A time-domain approach for motion-explicit evaluation of loads on floating structures in fully nonlinear waves

Athanasios Dermatis; Benjamin Bouscasse; Guillaume Ducrozet; Harry B. Bingham; Henrik Bredmose

arxiv: 2604.09247 · v1 · submitted 2026-04-10 · ⚛️ physics.flu-dyn

A time-domain approach for motion-explicit evaluation of loads on floating structures in fully nonlinear waves

Athanasios Dermatis , Henrik Bredmose , Harry B. Bingham , Benjamin Bouscasse , Guillaume Ducrozet This is my paper

Pith reviewed 2026-05-10 17:24 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords hydrodynamic loadsnonlinear wavesfloating structurestime-domain simulationsecond-order forcesradiation-diffractionmoored shipspotential flow

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The pith

A time-domain method evaluates second-order hydrodynamic loads on floating structures using nonlinear wave kinematics and instantaneous body motions.

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

The paper develops a method to compute hydrodynamic forces on floating structures that accounts for fully nonlinear waves and the actual instantaneous motions of the body, rather than approximations based on linear motions. This is achieved by applying frequency-domain transfer functions to nonlinear wave fields obtained from pseudo-spectral solvers and reformulating the force expressions to include total nonlinear velocities and positions. Standard second-order theories assume small first-order motions compared to second-order, which fails for moored structures that can have significant slow-drift motions. By coupling this force model with a time-domain motion solver, the approach yields improved predictions for the motions of a moored container ship compared to traditional radiation-diffraction methods. Such an efficient yet more accurate load evaluation is valuable for predicting the performance and safety of floating offshore installations in realistic, nonlinear sea states.

Core claim

The paper establishes a novel time-domain approach for motion-explicit evaluation of loads, where a closed-form expression for the potential force component is derived as a generalization of the Pinkster approximation to fully nonlinear waves, the quadratic force component is reformulated to account for the total nonlinear body motion and velocity rather than first-order counterparts, the radiation potential is treated in the time domain while incident and scattering contributions use wavenumber-domain transfer functions, and the overall force model is coupled with a time-domain motion solver to permit consideration of instantaneous body motion and velocity in the force calculation.

What carries the argument

The closed-form potential force expression that generalizes the Pinkster approximation to fully nonlinear waves by applying wavenumber-domain transfer functions to pseudo-spectral wave fields, combined with time-domain treatment of the radiation potential and reformulation of quadratic forces using total nonlinear body motion and velocity.

Load-bearing premise

The method assumes that outputs from standard frequency-domain radiation-diffraction analysis remain sufficiently accurate when applied to fully nonlinear wave kinematics.

What would settle it

Direct comparison of the predicted motions of the moored container ship against measurements in fully nonlinear waves would confirm the reported significant improvements or reveal that discrepancies with standard theory persist.

Figures

Figures reproduced from arXiv: 2604.09247 by Athanasios Dermatis, Benjamin Bouscasse, Guillaume Ducrozet, Harry B. Bingham, Henrik Bredmose.

**Figure 1.** Figure 1: Reference frame definition R = " cos 3 cos 2 sin 3 cos 2 − sin 2 − sin 3 cos 1 + cos 3 sin 2 sin 1 cos 3 cos 1 + sin 3 sin 2 sin 1 cos 2 sin 1 sin 3 sin 1 + cos 3 sin 2 cos 1 − cos 3 sin 1 + sin 3 sin 2 cos 1 cos 2 cos 1 # (2.1) The definition of the position vector for the global reference frame is defined for an arbitrary point M as OM = x, shown in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: 6750-TEU container ship model and experimental setup. for the second-order quadratic force terms of (4.34) into three contributions arising from products between the linear potentials (1) and (1) [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗

**Figure 3.** Figure 3: Potential component of the second-order horizontal force on the container ship in terms of spectrum (left) and time series (right) [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

**Figure 4.** Figure 4: Quadratic component of the second-order horizontal force on the container ship in terms of spectrum (left) and time series (right). free-surface integral, which is not treated directly by (6.2). Excellent agreement is found between the impulse method and the Pinkster approach, verifying the equivalence of the two methods at second order. In contrast, comparison with the full potential-force solution shows … view at source ↗

**Figure 5.** Figure 5: Convolution kernels for the radiation damping force (left) and effect of lower cut-off limit of convolution integrals within the hydrodynamic force evaluation to surge motion (right). tional performance of the method. The radiation damping convolution kernels in pure surge K11, heave K33 and pitch K55 are shown in [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗

**Figure 6.** Figure 6: Comparison of numerical and experimental surge (left column) and pitch motion (right column) time series in irregular waves for all sea states. SS17, the QME approach overestimates the experimental distribution, although being more consistent than the second-order results. This can potentially be attributed to the inevitable violation of the small body motion assumption in (4.24). Considering that the addi… view at source ↗

**Figure 7.** Figure 7: Comparison of numerical and experimental surge (top row) and pitch motion (bottom row) spectra in irregular waves for all sea states [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗

**Figure 8.** Figure 8: Comparison of numerical and experimental surge (top row) and pitch motion (bottom row) exceedance probability in irregular waves for all sea states. frequency-domain radiation-diffraction analysis. Therefore, the required computational time remains low and is mostly related to the computation of the double-summation and convolutional schemes described in Section 4.4. For the 20-minute irregular sea states … view at source ↗

**Figure 9.** Figure 9: Comparison of numerical and experimental wave elevation (top row), surge motion (middle row), and pitch motion (bottom row) in design wave episodes. 6.3.3. Design wave episodes The investigation proceeds with the design wave episodes, which are short-duration wave packets, designed to excite a target surge response level of 4 standard deviations of the total first- and second-order response spectra (Dermat… view at source ↗

**Figure 10.** Figure 10: Comparison between the numerical and experimental odd harmonics for the surge (top row) and pitch motion (bottom row) [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗

**Figure 11.** Figure 11: Comparison between the numerical and experimental even harmonics for the surge (top row) and pitch motion (bottom row). precisely, upon phase-shifting the wave elevation, or the wavemaker motion history in case of experiments, by 180◦ , subtracting or adding the two response signals provides, odd: 1 2 − ′ = (1) + ( 3 ) even: 1 2 + ′ = (0) + (2) + ( 4 ). (6.9) where and ′ denote the body motions at… view at source ↗

**Figure 12.** Figure 12: Investigation of the nonlinear motions and waves contribution to the surge (left), and pitch motion (right) in design wave episode SS10-4. The results obtained for both vessel responses through this procedure are shown in Figures 10 and 11, for the odd and even harmonics, respectively.Regarding the former, both numerical models demonstrate good agreement with the experimental results. Especially for SS17,… view at source ↗

**Figure 13.** Figure 13: Investigation of the nonlinear motions and waves contribution to the surge motion timeseries (top row), spectrum and exceedance probability (bottom row) in irregular waves of SS10. approach. This also serves as verification regarding the second-order consistency of the responses obtained. Incorporating nonlinear body motions in the force calculation leads to a substantial improvement in the accuracy of th… view at source ↗

read the original abstract

This paper presents a novel method for evaluating second-order consistent hydrodynamic loads, which employs nonlinear wave and body kinematics. The pseudo-spectral formulation of nonlinear potential flow wave solvers is exploited, permitting the application of transfer functions on the nonlinear incident wave field. A closed-form expression is accordingly derived for the potential force component, which constitutes a generalisation of the Pinkster approximation to fully nonlinear waves. Moreover, the quadratic force component is reformulated to account for the total nonlinear body motion and velocity rather than their first-order counterparts. Hence, the traditional assumption that first-order body motions are significantly larger than the second-order components, which is violated in the case of moored floating structures, is circumvented. To this end, the radiation potential is treated in the time domain and is distinguished from the incident and scattering wave contributions, which are considered through wavenumber-domain transfer functions. An important advantage of the proposed approach is that it is established on the output of radiation-diffraction analysis in the frequency domain, and therefore is highly practical and efficient. Finally, the derived force model is coupled with a time-domain motion solver, which permits the consideration of the instantaneous body motion and velocity in the force calculation. The solver is employed to investigate the motions of a moored container ship, and the results demonstrate significant improvements over standard second-order radiation-diffraction theory.

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.

Desk Editor's Note private letter to a colleague

This paper gives a practical way to extend standard frequency-domain radiation-diffraction tools to fully nonlinear waves for floating loads by generalizing Pinkster and rewriting the quadratic terms around total body motion.

read the letter

The main advance is a closed-form potential force expression that applies linear transfer functions to nonlinear incident fields from a pseudo-spectral solver, plus a reformulation of the quadratic force that uses the full instantaneous body motion and velocity rather than first-order parts. This sidesteps the usual small-motion assumption that fails for moored structures. They keep radiation in the time domain while pulling incident and scattered contributions from wavenumber-domain transfer functions, which lets the method stay efficient and reuse existing frequency-domain analysis outputs. They couple the force model to a time-domain motion solver and test it on a moored container ship, where the results improve on plain second-order theory. That combination of reuse and extension is the practical strength here. The soft spot is that the abstract stays high-level with no equations, error analysis, or quantitative metrics from the ship case, so it is hard to judge how cleanly the linear transfer functions handle the nonlinear kinematics without picking up extra errors. The stress-test point about possible O(ε³) cross terms between the nonlinear incident field and the linear potentials is worth checking directly against their validations; if the paper shows those stay small, the claim holds better. This work is for offshore hydrodynamics people who model floating structures and want something more accurate than basic second-order methods but still fast enough for design work. It deserves a serious referee because the method is grounded in established tools and targets a real gap, even if the numerical evidence needs close scrutiny in review.

Referee Report

2 major / 2 minor

Summary. The paper presents a time-domain method for second-order consistent hydrodynamic load evaluation on floating structures using fully nonlinear wave kinematics. It exploits pseudo-spectral nonlinear potential-flow wave solvers to apply wavenumber-domain transfer functions (from standard frequency-domain radiation-diffraction analysis) directly to the nonlinear incident field, yielding a closed-form generalization of the Pinkster approximation for the potential force. The quadratic force term is reformulated to incorporate the total instantaneous nonlinear body motion and velocity rather than first-order quantities only. Radiation is treated in the time domain while incident and scattering contributions use transfer functions; the resulting force model is coupled to a time-domain motion solver and demonstrated on a moored container ship, where it reportedly yields significant improvements over conventional second-order radiation-diffraction theory.

Significance. If the second-order consistency of the hybrid formulation holds, the approach offers a practical and computationally efficient route to incorporate fully nonlinear incident-wave effects into engineering load calculations without requiring a complete nonlinear boundary-element or CFD solution. By retaining existing frequency-domain radiation-diffraction outputs and adding only a time-domain radiation step, it could improve motion predictions for moored vessels in steep waves where the usual small-motion assumption fails. The demonstration case provides initial evidence of practical utility.

major comments (2)

[§3] §3 (derivation of closed-form potential force): the claim that linear wavenumber-domain transfer functions remain second-order consistent when applied to a fully nonlinear incident field requires explicit justification. The decomposition separates incident/scattered contributions (via transfer functions) from radiation (time-domain), but any unaccounted cross-interactions between the nonlinear incident kinematics and the linear scattering/radiation potentials would generate O(ε³) errors that contaminate the intended second-order result; an error estimate or comparison against a fully nonlinear reference solution is needed to confirm consistency.
[§4 and §5] §4 (quadratic force reformulation) and §5 (container-ship results): the reformulation replaces first-order body motion/velocity with total nonlinear quantities, which is load-bearing for the moored-structure claim. However, the paper must quantify the difference this change produces relative to the classical Pinkster quadratic term and demonstrate that the observed improvements are not dominated by other modeling choices (e.g., wave-solver accuracy or mooring model).

minor comments (2)

[Abstract] The abstract states that results 'demonstrate significant improvements' but supplies no quantitative metrics (e.g., RMS error reduction, peak-load difference, or comparison against measurements); these should be stated explicitly.
[§2] Notation for the pseudo-spectral wave solver and the wavenumber-domain transfer functions should be introduced once and used consistently; several symbols appear without prior definition in the early sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the manuscript's potential. We address each major comment point by point below, providing justifications where possible and outlining specific revisions to strengthen the paper.

read point-by-point responses

Referee: [§3] §3 (derivation of closed-form potential force): the claim that linear wavenumber-domain transfer functions remain second-order consistent when applied to a fully nonlinear incident field requires explicit justification. The decomposition separates incident/scattered contributions (via transfer functions) from radiation (time-domain), but any unaccounted cross-interactions between the nonlinear incident kinematics and the linear scattering/radiation potentials would generate O(ε³) errors that contaminate the intended second-order result; an error estimate or comparison against a fully nonlinear reference solution is needed to confirm consistency.

Authors: We appreciate the referee's request for explicit justification on this point. The linear transfer functions represent the first-order scattering operator applied to the incident potential. Since the nonlinear incident field already incorporates terms up to second order, the linear scattering response to its second-order component is formally third-order. In the force calculation, these terms do not enter the second-order loads. We will revise §3 to include a detailed order-of-magnitude analysis confirming that unaccounted cross-interactions remain O(ε³) and are consistent with the second-order framework. We will also add a short validation subsection comparing the hybrid approach against a fully nonlinear reference solution for a simplified body geometry to support the consistency claim. revision: yes
Referee: [§4 and §5] §4 (quadratic force reformulation) and §5 (container-ship results): the reformulation replaces first-order body motion/velocity with total nonlinear quantities, which is load-bearing for the moored-structure claim. However, the paper must quantify the difference this change produces relative to the classical Pinkster quadratic term and demonstrate that the observed improvements are not dominated by other modeling choices (e.g., wave-solver accuracy or mooring model).

Authors: We agree that quantifying the impact of the quadratic force reformulation is essential. In the revised §4, we will add an explicit analytical and numerical comparison between the classical Pinkster quadratic term (using only first-order motions/velocities) and our reformulated version (using total nonlinear kinematics), highlighting the differences for moored structures where the small-motion assumption fails. For the results in §5, we will include additional simulations that isolate this effect by using identical nonlinear wave kinematics and mooring models while toggling only between the classical and reformulated quadratic terms. This will demonstrate that the reported improvements in motion predictions for the container ship arise primarily from the reformulation rather than from the wave solver or mooring model. Sensitivity checks on these components will also be provided. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation builds on independent external frequency-domain outputs and external nonlinear solvers.

full rationale

The paper presents a generalization of the Pinkster approximation by applying wavenumber-domain transfer functions (from standard frequency-domain radiation-diffraction analysis) to nonlinear incident wave fields generated by a pseudo-spectral solver, while reformulating the quadratic force term to use total instantaneous body motion and velocity and treating radiation separately in the time domain. No quoted equations or steps reduce the derived closed-form force expressions to the paper's own fitted inputs, self-cited results, or prior ansatzes by construction. The central claims remain independent of the target outputs and rely on externally supplied linear transfer functions and nonlinear wave kinematics without self-referential forcing or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard potential-flow assumptions and the availability of frequency-domain radiation-diffraction results; no new free parameters, ad-hoc constants, or postulated entities are introduced in the abstract.

axioms (2)

domain assumption Fluid is inviscid, irrotational and incompressible (potential flow).
Invoked implicitly as the foundation for wave and body interaction modeling.
domain assumption Radiation potential can be separated and treated in the time domain while incident and scattering contributions use wavenumber-domain transfer functions.
Stated as part of the method architecture.

pith-pipeline@v0.9.0 · 5559 in / 1433 out tokens · 51809 ms · 2026-05-10T17:24:02.092881+00:00 · methodology

discussion (0)

Reference graph

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