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1. General

Offshore operations involve diverse types of structures. Nowadays, more than 3 vessels can

commonly operate simultaneously. These structures interact with each other by means of

mechanical links (lines, yokes, fenders, …), hydro-mechanical coupling (wave diffraction and

wave radiation), and shielding effects (wind, current). Especially when operating in exposed

areas, the dynamic behaviour of the multi-body system is of prime importance from the point

of view of feasibility, structural loading, workability and safety. Excitation due to first and

second order wave forces, wind gusts and current, the hoisting action, ballast procedures etc.

on the one hand and subsequent forces due to fluid reaction, mooring and mechanical

reactions on the other hand, form a complex system basically with non-linear characteristics.

This system features both stochastic and deterministic components of behaviour. Wind and

waves are of random nature and response to this type of excitation can be evaluated by

statistical analysis. Manually and automatically controlled actions such as hoisting, ballasting

and various winch operations result in deterministic transient behaviour. The overall

behaviour of the system contains components of both types and therefore depends on the

phase relations between stochastic and deterministic aspects.

The computer program aNySIM provides a time domain description of the motions of a

system involving several floating bodies and dry bodies. In total a system with Nbody*6

degrees of freedom is described in this way. Though basically deterministic, this approach

may be used for a stochastical approach. Motions and diverse loads may be simulated for

given environmental loadings. Statistical analysis will provide the required information.

With version 8.0 of aNySIM it is possible to perform the following simulations:

A general N body simulation with 6*N degrees of freedom.

A simulation of the ship moored to a jetty (a quay or a gravity based structure).

A side-by-side simulation involving 2 or more vessels.

A simulation of one or more moored vessels account for static and/or dynamic

mooring loads.

A simulation of one or more vessels under dynamic positioning control.

A simulation of a free hanging load hanging at the crane(s) of a crane vessel (without

hoisting action).

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The mathematical model can be outlined as follows:

the time domain analysis with 6*N degrees of freedom is based on the equations of motion

according to Cummins [1], with convolution integrals which cope with the frequency

dependency of the added mass and damping coefficients of all bodies

the environmental excitation forces may be arbitrary, which means that first order wave

forces, slowly varying drift forces, current and wind loading can be included

since retardation functions and first and second order wave force records can be generated

from the results of frequency domain analysis, on a basis of three-dimensional potentional

theory, arbitrary hull shapes are allowed

all transformations done in aNySIM are still valid for large rotation angles

additional arbitrary forces may be specified as functions of all state variables (including time)

through a special user program interface

The basic assumptions taken into account in the development of aNySIM are as follows:

Fluid reactive forces may be treated separately from first and second order wave

forces;

Retardation functions and added mass are considered with complete mutual

interaction;

First order and second order transfer functions, retardation functions and added mass

are fixed for the whole simulation duration (they are not updated when the positions

and the orientations of the bodies change);

Frequency dependent fluid reactive forces may be described within the potential

theory, viscous effects may be treated separately

The bodies involved in the simulation are large and heavy bodies like usually met in

offshore operations.

2. Theory

aNySIM is a time domain simulation program. This section explains what’s the procedure

followed by aNySIM to carry out simulations. It also explains how impulse response functions

are applied to the motions of floating structures.

2.1. Computational procedure in time domain

In order to describe the response of a moored floating structure to arbitrary environmental

loadings in the time domain, the simple mass-damper-spring analysis used in classic ship

motion theories can, in general, not be used. This analysis represents the equations of

motions as:

6

kj kj j kj j kj j k

j 1

(M a )x b x c x F k 1, 2 ... 6=

+ + + = =∑ && & (2.1)

where:

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M = mass of structure

a = added mass matrix

b = damping matrix

c = hydrostatic restoring matrix

Fk = external force in the k-th mode

kj = modes of motion.

The dependence of the hydrodynamic coefficients on the frequency of motion and non-linear

restoring force characteristics provided by the mooring system do not allow the linear

frequency domain approach nor the use of equation (2.1) in the time domain. To overcome

these problems use can be made of the impulse response theory to describe the fluid

reactive forces. A brief description of this approach is given in Section 2.2.

The equations of motion derived within potential theory describe the fluid reactive forces on a

floating structure under arbitrarily external loads varying in time. For six degrees of freedom

these equations can be written as follows:

t6

kj kj j kj kj j kj 1

(M m )x R (t )x( )d C x F (t) k 1, 2 ... 6= −∞

+ + − τ τ τ + = =∑ ∫&& & (2.2)

where:

xj = motion in j-th mode

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

M = inertia matrix

m = added inertia matrix (frequency independent)

R = matrix of retardation functions

C = matrix of hydrostatic restoring forces.

The only basic assumptions in this approach, viz. the separate treatment of the hydrodynamic

reactive forces and all other external loads, may be justified by the results of formerly

conducted research. The applications of the MARIN time domain simulation program

MOORSIM, which is based on the approach outlined above, have shown satisfactory

agreement with model test results [2].

In ANYSIM a general 1, 2 or N body system of motions is simulated with 6, 12 or 6*N

degrees of freedom respectively. All N bodies are (optionally) subject to wave induced forces,

hydrodynamic reaction forces and wind loads. Interaction forces between the rigid bodies

may either be linear or non-linear. Also mooring line forces may be included for the three

bodies.

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For instance, in the case of three bodies, the motions of the three coupled rigid bodies can be

described by 18 coupled differential equations of motion.

M11 M12 M13 1x&&

t

11

0

R (t )− τ∫ t

12

0

R (t )− τ∫ t

13

0

R (t )− τ∫ 1x&

M21 M22 M23 x 2x&& +

t

21

0

R (t )− τ∫ t

22

0

R (t )− τ∫ t

23

0

R (t )− τ∫ x 2x& +

M31 M32 M33 3x&&

t

31

0

R (t )− τ∫ t

32

0

R (t )− τ∫ t

33

0

R (t )− τ∫ 3x

c11 0 0 1x 1F

0 c22 0 x 2x = 2F 1 1 2 2 3 3(x ,x ,x ,x ,x ,x ,t)& & & (2.3)

0 0 c33 3x 3F

where:

M = inertia and added inertia matrix

R = matrix of retardation functions

C = matrix of hydrostatic restoring forces

x = motion vector

F = vector of external forces such as: - first order wave forces

- low frequency drift forces

- wind forces

- non-linear viscous damping forces

- interaction forces between three rigid bodies

- current forces

- mooring line forces

The numerical solution of the 18 second order differential equations (2.3) is carried out

according to the following global procedure. Suppose the simulation has arrived at the

moment t, while ∆t is the time increment applied, then the equations of motion have to be

solved for the moment t+∆t. First, the velocities for t+∆t’ are predicted, in which ∆t’ is a

fraction of ∆t depending on the chosen integration method (see Section 4.1).

Subsequently, the new position and orientation are predicted by numerical integration. For

these new coordinates and time (t+∆t') all forces can be calculated. After substitution of these

forces in (2.3), 18 linear equations are obtained from which the accelerations x&& (t+∆t) can be

found. Finally, the predicted velocities are checked by integration of the accelerations. In

case the difference is acceptable, the computation continues for the next time step; if not, the

time increment has to be decreased. This process continues until the equation of motion at

t+∆t are solved correctly.

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Based on the mathematical description of motions and forces as outlined above, the

computational procedure of ANYSIM has been developed.

As already indicated by equation (2.3) the computer model describes the motions of the N

individual bodies including the mechanical coupling due to crane systems and impacts.

Computation of fluid reactive forces by means of retardation functions may be considered as

linear filtering of the structure's velocities history. First and second order wave forces are

obtained from linear and quadratic transfer functions and the wave time record by means of

the internal computation based on the same approach as the program FORSIM. Refer to the

section [Wave] for futher explanations.

2.2. Impulse response function

When the response (R(t-τ) of a linear system to a unit impulse at the time t = τ is known, the

response of the system due to an arbitrary impulse δ is:

x(t) R(t ) F( ) R(t )= δ − τ = τ ∆τ − τ

Owing to the linearity of the system the response to an arbitrary force F(t) may be found by

superposition:

n

0x(t) lim F( ) R(t )

∆τ→= τ − τ ∆τ∑

or:

t

x(t) F( ) R(t ) d−∞

= τ − τ τ∫

This formulation is known as the Duhamel, Faltung or convolution integral.

The impulse response theory has been used by Cummins [1] to formulate the equations of

motion for floating structures by isolating the free-floating body in still water for which the

hydrodynamic restoring forces are assumed to be linear. According to Cummins the reaction

forces due to the water velocity potential may be derived by the impulse response theory by

considering the vessel’s velocity as system input. An impulsive displacement at t = 0 by a

constant velocity v, during a short time ∆t yields:

t 0x v t, with : lim v x

∆ →∆ = ∆ = &

During the impulse the water velocity field may be described by:

vφ = ψ

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where ψ is a normalised velocity potential fulfilling the fluid boundary conditions. After the

impulse the radiated disturbing wave may be described by a potential proportional to the

displacements because of the assumed linearity of the system:

(t) Xφ = χ ∆

For the total impulse at an arbitrary time t, due to an arbitrary motion it now follows:

t

(t) x (t ) x( ) d−∞

φ = ψ + χ − τ τ τ∫& &

By integration of the linearized hydrodynamic pressure, obtained from Bernouilli’s theory over

the wetted surface, the hydrodynamic reaction quantities are found:

S w

S w

m n ds

(t)R(t) n ds

t

= ρ ψ

∂χ= ρ

∂

∫ ∫

∫ ∫

where n is the normal vector.

By applying Newton’s law to these quantities together with the hydrostatic restoring forces

and the external loads, the equations of motion for six degrees of freedom can be derived:

t66

kj kj j kj j kj j k

j 1

(M m )x R (t )x ( ) d C x F (t)= −∞

+ + − τ τ τ + =∑ ∫&& &

where:

xj = motion in j-direction

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

Mkj = inertia matrix

Ckj = matrix of hydrostatic restoring forces

Rkj = matrix of retardation functions

mkj = added inertia matrix

k,j = model of motion.

As derived above, the frequency independent coefficients of inertia and the retardation

functions can be computed from the velocity potential. By substitution of a harmonic motion

into the time domain equations and comparison with the frequency domain equations Ogilvie

[3] has derived the relationship between the time domain and frequency domain quantities:

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kj kj kj

0

kj kj

0

1a m R (t) sin t dt

b R (t) cos t dt

∞

∞

= − ωω

= ω

∫

∫

where:

akj = frequency dependent added mass coefficient

bkj = frequency dependent damping coefficient

ω = circular frequency.

From these relationships the retardation function and the matrix of added inertia coefficients

may be found by inverse Fourier-transformation:

kj kj

0

kj kj kj

0

2R (t) b ( ) cos t d

1m a ( ') R ( ) sin ' t d

'

∞

∞

= ω ω ωπ

= ω + τ ω τω

∫

∫

Apparently the retardation functions and the coefficients of added inertia can be derived from

frequency dependent damping values and the added mass at one frequency.

Once the system of coupled differential equations is obtained, arbitrary in time varying

loading such as wave excited forces, current forces, non-potential fluid reactive forces and

non-linear mooring forces may be incorporated as external force contributions.

The solution may be approximated by numerical methods such as the finite difference

technique. Knowing the displacement and its 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 integration of velocities.

3. Solver

The program uses the DVERK suboutine to solve the equation of motions. This subroutine is

based on Verner’s fifth and sixth order pairs of formulas. This subroutine has been

developped by the University of Toronto.

The Runge-Kutta-Verner method is an adaptive procedure for approximating the solution of

the differential equation y’(x) = f(x,y) with initial condition y(x0) = c. The implementation

evaluates f(x,y) eight times per step using embedded fifth order and sixth order Runge-Kutta

estimates to come up with the solution and also the error. The next step size is then

calculated using the preassigned tolerance and error estimate. It attempts to keep the error

proportional to the tolerance specified by the user.

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The program is efficient for non-stiff systems.

[1] HULL, T. E., ENRIGHT, W. H., AND JACKSON, K.R. User's guide for DVERK--a

subroutine for solving non-stiff ODE's. Computer Science Tech. Rep. 100, Univ. of Toronto,

Oct. 1976.

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