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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 L∈R+,

in a physical domain

D∈R3

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)

∂s−1 , (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

2ρwD√1+e(Cdt |vr,t|vr,t+Cdn |vr,n|vr,n), (5b)

and the effect of weight and buoyancy computed using Equation (6)

fb=mlρc−ρw

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(rz−zg)−2ξpmlKD max(vz, 0), if rz≤zg,

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(t−τ)˙

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

Ω

, in the form of events denoted by

F

, with an associated probability

P

. This is denoted

by

Z:Ω→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,

d∈N

,

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:Ω→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:Ω→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)

∂s−1 , (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(t−τ)˙

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)

0 0.5 1 1.5 2 2.5 3

Horizontal position (m)

-1

-0.5

0

0.5

10-4

10-3

10-2

10-1

100

101

(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 ×10−2m

, and the variance is

1.26 ×10−1m2.

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

Vertical position (m)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

p = 1

p = 2

p = 3

p = 4

p = 5

p = 6

(a)

100101102103104105

log10(gPC Order) | log10(# MC samples)

10-6

10-5

10-4

10-3

10-2

10-1

log10(error)

Mean gPC

Mean MC

Variance gPC

Variance MC

(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 ×10−3m

ml8.18 ×10−2kg/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 ×10−3m

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 10–12.

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 10–12. 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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