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Abstract

Mooring systems exhibit high failure rates. This is especially problematic for offshore renewable energy systems, like wave and floating wind, where the mooring system can be an active component and the redundancy in the design must be kept low. Here we investigate how uncertainty in input parameters propagates through the mooring system and affects the design and dynamic response of mooring and floaters. The method used is a non-intrusive surrogate based uncertainty quantification (UQ) approach based on generalized Polynomial Chaos (gPC). We investigate the importance of the added mass, tangential drag, and normal drag coefficient of a catenary mooring cable on the peak tension in the cable. It is found that the normal drag coefficient has the greatest influence. However, the uncertainty in the coefficients play a minor role for snap loads. Using the same methodology we analyse how deviations in anchor placement impact the dynamics of a floating axi-symmetric point-absorber. It is shown that heave and pitch are largely unaffected but surge and cable tension can be significantly altered. Our results are important towards streamlining the analysis and design of floating structures. Improving the analysis to take into account uncertainties is especially relevant for offshore renewable energy systems where the mooring system is a considerable portion of the investment.
Journal of
Marine Science
and Engineering
Article
Uncertainty Quantification in Mooring Cable
Dynamics Using Polynomial Chaos Expansions
Guilherme Moura Paredes 1,, Claes Eskilsson 1and Allan P. Engsig-Karup 2,3
1Department of Civil Engineering, Aalborg University, DK-9220 AalborgØ, Denmark; cge@civil.aau.dk
2Department of Applied Mathematics and Computer Science, Technical University of Denmark,
DK-2880 Kgs. Lyngby, Denmark; apek@dtu.dk
3Center for Energy Resources Engineering, Technical University of Denmark,
DK-2880 Kgs. Lyngby, Denmark
*Correspondence: gmp@civil.aau.dk
Received: 1 February 2020; Accepted: 24 February 2020; Published: 2 March 2020


Abstract:
Mooring systems exhibit high failure rates. This is especially problematic for offshore
renewable energy systems, like wave and floating wind, where the mooring system can be an active
component and the redundancy in the design must be kept low. Here we investigate how uncertainty
in input parameters propagates through the mooring system and affects the design and dynamic
response of mooring and floaters. The method used is a nonintrusive surrogate based uncertainty
quantification (UQ) approach based on generalized Polynomial Chaos (gPC). We investigate the
importance of the added mass, tangential drag, and normal drag coefficient of a catenary mooring
cable on the peak tension in the cable. It is found that the normal drag coefficient has the greatest
influence. However, the uncertainty in the coefficients plays a minor role for snap loads. Using the
same methodology we analyze how deviations in anchor placement impact the dynamics of a floating
axi-symmetric point-absorber. It is shown that heave and pitch are largely unaffected but surge
and cable tension can be significantly altered. Our results are important towards streamlining the
analysis and design of floating structures. Improving the analysis to take into account uncertainties is
especially relevant for offshore renewable energy systems where the mooring system is a considerable
portion of the investment.
Keywords:
mooring system dynamics; mooring cables; floating structure dynamics; uncertainty
quantification; generalized polynomial chaos
1. Introduction
Mooring systems play an important role in assuring the safe operation (to life and to the
environment) of floating structures like oil and gas (O&G) platforms. The failure rates of mooring
cables for O&G installations are surprisingly high. Brindley and Comley [
1
] analyzed the times to
failure of mooring cables for semisubmersible mobile offshore drilling units (MODUs) operating in
the North Sea during the years 1996–2005, concluding that there is an average time of 24 operational
years per failure of single lines and 112 years for multiple lines. These numbers should be compared
with the industry target times of 10,000 and 100,000 operational years [
2
]. The reasons for the failures
are widespread. Kvitrud [
3
] reports on 16 line failures in the Norwegian Continental Shelf between
2010 and 2014. The failures were due to fatigue (four cases), overload (six cases), mechanical damage
(four cases), and manufacturing errors (two cases). In other words, more than 60% of the failures were
due to design errors. While a mooring failure of an O&G platform can cause severe consequences,
often the catastrophic effects are prevented due to high redundancy in the mooring design.
J. Mar. Sci. Eng. 2020,8, 162; doi:10.3390/jmse8030162 www.mdpi.com/journal/jmse
J. Mar. Sci. Eng. 2020,8, 162 2 of 23
In floating renewable energy technologies, due to their nature, mooring systems are not as critical
to life and environment. They are, however, a significant part of the total budget (18%–30% [
4
,
5
],
compared to 2% [
4
] for O&G). Due to the large portion of cost associated with the mooring system,
the level of redundancy in the mooring design is kept low. Hence, failures of mooring systems have
caused the loss of several offshore renewable energy prototypes [6].
To reduce the cost and optimize the performance of mooring systems, it is of interest to know
which parameters have the greatest contribution to the final design and to the dynamics of the structure
they keep on station. We also need to understand how uncertainty in input parameters propagates
through the mooring system and affects the design and dynamic response. Two parameters that are
subjected to uncertainty and can have a large influence on the dynamics of mooring systems are the
hydrodynamic coefficients of mooring cables and the position of the anchors. It is hard, and most of the
time impossible, to determine the actual values of hydrodynamic coefficients in full scale prototypes;
as for the anchors, it is difficult to ensure that they are positioned exactly as specified during design,
usually being allowed an installation tolerance around the expected position.
Hydrodynamic coefficients, being determined in small scale models, have the additional issues
of their values suffering from scale effects and of depending on the experimental methods applied.
For example, for the added mass coefficient of a cable used in Morison’s formulation for hydrodynamic
forces, the software Orcaflex [
7
] uses the value
1.0
for chains, while in Marin’s aNySIM model [
8
] the
default value is 0.6.
It is on the impact of hydrodynamic coefficients and anchor position on the dynamics of mooring
cables and floating structures that we will focus our study. In Section 4.2 we will analyze the impact
of uncertainty in the hydrodynamic coefficients in the peak tension of a catenary mooring system.
Afterwards, in Section 4.3, we will analyze the impact that errors in anchor placement have on the
dynamics of a floating buoy and respective mooring cable tension. To simulate the dynamics of cables
and floating structures we will use numerical models, described in Section 2, while forward uncertainty
propagation will be based on generalized Polynomial Chaos (gPC).
Generalized Polynomial Chaos, described in Section 3, provides a surrogate model, based on
polynomial expansions, for stochastic processes. It was introduced by Norbert Wiener in 1938 as
Homogeneous Chaos [
9
], using expansions based on the Normal Distribution and expanded by Xiu
and Karniadakis in 2002 to generalized Polynomial Chaos [
10
] to include several other probability
distribution laws. Its strength is the ability to, for the same inputs, provide almost the same results as
the original model but at a fraction of the computational cost and time. Furthermore, for problems with
low dimensionality (small number of uncertain inputs) gPC is much faster in converging to the solution
than traditional methods, such as the Monte Carlo (MC) method. Generalized Polynomial Chaos is a
well established procedure in uncertainty quantification and sensitivity analysis of computational fluid
dynamics (CFD) simulations, see e.g., [
11
,
12
] and the references therein. It has been applied to study,
for example, the dynamics of vehicles [
13
] and train wagons [
14
]. For renewable energy we note the
study of wind turbines [
15
]. In coastal engineering gPC has been used to study the sensitivity of coastal
inundation to parameters like Manning’s number [
16
,
17
], while in ocean engineering the method
has been applied to phase-averaging spectral wave modeling [
18
], water wave propagation and
transformation [
19
], wave scattering from an ice floe [
20
], and the motions of a heaving cylinder [
21
]
in irregular waves. It is also seeing applications in the field of marine renewable energy and moored
structures, in the prediction of extreme loads on wave energy converters [
22
], and long-term extreme
responses of moored floating structures [23] .
For completeness, we will first illustrate the use of gPC to the case of a simple taut linear string in
Section 4.1. The most important results of our research will be summarized in Section 5.
J. Mar. Sci. Eng. 2020,8, 162 3 of 23
2. Numerical Models
2.1. Mooring Equations
Let us consider the equation of perfectly flexible cables [24], applied to a cable of length LR+,
in a physical domain
DR3
and a time domain
t[
0,
tend]
,
tend >
0, subject to initial and Dirichlet
boundary conditions, Equation (1):
ml(s)2r(s,t)
t2=
sT(e(s,t))
1+e(s,t)
r(s,t)
s+fe(s,t), (1a)
e(s,t) =
r(s,t)
s1 , (1b)
r(s,t=0) = r0(t,r), (2)
r(s=0, t) = f1(t), (3)
r(s=L,t) = f2(t), (4)
where
ml
is the mass per unit length of the cable,
r(s
,
t)
is the position of the cable point
s
,
s[
0,
L]
is
the unstretched line coordinate, T is the tension magnitude,
e
is the extension, and
fe
are the external
forces acting on the cable.
r0(t
,
r)
is the initial deformation of the cable and
f1(t)
and
f2(t)
represent
the boundary conditions at the ends of the cable. The external forces,
fe
, include the effects of buoyancy,
weight, hydrodynamic forces (added mass and viscous drag), and ground forces (contact forces and
Coulomb drag). The added mass and the drag forces are computed via Morison’s equations [
25
] based
on the relative acceleration and velocity between the fluid and the cable, Equation (5):
fm=Acρwarf(1+Cm), (5a)
fd=1
2ρwD1+e(Cdt |vr,t|vr,t+Cdn |vr,n|vr,n), (5b)
and the effect of weight and buoyancy computed using Equation (6)
fb=mlρcρw
ρcg, (6)
where
fm
is the added mass force,
Ac
is the cable cross section area,
ρw
is the fluid mass density,
arf
is the relative cable-fluid acceleration,
Cm
is the cable added mass coefficient,
fd
is the viscous drag
force,
D
is the cable diameter,
Cdt
is the cable tangential drag coefficient,
Cdn
is the cable normal
drag coefficient,
vr,t
is the relative fluid-cable tangential velocity,
vr,n
is the relative fluid-cable normal
velocity,
fb
is the submerged weight of the cable,
ρc
is the mass density of the cable material, and
g
is
the acceleration of gravity.
Interaction of the cable with the ground is modelled using a spring-damper system in the normal
direction, Equation
(7a)
, and a Coulomb friction model in the tangential direction, Equation
(7c)
.
The spring-damper system applies stiffness and damping forces when the cable is setting down,
but when the cable is being lifted it applies only stiffness forces.
fc,z=(KD(rzzg)2ξpmlKD max(vz, 0), if rzzg,
0, otherwise, (7a)
fc,xy =−||fb||µsin vxy π
2, (7b)
vxy =(vx,vy)
max(vc,
(vx,vy)
), (7c)
J. Mar. Sci. Eng. 2020,8, 162 4 of 23
in which
fc,z
is vertical ground force,
fc,xy
is the horizontal ground force,
K
is the soil’s stiffness per
unit area,
rz
is the height of the cable,
zg
is the height of the sea-bottom,
ξ
is the soil’s normal damping
ratio,
vz
is the vertical component of the velocity of the cable,
vx
is the component of the velocity of
the cable in the
x
direction,
vy
is the component of the velocity of the cable in the
y
direction,
vc
is the
speed of fully developed friction force magnitude, and µis the soil’s Coulomb friction coefficient.
2.2. Numerical Mooring Model
The dynamics of mooring cables are solved using the numerical code MooDy, introduced in
Palm et al. [26]
. MooDy solves Equation
(1)
using an
hp
-adaptive discontinuous Galerkin (DG) method.
In doing so, the equation is first reformulated to be cast in conservative form. The DG method allows
discontinuities across the element boundaries, using a numerical flux to couple elements together.
This makes the DG method locally conservative and a good candidate for problems involving shocks,
such as snap loads. A DG cable is illustrated in Figure 1, where the elements are approximated using a
basis made up of Legendre polynomials (top right corner), and the numerical flux is made up of an
approximate Riemann solver (bottom right corner), which in MooDy is the local Lax–Friedrich flux.
For smooth solutions, MooDy exhibits convergence rates of (
p+
1
/
2) [
26
]. This allows high-resolution
solutions using few degrees-of-freedom. The model advances in time using the explicit third-order
strong-stability-preserving Runge–Kutta scheme.
Figure 1.
Outline of the high-order DG modeling approach. The cable is discretised into finite elements
of size hwith approximation order p. The jumps are exaggerated for illustrative purposes.
When coupled to external codes, such as floating body solvers, at each rigid body time step,
MooDy updates the position of the mooring fairlead and returns the corresponding mooring force.
As the time step required for the cable dynamics is smaller than for the body motion, several substeps
are performed in the mooring solver between each rigid body time step. Please note that in MooDy,
the hydrodynamic forces of added mass and drag, Equation
(5)
, are presently computed under the
assumption of quiescent water. For details of MooDy please see [26,27].
2.3. Floating Body Model
In the cases presented here concerning moored floating structures, the dynamics of the floating
structure are modeled using linear potential flow theory and Cummins’s Equation [
28
], Equation
(8)
:
(M+A)¨
X(t)+Zt
K(tτ)˙
X(t)dτ+CX (t)=fext (t), (8)
with
M
being the generalized mass matrix of the the floating body,
A
the added mass matrix at
infinity frequency,
K
the radiation impulse response function,
C
the hydrostatic stiffness matrix,
fext
the vector of external forces, and
X
,
˙
X
, and
¨
X
, respectively, the displacement, velocity, and acceleration
vectors. The vector of external forces,
fext
, contains any wave excitation forces,
fexc
and mooring forces,
J. Mar. Sci. Eng. 2020,8, 162 5 of 23
fmoor
. The hydrodynamic coefficients are typically computed using a boundary element method (BEM)
code for potential flow, such as WAMIT [29] or Nemoh [30].
The code used to solve Cummins’s Equation, or the dynamics of the floating structure,
was WEC-Sim, a time-domain multibody dynamics model, developed by NREL and Sandia [
31
].
To simulate nonlinear mooring dynamics, WEC-Sim has to be coupled to external mooring simulation
codes. The standard code coupled to WEC-Sim for mooring dynamics is the lumped-mass model
MoorDyn [
32
]. However, the code we use for mooring dynamics is MooDy. The coupling of MooDy to
WEC-Sim was introduced in [26] and further validated in [33].
3. Uncertainty Quantification
3.1. Generalized Polynomial Chaos
Consider a mathematical model
f
with uncertainty in one of its input variables. We can represent
f
by
f(x
,
Z)
, where
x
is the vector of deterministic input variables and
Z
is the input subjected to
uncertainty.
Z
is a random variable that can take values in
R
, from the sample space of possible
outcomes
, in the form of events denoted by
F
, with an associated probability
P
. This is denoted
by
Z:R
. Using the property that continuous function in
L2
with bounded variation can be
expressed as an infinite series, gPC provides a polynomial expansion surrogate model to f(x,Z):
fgPC (x,Z) =
k=0
ˆ
fk(x)φk(Z), (9)
in which
ˆ
fk(x)
are the polynomial coefficients and
{φk(Z)}
k=0
is the set of polynomial basis functions.
For optimal convergence of
fgPC (x
,
Z)
to the results of the original model
f(x
,
Z)
, the basis functions
should be selected based on the probability distribution of
Z
according to the Wiener–Askey
scheme [
10
]. Because Equation
(9)
is an infinite sum, in practical applications it must be truncated at
specified polynomial degree p, being represented by a series approximation:
fgPC (x,Z)
p
k=0
ˆ
fk(x)φk(Z). (10)
Usually, there is no prior knowledge about the minimum required degree of the polynomial
approximation, which needs to be tuned by trial and error. For the particular case of smooth solutions,
the coefficients decay rapidly with increasing order
p
, providing a means to determine the optimal
value for
p
. An important feature of gPC is that the coefficients
ˆ
fk(x)
encode information about the
moments of the probability distribution of the results. For example,
ˆ
f0(x)
is a scaled value of the
mean, and the variance can be obtained by summing the squares of the set of coefficients
ˆ
fi(x)p
i=1
in
the expansion.
For problems with
d
independent input random variables,
dN
,
φ(Z)
in Equations
(9)
and
(10)
is replaced by a tensor product of polynomials corresponding to each random variable
fgPC (x,Z)
p
|k|=0
ˆ
fk(x)Φk(Z) =
p
|k|=0
ˆ
fk(x)φk1(Z1)φk2(Z2)...φkd(Zd), (11)
where
Z:Rd
is the vector of input random variables,
k
is a multi-index such that
k=
(k1
,
k2
,
. . .
,
kd)N0
and
|k|=k1+k2+
...
+kd
, and
φki(Zi)
is the polynomial basis function of the
variable Ziof degree ki.
J. Mar. Sci. Eng. 2020,8, 162 6 of 23
3.2. Stochastic Collocation Method
The application of gPC to a mathematical model
f(x
,
Z)
can take two different forms: the
Stochastic Galerkin Method and the Stochastic Collocation Method [
34
]. We will be using the stochastic
collocation method because it is a nonintrusive method, so it does not require any knowledge of the
mathematical model
f(x
,
Z)
under study. For the computation of the gPC model
fgPC (x
,
Z)f(x
,
Z)
,
we only need to postprocess the results of simulations using the mathematical model at preselected
values
z(j)
of the uncertain input
Z
. The points
z(j)
where
f(x
,
Z)
is to be evaluated depending on
the method chosen to determine the coefficients
ˆ
fk
. The two methods we will use are the projection
method and the regression method.
In the projection method, the coefficients
ˆ
fk(x)
are determined by the inner product of
f(x
,
Z)
and the polynomial basis,
Φk(Z)
, with respect to the probability density function (PDF) of the random
variable, ρ(Z), Equation (12):
ˆ
fk(x) = hfk(x,z),Φk(z)i=Rf(x,z)Φk(z)ρ(z)dz
RΦ2
k(z)ρ(z)dz. (12)
Equation
(12)
is solved using quadrature rules, such as Gauss quadrature, which provide the
points
z(j)
where the model is to be evaluated and the quadrature weights,
w(j)
. Because multivariable
gPC is formulated as a tensor product, the number of points
z(j)
where the numerical model needs
to be evaluated grows exponentially with the number
d
of random input variables: the dimension
of the problem. This is known as the curse of dimensionality. For
d>
4 it is convenient to employ
sparse quadratures, such as Smolyak’s quadrature [
34
]; however, for
d>
5, even sparse quadratures
can require hundreds or thousands of costly model evaluations. In this situation it is more efficient to
apply a regression method or methods designed for high scalability with dimensions, e.g., [35].
In the regression method, the coefficients
ˆ
fk
are determined by minimizing the error of fitting
the values of the coefficients to the results of the numerical model, at randomly sampled values of
z(j)
. One of the procedures to fit the values of the coefficients is the Least Angle Regression (LAR)
algorithm [
36
], which computes a set of candidate solutions and chooses the one with the smallest
error. For the best results, the random
z(j)
should be sampled with a method which tries to cover the
sample space evenly.
In multivariate models, we can limit the interaction between the different univariate polynomials
in the tensor product by setting the value of the
q
-norm. A
q
-norm of 1 allows the tensor product
of any set of univariate polynomials to reach the maximum selected polynomial order; decreasing
the
q
-norm, until the minimum value of zero, decreases the maximum polynomial order allowed for
products of univariate polynomials, reducing the total number of polynomial terms.
3.3. Model Equations with Random Input Variables
The stochastic equation of cable dynamics follows directly from Equation
(1)
, by introducing a set
of random variables Z:Rd, Equation (16):
ml(s,Z)2r(s,t,Z)
t2=
sT(e(s,t,Z))
1+e(s,t,Z)
r(s,t,Z)
s+fe(s,t,Z), (13a)
e(s,t,Z) =
r(s,t,Z)
s1 , (13b)
r(s,t=0, Z) = r0(s,Z), (14)
r(s=0, t,Z) = f1(t,Z), (15)
r(s=L,t,Z) = f2(t,Z). (16)
J. Mar. Sci. Eng. 2020,8, 162 7 of 23
We obtain the stochastic Cummins’s equation, Equation
(17)
, from Equation
(8)
in a similar way
as Equation (16):
(M+A)¨
X(t,Z)+Zt
K(tτ)˙
X(t,Z)dτ+CX (t,Z)=Fext (t,Z). (17)
3.4. Methodology
We applied the following methodology throughout all the cases presented in the article:
1. select the process to be studied;
2. select a physics-based mathematical model for the process;
3. choose which parameters of the model are deterministic and which are random variables;
4. define the values of the deterministic parameters;
5. choose appropriate probability distributions for the random variables;
6. choose the method to determine the gPC model coefficients (quadrature or LAR)
7.
compute (for the case of quadrature) or sample (for the case of LAR) the values of the input
random variables where the physics-based model is to be evaluated;
8. evaluate the physics-based model at the previously defined values of the random variables;
9.
apply the chosen method (quadrature or LAR) to the results of point 8 to obtain the coefficients of
the gPC model;
10.
use the gPC model of the physics-based model to evaluate large samples of the random variables
and build probability density functions.
For the computation of the gPC model we use UQLab’s version 1.3.0, Polynomial Chaos
Expansions Module [
36
]. The probability density functions are estimated with the Kernel Density
Estimation method with a Gaussian kernel, using MATLAB’s ksdensity function, with a logarithmic
boundary correction. For solving the Cummin’s Equation
(8)
we employ WEC-Sim version 3.0.0,
commit 27cfb3b [31].
4. Case Studies
4.1. Linear String
In this case we will study the effect of uncertainty in the tension and in the mass per unit
length of a linear string in its geometry, at a certain time after being released from an unsteady
position. The equation of motion of a taut linear string, Equation
(18)
, is obtained from Equation
(1)
under the assumptions of constant extension, tension, mass per unit length, and small transverse
displacements [37]:
2y
t2=T
ml
2y
x2, (18)
with
x
the abscissa of the cable and
y
the ordinate. For a cable with length
L
, fixed at both ends,
y(0, t) = y(L,t) = 0, and zero initial velocity, y(x,t)
tt=0=0, the solution of Equation (18) is [37]:
y(x,t) =
n=0
bnsin nπ
Lxcos nπ
Lct, (19)
where
bn
are the Fourier series coefficients of the function describing the initial displacement
y(x
, 0
)
,
and
c
is the transverse wave celerity,
c=p(T/ml)
. For an initial displacement in the shape of a half
sine, Figure 2a, we have y(x, 0) = Asin nπ
Lx, and Equation (19) becomes:
y(x,t) = Asin π
Lxcos π
Lct. (20)
J. Mar. Sci. Eng. 2020,8, 162 8 of 23
There are six variables influencing the results of Equation
(20)
A
,
L
,
x
,
T
,
ml
, and
t
—of which
only two are subjected to uncertainty:
T
and
ml
. For the reference deterministic case, against which
we will compare the results of uncertainty quantification, we choose for the variables the values
presented in Table 1. The reference values for
L
,
T
, and
ml
were selected so that the cable would have
an oscillation period of
1 s
and that at odd multiples of
0.25 s
it would be on a perfectly straight line,
as illustrated in Figure 2a. For simplicity, we assign uniform distributions to
T
and
ml
:
T∼ U(
100, 116
)
,
ml∼ U(1, 5).
Because the two input distributions are uniform, following the Wiener–Askey scheme [
10
] for
optimal convergence, we use Legendre polynomials for the basis functions. Since the uncertainty
dimension is
d=
2
<
4, we use the full quadrature to solve Equation
(12)
. In addition, with
d=
2,
there is no need to limit the number of polynomial coefficients, so we choose a
q
-norm of 1. Thus,
we use
(p+
1
)d
quadrature points to construct the gPC. All the results presented below are for a time
instant t=0.25 s.
Table 1. Linear cable properties.
Property Value
A1 m
L3 m
T108 N
ml3 kg/m
0 0.5 1 1.5 2 2.5 3
x position (m)
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Initial position
Position at t = 0.25 s
(a)
(b)
Figure 2.
Example of a taut linear cable. (
a
) Deterministic displacements of the cable. (
b
) Representation
of the probability density of the vertical displacement of the cable, at t=0.25 s.
Now we will use gPC to estimate the probability density function of the position of the cable.
For that, we generated 500,000 random samples of
T
and
ml
and evaluated them in the gPC model.
Even though we are not really concerned with the physics of the problem, let us analyze the results
of the gPC computations. In Figure 2b, we illustrate the density function for the whole cable using
p=
6. We can immediately notice two things: first, in spite of the simple distributions of
T
and
ml
, the distribution of the cable position is far from simple. Second, there is a wide range of cable
positions significantly different from the deterministic one, which have a reasonable high probability
of occurrence.
We use this simple case to illustrate the convergence of the gPC mooring cable surrogate model.
Figure 3a illustrates how the PDF of position evolves as
p
is increased, for the midpoint of the cable
(
x=1.5 m
). For
p=
1 we obtain a quite crude PDF, but for
p=
2, the approximation is reasonable.
J. Mar. Sci. Eng. 2020,8, 162 9 of 23
A polynomial order
p=
3 has practically converged to the final PDF. For polynomial
p>
4 it is almost
impossible to tell the plots apart, and so the PDF has converged. We find that the expected value of
the position of the midpoint of the cable, computed using gPC, is
9.39 ×102m
, and the variance is
1.26 ×101m2.
(a)
(b)
Figure 3.
Convergence of the gPC method for the taut linear cable case. (
a
) Probability density functions
of the position of the midpoint of the cable, at
t=0.25 s
, determined using different polynomial orders.
(
b
) Convergence of the statistics of the position of the midpoint of the cable for
t=0.25 s
: generalized
Polynomial Chaos versus Monte Carlo method.
In Figure 3b we can see the evolution of the error in the mean and in the variance of the solution
of Equation
(20)
at the midpoint, as a function of the polynomial order and number of Monte Carlo
samples. The error is defined as the absolute value of the difference between the mean or variance
at a certain order (or number of samples) and the value at a polynomial order
p=
10, for gPC, or at
500,000 samples, for the MC method. Using gPC, the error decreases exponentially with increasing
order
p
, whereas using the MC method the error decreases algebraically. Moreover, the error in the
gPC method decreases monotonically, while in the MC method it does not. The large error in the first
point of the variance plot in gPC occurs because the variance can only be computed for polynomial
order p>2.
4.2. Oscillating Mooring Cable
In this case we analyze the influence of uncertainty in the hydrodynamic coefficients of a mooring
cable (added mass coefficient,
Cm
, normal drag coefficient,
Cdn
, and tangential drag coefficient,
Cdt
) on
the tension at its upper end. The simulations reproduce experiments reported by Lindahl [
38
], whose
data is available through ref. [
39
]. The experimental set-up is illustrated in Figure 4. It is composed of
a submerged chain, with one end anchored to the bottom of a concrete tank and the other attached
to a disk slightly above the water. The disk moves the top end of the chain in a circular motion,
with a radius
rm=0.20 m
, for two different periods:
Tr=1.25 s
and
Tr=3.50 s
. A summary of the
relevant properties of the cable and of the numerical model is presented in Table 2. The cable was
discretised using
Nel
= 10 elements of order
p=
5, with a limitation on the time-stepping that the
Courant–Friedrichs–Lewy value should not exceed 0.45.
J. Mar. Sci. Eng. 2020,8, 162 10 of 23
3,3 ± 0,003 m
32,554 ± 0,005 m
Motor with load cell
0,0 m
3,0 m Chain 33 ± 0,005 m
Anchoring point
rm
Figure 4. Experimental set-up for the oscillating mooring cable case.
Table 2. Deterministic or constant parameters of the cable and of the experimental set-up.
Parameter Value
K3×109Pa/m
vc0.01 m/s
µ0.03
ξ1
ρw1000 kg/m3
ρc7800 kg/m3
D2.2 ×103m
ml8.18 ×102kg/m
EA 1×104Pa
Figure 5shows a comparison between numerical and experimental tension at the fairlead. There is
a general good agreement. The oscillations in low tension regions are exaggerated in the numerical
model. This is due to (i) the problem being ill-posed at slack conditions [
40
] and (ii) the ground
interaction being modeled with the traditional elastic seabed approach (the so-called spring-mattress
method), which is known to introduce spurious oscillations [41].
2 2.5 3 3.5 4 4.5 5 5.5 6
Time (s)
-20
-10
0
10
20
30
40
50
60
70
80
90
Deterministic simulation
Experimental measurments
(a)Tr=1.25 s
2 3 4 5 6 7 8 9 10 11 12
Time (s)
-10
0
10
20
30
40
50
60
Deterministic simulation
Experimental measurments
(b)Tr=3.50 s
Figure 5.
Numerical simulation of the tension at the fairlead compared to experimental data from [
39
].
For each period, we analyze four different scenarios: one analyzing the effects of uncertainty
on all the coefficients at the same time, and three which analyze the effects of uncertainty in each
coefficient individually. The first scenario gives us insight about how general uncertainty in cable
properties affects the cable dynamics, while the other scenarios give us information about the most
relevant coefficients affecting the cable dynamics.
J. Mar. Sci. Eng. 2020,8, 162 11 of 23
As there is very little data on the distribution of the hydrodynamic coefficients, we assume a
uniform distribution centered around the deterministic values reported in [
38
], with a parameter
range of half the mean value, Table 3. The values of
Cm
,
Cdn
and
Cdt
relate to the cable’s geometry
and surface roughness and these coefficients are, therefore, dependent. However, because there is no
data describing the correlation between the different coefficients to enable a multivariate probability
modeling, and to simplify the analysis, we assumed that they are independent.
Table 3. Parameters of the probability distributions of the hydrodynamic coefficients.
Coefficient Deterministic Distribution Lower Upper
Value Bound Bound
Cm3.8 Uniform 1.90 5.70
Cdn 2.5 Uniform 1.25 3.75
Cdt 0.5 Uniform 0.25 0.75
The polynomial coefficients for the gPC models of the cable dynamics were computed using
a polynomial degree
p
of 5 and the full quadrature method (i.e.,
(p+
1
)d
quadrature points are
used to construct the gPC). Then, for each of the four scenarios,
3000
random samples of coefficient
values were drawn from their distributions and evaluated in the gPC model, to build the probability
density functions.
The results of the simulations are presented in Figure 6for the case of
Tr=1.25 s
and in Figure 7
for the case of
Tr=3.50 s
. The most striking result is that, for both oscillating periods, despite the
relatively large uncertainty in the coefficient values (50%), the range of variation in the tension cycles
and maximum tensions is not very large. It is mostly the spread of tension values in the lower tension
regions that is affected. This is more surprising in the case with an oscillating period
Tr=1.25 s
,
which shows snap loads. Snap loads, being generated by quick cable motions, are more dependent on
the hydrodynamic coefficients, but, they nevertheless seem to be little influenced by the uncertainty.
In all plots it is visible that when the tension is close to zero, the probability density and the
confidence intervals extend slightly to negative values. This is not possible in a flexible cable, as it
cannot sustain compression. However, it is well-known that in the absence of bending stiffness,
Equation
(1)
is ill-posed in slack conditions [
40
]. This gives rise to numerical oscillations when the
tension is close to zero, leading to a large variability of the results in this region. Because of this
variability, it is difficult for any regression or surrogate model (like the gPC method) to accurately
predict the trend of the original model.
For both oscillation periods—
Tr=1.25 s
and
Tr=3.50 s
—the maximum tensions happen when
all the coefficients—
Cm
,
Cdn
and
Cdt
—are varied simultaneously: around
79 N
for
Tr=1.25 s
and
around
69 N
for
Tr=3.50 s
. When the coefficients are varied individually, changes in
Cdn
lead to the
highest peak tensions, which are equal to, or only slightly lower than, when the coefficients are varied
simultaneously: around
75 N
for
Tm=1.25 s
and around
69 N
for
Tm=3.50 s
. Varying
Cdt
and
Cm
leads
to the lowest peak tensions:
73 N
to
74 N
for
Tm=1.25 s
and around
52 N
for
Tm=3.50 s
. Another
result that is common to both oscillating periods is the spread of tension values, which is largest when
all the coefficients are varied simultaneously, followed by when only
Cdn
is varied. Variations in
Cdt
and
Cm
cause significantly lower spread of the tension values. In fact, for
Tr=3.50 s
, varying
Cdt
causes
no noticeable changes in the tension values, so it is not possible to compute the PDF.
J. Mar. Sci. Eng. 2020,8, 162 12 of 23
2 2.5 3 3.5 4 4.5 5 5.5 6
Time (s)
-20
-10
0
10
20
30
40
50
60
70
80
90
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
(a) Effect of Cm,Cdn, and Cdt
2 2.5 3 3.5 4 4.5 5 5.5 6
Time (s)
-20
-10
0
10
20
30
40
50
60
70
80
90
10-5
10-4
10-3
10-2
10-1
100
101
(b) Effect of Cdn
2 2.5 3 3.5 4 4.5 5 5.5 6
Time (s)
-20
-10
0
10
20
30
40
50
60
70
80
90
10-5
10-4
10-3
10-2
10-1
100
101
(c) Effect of Cdt
2 2.5 3 3.5 4 4.5 5 5.5 6
Time (s)
-20
-10
0
10
20
30
40
50
60
70
80
90
10-5
10-4
10-3
10-2
10-1
100
101
102
(d) Effect of Cm
Figure 6.
Representation of the probability density of cable tension as a function of time, for
Tr=1.25 s
,
together with the mean tension (black line).
Focusing on the case with oscillating period
Tr=1.25 s
, we can see that there is a higher dispersion
of the tension when its value is close to
0 N
and increasing, than when at its maximum and decreasing.
This is probably caused by the instability of the equation of perfectly flexible cables, which becomes
ill-posed when the tension is zero, leading to numerical oscillations. In spite of, as mentioned above,
there being a larger dispersion of tension values when all the coefficients are varied simultaneously,
looking at Figure 6, we see that, in the low tension region, the probability densities are generally
higher when the coefficients are varied individually. This means that, although a wider range of
tension values is possible when all the coefficients are varied simultaneously, extreme values are
less likely to happen in this case than when the coefficients are varied individually. This points to a
possible smoothing effect of the variation of the coefficients over one another, something that requires
a deeper investigation.
J. Mar. Sci. Eng. 2020,8, 162 13 of 23
2 3 4 5 6 7 8 9 10 11 12
Time (s)
-10
0
10
20
30
40
50
60
10-4
10-3
10-2
10-1
100
(a) Effect of Cm,Cdn, and Cdt
2 3 4 5 6 7 8 9 10 11 12
Time (s)
-10
0
10
20
30
40
50
60
10-3
10-2
10-1
100
101
(b) Effect of Cdn
2 3 4 5 6 7 8 9 10 11 12
Time (s)
-10
0
10
20
30
40
50
60
(c) Effect of Cdt
2 3 4 5 6 7 8 9 10 11 12
Time (s)
-10
0
10
20
30
40
50
60
10-4
10-3
10-2
10-1
100
101
102
(d) Effect of Cm
Figure 7.
Representation of the probability density of cable tension as a function of time, for
Tr=3.50 s
,
together with the mean tension (black line).
For the oscillating period
Tm=3.50 s
, the differences between the four scenarios are more marked
than for
Tm=1.25 s
. Variations in
Cm
have limited influence in the tension, and changes in
Cdt
have
no effect at all; therefore, it was not possible to build a PDF for this case. It is variations in
Cdn
that
cause most of the variations in the tension, as can be seen in Figure 7a,b. In contrast to the case with
the oscillation period
Tm=1.25 s
, for
Tm=3.50 s
, the probability density and the mean have a smooth
evolution in time.
Another way to study the impact of the hydrodynamic coefficients is through variance-based
sensitivity analysis, which provides a global sensitivity analysis. A graphical representation of this is
illustrated in Figure 8, for the scenario where all the coefficients are varied simultaneously. This figure
shows the approximate contribution of each hydrodynamic coefficient to the variance of the tension as
a function of time. It is based on the Total Sensitivity Indexes (TSI) suggested by Sobol, computed using
the gPC coefficients, as described in [
14
,
42
]. In Figure 8, the TSIs are represented by their magnitude
relative to each other like in [
43
], instead of their absolute value. For both rotation periods,
Cm
and
Cdt
have the greatest influence when the tension is close to
0 N
.
Cm
has similar contributions in
Tr=1.25 s
and in
Tr=3.50 s
. For
Tr=1.25 s Cdt
dominates in the low tension region, while
Cdn
dominates
J. Mar. Sci. Eng. 2020,8, 162 14 of 23
during the high tension part of the cycle. For
Tr=3.50 s Cdn
dominates almost the entire time, with
the exception of very short periods, right before the cable goes slack, when Cmdominates briefly.
2 2.5 3 3.5 4 4.5 5 5.5
Time (s)
0
10
20
30
40
50
60
70
80
90
100
TSI contribution (%)
0
10
20
30
40
50
60
70
80
Tension (N)
TSI CM
TSI Cdn
TSI Cdt
Tension
(a)Tr=1.25 s
3 4 5 6 7 8 9 10 11
Time (s)
0
10
20
30
40
50
60
70
80
90
100
TSI contribution (%)
0
10
20
30
40
50
60
Tension (N)
TSI CM
TSI Cdn
TSI Cdt
Tension
(b)Tr=3.50 s
Figure 8.
Approximate relative contribution of each coefficient to the variance of the tension, together
with the plot of the mean tension.
This analysis is in line with what was seen before. We have
Cdn
playing a major role when
the tension is higher, because the cable is moving faster. When the tension is lower,
Cm
and
Cdt
have a greater, or even the greatest, contribution to the results. In both cases,
Cdn
seems to be the
leading coefficient contributing to the value of the maximum tension. Since
Cm
and
Cdt
are of greatest
importance when the tension is close to
0 N
, it might be that their contribution is mostly to the stability
of the numerical model, rather than to the tension, in a physical sense. Another hypothesis is the
geometry of the catenary itself. In slack mooring cables, tangential motions happen mostly either in
the very upper parts of the cable near the fairlead or on the portions of the cable that are lying on
the sea-floor and dragging along it. So these results might also mean that when the tension is low,
the dynamics of the cable might be dominated by the portions of the cable suspended immediately
after the fairlead or those portions interacting with the sea-floor.
From what was presented above we can draw some important results, at least for the cases
analyzed. First, in the simulation of cable dynamics, it seems to be more important to select an
appropriate value of the normal drag coefficient,
Cdn
, than any other coefficient. Although it has
been found in [
44
,
45
] that the hydrodynamic coefficients control the dynamics of submerged cables,
the extent to which each of them does is not so well understood. This has now been made possible
using gPC, which allowed thousands of values to be tested quickly. The relative importance of the
tangential drag coefficient,
Cdt
, and of the added mass coefficient,
Cm
, depends on the period of the
excitation: for shorter periods,
Cdt
has more influence than
Cm
, while for longer periods, the opposite
happens. In other words,
Cdt
has more importance for fast motions, whereas
Cm
has more importance
for slow motions. This happens because the drag forces depend on the square of the speed of the cable,
so for slow motions the tangential drag force will be small and grow quickly as the speed increases.
4.3. Moored Cylinder
Here we analyze the influence that deviations in anchor placement have on the dynamics of a
moored cylindrical buoy. The case study is based on physical model experiments of a vertical cylinder
moored with spread mooring system composed of three catenaries, Figure 9(see [
46
] for a close
description of the physical experiments). This case has been investigated numerically by coupled
mooring analysis using CFD [25] and using linear potential theory [33].
J. Mar. Sci. Eng. 2020,8, 162 15 of 23
The properties of the buoy used here follow [
25
], with a slightly modified mass and inertia to
include styrofoam lid and metal support of the load cells. The properties of the model are presented
in Table 4for the buoy, Table 5for the chain, and Table 6for the soil and water. For the analysis,
we selected three regular wave tests with a constant wave height
H=0.04 m
and three different
periods:
T=1.00 s
,
T=1.20 s
, and
T=1.40 s
. The dynamics of the moored floating cylinder were
modeled in a coupling of MooDy and WEC-Sim. The cables were modeled in MooDy, discretised
Nel =
10 elements of order
p=
5, and a time-step such that the CFL condition was always smaller
than 0.9. The buoy motion was modeled in WEC-Sim, using a time-step of 0.01 s.























(a) Side view
Anchor 1 Anch
or 3
Anchor 2
(b) Top view
(c) Photograph
Figure 9. Experimental set-up reproduced in the moored cylindrical buoy case.
J. Mar. Sci. Eng. 2020,8, 162 16 of 23
Table 4.
Properties of the buoy. D—diameter; h—height;
Ixx
—inertia around the horizontal axis
through the center of gravity; Cg—center of gravity (distance from the top).
Mass D h Ixx Cg
35.85 kg 0.515 m 0.401 m 0.87 kg m20.3247 m
Table 5. Chain properties.
Parameter Value
D4.786 ×103m
ml0.1447 kg/m3
EA 1.6 MN
Cdn 2.5
Cdt 0.5
Cm3.8
Cable length 6.95 m
Table 6. Water and soil properties.
Parameter Value
K3×108Pa/m
vc0.01 m/s
µ0.3
ξ1
ρw1000 kg/m3
We modeled the uncertainty in the anchor position using the one-dimensional Gaussian
distribution for each of the horizontal anchor coordinates:
xAi
and
yAi
, corresponding to cable
i
.
The data relative to the probability distributions of the uncertainty is presented in Table 7. For the
mean of the distribution of each coordinate we selected the deterministic value of the position of
the anchor in the experimental set-up. For the standard deviation we chose the value of
0.025 m
,
based on reasonable estimates of the maximum error when installing the anchor in the physical model,
given the dimensions of the set-up and the difficulty in handling the anchors. To obtain the gPC
model of the moored buoy, we generated 150 samples of the positions of the anchors using the Latin
Hyper Cube method and ran the numerical model for each of these sampled positions. Afterwards,
we used the LAR algorithm to fit the coefficients to a gPC expansion of order
p=
5, using Hermite
polynomials and q-norm of 1. With the gPC model we evaluated
3000
random samples of anchor
positions to build confidence intervals for the tension in the cables and for the surge, heave and pitch
motions. These were then compared with the physical model measurements. The results are presented
in Figures 1012.
Table 7. Parameters of the probability distributions of the anchors’ positions.
Anchor Deterministic Distribution Mean Standard
Coordinate Value Value Deviation
xA1 3.4587 m Normal 3.4587 m 0.025 m
yA1 5.9907 m Normal 5.9907 m 0.025 m
xA2 3.4587 m Normal 3.4587 m 0.025 m
yA2 5.9907 m Normal 5.9907 m 0.025 m
xA3 6.9175 m Normal 6.9175 m 0.025 m
yA3 0.0000 m Normal 0.0000 m 0.025 m
J. Mar. Sci. Eng. 2020,8, 162 17 of 23
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-1
0
1
2
3
4
5
6
95% C.I. Deterministic Experimental
(a) Tension in cable 1
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
95% C.I. Deterministic Experimental
(b) Surge displacement
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-1
0
1
2
3
4
5
6
95% C.I. Deterministic Experimental
(c) Tension in cable 2
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
95% C.I. Deterministic Experimental
(d) Heave displacement
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-1
0
1
2
3
4
5
6
95% C.I. Deterministic Experimental
(e) Tension in cable 3
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-15
-10
-5
0
5
10
15
95% C.I. Deterministic Experimental
(f) Pitch displacement
Figure 10.
95% confidence intervals together with deterministic simulation results using the mean
value of the input random variables, for T=1.00 s, H=0.04 m.
J. Mar. Sci. Eng. 2020,8, 162 18 of 23
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-1
0
1
2
3
4
5
6
95% C.I. Deterministic Experimental
(a) Tension in cable 1
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
95% C.I. Deterministic Experimental
(b) Surge displacement
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-1
0
1
2
3
4
5
6
95% C.I. Deterministic Experimental
(c) Tension in cable 2
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
95% C.I. Deterministic Experimental
(d) Heave displacement
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-1
0
1
2
3
4
5
6
95% C.I. Deterministic Experimental
(e) Tension in cable 3
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-15
-10
-5
0
5
10
15
95% C.I. Deterministic Experimental
(f) Pitch displacement
Figure 11.
95% confidence intervals together with deterministic simulation results using the mean
value of the input random variables, for T=1.20 s, H=0.04 m.
J. Mar. Sci. Eng. 2020,8, 162 19 of 23
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-1
0
1
2
3
4
5
6
95% C.I. Deterministic Experimental
(a) Tension in cable 1
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
95% C.I. Deterministic Experimental
(b) Surge displacement
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-1
0
1
2
3
4
5
6
95% C.I. Deterministic Experimental
(c) Tension in cable 2
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
95% C.I. Deterministic Experimental
(d) Heave displacement
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-1
0
1
2
3
4
5
6
95% C.I. Deterministic Experimental
(e) Tension in cable 3
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29
Time (s)
-15
-10
-5
0
5
10
15
95% C.I. Deterministic Experimental
(f) Pitch displacement
Figure 12.
95% confidence intervals together with deterministic simulation results using the mean
value of the input random variables, for T=1.40 s, H=0.04 m.
J. Mar. Sci. Eng. 2020,8, 162 20 of 23
Before analyzing the results of the gPC model we note that there are differences between the
physical model results and the deterministic simulations, already analyzed in [
25
,
33
]. The main cause
for the difference between the mean surge displacement in the simulations and the one measured in
the experiments is the WEC-Sim simulations not accounting for second order drift forces. As the mean
surge displacement and the tension in the cables are linked, more accurate results in the simulation of
surge, such as for T=1.40 s, lead to more accurate results in tension.
The results of the simulations are presented in Figures 1012. The confidence intervals represent
fairly well the tension in the cables and the heave and pitch motions. In surge, there is a long period
variation in the amplitude of the confidence intervals, as is clearly seen in Figure 12b. This is caused
by transient motions of the buoy in the beginning of the simulations: although the anchors’ position
change in each simulation, the buoy’s initial position is kept constant (in the one corresponding to the
mean anchor position), causing it to start its motion slightly out of equilibrium.
Within the range applied in this work, the positions of the anchors have very little influence in
the heave and pitch behavior. The surge motion and the tension, on the contrary, show a significant
influence from the position of the anchors. With a large confidence interval, surge seems to be the
most susceptible quantity to the effect anchor placement, which is partially due to having the mooring
tension, which displays a notable uncertainty, as the only restoring force. These are important results
for floating structures with working principles depending on heave and pitch dynamics, as neither the
design nor the installation need to be too demanding when it comes to anchor placement accuracy.
However, the influence played by the anchor position in surge and in the tension shows the need to
account for installation errors during the design phase. This can be done using gPC, or another UQ
method, to determine bounds of acceptable deviations in anchor positioning, which might relax design
and deployment tolerances.
5. Conclusions
We applied generalized Polynomial Chaos (gPC) to build cost-effective surrogate models to
perform uncertainty quantification of mooring cable dynamics. Two cases were considered: (i) the
sensitivity of hydrodynamic coefficients on peak tension loads and (ii) the sensitivity of anchor position
on tension loads and the resulting floater motion.
The study on hydrodynamic coefficients was based on numerical simulations of a single mooring
cable subjected to forced oscillatory motion of its top end. The results showed that the normal drag
coefficient has the greatest influence in the simulation results, while the tangential drag and the mass
coefficients have almost negligible influence. As such, when simulating mooring cables, it is more
important to get an accurate value of the normal drag coefficient than of the tangential drag or of the
mass coefficients. However, a large uncertainty in the value of the coefficients (50%) causes only small
changes in the value of the peak tension, even when snap loads are present. The results also show that
when the tension is low, tangent motions along the ground might be relevant, but further investigation
is required.
The test case for the impact of uncertainty in anchor position used numerical simulations
to reproduce physical model experiments of a cylindrical buoy moored by three catenary chains.
It was shown that deviations in the anchor position, relative to the expected one, have very little
influence in heave or pitch dynamics. However, the surge motion and mooring cable tension can be
significantly affected.
Through the use of cost-effective surrogate models (here based on gPC) it is possible to perform
sensitivity analysis to better understand how some mooring system parameters are more relevant to the
design than others. This will help designers and researchers choose where to focus their efforts when
analyzing mooring systems, ultimately contributing to improved and more reliable mooring designs.
Author Contributions:
Conceptualization, G.M.P., C.E. and A.P.E.-K; methodology, G.M.P., C.E. and A.P.E.-K.;
simulation and analysis, G.M.P.; writing—original draft preparation, G.M.P.; writing—review and editing, C.E.
and A.P.E.-K. All authors have read and agreed to the published version of the manuscript.
J. Mar. Sci. Eng. 2020,8, 162 21 of 23
Funding:
This project received funding from the European Union’s Horizon 2020 research and innovation
programme under grant agreement No. 752031.
Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the
study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to
publish the results.
Abbreviations
The following abbreviations are used in this manuscript:
BEM Boundary Element Method
CFL Courant–Friedrichs–Lewy
DG Discontinuous Galerkin
gPC generalized Polynomial Chaos
LARS Least Angle Regression
MC Monte Carlo
MODU Mobile Offshore Drilling Unit
O&G Oil and Gas
PDF Probability Density Function
TSI Total Sensitivity Index
UQ Uncertainty Quantification
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c
2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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