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arxiv: 2604.22882 · v1 · submitted 2026-04-24 · 💻 cs.LG · physics.comp-ph· physics.data-an

Predicting Wind Loads on Container Ships in Harbor Environments through Multi-Fidelity Modeling

Matilde Fiore , Andrea Bresciani , Miguel Alfonso Mendez , Jeroen van Beeck This is my paper

Pith reviewed 2026-05-08 12:21 UTC · model grok-4.3

classification 💻 cs.LG physics.comp-phphysics.data-an
keywords multi-fidelity modelingwind loadscontainer shipsco-krigingsurrogate modelingharbor environmentsflow simulations
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The pith

Multi-fidelity surrogate models using recursive co-kriging predict wind loads on modern container ships in harbors more accurately and with less computation than single-fidelity or empirical methods.

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

This paper develops a framework that combines simple empirical formulas, basic flow simulations, and detailed flow simulations to forecast how wind pushes on large container ships when they are docked near other structures. The approach uses recursive co-kriging to blend these different levels of detail into a single fast predictor. If it works as claimed, engineers could design safer mooring systems for today's bigger vessels without running thousands of expensive high-resolution simulations. The study shows the method works across different cargo loads and two example harbor layouts while beating older rule-of-thumb formulas. It also identifies which ship shape features matter most through sensitivity checks.

Core claim

The authors establish that their recursive co-kriging multi-fidelity surrogate, built from empirical correlations plus simplified and detailed flow simulations for open-sea and harbor settings, produces accurate wind-load coefficient predictions that depend correctly on key geometric parameters and require far fewer high-fidelity runs than direct simulation.

What carries the argument

recursive co-kriging, which systematically merges data from three levels of model fidelity to create a single surrogate that respects the relationships between cheap approximations and expensive accurate ones.

If this is right

  • The surrogate models accurately predict wind loads for various loading configurations in two harbor environments.
  • The framework captures dependence on geometric parameters identified by sensitivity analysis.
  • It consistently outperforms traditional empirical correlations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Engineers could apply the same framework to explore many ship designs quickly during early stages.
  • The approach may extend to other problems involving wind or fluid forces where models of varying accuracy exist.
  • Adjusting the low-fidelity models for specific harbor features could make predictions even more reliable for complex port layouts.

Load-bearing premise

The method assumes that the low-fidelity models, originally developed for open-sea conditions, transfer reliably to harbor geometries when fused through recursive co-kriging.

What would settle it

Running the multi-fidelity predictor on a container ship geometry and harbor configuration never seen during training and then comparing its output directly to a new set of high-fidelity flow simulation results or physical measurements; large discrepancies would indicate the fusion step fails.

Figures

Figures reproduced from arXiv: 2604.22882 by Andrea Bresciani, Jeroen van Beeck, Matilde Fiore, Miguel Alfonso Mendez.

Figure 1
Figure 1. Figure 1: Definition of the reference wind direction and of the local coordinate view at source ↗
Figure 2
Figure 2. Figure 2: Container ship considered in the present work. view at source ↗
Figure 3
Figure 3. Figure 3: The three loading configurations and four environments simulated: (a) empty ship, (b) intermediate load, (c) full load, (d) open sea, (e) empty quay, (f) view at source ↗
Figure 4
Figure 4. Figure 4: (a) Containers environment and (b) tanks environment. view at source ↗
Figure 5
Figure 5. Figure 5: Mesh used for the high-fidelity CFD approach for the open sea case and intermediate loading configuration. (a) View of the entire cylindrical domain, (b) view at source ↗
Figure 6
Figure 6. Figure 6: Comparison between measurements and the high-fidelity CFD approach for the tanks environment. (a) Longitudinal force coe view at source ↗
Figure 7
Figure 7. Figure 7: Geometry of the simplified ship employed in the setup of medium fidelity. view at source ↗
Figure 8
Figure 8. Figure 8: Domain employed to carry out precursor simulations of the flow within the gaps, to estimate the coe view at source ↗
Figure 9
Figure 9. Figure 9: Velocity-pressure drop relationship for the flow within the gaps ob view at source ↗
Figure 10
Figure 10. Figure 10: Snapshots of the mesh used in the simplified CFD setup, highlighting the rotating part of the mesh enabling to change the angle of attack. view at source ↗
Figure 11
Figure 11. Figure 11: Schematics of the methodology adopted to train the multi-fidelity surrogate model for wind loads with three fidelity levels. The data-sources are here view at source ↗
Figure 12
Figure 12. Figure 12: Schematics of the methodology adopted to train the environment-specific multi-fidelity surrogate models for wind loads. The data-sources are here view at source ↗
Figure 13
Figure 13. Figure 13: Results of the active subspace analysis: (a) cumulative sum of the eigenvalues of view at source ↗
Figure 11
Figure 11. Figure 11: The algorithm performs a sequential exploration of the pa￾rameter space over 500 iterations. The sample configurations selected at the different fidelity levels are shown in view at source ↗
Figure 14
Figure 14. Figure 14: Wind load coefficients predicted by the empirical correlation (LF), the simplified CFD (MF) and the detailed CFD (HF) for the full load configuration (left) and the intermediate load configuration (right). 13 view at source ↗
Figure 15
Figure 15. Figure 15: Sample configurations analysed with the data-sources of low, medium and high fidelity throughout the sequential learning process. Each polyline view at source ↗
Figure 16
Figure 16. Figure 16: Data distribution at the different fidelity levels at the initialization (top) and at the end of the training process (bottom). Data of lowfidelity (empirical correlation, LF), medium fidelity (simplified CFD, MF), high fidelity (detailed CFD in open sea, HF). The data related to the detailed CFD in container and tanks environments are denoted as HF-CE and HF-TE, respectively. 16 view at source ↗
Figure 17
Figure 17. Figure 17: Streamwise velocity contours obtained with sample configurations explored by the training algorithm with the medium fidelity data-source (simplified view at source ↗
Figure 18
Figure 18. Figure 18: Streamwise velocity contours obtained with sample configurations explored by the training algorithm with the medium fidelity data-source (simplified view at source ↗
Figure 19
Figure 19. Figure 19: Streamwise velocity contours obtained with sample configurations explored by the training algorithm with the medium fidelity data-source (simplified view at source ↗
Figure 20
Figure 20. Figure 20: Streamwise velocity contours obtained with sample configurations explored by the training algorithm with the medium fidelity data-source (simplified view at source ↗
Figure 21
Figure 21. Figure 21: Streamwise velocity contours obtained with sample configurations explored by the training algorithm with the medium fidelity data-source (simplified view at source ↗
Figure 22
Figure 22. Figure 22: Loading configurations considered for the validation of the surrogate models. view at source ↗
Figure 23
Figure 23. Figure 23: Comparison between the wind load coefficients predicted by the multi-fidelity (green) and single-fidelity (red) surrogate models for the loading configu￾ration depicted in 23d. Numerical data from the detailed CFD (open sea) are represented with black dots. 20 view at source ↗
Figure 24
Figure 24. Figure 24: Comparison between the wind load coefficients predicted by the multi-fidelity (green) and single-fidelity (red) surrogate models for the loading configu￾ration depicted in 24d. Numerical data from the detailed CFD (open sea) are represented with black dots. 21 view at source ↗
Figure 25
Figure 25. Figure 25: Comparison between predictions of the multi-fidelity surrogate models and simulation data of high-fidelity (detailed CFD) for the load configurations view at source ↗
Figure 26
Figure 26. Figure 26: Mean absolute errors (%) of the multi-fidelity surrogate model with respect to the detailed CFD in open sea for the load configurations depicted in Figure view at source ↗
Figure 27
Figure 27. Figure 27: Comparison between the wind load coefficients predicted by the multi-fidelity (green) and single-fidelity (red) surrogate models for the loading configu￾ration depicted in 27d (right). Numerical data from the detailed CFD (container environment) are represented with black dots. the proposed framework is able to capture the influence of har￾bour structures, despite the limited availability of high-fidelity… view at source ↗
Figure 28
Figure 28. Figure 28: Comparison between the wind load coefficients predicted by the multi-fidelity (green) and single-fidelity (red) surrogate models for the loading configu￾ration depicted in 28d (right). Numerical data from the detailed CFD (container environment) are represented with black dots. Technology Division of Ghent University for the ship and envi￾ronment conceptual design. References [1] N. K. Park, S. C. Suh, Te… view at source ↗
Figure 29
Figure 29. Figure 29: Comparison between the wind load coefficients predicted by the multi-fidelity (green) and single-fidelity (red) surrogate models for the loading configu￾ration depicted in 29d (right). Numerical data from the detailed CFD (container environment) are represented with black dots. [9] W. Blendermann, Parameter identification of wind loads on ships, Journal of Wind Engineering and Industrial Aerodynamics 51 (… view at source ↗
Figure 30
Figure 30. Figure 30: Comparison between the wind load coefficients predicted by the multi-fidelity (green) and single-fidelity (red) surrogate models for the loading configu￾ration depicted in 30d (right). Numerical data from the detailed CFD (container environment) are represented with black dots. [19] R. Pache, T. Rung, Data-driven surrogate modeling of aerodynamic forces on the superstructure of container ves￾sels, Enginee… view at source ↗
Figure 31
Figure 31. Figure 31: Comparison between predictions and simulation data of high-fidelity (detailed CFD) on the test dataset. Predictions of the multi-fidelity surrogate model view at source ↗
Figure 32
Figure 32. Figure 32: Comparison between predictions and simulation data of high-fidelity (detailed CFD) on the test dataset. Predictions of the multi-fidelity surrogate model view at source ↗
Figure 33
Figure 33. Figure 33: Sobol sensitivity indices of the three wind load coe view at source ↗
Figure 34
Figure 34. Figure 34: Response surface of the multi-fidelity surrogate model showing the variability of the longitudinal wind force coe view at source ↗
Figure 35
Figure 35. Figure 35: Response surface of the multi-fidelity surrogate model showing the variability of the lateral wind force coe view at source ↗
Figure 36
Figure 36. Figure 36: Response surface of the multi-fidelity surrogate model showing the variability of the yaw moment coe view at source ↗
read the original abstract

Modern container ships face higher wind loads due to increased windage areas, making accurate predictions of wind loads essential for mooring design. Existing empirical models, largely developed for container ships with smaller windage areas and simpler geometrical configurations than those of modern large-scale vessels, often lack accuracy and do not account for the influence of nearby structures. This study proposes a multi-fidelity surrogate modelling framework for the prediction of wind-load coefficients, combining empirical correlations with simplified and detailed CFD models for ships in open-sea and harbor environments. The approach relies on recursive co-kriging to consistently fuse information across fidelity levels, enabling accurate predictions at a reduced computational cost. A sensitivity analysis is used to identify the most influential geometric parameters, and the resulting reduced parameter space is explored through sequential sampling to efficiently construct the training database. The surrogate models are validated over a wide range of loading configurations and for two distinct harbor environments. The results demonstrate that the multi-fidelity approach significantly improves prediction accuracy compared to single-fidelity models, while substantially reducing the reliance on high-fidelity simulations. In particular, the proposed framework captures the dependence of wind loads on key geometric parameters and consistently outperforms traditional empirical correlations, providing a robust and efficient tool for engineering applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript presents a multi-fidelity surrogate modeling framework for predicting wind-load coefficients on modern container ships in harbor settings. It integrates empirical correlations, simplified CFD, and high-fidelity CFD using recursive co-kriging, incorporates sensitivity analysis for parameter reduction, and uses sequential sampling. The models are validated across loading configurations and two harbor environments, with claims of superior accuracy to single-fidelity approaches and empirical models, alongside reduced computational demands.

Significance. Should the quantitative validations confirm the reported accuracy improvements and computational savings, this framework would offer a practical advancement for naval architecture and mooring system design. By efficiently capturing geometric parameter dependencies and harbor effects, it could lower the barrier to accurate wind load predictions for large vessels where traditional empirical models fall short.

major comments (3)
  1. [Abstract] Abstract: The central claim that the multi-fidelity approach 'significantly improves prediction accuracy' and 'consistently outperforms traditional empirical correlations' is stated without any supporting quantitative metrics, such as specific error reductions, R² values, or references to validation tables/figures. This absence prevents assessment of whether the improvements are substantial enough to justify the framework's adoption.
  2. [Methodology] Methodology (recursive co-kriging description): The recursive co-kriging fusion of low-fidelity open-sea models with harbor high-fidelity data assumes the hierarchical discrepancy function adequately models additional physics like shielding and reflections from nearby structures. However, the manuscript provides no explicit check or validation that the correlation structure holds under this domain shift, raising the possibility of systematic bias in predictions for certain wind directions or ship positions, which directly impacts the claimed reduction in high-fidelity simulations.
  3. [Validation] Validation: The validation over a wide range of loading configurations and two harbor environments is described, but without details on how the post-hoc parameter reduction from sensitivity analysis interacts with the outperformance claims or the cross-validation strategy, it remains unclear whether the results robustly support reduced reliance on high-fidelity data.
minor comments (1)
  1. [Introduction] Introduction: The discussion of increased windage areas in modern ships could benefit from a brief quantitative comparison to older vessels to better contextualize the problem scale.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify key aspects of the work. We address each major point below and will revise the manuscript accordingly to strengthen the presentation of results and methodology.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that the multi-fidelity approach 'significantly improves prediction accuracy' and 'consistently outperforms traditional empirical correlations' is stated without any supporting quantitative metrics, such as specific error reductions, R² values, or references to validation tables/figures. This absence prevents assessment of whether the improvements are substantial enough to justify the framework's adoption.

    Authors: We agree that the abstract's brevity omits specific metrics. The full manuscript reports these in Section 4 (e.g., R² > 0.92 and 15-25% error reduction vs. empirical models in Tables 3-5 and Figures 6-8). We will revise the abstract to include representative quantitative values and cross-references to the validation results. revision: yes

  2. Referee: [Methodology] Methodology (recursive co-kriging description): The recursive co-kriging fusion of low-fidelity open-sea models with harbor high-fidelity data assumes the hierarchical discrepancy function adequately models additional physics like shielding and reflections from nearby structures. However, the manuscript provides no explicit check or validation that the correlation structure holds under this domain shift, raising the possibility of systematic bias in predictions for certain wind directions or ship positions, which directly impacts the claimed reduction in high-fidelity simulations.

    Authors: The recursive co-kriging is intended to capture domain-shift effects via the discrepancy term, and overall validation across wind directions and positions supports its effectiveness. However, we acknowledge the lack of an explicit correlation-structure diagnostic for the discrepancy function. We will add this analysis in the revised methodology section, including plots of predicted vs. observed discrepancies for representative wind angles and positions. revision: yes

  3. Referee: [Validation] Validation: The validation over a wide range of loading configurations and two harbor environments is described, but without details on how the post-hoc parameter reduction from sensitivity analysis interacts with the outperformance claims or the cross-validation strategy, it remains unclear whether the results robustly support reduced reliance on high-fidelity data.

    Authors: The sensitivity analysis (Section 3.2) reduces the parameter space before sequential sampling, and cross-validation is performed on the resulting design. To clarify the interaction, we will expand Section 4 with additional discussion and metrics showing accuracy retention after reduction and details of the k-fold cross-validation procedure used to quantify the savings in high-fidelity evaluations. revision: yes

Circularity Check

0 steps flagged

No circularity: multi-fidelity fusion and validation are independent of target outputs

full rationale

The paper applies standard recursive co-kriging to fuse three fidelity levels (empirical correlations, simplified CFD, detailed CFD) for wind-load coefficients on container ships. Training data are generated via sequential sampling in a reduced parameter space identified by sensitivity analysis, and models are validated on held-out configurations across two harbor environments. No equation or claim equates the final prediction to a parameter fitted from the same target data; the outperformance versus single-fidelity baselines is shown empirically. The derivation chain therefore remains self-contained and does not reduce by construction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is limited to the high-level assumptions stated or implied in the summary; full paper would likely add more.

axioms (1)
  • domain assumption Recursive co-kriging can consistently fuse multi-fidelity information without systematic bias across open-sea and harbor geometries.
    Invoked by the claim that the framework enables accurate predictions at reduced cost.

pith-pipeline@v0.9.0 · 5528 in / 1234 out tokens · 51427 ms · 2026-05-08T12:21:19.299100+00:00 · methodology

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Reference graph

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