text:
## Abstract

In this study, effects of damage levels of fiber ropes on the performance of a hybrid taut-wire mooring system are investigated. The analysis is performed using a numerical floating production storage and offloading (FPSO) model with a hybrid mooring system installed in 3000 m of water depth. An in-depth study was conducted using the numerical model, the dynamic stiffness equation of damaged fiber ropes, the time-domain dynamic theory, the rainflow cycle counting method, and the linear damage accumulation rule of Palmgren-Miner. Results indicate that, in a mooring line with an increasing damage level, the maximum tension decreases, while the offset of the FPSO increases. Particularly, when a windward mooring line failure occurs, in addition to the significant increase in the offset of the FPSO, the maximum tension, tension range, and annual fatigue damage levels of the remaining lines adjacent to the failed also increase significantly. The present work can be of great benefit to the evaluation of the offset of the floating platform, the tension response, and the service life of the hybrid mooring systems.

## 1 Introduction

Station-keeping of offshore large floating structures is a key technology for the exploitation of marine resources. For any type of offshore large floating structures, a reliable and economic positioning technology is of great necessity [13]. Note that, for a permanent positioning of floating structures in the field, catenary, and taut-wire mooring systems are used, as illustrated in Fig. 1 [4]. Catenary mooring systems often use steel chains or wire ropes as mooring lines and rely on the weight of the mooring lines to provide part of the restoring forces. However, the weight (and cost) of the mooring lines increases as their length increases. This relationship causes catenary systems to become less efficient and less economical in deeper water. Therefore, the use of synthetic fiber ropes as main sections of the mooring lines was proposed in the 1960s [5]. In 1997, the first polyester taut-wire mooring system was successfully installed in the Campos Basin, offshore Brazil [6]. Since then, the polyester ropes used in taut-wire mooring systems have become the preferred option for depths up to 1500 m. However, with numerous discoveries of natural resources in deeper waters, the question of whether the polyester ropes can provide sufficient stiffness to maintain acceptable platform offsets at various depths has been raised [79]. Based on the observation that the stiffness of high modulus polyethylene (HMPE) ropes is higher than that of the polyester ropes at the same minimum breaking strength, the HMPE ropes were considered as alternatives to the polyester ropes [7,9]. However, compared to polyester, HMPE shows a lower elongation at breakpoint as well as a higher creep rate and a higher Young's modulus, which causes a larger load range in a storm [1012]. Considering these aforementioned factors, a preferred hybrid rope configuration was proposed [12,13].This configuration uses the stiffer HMPE rope in cooler water closer to the seabed and the polyester rope in warmer water closer to the vessel or platform. Consequently, taut-wire mooring systems utilizing various hybrid mooring lines have the capability to keep the position of the floating structures in deepwater locations [14,15].

Fig. 1
Fig. 1
Close modal

Besides the function of positioning, a point worth emphasizing is that mooring systems also play a vital role in security assurance of floating structures in the harsh environment. According to the statistics of engineering practices, the occurrence of catastrophic failures, such as capsizing, fracture, leakage, uncontrolled drifting, and collision, is more frequent in the extreme marine environment than anticipated [2,3,16,17]. According to an investigation of 69 cases of failure of floating platforms since 1964, the number of failures of the mooring system is 47, reaching a high level of 68%, while the remaining 32% represent the other factors, as listed in Table 1 [3]. This shows that the performance of the mooring systems is of significance for the safe operation of the floating structures [2,3].

Table 1

Cases of severe accident, failure, and capsizing of floating platforms (69 cases)

CauseFailure of mooring systemBlowout, leakage, fire, blastDestruction of the floating structureHolistic inclination or capsizingAccidents in haul and installation
Number4710642
CauseFailure of mooring systemBlowout, leakage, fire, blastDestruction of the floating structureHolistic inclination or capsizingAccidents in haul and installation
Number4710642

In taut-wire mooring systems, fiber ropes serve as load transmitters between the floating structures and their anchors. In engineering applications, fiber ropes may be damaged due to the following reasons: (1) rope handling during the installation, (2) wear experienced during the service, (3) ingress of sand and marine growth, (4) material and manufacturing defects, and (5) local sub-rope or element rupture during the service [1720]. The damage to the mooring lines over time intensifies or even causes the failure of the mooring lines. Once mooring line damage or failure occurs, it will affect the dynamic response of the mooring system or even lead to loss of station-keeping capability. In addition, the cost of a single mooring line failure and replacement can be enormous. Thus, it is necessary to understand the mechanical behaviors of the damaged fiber ropes and the sensitivity of the mooring system’s performance to the damage levels of the fiber ropes.

In the aspect of the mechanical behavior of the damaged fiber ropes, the residual strength of damaged fiber ropes was experimentally investigated by Williams et al. [18], Ward et al. [19,20], and Flory [21]. All these studies adopted the method of cutting some sub-ropes or elements to create the initial damage state of specimens. In a study by Liu et al. [22], an empirical equation that accounts for the dynamic stiffness of damaged fiber ropes was proposed based on experimental investigations. The aforementioned research works are helpful in estimating the positioning performance of the mooring system with the damaged fiber ropes.

In terms of dynamic response of the mooring systems with the damaged mooring lines, Huang et al. [23] performed a fatigue analysis of a polyester taut-wire mooring system of a floating production storage and offloading (FPSO) model and investigated the response of the mooring system with an one-line failure. Qiao et al. [24] investigated the corrosion effect on fatigue life of two mooring systems. Ahmed et al. [25] was apparently the first to evaluate the dynamic response of truss spar platforms for various catenary mooring configurations with the damage levels of the mooring lines and simplified these lines as springs. However, to investigate the effect of the damage levels of the fiber ropes on the mooring performance, it is important that the nonlinear behavior of the fiber ropes should be considered, for example, by introducing the empirical equation of dynamic stiffness [15,23,26].

However, little work has been devoted so far on examining the effects of the damage levels of the fiber ropes on the dynamic response of a taut-wire mooring system. The reason may be that the damage effect on the fiber mooring lines has not been understood well. Hence, the present study may be of great necessity in the design hybrid taut-wire mooring systems at the preliminary stage.

## 2 Numerical Model of a Hybrid Taut-Wire Mooring System

To investigate the effects of the damage levels of fiber ropes on the performance of a taut-wire mooring system, a numerical model of the FPSO, and the details of the mooring system are presented as follows.

### 2.1 Numerical Model of an FPSO.

In the present study, an FPSO numerical model installed in 3000 m of water depth is developed. A procedure for establishing the numerical model is presented as follows. First, based on the ship shape diagram (as shown in Fig. 2(a)) and the principal dimensions of the FPSO (as listed in Table 2), using a mesh generator in hydrostar (software) to generate the numerical model mesh, as shown in Fig. 2(b). After creating the model mesh, the diffraction and radiation coefficient, the motion of the ship, the second-order loads using near field, and the middle field formulations are derived [27,28]. Then, the response amplitude operators (RAOs) transfer functions are constructed using hydrostar [28]. The RAOs as functions of wave frequency are shown in Fig. 3 for five different incident wave angles $θ=0deg,22.5deg,45deg,67.5deg,90deg$. The wave heading is defined as the angle between the propagation direction and the direction of the axis Ox, which is positive in the forward direction of the vessel. The detailed procedure for developing the numerical model and implementation can be obtained from the hydrostar manual [28].

Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal
Table 2

Main particulars of the moored FPSO

DescriptionSymbolUnitQuantity
Vessel sizeLOA$m$244.8
Length between perpendicularLPP$m$234.0
BreadthB$m$41.2
DepthH$m$21.6
Draft (in full load)T$m$13.1
Displacement (in full load)d$m3$100,550
Roll radius of gyrationKxx$m$13.7
Pitch radius of gyrationKyy$m$58.5
Yaw radius of gyrationKψψ$m$58.5
DescriptionSymbolUnitQuantity
Vessel sizeLOA$m$244.8
Length between perpendicularLPP$m$234.0
BreadthB$m$41.2
DepthH$m$21.6
Draft (in full load)T$m$13.1
Displacement (in full load)d$m3$100,550
Roll radius of gyrationKxx$m$13.7
Pitch radius of gyrationKyy$m$58.5
Yaw radius of gyrationKψψ$m$58.5

### 2.2 Description of Mooring Configurations.

The mooring system is composed of four groups, and each group contains three hybrid mooring lines, as illustrated in Fig. 4. The configuration of the hybrid lines comprises the bottom chain linked to an anchor, the HMPE rope in the cooler water closer to the seabed, the polyester rope in the warmer water closer to the vessel and the hang-off chain linked to the fairlead. The hybrid mooring line has the following benefits. First, the bottom and hang-off chains have good resistance to abrasion of the seabed and wear at trumpet welds, respectively. Second, the hybrid mooring line including the HMPE and the polyester ropes can provide adequate stiffness needed to handle maximum loads during the station-keeping in a storm, while ensuring sufficient elasticity to control the offset of floating platforms induced by environmental loads. Third, the lower environment temperature in cooler water closer to the seabed decreases the HMPE creep rate and assures the survivability of the HMPE ropes. Furthermore, the hybrid line allows the mooring designer to engineer the stiffness of the mooring system by controlling the length of the HMPE rope components.

Fig. 4
Fig. 4
Close modal

In Table 3, detailed parameters of the hybrid mooring line, including the length, diameter, and minimum breaking strength (MBS) of rope segments, are listed. In addition, the dynamic stiffness of the fiber ropes is introduced to obtain the dynamic response of fiber mooring systems [1,7,23]. The “dynamic stiffness” is defined as that the ratio of the rope load to the strain between the trough and peak loads imposed during cyclic test, typically normalized by the minimum breaking strength. In the present study, to investigate the effects of the damage levels of the fiber ropes on the performance of the mooring system, the next paragraphs contain a more precise explanation of the dynamic stiffness of initially damaged fiber ropes.

Table 3

Main particulars of the hybrid taut-wire mooring line

DescriptionUnitQuantity
Length of mooring linem3870.96
Segment 1: (ground position): R4k4 studlink chain
Length at anchor pointm121.92
Diametermm150
Weight in airkg/m492.75
Weight in waterkg/m428.35
Stiffness, AEkN73,339.07
Minimum breaking strengthkN19,728
Segment 2: HMPE ropes
Lengthm1828.8
Diametermm194
Weight in airkg/m17.2
Weight in waterkg/m0.28
Minimum breaking strengthkN19,600
Segment 3: polyester ropes
Lengthm1828.8
Diametermm259
Weight in airkg/m46.8
Weight in waterkg/m12
Minimum breaking strengthkN19,600
Segment 4: (hang-off position): chain
Lengthm91.44
Diametermm150
Weight in airkg/m492.75
Weight in waterkg/m428.35
Stiffness, AEkN73,339.07
Minimum breaking strengthkN19,728
DescriptionUnitQuantity
Length of mooring linem3870.96
Segment 1: (ground position): R4k4 studlink chain
Length at anchor pointm121.92
Diametermm150
Weight in airkg/m492.75
Weight in waterkg/m428.35
Stiffness, AEkN73,339.07
Minimum breaking strengthkN19,728
Segment 2: HMPE ropes
Lengthm1828.8
Diametermm194
Weight in airkg/m17.2
Weight in waterkg/m0.28
Minimum breaking strengthkN19,600
Segment 3: polyester ropes
Lengthm1828.8
Diametermm259
Weight in airkg/m46.8
Weight in waterkg/m12
Minimum breaking strengthkN19,600
Segment 4: (hang-off position): chain
Lengthm91.44
Diametermm150
Weight in airkg/m492.75
Weight in waterkg/m428.35
Stiffness, AEkN73,339.07
Minimum breaking strengthkN19,728

### 2.3 Dynamic Stiffness of the Initially Damaged Fiber Ropes.

As observed in the aforementioned studies [1,7,22], the dynamic stiffness of fiber ropes is a function of the initial damage level, mean load, tension or strain amplitude, load periods, and load cycles. The initial damage level (ϕ) is the ratio of the loss in rope cross-sectional area to its original cross-sectional area [1820]. Utilizing the definition of initial damage level of ropes, Liu et al. [22] cut strands or sub-ropes to inflict the desired damage level of the rope specimens to simulate the damage state of the HMPE and the polyester ropes. The advantage of this method is that the predetermined artificial damage can be directly inflicted on the test specimen in a well-controlled manner. Using the damaged rope samples, the experimental results of the cyclic tests were obtained. From the experimental data obtained, the dynamic stiffness is calculated as follows:
$Kr=(Tnp−Tn−1t)/MBSεnp−εn−1t$
(1)
where, the dynamic stiffness Kr is dimensionless, $Tnp$ and $εnp$ are the peak tension and its corresponding strain of the nth tension-elongation hysteresis loop, respectively; $Tn−1t$ and $εn−1t$ are the trough tension and its corresponding strain of the (n−1)th tension-elongation hysteresis loop, respectively; n ≥ 2. The detailed experimental method and results can be found in Refs. [22,2931].
Based on experiments of dynamic stiffness of the damaged fiber ropes, Liu et al. [22] proposed an empirical expression of dynamic stiffness considering the effects of the initial damage level, mean load, load amplitude, and the number of load cycles. Even though this equation of dynamic stiffness includes many factors, it is not easy to include all these factors in the mooring analysis. Therefore, the equation of dynamic stiffness of the fiber ropes has to be simplified. Davies et al. [7] proposed the equation of dynamic stiffness, which includes only the main factor, i.e., the mean tension. In addition, the damage levels of the fiber ropes significantly affect the dynamic stiffness [22]. Hence, in this study, we propose an equation of the dynamic stiffness, which includes the effects from the damage level and the mean tension of the fiber ropes, as shown in Eq. (2)
$Kr=δ(1−ϕ)ω+β(1−ϕ)ψTm$
(2)
where Kr is the dimensionless dynamic stiffness; ϕ is the initial damage value of the fiber ropes, which changes from 0 to 1; Tm is the mean tension of the fiber ropes; and δ, ω, β, ψ are the coefficients related to the material and the structure of the fiber ropes. The term $δ(1−ϕ)ω$ reflects the initial dynamic stiffness of the fiber rope, whether it is damaged or not. It is important to add the damage effect to the term $δ(1−ϕ)ω$, because the degree of damage will affect the initial state of the mooring lines. The term $β(1−ϕ)ψ$ indicates the effects of both the damage level and the mean load level on the dynamic stiffness. To determine the coefficients in Eq. (2), the experimental data of 6-mm-diameter polyester and HMPE ropes with different initial damage levels presented by Liu et al. [22] are used. The experimental values of the polyester ropes for load conditions (mean tension $Tm=20%MBS$ and $40%MBS$, strain amplitude $εa=0.16%$, and cycle number N = 700–800th) are listed in Table 4. The experimental data of the HMPE ropes for load conditions (mean tension $Tm=20%MBS$ and $40%MBS$, strain amplitude $εa=0.13%$, and cycle number N = 700–800th) are listed in Table 5. The matlab function, nlinfit, which can solve the nonlinear least-square problems, is used to analyze the experimental data in Tables 4 and 5. Then, the coefficients of the HMPE and the polyester ropes are obtained and listed in Table 6. The numerical results of the dynamic stiffness are calculated using Eq. (2), as listed in Tables 4 and 5. Error analysis of the dynamic stiffness between the experimental data and numerical results is performed. The average relative errors of all cases for the polyester and the HMPE ropes are 3.23% and 2.70%, respectively. These results indicate that generally a good agreement between the experimental data and the numerical results is obtained using the empirical equation.
Table 4

Test data of 6 mm polyester ropes for load conditions (strain amplitude ɛa = 0.16% and cycle number N = 700–800th)

CaseInitial damage ϕ (%)Tm (% MBS)Kr (Exp.)Kr (Num.)Relative error (%)
102018.519.34.2
24023.523.92.1
313.332016.617.02.7
44020.321.14.1
526.672014.314.73.1
64017.618.23.7
7402011.912.44.0
84015.015.32.1
CaseInitial damage ϕ (%)Tm (% MBS)Kr (Exp.)Kr (Num.)Relative error (%)
102018.519.34.2
24023.523.92.1
313.332016.617.02.7
44020.321.14.1
526.672014.314.73.1
64017.618.23.7
7402011.912.44.0
84015.015.32.1
Table 5

Test data of 6 mm HMPE ropes for load conditions (strain amplitude ɛa = 0.13% and cycle number N = 700–800th)

CaseDamage (%)Initial damage ϕ (%)Tm (% MBS)Kr (Num.)Relative error (%)
102065.165.20.3
24073.776.74.2
38.332058.561.24.6
44067.970.53.8
516.672057.957.31.0
64062.464.73.6
733.332048.049.94.1
84053.254.21.9
CaseDamage (%)Initial damage ϕ (%)Tm (% MBS)Kr (Num.)Relative error (%)
102065.165.20.3
24073.776.74.2
38.332058.561.24.6
44067.970.53.8
516.672057.957.31.0
64062.464.73.6
733.332048.049.94.1
84053.254.21.9
Table 6

Coefficients in the empirical expression of the dynamic stiffness for the fiber ropes

Materialδωβψ
Polyester14.66400.86030.23200.9002
HMPE53.69360.39430.57702.4786
Materialδωβψ
Polyester14.66400.86030.23200.9002
HMPE53.69360.39430.57702.4786

## 3 Environmental Conditions

To investigate the effect of damaged lines on dynamic response of the hybrid taut-wire mooring systems, a representative extreme environmental load of combined wind–wave–current forces is used in the area of South China Sea S4, as shown in Table 7 [23,32]. The wave and wind condition is for a 100-year return period [32]. A constant profile for the current condition is for a 10-year return period [32]. In addition, to investigate the effect of one-line failure on the fatigue life of the adjacent lines, the long-term environmental conditions in the area of Nanhai S4 of China are selected and represented by 64 short-term environmental states. The corresponding environmental parameters are listed in Table 8 [23]. It should be noted that the wave condition is given by the JONSWAP spectrum with the peak enhancement factor γ = 2.5, the wind load is generated using the API wind spectrum, and the current is represented with the constant current spectrum [33]. In Tables 7 and 8, HS, TS, VW, VC, and P denote the significant wave height, the peak period, the mean wind velocity, the mean current velocity, and the occurrence probability of each sea state, respectively. In the present study, the analysis is performed only for the collinear direction of the wave, wind, and current case. The collinear direction is selected as $90deg$, which is measured counterclockwise from the East, as shown in Fig. 4. The reason that all loads are applied in the same direction is based on the results of the following studies. In the fatigue analysis example of the API-recommended practice 2SK [34], the wind, wave, and current are assumed colinear. In addition, Ahmed et al. [25,35] investigated the response of a truss spar under the unidirectional waves, current, and wind loads based on the observation that waves are accompanied by the current and the wind in practical situations.

Table 7

The typical 100-year return period of extreme environmental conditions at South China Sea

ParametersHS (m)TS (s)VW (m/s)VC (m/s)
Value13.615.156.32.05
ParametersHS (m)TS (s)VW (m/s)VC (m/s)
Value13.615.156.32.05
Table 8

Parameters of the 64 short-term sea states

Sea stateHS (m)TS (s)VW (m/s)VC (m/s)P (%)Sea stateHS (m)TS (s)VW (m/s)VC (m/s)P (%)
10.253.53.40.1020.120.2543.80.1141
30.67545.60.1680.841.0547.00.210.7
51.5548.50.2550.562.175410.10.3030.2
70.2554.20.1262.980.67556.00.183.1
91.0557.60.2283.4101.5559.00.273.7
112.175510.80.3241.9122.875511.60.3480.7
133.625513.00.390.1140.2564.30.1292.9
150.67566.40.1923.7161.0567.80.2345
171.5569.40.2827.2182.175611.20.3365.5
192.875612.00.363.2203.625613.20.3961.1
214.5614.50.4350.2220.2574.50.1351.5
230.67576.60.1982.1241.0578.20.2463.3
251.5579.80.2945.7262.175711.60.3485.7
272.875712.40.3724.6283.625713.70.4112.4
294.5715.00.451.3305.5716.10.4830.3
310.2584.80.1440.5320.67586.80.2040.8
331.0588.40.2521.3341.55810.00.32.5
352.175812.00.363362.875812.80.3843.1
373.625814.00.422.2384.5815.40.4621.7
395.5816.50.4950.6406.75818.20.520.2
410.2595.00.150.1420.67597.00.210.2
431.0598.60.2580.4441.55910.60.3180.8
452.175912.60.3781462.875913.20.3961.2
473.625914.80.4441484.5916.00.481
495.5916.80.5040.6506.75918.00.540.4
510.675107.50.2250.1521.05109.20.2760.1
531.551012.00.360.2542.1751013.40.4020.2
552.8751014.00.420.3563.6251015.60.4680.3
574.51016.70.5010.3585.51017.40.5220.2
596.751019.10.5730.3602.8751115.20.4560.1
613.6251116.40.4920.1624.51117.20.5160.1
635.51118.00.540.2643.62512200.60.1
Sea stateHS (m)TS (s)VW (m/s)VC (m/s)P (%)Sea stateHS (m)TS (s)VW (m/s)VC (m/s)P (%)
10.253.53.40.1020.120.2543.80.1141
30.67545.60.1680.841.0547.00.210.7
51.5548.50.2550.562.175410.10.3030.2
70.2554.20.1262.980.67556.00.183.1
91.0557.60.2283.4101.5559.00.273.7
112.175510.80.3241.9122.875511.60.3480.7
133.625513.00.390.1140.2564.30.1292.9
150.67566.40.1923.7161.0567.80.2345
171.5569.40.2827.2182.175611.20.3365.5
192.875612.00.363.2203.625613.20.3961.1
214.5614.50.4350.2220.2574.50.1351.5
230.67576.60.1982.1241.0578.20.2463.3
251.5579.80.2945.7262.175711.60.3485.7
272.875712.40.3724.6283.625713.70.4112.4
294.5715.00.451.3305.5716.10.4830.3
310.2584.80.1440.5320.67586.80.2040.8
331.0588.40.2521.3341.55810.00.32.5
352.175812.00.363362.875812.80.3843.1
373.625814.00.422.2384.5815.40.4621.7
395.5816.50.4950.6406.75818.20.520.2
410.2595.00.150.1420.67597.00.210.2
431.0598.60.2580.4441.55910.60.3180.8
452.175912.60.3781462.875913.20.3961.2
473.625914.80.4441484.5916.00.481
495.5916.80.5040.6506.75918.00.540.4
510.675107.50.2250.1521.05109.20.2760.1
531.551012.00.360.2542.1751013.40.4020.2
552.8751014.00.420.3563.6251015.60.4680.3
574.51016.70.5010.3585.51017.40.5220.2
596.751019.10.5730.3602.8751115.20.4560.1
613.6251116.40.4920.1624.51117.20.5160.1
635.51118.00.540.2643.62512200.60.1

## 4 Dynamic and Fatigue Analysis

### 4.1 Time-Domain Dynamic Analysis.

For the water depth exceeding 100 m, a dynamic analysis of the mooring system should be performed [32]. The present dynamic analysis is based on the time-domain dynamic theory (Cummins' equation) [36]. The governing equation is
$[M+Ma∞]{x¨}−∫−∞tx˙(τ)R(t−τ)dτ+[B]{x˙}+[K]{x}=[F]$
(3)
where ${x¨}$, ${x˙}$, and {x} represent the motion acceleration, velocity, and displacement of a floating platform, respectively, M is the structure mass matrix, $Ma∞$ is the added mass matrix at the infinite frequency, R(tτ) is the retardation function, [K] includes both hydrostatic stiffness and stiffness from the mooring lines, [B] is the damping matrix, and [F] is the load vector, including the wave, wind, current, damping, and mooring forces. In the present study, the time-domain response of the moored floating structures under environmental condition is obtained using the ariane 7.0 software, which is based on Cummins' equation. To solve Eq. (3), all the hydrodynamic coefficients including added mass matrix, radiation damping matrix, and the RAOs of the FPSO are obtained by using hydrostar [27], which requires the input of the geometry of the FPSO, as described in Sec. 2.1. The analysis procedure mainly includes creating the mesh model of the FPSO, calculating the hydrodynamic coefficients according to the potential theory, and using the ariane 7.0 software to calculate the response of the moored FPSO. Equation (3) is based on Cummins' equation and the hydrodynamic coefficients, and the forces are derived from the linear potential flow theory. The hypothesis of the potential flow theory is that the external flow around bodies is inviscid and irrotational. Hence, the flow potential functions satisfy the Laplace equation. The limitation of the potential flow theory is that the viscous and rotational effects are neglected.

### 4.2 Fatigue Analysis.

To estimate the fatigue life of the mooring systems, a linear damage accumulation rule (Palmgren-Miner), T-N curves, the rainflow cycle counting method, and time-domain dynamic theory are employed. This fatigue analysis method is used because the rainflow counting method, first proposed by Matsuishi and Endo [37], has been regarded as the classic method to estimate the fatigue life of structure elements [38]. The method is based on the counting of peaks and valleys, which is similar to the rain flow falling on a pagoda and running down the edges of the roof, to determine the equivalent constant amplitude cycles [39]. This fatigue analysis method has been widely applied in the designs of the marine structures [34,4042].

The fatigue life of the mooring system is given as [41] follows:
$L=1/(αD)$
(4)
where L is the fatigue life and α is the fatigue safety factor, which is recommended to be 10 for fiber ropes [41]. D is the total annual accumulated fatigue damage, which is calculated as follows:
$D=∑jDj=∑j∑kpj3.15576×107djnjkNk$
(5)
where Dj is the annual fatigue damage under the environmental state j, pj is the occurrence probability of the environmental state j (the sum of the probabilities of all the selected environmental conditions should be equal to 1), dj is the dynamic simulation duration under the environmental state j, njk is the number of cycles within the tension range R under the environmental state j, and Nk is the number of cycles to failure at the normalized tension range R, according to the selected T-N curve of the mooring lines. Here, the cycle number, njk, and the corresponding range, R, are obtained using the rainflow counting method to analyze the load time history. In addition, Nk depends on the T-N curve of the mooring lines, which is given as follows [34,4143]:
$(Nk)RG=K$
(6)
where R is the ratio of tension range to minimum breaking strength, G and K are the coefficients of the fiber ropes, which is obtained from the results of the cyclic tests. In addition, the annual fatigue damage is calculated using the linear damage accumulation rule of Palmgren-Miner.

The detailed calculation procedure for the fatigue analysis is delineated as follows. First, the long-term environmental events are represented as discrete cases. Second, a time-domain dynamic analysis of the mooring system is performed to obtain the tension time history of the mooring lines of each case. Third, the annual fatigue damage, Dj, is calculated using the rainflow method and T-N curve. Finally, the total annual fatigue damage, D, and the fatigue life, L, are obtained using the Palmgren-Miner rule.

## 5 Analytical Cases

In the present study, a case of using undamaged mooring lines is selected as the reference case in designing a representative analysis set. Based on the method described above, the response of the intact mooring system under the environmental loads from Table 7 is calculated. Here, the collinear direction of the wave, wind, and current is defined as $90deg$, which is measured counterclockwise from the East, as shown in Fig. 4. Hence, Lines 4–6 are on the windward side of the FPSO and Lines 10–12 are on the leeward side of the FPSO. The axial tensions of the undamaged mooring lines at the fairlead are obtained, as shown in Table 9. As indicated in the table, the maximum load and the load range of the windward Line 5 is larger than those of the other mooring lines. These results are consistent with those specified in the guideline of API-RP 2SM [41], which shows that the windward lines withstand higher environmental loads than the leeward lines. According to the tension results of the intact mooring system, the windward Line 5 is selected as the reference line to design the analytical cases. Hence, nine cases are designed to investigate the effect of the damaged lines on dynamic response of the hybrid taut-wire mooring system, as shown in Table 10. Case 1 is the reference case of the intact mooring system. Cases 2–4 are designed to investigate the dynamic response of the mooring system with the initial damage level of Line 5 increasing until failure. Case 5 is used to investigate the dynamic response of the mooring system with the failure of Line 5, and 30% of the initial damage level of Line 4 occurs. Case 6 is designed to investigate the dynamic response of the mooring system considering the failures of both Lines 4 and 5. Case 7 examines the dynamic response of the mooring systems considering the effects of the damaged windward Lines 4–6. Case 8 examines the dynamic response of mooring systems under the effects of the damaged leeward Lines 10–12. Case 9 examines the dynamic response of the mooring system when all the lines suffer 5% damage. In addition, to investigate the effect of a failed windward line on the fatigue life of the remaining windward rope, which is adjacent to the failed one, fatigue analyses of Line 6 in Cases 1 and 4 are performed.

Table 9

Axial tensions of the intact mooring lines at the fairlead for Case 1

LineMean (% MBS)Standard deviation (% MBS)Maximum (% MBS)Minimum (% MBS)Load range (% MBS)
119.781.7726.6413.2913.35
219.951.7627.0113.8013.21
320.121.7827.3914.3113.08
421.983.3535.9410.2325.71
521.983.3635.9910.2125.78
621.973.3535.9410.2325.71
720.111.7827.3714.2813.09
819.941.7626.9913.7613.23
919.771.7826.6213.2713.35
1018.073.0831.039.0122.02
1118.073.0831.058.9922.06
1218.073.0731.039.0122.02
LineMean (% MBS)Standard deviation (% MBS)Maximum (% MBS)Minimum (% MBS)Load range (% MBS)
119.781.7726.6413.2913.35
219.951.7627.0113.8013.21
320.121.7827.3914.3113.08
421.983.3535.9410.2325.71
521.983.3635.9910.2125.78
621.973.3535.9410.2325.71
720.111.7827.3714.2813.09
819.941.7626.9913.7613.23
919.771.7826.6213.2713.35
1018.073.0831.039.0122.02
1118.073.0831.058.9922.06
1218.073.0731.039.0122.02
Table 10

Analytical cases considering different initial damage values

CaseCondition
1Intact
220% Initial damage $(ϕ=20%)$ of Line 5
330% Initial damage $(ϕ=30%)$ of Line 5
4Failure $(ϕ=100%)$ of Line 5
530% Initial damage $(ϕ=30%)$ of Line 4 and failure $(ϕ=100%)$ of Line 5
6Failures $(ϕ=100%)$ of Lines 4 and 5
720% Initial damage $(ϕ=20%)$ of Lines 4–6
820% Initial damage $(ϕ=20%)$ of Lines 10–12
95% Initial damage $(ϕ=5%)$ of all lines
CaseCondition
1Intact
220% Initial damage $(ϕ=20%)$ of Line 5
330% Initial damage $(ϕ=30%)$ of Line 5
4Failure $(ϕ=100%)$ of Line 5
530% Initial damage $(ϕ=30%)$ of Line 4 and failure $(ϕ=100%)$ of Line 5
6Failures $(ϕ=100%)$ of Lines 4 and 5
720% Initial damage $(ϕ=20%)$ of Lines 4–6
820% Initial damage $(ϕ=20%)$ of Lines 10–12
95% Initial damage $(ϕ=5%)$ of all lines

## 6 Results and Discussion

### 6.1 Results of Time-Domain Analysis

#### 6.1.1 Horizontal Offsets of the FPSO.

Based on the dynamic analysis method presented above, the maximum horizontal offsets of the FPSO for every case are calculated and presented in Table 11. It can be observed from Table 11 that the maximum offset of the FPSO changes with the damage levels of the mooring lines under the same environmental loads from Table 7. Cases 2–4 in Table 11 indicate that with the damage to Line 5 increasing, the maximum horizontal offset of the FPSO increases accordingly. Specifically, once the failure of Line 5 occurs, the maximum horizontal offset of the FPSO significantly increases from 23.37 m to 31.86 m. Case 5 in Table 11 indicates that the 30% damage level of Line 4 adjacent to the failed one leads to ((38.85–22.42)/22.42) 73.28% increase compared with Case 1. When Line 4 also fails as shown in Case 6, the increase in the maximum horizontal offset is quite significant, i.e., 185.28% of the Case 1 offset. When the windward Lines 4–6 are at 20.0% damage level (Case 7), the increase in the maximum offset is ((26.90–22.42)/22.42) 19.98% of the Case 1 offset. When the leeward Lines 10–12 are at 20.0% damage (Case 8), the increase in the maximum offset is 17.71% of the Case 1 offset. A comparison of the horizontal offset between Cases 7 and 8 shows that the maximum horizontal offset of Case 7 is slightly larger than that of Case 8, as shown in Fig. 5. Comparing Case 9 with Case 1 in Table 11, the uniform 5.0% damage to all lines causes a 1.63-m increase in the maximum horizontal offset.

Fig. 5
Fig. 5
Close modal
Table 11

Maximum offsets of the FPSO under different initial damage levels

Case (condition)Offset heading of the FPSO (deg)Maximum offset of the FPSO (m)Increase in offset compared with the intact one (%)
1 (Intact)0.022.420.00
2 (ϕ = 20% of Line 5)0.00923.374.24
3 (ϕ = 30% of Line 5)0.01124.449.01
4 (Failure of Line 5)0.03431.8642.11
5 (ϕ = 30% of Line 4 and failure of Line 5)0.18638.8573.28
6 (Failures of both Lines 4 and 5)0.48663.96185.28
7 (ϕ = 20% of Lines 4–6)0.01626.9019.98
8 (ϕ = 20% of Lines 10–12)0.01326.3917.71
9 (ϕ = 5% of all lines)0.00524.057.27
Case (condition)Offset heading of the FPSO (deg)Maximum offset of the FPSO (m)Increase in offset compared with the intact one (%)
1 (Intact)0.022.420.00
2 (ϕ = 20% of Line 5)0.00923.374.24
3 (ϕ = 30% of Line 5)0.01124.449.01
4 (Failure of Line 5)0.03431.8642.11
5 (ϕ = 30% of Line 4 and failure of Line 5)0.18638.8573.28
6 (Failures of both Lines 4 and 5)0.48663.96185.28
7 (ϕ = 20% of Lines 4–6)0.01626.9019.98
8 (ϕ = 20% of Lines 10–12)0.01326.3917.71
9 (ϕ = 5% of all lines)0.00524.057.27

These results show that the maximum horizontal offset of the FPSO increases with the increasing damage levels of the windward mooring lines because the damage causes the degradation of dynamic stiffness of the fiber ropes. Furthermore, with the stiffness degradation of the fiber ropes, the entire stiffness of the mooring system decreases. Particularly, when the failure of one windward line occurs, the increase in the maximum horizontal offset is about 42.11% of the Case 1 offset. If the failure of two windward lines occurs, the increase in the maximum horizontal offset is about four times the offset of the case of the one-line failure. This indicates that the offset of the FPSO is sensitive to the failure of the windward lines. Hence, capturing and monitoring the offset of the floating structures may help to assess the integrity of the mooring systems. In addition, Table 11 also shows that the offset heading of the FPSO is affected by the damaged mooring lines. From the results of Case 5 in Table 11, it can be observed that the damage level of Case 5 (30% damage level of Line 4 and the failure of Line 5) leads to an increase of 0.186 deg compared with that of Case 1. This indicates that when the damaged lines changes the symmetry of the mooring arrangement under the unidirectional environmental loads, the offset heading increases. Moreover, the abrupt change in heading due to the sudden mooring line failure may increase the roll motion of the FPSO and affect the operational safety [44]. Hence, the present study helps to understand the effect of the damaged mooring line on the offset and heading of the FPSO.

#### 6.1.2 Tension Response of the Mooring Lines.

In this section, the line tensions at fairleads of the mooring ropes are reported. In Tables 9 and 12, the MBS is 19600 kN for the polyester and the HMPE mooring ropes. Table 9 shows the line tensions of the intact mooring system. In addition, Table 12 shows the maximum tension and tension range values of Lines 5 and 6 for each case. It is observed that when Line 5 is at 20% damage (Case 2), the decrease in the maximum tension of Line 5 is (35.99–33.59) 2.40% MBS, compared with Case 1. In terms of tension response of Line 5, the difference between Cases 1 and 2 is small. When Line 5 suffers 30% damage (Case 3), the decrease in the maximum tension of Line 5 is (35.99–32.53) 3.46% MBS. The tension response of Line 5 in Case 3 is similar to that in Case 2. When failure of Line 5 occurs (Case 4), the increase in the maximum tension of Line 6 is (38.68–35.74) 2.94% MBS, compared with Case 1. The tension response of Line 6 for Cases 1 and 4 are presented in Fig. 6. When the failure of Line 5 occurs and Line 4 incurs 30% damage (Case 5), the increase in the maximum tension of Line 6 is (42.67–38.68) 3.99% MBS, compared with Case 4. This shows that the increase in the damage level of Line 4 causes the maximum tension of Line 6 to increase. When both Lines 4 and 5 fail (Case 6), the increase in the maximum tension of Line 6 is (56.79–35.74) 21.05% MBS compared with Case 1, which is a noticeable increase, as shown in Fig. 7. Furthermore, to analyze the sensitivity of the tension response of Lines 5 and 6 to the damage levels of mooring ropes, the maximum tensions of both Lines 5 and 6 at the fairlead from Cases 1 to 6 are shown in Fig. 8. This figure shows that the maximum tension of Line 5 decreases with the increasing damage level of Line 5. At the same time, the maximum tension of Line 6 increases with the increasing damage levels of both Lines 4 and 5. In addition, the tension ranges of both Lines 5 and 6 have a similar trend as their maximum tension, as shown in Fig. 9, which also shows that the tension ranges of Line 6 increase with the increasing damage levels of both Lines 4 and 5.

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal
Table 12

Axial tensions of the mooring Lines 5 and 6 at the fairlead

Case (condition)Line 5 maximum (% MBS)Line 5 range (% MBS)Line 6 maximum (% MBS)Line 6 range (% MBS)
1 (Intact)35.9925.7835.7425.51
2 (ϕ = 20% of Line 5)33.5922.2435.7625.57
3 (ϕ = 30% of Line 5)32.5320.5735.8425.80
4 (Failure of Line 5)38.6826.42
5 (ϕ = 30% of Line 4 and failure of Line 5)42.6731.95
6 (Failures of both Lines 4 and 5)56.7942.14
7 (ϕ = 20% of Lines 4–6)35.4923.8635.4523.8
8 (ϕ = 20% of Lines 10–12)38.1528.138.1128.04
9 (ϕ = 5% of all lines)35.3325.0235.2824.95
Case (condition)Line 5 maximum (% MBS)Line 5 range (% MBS)Line 6 maximum (% MBS)Line 6 range (% MBS)
1 (Intact)35.9925.7835.7425.51
2 (ϕ = 20% of Line 5)33.5922.2435.7625.57
3 (ϕ = 30% of Line 5)32.5320.5735.8425.80
4 (Failure of Line 5)38.6826.42
5 (ϕ = 30% of Line 4 and failure of Line 5)42.6731.95
6 (Failures of both Lines 4 and 5)56.7942.14
7 (ϕ = 20% of Lines 4–6)35.4923.8635.4523.8
8 (ϕ = 20% of Lines 10–12)38.1528.138.1128.04
9 (ϕ = 5% of all lines)35.3325.0235.2824.95

When all the windward Lines (4–6) are at 20% damage (Case 7), the decrease in the maximum tension of windward Line 5 is (35.99–35.49) 0.50% MBS, while the increase in the maximum tension of the leeward Line 11 is (32.04–31.05) 0.99% MBS compared with Case 1. Similarly, when all the leeward Lines (10–12) are at 20% damage (Case 8), the increase in the maximum tension of windward Line 5 is (38.15–35.99) 2.16% MBS, while the decrease in the maximum tension of the leeward Line 11 is (31.05–30.17) 0.88% MBS compared with the intact mooring (Case 1). When the uniform 5% damage to all lines (Case 9) occurs, the tension response of Line 5 for Cases 1 and 9 is similar, but Case 9 results in (35.99–35.33) 0.66% MBS decrease in the maximum tension of Line 5, compared with Case 1.

Observing that the line with the maximum load is usually the windward mooring line, therefore, the windward line should receive more attention. The damage to the windward line over time can lead to an increase in the broken danger. If the damage to a single mooring line is not detected, the tension of the remaining lines will be increased, which may cause the line to fail in the next instance of severe weather. Even though an FPSO is designed to cope with the failure of a single mooring line, in the event of subsequent line failures, the increased loads tend to cause additional lines to fail. The failure of multiple lines may result in an FPSO breaking away from the moorings and lead to catastrophic failures. The present study helps to understand the effects of the damage levels of the fiber ropes on the performance of the mooring system.

### 6.2 Results of the Fatigue Analysis.

The above results of the tension response show that the maximum tension of the remaining line, adjacent to the failed one, significantly increases. Line 6, which is adjacent to the failed Line 5, is the line carrying the maximum load and is thus selected as the line on which fatigue analysis is performed. By comparing the fatigue life of Line 6 between Cases 1 and 4, the effect of failure of Line 5 on the fatigue life of Line 6 can be obtained. The tension time curves of Line 6 for Cases 1 and 6 under the 64 short-term environmental loads are utilized to obtain the damage levels of the mooring lines based on the rainflow method and T-N curves. A T-N curve of the polyester ropes is obtained from the guidance note of API in which G is 9 and K is 7.5 [42]. However, the T-N curve of the HMPE ropes has not yet been suggested in guidance notes of ocean engineering [34,4143]. Only the DSM Dyneema Company has published some experimental fatigue data of the HMPE ropes [45]. Using these data of the HMPE ropes, the parameters of T-N curve were computed and found that G and K are equal to 1.26 and 28,704,210.73, respectively.

Based on the method described above, the fatigue damage and fatigue life of Line 6, which is adjacent to Line 5, are obtained for Cases 1 and 4, as listed in Table 13. This shows that when the failure of Line 5 (Case 4) occurs, the decrease in the fatigue life of the polyester rope and the HMPE rope in Line 6 is about 50.74% and 19.90% of the Case 1 fatigue life, respectively. It can be concluded that a mooring line failure leads to the change of the mooring configuration, which could increase the maximum load and load range in the remaining lines adjacent to the failed one. Furthermore, the increasing tension range of the line adjacent to the failed one induces more fatigue damage. The results presented in this study help to understand the effect of the failed windward mooring line on the fatigue life of the remaining lines, adjacent to the failed one, in the hybrid mooring system.

Table 13

Comparison of fatigue life of Line 6 for Cases 1 and 4

CaseMain components of mooring Line 6Annual fatigue damageFatigue life (year)
Case 1Polyester1.05 × 10−12$9.52×1010$
HMPE2.06 × 10−5$4.85×103$
Case 4Polyester2.13 × 10−12$4.69×1010$
HMPE2.47 × 10−5$4.05×103$
CaseMain components of mooring Line 6Annual fatigue damageFatigue life (year)
Case 1Polyester1.05 × 10−12$9.52×1010$
HMPE2.06 × 10−5$4.85×103$
Case 4Polyester2.13 × 10−12$4.69×1010$
HMPE2.47 × 10−5$4.05×103$

## 7 Concluding Remarks

Over the service life of the taut-wire mooring systems, rope damage and mooring incidents have been occurring at a high rate. The dynamic response of the mooring systems with the damaged or failed lines has been of high concern to the offshore engineers and platform operators. Hence, it is important to investigate the effects of damaged ropes on the dynamic response of the taut-wire mooring systems.

In this study, based on the dynamic stiffness equation of damaged fiber ropes, the time-domain dynamic theory, the rainflow cycle counting method, and the linear damage accumulation rule of Palmgren-Miner, an analysis is presented to investigate the effects of the damage levels of fiber ropes on the safety performance of a hybrid taut-wire mooring system. The dynamic analyses of the FPSO with the damaged fiber mooring system under the typical extreme environmental loads were performed. The numerical results indicate that the offset of FPSO and the tensions of the remaining windward lines increase with the increasing damage levels of the windward ropes. For example, when the failure of one windward line occurred (Case 4), the increase in the maximum horizontal offset was 42.11% of the offset of the intact system. In addition, the increase in the tension of the remaining windward line was 8.2% of the line tension of the undamaged mooring system. Furthermore, once a windward mooring line fails, the tension range of other windward mooring lines increases. In addition, fatigue analyses of the intact and damaged mooring systems demonstrated that the failure of a windward line causes a change of the mooring configuration, which could remarkably increase the maximum tension, tension range, and the annual fatigue damage levels of the remaining windward lines, adjacent to the failed one. Hence, once a mooring line has serious damage phenomena, the displacement of the floating platform, the maximum tension, and the fatigue life of the damaged line need to be evaluated to ensure safe operation of the mooring system. The results also show that when the damage of all the windward lines occurs, the maximum tension of the windward lines decreases and the maximum tension of the leeward ones increases, and vice versa. These conclusions are valid only for a typical extreme environmental condition. However, the present work provides a method for investigating the effects of damaged fiber ropes on the performance of the hybrid taut-wire mooring system. This study, therefore, helps to ensure safe operation of the mooring system.

## Acknowledgment

The first and second authors greatly appreciate the support of the Oregon State University School of Civil and Construction Engineering for the first author's two-year study.

## Funding Data

• National Natural Science Foundation of China (Grant Nos. 51609079 and 51539008; Funder ID: 10.13039/501100001809).

• Fundamental Research Funds for the Central Universities (Grant No. 2017B11314; Funder ID: 10.13039/501100002942).

• International Postdoctoral Exchange Fellowship Program 2017 (Grant No. 20170013; Funder ID: 10.13039/100011040).

## Nomenclature

• D =

the total annual accumulated fatigue damage

•
• L =

the calculated fatigue life

•
• M =

the structure mass matrix

•
• P =

the occurrence probability of each sea state

•
• R =

the ratio of tension range to MBS (minimum breaking strength)

•
• dj =

the dynamic simulation duration under the environmental state

•
• njk =

the number of cycles within the tension range R under the environmental state j

•
• pj =

the occurrence probability of the environmental state j

•
• Dj =

the annual fatigue damage under the environmental state j

•
• HS =

the significant wave height

•
• Kr =

the dimensionless dynamic stiffness

•
• $Ma∞$ =

the added mass matrix at the infinite frequency

•
• Nk =

the number of cycles to failure at normalized tension range R

•
• Tm =

the mean tension of fiber ropes

•
• TS =

the peak wave period

•
• VC =

the mean current velocity

•
• VW =

the mean wind velocity

•
• G, K =

the coefficients of T-N curves of fiber ropes

•
• MBS =

minimum breaking strength of mooring lines (kN)

•
• R(tτ) =

the retardation function

•
• {x} =

the motion displacement of floating platform

•
• ${x¨}$ =

the motion acceleration of floating platform

•
• ${x˙}$ =

the motion velocity of floating platform

•
• [B] =

the damping matrix

•
• [F] =

the loads, including the wave, wind, current, damping, and mooring forces

•
• [K] =

the stiffness matrix including hydrostatic stiffness and stiffness from mooring lines

•
• α =

the fatigue safety factor

•
• δ, ω, β, ψ =

coefficients of dynamic stiffness of damaged fiber ropes

•
• ϕ =

the initial damage value of fiber ropes

## References

1.
Del Vecchio
,
C. J. M.
,
1992
, “
Light Weight Materials for Deep Water Moorings
,” Ph.D. thesis,
.
2.
Ma
,
K. T.
,
Duggal
,
A.
,
Smedley
,
P.
,
L’Hostis
,
D.
, and
Shu
,
H. B.
,
2013
, “
A Historical Review on Integrity Issues of Permanent Mooring Systems
,”
Proceeding of the 45th Annual Offshore Technology Conference
, OTC 24025.
3.
Liu
,
H. X.
,
Li
,
Z.
, and
Zhang
,
Y. M.
,
2015
, “
An Overview of Positioning Technologies for Large Floating Structures in Deep Waters
,”
Proceedings of the International Conference on Underwater Science, Technology and Education
,
Hong Kong
,
Aug. 20
, pp.
46
51
.
4.
Lian
,
Y. S.
,
Zheng
,
J. H.
,
Liu
,
H. X.
, and
Xu
,
P. F.
,
2018
, “
A Study of the Creep-Rupture Behavior of HMPE Ropes Using Viscoelastic-Viscoplastic-Viscodamage Modeling
,”
Ocean Eng.
,
162
(
8
), pp.
43
54
. 10.1016/j.oceaneng.2018.05.003
5.
Banfield
,
S.
, and
Casey
,
N.
,
1998
, “
Evaluation of Fiber Rope Properties for Offshore Mooring
,”
Ocean Eng.
,
25
(
10
), pp.
861
879
. 10.1016/S0029-8018(97)10017-8
6.
Petruska
,
D. J.
,
Kelly
,
P.
,
Stone
,
B.
,
Ahjem
,
V.
,
Zimmerman
,
E. H.
,
Garrity
,
R.
, and
Veselis
,
T.
,
2010
, “
Fiber Moorings, Recent Experiences and Research: Updating API RP 2SM on Synthetic Fiber Rope for Offshore Moorings
,”
Proceedings of the 42nd Annual Offshore Technology Conference
, OTC 20836.
7.
Davies
,
P.
,
Francois
,
M.
,
Grosjean
,
F.
,
Baron
,
P.
,
Salomon
,
K.
, and
Trassoudaine
,
D.
,
2002
, “
Synthetic Mooring Lines for Depths to 3000 Meters
,”
Proceedings of the 34th Offshore Technology Conference
,
Houston, TX
, OTC 14246.
8.
Chi
,
C. H.
,
Lundhild
,
E. M.
,
Veselis
,
T.
, and
Huntley
,
M. B.
,
2009
, “
Enabling Ultra-Deepwater Mooring With Aramid Fiber Rope Technology
,”
Proceedings of the 41st Annual Offshore Technology Conference
,
Houston, TX
, OTC 20074.
9.
Vlasblom
,
M. P.
,
Boesten
,
J.
,
Leite
,
S.
, and
Davies
,
P.
,
2012
, “
Development of HMPE Fiber for Permanent Deepwater Offshore Mooring
,”
Proceedings of the 44th Offshore Technology Conference
,
Houston, TX
, OTC 23333.
10.
Lian
,
Y. S.
,
Liu
,
H. X.
,
Huang
,
W.
, and
Li
,
L. A.
,
2015
, “
A Creep-Rupture Model of Synthetic Fiber Ropes for Deepwater Moorings Based on Thermodynamics
,”
Appl. Ocean Res.
,
52
(
8
), pp.
234
244
. 10.1016/j.apor.2015.06.009
11.
Lian
,
Y. S.
,
Zheng
,
J. H.
, and
Liu
,
H. X.
,
2018
, “
An Investigation on Creep and Creep-Rupture Behaviors of HMPE Ropes
,”
ASME J. Offshore Mech. Arct. Eng.
,
140
(
2
), p.
021401
. 10.1115/1.4038345
12.
Garrity
,
R.
, and
Fronzaglia
,
W.
,
2008
, “
The Use of HMPE Mooring Lines in Deepwater MODU Mooring Systems
,”
Proceedings of the OCEANS’08 MTS/IEEE Conference
,
,
Sept. 15–18
, pp.
620
624
.
13.
Leite
,
S.
, and
Boesten
,
J.
,
2011
, “
HMPE Mooring Lines for Deepwater MODUs
,”
Proceedings of the 43rd Offshore Technology Conference
,
Houston, TX
, Paper No. 22486.
14.
Yuan
,
Z. M.
,
Incecik
,
A.
, and
Ji
,
C.
,
2014
, “
Numerical Study on a Hybrid Mooring System With Clump Weights and Buoys
,”
Ocean Eng.
,
88
(
1
), pp.
1
11
. 10.1016/j.oceaneng.2014.06.002
15.
Lian
,
Y. S.
,
Liu
,
H. X.
, and
Hu
,
L. M.
,
2015
, “
Feasibility Analysis of a New Hybrid Mooring System Applied for Deep Waters
,”
Proceedings of the 25th International Offshore and Polar Engineering Conference
,
Kailua-Kona
, Paper No. ISOPE-I-15-297.
16.
Lugsdin
,
A.
,
2012
, “
Real-Time Monitoring of FPSO Mooring Lines, Risers
,”
Sea Technol.
,
53
(
7
), pp.
21
23
.
17.
Gordon
,
R. B.
,
Brown
,
M. G.
, and
Allen
,
E. M.
,
2014
, “
Mooring Integrity Management: A State-of-the Art Review
,”
Proceedings of 46th Offshore Technology Conference
,
Houston, TX
, Paper No. OTC 25134-MS.
18.
Williams
,
J. G.
,
Miyase
,
A.
,
Li
,
D.
, and
Wang
,
S. S.
,
2002
, “
Small-Scale Testing of Damaged Synthetic Fiber Mooring Ropes
,”
Proceedings of 34th Offshore Technology Conference
,
Houston, TX
, Paper No. OTC 14308MS.
19.
Ward
,
E. G.
,
Ayres
,
R. R.
,
Banfield
,
S. J.
,
O’Hear
,
N.
, and
Laurendine
,
T.
,
2006
, “
The Residual Strength of Damaged Polyester Ropes
,”
Proceedings of 38th Offshore Technology Conference
,
Houston, TX
, Paper No. OTC 18150.
20.
Ward
,
E. G.
,
Ayres
,
R.
,
Banfield
,
S. J.
, and
O’Hear
,
N.
(
2006
). “
Full Scale Experiments on Damaged Polyester Rope
” Technical Report, JIP-FP-1,
Offshore Technology Research Center
.
21.
Flory
,
J.
,
2008
, “
Assessing Strength Loss of Abraded and Damaged Fiber Rope
,”
Proceedings of the OCEANS 2008—MTS/IEEE Conference
,
Kobe, Japan
,
Sept. 15–18
, pp.
126
138
.
22.
Liu
,
H. X.
,
Lian
,
Y. S.
,
Li
,
L. A.
, and
Zhang
,
Y. M.
,
2015
, “
Experimental Investigation on Dynamic Stiffness of Damaged Synthetic Fiber Ropes for Deepwater Moorings
,”
ASME J. Offshore Mech. Arct. Eng.
,
137
(
6
), p.
061401
. 10.1115/1.4031392
23.
Huang
,
W.
,
Liu
,
H. X.
,
Shan
,
G. M.
, and
Hu
,
C.
,
2011
, “
Fatigue Analysis of the Taut-Wire Mooring System Applied for Deep Waters
,”
China Ocean Eng.
,
25
(
3
), pp.
413
426
. 10.1007/s13344-011-0034-5
24.
Qiao
,
D. S.
,
Yan
,
J.
, and
Ou
,
J. P.
,
2014
, “
Fatigue Analysis of Deepwater Hybrid Mooring Line Under Corrosion Effect
,”
Pol. Marit. Res.
,
21
(
3
), pp.
68
76
. 10.2478/pomr-2014-0032
25.
Ahmed
,
M. O.
,
Yenduri
,
A.
, and
Kurian
,
V. J.
,
2016
, “
Evaluation of the Dynamic Responses of Truss Spar Platforms for Various Mooring Configurations With Damaged Lines
,”
Ocean Eng.
,
123
(
1
), pp.
411
421
. 10.1016/j.oceaneng.2016.07.004
26.
Kim
,
M. S.
,
2004
, “
Dynamic Simulation of Polyester Mooring Line
,” Master thesis,
Texas AM University
,
Prairie View, TX
.
27.
Bureau Veritas (BV)
,
2007
,
Theoretical Manual of HydroStar
, 1st ed.,
BV Press
,
Paris
.
28.
Bureau Veritas (BV)
,
2016
,
HydroStar for Experts User Manual
, 3rd ed.,
BV Press
,
Paris
.
29.
Liu
,
H. X.
,
Huang
,
W.
,
Lian
,
Y. S.
, and
Li
,
L. A.
,
2014
, “
An Experimental Investigation on Nonlinear Behaviors of Synthetic Fiber Ropes for Deepwater Moorings Under Cyclic Loading
,”
Appl. Ocean Res.
,
45
, pp.
22
32
. 10.1016/j.apor.2013.12.003
30.
Lian
,
Y. S.
,
Liu
,
H. X.
,
Li
,
L. A.
, and
Zhang
,
Y. M.
,
2017
, “
An Experimental Investigation on Fatigue Behaviors of HMPE Ropes
,”
Ocean Eng.
,
139
, pp.
237
249
. 10.1016/j.oceaneng.2017.05.007
31.
Lian
,
Y. S.
,
Liu
,
H. X.
,
Li
,
L. A.
, and
Zhang
,
Y. M.
,
2018
, “
An Experimental Investigation on the Bedding-in Behavior of Synthetic Fiber Ropes
,”
Ocean Eng.
,
160
, pp.
368
381
. 10.1016/j.oceaneng.2018.04.071
32.
Det Norshke Veritas (DNV)
,
2008
,
Position Mooring Offshore Standard OS-E301
, 1st ed.,
DNV Press
,
Oslo
.
33.
Bureau Veritas (BV)
,
2007
,
Theoretical Manual of Ariane 7
, 1st ed.,
BV Press
,
Paris
.
34.
American Petroleum Institute (API)
,
2015
,
Design and Analysis of Stationkeeping Systems for Floating Structures
, API Recommended Practice 2SK, 3rd ed.,
API Press
,
Washington, DC
.
35.
Ahmed
,
M. O.
,
Yenduri
,
A.
, and
Kurian
,
V. J.
,
2015
, “
Investigation on the Dynamic Responses of a Truss Spar Platform for Different Mooring Line Groups
,”
J. Marine Sci. Appl.
,
14
(
2
), pp.
189
195
. 10.1007/s11804-015-1301-2
36.
Cummins
,
W.
,
1962
,
The Impulse Response Function and Ship Motions
,
David Taylor Model Dasin (DTNSRDC)
,
Bethesda, MD
.
37.
Matsuishi
,
M.
, and
Endo
,
T.
,
1968
, “
Fatigue of Metals Subjected to Varying Stress
,”
Paper presented at Japan Society of Mechanical Engineers
, No.
68
(
2
),
Fukuoka
, pp.
37
40
.
38.
Rychlik
,
I.
,
1987
, “
A New Definition of the Rainflow Cycle Counting Method
,”
Int. J. Fatigue
,
9
(
2
), pp.
119
121
. 10.1016/0142-1123(87)90054-5
39.
Lee
,
Y. L.
, and
Tana
,
T.
,
2011
,
Metal Fatigue Analysis Handbook: Practical Problem-Solving Techniques for Computer-Aided Engineering
,
Butterworth-Heinemann Press
,
Waltham, MA
.
40.
Ragan
,
P.
, and
Manuel
,
L.
,
2007
, “
Comparing Estimates of Wind Turbine Fatigue Loads Using Time-Domain and Spectral Methods
,”
Wind Eng.
,
31
(
2
), pp.
83
99
. 10.1260/030952407781494494
41.
American Petroleum Institute (API)
,
2014
, “Recommended Practice for Design, Manufacture, Installation, and Maintenance of Synthetic Fiber Ropes for Offshore Mooring,”
API Recommended Practice 2SM
, 2nd ed.,
API Press
,
Washington, DC
.
42.
Bureau Veritas (BV)
,
2007
,
Guidance Notes Certification of Fibre Ropes for Deepwater Offshore Services
,
BV Press
,
Paris
.
43.
American Bureau of Shipping (ABS)
,
2011
,
Guidance Notes on the Application of Fiber Rope for Offshore Mooring
,
Houston, TX
.
44.
Paik
,
J. K.
, and
Thayamballi
,
A. K.
,
2007
,
Ship-Shaped Offshore Installations: Design, Building, and Operation
,
Cambridge University Press
,
Cambridge
.
45.
DSM Dyneema
,
2009
,
Technical Brochure: Dyneema in Marine and Industrial Applications
,
DSM Dyneema Press
,
Geleen
.


text:
# naginterfaces.library.lapacklin.ztbcon¶

naginterfaces.library.lapacklin.ztbcon(norm, uplo, diag, kd, ab)[source]

ztbcon estimates the condition number of a complex triangular band matrix.

For full information please refer to the NAG Library document for f07vu

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/f07/f07vuf.html

Parameters
normstr, length 1

Indicates whether or is estimated.

or

is estimated.

is estimated.

uplostr, length 1

Specifies whether is upper or lower triangular.

is upper triangular.

is lower triangular.

diagstr, length 1

Indicates whether is a nonunit or unit triangular matrix.

is a nonunit triangular matrix.

is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be .

kdint

, the number of superdiagonals of the matrix if , or the number of subdiagonals if .

abcomplex, array-like, shape

The triangular band matrix .

Returns
rcondfloat

An estimate of the reciprocal of the condition number of . is set to zero if exact singularity is detected or the estimate underflows. If is less than machine precision, is singular to working precision.

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: , or .

(errno )

On entry, error in parameter .

Constraint: or .

(errno )

On entry, error in parameter .

Constraint: or .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

Notes

ztbcon estimates the condition number of a complex triangular band matrix , in either the -norm or the -norm:

Note that .

Because the condition number is infinite if is singular, the function actually returns an estimate of the reciprocal of the condition number.

The function computes or exactly, and uses Higham’s implementation of Hager’s method (see Higham (1988)) to estimate or .

References

Higham, N J, 1988, FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation, ACM Trans. Math. Software (14), 381–396

format: plain


text:
Vertical ideal spring, streched by weight, different results working with forces and energy [duplicate]

Imagine we have a vertical spring with constant $$k$$ and load on it (like a weighing machine) a mass $$m$$.

Initially, the energy of the system is $$0$$, since the spring it's on it's original position and there's no speed (so kinetic energy equals $$0$$) and we choose potential energy to be null at that point. Once released, the mass reaches a new equilibrium position, where the spring is streched a distance $$-\Delta y$$. At this new point, the mass has no speed (kinetic energy equals $$0$$) and since the spring is streched there's a potential energy $$E_p=(1/2)k(-\Delta y)^2=(1/2)k\Delta y^2$$ and also a gravitational energy $$E_p=mg(-\Delta y) = -mg\Delta y$$, so since energy is conserved $$\Delta y=\frac{2mg}{k}$$

If we apply forces at the new rest point, the object has no acceleration, then no net force and the unique forces acting on it are weight (downwards, $$mg$$) and the elastic force ($$k\Delta y$$, upwards) and this leads us to $$\Delta y = \frac{mg}{k}$$

Both resualts are different so, what I'm missing when working with energies? I know that with forces it's Ok.

I guess that is something related about no conservative or external forces (the force done by my hand for dropping the object over the plate) but I have no confirmation

I know there's some related question like If string is stretched just by weight, where does the gravitational potential energy goes if only half is converted to elastic potential energy? or Inconsistencies in Vertical Spring however I think that both cases are a little different with my case

• How do you think your question differs from the ones you linked? Both of them have the correct answer in them.
– noah
Oct 4 '21 at 15:37
• I don't get why I have to take into account velocity in my case. There's no initial speed and at the end is static. In a spring HSM there's no quasi-statically movement and energies are conserved Oct 4 '21 at 15:39
• – noah
Oct 4 '21 at 16:30
• ”Once released, the mass reaches a new equilibrium position...At this new point, the mass has no speed” If you try this experiment, you’ll find that the mass has a nonzero speed. The only other explanations are that something halted its motion, doing its own work, or that the damping is so high that it can’t be ignored in your conservation of energy calculations. This question has been similarly asked and answered about a dozen times on this site. Oct 4 '21 at 18:39
• Yes, now I just realise and I fell so stupid. I was always thinking with a weight scale and there's should be a mechanism inside that when the spring reaches the maxium strech locks the system (and, therefore, i can't apply conservation of energy). Thank you so much Oct 4 '21 at 19:11

If you release the mass from $$\Delta y = 0$$ and do not stop it at its equilibrium point $$\Delta y = \frac {mg} k$$ then it accelerates until it reaches its equilibrium point and then decelerates until it is momentarily stationary at $$\Delta y = \frac {2mg} k$$ (because it has inertia it cannot stop at its equilibrium point, even though the net force on it here is zero). The mass then repeats the same motion in reverse. If there is no loss of energy then it will oscillate indefinitely between $$\Delta y = 0$$ and $$\Delta y = \frac {2mg} k$$ - this is SHM.
If you want the mass to be stationary at its equilibrium point $$\Delta y = \frac {mg} k$$ then you must remove kinetic energy from the mass to do this, either by bringing it to a hard stop at this point, or by making it fall slowly so that its velocity is always small. In either case energy is removed from the system, so you cannot use a conservation of energy argument.


text:
# Math Help - Calculus help

1. ## Calculus help

I need to find the geodesic between 2 fixed points with respect to the integral of square root of(1+sin^2x(dg/dx)^2)dx where x is the parameter. It says to show that the geodesic has the form cos x=A sin x cos g+B sin x cos g

2. ## Re: Calculus help

Hey sean60630.

What does the g function refer to?

3. ## Re: Calculus help

Misinterpretation.


text:
dream vignette says limma-voom repeated measures two-group comparison needs two duplicateCorrelation rounds which differs from limma user's guide and manual
3
0
Entering edit mode
hermidalc ▴ 10
@hermidalc-21769
Last seen 8 months ago
National Cancer Institute and Universit…

In the dream vignette it's described that for an RNA-seq repeated measure design the limma analysis needs two duplicateCorrelation rounds, though this differs from the limma user's guide and manual. Below for convenience is the design:

> metadata
Individual Disease Experiment Sex DiseaseSubtype
sample_01               ID1       0  sample_01   F              0
sample_02               ID1       0  sample_02   F              0
sample_03               ID2       0  sample_03   M              0
sample_04               ID2       0  sample_04   M              0
sample_05               ID3       0  sample_05   F              0
sample_06               ID3       0  sample_06   F              0
sample_07               ID4       0  sample_07   M              0
sample_08               ID4       0  sample_08   M              0
sample_09               ID5       0  sample_09   F              0
sample_10               ID5       0  sample_10   F              0
sample_11               ID6       0  sample_11   M              0
sample_12               ID6       0  sample_12   M              0
sample_13               ID7       1  sample_13   F              1
sample_14               ID7       1  sample_14   F              1
sample_15               ID8       1  sample_15   M              1
sample_16               ID8       1  sample_16   M              1
sample_17               ID9       1  sample_17   F              1
sample_18               ID9       1  sample_18   F              1
sample_19              ID10       1  sample_19   M              2
sample_20              ID10       1  sample_20   M              2
sample_21              ID11       1  sample_21   F              2
sample_22              ID11       1  sample_22   F              2
sample_23              ID12       1  sample_23   M              2
sample_24              ID12       1  sample_24   M              2

and the limma two-round duplicateCorrelation analysis code from the vignette, where the DE test is an ordinary two-group comparison ~Disease:

# apply duplicateCorrelation is two rounds
design = model.matrix( ~ Disease, metadata)
vobj_tmp = voom( geneExpr, design, plot=FALSE)
dupcor <- duplicateCorrelation(vobj_tmp,design,block=metadata$Individual) # run voom considering the duplicateCorrelation results # in order to compute more accurate precision weights # Otherwise, use the results from the first voom run vobj = voom( geneExpr, design, plot=FALSE, block=metadata$Individual, correlation=dupcor$consensus) # Estimate linear mixed model with a single variance component # Fit the model for each gene, dupcor <- duplicateCorrelation(vobj, design, block=metadata$Individual)

# But this step uses only the genome-wide average for the random effect
fitDupCor <- lmFit(vobj, design, block=metadata$Individual, correlation=dupcor$consensus)

# Fit Empirical Bayes for moderated t-statistics
fitDupCor <- eBayes( fitDupCor )

This differs from the limma user's guide which, for similar designs and DE test questions, describes only one duplicateCorrelation round. Also, the limma manual says that specifying parameters like block and correlation in voom only passes them to lmFit and voom doesn't use it for precision weights calculation. This would also make sense for only needing one round. Which way is the correct way?

I compared the vignette example with two rounds and the limma guide with one round and the p-values are not identical, the one-round has slightly higher p-values and slightly lower log-odds.

2
Entering edit mode
@james-w-macdonald-5106
Last seen 19 hours ago
United States
1
Entering edit mode

Might be a good idea for limma team, when they have the time, to update the relevant areas in the limma user's guide with this new recommended double approach and also in the limma reference manual voom section regarding ... additional parameters, because it made it seem setting parameters like block and correlation would only get them passed to lmFit not actually being used in precision weights calc in voom.

1
Entering edit mode

I can't speak for the limma team and what they will or will not update, but you seem confused.

The fact that they are passed to lmFit, and hence use a linear mixed model instead of a conventional ANOVA is how the block and correlation arguments affect the estimation of precision weights. That's the whole point of the two-step approach (see the post by Bernd Klaus in the thread I reference above). So when the help page says

     ...: other arguments are passed to lmFit.

That's completely accurate. One could argue that there could be other information about why you might want to do the two-step thing when using a linear mixed model, but there's always the tension between adding all possible relevant information in the age of TL;DR, and being concise because people won't read things.

0
Entering edit mode

My apologies yes I found that phrase confusing, since I thought it meant that passing block and correlation options to voom would mean they weren't actually used there and simply passed to the subsequent call of lmFit(v, ...) with the voom result so you wouldn't need to specify block and correlation again. But I see in the examples you linked above you pass block and correlation to both voom and lmFit for the second round so voom is using them not simply passing them.

1
Entering edit mode

Yes. If you want lmFit to use a mixed model you have to supply a block and correlation argument. That's just how it works (e.g., that's how you tell lmFit that you want a mixed model).

The voom function is only used to estimate observation-level weights so you can use conventional linear models (which is what lmFit does), and the two-step procedure is simply done to update the weights to be more amenable to the mixed model rather than based on a fixed effects only model.

It's true that if you have values in the 'weights' component of an EList then lmFit will automatically fit a weighted linear model, so one might assume that paradigm extends to other things like fitting a mixed model because of something that voom might return.

However, what lmFit does is based on what getEAWP provides it, and there is documentation for that function (and the help page for the EList class, which is what voom returns), so there's no need to assume anything - you can just read those help pages along with the one for lmFit and know for sure what's up.

1
Entering edit mode

Leandro, it is a good start that you are reading the manual help page for voom, but you need to read the "Value" section of the page as well. You can see from this section that voom() only returns expression values and weights. It does not return blocks or correlations, so these could not possibly be passed on to any downstream functions. There is no way in R to pass on any information that is not stored in the output object.

0
Entering edit mode

Very sorry, I do usually read the return value description I simply forgot in this case. Thank you for the help!

1
Entering edit mode
@gabrielhoffman-8391
Last seen 5 months ago
United States

When I referred to two rounds of duplicateCorrelation(), I was referring to Gordon Smyth's suggestion rather than the original limma manual. The discrepancy is not ideal.

Since voom() uses lmFit() internally to get the residuals, specifying a block will be used to compute the residuals and thus the precision weights. The confusion is that following the call to voom() the user must call lmFit() directly. This call must explicitly specify block and correlation if desired; it will not inherit it from he previous voom() call as far as I can tell.

0
Entering edit mode

Thank you both James and Gabriel. It's much clearer now.

0
Entering edit mode
@gordon-smyth
Last seen 3 hours ago
WEHI, Melbourne, Australia

James Macdonald's comments are on the mark but I think it may be helpful if I give a summary of the situation.

The limma Manual and User's Guide do not actually give any explicit advice about how to use duplicateCorrelation in conjunction with voom, except from carefully documenting the input arguments and output object from each of the functions. The documentation does not give any recommendation regarding how many times to run voom and duplicateCorrelation, so there can't be any discrepancy between the limma documentation and anything else.

We have however given advice on this support forum, and we have been giving the same advice for over six years:

The reason why the limma User's Guide gives examples with only one duplicateCorrelation round is because only one round is needed in the contexts covered by the examples. In the examples, no estimated weights are involved.

voom produces an EList data object with the components documented on the help page. You can read more information about EList objects by typing help("EList-class"). You can see that EList objects do not store correlations or blocks, so no information about correlations can be passed from voom to subsequent calls of lmFit.

It is clear from the structure of the functions that duplicateCorrelation has to be run before lmFit and the correlation stored. It is also clear that duplicateCorrelation can't use the weights from voom unless voom is run first. So the only question really is whether voom itself benefits from knowing the blocking structure, so does it need to be run once or twice? It doesn't make a huge difference whether voom is run once or twice, so both approaches are "correct". Originally we recommended one round, just for simplicity, but for the past 6 years we have been recommending two rounds as a compromise between safety and computational load.

0
Entering edit mode

Thank you very much, that makes complete sense. Apologies for not finding it when I searched.


text:
## Language-based security for mobile code, with applications to smart cards

A 7-hour lecture given at TECS Week 2005 by Xavier Leroy.

This detailed set of slides explains various security models, and related attacks.

It has been awhile since we discussed mobile code and its security implications, but I am sure everyone realizes this is a major issue, that will only become more important as network applications become more and more pervasive (e.g., via mobile phones).

## Comment viewing options

### proof-carrying code patented

On a related note, it seems that proof-carrying code' (page 126) has already been patented (see espacenet) - though it talks about source code level', so applying it to bytecode is probably out of its reach.

Now this might be an invalid patent, but getting it recognised as invalid and removed will be both costly and slow. Until that time it might be a powerful means of intimidation...

### From the information on espac

EDIT: From the information on espacenet, I can't tell if this is a granted patent, as it was published in 2005, and has no INPADOC legal information available, although the application is from 2003. Of course, USPTO online services are largely unavailable this weekend, as they are moving to a new datacentre. Moreover, this application has only been made in the USA, leaving most people free to enjoy the security that US residents may be denied if this is granted.


text:
Boundary conditions / uniqueness of the propagators / Green's functions

My question(s) concern the interpretation and uniqueness of the propagators / Green's functions for both classical and quantum fields.

It is well known that the Green's function for the Laplace equation $$\Delta_x G(x,x') = \delta^{(3)}(x-x')$$ with boundary conditions $$G(x,x') = 0 \text{ for } x\in\partial D, x'\in D$$ is determined uniquely. This is because any harmonic function that vanishes on the boundary of domain must vanish inside as well.

In contrast, I am confused about the uniqueness of the Green's function for the wave equation $$(∂_t^2 - \Delta)G(x,t;x',t') = \delta^{(3)}(x-x')\delta(t-t')$$ Without specifying boundary conditions, there exist many homogenuous solutions to the wave equation and thus many different Green's functions.

Physicists usually offer me several canonical choices, like the retarded, the advanced and also the Feynman propagator, but I don't understand what makes each choice unique compared to the others, or which boundary conditions correspond to which choice. Hence, my question is

Which boundary conditions correspond to retarded, the advanced and the Feynman propagator? What other possibilities for boundary conditions and propagators are there?

I am also confused by the situation in quantum field theory, where we have various conventions for propagators and time-ordering like $$\langle0|T_t\lbrace a(x,t)a^\dagger(x',t')\rbrace |0\rangle$$ Apparently, the ground state is very important for selecting the right Green's function (see also my previous question), but I still don't understand why that is.

How does the vacuum state act as a boundary condition in the spirit of the previous question?

There is also the issue of imaginary time. The point is that imaginary time turns the wave equation $(\partial_t^2 -\Delta)\phi=0$ into the Laplacian $(\partial_\tau^2 + \Delta)\phi = 0$, but I don't understand how the usual analytic continuation $\tau \to it \pm i\eta$ for various propagators depends on the boundary conditions. For instance, Tsvelik writes in equation (22.4) that the imaginary time Green's function for the Laplacian in 2D is "obviously" the one that vanishes when $\tau$ and $x$ go to infinity, but I don't understand the reasoning behind this choice.

What are the right boundary conditions for Green's functions in imaginary time? Is it clear that the path integral formalism selects the right ones?

-
 Indeed, the wave equation solely does not fix its solution in a unique way. So for different boundary conditions you get different unambiguous solutions. Vacuum expectation of a product of two field operators (filed correlator) is a Green's function because the Green's function has a spectral bi-linear representation via eigenfunctions $\sum_n \psi_n (\vec{x})\psi_n (\vec{x'})e^{-iE_n (t-t')}$ and $\theta (t-t')$. – Vladimir Kalitvianski Jan 18 '12 at 15:55

Retarded propagators are those with $G(\dots, t,t')=0$ for all $t<t'$. They're vanishing before $t=t'$, the delta-function "stimulates" the field at $t=t'$, and the Green's function for positive $t-t'$ measures the response of the field. One may view this description as a construction of the Green's function which also proves that it's unique.

Advanced propagators are the time reversal of the retarded ones, with the sign of $t-t'$ interchanged in the description above. (Recheck whether I haven't permuted the two propagators by a mistake: beware of sign errors.) Feynman propagators are the arithmetic averages of the corresponding retarded and advanced ones.

The (non-thermal) Green's functions are defined as correlation functions in the vacuum state. This is useful because the ground state is the simplest state and all other multiparticle states may be built from the ground state by adding linear combinations of fields (check the LSZ formula etc.). For this reason, the structure of the vacuum including all the correlations in it "knows" about all dynamical questions.

In thermal field theory, the expectation values in the vacuum are replaced by ${\rm Tr}(\rho \dots)$ where $\rho=\exp(-\beta H)$ is the thermal density matrix.

The latter point also clarifies why the vacuum statet acts like a "boundary condition": the $T=0$ (non-thermal) correlators may be calculated for $t=\infty (1+i\epsilon)$ which has an infinite real part but a relatively smaller imaginary part, too. By analyticity, the small imaginary "angle" doesn't change the Green's function much. However, it simplifies the calculation because the evolution operator will contain an extra factor of $\exp(-\epsilon \infty H)$ which is still enough to exponentially suppress all excited states relatively to the ground state. When $\infty$ is really sent to infinity, only the matrix elements evaluated relatively to the ground state survive. The same comments and choices apply to the infinite future. For this reason, the vacuum matrix elements are automatically picked from the simplest boundary conditions one may assume at $t=\pm\infty$.

The small but nonzero value of $\epsilon$ gets translated to the $i\epsilon$ in Feynman's propagator: a procedure like this one also shows why Feynman's propagators are the correct ones to use in the Feynman diagrams.

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 The equation $G(\vec{x},\vec{x'}|t,t')=0$ for all \$t


text:
# Have new line between paragraphs, no indentation [duplicate]

I am sure that this must be a dup, but I can't find the right keywords to search for. So I am posting.

Right now my latex paragraphs look like this:

The paragraph does not start out with
an indent.  But when you make a new
paragraph there is an indent.
This is a new paragraph.  Can
you see the indent?
This is another paragraph.

I would like to have no indenting but and an empty line between paragraphs.
This is what I am using to remove the indents:

\setlength{\parindent}{0}

But I am not sure how to add in new lines. I think it is something like this:

\setlength{\parskip}{1}

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## migrated from stackoverflow.comSep 26 '12 at 12:58

This question came from our site for professional and enthusiast programmers.

## marked as duplicate by lockstep, egreg, barbara beeton, clemens, diabonasSep 26 '12 at 17:41

I asked the very same question! See here : stackoverflow.com/questions/2195908/latex-multiple-linebreaks –  rlb.usa Jul 1 '10 at 16:19
@rlb.usa: I've voted to close this qn as a duplicate, although I'm not sure that the answers here aren't better than there. Any firm views on the matter? –  Charles Stewart Jul 4 '10 at 20:21

Just include the following before \begin{document}

\usepackage[parfill]{parskip}

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+1 for this, much better to include a package than to fiddle round with parskip or noindent. –  pwaring May 7 '11 at 10:05

Use the \noindent command.

Example:

This is my paragraph.

\noindent My second paragraph

Check the following out for more detail: LaTeX:Layout guide

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Note that \noindent is only for occasional special effects. The correct way to do this is by using the \parindent and \parskip parameters mentioned elsewhere in this thread and in the duplicate answer. –  Norman Gray Jul 2 '10 at 7:56

Single line

\parskip = \baselineskip

Two lines

\parskip = \baselineskip
\multiply \parskip by 2

or

\parskip = 2\baselineskip % no shrink and no stretch

Fixed vertical space

\parskip = 1cm % Or 1 mm or 1in, for example

Stretchable and shrinkable space

\parskip = 12pt plus 6pt minus 4pt

Zero space

\parskip = 0pt

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One solution if it's only locally you want this is to add \newline at the end of a paragraph to get the empty line between the paragraphs and \noindent at the beginning of the next i.e.

The first paragraph start out without an indent. \newline

\noindent This is a new paragraph without indent.

However, if you want the new lines between the paragraphs throughout the document this solution is tedious...

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text:
Plot One-Way-Anova table sum of squares (SS) of each factor level (group) against the dependent variable. The SS of the factor variable against the dependent variable (variance within and between groups) is printed to the model summary.

sjp.aov1(
var.dep,
var.grp,
meansums = FALSE,
title = NULL,
axis.labels = NULL,
rev.order = FALSE,
string.interc = "(Intercept)",
axis.title = "",
axis.lim = NULL,
geom.colors = c("#3366a0", "#aa3333"),
geom.size = 3,
wrap.title = 50,
wrap.labels = 25,
grid.breaks = NULL,
show.values = TRUE,
digits = 2,
y.offset = 0.15,
show.p = TRUE,
show.summary = FALSE
)

## Arguments

var.dep Dependent variable. Will be used with following formula: aov(var.dep ~ var.grp) Factor with the cross-classifying variable, where var.dep is grouped into the categories represented by var.grp. Logical, if TRUE, the values reported are the true group mean values. If FALSE (default), the values are reported in the standard way, i.e. the values indicate the difference of the group mean in relation to the intercept (reference group). character vector, used as plot title. Depending on plot type and function, will be set automatically. If title = "", no title is printed. For effect-plots, may also be a character vector of length > 1, to define titles for each sub-plot or facet. character vector with labels used as axis labels. Optional argument, since in most cases, axis labels are set automatically. Logical, if TRUE, order of categories (groups) is reversed. Character vector that indicates the reference group (intercept), that is appended to the value label of the grouping variable. Default is "(Intercept)". Character vector of length one or two (depending on the plot function and type), used as title(s) for the x and y axis. If not specified, a default labelling is chosen. Note: Some plot types may not support this argument sufficiently. In such cases, use the returned ggplot-object and add axis titles manually with labs. Use axis.title = "" to remove axis titles. Numeric vector of length 2, defining the range of the plot axis. Depending on plot type, may effect either x- or y-axis, or both. For multiple plot outputs (e.g., from type = "eff" or type = "slope" in plot_model), axis.lim may also be a list of vectors of length 2, defining axis limits for each plot (only if non-faceted). user defined color for geoms. See 'Details' in plot_grpfrq. size resp. width of the geoms (bar width, line thickness or point size, depending on plot type and function). Note that bar and bin widths mostly need smaller values than dot sizes. numeric, determines how many chars of the plot title are displayed in one line and when a line break is inserted. numeric, determines how many chars of the value, variable or axis labels are displayed in one line and when a line break is inserted. numeric; sets the distance between breaks for the axis, i.e. at every grid.breaks'th position a major grid is being printed. Logical, whether values should be plotted or not. Numeric, amount of digits after decimal point when rounding estimates or values. numeric, offset for text labels when their alignment is adjusted to the top/bottom of the geom (see hjust and vjust). Logical, adds significance levels to values, or value and variable labels. logical, if TRUE (default), a summary with chi-squared statistics (see chisq.test), Cramer's V or Phi-value etc. is shown. If a cell contains expected values lower than five (or lower than 10 if df is 1), the Fisher's exact test (see fisher.test) is computed instead of chi-squared test. If the table's matrix is larger than 2x2, Fisher's exact test with Monte Carlo simulation is computed.

A ggplot-object.

## Examples

data(efc)
# note: "var.grp" does not need to be a factor.
# coercion to factor is done by the function
sjp.aov1(efc$c12hour, efc$e42dep)
#> Warning: Continuous limits supplied to discrete scale.
#> Did you mean limits = factor(...) or scale_*_continuous()?