SeaFEM Theory Ma SeaFEM Theory Manual Theory ... - Compass

1. Table of contents
1. Table of contents ....................................................................................................... 2
2. Introduction ............................................................................................................... 3
3. Problem statement ..................................................................................................... 4
Governing equations ..................................................................................................... 4
Velocity potential decomposition ................................................................................. 4
Radiation condition and wave dissipation .................................................................... 5
Free surface potential flow problems with currents ..................................................... 6
4. Numerical formulation .............................................................................................. 8
Free surface boundary condition .................................................................................. 8
Stremalines scheme for the free surface boundary condition ....................................... 9
SUPG stabilization scheme for the free surface boundary conditions ....................... 12
Free surface kinematic boundary condition................................................................ 15
Free surface boundary condition with absorption ...................................................... 19
Radiation boundary condition .................................................................................... 19
Body boundary condition ........................................................................................... 20
Boundary condition for limit height (Hfs) .................................................................. 20
Boundary condition for transom stern ........................................................................ 21
5. Multi-body dynamics .............................................................................................. 22
Body dynamics ........................................................................................................... 22
Non-linear hydrostatics............................................................................................... 23
Body links ................................................................................................................... 23
7. Body matrices referred to an arbitrary generic point .............................................. 25
8. Statistical description of an irregular sea ................................................................ 27
Spectrum discretization .............................................................................................. 27
Convergence ............................................................................................................... 28
Spectral moments ....................................................................................................... 28
Waves spectrum .......................................................................................................... 29
9. Mooring system modelling...................................................................................... 31
Catenary equations ..................................................................................................... 31
Elastic catenary formulation ....................................................................................... 33
Dynamic cable formulation ........................................................................................ 36
10. Morison’s forces .................................................................................................. 41
11. Response Amplitude Operators (RAOs) ............................................................. 43
12. Fluid-Structure interaction algorithm .................................................................. 44
13. References ........................................................................................................... 45
14. Glossary ............................................................................................................... 47

3 Introduction
2. Introduction
SeaFEM deals with real problems regarding ocean waves interacting with floating
structures. These sorts of problems can be simulated using potential flow theory along
with Stokes perturbation approximation. In this sense, SeaFEM is a time domain solver
based on the finite element method which is capable of solving multi-body seakeeping
problems on unstructured meshes. Since it is based on Stokes wave theory, no remeshing
or moving mesh technique are needed, which keeps computational costs and
computational times low. The algorithm has been adapted to include non-linear external
forces, like those used to define mooring systems.

4 Problem statement
3. Problem statement
Governing equations
In SeaFEM, the first order diffraction-radiation problem of a set of floating bodies is
considered:
∇ = 0
+ + = 0
=
− = 0 = (1)
∇ ∙ n = v ∙ n
φ = 0
= −
where ∇ = ( , ) is the gradient in the horizontal plane, Ω is the fluid domain,
Γ represents the wetted surface of the ship, is the water depth, v is the local ship
velocity of a point over the wetted surface, n is the normal of the ship wetted surface
pointing outwards the ship, and is the free surface pressure. The domain is assumed to
be infinite in the horizontal directions.
Velocity potential decomposition
The aim of SeaFEM is to simulate the dynamics of a set of floating bodies subjected to
the action of waves. To do so, the environment is modelled as the sum of a number of
airy waves. This can be expressed in terms of a velocity potential given by:
cosh(| |( + ))
= cos(|
cosh(| |)
|( + − + )
(2)
where are the wave amplitudes; are the wave frequencies; are the wave
numbers; are the wave directions; are the wave phases. From this point on, we
will refer to as the incident potential. This potential, along with the dispersion
relation = | | tanh(| |) , fulfils eq. (1), and therefore is solution of the
mathematical model in the absence of bodies.
In order to obtain the solution to the governing equations, a velocity potential
decomposition is used. Let be the solution to the governing equations. Then can be
decomposed as = + and = + , where represents the velocity potential of
waves diffracted and radiated by the bodies.
Introducing the velocity potential decomposition into the governing equations the
system of equations to be fulfilled by is obtained:
∇ = 0
+ + = 0
=

5 Problem statement
− = 0 = (3)
∇ ∙ n = (v − ∇) ∙ n
= 0
= −
Hence, the purpose is to find for a given incident potential and for a given v .To this
aim, eqs. () will be solved in a finite domain by means of the finite element method.
Once the velocity potential has been obtained, the pressure at any point can be
calculated from:
= −( + ∙ ∇ + ∇ ∙ ∇ + ∇ ∙ ∇ + ) (4)
Radiation condition and wave dissipation
Waves represented by φ are born at the bodies and propagate in all directions away
from the bodies. These waves have to either be dissipated or to be let go out the domain
so they will not come back and interact with the bodies. Because of this, a Sommerfeld
radiation condition at the edge of the computational domain is introduced:
∂
tφ
+ c∇φ
⋅ n
R
= 0 on Γ
R
(5)
where Γ
R
is the surface limiting the domain in the horizontal directions, and c is a
prescribed wave velocity. eq. (5) will let waves moving at velocity c to escape out the
domain. However, waves with very different velocities will not be leaving the domain.
Hence, wave dissipation is also introduced into the dynamic free surface boundary
condition by varying the pressure such that:
P / ρ = κ(
x) ∂ φ
(6)
a
z
where κ ( x) is a damping coefficient. eq. (6) increases pressure when the free surface is
moving upwards, while decreases the pressure when the free surface is moving
downwards. By doing this, energy is transferred from the waves to the atmosphere and
waves are damped. However, the coefficient κ ( x) will be set to zero in the area nearby
the bodies so that damping will have no effect on the solution of the wave structure
interaction problem.
Combining the dynamic and kinematic boundary conditions, introducing eq. (5), and
adding eq. (6), and choosing C ' = 0 , the governing equations for φ become:
φ
2
∇ = 0
in Ω
∂ φ = −g∂ φ −κ(
x) ∂ ∂ φ
on z = 0
tt z t z
∂ zφ = 0
on z = − H
( ψ )
∇φ
⋅ = −∇ ⋅
nB vB n
B
on Γ
B
∂
tφ
+ c∇φ
⋅ n
R
= 0
on Γ
R
(7)

6 Problem statement
η = − 1 Pa
t
g
∂ φ − ρ g
on z = 0
where the free surface elevation has been decoupled from the problem of obtaining the
velocity potential.
Free surface potential flow problems with currents
SeaFEM is also capable of solving free surface potential flow problems in the presence
of water currents. Within this context, the governing equations for the first order
diffraction-radiation wave problem are:
∇ = 0
+ · ∇ + 1 2
∇ · ∇ + + = 0
+ ( · ∇ ) · ∇ − = 0
( + ∇) · = ·
= 0
in Ω
in z = 0
in z = 0
in Γ b
in z = -H
(8)
where ∇ is the gradient in the horizontal plane, is the local body velocity of a point
over the wetted surface, and is the water current. In this case, once the velocity
potential is obtained the pressure field can be calculated straightforward using the
Bernouilli equation:
= − + · ∇ + ∇
+ (9)
2
As in the case without currents, the problem is solved by first decomposing the velocity
potential and the free surface elevation as follows:
= +
= + (10)
where the first terms in the right hand side account for scattered wave components and
the second terms concern incident waves.
By doing this, it is possible to split the governing equations into two sets of equations.
The first set of equations reads as follows:
∇ = 0
+ · ∇ + = 0
+ · ∇ − = 0
= 0
in Ω
in z = 0
in z = 0
in z = -H
(11)
and has Airy waves transported by a uniform current as analytical solution.

7 Problem statement
cosh(| |( + ))
= cos(
cosh(| |)
( − ) − + )
(12)
= ( · ( · ) − + )
The second set of equations concerns the scattered wave’s potential and, after
neglecting second order terms, reads as follows:
∇ = 0
+ · ∇ + 1 2
∇ · ∇ + ∇ · ∇ + + = 0
+ ( · ∇ ) · ∇ + ∇ · ∇ − = 0
∇ · = ( − − ∇) ·
= 0
in Ω
in z = 0
in z = 0
in Γ b
in z = -H
(13)
Note that in the presence of water currents, the velocity potential and the free surface
elevation problems cannot longer be decoupled. Notice also that the terms ∇ · ∇
and ∇ · ∇ account for the deviation of the incident Airy waves due to the fact that
incident waves are transported by a non-uniform flow field.
Within the approach used in SeaFEM, it is convenient from a numerical point of view to
solve the governing equations in a frame of reference fix to the floating body rather than
on the global frame of reference. For an observer moving with the floating body, the
flow field around the body will be:
() = − () (14)
and the governing equations in the local frame of reference become:
∇ = 0
+ · ∇ + 1 2
∇ · ∇ + ∇ · ∇ + + = 0
+ ( · ∇ ) · ∇ + ∇ · ∇ − = 0
∇ · = (− − ∇) ·
= 0
in Ω
in z = 0
in z = 0
in Γ b
in z = -H
(15)
where the incident wave potential and incident wave elevation must be transformed to
the local frame of reference.

8 Numerical formulation
4. Numerical formulation
This section presents the formulation based on the finite element method (FEM) to
solve the system of equations presented in sections 2.1 – 2.3. This formulation has been
developed to be used in conjunction with unstructured meshes. The use of unstructured
meshes enhances geometry flexibility and speed ups the initial modelling time.
*
Let Q
h
be the finite element space to interpolate functions, constructed in the usual
manner. From this space, we can construct the subspace Qh,
φ
, that incorporates the
Dirichlet conditions for the potential φ . The space of test functions, denoted by Q
h
, is
constructed as Qh,
φ
, but with functions vanishing on the Dirichlet boundary. The weak
form of the problem can be written as follows:
Q φ
φ ∈ , by solving the discrete variational problem:
Find [ h ] h,
∇ · ∇ Ω =
Ω
= · Γ + · Γ +
·
Γ +
∈
where
) φ ,
) Z0
φ , ) )
H
φ and φ Z−
B
n
R
n
n
n
·
Γ
∀
(16)
are the potential normal gradients corresponding to the
Neumann boundary conditions on bodies, radiation boundary, free surface and bottom,
respectively.
At this point, it is useful to introduce the associated matrix form
L b b b b
B R Z0 Z−
H
φ = + + +
(17)
B R Z0
Z H
where L is the standard laplacian matrix, and b , b , b and b −
are the vectors
resulting of integrating the corresponding boundary condition terms. Regarding the
bottom boundary for the refracted and radiated potential, it is imposed naturally in FEM
H
by b Z − = 0 .
Free surface boundary condition
Solving the velocity potential free surface boundary condition efficiently is the most
important point of the problem stated since this is where a difference is made when
solving the mathematical model in eqs. (7) using the FEM.
The free surface conditions can be rewritten as:

9 Numerical formulation
∂
tη + U ⋅∇
hη + ∇hφ ⋅∇hξ − φz
= 0
123
Convective term
1
∂
tφ + U⋅∇hφ −
2
∇hφ ⋅∇
hφ + ∇hψ ⋅∇
hφ + P / ρ + gη
= 0
123
Convective term
(18)
where = ( + ∇ ) is the base flow. Linearization respect to this based flow is
quite common. Linearizations most commonly used are: the Kelvin linearization, which
assumes that ∇ hφ ~ ε and hence = and the double body, which assumes that
DB *
DB
∇
hφ = ∇
hφ + ∇
hφ
, where ∇
hφ
~ O(1)
and
* DB
∇
hφ
~ O( ε ) , being ∇ hφ the
scattered velocity potential obtained when solving the double body problem. As we
mentioned before, in this work no linearization is assumed since it is considered that
∇
hφ
~ O(1)
. Therefore, the convective velocity U will be different in each time step,
and the numerical scheme will be adapted accordingly. Three main issues must be kept
in mind:
1. Convective terms are likely to introduce numerical dispersion leading to
unrealistic free surfaces.
2. Pressure field over the free surfaces must be imposed in order to be capable of
reproducing aircushion effects.
3. Scattered waves must be absorbed in the far field in order to mimic an infinite
domain in the horizontal directions.
The numerical schemes adopted for solving the kinematic-dynamic free surface
boundary conditions are based on Adams-Bashforth-Moulton schemes, using an explicit
scheme for the kinematic condition, and implicit one for the dynamic condition. Then
n 1
can be imposed as a Dirichlet Boundary condition. The schemes read as follows:
φ +
( U )
n+
1 n n
n n n
= − ∆t ⋅∇h − ∆t∇h ⋅∇
h
+ ∆t
z
η η η φ ξ φ
n+
1/ 2
1
n
( U
h ) 2 ( h h ) h h ( / )
φ = φ − ∆t ⋅∇ φ + ∆t ∇ φ ⋅∇ φ − ∆t∇ ψ ⋅∇ φ − ∆ t P ρ + gη
n+ 1 n n n n+ 1 n+
1
(19)
It is noted that conditions of pressure acting on the free surface can be introduced in eq.
(19) straightforward.
Stremalines scheme for the free surface boundary condition
It was pointed out before that care must be taken when solving the free surface
conditions, especially when evaluating the convective terms. In this work, the
convective term is obtained by differentiating along streamlines:
( U η )
h
n n n
U
Lη
⋅∇ = ∂
n+ ( ) 1/ 2 n+ 1 n n+
1
U ⋅∇ φ = V + ∇ φ ∂ φ
h b h L
(20)
where ∂
L
denotes the derivative along the streamline. This streamline derivatives is
estimated using a two points upstream and one point downstream differential operator

10 Numerical formulation
inspired by the quickest scheme [18]. Figure 1 shows the tracing of the streamline at
node C. The left (L) and forward left (FL) points are the upstream points, while the right
(R) point corresponds to the downstream point. The values of the scattered velocity
potential φ and scattered free surface elevation η at L, FL and R points are obtained by
linear interpolation between the nodes of the edges where they lie on. The stream line
differential operator reads as:
∂ φ ≈ δ φ = α φ + α φ + α φ + α φ
n n
L L R R C C L L FL FL
∂ η ≈ δ η = α η + α η + α η + α η
n n
L L R R C C L L FL FL
(21)
where the stencils are:
α
α
2 ⎛ 1 1 1
∆x
R
R
= ⎜ − CrR
− −
∆x R
+ ∆xL 2 2 3 ∆ xR + ∆xL
⎝
2
( 1 CrR
)
2 ⎛ 1 1 1 ∆x
1
∆x
2 L
2
( ) ( )
R
C
= ⎜ CrR + CrL + 1- CrR + 1-CrL
∆x R
+ ∆xL 2 2 3 ∆x L
3 ∆x L
+ ∆xFL
⎝
2 ⎛ 1 1 1 ∆x
1 ∆x
αL
= − + + +
∆x + ∆x ⎝ 2 2 3 ∆x 3 ∆x ( ∆ x + ∆x )
2
L
⎜ CrL CrL CrL
R L FL L R L
αFL
2
2 ⎛ 1 ∆xL
⎜
x
R
+ xL ⎝ 3 x
FL( x
L
+ ∆x FL
)
Cr = U ∆t / ∆x
α
2
= ∆ ∆ ∆ ∆
( 1-Cr L )
α
⎞
⎟
⎠
2 R
2
( 1- ) ( 1- )
⎞
⎟
⎠
⎞
⎟
⎠
⎞
⎟
⎠
(22)
Stream line
FLx
o
fl1
o
fl2 l2
r2
V
V
o
Lx
o
l1
V
V
o
o
C
o
x
V
o
x
R
o
r1
V
V
∆XFL
∆XL
φFL
φL
φC
Figure 1: Streamline discretization.
∆XR
φR

11 Numerical formulation
Above equations are used to integrate the free surface boundary conditions when the
convective terms are dominant. However, when these terms can be neglected, the free
surface condition can be simplified. For the sake of clarity, in the following, details of
the integration of the free surface boundary conditions are given for this simplified case.
Neglecting higher order terms, eq. (18) can be rewritten as:
− = 0
+ = 0
(23)
Or equivalently,
1
− = 0
+ = 0
(24)
For efficiency and flexibility reasons, SeaFEM uses a forth order compact Padé scheme
in order to solve this problem. This scheme is implicit with symmetric stencils, which
provides good stability properties and requires solving the linear system in eq. (17) once
per time step.
Although the free surface boundary condition is usually implemented as a Dirichlet
boundary condition by imposing the value of the velocity potential at the time step to be
calculated, in SeaFEM it is implemented as a Neumann boundary condition that fulfils
the flux boundary integral:
b = M φ (25)
Z0 Z
Z 0
0
Γ z
where
Z
M Γ 0 is the corresponding boundary mass matrix. Rather than obtaining the
Z0
Z0
vector φ
z
and calculating the values of b , SeaFEM proceeds in a different manner.
Let’s consider the free surface boundary condition outside the absorbing zone (where
the absorbing factor equals zero, which is inside the analysis area). The forth order
compact Padé scheme reads, for the second eq. in (17) as:
n+ 1 n n−1
φ − 2φ + φ
n 1
n n n
= −g∂ 2
zφ − g ∂
zφ − ∂
zφ + ∂
zφ
∆t
12
+ 1 −1
( 2
)
(26)
Introducing Taylor series expansion around time t in eq. (26) and using eq. (17), we
4
recover the free surface boundary condition with O( ∆ t ) . Eq. (26) is an implicit scheme
which has to be solved along with the linear system given in eq (17). At first sight, it
seems like an iterating procedure should be used requiring solving the linear system
n 1
several times per time steps. However, this can be avoided by decoupling and
n 1
n 1
∂ . To this aim, from eq. (26) φ +
z
zφ +
n 1
is written as a function of φ + :
φ +
12
∂ φ = −10∂ φ − ∂ φ − φ − 2φ + φ
g∆t
( )
n+ 1 n n− 1 n+ 1 n n−1
z z z
2
(27)

12 Numerical formulation
Z0
This approximation is used to evaluate ) φ
n 1
t +
can be calculated as follows:
z
at
n 1
t + Z0
, and therefore, the integral of
b at
⎡
12
⎤
φ ⎢ φ φ φ φ φ
g∆t
⎥ (28)
⎣
⎦
Z
n+ 1
0 Z
n+ 1
0 Z
n
0 Z
n−1
0
n+ 1 n n−1
Z
( ) ( ) 10( ) ( ) ( 2 )
0 Z
b = MΓ
φz = MΓ
− φ 0
z
− φz
− φ − φ +
φ
2
Introducing eq. (28) into Eq. (17) we obtain:
+ 12
∆
= 12
∆ (2 − ) − 10
−
+ ( ) + ( )
(29)
Eq. (29) imposes a strong coupling between the free surface boundary condition and the
Laplace equation. This is carried out by modifying the system matrix L .
n 1
Once the system is solved, ∂ zφ + at the free surface is obtained using eq. (27) Then,
whenever the velocity potential is solved at the present time step, the free surface
elevation is computed by means of η = −(1/ g) ∂ t
ϕ using the following fourth order
finite difference scheme:
1 ⎛ 25 4 1
η = − ⎜ ϕ − 4ϕ + 3ϕ − ϕ + ϕ
g∆t
⎝ 12 3 4
n+ 1 n+ 1 n n−1 n−2 n−3
⎞
⎟
⎠
(30)
SUPG stabilization scheme for the free surface boundary conditions
Alternatively, a SUPG stabilization scheme is also available in SeaFEM for the
integration of the free surface boundary conditions. By using this FEM based
stabilization scheme, the dynamic and kinematic free surface boundary conditions are
solved as follows.
P
v 0
(31)
2 ρ
∂ φ
1
+ 2
( + ∇ hψ + ∇ hφ ) ⋅∇ hφ − ∇ hφ + + g ζ =
∂t
Introducing V = v + vψ + v
φ
P
V 0
(32)
2 ρ
∂ φ 1
+ ⋅∇ 2
hφ − ∇ hφ + + g ζ =
∂t
Weak formulation:

13 Numerical formulation
∫
⎛ ∂ φ 1 2 P ⎞
W ⎜ + ⋅∇
hφ − ∇
hφ + + gζ ⎟ dσ
= 0
⎝ ∂t
V 2 ρ ⎠
(33)
Ω
Discrete Galerkin method:
⎛ ∂φ
j
1 2 1
⎞
∀ i ∫ i ⎜ j + ⋅∇ j j j
h φ
j
− ∇
hφ j
+ Pj + g ζ
j ⎟dσ
= 0
∂t
V 2 ρ
Ω ⎝
⎠
(34)
Time marching scheme:
∀ −
+ ( · ∇
Δ
)
Ω
+ − 1 2 ∇ +
+ = 0
Ω
(35)
SUPG Stabilization:
∀ −
+ (( ) ·
)
+ − 1 2 +
+
Ω
+ h ( )
2|( )
|
−
(36)
+ (( ) ·
)
+ − 1 2 +
+ = 0
n+
α
e
ke
Where ( )
α
V = V .
n+
ke

14 Numerical formulation
∀
+ h ( )
2|( ) | ∇
−
, ,
+ h ( )
2|( ) |
= −Δ ·
+ h ( )
2|( ) | ·
(37)
−
+ h ( )
2|( ) |
− 1 2 +
In matrix form:
r
n α
n+ 1 r
+ n+ α n+
α
n+
1
( M + M
S ) φ + ∆ t ( C + CS
) αφ =
r
n α
n
r
+ n+ α n+
α
n
M + M φ − ∆ t C + C 1−α φ
( S ) ( S )( )
n α ⎛ 1 r ur ur
+ ( M + M
S ) ⎜ ( ∇hφ
) − P − ∆tg
⎝ 2
n+ α
2
+
n+ α n+
α
∆t
ρ
ζ
⎞
⎟
⎠
(38)
Where:
= ;
Ω
= h ( )
2|( ) | ;
Ω
(39)
= · ∇ ;
Ω

15 Numerical formulation
= h ( )
2|( ) | ∇ · ∇ ;
Ω
Free surface kinematic boundary condition
∂ ζ + ( v + ∇
hφ ) ∇
hζ + ∇
hφ ∇
hη − φz
= 0
(40)
∂t
introducing
v
φ
= ∇
hφ
and U = v + vφ
= v + ∇
h
φ
∂ ζ + U ∇
hζ + v
φ ∇
hη − φz
= 0
(41)
∂t
Weak formulation
∫
⎛ ∂ζ
⎞
W ⎜ + ⋅∇
hζ +
φ
⋅∇hη − φz
⎟dσ
= 0
⎝ ∂t
U v ⎠
(42)
Ω
Discrete Galerkin method:
∫
⎛ ∂ζ
⎞
⎜ ( h ) ζ
j hη h
φ
j
φz j ⎟ σ 0
∂t
U Ω ⎝
⎠
(43)
i j j
∀ i + ⋅∇ j + ∇ ⋅∇ j − j d =
Time marching scheme:
∀
−
+ ( · ∇
Δ
)
Ω
(44)
+ ((∇ ) · ) −
= 0
SUPG Stabilization:

16 Numerical formulation
∀ −
+ (( ) ·
)
+ (( ) ) ·
Ω
−
+ h ( )
2|(
)|
−
(45)
+ (( ) ·
)
+ (( ) ) ·
− = 0
n+
α
e
ke
Using ( )
U = U
n+
ke
α

17 Numerical formulation
∀ + h( )
2|( ) |
− + h( )
2|( ) |
= −Δ ·
− h( )
2|( ) | ·
(46)
− ( ) ·
+ h( )
2|( ) | ( ) ·
+ + h( )
2|( ) |
In matrix form:
ur
n α
n+ 1 ur
+ n+ α n+
α
n+
1
( + S
) ζ + α∆ t ( D + DS
) ζ =
ur
n
n
ur
+ α n+ α n+
α
n
( + S
) ζ − (1 −α ) ∆ t ( D + DS
) ζ
uur
n α
n α
r
+ n+ α n+
α
n
+ ∆ t ( + S ) φz − ∆ t ⎡
⎣E + E ⎤
S ⎦φ
+ + α
(47)
Where the matrices are defined as follows:

18 Numerical formulation
=
n+
α
S
∑ ∫
e e
Ω
ie
je
d
e
n+
α
he ( U )
ie
je
∑
h
e
n+
α ∫
e
e
2 ( U ) Ω
ke
α
∑ ∫ ( Uk
)
e h
n+ α
i n+
j
D = ⋅∇ d
e e
Ω
σ
= ∇ dσ
e
n+
( U )
e
n+
( U )
he
⎡
⎤
n+ α
i ke
n+
α j
DS = ∑
h (
k )
e h dσ
α ⎢ ∫ ∇ U ⋅∇ ⎥
e
e
2 ⎢⎣
Ω
⎥⎦
∑ ∫
k
n+
α
e
( ( hη
)
k h )
= ∇ ⋅∇
n+
α
i j
E d
e
e e
Ω
e
n+
( U )
e
n+
( U )
α
α
k
n+
α
e
( ( η )
k )
he
⎡
⎤
n+
α
i j
ES = ∑
∇h ∇h ⋅∇h dσ
α ⎢ ∫
⎥
e
e
e
2 ⎢⎣
Ω
⎥⎦
σ
σ
(48)

19 Numerical formulation
Free surface boundary condition with absorption
In order to include the wave damping effect, SeaFEM discretizes eq. (16) as follows:
n+ 1 n n−1
n+ 1 n−1
φ − 2φ + φ
n 1
n+ 1 n n−1
∂
zφ
− ∂
zφ
= −g∂ 2
zφ − g ( ∂
zφ − 2 ∂
zφ + ∂zφ ) −κ(
x) (49)
∆t
12 2∆t
Eq. (49) is simply eq. (27) plus a second order finite difference in time for the absorbing
n 1
term. Then ∂ zφ + can be obtained as:
10g∆t
⎛ g∆t
−6 κ( x) ⎞
12
∂ φ = − ∂ φ −⎜
⎟∂ φ − φ − φ + φ
g∆ t + 6 κ( x) ⎝ g∆ t + 6 κ( x) ⎠ g∆ t + 6 κ(
x) ∆t
( 2 )
n+ 1 n n− 1 n+ 1 n n−1
z z z
2
Proceeding like in the previous section, we obtain:
⎛ 12 ⎞ n+ 1 ⎛ 12
n n−1
⎞
Z0 Z0
⎜L + M
2
Γ
Γ
2 ( 2 )
g t 6 κ ( t
⎟ φ = M ⎜ φ −
φ
∆ + ∆ g∆ t + 6 κ ( ∆t
⎟
⎝ x) ⎠ ⎝ x)
⎠
⎛ 10g∆t
⎛ g∆t
− 6 κ ( x) ⎞
⎜
g∆ t + 6 κ (
⎜
g∆ t + 6 κ (
⎟
⎝ x) ⎝ x) ⎠
Z0 Z0
B R
( φ
z ) ( φ
z ) ⎟ ( ) ( )
(50)
n n− 1 n+ 1 n+
1
Z
− MΓ
0
+ + b + b
This implementation has several advantages over traditional. The linear system has to
be solved only once per time step; the free surface numerical scheme is implicit; only
information from two previous times at the free surface is required; there is no
restriction about the grid structure, hence unstructured meshes can be used with no
restriction; and the vertical fluid velocity at the free surface is easily computed using eq.
(50).
Radiation boundary condition
When the fluid domain is bounded, an implementation of a radiation boundary
condition is recommended to avoid artificial wave reflexions at the edges of the
computational domain, where waves are supposed to go through without reflection.
SeaFEM makes use of the Sommerfeld radiation condition given in eq. (5). In this
sense, it is assumed that the waves scattered by the bodies hit the outlet boundary in
perpendicular. Hence, the radiation condition is approximated by:
⎞
⎠
(51)
)
R
( φn
)
n+
1
n+ 1 φ −
= −
n
φ
c∆t
(52)
Then the term
R
b can be calculated through:

20 Numerical formulation
1
φ φ φ (53)
∆t
R
n+ 1
R
n+
1
n n−1
R
R
( b ) = MΓ
( φn
) = MΓ
( φ −
φ )
R
where Γ represent the outlet boundary that has been assumed to be vertical.
Introducing eq. (53) into eq. (51) we get:
⎛ 12 c ⎞ n+ 1 ⎛ 12
n n−1
⎞
Z0 R
Z0
⎜ L + MΓ + MΓ Γ
2 2 ( 2 )
g t 6 κ ( t t
⎟ φ = M ⎜ φ −
φ
∆ + ∆ ∆ g∆ t + 6 κ ( ∆t
⎟
⎝ x) ⎠ ⎝ x)
⎠
M
⎛ 10g∆t
⎛ g∆t
− 6 κ ( x) ⎞
⎜
g∆ t + 6 κ (
⎜
g∆ t + 6 κ (
⎟
⎝ x) ⎝ x) ⎠
Z
− Γ 0
+
c
∆t
n
R
+ MΓ
φ +
B
( b )
n+
1
Z
n
0 Z0
( φ
z ) ( φ
z )
(54)
A strong coupling between the Sommerfeld radiation condition and the Laplace
equation has been defined, as it was done with the free surface in the previous section.
Then the boundary condition is integrated within the system matrix, avoiding iterating
among the equations.
n−1
⎞
⎟
⎠
Body boundary condition
The boundary condition to be imposed over the surface of the bodies is given by Eq.
(11). The movements of the bodies are assumed to be small enough so that the
computational domain can remind steady, as well as the normals to the bodies’ surface.
B
Hence, the term b is calculated by:
B
n
B
B
( ) = Γ ( n )
+ 1 n+
1
b M φ
(55)
Boundary condition for limit height (Hfs)
The limit height boundary condition is formulated as finding a pressure field to be
applied over the free surface such that the elevation of this one is limited by the location
of the given surface. That is to say, the given surface act as an upper limit for the free
surface elevation.
The free surface boundary conditions are applied in different ways depending on
whether the free surface is in contact with the surface or not. If the free surface is not in
contact, the boundary conditions are applied as if there is no condition, but if there is
contact, the implementation will be different in order to ensure that the free surface does
not penetrate de surface and the necessary pressure to fulfil this condition is calculated.
It will be said that the free surface node where the algorithm is to be applied is dry if the
seal is not in contact with the free surface at that location, and wet if it is.
The main challenge for an algorithm like this is to be capable of capturing when a node
goes from dry to wet and vice versa, as well as estimating the pressure field on the wet
nodes. For a dry node, the implementation of both, the kinetic and dynamic boundary

21 Numerical formulation
condition is the same as for any other node not interacting with the surface. However,
for wet nodes, the free surface boundary condition is imposed via imposing that the free
surface elevation matches the surface elevation, and ensuring that there is no flow
across the seal. These two conditions are represented by the following equations:
+ = → = −
= · ∇ + ∇ ∇ + 1 ∆ ( − )
(56)
The change from being a dry node to become a wet node is identified via the kinematic
BC though the condition + = . On the other hand, the switch from being a wet
node to become a dry one is carried out by comparing the dynamic pressure with the
reference pressure plus a detachment condition. This detachment condition requires for
the free surface to detach at a specific node that this node is not completely surrounded
by attached nodes. Should this be the case, there would be not path for the air to move
in. Therefore, if there is no connection with the air, pressure might drop below the
reference pressure. The dynamic pressure on wet nodes is obtained by applying the
second eq. in (18).
Boundary condition for transom stern
The boundary condition to be applied on the body subject to study is the usual no flux
across the body surface. For this purpose, a normal flux is induced via a Newman
boundary condition in order to cancel out the normal component of the incident velocity
plus the velocity of the body at the specific location.
However, when considering transom sterns, flow detachment happens at the lower edge
of the stern. While potential flow is incapable of predicting this sort of detachment, a
transpiration model will be used to enforce it. To do so, the null normal flux boundary
condition is not used. On the contrary, a flux is allowed in order to enforce that the
detachment edge belong to the free surface stream surface.
Figure 2 explains the transpiration model idea.
2 2
=U 8U +0.5gh
U
8
U
h U
8
Figure 2: Transpiration model for transom stern Boundary condition with flow detachment.

22 Multi-body dynamics
5. Multi-body dynamics
Once the velocity potential has been obtained, the pressure at any point can be
calculated from:
P = −ρgz
− ρ∂ ϕ
(57)
t
Eq. (57) requires estimating the value of ∂ tϕ . The same fourth order finite difference
scheme that has been used for the free surface elevation is used here:
P
ρ ⎛ 25 4 1
= −ρgz
− ⎜ ϕ − 4ϕ + 3ϕ − ϕ + ϕ
∆t
⎝ 12 3 4
n + 1 n + 1 n n − 1 n − 2 n − 3
⎞
⎟
⎠
(58)
Body dynamics
Integrating the pressure over the bodies’ surface, the resulting forces and moments are
obtained. On the other hand, the body dynamics is given by the equation of motion:
+ = (59)
where is the mass matrix of the multi-body system; is the hydrostatic restoring
coefficient matrix of the multi-body system; F are the hydrodynamic forces induced
over the bodies plus any other external forces; and X represent the vector containing the
movements of the six degrees of freedom of each body. Both and are assumed to
be constants.
In the specific case where the bodies are fixed, only refracted waves are calculated and
the linear system given in eq. (54) is to be solved just once per time step. However,
when solving the body dynamics along with the wave problem requires an iterative
procedure since interaction between the waves and the movements of the structure exist,
giving birth to waves radiated by the bodies.
In order to solve Eq.(63), SeaFEM uses an implicit Bossak-Newmark´s algorithm [17]:
(1 − ) + = − (60)
= + ∆(1 − ) + (61)
= + ∆ + ∆
2 (1 − 2)
+ 2
In the above integration algorithm, α is the parameter defining the numerical damping
added in the integration. This damping created a desirable stabilizing effect in the body
dynamics integration. α is usually set to α =-0.1, and in SeaFEM, γ and β are calculated
as γ= 0.5- α and β=0.5γ +0.025α.
Within each time step, the system of eqs. (60)-(62) is solved iteratively along with eq.
(54). This requires predicting the body velocities by solving eq. (61), introducing this
(62)

23 Multi-body dynamics
velocities into eq. (55), and solving the linear system in eq. (54), introducing the
pressure forces into eq. (60), and repeating the process until convergence is reached. In
order to accelerate convergence, SeaFEM uses a second order prediction method for
estimation of the bodies’ velocities and positions for the next iteration.
The algorithm implemented also allows considering non-linear external forces acting on
the bodies such as mooring forces. In this implementation they are evaluated for every
iteration of the solver and added to the right hand side of eq. (59).
Non-linear hydrostatics
Eq. (59) is valid for small rotations of the bodies. However, in the case of switching on
the non-linear hydrostatics option, the rigid body dynamics solver of SeaFEM is
extended for taking into account large motions. For this purpose, the Euler equations are
used to integrate the body dynamics. Equation (59) is now rewritten as:
= ∗ (63)
with
∗ =
−
∧ · (64)
Where , are the 3x1 vectors containing the external forces and moments acting on
the body, is the instantaneous angular velocity of the body and the inertia tensor.
The Euler equations are derived in a rotating reference frame fixed to the body, and
therefore, the inertia tensor is assumed to be constant.
It is important to remark that, in equation (64), must be evaluated in the local
reference frame.
Bossak-Newmark´s algorithm shown in eqs. (60)-(62) is then applied to the integration
of eq. (64) as follows:
(1 − ) + = ∗ (65)
= + ∆(1 − ) + (66)
= + ∆ + ∆
2 (1 − 2)
+ 2
Body links
When links exist between bodies, eq. (59) must be solved along with the algebraic
system of equations
() = ∑ + = 0
(68)
which represents the different restrictions existing between various degrees of freedom
x j of two bodies.
(67)

24 Multi-body dynamics
Within this context, let ( ) be such that ( ) = − and ( ) =
− h ( ) = () = ∑
.
Then, let’s formulate the following optimization problem:
Maximize ( ) subject to: h ( ) = 0 i=1, 2, …, m. The solution to the problem
can be obtained using Lagrange multipliers. Then, the above problem is reformulated
as:
∇() + () = − + = 0
() = = 0
(69)
Where = and = … .

25 Body matrices referred to an arbitrary generic point
7. Body matrices referred to an arbitrary generic point
In this section we briefly introduce the formulation used to relate the body’s matrices
(e.g. mass, damping or stiffness matrices) from two different points of reference. This is
useful when body properties are known from a reference point which is not coincident
with the gravity centre of the body or more generally with the point to which all output
results (i.e. displacements, forces and moments, etc.) want to be referred.
Let A be a matrix relating forces and momentums as a function of displacements and
rotations respect to a point of reference I .
⎛ FI ⎞ ⎛ xI ⎞ ⎛ AFx
AF
θ ⎞⎛ xI
⎞
⎜ ⎟ = A⎜ ⎟ = ⎜ ⎟⎜ ⎟
⎝ M
I ⎠ ⎝θI ⎠ ⎝ AMx
AM
θ ⎠⎝θI
⎠
(70)
Let B be a matrix relating forces and momentums as a function of displacements and
rotations respect to a point of reference R .
⎛ FR ⎞ ⎛ xR ⎞ ⎛ BFx
BFθ
⎞⎛ xR
⎞
⎜ ⎟ = B⎜ ⎟ = ⎜ ⎟⎜ ⎟
⎝ M
R ⎠ ⎝θR ⎠ ⎝ BMx
BMθ
⎠⎝θR
⎠
(71)
Let r be the vector r
= uur IR . The following relationships hold:
F = F ; M = M + F × r
R I R I I
x = x − θ × r;
θ = θ
I R R I R
(72)
Here, it must be noted that the vector cross product can always be written as the product
of a skew-symmetric matrix and a vector in the form:
× = (73)
where the skew-symmetric matrix is given in terms of the vector components as
follows:
0 −
= − 0 (74)
− 0
Then:
F = F ; M = M + R F
R I R I I
x = x − Rθ ; θ = θ
I R R I R
(75)
and using F = F ; M = M + R F we can write
R I R I I

26 Body matrices referred to an arbitrary generic point
F = F = A x + A
θ
R I Fx I Fθ
I
( θ )
M = M + R F = A x + A θ + R A x + A
R I I Mx I Mθ
I Fx I Fθ
I
(76)
Inserting x = x − Rθ ; θ = θ
I R R I R
F = A x − A Rθ
+ A
θ
R Fx R Fx R Fθ
R
( )
M = A x − A Rθ + A θ + R A x − A Rθ + A θ
R Mx R Mx R Mθ
R Fx R Fx R Fθ
R
(77)
Reordering terms:
F = A x + A θ − A Rθ
R Fx R Fθ
R Fx R
M = A x + A θ + R A x − A Rθ + R ⋅ A θ − R A Rθ
R Mx R Mθ
R Fx R Mx R Fθ
R Fx R
(78)
In matrix form:
R R ⎛ 0 −AFx
R ⎞ R
⎛ F ⎞ ⎛ x ⎞ ⎛ x ⎞
⎜ ⎟ = A⎜ ⎟ + ⎜ ⎟⎜ ⎟
⎝ M
R ⎠ ⎝θR ⎠ ⎝ R AFx − AMx R + R AF θ
− R AFx
R ⎠⎝θR
⎠
(79)
Hence, matrices and are finally related by the following expression:
=
+ 0 −
(80)
− − +

27 Statistical description of an irregular sea
8. Statistical description of an irregular sea
Spectrum discretization
Let be S( ω, α ) an energy density spectrum describing a sea state in terms of the wave
frequency and direction of propagation. The discretization procedure to obtain a
stationary and ergodic realization based on monochromatic waves is as follows:
Let be ωmin
the minimum frequency to be considered, ωm
ax
the maximum frequency to
be considered, αmin
the lower direction of propagation to be considered, α
m ax
the larger
w
direction of propagation to be considered, the number of wave frequencies, and α
the number of wave directions to be considered. Then, the frequency and direction
discretization sizes are given by:
∆ = ( − )/
Δ = ( − )/( − 1) (81)
Then, the wave elevation is given by:
w
α
i= 1 j=
1
( )
η = ∑∑ Aij cos kij cos( α
j
) x + kij sin( α
j
) y − Ω
ijt
+ δij
(82)
Where
Ωij
is the wave
δ is the wave phase, t represents time, and
Aij
is the monochromatic wave amplitude, kij
is the wave number,
angular velocity, α
j
is the wave direction,
ij
x,
y are the horizontal Cartesian coordinates. Each parameter is obtained as follows:
Ω is a random variable with uniform distribution in [ ω ω / 2, ω ω / 2]
ij
i
−∆ + ∆ .
i
= + ( − 1/2)Δ
= + ( − 1)Δ
Ω = tanh
(83)
δij
is a random variable with uniform distribution in [ 0,2π ].
A
= 2 ∆ω∆α
S ,
( ω α )
∑
ij i j
l,
m
1
16
H
2
S
( ω α )
2 ∆ω∆αS
,
where H = 4 m0
is the significant wave height, and m0
S( ω, α)
dωdα
S
spectrum wave energy.
l
m
∞ π
0 −π
(84)
= ∫ ∫ is the

28 Statistical description of an irregular sea
Convergence
Convergence of the discretized spectrum will happen as ωmin → 0, ωmax
→ ∞ , ∆ω
→ 0
, αmin → 0, αmax → 2π
, and ∆α
→ 0 .
The rate of convergence with
integration.
∆ ω and ∆ α is that of the rectangle rule of numerical
Spectral moments
1. Zero order moment : m
0
The spectral energy of a wave spectrum is given by:
∞ π
1
0
= ∫ ∫ ω α ω α =
16
0 −π
2
m S( , ) d d HS
The discrete spectrum is scaled such that the spectral moment m
0
is conserved.
Therefore:
1 1
2 2
∑ Aij
= HS
(85)
i,
j 2 16
2. First order moment: m1
m = S( ω , α ) Ω ∆ω∆ α = S( ω , α ) ω ∆ω∆ α + S( ω , α ) ε ∆ω∆ α = m + m
m
m
*
1 i j ij i j ij i j ij
D
1
P
1
i, j i, j i,
j
D
1
P
1
∑ ∑ ∑
∑
= S( ω , α ) ω ∆ω∆α
i,
j
∑
= S( ω , α ) ε ∆ω∆α
i,
j
i j ij
i j ij
(86)
Where
ε is uniform distributed between [ ω / 2, ω / 2]
ij
P
component of the first moment, and m1
ωmax
→ ∞ , ω
min
= 0 , α
min
= 0, αmax 2π
D
−∆ ∆ , m
1
is a deterministic
is a random component. Assuming that
= , the deterministic component converges to:
lim
ω
m m ωS( ω, α)
dωdα
α π
D
∆ →0 1 1
∆ →0
∞ π
= = ∫ ∫ (87)
0 −
On the other hand, for large values of ω P
, the probabilistic part m1
is a random variable
with normal distribution. The mean and variance of this distribution are:
∆ω
/ 2
1
µ = S( ω , α ) ∆ω∆ α ω dω
= 0
∆ω
∑ ∫ (88)
i j
i, j
−∆ω
/ 2

29 Statistical description of an irregular sea
/
= , ΔΔ 1
Δ
=
Δ
, ΔΔ
12 (89)
,
/
Hence, the probabilistic component converges to a random variable with zero mean and
zero variance. Therefore:
,
lim m = lim m + lim m = m + 0
(90)
*
D
P
∆ω→0 1 ∆ω→0 1 ∆ω→0 1 1
∆α →0 ∆α →0 ∆α
→0
Waves spectrum
Several standard waves spectrum are predefined in SeaFEM:
Pearson Moskowitz spectrum:
This is probably the simplest idealized spectrum, obtained by assuming a fully
developed sea state, generated by wind blowing steadily for a long time over a large
area [19]. The resulting spectrum is [10]:
() = (0.11/2)( /) .(/) (91)
where is the wave period; is the significant wave height; is the mean wave
period, which is obtained via = 2 / , with and the zero and first
moments of the wave spectrum.
Jonswap2 spectrum:
The JONSWAP spectrum was established during a joint research project, the "JOint
North Sea WAve Project" [20]. This is a peak-enhanced Pierson-Moskowitz spectrum
given on the form [21]:
() = 5
32π T
T · ε · e .
. ()
= e (. )/(√) (92)
where = 2/ , = 0.07 for ≤ 6.28/ , = 0.09 for > 6.28/ , is the
wave period; is the significant wave height, T is the peak wave period and is the
peakedness parameter.
Jonswap spectrum:
An alternative definition of the JONSWAP spectrum is given by [10]:
() = 155
T ω · 3.3 · e
= e (. )/(√) (93)

30 Statistical description of an irregular sea
where = 2/ , = 0.07 for ≤ 5.24/ , = 0.09 for > 5.24/ , is the
wave period; is the significant wave height; is the mean wave period, which is
obtained via = 2 / , with and the zero and first moments of the wave
spectrum.

31 Mooring system modelling
9. Mooring system modelling
SeaFEM can handle complex mooring systems made up of various mooring lines
attached to the floating structure. Each mooring line can be in turn composed of various
segments each one resembling a chain, a steel cable or even a synthetic fiber. Forces
resulting from the action of buoys and sinkers acting at the junctions between mooring
line segments can also be considered. Hence, SeaFEM can deal with a wide variety of
multi-segmented mooring line systems.
Cable tensions depend on the buoyancy and lateral displacements of the floating
structure, the cable weight in water, the elasticity in the cable and the geometrical layout
of the mooring system. Hence, as the floating structure moves in response to unsteady
environmental loadings, the mooring restraining forces change with the changing cable
tension. This means that the mooring system has an effective compliance whose
response is in general non-linear. Within SeaFEM, mooring inertia and damping are
ignored, but the non-linear response is accounted for in the mooring system dynamics.
Mooring systems within SeaFEM are solved using a quasi-static approach in the sense
that at any step of the calculation, and once the floating body displacements are known,
the mooring system solver calculates the tensions and the geometrical configuration of
each mooring line segment assuming that each cable is in static equilibrium at that
instant. The mooring loads resulting from the calculation are added to the total load
acting on the floating body, and the resulting dynamic equations of motion of the
structure are solved again until convergence.
At each time step the implicit non-linear system of equations describing the mooring
system is solved using a classical Newton-Raphson scheme.
The formulation implemented within SeaFEM for dealing with catenary based mooring
systems is outlined in what follows. Additionally, cables behaving as springs (both in
tension and/or compression) can also be modeled.
Catenary equations
In a local coordinate system with its origin located at the lower cable point, the mooring
equations for the catenary read as follows:
=
h
+ + (94)
=
h
+ +
(95)
where z is the vertical position, s is the catenary arc length, is the catenary weight per
unit length in water, and is the horizontal component of the cable tension which is
constant everywhere. If the conditions ( = 0) = 0 and ( = 0) = 0 are applied, the
equations result to be:

32 Mooring system modelling
=
h
+ − cosh () (96)
=
h
+ − sinh ()
(97)
The action of any part of the line upon its neighbor is purely tangential, and the
tangential direction can be written as:
=
= sinh
+ (98)
The equilibrium equations can be written as:
· = · =
· − · = ·
(99)
From the horizontal component of the tension at any point of the catenary, it is possible
to write the modulus of the cable tension at any point of the catenary as:
=
= · 1 + = · h
+ (100)
Hence, the vertical component of tension at any point of the catenary can be written as:
= · = · h
+ (101)
In particular, at the upper and bottom vertex of the catenary Eq.(83) reads respectively:
= · h
+ (102)
= · h() (103)
And the vertical equilibrium equation will result in:
=
h
+ − h () = ( = )
(104)
For a given length of the catenary (L) and for given horizontal and vertical distances
(l,h) between the initial and end points of the catenary, it is possible to solve the
equations by using a classical Newton-Raphson iterative process.
If the catenary has a seabed contact in its lower point, the following condition must be
added
= 0
(105)

33 Mooring system modelling
and the catenary equations result in:
=
h
− 1
=
h
(106)
Elastic catenary formulation
The elastic catenary formulation used within SeaFEM is similar to that presented in
[12].
Each mooring line is analyzed in a local coordinate system that originates at the anchor.
The local z-axis of this coordinate system is vertical and the local x-axis is directed
horizontally from the anchor to the instantaneous position of the fairlead. When the
mooring system module is called for a given structure displacement, SeaFEM first
transforms each fairlead position from the global frame to this local system to determine
its location relative to the anchor, x F and z F .
z
V F
H F
Fairlead
V A
Anchor
H A
Figure 3: Catenary system of reference.
x
In the local coordinate system, the analytical formulation is given in terms of two
nonlinear equations in two unknowns—the unknowns are the horizontal and vertical
components of the effective tension in the mooring line at the fairlead, H F and V F ,
respectively.
( , ) =
+ 1 +
− −
( , ) =
1 +
+ 1 +
−
+
(107)
− 1 +
−
+ 1
−
2
The analytical formulation of two equations in two unknowns is different when a
portion of the mooring line adjacent to the anchor rests on the seabed:

34 Mooring system modelling
( , ) = −
+
+ 1 +
+
( , ) =
1 +
+
2 − −
+ −
−
−
−
, 0
− 1 +
−
+ 1
−
2
(108)
= −
The last term on the right-hand side of the x F equation, which involves C B , corresponds
to the stretched portion of the mooring line resting on the seabed that is affected by
static friction. The seabed static friction was modeled simply as a drag force per unit
length. The MAX function is needed to handle cases with and without tension at the
anchor. Specifically, the resultant is zero when the anchor tension is positive; that is, the
seabed friction is too weak to overcome the horizontal tension in the mooring line.
Conversely, the resultant of the MAX function is nonzero when the anchor tension is
zero. This happens when a section of cable lying on the seabed is long enough to ensure
that the seabed friction entirely overcomes the horizontal tension in the mooring line.
These equations consider that slack catenary is always tangent to the seabed at the point
of touchdown.
The mooring system module uses a Newton-Raphson iteration scheme to solve
nonlinear for the fairlead effective tension (H F and V F ), given the line properties (L, ,
EA, and C B ) and the fairlead position relative to the anchor (x F and z F ). The mooring
system module uses the values of H F and V F from the previous time step as the initial
guess in the next iteration of the Newton-Raphson scheme. As the model is being
initialized, the following starting values,
and , are used [12]:
=
2
= 2 (109)
h( ) +
where the dimensionless catenary parameter, λ 0 , depends on the initial configuration of
the mooring line:
1000000 = 0
0.2 + ≥
=
(110)
3
−
− 1 h
Once the effective tension at the fairlead has been found, determining the horizontal and
vertical components of the effective tension in the mooring line at the anchor, H A and
V A , respectively, is simple. From a balance of external forces on a mooring line, one can
easily verify that

35 Mooring system modelling
=
= −
(111)
When no portion of the line rests on the seabed, and
= ( − , 0)
= 0
(112)
The mooring system module solves the configuration of, and effective tensions within,
the mooring line. When no portion of the mooring line rests on the seabed, the equations
for the horizontal and vertical distances between the anchor and a given point on the
line, x and z, and the equation for the effective tension in the line at that point, Te, are as
follows:
() =
+
+ 1 +
+
−
+ 1 +
+
() =
1 + +
− 1 +
+ 1
+
2
(113)
() = + ( + )
where s is the unstretched arc distance along the mooring line from the anchor to the
given point. The equations with seabed interaction are:
0 ≤ ≤ −
+
2 − 2 −
+ −
−
, 0 −
≤ ≤
() =
+
( − )
+ 1 + ( − )
+
+
2 − + −
−
, 0 ≤ ≤
0 0 ≤ ≤
() =
1 + ( −
)
− 1 + ( − )
2
≤ ≤
( + ( − ), 0)
() =
+ ( − )
≤ ≤
(114)
The final calculation is a computation of the total load on the system from the
contribution of all mooring lines. This mooring system-restoring load is found by first
transforming each fairlead tension from its local mooring line coordinate system to the
global frame, then summing up the tensions from all lines.

36 Mooring system modelling
Dynamic cable formulation
SeaFEM includes a finite element model (FEM) for solving mooring line dynamics. For
this purpose, the line is divided into a series of straight segments modeled by nonlinear
truss elements, with three translational degrees of freedom per node. Lagrangian
formulation is used to describe the dynamics of the mooring line.
Figure 4: Scheme of the FEM cable model.
Applying FEM formulation to this nonlinear elastodynamics problem, the equations of
the dynamics of the line can be written as follows,
+ + + + = (115)
Where is the vector of nodal translational degrees of freedom, the vector of external
loads, M the inertia matrix of the line, the added mass matrix, is the pretension
vector in the initial configuration, and , the vector of internal forces of the cable.
Damping effects of mooring cable can be inserted through a Rayleigh-type damping
matrix C.
So, considering the standard linearization of the internal forces, above equations can be
expressed as,
+ + + + + + = (116)
where and are the so called material stiffness matrix and geometric stiffness
matrix, respectively [22]. The corresponding elemental material stiffness matrix , and
geometric stiffness matrix, , can be obtained in terms of the strain of the truss
element, ,

37 Mooring system modelling
+ =
=
⊗
+
(117)
Where is the volume of the element, the axial stress and the axial elastic
modulus. The FEM allows calculating those matrixes as follows [22]:
=
⊗
= B B d
=
= B B d
The elemental inertia matrixes are evaluated as:
= N Nd
And the added mass matrix is calculated as:
(118)
(119)
= N N d
(120)
In the above equations, is the general matrix of shape functions [23], is the section
area of the cable, and is the added mass coefficient.
Finally, the damping matrix is evaluated using the Rayleigh formulation:
= 1 + 2 M (121)
The coefficients 1 , can be calculated as:
= 1 2 1
+ 2 (122)
where
is the damping ratio corresponding to the natural frequency of the structure.
The external forces vector is evaluated by assembling the contribution of the different
forces acting on the element:
= + + + +
(123)
where is the weight, h is the buoyancy force, is the drag force related to the
currents, is the force due to the seabed interaction and is the drag force due to the
waves.
The weight and buoyancy force are calculated as follows:

38 Mooring system modelling
+ = h
N d,
0
= + = ( − )
(1 + )
(124)
where
is the density of the cable,
is the water density, weight per meter and
the strain of the considered element.
On the other hand, the drag force acting on the element, is evaluated as:
+ =
N d,
0
=
1
2 |
| d +
1 2 |
| d
(125)
where D is the characteristic diameter of the element, and the normal and
tangential drag coefficients and is the velocity relative to the element.
The seabed interaction is modeled with the spring and normal and tangential damping
terms given by,
= N d + N d + N d
(126)
where the vertical forces per unit length, due to the stiffness of the seabed
, are
expressed as:
=
− ( )
, if − ( ) ≤
, if − ( ) >
(127)
where is the vertical coordinate of the corresponding node, is the length of the cable
associated to the node, and
is a coefficient of the model. is the coefficient of the
seabed normal stiffness force which has the magnitude:
= (128)
Where is the diameter of the line, and is the ground normal stiffness per unit
length. The term is then evaluated as:
= +
(129)
The seabed damping forces are applied in the normal and tangential directions. The
normal damping force is only applied when the penetrating object is travelling into the
seabed. It is applied in the seabed outwards direction and has a magnitude given by:
= , if < 0
0 , if ≥ 0
(130)

39 Mooring system modelling
and is formulated as a fraction, , of the critical damping:
= 2,0
( )
(131)
Finally, the tangential damping force is again evaluated as a fraction, , of the critical
damping, and is applied in the direction opposing the tangential component of the
velocity of the penetrator:
=
(1321)
= 2,0
( )
An implicit time integration scheme based on called Bossak-Newmark method [7] is
applied to solve the system. It will lead to a system of algebraic equations to be solved
in iterative manner.
(1 − ) , + , + ,
+ , + , ,
= , + , , + , − +
− , − ( + 1) − , + ,
(133)
where is time step, i denotes iteration, is a parameter related with Bossak-
Newmark implicit method, and and are parameters related to Newmark time
integration scheme.
Finally, the new position and velocity of each node can be evaluated as,
Δ = +
+ 2
2 (1 − 2) + 2
(134)
= + d(1 − ) +
The dynamic cable solver is integrated within SeaFEM dynamic solver. The scheme of
this integration is presented in Figure 5. In order to accelerate the scheme, the mooring
forces are linealised within the body dynamics group, by evaluating the stiffness matrix
of the line.

40
Mooring system modelling
Figure 5: Scheme of the dynamic integration solver of SeaFEM.
A detailed description of the FEM cable model implemented in SeaFEM can be found
in [23].

41 Morison’s forces
10. Morison’s forces
When viscous effects may be advanced to have a significant effect on the dynamic
behavior of an offshore structure, Morison's equation can be used to evaluate wave
loads on slender cylindrical elements of the structure [14-16]. In SeaFEM, force
corrections due to viscous effects can be also taken into account by using the Morison's
equation. For this purpose, an auxiliary framework structure, associated to a body must
be defined. See the SeaFEM user manual for details on how to define the auxiliary
framework structure elements using the Tcl interface of SeaFEM.
Based on the information provided by the user, SeaFEM evaluates Morison's forces per
unit length acting on the framework structure. After integration along the different
elements, the resultant forces are incorporated to the dynamic solver of the rigid body to
which the idealized framework structure has been associated.
It is useful to write the Morison’s equation in a vectorial formulation that automatically
takes into account the actual orientation of structural elements and force components.
Considering a segment of a long slender structural element submerged into water its
local orientation is given by a unit vector
= + + (135)
being l, m, n the directional cosines and (l,j,k) the unit vectors of the global coordinate
system.
Similarly, the relative fluid velocity vector and the relative acceleration vector of the
submerged body are given by:
= v + v + v
= a + a + a
(136)
Remark: The relative fluid velocity vector and the relative acceleration vector are
evaluated based on the undisturbed wave potential equations.
The force per unit length on a slender cylindrical element may be written as the sum of
inertia, drag, friction and lift forces:
= + + + + (137)
where the inertia force is oriented along the acceleration vector component normal to
the element member, and its magnitude is proportional to the acceleration component. Lift
force is oriented normal to the velocity vector and normal to the axis of the element, and
its magnitude is proportional the velocity squared. Drag force is proportional to the
squared velocity component normal to the element and normal to the lift force, while the
linear drag force is proportional to the velocity component normal to the element.
Finally, friction force is aligned along the axis of the element and proportional to the

42 Morison’s forces
squared velocity components tangential to the element axis. All these are satisfied if the
various force components are defined as follows:
= ( × × )
= 1 2 | × × |( × × )
= 1 2 ( × × )
= 1 2 | · |( · ) ·
(138)
= 1 2 | × |( × )
where D is a linear dimension (the diameter in the case of a cylinder), S is the cross
section area, C M is the added mass coefficient, C D is the non-linear drag coefficient, C V
is the linear drag coefficient, C F is the friction coefficient, and C L is the lift coefficient.
Remark: If the defined C M coefficient is greater than 1, then F M component is evaluated
in the following alternative way:
= ( − 1)( × × ) + ( × × ) (139)
where is the fluid acceleration of the incident wave, and therefore, the second term in the
right hand side of the above equations represents the Froude-Kriloff force.
As stated above, equations (104-108) can estimate the different components of the force
per unit length on a long (slender) structural element. Therefore, they can be integrated
along the element axis, to obtain the additional forces an moments acting on the center
of gravity of the associated body.

43 Response Amplitude Operators (RAOs)
11. Response Amplitude Operators (RAOs)
RAOs are transfer functions of the relation between the wave exciting forces and ship
movements, used to determine the effect that a sea state will have upon the motion of a
ship through the water.
Calculation of Response Amplitude Operator (RAO) in SeaFEM is done by analysing
the time series response of the ship, using a discretized white noise spectrum. This
spectrum is defined by a number = 2 of waves of equal amplitude and periods
varying between the maximum and minimum values defined by the user. The value of
these se periods are selected to match the discrete Fourier transform of the output
signal, give by:
= · ∆∗
(140)
Given and , the minimum and maximum periods of the analysis, the
frequency increment is:
∆ = ( − ) ⁄ =
−
⁄
(141)
The well known Fast Fourier Transform algorithms give a procedure to obtain an exact
evaluation of the transfer functions defined above. This way, the time step and the total
computing time are internally fixed to match the required sampling time and total
sampling points. Then, the holding frequency ∆ ∗ is evaluated as
∆ ∗ = (∆ , ) ⁄
And the discrete frequencies are:
= · ∆ = 0, 1, 2, … , − 1
The required sampling frequency defines the time step as:
(142)
(143)
∆ =
1
2∆ ∗ · (144)
The required number of sampling points defines the total calculation time as:
= 1
∆ ∗ (145)

44 Fluid-Structure interaction algorithm
12. Fluid-Structure interaction algorithm
SeaFEM features a fluid-structure interaction algorithm, able to perform coupled
analysis with the structural solver of the Tdyn’s suite Ramseries.
The fluid-structure coupling is performed by an implicit iterative algorithm. This way,
the pressure field computed in every iteration of the diffraction-radiation solver is sent
to the structural solver to compute the body deformation. The resulting displacements
are sent back to the fluid solver, and used as boundary condition to compute the new
pressure field for the following iterations. The iterative process continues until the
convergence norm condition is fulfilled; the relative distance of the pressure vectors
between two successive iterations is below a given value. The iterative algorithm is
accelerated using the Aitken method [21].
Since the diffraction-radiation and structural solvers are independent, the strategy used
to communicate both solvers is based on the interchange of information at memory level
by means of TCP-IP sockets. The library developed for this purpose, also takes care of
the data interpolation from one mesh to another mesh.

45 References
13. References
[1] Oñate E. & García, J., A finite element method for fluid-structure interaction with
surface waves using a finite calculus formulation, Comp. Methods Appl. Mech. and Eng.
2001; 191: 635-660.
[2] García, J., Valls A., & Oñate, E., ODDLS: A new unstructured mesh finite element
method for the analysis of free surface flow problems, Int. J. umer. Meth. Fluids
2008; 76 (9): 1297-1327.
[3] Wu, G.X. & Eatock Taylor, R., Finite element analysis of two dimensional nonlinear
transient water waves, Appl. Ocean Res. 1994 ; 16: 363-372.
[4] Wu, G.X. & Eatock Taylor, R., Time stepping solution of the two dimensional nonlinear
wave radiation problem, Ocean Eng. 1995; 22: 785-798.
[5] Greaves, D.M., Borthwick, A.G.L., Wu, G.X. and Eatock Taylor, R., A moving
boundary finite element method for fully nonlinear wave simulation, J. Ship Res. 1997;
41: 181-194.
[6] Ma, Q.W., Wu, G.X. and Eatock Taylor, R., Finite element simulation of fully
nonlinear interaction between vertical cylinders and steep waves--part 1 methodology
and numerical procedure, Int. J. umer. Meth. Fluids 2001; 36: 265-285.
[7] Ma, Q.W., Wu, G.X. and Eatock Taylor, R., Finite element simulation of fully
nonlinear interaction between vertical cylinders and steep waves--part 2 numerical
results and validation, Int. J. umer. Meth. Fluids 2001; 36: 265-285.
[8] Westhuis, J.H., The numerical simulation of nonlinear waves in the hydrodynamic
model test basin, Ph.D. Thesis 2001, Universiteit Twente, The Netherlands.
[9] P.X. Hu, G.X. Wu and Q.W. Ma, Numerical simulation of nonlinear wave radiation
by a moving vertical cylinder, Ocean Engn . 2002; 29: 1733–1750.
[10] Faltinsen O.M., Sea loads on ships and offshore structures, Cambridge Ocean
Technology Series. 1998.
[11] Clauss, G. F. , Schmittner, C., and Stutz, K., Time-domain investigation of a
semisubmersible in rouge waves, 21st International Conference on Offshore Mechanics
and Arctic Engineering June 23-28, 2002, Oslo, Norway. OMAE2002-28450.
[12] Jonkman, J.M., Dynamic modelling and loads analysis of an offshore floating wind
turbine, Technical report REL/TP-500-41958; November 2007.
[13] Serván, B. and García, J., Advances in the Development of a Time-domain
Unstructured Finite Element Method for the Analysis of Waves and Floating Structures
Interaction. Marine 2011. Lisbon, Portugal.

46 References
[14] Morison J. R., O’Brien M. P., Johnson J. W., and Schaaf S. A., 1950, The force
exerted by surface waves on piles. Petroleum Transactions, American Institute of
Mining Engineers, 189, pp. 149–154.
[15] Recommended practice DNV-RP-C205. Environmental conditions and environmental
loads. April 2007.
[16] Det Norske Veritas. A Course in Ocean engineering. Available at:
.
[17] Wood W. L., Bossak M., and Zienkiewicz O. C. An alpha modication of
Newmark's method. International Journal for Numerical Methods in Engineering, 1980,
15(10):1562-1566.
[18] Leonard, B. P. A stable and accurate convective modelling procedure based on
quadratic upstream interpolation. Computer methods in applied mechanics and
engineering 19 (1979) 59-98.
[19] Pierson J.P. and Moskowitz L. A Proposed Spectral Form for Fully Developed
Wind Seas Based on the Similarity Theory of S.A.Kitaigorodskii. Journal of
Geophysical Research, 1964, Vol. 69, No.24.
[20] Hasselmann K., Barnett R.C., Bouws E., Carlson H., Cartwright D.E., Enke K.,
Ewing J.A., Gienapp H., Hasselmann D.E., Kruseman P., Meerburg A., Müller P.,
Olbers, D.J., Richter K., Sell W., Walden H. Measurements of Wind-Wave Growth and
Swell Decay during the Joint North Sea Wave Project (JONSWAP). Deutches
Hydrographisches Institut, 1973, No.12.
[21] Irons B. M., Tuck R.P. A Version of the Aitken Accelerator for Computer Iteration.
International Journal for Numerical Methods in Engineering, 1969, 1:275-277.
[22] Bathe, K.H. Finite element procedures. Prentice Hall, 1996.
[23] Gutiérrez, J.E. Desarrollo de herramientas software para el análisis de
aerogeneradores “offshore” sometidos a cargas acopladas de viento y oleaje. PhD
dissertation, Universidad Politécnica de Cartagena (2014).

47 Glossary
14. Glossary
∇ = ( , ) : Horizontal gradient
Ω : Fluid domain
Γ : Body wet surface
H : Water depth
ρ : Water density
v : Local body velocity
n : Normal of the body wetted surface
u : Water current
U = − : Flow field from the point of view of the floating body
: Free surface pressure
: Wave potential
: Incident wave potential
: Difracted/radiated wave velocity potential
: Free surface elevation
: Incident wave free surface elevation
: Free surface elevation of diffracted and radiated waves
: Wave amplitudes
: Wave frequencies
: Wave number vectors
: Wave directions
: Wave phases
Γ : Outlet boundary of the computational domain
c : Prescribed wave velocity
n : Normal of the outlet boundary surface of the domain

48 Glossary
(): Wave dissipation damping coefficient