# Uncertainty Quantification in Mooring Cable Dynamics Using Polynomial Chaos Expansions

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Models

#### 2.1. Mooring Equations

_{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):

**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.

**f**

_{c,z}is vertical ground force,

**f**c,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

#### 2.3. Floating Body Model

## 3. Uncertainty Quantification

#### 3.1. Generalized Polynomial Chaos

#### 3.2. Stochastic Collocation Method

#### 3.3. Model Equations with Random Input Variables

#### 3.4. Methodology

- 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.

## 4. Case Studies

#### 4.1. Linear String

_{1}, and t—of which only two are subjected to uncertainty: T and m

_{1}. For the reference deterministic case, against which we will compare the results of uncertainty quantification, we choose for the variables the values presented in Table 1. The reference values for L, T, and m

_{1}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}: $T\sim \mathcal{U}(100,116)$, ${m}_{l}\sim \mathcal{U}(1,5)$.

_{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.

^{−2}m, and the variance is 1.26 × 10

^{−1}m

^{2}.

#### 4.2. Oscillating Mooring Cable

_{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. 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.

_{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.

_{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.

_{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.

_{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.

_{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.

_{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.

_{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.

_{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

_{Ai}and y

_{Ai}, corresponding to cable i. The data relative to the probability distributions of the uncertainty is presented in Table 7. For the mean of the distribution of each coordinate we selected the deterministic value of the position of the anchor in the experimental set-up. For the standard deviation we chose the value of 0.025 m, based on reasonable estimates of the maximum error when installing the anchor in the physical model, given the dimensions of the set-up and the difficulty in handling the anchors. To obtain the gPC model of the moored buoy, we generated 150 samples of the positions of the anchors using the Latin Hyper Cube method and ran the numerical model for each of these sampled positions. Afterwards, we used the LAR algorithm to fit the coefficients to a gPC expansion of order p = 5, using Hermite polynomials and q-norm of 1. With the gPC model we evaluated 3000 random samples of anchor positions to build confidence intervals for the tension in the cables and for the surge, heave and pitch motions. These were then compared with the physical model measurements. The results are presented in Figure 10, Figure 11 and Figure 12.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BEM | Boundary Element Method |

CFL | Courant–Friedrichs–Lewy |

DG | Discontinuous Galerkin |

gPC | generalized Polynomial Chaos |

LARS | Least Angle Regression |

MC | Monte Carlo |

MODU | Mobile Offshore Drilling Unit |

O&G | Oil and Gas |

Probability Density Function | |

TSI | Total Sensitivity Index |

UQ | Uncertainty Quantification |

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**Figure 1.**Outline of the high-order DG modeling approach. The cable is discretised into finite elements of size h with approximation order p. The jumps are exaggerated for illustrative purposes.

**Figure 5.**Numerical simulation of the tension at the fairlead compared to experimental data from [39].

**Figure 6.**Representation of the probability density of cable tension as a function of time, for T

_{r}= 1.25 s, together with the mean tension (black line).

**Figure 7.**Representation of the probability density of cable tension as a function of time, for T

_{r}= 3.50 s, together with the mean tension (black line).

**Figure 8.**Approximate relative contribution of each coefficient to the variance of the tension, together with the plot of the mean tension.

**Figure 10.**95% confidence intervals together with deterministic simulation results using the mean value of the input random variables, for T = 1.00 s, H = 0.04 m.

**Figure 11.**95% confidence intervals together with deterministic simulation results using the mean value of the input random variables, for T = 1.20 s, H = 0.04 m.

**Figure 12.**95% confidence intervals together with deterministic simulation results using the mean value of the input random variables, for T = 1.40 s, H = 0.04 m.

Property | Value |
---|---|

A | 1 m |

L | 3 m |

T | 108 N |

m_{l} | 3 kg/m |

Parameter | Value |
---|---|

K | 3 × 10^{9} Pa/m |

v_{c} | 0.01 m/s |

μ | 0.03 |

ξ | 1 |

ρ_{w} | 1000 kg/m^{3} |

ρ_{c} | 7800 kg/m^{3} |

D | 2.2 × 10^{−3} m |

m_{l} | 8.18 × 10^{−2} kg/m |

EA | 1 × 10^{4} Pa |

Coefficient | Deterministic Value | Distribution | Lower Bound | Upper Bound |
---|---|---|---|---|

C_{m} | 3.8 | Uniform | 1.90 | 5.70 |

C_{dn} | 2.5 | Uniform | 1.25 | 3.75 |

C_{dt} | 0.5 | Uniform | 0.25 | 0.75 |

**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 m^{2} | 0.3247 m |

Parameter | Value |
---|---|

D | 4.786 × 10^{−3} m |

m_{l} | 0.1447 kg/m^{3} |

EA | 1.6 MN |

C_{dn} | 2.5 |

C_{dt} | 0.5 |

C_{m} | 3.8 |

Cable length | 6.95 m |

Parameter | Value |
---|---|

K | 3 × 10^{8} Pa/m |

v_{c} | 0.01 m/s |

μ | 0.3 |

ξ | 1 |

ρ_{w} | 1000 kg/m^{3} |

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 |

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## Share and Cite

**MDPI and ACS Style**

Moura Paredes, G.; Eskilsson, C.; P. Engsig-Karup, A.
Uncertainty Quantification in Mooring Cable Dynamics Using Polynomial Chaos Expansions. *J. Mar. Sci. Eng.* **2020**, *8*, 162.
https://doi.org/10.3390/jmse8030162

**AMA Style**

Moura Paredes G, Eskilsson C, P. Engsig-Karup A.
Uncertainty Quantification in Mooring Cable Dynamics Using Polynomial Chaos Expansions. *Journal of Marine Science and Engineering*. 2020; 8(3):162.
https://doi.org/10.3390/jmse8030162

**Chicago/Turabian Style**

Moura Paredes, Guilherme, Claes Eskilsson, and Allan P. Engsig-Karup.
2020. "Uncertainty Quantification in Mooring Cable Dynamics Using Polynomial Chaos Expansions" *Journal of Marine Science and Engineering* 8, no. 3: 162.
https://doi.org/10.3390/jmse8030162