wind and current forces. In this case the behaviour of the tanker and the buoy has to be determined to ascertain smooth operation in the operating conditions. More-over, the safety of the system during the most severe condition (the survival condition) has to be established. An important phenomenon inherent to this type of mooring is the possible occurrence of unstable motion (galloping) due to interaction of the non-stationary external

forces and the dynamics of the system.

- Vessel kept on station by dynamic positioning

The behaviour of a floating structure exposed to

5.

fixed point it is kept in position by means of

hydrodynamic forces generated by purposely installed thrusters and the propellers. The reference point is usually on the sea floor and the vessel's deviation from the reference point may be measured by means of an acoustic device (transponder) which activates the control system. The purpose of the dynamic positioning system is to minimize the low frequency deviation with respect to the reference point. Typical aspects are the establishment of the optimum control algorithm and the determination of the minimum power required to meet the design objectives in all relevant conditions.

The problems described above may of course be solved by a variety of methods, both experimental and theoretical. For instance full scale measurements of similar structures

in similar circumstances may be used to derive the required data. Alternatively, model testing has the

advantage of a fairly wide range of applicability and of controlled testing conditions. Depending on the

sophistication of the experimental facility the combined effect of wind, waves and current on complex compound

structures may be established. There are some disadvantages

to this method. Not all phenomena that may influence

the behaviour of the structure can be included always. Moreover, model testing may be quite expensive. This often prohibits the analysis of alternative designs

which are of vital importance to achieve the optimum one. Theoretical methods are often applied when their

validity is sufficient for the aspects of behaviour

under investigation. For this purpose mathematical models have to be developed and selected to describe these

physical aspects while other parts of the real world are simplified or neglected as much as possible. It will be clear that both environmental conditions and behaviour characteristics of the floating structure are of prime importance for the selection and development of useful theoretical methods.

ERGODIC STATIONARY RANDOM NON-ERGODIC NON -STATIONARY PROCESS PERIODICALLY

FIG 1 Classification of dynamic processes

DETERMINISTIC

6.

environmental loading may be classified according to

Figure 1. Some of the most important differences will be

defined here. Unlike a deterministic process, random or

stochastic processes cannot be described by exact

mathematical formulations but only in terms of probability

and statistical quantities. The random process is called stationary when its probability properties are invariant with respect to an arbitrary shift of the time scale. When one realization determines the ensemble averages of the statistical properties, the stationary process is said to be ergodic.

The response of a dynamic system to an arbitrary input may be given by

frequency domain methods, probabilistic methods, time domain methods.

The marine environment, characterized by its random-ness and often assumed to be ergodic, requires a description of stochastic origin such as frequency

domain and probability domain. The use of the frequency

domain has shown to provide quite accurate motion

estimates in irregular waves. The response of the vessel to regular waves is determined and the statistical

properties of the motions to be expected in irregular waves can be obtained using spectral analysis techniques. Because of the assumed validity of the superposition

principle the method is, however, restricted to linear processes.

Probabilistic methods setting the probability

distri-butions of characteristic quantities (e.g. wave forces)

may also be used to describe the random environment for

important classes of marine structures.

Time domain methods describing the process in

time-history traces are, according to their nature, deterministic methods. The prime importance of these methods is that

non-linearities of the processmay be treated.

FREQUENCY DOMAIN METHODS probabilistic functions spectral density PROBABILISTIC METHODS

7.

natural representation of the behaviour of floating structures. Therefore an increasing number of advanced engineering methods rely on this technique. This tendency

is enhanced by the growing availability and accessibility of sophisticated computer systems, counteracting the

relatively great computational efforts linked to the use of the time-domain.

Mathematical time domain representation of a process, in the following called "simulation" may be defined as:

"The formulation and use of a set of mathematical relations

that describe the process in the time ("model"),based on a

number of selected parameters, in order to calculate one or more quantities of the process. The parameters

that are relevant to the process and the set of mathemati-cal relations have to be selected keeping in mind the

purpose of the simulation. It then follows that more than one mathematical model may emerge from one

physical reality. Because of their advantages with

respect to the random environment and computational techniques, frequency domain and probabilistic methods may be used for different parts of the mathematical time-domain model construction as long as the pre-requisites linked to these methods are fulfilled. The transformations necessary for the combined use of different domains are illustrated in Figure 2. It

must be emphasized that time-traces obtained from such

transformations are just possible realizations. Only when the ergodic property is sufficiently satisfied, the produced time-record may be considered representative of the entire process part.

From the foregoing it will be clear that many different approaches can be followed to simulate the behaviour of floating structures. For instance a simulation method

has been developed by Van Oortmerssen

_1:1]

the motions of a vessel moored to _{a jetty. The method}

is capable of handling non-linear fender and mooring

8.

should be restrained fairly stiffly in order to minimize the low frequency deviation in the horizontal plane. The restriction has to be applied because the wave

and wind direction are assumed to be constant during the entire simulation.

A simulation method that deals with (part of) the problems of the second example has been reported by

a number of authors (e.g. Wichers

1:2]

These methods simulate the low frequency motions in a homogeneous wind and current field and were developed to study the problem of unstable motions of a tanker moored to a SPM.

Simulation methods that take into account the dynamic positioning were amongst others developed at TNO-IWECO

1-47

_ _

_ _

positioned vessel under influence of wind, waves and current can be simulated.

It is the purpose of the present investigation to analyse the physical aspects of the motions of a moored structure in a systematical way and if possible, to indicate

methods to describe the physical phenomena in a mathemat-ical way, suitable for the simulation process.

Furthermore the main shortcomings in the application of simulation models will be discussed and recommendations for further research will be given.

Based on this general description a future designer of a simulation technique or the user of such a technique may decide whether a particular physical phenomenon has to be included. It is hoped that this will enable them to design an optimum technique or to judge whether a given tech-nique is suitable for the particular application.

As will be clear from the examples above, the word

"moored" is used in a wide sense in this investigation: apart from structures moored in a passive way it also applies to dynamically positioned structures using either thrusters or anchor winches.

9.

The investigation will, however, be restricted to a single

moored structure and this structure will be assumed to

be completely rigid.

A general description of the behaviour of moored

structures and their environmental conditions is presented in section 2. A survey of the mathematical descriptions and detailed treatments of the basic

parameters is given in section 3. In section 4 one

existing simulation method is discussed in view

of the descriptions given in section 3. Finally, possible improvements and recommendations with respect to future research and development are presented in section 5.

2. GENERAL DESCRIPTION OF THE BEHAVIOUR OF MOORED AND

DYNAMICALLY POSITIONED STRUCTURES

Environment

Mooring and dynamic positioning

Vessel moored to a jetty

Vessel moored to a single point mooring Dynamically positioned vessel

In general terms the mechanisms of motion are discussed in relation with the environmental conditions and the mooring system. Some typical physical phenomena are illustrated by detailed descriptions of the three mooring cases presented in section 1.

2. GENERAL DESCRIPTION OF THE BEHAVIOUR OF MOORED AND

DYNAMICALLY POSITIONED STRUCTURES

Li.

In the following some typical physical phenomena related to the behaviour of moored floating objects will be dis-cussed. Moreover, the three mooring systems mentioned in section I will be described in detail because these examples cover a great part of the spectrum of possible mooring systems.

The instantaneous velocity fields in water and air as governed by the existing wind, wave and current condition will result in aerodynamic and hydromechanic forces and moments acting on the

structure. Apart from the geometry, the magnitude of these forces depends on the orientation (heading) of the struc-ture and of course on the two velocity fields disturbed by the presence of the structure. As a result of the ex-ternal aero- and hydrodynamic forces the structure will experience accelerations and consequently velocities and displacements. Due to the displacements of the structure the mooring forces will alter in magnitude and direction either directly in case of passive mooring or by means of a control system. These mooring forces, however, are in general also affected by the environmental conditions

directly.

As soon as the structure is moving the aerodynamic and hydromechanic forces will also depend on the structure's excursions and their time derivatives. The resulting

acceleration will now be determined by the combined effect of aerodynamic, hydromechanic and mooring forces.

Environment

The response of the structure to the environmental exci-tations is ruled by its geometry but also by the

character of the external loads. The water velocity field may be built up by current and wave fields. The current component caused by ocean circulation, tidal effects, wind

action and river mouths is characterized by relatively slow

variation in velocity and direction. Swell originating from remote wave fields together with local wind induced waves

MOORING LINE MOORING HOOK SYNTHETIC TAIL MOORING PULLY JETTY

o

-I]

FIG. 3

Typical

arrangement for a

vessel moored to a jetty

12.

provide random time and space dependent velocity fields,

containing a range of high frequency oscillations.

The aerodynamic velocity field is characterized by a constant velocity component and a varying part that may cause high gust loads. The short term variations of wind direction are small.

The load on the moored structure resulting directly from

the wind, wave and current action may be split into a steady part due to both velocity fields, a slowly varying part with typical periods of 50 seconds and longer, due to wave and wind action, and a high frequency part due to waves with a typical range of periods from 5 to 20 seconds. In general the high frequency wave force will be an order of magnitude larger than the steady and slowly varying

components. Unlike the high frequency wave forces the steady and slowly varying components are mainly effective in the horizontal modes of motion. In some cases, however, also vertical components may be observed (e.g. the steady tilt of semi-submersibles).

Mooring and Dynamic Positioning

The purpose of the mooring or D.P.-system is to provide forces acting on the structure, in order to keep the vessel

in a desired position with sufficient accuracy so that operational activities can be executed. In general their great magnitude and high frequency of oscillation make

it impossible to balance the forces induced by the high frequency components of the wave velocity field by the mooring. Fortunately the motions induced by these forces are often within the allowable limits of excursions. Owing

to this only the mean and low frequency parts of the

environmental loads, mainly acting in the horizontal plane, are left to be counteracted by the mooring. It must be

noted,however,that the first order wave motions generally handicap the mooring by reducing the allowable low

frequency excursions, disturbing the control system and heavy high frequency loading of the mooring equipment.

13.

How much the structure is allowed to move depends on the

nature of the operation carried out during the mooring. This specification and the environmental loads together with the type and size of the structure, determine the type and dimensions of the mooring system. While the

restoring forces in vertical direction are mainly provided

by gravity and buoyancy, the mooring system determines the horizontal restoring forces. The range of natural periods for the horizontal modes, as governed by the mooring

stiffness, generally coincides with the periods of low

frequency excitations. Because of the low potential damping at these frequencies, slowly varying external loads,even of small magnitude,may induce large amplitude oscillations. Non-symmetry and non-linearity of the restoring-excursion

relationship of the mooring cause coupled non-harmonic

motions in the horizontal modes.

In case of active position-keeping artificial damping and couplings may be introduced by the control system.

From the foregoing, it is clear that three components of behaviour may be distinguished in case of moored floating

structures

wave frequency oscillatory motions,

- mean excursions due to mena exciting and restoring

forces,

low frequency behaviour due to slowly varying exciting forces and the mooring forces.

To illustrate some typical aspects involved with the

behaviour of moored structures the three examples from

section I will be discussed here in more detail.

Vessel moored to a jetty

As illustrated by Figure3 the connection between the vessel and the jetty is made by mooring lines (either steel or

nylon) that provide tension forces which keep the vessel

14.

compression forces which prevent damage due to shockloads.

The number and geometry of mooring lines and fenders together

with the elastic characteristics of each component determine

the mooring characteristic of the whole system. In general terms this mooring arrangement yields a fairly stiff

system, especially in the sway (transverse) and yaw

(rotation about the vertical axis) direction. In surge (longi-tudinal) direction the system is softer because of the

greater length of mooring lines in that direction. Due to

the large mass involved the natural periods of such mooring

systems will typically be in the order of magnitude of 60 - 100 seconds. The high stiffness of the mooring in sway and yaw direction is typical for this type of mooring arrangements. Consequently the direction and through that the magnitude of the external loads only depends on

changes of the environment while the heading of the structure remains almost the same.

The disturbances caused by wave action are in the frequency range between 5 and 20 seconds. Therefore the motions of the vessel are dominated by mass effects and are hardly affected by the mooring lines and fenders which simply have to

comply with any motion that is imposed by the waves.

The sole purpose of the mooring system is to restrict the mean excursion of the moored vessel.

External forces oscillating with low frequencies will cause motions (and thus mooring line forces) that are dynamically amplified. For example if the elasticity of

the mooring system amounts to 0.01m/ton and if an oscil-lating force of 10 tons is applied the resulting motion may vary between 0.1 metre for a frequency approaching

zero, and as much as approx. 2.5 metres close to the natural frequency of the system. In case of a moored vessel exposed to wind, waves, and current all forces

caused by these phenomena contain low frequency components leading to this dynamic behaviour. Moreover, non-linear restoring force-excursion relationships of the mooring

15.

system may induce low frequency motions even in regular

waves when the wave frequency is a multiple of the

"natural frequency" of the system. These are the so-called "sub-harmonic motions".

ssel moored to a single point mooring

Let us assume for the sake of simplicity that the vessel is moored to a fixed point by means of a non-linear

bow-hawser (see Figure 4 ). In the absence of wind, waves

and current the vessel may of course take up any position provided that the bow-hawser is slack. If wind, waves, current or any combination of these are present, the

vessel will search for a position in which all forces

are in equilibrium and the bow-hawser will be tightened. Generally a natural frequency pertaining to one mode of mption cannot be established. Because of the non-linearit of the bow-hawser the spring coefficient is not constant and depends on the loads in the bow-hawser. For a given

(mean) external load the (mean) bow-hawser load may be determined and the spring coefficient linearized to establish the natural frequency. It is more interesting however to consider the coupled horizontal motions

(surge, sway, yaw) of a vessel moored to an S.P.M.

Both from model test and in situ observations it is known

that non-co-linear action of the excitation forces may

cause unstable conditions resulting in coupled low-frequency horizontal motions, so-called "galloping" and "fishtailing".

These large amplitude motions may even occur when the vessel is only under the influence of steady current or wind velocity fields. The appearance of this instability depends on a number of factors such as wind speed, current

speed and ship geometry but it is known that the length of the bow-hawser and astern propulsion are important

factors which may be used to reduce the phenomena. If the

bow-hawser is to long the vessel will exhibit large

deflections in the horizontal plane in surge, sway and yaw direction with typical periods of 100 - 300 seconds. Astern propulsion may tighten the bow-hawser and reduce the

PROPELLER AFT THRUSTERS TRANSMITTER

6/

it

1=

gig

FIG. 5 Typical

_{dynamic positioning arrangement}

_{for a}

_{drilling ship}

o

INCLINOMETER

The low-frequency behaviour of a structure moored to an S.P.M. is furthermore influenced by slowly-varying excitation loads, the non-linearity of the overall restoring force-displace-ment relationship and the dynamics of the mooring itself.

Coupling of motions may arise from non-symmetrical mooring arrangements.

Dynamically posiuioned vessel

The purpose of dynamic positioning is to minimize the low-frequency deviations in the horizontal plane and to

station the vessel as close as possible to the required

position where conventional mooring systems are impossible, impractical or even prohibited. The control forces may

be either of mechanical nature by means of controllable winches or of hydrodynamic nature using variable thrusters or propellers (see Figure 5). The restoring forces that

result from the mooring lines in a passive system are now produced by winches or thrusters.

If the magnitude of the restoring forces is directly

related to the deviation from the desired position a simi-lar system as the passive one is obtained and no improve-ment with respect to the low-frequency behaviour may

be expected. Such improvements may result from the possi-bility to provide the system with an artificial form of damping and to include so-called feed forward control loops. The artificial damping is created by applying a control force proportional to the velocity of the

low-frequency motion. By changing the constants in the equations that determine the control forces as a function of

low-frequency excursion and velocity the natural low-frequency and damping of the system may be selected to suit the requirements. Limitations are imposed by the required

power, the thruster characteristics and the required command characteristics. The feed forward control loops may success-fully be used to counteract disturbing forces, if these can

be linked to one or more parameters that can be measured

ENVIRONMEN

-TAL

FLOWFIELDS

AERODYNAMIC FLOW/ LOAD TRANSF.

HYDRODYNAMIC

FLOW/ LOAD TRANSF. -0-MECHANIC

LOADING

MOORING CONTROL SYSTEM

FOR

/BEHAVIOUR TRANSFER

FIG. 6 Block diagram behaviour mechanism

for moored floating structures

BEHAVIOUR

*

17.

fairly accurate by measuring the wind velocity and relative direction. If a counteracting force is applied at once

the ultimate result will be a small deviation from the

original position (ideally zero). The importance of the

high-frequency wave-induced oscillations, which cannot

be counteracted by the system, is their disturbing influence on the position reference system and the control system.

For the control algorithm only low-frequency excursions are

of interest, hence the high-frequency disturbances have to be removed by filtering to determine the correct control commands. This filtering is connected with phase-shifting

(time-delay) and consequently reduces the performance of

the D.P.-system.

Apart from analysis of moored structures comparable with those described above, a growing demand is felt for esti-mating the behaviour of floating structures during arbi-trary transient motions. Peak loads occurring with transient behaviour, for example caused by periodical changes of

the mooring forces (e.g. pulling forward of a pipe-laying barge) or by failing of a part of the mooring system

(e.g. breaking of an anchor line), may be of prime

importance for the operation and survival of many struc-tures. The response of the structure to the loads involved with this type of operation may be used to optimize the

structure'soperations. This behaviour may be estimated by mathematical simulation both during the design and operation

stage of the structure.

In view of the brief description given in this section the purpose of mathematical simulation of moored floating

structures may be defined as: "The determination of the

motions of the moored structure in order to analyse the behaviour

of both structure and mooring system in a number of speci-fied conditions". The information obtained can be used to set the operational and survival limitations and to judge the adequacy and overall feasibility of the system. In this

18.

way mathematical simulation is a powerful tool to

estimate the importance of the parameters involved with

respect to the behaviour of the structure and may therefore be of extreme importance for design and operation purposes.

Basically the parameters of importance may be split up into three groups (Figure 6):

Environmental conditions, such as the water depth, the presence of other fixed or floating structures and the

exciting forces due to wind, waves and currents. Hull form with appendages.

- Equipment, the mooring geometry and a possible mooring control system are of prime importance. Furthermore special operational equipment such as pipes, stingers,

risers, digging devices and swell compensators may

either affect the structure's behaviour or be affected

by it.

It will be clear that the interactions between these groups

of parameters cannot be neglected when the behaviour of the

struc-ture has to be determined (e.g. when the mooring is provided by means of controllable thrusters the thrust

inter-actions with hull form and current velocity fields are

of prime importance). Generally, however, each

mathe-matical simulation is made in order to analyse only a limited number of aspects of the total system. Therefore the mathematical model does not need to contain all para-meters involved in the system's behaviour but only the most characteristic ones.

In section 3 the general approaches which may be followed to derive mathematical models to simulate moored

floatina objects will be aiven. noreover, the main

contributions of the fore-mentioned _{behaviour aspects}

STRUCTURES

3.1 HYDRODYNAMIC APPROACHES

Solving the time domain equations of motion

3.2 EXCITATION LOADS

3.2.1 Hydrodynamic loads Flowfield

Flow-load transfer

Large volume structures Small volume structures

Application of fluid excited loads 3.2.2 Aerodynamic loads

Flowfield

Flow-load transfer

Application of wind excited loads

3.3 HYDROMECHANIC REACTION FORCES Linear fluid reactive forces Non-linear fluid reactive forces

3.4 POSITION KEEPING FORCES 3.4.1 General

3.4.2 Passive position keeping Anchor line systems Pile and leg moorings

Single Point Moorings

Application of mooring system analysis 3.4.3 Active position keeping

Position reference system Control System

Controlled mooring system

3.5 SPECIAL LOADS

Compound structures and mechanical interactions

Hydrodynamic interactions

19.

From the classical ship theory, typical classes of hydrodynamic approaches to the subject are discussed. Starting with linear-frequency-domain originated methods, the development of more suitable and complex models is shortly reviewed. Based on the hydrodynamic approaches, the vessel loads are separately discussed as excitation, fluid-reactive and position keeping forces. Furthermore some special loads are discussed.

The excitation loads are treated in relation to the en-vironmental conditions and the structure's dimensions

(viz, the flow field characteristics and the flow-load

transfer). Moreover the application in simulation models

is given. A separate treatment of hydro- and aero-dynamic loading is given after a discussion of the mutual inter-ferences. The fluid excited 'oads are discussed for large volume structures, where use can be made of linear potential

theories for wave loading and empirical current load

formulations, and small volume structures where non-linear effects in wave loading may be more significant.

The hydromechanic reaction forces are discussed in relation to the typical classes of simulation. Methods to determine these loads are discussed with regard to the fluid excited loads. Special attention is given to the incorporation of non-linear fluid reactive forces.

The position keeping forces are discussed in relation to

the ultimate structure's behaviour. Typical aspects of investigation are given to illustrate the importance of

time-domain-simulation and mooring. system. Passive

position keeping is treated by three examples viz,

anchor-line systems, pile and leg moorings and single point

moorings. Active position keeping is treated by discussing

the three main components viz, position measurement system manual or automatic control system and thruster or winch system. Special attention is given to thruster-hull-current interaction effects in case of D.P.-propulsion systems.

For multi-body systems mechanical and hydrodynamic

xi

X3

FIG. 7

The coordinate system

X

3. MATHEMATICAL DESCRIPTION OF MOORED AND DYNAMICALLY

POSITIONED STRUCTURES

3.1 HYDRODYNAMIC APPROACHES

To describe the behaviour of a moored floating body in a mathematical way, the dynamic interactions between the

environmental velocity fields, structure motions and the

mooring system have to be considered. The contributions of these aspects to the behaviour of the structure

under investigation should be described by a mathematical

model in a physically sound way. The type of motion of special interest will determine the validity of the

mathematical description. Because of the great variety

of physical phenomena involved in the motions of floating structures a generally valid model can hardly be developed

and each model, however complex it may be, is just an

approximative reflection of the "real world". However, even

approximate mathematical descriptions which reflect the typical

behaviour of the vessel under investigation may improve the user's insight and can be extended and adapted according to prototype or physical model test observations.

The basis of the mathematical models will be Newton's law

of dynamics:

- F _(3.1)

Since the inertia properties of the vessel may be regarded

as constant for motion analysis,

it follows:

MK = F (3.2)

All conventional descriptions consider the structure's hull as system and the loads due to environment and mooring system as external forces described by F.

To prevent transformations between different systems of coordinates the ship motions and external forces are described in a right handed coordinate system with the origin in the centre of gravity of the ship (Figure 7).

21.

dMS

FIG. 8 Classification of mathematical simulation models DESCRIPTION OF HYDRODYNAMIC LOADS HYDRO-DYNAMIC LOADING INTERACTIONS (mooring etc.) FLUID-DOMAIN VISCOUS EFFECTS EXCITING LOADS

I-- - - - -1

POTENTIAL LOW FREQ.

EFFECTS I COMP.

r---i

I HIGH / LOW! CONSTANT

CUMMINS

I FREQ. 1 COEFF.

22.

In case of small amplitudes of motion it may be assumed that the system of axes is space fixed.

The mathematical treatments of moored structures are, up to now, focussed on the description of the structure -fluid interactions. Initially these treatments were used to describe the structure's response to simple harmonic waves. Such descriptions were based on the

classical ship motion theory, isolating the fluid reactive forces due to the vessel's motions from the exciting

forces (see Figure 8). In this way the vessel was represented by a simple mass-damper-spring system:

6

(M .+a .* +c .x =F k=1,2,...6 (3.3)

kj k3 j kj 3 k3 k

3=1

Where a and b are coefficients which describe the com-ponents of the hydrodynamic force in- and quadrature-phase with the motion, respectively called "added mass" and "damping". The coefficient c presents the hydro-static restoring force due to buoyancy.

The isolation of the fluid reactive forces, which is a

con-sequence of the assumption of linearity, may be justified

by the dominating influences of fluid inertia and gravity

on the small motions of large bodies in small amplitude

waves. Moreover this approach is widely validated for free-floating ship-like hulls by numerous experiments.

The equation of motion pointed out above, cannot be considered as an instantaneous relationship between the motion variables and arbitrarily varying external forces because the hydrodynamic coefficients for added mass and damping depend on the frequency of motion. Therefore it can only be used in the frequency domain considering steady harmonic motions due to pure sinusoidal excitation such as first order wave forces. Only restoring forces propor-tional to the vessel's displacements and their time

derivatives may be involved by adding these to the hydromechanic reaction terms.

23.

The behaviour of many offshore systems may be

significant-ly influenced by strong non-linear effects in the restoring

forces:

- The fluid reactive forces may be non-linear due to large amplitudes of motion and viscous effects. Hydro-static restoring forces may be non-linear especially for complex hull forms and large excursions.

Moreover, second order pressures resulting from the motions contribute to the wave drift forces.

- The restoring force characteristics are in general

strongly non-linear. Cable and cable-buoy moorings will provide non-linear mooring forces due to their geometry, catenary effects and material properties. In case of active positioning non-linearities may also arise from the measuring and control system. Moreover, the forces delivered by thrusters may be of non-linear origin

due to propulsion-thrust characteristics and interaction effects.

Many investigators have used the frequency domain

description to treat the dynamic behaviour of vessels due to the non-linear forces mentioned above. Restoring

forces were taken into account by linearization or

extending the equations of motion with non-linear terms and solving them by methods of equivalent linearization. Non-harmonic components of motion were studied by

regarding the frequency domain description as actual differential equations of motion thus neglecting the frequency dependency of the hydrodynamic coefficients.

As it appears from the description given in section 2

the low frequency behaviour coherent with the non-linearities previously mentioned govern the operation and, by doing

so, the design of many moored systems. Moreover, model test programs and numerical calculations have clearly shown that the hydrodynamic coefficients describing the acceleration and velocity components of the hydrodynamic reaction forces are sensitive to changes in frequency

of motion especially in waters of restricted depth. Consequently, in general frequency domain descriptions will not be valid to describe the fluid reaction forces due to arbitrary motions of moored structures.

To overcome the problems involved with the frequency-dependency of the hydrodynamic reaction forces, it is suggested by several investigators to distinguish basically two components of motion which may be based on the nature of the exciting loads:

high frequency oscillatory motions due to loads with frequencies equal to the first order wave frequency, low frequency oscillations due to slowly varying loads which may be caused by wind and current, but also by higher order wave forces.

This approach is based on the following assumptions: the restoring forces for high frequency motions have pure linear characteristics,

the fluid reactive forces for low frequency motions are frequency-independent,

no interaction effects between low- and high-frequency behaviour are present.

The high frequency components of motion may be described by the conventional frequency domain theory. The low

frequency components may be treated by describing the fluid reactive forces with constant coefficients because the frequency range of these motions is sufficiently low and narrow. The viscous reaction forces may be handled together with possible current loads, while other mean and slowly varying external loads due to wave- and wind action and mooring arrangements may be described as

functions of the structure's motions and the time. In this way the low frequency behaviour may be described in the time domain.

25.

By superimposing a high frequency motion time domain

realization on the low frequency components, a non-linear time domain description of the overall structure's behav-iour may be obtained.

When the assumptions mentioned above are not fulfilled or

when the behaviour of the system is characterized by

non-steady-state or transient behaviour the approaches lined out above cannot be justified. Strong interaction effects on the behaviour of the total structure may be caused by non-linear mooring characteristics, resulting in subharmonic

motions even in regular waves. Furthermore, high frequency

motions contribute to the wave drift forces and sometimes also

to low frequency characteristics of the mooring system (e.g. time delay caused by filtering in case of active positioning). Shock loads and transient motions may be

observed when the vessel is connected to earth or to another structure by means of a more or less rigid construction (e.g. the fender-mooring line connection to a jetty, the ladder of a cutter-suction dredger). Moreover, the analysis of transient behaviour due to controlled changing, or failing of the mooring system is an important aspect of many vessels moored offshore.

A flexible approach of this type of dynamic processes may be, the use of the impulse response function technique to describe the fluid-reactive forces. This technique, which is based on the linearity of the system, describes the response to an arbitrarily varying force F(t) when the

response to a unit impulse at the time t=T, is known

(R(t-T)):

x(t) = I

F(T)

(3.4)

-CO

The impulse response theory has been used by Cummins [6] to formulate the fluid reactive forces due to ship

motions by considering the vessellsvelocity as system

26.

The equations of motion in the time domain, obtained in

this way (see Appendix) have the following form:

6

X (Mki+ mki)Rj +f Rkj(t-T)Xj(T)dT + Ckjxj = Fk(t)

j=1 j

k = 1, 2 , 6 (3.5)

Where: x. = motion in j-direction

]

Fk(t) = arbitrarily in time varying external force in the k-mode of motion

= inertia matrix

Mkj

Ckj = matrix of hydrostatic restoring forces

Rkj = matrix of retardation functions

mkj = added-inertia matrix.

The only basic assumption according to Cummins formulation, viz, the separate treatment of the linear hydrodynamic

reaction forces and all other external forces, may be justified by known experimental techniques and

numerical computations. Non-linearities of hydrodynamical reaction forces caused by viscous effects and high waves may often be treated in a separate way. Moreover, recent

simulation studies based on this formulation by Van

Oortmerssen [1], Muga and Freeman [7] and Shinozuka et al.

[8] have shown its validity and flexibility for a wide

scope of mooring cases.

In recent years, more rigorous attempts have been made to solve the complete non-linear fluid-structure dynamics without separating the hydromechanic reaction forces. To

illustrate the approach a method developed by Bourianoff

and Penumalli [9] is shortly discussed. This method yields the coupling of the incompressible viscous flow equation with the equations of ship motion. A kinematical boundary

condition is formed by the structure-fluid interface where the fluid velocity is equal to the body velocity

(reactive boundary). Furthermore the fluid is bound by

rigid-, stationary- and free-surface, together with outflow and inflow (wave action) conditions.

27 .

The fluid region may be represented by finite difference

meshes for velocities and pressures. The solution

tech-nique yields the approximation of the time boundary by a

boundary of mesh cells for the free surface and reactive

boundaries. After setting of the boundary conditions the

right flow field may be found from iterating velocities and pressures until the incompressibility condition is

satisfied. Then the free surface and reactive boundaries

are updated in preparation for the next time step.

Basically this method is very flexible, and preliminary

calculations have shown encouraging results. Unfortunately,

the suitability for non-linear processes in general has

not been proven yet, and the physical insight gained from

the use of these complex methods seems to be considerably less, compared with the previous methods. Moreover, the computer

time needed for these techniques is extremely long.

There-fore, up to now, hardly any use is made of these

tech-niques and the interest is focussed on the application

of the constant coefficient approaches and the Cummins formulation.

Soloing the time domain eql4ations

Once the system of coupled differential equations is obtained the solution may be approximated by numerical methods such as the finite difference technique. Knowing

the displacements and their time derivatives until a certain time, the simulation may be continued with a small time step predicting the velocity from the acceleration and the known time histories by use of time series expansions. The new position may then be predicted by numerical inte-gration of the velocities.

The computer time necessary to solve the equations of

motion is proportional to the number of time _{steps to be}

taken. Consequently the overall simulation time and the time step size, which is upperbounded by the system's stability, determines the computation time. Especially

28.

in cases of high "stiffness", viz, systems with very

high restorinj force-disnlacement ratios, snecial

predictor-corrector methods have to be applied to improve the convergence and to reduce the computer time.

In order to obtain the equations of motion in the time domain, the coefficients which describe the hydromechanic reaction forces have to be determined. Moreover, all

external forces which contribute to the aspects of the behav-iour of the structure under consideration have to be known

as functions of the vessel's position, their time derivatives

and the time itself. These loads, which may be named

according to each nature; excitation _{fluid reaction and}

position keeping forces, will be discussed in the _following

sections together with their most important interaction

effects.

3.2 _{EXCITATION LOADS}

The presence of a structure in the hydro-and aerodynamic velocity fields results in exciting loads due to the disturbances of the flow. The contributions of these

loads, generally referred to as wind, wave and current loads, to the total external loads on the structure should be determined in terms of time histories.

To obtain the ultimate loading quite different approaches may be followed. The choice and application of the

approach depends on the simulation goal and the environ-ment under investigation. Deterministic analysis of the structure's behaviour demands an actual representation of the desired environment. However, the environmental phenomena are characterized by a stochastic nature which greatly depends on time and location. Consequently,proba-bilistic approaches are often favoured. In general terms

the description of the flow fields should cover the structure's area as a function of time.

29.

The flow fields used for simulation could be taken equal to the conditions at the actual location under investi-gation. Unfortunately the measurements involved in the collection of this information are time-consuming and expensive. Moreover the statistical approach needed to describe the long term process requires extremely long observation times. Therefore the environmental conditions

used for simulation are often based on mathematical and physical models of the flow fields taking into account the

local circumstances such as the seafloor topography,

waterdepth, current and wind. The operational and

design conditions are selected in terms of the average number of days per year that the environment permits operation ("workability") or the average number of years that passes between the occurrence of a certain condition

("return period").

Hydrodynamic-aerodynamic interactions

The atmospheric flow, resulting from pressure gradients caused by unequal heating of the earth's surface, basically determines the greatest part of the exciting loading of

marine structures. The aerodynamic flow field _exerts

forces on the part of the structure above the water

sur-face, on the other hand it generates wave action and wind currents. Apart from reflection, cloud cover,

preci-pitation and rotation of earth, the flow system is also influenced by topography and surface roughness. For exposed offshore locations this means that the wind

field is affected by the water surface conditions in a

large area across the location.

The local water surface elevation is partly due to local

wind action inducing irregular, directionally _{spread waves}

by pressure action and friction, the _{so-called "sea state".}

This process is governed by the wind velocity, _-direction,

-fetch and duration. Furthermore, the local wave action

Waves

As mentioned in the previous part of this section, the

local water surface elevation is due to the irregular,

directionally spread sea state basically induced by the local

wind condition, and also due to regular _{unidirectional swell.}

In general terms the wave action is characterized by high frequency orbital motions of the water particles,

asymptotically decreasing with the depth beneath the

surface. The orbital paths of the water particles tend to circles for deep water. The path radius and period at the surface are equal to the surface elevation and wave period respectively. Typical periods of the irregular seas are

30.

waves radiated from remote wave fields. This is the

so-called "swell". The wave system is also influenced by

currents partly generated by wind (up to 3 percent of the

wind velocity at 10 m height).

Although the wind, wave and current actions are

character-ized by mutual interferences, an integral approach to the velocity fields and the ensuing structure loads is

obviously unsuitable and not necessary. Therefore the

environmental loadings are dealt with more or less

separately. In general, the wind and current are defined according to the assumed local conditions. The wave action

definition is often based on the aerodynamic velocity

field generally characterized by velocity, direction, duration and fetch. Furthermore the local seafloor

topo-graphy, current and swell are taken into account to simulate the hydrodynamic flow field accurately.

Unlike the separate treatment of the aero- and hydrodynamic loadings, a division into wave and current components of the fluid loading cannot always be justified because of non-linear flow load transfer characteristics of some

struc-tures.

3.2.1 Hydrodynamic loads Flow field

31.

wiLhin the range of 5 to 20 seconds while swell periods

range from approximately 15 to 20 seconds. Apart from

the initiating local and distant wind field, the wave

condition is influenced by possible currents, the

water-depth and the surrounding sea-floor topography.

Current

The occurrence of currents may be caused by ocean

cir-culation, tidal effects, river flows and density differences, but also by wind. Because of the minor influence of wind,

which exerts only upper flow velocities less than a

few percent of the wind speed, the current velocity field is characterized by a steady flow with very low frequen-cies of changing. Typical current velocities range up to 4 knots. The current flow is strongly affectedby the sea-floor condition and the water depth. The vertical velocity distribution caused by bottom disturbances at the one hand, and wind and wave influences on the other, may be of

importance, especially in areas of restricted water depths.

Combined wave and current action results in oscillatory orbital flows with non-zero mean. Positive currents, in

the direction of the waves, tend to lengthen the waves

and thus reduce the energy of the waves. On the other hand,

adverse currents steepen the waves and increase the energy

density of the wave system. Moreover, _{wave height and length}

but also

phenomenon when a wave system encounters an area of

different current speed. Therefore the surface wave

spectrum, defining the energy distribution over the frequency components of the wave system, is affected by the magnitude and the direction of the current.

Subsequent-ly this will result in different fluid _loading.

A great number of moored structures have to operate in

intermediate and shallow waters, sometimes even near the

shore. Hence the influence of the _{sea floor topography}

viscosity force 1JUL

drag force amplitude _UT

Keulegan-Carpenter

Kc (3.8)

inertia force amplitude L

L.f

Strouhal S, = hydrodynamic resonance = (3.9)

32.

Wave action is changed in shallow water by reduced wave

lengths caused by prevention of fully developed orbital

motions. This occurs when the wave length-depth ratio is

smaller than approx. 2 to 4. The wave height is

upper-bounded by the steepness (maximum approx. 1/7). Therefore long waves will often break when encountering shallowing

waters. The wave direction is changed when the wave system progresses through areas of different water depth by

changes of the wave celerity. Depending on the refraction pattern theenergy of the wave system will either increase or decrease. Near shore the wave system will be affected by reflection and diffraction effects.

Flow-load transfer

The fluid excited loads may be described by the effects of gravity, inertia and viscosity, which mainly govern the dynamics of fluid motion. Other phenomena as com-pressibility, non-homogeneity, surface tension and vapour pressure of the fluid, are usually of minor importance with respect to the motions of floating structures. The relative importance of the three contributions are determined Joy the

characteristic dimensions of the structure, the fluid velocities and accelerations and the fluid physical properties. To judge the fluid dynamics use may be made of the wellknown characteristic numbers:

inertia force pU2L21

Froude _{Fn - (3.6)}

gravity force

_pLg

inertia force pU2L2 UL

In which:

characteristic dimension characteristic velocity

velocity amplitude of oscillating flow

period of oscillation

frequency of vortex shedding p/p kinematic viscosity of fluid acceleration due to gravity.

Both Reynolds and Keulegan Carpenter number indicate that for typical wave-current flow problems involved with large floating structures the contribution of viscous forces to the total

hy-drodynamic force is negligibly small compared to the inertia force contributions. Moreover, the Froude number shows that grav-ity effects may be of importance in case of large characteristic

structure dimensions. The forces due to inertia and

gravity effects may be described by potential theory.

The effect of viscosity resulting in frictional drag due to shear stress between structure and fluid and pressure drag due to flow separation, may be of importance in case of steady currents (friction) and in cases of blunt or

bluff bodies such as spheres, circular cylinders and flat

plates perpendicular to the fluid motion (boundary layer separation). The rolling up of unstable shear layers

results in the shedding of eddies. Alternating shedding

causing the well-known Von Karman vortex street, may produce lift forces perpendicular to the incident flow. This phenomenon may occur at a particular value of the Strouhal number and may lead to structural vibrations

and an increase of drag forces in case of slender flexible bodies; the so-called "strumming".

From the short description given above it will be clear that the relative magnitude of structural dimensions with regard to the wave height and length and the current velocity

34.

governs the loading and consequently the choice of the

method of calculation.

Large volume structures

The inertia and gravity effects dominating the high frequency oscillating wave forces on these structures are generally assumed to be proportional to the wave height in case of small amplitude regular waves.

Conse-quently these wave forces may be treated separately from

possible non-linear wave and current forces.

Waves

The assumed linearity and the dominating effects of inertia and gravity justify estimates of first order wave forces by numerical methods based on the linear potential theory. This approach supposes small amplitude sinusoidal waves while the fluid is assumed to be ideal and irrotational. When these prerequisites are fulfilled, the flow field around an arbitrary structure may be defined by a velocity

potential representing the undisturbed _{incident wave}

("Froude-Kriloff"-part) and the contribution of the flow

disturbance by the structure ("Diffraction"-part). _All

potential contributions have to fulfill _{the Laplace}

equation and the boundary conditions _{at the free surface, on}

the sea floor, on the body's _{surface and at infinity.}

The linearization of the free surface condition may be

justified in these cases by the _{small amplitude of the waves}

and the minor influences of current. When the time and space dependent potential function is known, the wave excited

forces may be determined by integrating the hydrodynamic pressure, following from Bernoulli's theorem, along the wetted surface of the vessel.

Analytical solutions of the potential function are

restrict-ed to bodies with simple geometry which can be describrestrict-ed by simple mathematical functions, such as spheres, ellips-oids and cylinders. For that purpose use has to be made of

numerical methods (see Bai and Yeung [0).

The velocity potentials may be determined by means of a distribution of singularities such as pulsating sources

over the wetted surface of the structure. The source

potential may be described by a Green's function satisfying

the Laplace equation and the boundary conditions

_0-11:

F141).

approaches are valid for both deep and shallow water and arbitrary hull shapes.

The diffraction problem around a 3-dimensional structure

can also be solved by finite element methods. These methods are based on a subdivision of the fluid domain into a mesh of elements for which the potential may be approximated. The subdivision may range over the complete fluid domain or be localized by using analytic solutions in the outer

flow field.

When the longitudinal dimensions of the floating _structure

are large relative to the lateral and vertical dimensions, use can be made of slender body theories. Unfortunately the

assumptions to be made and numerical procedures _{to solve}

the problem restrict the region of application. _The

well-known strip theory reduces the 3-dimensional problem to a two-dimensional one. The wave forces are obtained

from the sectional values derived by conformal transformation or two-dimensional linear potential methods. Excellent

agreement with model test results have been found, but the

method is restricted to deep water and _{slenderness ratios}

above approx. 3. Moreover, no information is obtained for

surge.

Inherent to the linear approach followed _{at the determination}

of the first order wave loads according _{to the methods}

discussed above, is the preferable calculation in the frequency domain. Irregular and high amplitude waves may be split up in a number of harmonic components so that the