Uncertainty Quantification in Mooring Cable Dynamics Using Polynomial Chaos Expansions

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.

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.

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.

2. Numerical Models

2.1. Mooring Equations

Let us consider the equation of perfectly flexible cables [24], applied to a cable of length $L\in {\mathbb{R}}^{+}$, in a physical domain $D\in {\mathbb{R}}^3$ and a time domain $t\in [0,t_{\mathrm{end}}],t_{\mathrm{end}}>0$, subject to initial and Dirichlet boundary conditions, Equation (1):

$$\begin{array}{}\hfill \mathrm{(1a)}& \hfill m_1\left(s\right)\frac{{\partial }^2\mathbf{r}(s,t)}{\partial t^2}& =& \frac{\partial }{\partial s}\left(\frac{\mathrm{T}\left(ϵ(s,t)\right)}{1+ϵ(s,t)}\frac{\partial \mathbf{r}(s,t)}{\partial s}\right)+{\mathbf{f}}_e(s,t),\hfill \hfill \mathrm{(1b)}& \hfill ϵ(s,t)& =& \left|\frac{\partial \mathbf{r}(s,t)}{\partial s}\right|-1,\hfill \hfill \mathrm{(2)}& \hfill \mathbf{r}(s,t=0)& =& {\mathbf{r}}_0(t,\mathbf{r}),\hfill \hfill \mathrm{(3)}& \hfill \mathbf{r}(s=0,t)& =& f_1\left(t\right),\hfill \hfill \mathrm{(4)}& \hfill \mathbf{r}(s=L,t)& =& f_2\left(t\right),\hfill \end{array}$$

where m_l is the mass per unit length of the cable, $\mathbf{r}(s,t)$ is the position of the cable point s, $s\in [0,L]$ is the unstretched line coordinate, $\mathrm{T}$ is the tension magnitude, ϵ is the extension, and ${\mathbf{f}}_e$ are the external forces acting on the cable. ${\mathbf{r}}_0(t,\mathbf{r})$ is the initial deformation of the cable and $f_1\left(t\right)$ and $f_2\left(t\right)$ represent the boundary conditions at the ends of the cable. The external forces, ${\mathbf{f}}_e$, 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):

$$\begin{array}{}\hfill \mathrm{(5a)}& \hfill {\mathbf{f}}_m& =& A_c{\rho }_{\mathrm{w}}{\mathbf{a}}_{\mathrm{rf}}(1+Cm),\hfill \hfill \mathrm{(5b)}& \hfill {\mathbf{f}}_d& =& \frac{1}{2}{\rho }_{\mathrm{w}}D\sqrt{1+ϵ}(C_{dt}\mid {\mathbf{v}}_{\mathrm{r},t}\mid {\mathbf{v}}_{\mathrm{r},t}+C_{dn}\mid {\mathbf{v}}_{\mathrm{r},n}\mid {\mathbf{v}}_{\mathrm{r},n}),\hfill \end{array}$$

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

$${\mathbf{f}}_b=m_{\mathrm{l}}\left(\frac{{\rho }_c-{\rho }_w}{{\rho }_c}\right)\mathbf{g},$$

(6)

where f_m is the added mass force, A_c is the cable cross section area, ρ_w is the fluid mass density, a_rf is the relative cable-fluid acceleration, C_m is the cable added mass coefficient, f_d is the viscous drag force, D is the cable diameter, C_dt is the cable tangential drag coefficient, C_dn is the cable normal drag coefficient, v_r,t is the relative fluid-cable tangential velocity, v_r,n is the relative fluid-cable normal velocity, f_b is the submerged weight of the cable, ρ_c 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.

$$\begin{array}{}\hfill \mathrm{(7a)}& \hfill {\mathbf{f}}_{c,z}& =& \left\{\begin{array}{cc}KD(r_z-z_g)-2\xi \sqrt{m_{1}KD}\max (v_z,0),\hfill & \hfill \mathrm{if}{r}_z\le z_g,\\ 0,\hfill & \hfill \mathrm{otherwise},\end{array}\right.\hfill \hfill \mathrm{(7b)}& \hfill {\mathbf{f}}_{c,xy}& =& -\left|\right|{\mathbf{f}}_b\left|\right|\mu \sin \left({\mathbf{v}}_{xy}\frac{\pi }{2}\right),\hfill \hfill \mathrm{(7c)}& \hfill {\mathbf{v}}_{xy}& =& \frac{(v_x,v_y)}{max(v_{\mathrm{c}},∥(v_x,v_y)∥)},\hfill \end{array}$$

in which f_c,z is vertical ground force, fc,xy is the horizontal ground force, K is the soil’s stiffness per unit area, r_z is the height of the cable, z_g is the height of the sea-bottom, ξ is the soil’s normal damping ratio, v_z is the vertical component of the velocity of the cable, v_x is the component of the velocity of the cable in the x direction, v_y is the component of the velocity of the cable in the y direction, v_c is the speed of fully developed friction force magnitude, and μ is 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.

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):

$$\left(\mathbf{M}+{\mathbf{A}}_{\infty }\right)\ddot{\mathbf{X}}\left(t\right)+{\int }_{-\infty }^t\mathbf{K}\left(t-\tau \right)\dot{\mathbf{X}}\left(t\right)\mathrm{d}\tau +\mathbf{CX}\left(t\right)={\mathbf{f}}_{\mathrm{ext}}\left(t\right),$$

(8)

with $\mathbf{M}$ being the generalized mass matrix of the the floating body, ${\mathbf{A}}_{\infty }$ the added mass matrix at infinity frequency, $\mathbf{K}$ the radiation impulse response function, $\mathbf{C}$ the hydrostatic stiffness matrix, ${\mathbf{f}}_{\mathrm{ext}}$ the vector of external forces, and $\mathbf{X}$, $\dot{\mathbf{X}}$, and $\ddot{\mathbf{X}}$, respectively, the displacement, velocity, and acceleration vectors. The vector of external forces, ${\mathbf{f}}_{\mathrm{ext}}$, contains any wave excitation forces, ${\mathbf{f}}_{\mathrm{exc}}$ and mooring forces, ${\mathbf{f}}_{\mathrm{moor}}$. 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 $f(\mathbf{x},Z)$, where $\mathbf{x}$ 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 $\mathbb{R}$, from the sample space of possible outcomes Ω, in the form of events denoted by $\mathcal{F}$, with an associated probability P. This is denoted by $Z:\mathsf{\Omega }\to \mathbb{R}$. Using the property that continuous function in $L_2$ with bounded variation can be expressed as an infinite series, gPC provides a polynomial expansion surrogate model to $f(\mathbf{x},Z)$:

$$f_{gPC}(\mathbf{x},Z)=\sum _{k=0}^{\infty }{\widehat{f}}_k\left(\mathbf{x}\right){\varphi }_k\left(Z\right),$$

(9)

in which ${\widehat{f}}_k\left(\mathbf{x}\right)$ are the polynomial coefficients and ${\left\{{\varphi }_k\left(Z\right)\right\}}_{k=0}^{\infty }$ is the set of polynomial basis functions. For optimal convergence of $f_{gPC}(\mathbf{x},Z)$ to the results of the original model $f(\mathbf{x},Z)$, 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:

$$f_{gPC}(\mathbf{x},Z)\approx \sum _{k=0}^p{\widehat{f}}_k\left(\mathbf{x}\right){\varphi }_k\left(Z\right).$$

(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 ${\widehat{f}}_k\left(\mathbf{x}\right)$ encode information about the moments of the probability distribution of the results. For example, ${\widehat{f}}_0\left(\mathbf{x}\right)$ is a scaled value of the mean, and the variance can be obtained by summing the squares of the set of coefficients ${\widehat{f}}_i{\left(\mathbf{x}\right)}_{i=1}^p$ in the expansion.

For problems with d independent input random variables, $d\in \mathbb{N}$, $\varphi \left(Z\right)$ in Equations (9) and (10) is replaced by a tensor product of polynomials corresponding to each random variable

$$f_{gPC}(\mathbf{x},\mathbf{Z})\approx \sum _{\left|\mathbf{k}\right|=0}^p{\widehat{f}}_{\mathbf{k}}\left(\mathbf{x}\right){\mathsf{\Phi }}_{\mathbf{k}}\left(\mathbf{Z}\right)=\sum _{\left|\mathbf{k}\right|=0}^p{\widehat{f}}_{\mathbf{k}}\left(\mathbf{x}\right){\varphi }_{k_1}\left(Z_1\right){\varphi }_{k_2}\left(Z_2\right)\dots {\varphi }_{k_d}\left(Z_d\right),$$

(11)

where $\mathbf{Z}:\mathsf{\Omega }\to {\mathbb{R}}^d$ is the vector of input random variables, $\mathbf{k}$ is a multi-index such that $\mathbf{k}=(k_1,k_2,\dots ,k_d)\in {\mathbb{N}}_0$ and $\left|\mathbf{k}\right|=k_1+k_2+\dots +k_d$, and ${\varphi }_{k_i}\left(Z_i\right)$ is the polynomial basis function of the variable $Z_i$ of degree $k_i$.

3.2. Stochastic Collocation Method

The application of gPC to a mathematical model $f(\mathbf{x},\mathbf{Z})$ 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 $f(\mathbf{x},\mathbf{Z})$ under study. For the computation of the gPC model $f_{gpc}(\mathbf{x},\mathbf{Z})\approx f(\mathbf{x},\mathbf{Z})$, we only need to postprocess the results of simulations using the mathematical model at preselected values ${\mathbf{z}}^{\left(j\right)}$ of the uncertain input $\mathbf{Z}$. The points ${\mathbf{z}}^{\left(j\right)}$ where $f(\mathbf{x},\mathbf{Z})$ is to be evaluated depending on the method chosen to determine the coefficients ${\widehat{f}}_{\mathbf{k}}$. The two methods we will use are the projection method and the regression method.

In the projection method, the coefficients ${\widehat{f}}_{\mathbf{k}}\left(\mathbf{x}\right)$ are determined by the inner product of $f(\mathbf{x},\mathbf{Z})$ and the polynomial basis, ${\mathsf{\Phi }}_{\mathbf{k}}\left(\mathbf{Z}\right)$, with respect to the probability density function (PDF) of the random variable, $\rho \left(\mathbf{Z}\right)$, Equation (12):

$${\widehat{f}}_{\mathbf{k}}\left(\mathbf{x}\right)=\langle f_{\mathbf{k}}(\mathbf{x},\mathbf{z}),{\mathsf{\Phi }}_{\mathbf{k}}\left(\mathbf{z}\right)\rangle =\frac{\int f(\mathbf{x},\mathbf{z}){\mathsf{\Phi }}_{\mathbf{k}}\left(\mathbf{z}\right)\rho \left(\mathbf{z}\right)\mathrm{d}\mathbf{z}}{\int {\mathsf{\Phi }}_{\mathbf{k}}^2\left(\mathbf{z}\right)\rho \left(\mathbf{z}\right)\mathrm{d}\mathbf{z}}.$$

(12)

Equation (12) is solved using quadrature rules, such as Gauss quadrature, which provide the points ${\mathbf{z}}^{\left(j\right)}$ where the model is to be evaluated and the quadrature weights, ${\mathbf{w}}^{\left(j\right)}$. Because multivariable gPC is formulated as a tensor product, the number of points ${\mathbf{z}}^{\left(j\right)}$ 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 ${\widehat{f}}_{\mathbf{k}}$ 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 ${\mathbf{z}}^{\left(j\right)}$. 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 ${\mathbf{z}}^{\left(j\right)}$ 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 $\mathbf{Z}:\mathsf{\Omega }\to {\mathbb{R}}^d$, Equation (16):

$$\begin{array}{}\hfill \mathrm{(13a)}& \hfill m_1(s,\mathbf{Z})\frac{{\partial }^2\mathbf{r}(s,t,\mathbf{Z})}{\partial t^2}& =& \frac{\partial }{\partial s}\left(\frac{\mathrm{T}\left(ϵ(s,t,\mathbf{Z})\right)}{1+ϵ(s,t,\mathbf{Z})}\frac{\partial \mathbf{r}(s,t,\mathbf{Z})}{\partial s}\right)+{\mathbf{f}}_e(s,t,\mathbf{Z}),\hfill \hfill \mathrm{(13b)}& \hfill ϵ(s,t,\mathbf{Z})& =& \left|\frac{\partial \mathbf{r}(s,t,\mathbf{Z})}{\partial s}\right|-1,\hfill \hfill \mathrm{(14)}& \hfill \mathbf{r}(s,t=0,\mathbf{Z})& =& {\mathbf{r}}_0(s,\mathbf{Z}),\hfill \hfill \mathrm{(15)}& \hfill \mathbf{r}(s=0,t,\mathbf{Z})& =& f_1(t,\mathbf{Z}),\hfill \hfill \mathrm{(16)}& \hfill \mathbf{r}(s=L,t,\mathbf{Z})& =& f_2(t,\mathbf{Z}).\hfill \end{array}$$

We obtain the stochastic Cummins’s equation, Equation (17), from Equation (8) in a similar way as Equation (16):

$$\left(\mathbf{M}+{\mathbf{A}}_{\infty }\right)\ddot{\mathbf{X}}\left(t,\mathbf{Z}\right)+{\int }_{-\infty }^t\mathbf{K}\left(t-\tau \right)\dot{\mathbf{X}}\left(t,\mathbf{Z}\right)\mathrm{d}\tau +\mathbf{CX}\left(t,\mathbf{Z}\right)={\mathbf{F}}_{\mathrm{ext}}\left(t,\mathbf{Z}\right).$$

(17)

3.4. Methodology

We applied the following methodology throughout all the cases presented in the article:

• select the process to be studied;
• select a physics-based mathematical model for the process;
• choose which parameters of the model are deterministic and which are random variables;
• define the values of the deterministic parameters;
• choose appropriate probability distributions for the random variables;
• choose the method to determine the gPC model coefficients (quadrature or LAR)
• 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;
• evaluate the physics-based model at the previously defined values of the random variables;
• apply the chosen method (quadrature or LAR) to the results of point 8 to obtain the coefficients of the gPC model;
• 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]:

$$\frac{{\partial }^{2}y}{\partial t^2}=\frac{T}{m_{\mathrm{l}}}\frac{{\partial }^{2}y}{\partial x^2},$$

(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, ${\left.\frac{\partial y(x,t)}{\partial t}\right|}_{t=0}=0$, the solution of Equation (18) is [37]:

$$y(x,t)=\sum _{n=0}^{\infty }{b}_n\sin \left(\frac{n\pi }{L}x\right)cos\left(\frac{n\pi }{L}ct\right),$$

(19)

where $b_n$ are the Fourier series coefficients of the function describing the initial displacement $y(x,0)$, and c is the transverse wave celerity, $c=\sqrt{(T/m_{\mathrm{l}})}$. For an initial displacement in the shape of a half sine, Figure 2a, we have $y(x,0)=Asin\left(n\frac{\pi }{L}x\right)$, and Equation (19) becomes:

$$y(x,t)=A\sin \left(\frac{\pi }{L}x\right)\cos \left(\frac{\pi }{L}ct\right).$$

(20)

There are six variables influencing the results of Equation (20)—A, L, x, T, m₁, and t—of which only two are subjected to uncertainty: T and m₁. 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. Linear cable properties.

Property	Value
A	1 m
L	3 m
T	108 N
ml	3 kg/m

. The reference values for L, T, and m₁ 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 m_l: $T\sim \mathcal{U}(100,116)$, $m_l\sim \mathcal{U}(1,5)$.

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.

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 m_l 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 m_l, 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). For p = 1 we obtain a quite crude PDF, but for p = 2, the approximation is reasonable. 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 × 10⁻² m, and the variance is 1.26 × 10⁻¹ m².

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, C_m, normal drag coefficient, C_dn, and tangential drag coefficient, C_dt) 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 r_m = 0.20 m, for two different periods: T_r = 1.25 s and T_r = 3.50 s. A summary of the relevant properties of the cable and of the numerical model is presented in

Table 2. Deterministic or constant parameters of the cable and of the experimental set-up.

Parameter	Value
K	3 × 109 Pa/m
vc	0.01 m/s
μ	0.03
ξ	1
ρw	1000 kg/m3
ρc	7800 kg/m3
D	2.2 × 10−3 m
ml	8.18 × 10−2 kg/m
EA	1 × 104 Pa

. The cable was discretised using N_el = 10 elements of order p = 5, with a limitation on the time-stepping that the Courant–Friedrichs–Lewy value should not exceed 0.45.

Figure 5 shows 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].

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.

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. Parameters of the probability distributions of the hydrodynamic coefficients.

Coefficient	Deterministic Value	Distribution	Lower Bound	Upper Bound
Cm	3.8	Uniform	1.90	5.70
Cdn	2.5	Uniform	1.25	3.75
Cdt	0.5	Uniform	0.25	0.75

. The values of C_m, C_dn and C_dt 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.

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 6 for the case of T_r = 1.25 s and in Figure 7 for the case of T_r = 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 T_r = 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—T_r = 1.25 s and T_r = 3.50 s—the maximum tensions happen when all the coefficients—C_m, C_dn and C_dt—are varied simultaneously: around 79 N for T_r = 1.25 s and around 69 N for T_r = 3.50 s. When the coefficients are varied individually, changes in C_dn 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 T_m = 1.25 s and around 69 N for T_m= 3.50 s. Varying C_dt and C_m leads to the lowest peak tensions: 73 N to 74 N for T_m = 1.25 s and around 52 N for T_m = 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 C_dn is varied. Variations in C_dt and C_m cause significantly lower spread of the tension values. In fact, for T_m = 3.50 s, varying C_dt causes no noticeable changes in the tension values, so it is not possible to compute the PDF.

Focusing on the case with oscillating period T_m = 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.

For the oscillating period T_m = 3.50 s, the differences between the four scenarios are more marked than for T_m = 1.25 s. Variations in C_m have limited influence in the tension, and changes in C_dt have no effect at all; therefore, it was not possible to build a PDF for this case. It is variations in C_dn 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 T_m = 1.25 s, for T_m = 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, C_m and C_dt have the greatest influence when the tension is close to 0 N. C_m has similar contributions in T_r = 1.25 s and in T_m = 3.50 s. For T_m = 1.25 s C_dt dominates in the low tension region, while C_dn dominates during the high tension part of the cycle. For T_m = 3.50 s C_dn dominates almost the entire time, with the exception of very short periods, right before the cable goes slack, when C_m dominates briefly.

This analysis is in line with what was seen before. We have C_dn playing a major role when the tension is higher, because the cable is moving faster. When the tension is lower, C_m and C_dt have a greater, or even the greatest, contribution to the results. In both cases, C_dn seems to be the leading coefficient contributing to the value of the maximum tension. Since C_m and C_dt 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, C_dn, 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, C_dt, and of the added mass coefficient, C_m, depends on the period of the excitation: for shorter periods, C_dt has more influence than C_m, while for longer periods, the opposite happens. In other words, C_dt has more importance for fast motions, whereas C_m 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].

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 4. Properties of the buoy. D—diameter; h—height; ${\mathrm{I}}_{\mathrm{xx}}$—inertia around the horizontal axis through the center of gravity; ${\mathrm{C}}_{\mathrm{g}}$—center of gravity (distance from the top).

Mass	D	h	${\mathbf{I}}_{\mathbf{xx}}$	${\mathbf{C}}_{\mathbf{g}}$
35.85 kg	0.515 m	0.401 m	0.87 kg m2	0.3247 m

for the buoy,

Table 5. Chain properties.

Parameter	Value
D	4.786 × 10−3 m
ml	0.1447 kg/m3
EA	1.6 MN
Cdn	2.5
Cdt	0.5
Cm	3.8
Cable length	6.95 m

for the chain, and

Table 6. Water and soil properties.

Parameter	Value
K	3 × 108 Pa/m
vc	0.01 m/s
μ	0.3
ξ	1
ρw	1000 kg/m3

for 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 $N_{\mathrm{el}}=10$ 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.

We modeled the uncertainty in the anchor position using the one-dimensional Gaussian distribution for each of the horizontal anchor coordinates: x_Ai and y_Ai, corresponding to cable i. The data relative to the probability distributions of the uncertainty is presented in

Table 7. Parameters of the probability distributions of the anchors’ positions.

Anchor Coordinate	Deterministic Value	Distribution	Mean Value	Standard Deviation
$x_{\mathrm{A}1}$	−3.4587 m	Normal	−3.4587 m	0.025 m
$y_{\mathrm{A}1}$	5.9907 m	Normal	5.9907 m	0.025 m
$x_{\mathrm{A}2}$	−3.4587 m	Normal	−3.4587 m	0.025 m
$y_{\mathrm{A}2}$	−5.9907 m	Normal	−5.9907 m	0.025 m
$x_{\mathrm{A}3}$	6.9175 m	Normal	6.9175 m	0.025 m
$y_{\mathrm{A}3}$	0.0000 m	Normal	0.0000 m	0.025 m

. 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 Figure 10, Figure 11 and Figure 12.

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 $T=1.40\mathrm{s}$, lead to more accurate results in tension.

The results of the simulations are presented in Figure 10, Figure 11 and Figure 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.