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Contract No. DE-AC36-08GO28308

Extending the Capabilities of the 

Mooring Analysis Program: A 

Survey of Dynamic Mooring 

Line Theories for Integration 

into FAST

Preprint

M. Masciola, J. Jonkman, and A. Robertson

National Renewable Energy Laboratory

To be presented at the 33rd International Conference on Ocean, 

Offshore and Arctic Engineering

San Francisco, California

June 8 – 13, 2014

Conference Paper

NREL/CP-5000-61159

March 2014
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EXTENDING THE CAPABILITIES OF THE MOORING ANALYSIS PROGRAM: A 

SURVEY OF DYNAMIC MOORING LINE THEORIES FOR INTEGRATION INTO FAST 

Marco Masciola 

National Renewable Energy Laboratory 

Golden, Colorado USA 

Jason Jonkman 

National Renewable Energy Laboratory 

Golden, Colorado USA 

Amy Robertson 

National Renewable Energy Laboratory 

Golden, Colorado USA 

ABSTRACT 

Techniques to model dynamic mooring lines take various forms. 

The most widely used models include a heuristic representation 

of the physics (such as a lumped-mass system), a finite-element 

analysis discretization of the lines (discretized in space), or a 

finite-difference model (which is discretized in both space and 

time). In this paper, the authors explore the features of the 

various models, weigh the advantages of each, and propose a 

plan for implementing one dynamic mooring line model into the 

open-source Mooring Analysis Program (MAP). MAP is 

currently used as a module for the FAST offshore wind turbine 

computer-aided engineering (CAE) tool to model mooring 

systems quasi-statically, although dynamic mooring capabilities 

are desired. Based on the exploration in this paper, the lumpedmass representation is selected for implementation in MAP 

based on its simplicity, low computational cost, and ability to 

provide physics similar to those captured by higher-order 

models. 

To begin, the underlying theories defining the three classes of 

dynamic mooring line models are identified and explored. This 

leads to insight into the capabilities of each representation. 

These capabilities are weighed against the current needs of the 

FAST wind turbine CAE tool, to which MAP will be coupled. 

Based on the assessment, a plan for integrating the dynamic 

mooring line theory into the current MAP structure is 

developed. Common problems arising from the determination 

of the model static equilibrium and known issues with 

numerical stability are addressed. Because MAP is a module 

that FAST can call, a plan consistent with the FAST 

modularization framework principles is described. Adding 

dynamic mooring line capabilities extends the features in MAP 

and also allows uncoupled analysis to be performed through 

MAP’s native Python bindings. 

NOMENCLATURE 

 cubic spline coefficients 

 damping force vector N 

 elasticity matrix N 

 cable axial stiffness N 

 cable bending stiffness Nm2

 time and space dependent force N 

nth longitudinal natural frequency Hz 

nth transverse natural frequency Hz 

hydrodynamic force vector [hx hy hz] N 

 curvature parameter 

unstretched cable length m 

mass matrix kg 

external applied moment Nm 

 internal element bending moment Nm 

 internal bending moment vector Nm 

3 × 3 rotation matrix 

 global position vector m 

 position vector on a line at (s,t) m 

unstretched distance along the line m 

tension in the ith element N 

tangential pretension at fairlead N 

 element tension force vector N/m 

 time s 

 inputs 

 axial cable displacement m 

gravitational force vector N 

 translational cable displacement m 

 continuous state equations 
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 continuous states 

 global frame coordinates m 

 outputs 

 constraint equations 

 constraint states 

 relaxation parameter 

 artificial force N 

 fluid and cable density, respectively kg/m3

 cable mass per length kg/m 

 Lagrange multiplier 

 interpolation (shape) function 

1 INTRODUCTION 

As a publicly disseminated, open-source program to model 

mooring systems, the Mooring Analysis Program (MAP) tool is 

designed for use in parallel with the FAST offshore wind 

turbine computer-aided engineering (CAE) tool developed by 

the National Renewable Energy Laboratory (NREL) [1]. The 

FAST tool is developed to model, simulate, and analyze a wide 

range of land-based, fixed-bottom offshore, and floating 

offshore wind turbines. Among its features, FAST is a coupled 

aero-hydro-servo-elastic tool that models the aerodynamic 

loads, turbine control system, tower and blade flexibility, and 

hydrodynamic loads from irregular waves and currents in 

offshore wind turbines. Historically, FAST has had the 

capability to model mooring systems with nonintersecting, 

homogenous cables using a quasi-static approximation, but in 

the new FAST modularization framework [2], MAP is now used 

as a replacement for the old model. MAP improves on the 

traditional quasi-static functionalities by adding the ability to 

model multisegmented mooring systems; however, a dynamic 

mooring line model is needed. This paper discusses the 

introduction of a dynamic mooring line model in MAP 

configured to the requirements of the FAST modularization 

framework. 

MAP currently can model multisegmented, quasi-static (MSQS) 

mooring systems [3]. The existing theory in MAP is based on 

conventional closed-form analytical cable models solved 

simultaneously with force-balance equations to ensure static 

equilibrium of the interconnected system. The theory is similar 

to that outlined in Peyrot and Goulois [4], but the solution 

procedure is fundamentally different because the problem is 

solved with a monolithic perspective. MAP is designed to be an 

independent library accessible from third-party software written 

in Python, Fortran, C, or C++. It has binaries released for 

Windows, OSX, and Linux platforms, and is constructed with 

FAST framework definitions in mind. This last feature offers 

the flexibility for MAP to be used as a design tool or a 

simulation program. Although MAP’s MSQS model has 

widespread utility for rapid prototyping, design, and modeling 

of mooring systems, it has well-known limitations because it 

lacks inertial effects, bending, torsion, shear stiffness, and 

hydrodynamic loads. The inability to represent contoured 

bathymetries with quasi-static models is another well-known 

limitation [5,6]. A dynamic mooring line model can resolve 

these limitations by capturing the true system physics more 

accurately and eliminating the geometric limitations that plague 

closed-form analytical models. In this context, “geometric 

limitations” refer to “S” shaped cable profiles that cannot be 

achieved by closed-form continuous analytical models [7] and 

mooring lines resting on an inclined seabed [8]. 

1.1 DYNAMIC MOORING LINE CAPABILITIES 

Dynamic mooring line models are distinct from quasi-static 

models because they can account for inertia effects, internal 

damping, and drag loading. In particular, dynamic mooring 

lines propagate longitudinal u(s,t) and transverse w(s,t) 

vibrations as shown in FIGURE 1 and Fig. 2. These vibrations 

are ignored in quasi-static models, although there are more 

advanced continuous analytical models that attempt to capture a 

limited vibration mode range [9,10]. As Fig. 1 suggests, the 

transverse cable vibrations impart a distinct set of internal cable 

loads that are not captured in quasi-static analysis. These 

frequencies are estimated as [11] 

, (1) 

where n is an integer representing the nth mode shape. Equally 

as important are the longitudinal vibrations propagating axially 

along the mooring line with frequency 

. (2) 

Equations (1) and (2) are regarded as estimates because the 

boundary conditions can be different from the fixed-fixed 

condition used in their derivation. 

The strength of the platform-cable coupling is dependent on 

many parameters [12], but the platform-mass to cable-mass 

ratio is one common proxy for identifying the strength of this 

coupling [5]. The severity of the sea state also implies the 

importance of using a dynamic mooring model in the simulation 

as demonstrated in the DeepCwind semisubmersible tank test 

campaign [12,13]. Because of the limitations with quasi-static 

analysis, a dynamic mooring formulation is viewed as a 

rigorous approach for measuring ultimate and fatigue loads in 

floating offshore wind turbine moorings. This is the primary 

motivation for adding this capability in FAST through the MAP 

module. Dynamic mooring models also feature the ability to 

easily model cable/seabed contact with irregularly contoured 

seafloors [14] and impact with the seafloor, as well as the 

capacity to capture fluid drag loading and vortex-induced 

vibrations [15]. 
3 

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FIGURE 1: MOTION OF A SINGLE TETHER UNDERGOING 

TRANSVERSE w(s,t) OSCILLATIONS IN A TENSION-LEG 

PLATFORM (TLP). FIGURE IS ADAPTED FROM [12]. 

1.2 SYNOPSIS OF THIS WORK 

This paper serves two purposes. The first is to investigate the 

various mooring dynamic models used in past offshore system 

modeling efforts. The second is to select one dynamic mooring 

model, and reshape its formulation to meet the requirements of 

the FAST modularized framework for integration into the FAST 

CAE tool. Although there are a number of dynamic mooring 

line representations ranging from finite-element analysis (FEA), 

finite-difference (FD), and lumped-mass (LM), most models 

achieve similar results as long as a sufficiently fine 

discretization is used. Simplifications can be introduced into the 

model to reduce computational expense, and a common 

omission is bending, torsion, and shear stiffness. Cable bending 

effects are important in low-tension towing maneuvers [16] and 

for fatigue analysis in taut systems, such as at the tendon roots 

in TLPs [17], but bending also augments the stability of the 

numerical model [18,19]. 

FIGURE 2: TRANSVERSE (TOP) AND LONGITUDINAL 

(BOTTOM) VIBRATIONS IN A CABLE ARE A FUNCTION OF 

BOTH TIME (t) AND DISTANCE ALONG THE LINE (s). 

FIGURE IS ADAPTED FROM [12]. 

This paper concentrates on the differences in formulation 

among the dynamic model families, limitations of the solution 

techniques, and expected computational efforts. Classifying the 

model in this fashion helps justify the use of a particular 

dynamic mooring line formulation based on what has been done 

in the past, the current capabilities of FAST, and the desired 

features. 

2 MOORING LINE THEORIES 

Mooring line dynamic theories are categorized into two main 

groups: FEA models and FD models. A third subcategory, the 

LM model, can be derived from the FEA process, so it is 

regarded as a simplification of a higher-order model. It is 

valuable, however, to describe the distinct features of all three 

models to understand the capabilities and cost each bring to a 

dynamics simulation. 

The earliest discretized cable models emerged in the late 1950s 

[20] as heuristic representations of the physical system. At this 

time, FEA was still being pioneered [21]. It was not in 

widespread use and not a candidate for cable system modeling. 

The LM representation matured over the next two decades [22], 

which naturally led to FEA concepts being applied to the cable 

dynamics problem [23]. As Ketchman and Lou [23] 

demonstrated, however, the LM methodology converges onto 

the same results as FEA representations as long as the 

discretization size is sufficiently small. Discretized cable 

systems comprise a system of nodes and elements. Take, for 

example, the structure illustrated in Fig. 3. This cable array is 

modeled as a system of N+1 nodes and N visco-elastic 

elements. The position of the ith node is defined in the global 

[x,y,z] coordinate frame using vector ri. Definitions of the 
4 

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element stiffness term t, damping term b, hydrodynamic force 

h, and weight term w depend on the formulation used, which 

will be introduced later when the LM model is discussed. 

When bending and torsion stiffness is included, the derivation 

becomes burdened, and a discretization of the continuous 

formulation must be adopted to obtain the equation of motion. 

The continuous cable with bending and torsion loads included, 

but with shear neglected, can be shown to be equivalent to 

[24,25] 

 (3) 

, (4) 

where () is a time derivative, (’) is a spatial derivative, and 

represents the per-unit-length equivalent to ( ); hence, the 

governing formulation is in the form of a partial differential 

equation (PDE). The spatial derivatives are needed to resolve 

the slope (curvature) of the line element between adjacent nodes 

to calculate bending moments. Equation (3) is a summation of 

forces along the line element, and Equation (4) states that the 

internal moment n must balance the applied external moment m

and the internal moment arm that results from strain and 

damping. Effects from bending are included in Equation (3) by 

rewriting Equation (4) in terms of (t’+b’) and substituting it 

into Equation (3). There are two methods to resolve Equations 

(3) and (4). One method applies Galerkin’s method to eliminate 

the spatial derivatives and reduce the PDE to an ordinary 

differential equation (ODE); this leads to FEA and higher-order 

LM formulations [26]. An FEA discretization of M leads to a 

diagonally dominant matrix populated with off-diagonal terms. 

If the mass matrix is reduced to a diagonal matrix, the 

formulation adopted is considered to be a LM matrix, which 

yields the LM formulation in [35]; this is regarded as a “highorder” LM model. Some LM models go even further and 

discretize the stiffness matrix and force terms without 

interpolation functions [27,34,44]; this is regarded as a “loworder” LM model. 

The second discretization method involves a Taylor series 

expansion of the differential terms to estimate derivatives about 

an operating point; this leads to the FD approach. From a 

simple heuristic cable model, the FEA and FD approaches take 

form. Equations (3) and (4) spawn the growth of most 

discretized cable models that include bending effects. 

2.1 LUMPED-MASS MODEL 

There are several variations of the LM model 

[8,16,18,27,28,29,30], each echoing a different formulation 

based on specific needs and derivation processes. A model is 

defined as an LM system if the mass matrix is strictly diagonal 

[26]. As was done in other works [31], the force terms can be 

discretized based on Galerkin’s expansion in an LM model, but 

the fundamental interest of preserving a diagonal mass matrix 

must be maintained. The benefit of an LM model is that the 

mass matrix does not need to be inverted, using either iterative 

(i.e., Krylov sequence) [32] or direct (i.e., LU factorization) 

methods, which can be cumbersome as the matrix size grows. 

This eliminates added computation to give results quickly. 

Lumped-mass approximations work well for cable systems 

because the chain of elements can easily (naturally) be arranged 

to lie in series (see Fig. 3). In comparison, consistent mass 

formulations achieve matrices with an upper and lower 

bandwidth of two [31,33]. Consequently, the LM formulation is 

an acceptable simplification for a single-line system with 

elements in series. The mass matrix bandwidth will increase, 

however, when three or more elements connect to a single node, 

such as nets and the web systems considered in [4]. In these 

scenarios, the LM approach could lose accuracy and the LM 

model might miss coupling terms. 

The LM representation traditionally has been derived 

heuristically using Newton-Euler methods, but it can also be 

obtained based on Lagrange and Hamiltonian mechanics [34]. 

Elements contain the spring and damper properties; nodes 

contain the system mass properties. Without bending or 

torsional effects, the equation of motion for the LM model is 

obtained based on a summation of forces at the nodes [18,28]: 

, (5) 

FIGURE 3: DIAGRAM OF THE LM MODEL. ELEMENTS 

CONTAIN THE SPRING AND DAMPING PROPERTIES OF 

THE SYSTEM; NODES CONTAIN THE MASS PROPERTIES. 
5 

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which is a second-order differential equation describing the 

motion of the ith node in the cable kinematic chain shown in 

Fig. 3. Note that Equation (5) was derived purely on the basis of 

Newton’s equation. No concepts from FEA were brought in, and 

the equation was obtained without the aid of Equations (3) and 

(4). The internal element forces bi and ti depend not only on the 

position of the ith node, but also on the motion relative to 

adjacent nodes i+1 and i-1. The gravitational force vector wi is 

the net weight of the node in fluid. The hydrodynamic force 

vector hi represents the viscous drag loading and other 

exogenous external loads applied at the node. Hydrodynamic 

viscous loads and inertial loads from wave kinematics (i.e., the 

Froude-Krylov force) are largely neglected in hi based on the 

assumption that line elements are far from the sea surface, but 

added-mass effects related to the acceleration of the 

surrounding fluid resulting from the line element motion are 

included in Mi in Equation (5). This is important for capturing 

the correct translational oscillation frequency akin to Equation 

(1), because the cable will have a larger apparent mass in 

seawater compared to air. 

Equation (5) is regarded as a low-order LM formulation 

because bending effects are omitted. Equations (3) and (4) were 

later used to derive a high-order LM formulation in [35] by 

introducing bending stiffness, which takes the form of 

. (6) 

The LM formulation in [35] adopts a similar interpretation to 

Equation (5), except an additional internal force term is 

included to account for bending effects. As a result, bending 

resistance in this specific instance of the LM model can be 

disabled without changing the solution approach of the original 

formulation. One notable feature of this interpretation is that 

only three equations need to be integrated to include axial 

forces and bending effects, although torsion is neglected. 

2.2 FINITE-ELEMENT MODEL 

FEA cable models have matured over the past four decades 

[25,36,37,38] to include several advanced features, including 

cable pay-out [39], hybrid models combining FEA with 

analytical solutions [14], and advanced numerical integration 

strategies to augment stability [40]. Notably, FEA models are 

capable of yielding higher quality results with lower 

discretization resolution compared to LM representations 

[20,38]. In theory, Equation (5) can be interpreted to represent 

the consistent formulation of an FEA cable model, but an 

alternative formulation will be examined for the purpose of 

appreciating the diversity of techniques used in practice. A 

specific modern FEA implementation with bending stiffness is 

given in Garrett [25,41] and Ran [33]. The derivation process 

initiated in [25,33,41] is similar to the comprehensive, highorder LM approach in [35], and is based on a discretization of 

Equations (3) and (4). The LM and FEA models, however, 

differ on several important points: 

 Mass matrix discretization 

 Boundary condition discretization 

 Discretization of the external and internal forces. 

The following integral equation describes the equation of 

motion for a generic system expressed in an FEA formulation: 

, (7) 

which is shown to be analogous to Equation (6) in [25,33,41] 

with internal cable damping omitted. The important concept to 

keep in mind is that an interpolation function, Φi, is needed to 

discretize the force, mass matrix, and stiffness matrix. For cable 

systems including bending, additional algebraic equations are 

needed to resolve the radius of curvature (slope) along the line: 

. (8) 

The combination of Equations (6)/(7) and (8) is known 

collectively as a differential-algebraic equation (DAE) when 

solved together [42]. The constraint in Equation (8) originates 

from solving the radius of curvature, where λ is a Lagrange 

multiplier, which is analogous to coefficients used in the cubic 

spline functions for the LM representation in [35]. The 

interpolation functions, Φi, are a function of λi. In practice, 

Equation (8) is usually solved explicitly (based on previous 

time-step information), thus reducing the DAE formulation into 

an ODE. Although the method in [25,33,41] is based on 

Galerkin’s method of weighted residuals, a different basis 

function can be used to obtain the equation of motion for the 

discretized cable system [43]. In [25,33,35,41], Φi is based on 

piece-wise cubic polynomials passing through the nodes to 

represent the suspended cable as a twisted spline. Unlike the 

LM formulations, the FEA cable representation uses the 

interpolation function Φi to assemble the consistent mass matrix 

M. Regardless of the discretization procedure, the FEA model 

and variations of the LM model can be expressed as Equation 

(6). 

2.3 FINITE-DIFFERENCE MODEL 

The FD approach is based on a Taylor series expansion of the 

governing PDEs to reduce the PDEs into ODEs and algebraic 

equation relations [45]. The FD approach differs from FEA by 

replacing piece-wise gradients with first-order difference 

functions. In other words, FEA computes derivatives of the 

basis function exactly; FD estimates the gradients. Equation (3) 

can be discretized in both space and time, with the FD 

formulation ostensibly defined as [46‒48] 
6 

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, (9)

where i is discretization with respect to time, j is discretization 

of the spatial derivatives, and f = (h+w). Equation (9) is not 

universally applicable for all FD implementations, and the 

discretization profile depends on the finite-differencing pattern 

(stencil) used. Formulations diverging from the abridged 

representation in Equation (9), though, are common 

[45,46,48‒50]. Equation (9) is defined for the box stencil, 

which is related to center finite differencing. Note that timederivative variables are differenced with respect to Δt and 

spatial derivatives are differenced with respect to Δs. Similar to 

computational fluid dynamics (CFD), stability depends on the 

stencil, the discretization size (both time and spatial steps), and 

the sizes of Δt and Δs. 

Unlike the FEA formulation, which guarantees L2

 stability [51], 

the FD approach is known to break down if finite-differencing 

schemes are paired with incompatible integration techniques 

[52]. This introduces a conundrum of options that might need 

special tailoring for individual problems. The FD model also 

does not guarantee the conservation of energy, which is an 

inherent characteristic supported by FEA models because of 

Galerkin orthogonality. As an end result, mesh convergence 

studies might have to be performed to determine sensitivities. 

This suggests that unlike LM and FEA models, it is less obvious 

with FD representations what the optimal time-step size Δt and 

spatial resolution Δs are. In other words, mesh resolution can be 

dictated by numerical needs rather than accuracy requirements. 

Gobat and Grosenbaugh [47] studied the numerical limitations 

of FD techniques, and this effort led to a comprehensive 

understanding of how integration methods contribute to 

stability. This work greatly advanced the state of the art in FD 

cable models, and is worth further study for interested readers. 

Although FD models have limitations in terms of numerical 

implementation, the main advantages of this model are that it 

can be implemented into a computer code with relative ease 

(Equation [9] compared to Equations [6] and [7) and the 

formulation can be reorganized to contain a diagonal mass 

matrix (but without a need for order reduction like the LM 

model). Once the numerical barriers are addressed, FD models 

provide high-fidelity modeling capabilities on par with FEA 

representations. To summarize, FD models are rarely used in 

commercial applications or in production codes because of 

numerical sensitivities and nuances, and more work is 

warranted to improve the approach. 

2.4 PROPOSED MODEL 

The LM and FEA models discussed previously offer examples 

of typical formulations used in dynamic simulation programs. 

Unlike the FD formulation, the LM and FEA models are 

inherently less prone to numerical errors (drift) because the 

formulation ensures that energy flux is conserved. Depending 

on the implementation used, the LM model can be based on an 

order reduction of the FEA process, but LM formulations may 

require more elements to approach the accuracy of FEA 

assemblies. Because of its coding simplicity, ability to capture 

the physics of a higher-order model, computational efficiency, 

and historical application as a design and analysis tool, the LM 

model is chosen for development in MAP. The development is 

based on the high-order LM methodology described by 

Buckham and colleagues [16,18,35].

3 MODEL DEVELOPMENT 

The LM model pictured in Fig. 4 represents a kinematic chain 

of mass-spring-damper elements and is used to describe the 

equation of motion for the system. In Fig. 4, a breakdown of 

one element is given. The vector ri describe the node position in 

the inertial frame. Each element has a damping coefficient, bi,

and a stiffness coefficient, ki. An orthogonal local coordinate 

system, xi, yi, zi, describes the element orientation relative to F0, 

an inertial global frame, where zi is parallel to the straight-line 

element and zi = xi × yi. Finally, the radius of curvature, , is 

used to compute the magnitude of the bending moment of the 

elements. The line curvature is calculated using third-order 

spline functions passing through sequential node points. The 

forces acting on a line element are viscous hydrodynamic loads, 

internal tension, internal damping, self-weight, buoyancy, and 

bending resistance loads, all of which are defined in the local 

element frame xi, yi, zi. The nodal equation of motion from 

Equation (6) is used, where the internal bending force is defined 

as [35] 

. (10) 
7 

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FIGURE 4: DEFINITION OF ELEMENT AND NODE 

COMPONENTS WITHIN THE LM MODEL. q(s) IS A CUBIC 

SPLINE FUNCTION INTERPOLATED THROUGH ALL THE 

NODE POINTS. 

3.1 LINE KINEMATICS 

Defining the line kinematics and system coordinate frame is the 

first step in obtaining the equation of motion. Alternative 

derivations can also be found in [16,18,27,28,31,35]. Let 

 (11) 

represent the transformation matrix relating the body-frame 

coordinates to a global inertial frame. As a result, 

 

, (12) 

where Li + Δi is the stretched length of the ith element. From 

trigometric relations, the Euler angles θi and φi are given by 

 (13) 

and 

 (14) 

if cosθi > sinθi, and 

 (15) 

otherwise. The inverse tangent calculation should capture 

angles across the full range of the unit circle (i.e., from 0–2π). 

From Equation (12), it is evident that the stretched line length is 

equivalent to the magnitude of the difference between ri and 

ri+1: 

. (16) 

3.2 INTERNAL FORCES 

Internal forces are derived from three sources: strain, damping, 

and bending resistance. Strain is proportional to the element 

stiffness, ki, which is the element axial stiffness EA/Li. Internal 

forces occur in the direction parallel to zi, so 

 (17) 

and 

, (18) 

where Δi is readily known from Equation (16). Equation (17) is 

set to zero if Δi < 0, because cables cannot support compressive 

loads. Applying internal damping in structural members is a 

delicate issue [53], but it is generally accepted to be 

proportional to the relative velocity between nodes: 

. (19) 

Deciding a coefficient for bi is a matter being debated within 

structural modeling circles, but a strong argument states that 

damping can be stiffness and mass proportional (i.e., Rayleigh 

damping) [11]. With certain integration schemes, though, 

structural damping is not recommended because this leads to 

conservative results [40]. This is especially true for most 

implicit integration techniques that include numerical 

dissipation. The effort of deriving the internal bending force 

was treated rigorously in [16,18,35]; readers are encouraged to 

refer to those texts for a detailed treatment of the bending 

stiffness formulation. As demonstrated by earlier works, the 

bending stiffness is given by the quantity 
8 

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, (20) 

where and equate the magnitude and axis (direction) at 

which the bending moment is applied in the local element 

frame. Equation (20) is expressed in an inertial reference frame 

consistent with Equation (6) once the Euler rotational sequence 

is performed. The spatial curvature parameter can be 

determined by [54] 

, (21) 

where is the magnitude of the radius of curvature, is the 

discrete form of q(s) to represent a three-dimensional cubic 

spline expressed in a Frenet-Serret frame, and is the second 

derivative of the spline function with respect to s. is a thirdorder polynomial function evaluated for the ith element and is 

valid between the two nodes bounding an element: 

. (22) 

It is essential for slopes of adjacent piece-wise elements to 

match, such that 

. (23) 

An algorithm describing the process to obtain the piece-wise 

spline function can be found in [55]. The X,Y components of the 

curvature parameter are determined using 

. (24) 

The slope of the cubic spline function passing through the 

nodes is the critical component to determine the bending force 

magnitude. When the spline function, , is a straight line, the 

bending moment vanishes and Equation (20) is zero. As an 

alternative, if a cable element can be represented by a straightline spline function, Ci=Di=0 in Equation (22), and in 

Equation (21) is zero. 

3.3 EXTERNAL FORCES 

External forces applied to the system arise from hydrodynamic 

loading, gravity, and seabed contact. To maximize code reuse 

and avoid duplicating actions performed in other FAST 

modules, hydrodynamic loads are calculated using a separate 

hydrodynamics module as exemplified in [56]. The 

hydrodynamic loads are based on the cable node motion and 

wave kinematics, which are used as input in a relative form of 

Morison’s equation to capture the element added-mass and 

quadratic drag forces. Buoyancy loads are also computed by the 

hydrodynamics module. The mapping of forces at discrete 

locations along the line occurs through the “mesh library,” 

which is a component defined in the FAST standard library 

[57]. Users only need to be concerned with providing the right 

inputs to the mesh library to obtain the desired outputs. More 

details on the mesh library are found in [64], but assuming the 

lumped hydrodynamic load, hx,y,z, is defined for the ith node, the 

force defined in vector form is 

. (25) 

The terms resulting from gravity are found using 

, (26) 

which is the net weight of the node in a vacuum. Seabed 

resistance forces are included through using an additional 

spring term. This spring is engaged when the Zi node position 

drops below a certain threshold. The formulation for this 

quantity resembles Equation (17), but requires Equation (18) to 

be rewritten to include this additional force. Some formulations 

include a two-parameter spring model with damping to relax the 

recoiling effect caused by a sudden onset of the contact force 

[14,48]. 

3.4 CABLE STATICS 

Cable statics are an important preliminary step in running a 

dynamics simulation, but numerical limitations drive this 

delicate matter. Because the objective of a dynamic mooring 

line model is to address a dynamics problem, the statics 

problem is largely viewed as an afterthought. The performance 

of the dynamic model, however, is strongly dependent on the 

quality of the initial starting point of the nodes and stretched 

element lengths. 

A statics solution is achieved by removing all time-dependent 

variables in Equation (6), then solving the resulting nonlinear 
9 

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equations. Minimizing these functions is an immensely difficult 

task because the Jacobian matrix used by the nonlinear rootfinding algorithm is poorly conditioned [44]. Several 

approaches to solving the statics problem exist, and the chosen 

path is a matter of preference. The simple approach is dynamic 

relaxation [58,59], which involves executing a dynamics 

simulation with large damping values until transients dissipate. 

Another approach implements an artificial force term into 

Equation (6) that is initially set equal to the (unbalanced) node 

forces [19,60,61] 

, (27) 

where the j subscript is the solve step. For the first solve step, 

, (28) 

βj ranges from {1,0}, where βi = 1 at the first solve, and βj = 0 at 

convergence. A third approach uses a shooting algorithm, which 

involves a rewrite of the equations being solved by converting 

Equation (6) from an initial value problem to a boundary 

condition problem [44,62,63]. The approach to be implemented 

in MAP will be selected from one of these methods. 

4 FAST FRAMEWORK AMALGAMATION 

The FAST modularization framework can be considered to be 

an application program interface that defines the guidelines and 

standard practices for modules being integrated into the latest 

version of the FAST wind turbine CAE tool [1,57]. This 

framework compartmentalizes different components modeled in 

the wind turbine to separate the physics of the system. Although 

for most modules, communication with other modules is 

essential (such as hydrodynamic force mapping on a structural 

model [56]), the information-passing corridor is managed by the 

FAST glue code. In this regard, module developers are 

responsible for defining the quantities exchanged between 

modules and the information passed to and from FAST. This 

leads to extensible modules that can be separated away from the 

core glue code, allowing modules to conduct independent 

studies without redefining the source code. 

To draw parallels between the LM model and the FAST 

framework, the single-point mooring in Fig. 3 is used to define 

a mooring anchored to the seabed at node N+1 and attached to a 

floating wind turbine at node 1. Figure 5 demonstrates the flow 

of information between FAST and the proposed LM component 

of the MAP module. The passage of information and program 

flow illustrated falls under the auspice of the FAST “loosecoupling” definition. Under the loose-coupling definition, each 

individual module is responsible for executing the solve 

routines and integration of differential equations (when 

necessary) for that module. The particular definition of each 

data structure is interpreted by the FAST glue code to describe 

the action taken on that variable. 

4.1 DATA STRUCTURES 

Data structures are used to define the interaction of variables 

between the FAST glue code and modules. For example, 

parameters p are constant, time-invariant values throughout the 

simulation, such as sea density, gravitational constant, or the 

anchor position of a line element. Likewise, continuous states, 

x(t), are those with which a differential equation is associated 

[57] 

. (29) 

Constraint states z are variables iterated by a root-finding 

algorithm to find the minimum of a function 

FIGURE 5: INPUTS, OUTPUTS, AND INTERNAL STATES FOR THE LM MODULE IN MAP. THIS FLOWCHART IS DEFINED FOR LOOSE 

COUPLING [1,56,57]. 
10 

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications. 

, (30) 

with a requirement that 

. (31) 

Otherwise, the matrix in Equation (31) is rank deficient and a 

bounded inverse does not exist around a solution. Equation (31) 

is defined as the Jacobian matrix determinant, where the 

Jacobian matrix is used to find the roots of Equation (30). It 

becomes evident from Equations (29) and (30) that FAST is 

formulated as a multibody dynamics system and the solution is 

structured as a DAE [42]. The formulation, though, reduces to a 

conventional ODE when constraints z are not present. 

In the context of a dynamic cable model for FAST, the inputs to 

MAP would be the position and velocity of the node attaching 

to the floating platform and hydrodynamic loads on all nodes: 

. (32) 

The values in Equation (32) are transferred from the glue code 

using a mesh, because the node position and velocity are related 

to the platform global motion as computed by FAST. Outputs 

from MAP consist of the force and moment at the node 

attaching to the platform, and the position and velocity of all 

intermediate nodes between the fairlead and anchor. The 

intermediate nodes are mapped as meshes so kinematic 

information can be received by an external hydrodynamics 

module to calculate viscous loads. As a result, 

, (33) 

where the moments mbx

1 and mby

1 at the cable root are 

calculated using the procedure in [35]. The output vector is 

information passed back to the FAST glue code to advance to 

the next time-step sequence. In this application, the constraint 

variable, z, consists of the coefficients’ cubic polynomial 

function, , in Equation (22). As a result, the constraints are 

. (34) 

The continuous state consists of nodes whose position and 

velocity are obtained by way of numerical integration. They are 

defined as the nodes lying between the anchor point and 

fairlead: 

. (35) 

The node attaching to the floating vessel at i=1 has prescribed 

motion supplied by the FAST glue code, and it does not have an 

associated differential equation. Likewise, the anchor node is 

fixed, so it is encapsulated within p. The remaining variables, 

such as , θi and φi , are preserved in a database named “other 

states.” Storage of these variables in the other states derived 

type is allowable because they are explicitly derived from x, z, 

p, or u. 

4.2 EQUATIONS 

In this application of the LM model, two sets of equations are 

necessary to complete the FAST definitions. The first, the 

continuous equations, are those that can be differentiated with 

time. This was previously defined in Equation (6), rewritten as 

the following ODE: 

.

 (36) 

The second state equations, the constraint equations, are those 

iterating on the constraint variables, z(t), to minimize a 

function. In this case, the functions minimized are those 

providing the solution to the cubic spline interpolation function 

in Equation (22). An example algorithm is given in [55]. 

5 SUMMARY 

This paper surveys the various dynamic modeling line theories 

that are currently used in offshore modeling applications. 

Advantages of each model are considered, and although each 

model has merit, the LM model is selected for incorporation in 

MAP. The LM model is selected for the following reasons: 

 Computational performance is optimized for diverse 

applications: 

 A diagonal mass matrix may omit acceleration 

coupling terms, but limitations can be rectified by 

increasing the mesh resolution. 
11

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• Sensitivities of the numerical approaches are well studied.

• Incorporation of bending stiffness is available as a user 

option: 

− Over the past two decades, the abilities of LM models 

have improved significantly to the extent that highorder bending stiffness models have been included.

− Bending can be disabled at run time. When bending 

stiffness is disabled, ni = 0 in Equation (6). 

Equations describing the LM model are reconstructed in the 

FAST framework context. Although this paper explores the 

framework from a loose-coupling standpoint, this model can be 

expanded to include essentials for tight coupling.

As a land-based and floating wind turbine modeling tool, FAST 

is responsible for combining several physics to aid in the 

simulation of these devices. To assist in the integration of 

various physics, the modularization framework has been 

developed to help encapsulate data, encourage code reuse, and 

enable state linearization (e.g., for eigen analysis) and tightcoupling dynamic analysis. This paper explains a new dynamic 

mooring line module in the context of this new framework. 

MAP is a module developed for FAST that is currently capable 

of quasi-static modeling. By adding a dynamic mooring line 

representation, MAP will enable coupled analysis in FAST.

Future work will involve implementation, verification, and 

validation of this LM model within MAP and application of the 

coupled code to the analysis of promising floating offshore 

wind concepts.

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