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REVIEW 4 major objections 6 minor 23 references

Fatigue reliability analysis of offshore wind turbines under combined wind-wave excitation via DPIM

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The direct probability integral method with 1000 representative points reproduces Monte Carlo fatigue reliability for a 5MW spar floating wind turbine under combined wind-wave loading at about one twentieth of the CPU time.

desk verdict A credible DPIM-to-MCS comparison for FOWT fatigue reliability, but the headline numbers rest on a single-seed assumption that isn't tested. read the letter →

arxiv 2502.09429 v1 pith:XJVPERKZ submitted 2025-02-13 math.NA cs.NAphysics.comp-ph

Jingyi Ding , Hanshu Chen , Xiaoting Liu , Youssef F. Rashed , Zhuojia Fu This is my paper · ORCID
classification math.NAcs.NAphysics.comp-ph MSC 65C0562N05
keywords offshorewindturbinescombinedwind-waveexcitationsdirectprobabilityintegralmethodfatiguereliabilityanalysisfloatingturbineMonteCarlovalidationSouthChinaSea
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that fatigue reliability for a floating offshore wind turbine under combined wind and wave loading can be computed with the direct probability integral method (DPIM) using only 1000 representative environmental states, and that this reproduces Monte Carlo estimates at roughly one twentieth of the computational cost. The test case is a 5MW spar floating wind turbine in South China Sea conditions, with fatigue damage evaluated at the tower base and blade root by rainflow counting, S-N curves, and a linear cumulative damage rule. If this holds, long-term fatigue reliability becomes feasible for large floating wind systems where full Monte Carlo simulation is too expensive. The paper also reports that aligned wind and waves are the harshest case: fatigue reliability at 20 years is 0.855 at the tower base and 0.864 at the blade root, falling to 0.704 and 0.746 at 25 years.

What carries the argument

The direct probability integral method (DPIM) is the central mechanism: it partitions the input probability space into 1000 representative points, runs one 600-second coupled time-domain simulation per point, and assembles the probability density of the fatigue response as a sum of smoothed contributions, with a Heaviside-function integral for reliability. Putting each representative point through the aero-hydro-servo-elastic simulation chain produces a stress time series, which is reduced to stress ranges by rainflow counting, corrected for mean stress, and converted to damage with standard S-N curves ($m=3$ for the steel tower and $m=8$ for the composite blades) and linear damage accumulation. The key advantage is that no generalized density evolution equation needs to be solved; the method needs only deterministic response samples at the representative points.

What would settle it

Take a subset of the 1000 representative states, run several independent turbulent wind and wave seed realizations per state, and compare the seed-averaged fatigue damage distribution with the single-seed DPIM result; if the short-term damage variance is large, the DPIM probability densities and the reported 0.855/0.864 reliability values will not survive seed averaging.

Watch

Extended reading notes

Core claim

The central claim is that DPIM converts the probability-density evolution of a nonlinear floating wind turbine into a weighted sum over representative points in the joint wind-wave probability space, so that the full fatigue damage distribution, not just a few moments, can be obtained from about 1000 fully coupled time-domain simulations. The paper validates this against Monte Carlo simulation with 10,000 samples: the stress and fatigue-damage probability density functions match, while the CPU time drops from roughly 611,770 seconds to 22,910 seconds, a factor above 20. Using the resulting damage distributions, the paper finds that under aligned wind and wave directions the 20-year fatigue reliability is 0.855 at the tower base and 0.864 at the blade root, and that both fall to 0.704 and 0.746, respectively, when the service life is extended to 25 years. The paper concludes that the design life is met but that reliability declines sharply beyond 20 years, especially for the tower base.

Load-bearing premise

The load-bearing premise is that a single 600-second time-domain simulation per representative environmental state captures that state's fatigue damage rate; if the damage rate varies strongly from one turbulent wind and wave seed to the next, the fitted damage distribution and the resulting reliability values will be overconfident.

Editorial extensions

If this is right

  • At the aligned wind-wave condition, the 20-year fatigue reliability is 0.855 at the tower base and 0.864 at the blade root; extending operation to 25 years lowers these values to 0.704 and 0.746.
  • Fatigue reliability increases as the angle between wind and wave directions grows, so the aligned case is the governing design condition for these components.
  • Because DPIM needs only 1000 representative states, an engineer can map fatigue reliability over service life and environmental direction with the same fidelity as a 10,000-sample Monte Carlo study but at roughly 1/20 of the CPU cost.
  • The tower base is the more fatigue-prone location: its damage distribution has a larger probability of exceeding the failure threshold $D=1$ than the blade root, which matches the higher mean stress found at tower-base Node 7.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the one-seed, 600-second damage estimates carry substantial turbulence-seed scatter, repeating each representative state over several seeds could shift the reported reliability values even though the DPIM-versus-Monte Carlo agreement would likely survive.
  • The same representative-point scheme could be transferred to other floater concepts or sites, but the number of points and the smoothing parameter would need to be re-tuned because the joint environmental distribution and the nonlinearity of the response change.
  • The sharp reliability drop between 20 and 25 years suggests the 20-year results sit close to the steep part of the damage tail; a formal sensitivity study of the S-N slope and mean-stress correction would show how much of the drop is material-model driven.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper applies the direct probability integral method (DPIM) to fatigue reliability analysis of an NREL 5MW OC3-Hywind spar floating offshore wind turbine under combined wind-wave excitation. Long-term joint environmental distributions are fitted to South China Sea reanalysis data, short-term fatigue damage is computed from OpenFAST time-domain simulations via rainflow counting, a Goodman correction, S-N curves, and Palmgren-Miner accumulation, and DPIM is compared with Monte Carlo simulation (MCS). The authors report that DPIM with 1000 representative points reproduces MCS stress and fatigue-damage distributions while using roughly 1/20 of the CPU time, and that under aligned wind-wave conditions the 20-year fatigue reliabilities are 0.855 for the tower base and 0.864 for the blade root, decreasing to 0.704 and 0.746 at 25 years.

Significance. If the numerical claims hold, the paper would demonstrate a practically useful acceleration of fatigue reliability assessment for floating offshore wind turbines, replacing 10,000 full OpenFAST simulations with 1000, and it would provide site-specific reliability estimates from long-term reanalysis data. The work has several strengths: it uses a well-established aero-hydro-servo-elastic tool (OpenFAST), anchors the environmental model to reanalysis data rather than calibrating it to the reliability outputs, and benchmarks DPIM against MCS. The reported speedup factor and specific reliability numbers are falsifiable and clearly stated. However, the absolute reliability values and the reproducibility of the calculation rest on parameter values and statistical assumptions that are not fully documented in the manuscript, as detailed below.

major comments (4)
  1. [Section 2.5, Eq. (5)] The S-N curve intercept parameter a is never reported. Equation (5) is lg N = lg a - m lg Δσ, so a is required to compute the number of cycles to failure N in Eq. (6) and hence every fatigue damage value D_j^ST. Only the slopes m = 3 for steel and m = 8 for composite are given. Without the a values (and their sources), the fatigue damage distributions in Figs. 7-9 and the reliability numbers 0.855, 0.864, 0.704, and 0.746 are not reproducible. This is a load-bearing omission.
  2. [Sections 2.4 and 2.6; Eqs. (6)-(8)] The short-term fatigue damage rate for each environmental state is estimated from a single 600-second OpenFAST time-domain simulation (T_j = 600 s in Eq. (7)), with no averaging over multiple turbulent wind or wave seeds and no reported seed-to-seed scatter. Fatigue damage is a nonlinear function of stress ranges, and a 600-s record for a spar-type floater may be dominated by a few large cycles or low-frequency platform motions. Because DPIM and MCS use the same one-seed protocol, their agreement does not validate the damage-rate estimator itself. The absolute reliability values therefore rest on an unverified statistical stability assumption. The authors should report multiple-seed statistics for at least a subset of environmental states, or otherwise quantify the sampling uncertainty of D_j^ST.
  3. [Section 3.1, Eq. (13)] The DPIM smoothing parameter σ in Eq. (13) is not reported, although the PDF comparisons in Figs. 6 and 7 depend on it. Moreover, the claimed agreement between DPIM and MCS is only qualitative: no error metric (e.g., relative L1/L2 error, difference in damage quantiles, or Kolmogorov-Smirnov distance) is given. Reporting σ and quantitative comparison errors is necessary to support the accuracy claim in Section 3.1.
  4. [Section 3.3, Eqs. (16)-(18)] The paper concludes that the turbine 'meets the design requirements' based on the computed fatigue reliabilities, but no target reliability level or design acceptance criterion is defined. The values 0.855 and 0.864 may or may not satisfy a code-based target such as those in IEC 61400 or DNV standards. The authors should state the acceptance threshold used for the design conclusion, or explicitly present the reliability values as unconditional estimates without a pass/fail claim.
minor comments (6)
  1. [Section 2.4, Eq. (4)] The notation in Eq. (4) is confusing: σ_i^RF, σ_i^R, ε, σ_ult, and σ_MF are not all defined precisely in the text, and the standard Goodman correction is usually written as a mean-stress correction involving the ultimate strength. Please clarify the formula and the meaning of the 'Goodman exponent'.
  2. [Fig. 2 caption] The caption 'PDF curves of wind and wave loading parameters under DPIM' is misleading because Fig. 2 shows the input environmental distributions, not DPIM results. Rephrase to 'fitted PDFs of the environmental random variables'.
  3. [Section 2.3, Eqs. (2)-(3)] The blade-root stress equations use subscripts xM, yM, zM and xF, yF, zF without clear definitions of the fixed coordinate system in Fig. 4. Specify the sign conventions and the axes used for the bending moments and shear forces.
  4. [Section 2.6] The text says 'when D > 1 suggests that the material has exhausted its fatigue life'; the conventional failure criterion is D ≥ 1 or D = 1. Please correct this wording.
  5. [Section 3.2] The choice of Node 7 as the 'danger point' is based on mean axial stress at the tower base, but fatigue damage depends on stress ranges, not mean stress alone. Clarify whether the maximum fatigue damage location was verified to coincide with Node 7 for all environmental states and wave angles.
  6. [Section 2.1] The OpenFAST version, TurbSim version, and turbulence seed management are not stated. Reporting these details would improve reproducibility, especially given the single-seed concern in the major comments.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: DPIM reliability estimates are genuine outputs validated against MCS with independently fitted environmental inputs.

full rationale

The paper derives fatigue-reliability estimates for a 5 MW spar FOWT by chaining independent ingredients: (1) a joint distribution of wind/wave parameters fitted to ERA-Interim reanalysis data (Section 2.2, Table 2); (2) a deterministic OpenFAST time-domain model; (3) rainflow counting, S-N curves, and Palmgren-Miner accumulation (Sections 2.4-2.6); (4) the DPIM as a numerical integration scheme (Eqs. 9-18, from Chen & Yang [11,12]); and (5) definition of fatigue reliability as P(D < 1) under the computed damage PDF. The reported reliability values 0.855 and 0.864 are outputs of this chain, not quantities used to calibrate any parameter. The DPIM results are benchmarked against 10,000-sample MCS using the same simulator and the same environmental distributions; agreement therefore validates the DPIM quadrature, not a fitted quantity. The only fitted model inputs are environmental distribution parameters, which are external and applied identically to both methods. No equation in the paper reduces the claimed reliability prediction to an input by construction. Reliance on the authors' prior DPIM work is a citation of the method, not a load-bearing self-citation: the method's use is supported by the internal MCS comparison and by the standard nature of the probability-integral formulas. The single 600-s simulation per environmental state and the absence of seed-to-seed scatter is a missing-support issue concerning statistical stability of the short-term damage estimate; it does not make the derivation circular, because it does not define the output in terms of the input. Overall circularity score: 0.

Assumptions & free parameters 9 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a chain of inputs: environmental distribution parameters fitted to reanalysis data, S-N curve parameters that are partly unreported, a DPIM smoothing parameter that is not stated, and a single-seed 600-second simulation protocol. Together these introduce several unverified degrees of freedom, though none of them is fitted to the target reliability values, so the circularity burden remains low.

free parameters (9)
  • Weibull scale and shape for mean wind speed (a=11.9799, b=2.8005) = a=11.9799, b=2.8005
    Fitted to ERA-Interim reanalysis data at 21N,113E and taken from the environmental model in Section 2.2; every fatigue damage value depends on this wind-speed distribution.
  • Lognormal parameters for significant wave height (mu=0.4887, sigma=0.4489) = mu=0.4887, sigma=0.4489
    Fitted to reanalysis data in Section 2.2; controls the wave height distribution that drives hydrodynamic fatigue loads.
  • Lognormal parameters for spectral peak period (mu=2.0759, sigma=0.1547) = mu=2.0759, sigma=0.1547
    Fitted to reanalysis data in Section 2.2; affects wave frequency and the number of stress cycles.
  • DPIM smoothing parameter sigma in Eq. (13) = not reported
    Controls the kernel width in the DPIM probability density estimate; the paper never states its value, yet the PDF and reliability results depend on it.
  • S-N curve intercept a in Eq. (5) = not reported
    The S-N curve lgN = lg a - m lg(delta-sigma) requires the intercept a; the paper gives m=3 for steel and m=8 for composite but never gives a, and fatigue damage values scale inversely with a.
  • Goodman exponent epsilon in Eq. (4) = 1.0
    Chosen to correct mean stress effects in the rainflow counting; a different exponent would change the equivalent stress ranges.
  • Simulation duration T_j = 600 s = 600 s
    Chosen in Section 2.6 for each time-domain simulation; a longer duration would reduce variance in the damage-rate estimate but increase cost.
  • Number of DPIM representative points N = 1000 = 1000
    Chosen in Section 3; no convergence study is reported, so the adequacy of 1000 points for the 3-dimensional environmental space is an unverified choice.
  • Monte Carlo sample size = 10,000 = 10000
    Benchmark choice in Section 3; not a fitted parameter for the central claim but affects the statistical precision of the MCS reference.
assumptions (6)
  • standard math The direct probability integral method equations (Eqs. 9-18) are valid for the MDOF nonlinear FOWT system under random wind-wave excitation.
    Invoked in Section 2.7; the paper relies on the DPIM framework from references [11] and [12] without re-deriving its validity for this application.
  • domain assumption OpenFAST with BEM aerodynamics, Kaimal turbulence, potential-flow and Morison hydrodynamics accurately predicts the fatigue loads of the 5MW OC3-Hywind spar.
    Section 2.1 describes the simulation toolchain but provides no validation against field measurements or independent code-to-code comparisons.
  • domain assumption The joint probability distribution of wind speed, wave height, and peak period from Song et al. [21], fitted to ERA-Interim reanalysis at 21N,113E, represents the long-term South China Sea site.
    Section 2.2 takes this model as given; the paper does not validate it against local buoy measurements or newer hindcast data, and all reliability values inherit its accuracy.
  • domain assumption Neglecting second-order wave forces is acceptable for this spar-type floater.
    Section 2.1 cites reference [18] for this simplification; if second-order forces are non-negligible, the fatigue damage estimates would change.
  • domain assumption The S-N curves from DNV-RP-C203 and the Palmgren-Miner linear damage rule apply to the tower steel and blade composite materials.
    Sections 2.5 and 2.6 use these standards, but the material curve intercept a is never specified, so the reader cannot independently verify the fatigue life calculations.
  • ad hoc to paper A single 600-second time-domain simulation per environmental state gives a statistically stable estimate of the fatigue damage rate for that state.
    Section 2.6 sets T_j=600 s and Section 2.4 describes the simulation procedure; no multiple-seed averaging or convergence check is reported, making this a load-bearing assumption for the damage distribution.

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Pith. "Pith review of Fatigue reliability analysis of offshore wind turbines under combined wind-wave excitation via DPIM." pith.science (2026). https://pith.science/paper/XJVPERKZ

@misc{pith2026250209429,
  author       = {Pith},
  title        = {Pith review of: Fatigue reliability analysis of offshore wind turbines under combined wind-wave excitation via DPIM},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XJVPERKZ}},
  note         = {Machine review of arXiv:2502.09429}
}
read the original abstract

As offshore wind turbines develop into deepwater operations, accurately quantifying the impact of stochastic excitations in complex sea environments on offshore wind turbines and conducting structural fatigue reliability analysis has become challenging. In this paper, based on long-term wind-wave reanalysis data from a site in the South China Sea, a novel direct probability integral method (DPIM) is developed for the stochastic response and fatigue reliability analyses of the key components for the floating offshore wind turbine structures under combined wind-wave excitation. A 5MW floating offshore wind turbine is considered as the research object, and a fully coupled dynamic response analysis of the wind turbine system is conducted to calculate the short-term fatigue damage value of tower base and blade root. The DPIM is applied to calculate the fatigue reliability of the wind turbine structure. The accuracy and efficiency of the proposed method are validated by comparing the obtained results with those of Monte Carlo simulations. Furthermore, the results indicate that the fatigue life of floating offshore wind turbine structures under combined wind-wave excitation meets the design requirements. Notably, the fatigue reliability of the wind turbine under aligned wind-wave condition is lower compared to misaligned wind-wave condition.

Figures

Figures reproduced from arXiv: 2502.09429 by the authors.

Figure 5
Figure 5. Comparison of mean axial stress under different wind and wave angle conditions The average stress values at different points at the tower base under the aligned wind-wave condition, calculated using the DPIM and MCS methods, are shown in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 7
Figure 7. When the wind and wave angle is 90°, the probability that the fatigue damage value at the [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

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