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ABSTRACT: This paper presents the development of an open-source object-oriented program, named OpenMOOR, for static and
dynamic analyses of mooring systems in ocean renewable energy applications, including offshore wind turbines and wave energy
devices. The program is developed for cross-platform applications. It can be used as a standalone software or a dynamic linking
library for developing coupled models of the moored structures/devices. A finite difference model of mooring cables is adopted
for solving the motion of a single cable which can deal with the hydrodynamic effect, cable bending/torsional stiffness and
nonlinear strain-tension relationship. Parallel computing is implemented for efficient analysis of a mooring system consisting of
multiple cables, as common in the practice. OpenMOOR has then applied to analyse a single mooring cable under forced harmonic
motions at the top end. The cable responses are found to agree well with the experimental data in the literature, which validates
OpenMOOR.
KEY WORDS: Mooring systems; Open-source program; Nonlinear cable mechanics; coupled analysis; object-oriented
framework; ocean renewable energy.
1 INTRODUCTION*
Mooring systems are of significant importance for station
keeping of ocean renewable energy applications to harness
offshore wind, wave and tidal current energies. Two main
challenges exist in design and analysis of the mooring systems
(i) the balance of design cost/complexity and the performance
[1], particularly for wave energy devices, and (ii) the numerical
simulation for assessing the safety, survivability, and coupled
analysis of the floating structures [2]. Recent research has
focused on relevant topics. For example, a novel mooring
configuration which combines the tension legs and catenary
chains for floating wind turbines was studied and found to be
able to improve the dynamic performance of the tension leg
platform in operational and extreme conditions [3], via wind-
wave tunnel tests on a scaled model. Cost optimization of
mooring systems for large scale wave energy converters was
performed by [4] to facilitate the commercialization of wave
energy concepts. Based on basin testing, extreme mooring
loads were investigated for an array of wave energy devices [5]
and [6] which analysed the peak mooring load in relation to the
environmental conditions including wave and current and it is
found that increasing pre-tension of cables can reduce the peak
mooring loads which however may reduce the device motion
and hence the energy able to be harvested.
For investigating mooring cables through model tests,
methods have been proposed for developing scaled models, as
in [7]. A method using truncated mooring cables was proposed
by [8]. However, most of the studies resort to mathematical
modelling and numerical simulations considering the large
scale and complexity of the mooring systems. Nonlinear
dynamic analyses have been conducted and compared with
linear and quasi-static analyses for typical offshore wind
turbine concepts with the mooring system, to illustrate the
importance of mooring cable dynamics in coupled analysis of
offshore wind turbines [9-12]. The nonlinear dynamic
behaviour of mooring cables could be more pronounced in the
station keeping of wave energy devices [13], because the
relative-small size devices can experience relatively larger
displacements. While the mooring systems for offshore wind
turbines are more or less similar to those in oil and gas industry,
the mooring systems for wave energy devices are quite
different [14]. Although the quasi-static analytical solution is
still dominantly used in the iterative design procedure for wave
energy devices [15], the design is required to consider the
reliability and survivability at the final step where a fully
nonlinear model is needed. Nonlinear dynamic analysis is also
recommended by DNV-OS-E301 POSMOOR [16]. The
importance of mooring line dynamics has also been illustrated
by experimental measurements [13].
There exist a number of commercial programs providing the
capacity of simulating mooring cable dynamics and many can
also perform coupled analysis of the moored structures. A
recent review on mathematical models and analysis tools can
be found in [17], and available tools have also been examined
in [18]. Apart of commercial software, many in-house codes
have been developed for specific research focus and for the
convenience of coupled analysis to varied extent. For example,
DOOLINES provides a finite element framework for dynamics
of offshore lines, which is currently a proprietary software [19].
Analytical catenary solution has been extended to deal with
multi-segment cables, especially for crowfoot mooring, and to
account for the cable-seabed friction [20], leading to a
comprehensive quasi-static mooring analysis program named
MAP++ [21]. A finite element code called FEAMooring has
been developed particularly for coupled offshore wind turbine
analysis by [22,23]. A lumped mass model, named MoorDyn,
has been developed and validated against experimental tests on
a floating wind turbine model [24]. Another lumped mass
Development of an open-source simulation tool for mooring systems
Lin Chen, Biswajit Basu
1School of Engineering, Trinity College Dublin, Dublin 2, Ireland
email: l.chen.tj@gmail.com, basub@tcd.ie
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model, OPASS code, is based on finite element formulation
while with linear interpolation of the element mass [25] and it
has been validated using experimental data on a single cable. A
hp-adaptive discontinuous Galerkin method was developed for
modelling snap loads in mooring cables in particular for wave
energy devices which are probably subjected to large
displacements due to the extreme wave loadings [26,27] and
the program is named MooDy. In addition, a nonlinear finite
element model based on bar elements with three translational
degrees of freedom per node is used for coupled analysis in
[28].
Among all the in-house codes, to the best knowledge of the
authors, three are made open-source, namely MAP++,
FEMooring, and MoorDyn. MAP++ is a well-documented
program while due to its quasi-static nature, it is receiving less
attention for further development. FEMooring was developed
in Fortran especially used as a module of FAST, a widely used
open-source wind turbine analysis tool [29]. MoorDyn has also
been coupled with FAST and is still under intensive
development. Recently, it has been improved to consider the
seabed friction and applied to connect multiple floaters [30].
Another option is building mooring models based on available
open-source general-purpose finite element libraries. For
example, one effort has been made by [31] using software
package Code_Aster. Other popular libraries such as DEAL.II
[32] and FEniCS [33] can also be explored. Nevertheless,
considerable effort is still needed to tailor those general-
purpose library to simulate mooring cables.
This study is aimed at the development of a cross-platform
open-source program for mooring cable simulation with
improved capacity, named OpenMOOR. The mathematical
model developed by [34,35] is used as the basis, which was also
used by WHOI Cable [34,35]. WHOI cable was developed for
towed cables and risers in ship and oil industry and it is a
proprietary software of Woods Hole Oceanographic Institution.
The mathematical model has some advantages as compared to
those implemented in available open-source programs,
including the capacity to include the current effect, cable
bending stiffness and torsional stiffness, and also the nonlinear
strain-stress relationship. Particularly, the nonlinear strain-
stress relationship could be important for optimize the design
of mooring cables for wave energy devices. For example, a
novel fibre rope mooring tether with lower axial stiffness was
studied by [36] to reduce the peak and fatigue loads in the rope.
The so-called "Exeter Tether" also provides capacity of
selectable axial stiffness and the experimental results clearly
showed a nonlinear strain-stress relationship.
The rest of the paper is organized as follows. Section 2
introduces the OpenMOOR implementation. Section 3 presents
validation and verification results. OpenMOOR is further
applied to study the wave and current effects on a floating
offshore wind turbine in Section 4. A brief summary is
provided in Section 5.
2 OPENMOOR*FRAMEWORK*
Station-keeping,problem,in,offshore,renewable,energy,
A typical mooring system is composed of multiple cables, each
of which exhibits strongly nonlinear behaviour and requires
careful treatment in numerical simulation. Fig. 1 shows one
cable of a floating structure. The mooring cables provide
restoring stiffness for the structure preventing it from drifting
far away from the desired spot. In operational/extreme
conditions, the cables are subjected to loads due to surface
waves and underlying current. OpenMOOR is aimed to
simulate such multi-cable mooring systems in ocean renewable
energy applications and also to be a framework to conveniently
implement other mooring models.
Figure 1. Station keeping using mooring cables in renewable
energy applications.
Mooring,cable,mechanics,
Several mathematical models for mooring cable mechanics
along with the numerical solving methods are available in the
literature. The most widely used model formulates the
governing equations of a cable without bending stiffness in
Cartesian coordinate, leading to second-order partial
differential equations, which have been often solved using
finite element methods [37,38]. A more comprehensive model
based on rod theory was derived by[39], able to take into
account both bending and torsional effects and it is also solved
using finite element method.
Another popular model establishes the balance equations in
the local Lagrangian coordinate along the cable, leading to
first-order partial differential equations [40-43]. It can consider
the bending stiffness, torsional stiffness and the nonlinear
strain-stress relationship easily. The box method is normally
used for solving the equations, which is able to achieve a
second-order accuracy. By using this model, the governing
equations are expressed in matrix form as
! " #"
#$ % &'"( #"
#) % * " + , (1)
where ! " and & " are stiffness and mass matrices
depending on cable state and " is the state vector. For three-
dimensional problems as described in [41] when the Euler
Angle formulation is applied the state vector is defined as " +
- ./.01 2 3 4 5 67689:with -= strain, ./
and .0 are shear forces, 1,92 and 3 are cable velocity
components in the local coordinate, 4 and 5 are Euler angles
and 67and 68 are material curvatures. The interested readers
are advised to [40] for details and the quaternion formulation.
Noteworthy is that in all the formulations Morison's equation
is used to account for hydrodynamic effect on cables [44],
including the drag force, inertia force and the Froude-Krylov
force, which is valid considering the slenderness of the mooring
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cables. The model described by Eq. (1) is implemented in
OpenMOOR. The equations are first discretized in space and
then the generalized-; method is implemented for time
stepping to add numerical dissipation for improving the
computation stability [45-47].
Implementation,
OpenMOOR is developed in C++ using object-oriented
programming. The Eigen Library is used for matrix and vector
manipulation [48], which is a C++ template library including
only header files and hence simple to use. The Eigen Library
provides a Matlab-like development environment, which
enables efficient implementation of novel models by structural
engineers. The odeint library [49] is used for integration in
solving the platform motion, which contains also only header
files. The source code of OpenMOOR is hosted by GitHub
along with documentation and examples
(https://github.com/chen-lin/OpenMOOR). The key classes of
OpenMOOR and their relationship are demonstrated in Fig. 2.
Figure 2. OpenMOOR main classes and their relationships.
The main classes of OpenMOOR are described as follows.
• Class Node. A Node is defined by its arc-length coordinate
originated from one end of the cable which it belongs to,
corresponding structural and hydrodynamic properties and
associated seabed parameters if it is in contact with the
seabed. Those properties are defined using StructProperty,
HydroProperty and SeabedProperty classes respectively.
This arrangement allows the consideration of nonuniform
properties along the cable length. Methods are defined to
formulate the nodal mass matrix and stiffness matrix and
the nodal force vector based on the present nodal state. For
boundary nodes, methods are provided to define the
constraint equations.
• Class Catenary: An extensible Catenary in two-
dimensional space is defined by its chord vector and its
unstretched length, weight per unit length and axial
stiffness. It is used to initialize the nonlinear solving
procedure. Analytical catenary solution is important for
reducing the time taken for dissipating out the transient
responses and also important for developing linearized
model for the mooring system. The linear stiffness matrix
is required to approximate the restoring stiffness of a
mooring system on the platform [50], which can further be
used to carry out a shooting procedure for solving the
coupled static problems. The two-dimensional elastic
catenary theory considering the seabed contact is
implemented [50,51]. In case of cables with non-uniform
structural property, the averaged cable weight per unit
length and axial stiffness are used for obtaining the
analytical catenary solutions.
• Class Cable: A Cable in OpenMOOR is defined by its
unstretched length and the initial positions of its two
connections in global coordinate system. A Cartesian
coordinate system is defined for each cable with the origin
placed at one end. Methods are defined for discretizing a
cable into uniformly distributed or non-uniformly
distributed nodes and initializing the nodal states using
catenary solutions. Initial cable nodal states can also be
read from input files. Other main methods include the
formulation of the discretized equations from relevant
nodes and solving the equations using a specified Solver
for updating cable state.
• Class Platform: A Platform is a rigid body of six-degree-
of-freedom for simulating the moored floating
structure/device. Its motion is defined by one reference
point. Each platform can have multiple fairleads each of
which is connected to a cable. At each time instant, the
platform displacement and velocity are given at the
reference point. The fairlead motions are calculated
accordingly and then used as instantaneous boundary
conditions in cable analysis. After the cable analysis, the
total mooring load is assembled from the cable fairlead
tension based on the instantaneous relative fairlead
position with respect to the reference point. In addition,
functions are provided to set up mass matrix and the
restoring stiffness due to hydrostatic effect for the purpose
of dynamic relaxation analysis and static analysis using
shooting method. Time integration is implemented to solve
the six-degree-of-freedom motion of the platform.
Noteworthy is that in the simulation once the fairlead states
are updated, the static/dynamic analysis of the cable is
independent of each other. Parallel computing is well-
suited and important since the cable analysis requires the
most computational effort, which is implemented using
OpenMP (http://www.openmp.org) in C++.
• Class Solver: A Solver is solves Eq. (1) after finite
difference discretization which is then expressed as
augmented matrix. A Newton-type iteration with
relaxation is used and the method for adjusting the
relaxation factor according to the error evolution is also
provided. In addition, parameters for using generalized-;
method to formulate the algebraic equations for each cable
from the nodes are also stored as members of the cable
solver. The development of the solver class is referred to
the numerical recipes [52] for two-boundary-point
problems. In OpenMOOR, each cable is assigned with one
solver and different solver can be used for the cables.
• Class Setting: the Setting class specifies the path to read
input files and output results. It also defines the analysis
type and prepares corresponding parameters.
• Class Simulation: A Simulation performs a particular
analysis according to the Setting. Currently, OpenMOOR
supports three types of analysis if used as a standalone
program, i.e. static analysis using shooting method,
dynamic relaxation [52] and dynamic analysis if the
motion of the platform is specified. In case of using
OpenMOOR as a dynamic linking library, only dynamic
analysis is supported, while the dynamic relaxation can be
easily implemented in coupled analysis.
Solver
Cable Platform
Node
Catenary
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Organization,of,input,and,output,
The input files include the setting file and the main input file
providing cable and platform geometry and structural and
hydrodynamic properties and so on. They are supposed to be
provided as XML files and handled using the rapidXML
(http://rapidxml.sourceforge.net) for extracting the parameters.
Other possible inputs including the initial cable state and the
current profile data can be supplied as simple text files. The
cable nodal state and platform state can be output upon
requirement as text files. They can be easily read into Matlab
or Python for post-processing. The Reader and Writer classes
are available to deal with file reading and writing in
OpenMOOR.
Solving,procedure,
After creating the platform and cables and finishing the
initialization, for a new platform state (updated displacements
and velocities), OpenMOOR solves the mooring load following
the steps described below:
• Update the fairlead displacements and velocities;
• Perform cable analysis in parallel and update the cable
tension at the fairlead;
• Transform cable tension into global coordinate system and
assemble the mooring load.
3 APPLICATION*AND*VAL IDATIO N*
In this section, OpenMOOR is validated using experimental
data from a scaled mooring cable model. The experiment was
carried out by [54] while the digitalized measurement data
provided by [7] is used. The experiment is briefly described
here for the sake of completeness. The experiment setup is
illustrated in Fig. 3. The tests were conducted in a manoeuvring
basin on a chain cable with an unstretched length of 33 m. The
cable was anchored to the basin bottom at one end and the other
cable end was attached, via a ball bearing, to a sheave on the
axis of the motor at a distance from its centre.
Figure 3. The experiment setup (adapted from [7]).
Table 1. Two of the test cases [7].
Case No.
Period (s)
Radius (m)
1
3.5
0.2
2
1.25
0.2
Several test cases have been carried out with the upper cable
end attached at different distances from the motor axis and at
varying motor rotating speeds. The water depth was 3.0 m. The
force at the upper cable end was measured by a force probe, and
the force data for two of the test cases is made publicly
accessible by [7]. The corresponding forced motions of the
cable upper end in those two cases are listed in Table 1.
In using OpenMOOR to simulate the cable motions, the input
model data derived in [7] is used, as listed in Table 2. The
present version of OpenMOOR ignores the friction effect of the
bottom and hence the related parameters are not listed. The
damping effect of seabed is marginal and is also ignored here.
Noting that the tangential drag coefficient in [7] was applied to
the cable diameter while in OpenMOOR it is applied to the
circumference and hence a value of < =>?@ is used instead.
Besides, a small bending stiffness, i.e. 1 NAm2 is considered to
avoid the ill-posed problem of a perfectly flexible cable in the
absence of positive tension [55]. This small magnitude of the
bending stiffness, however, has limited effect on the solution.
In addition, the seabed stiffness in the calculation is assumed to
be 3E5 Pa which is much smaller than the value presented in
the table. However, parametric studies have been conducted,
showing that this affect the upper end cable tension marginally
while a large stiffness may cause numerical instability and also
increase the computation time.
Table 2. Input data of the model for numerical simulation [7].
Parameter
Unit
Value
Environment
Water density
kg/m3
1000
Water depth
m
3
Bottom stiffness
Pa
3E9
Cable
Unstretched length
m
33
Horizontal span
m
32.554
Vertical span
m
3.3
Density
kg/m3
7800
Axial stiffness
N
10000
Mass per unit length
kg/m
0.0818
Wet weight per unit length
N/m
0.699
Steel diameter
m
0.0022
Normal drag coefficient
-
2.5
Tangential drag coefficient
-
0.5
Added mass coefficient
-
3.8
In the simulation, to deal with these challenging cases when
the cable tension could become zeros, a total of 200 nodes are
used for discretion. A time step of 0.002 second is used and the
generalized-; method is applied. In each case, the simulation
is conducted for 25 seconds. It is also assured that when the
cable nodes are above the still water level the hydrodynamic
effects are automatically excluded. The simulation begins with
the cable upper end at the lowest point of its motion trajectory,
as shown in Fig. 3. For the comparison with experimental
measurements, the transient responses are truncated. The
simulated results are plotted in Fig. 4 and Fig. 5 for the two
cases, respectively, along with the corresponding experimental
measurements of the tension at the upper cable end. Animations
of the full cable motion during numerical simulation can be
found in the GitHub repository.
Overall, the simulated cable tension is consistent with the
experimental measurement in each case. This validates the
capacity of OpenMOOR for modelling the mooring cables in
these challenging situations. In general, it also achieves a
comparable accuracy as MooDy which was used in [7]. The
inclusion of bending stiffness in OpenMOOR makes the
simulated responses smoother, while Moody seems to be able
to capture the high frequency oscillations quite well. The same
experimental data have also been used by [28] for validating
0.0 m
-3.0 m
32.554 m
3.3 m
radius
rotating sheave
chain cable of 33 m
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the finite element model developed thereof. By comparing to
the results presented in [28], it seems that OpenMOOR
achieves a better accuracy as compared to the cable model in
that study.
Figure 4. Comparison of the tension at the cable upper end in
case 1.
Figure 5. Comparison of the tension at the cable upper end in
case 2.
4 CONCLUSION*AND*FUTURE*STUDY*
This paper introduces the development, validation and
applications of OpenMOOR, an object-oriented framework for
nonlinear mechanical analysis of mooring systems in offshore
renewable energy applications. The program attempts to be
useful for carrying out nonlinear static and dynamic analysis of
mooring systems for offshore wind turbines and wave energy
devices in the concept design and also for cross verification in
developing novel models of mooring cables. The program can
also be a framework for the implementation of other
mathematical models and numerical schemes for mooring cable
simulation.
Currently, OpenMOOR is able to handle cables with
nonuniform property along the cable but ignores the seabed
friction and wave loading on the cable. It will be further
developed to include these effects and implement novel solving
schemes like harmonic balance method for periodic responses
analysis [56-59]. It will also be used for studying coupled
dynamics of typical floating wind turbine and wave energy
device concepts for the purpose of design optimization and
control [60-66].
ACKNOWLEDGM ENTS*
This work has received funding from the European Union’s
Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie EID project ICONN Grant
Agreement No. 675659 and the Irish Research Council (IRC)
via the Government of Ireland Postdoctoral Fellowship (Project
ID: GOIPD/2017/1260).
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CERI-ITRN2018
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