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

This paper presents a novel, efficient time-domain method for evaluating second-order hydrodynamic loads on floating structures in fully nonlinear waves by reformulating force components to account for total nonlinear body motion and velocity, thereby overcoming the limitations of traditional first-order assumptions and demonstrating significant improvements in predicting the motions of moored ships.

The Gist

Imagine you are trying to predict how a giant floating ship will bob, sway, and twist in a stormy ocean. For decades, engineers have used a "linear" approach to this problem. Think of this like predicting the path of a leaf floating in a gentle stream: you assume the water moves in simple, predictable waves, and the leaf just follows along, never getting pushed too hard or moving too wildly.

However, real oceans aren't gentle streams. They are chaotic, with waves crashing into each other, piling up, and creating "rogue" waves. Furthermore, when a massive ship hits a big wave, it doesn't just bob gently; it heaves, rolls, and pitches violently. The old "leaf in a stream" math breaks down here because it assumes the ship's movement is small and the waves are simple.

This paper introduces a new, smarter way to calculate the forces on these floating giants, specifically for ships that are anchored (moored) in deep water, like floating wind farms or oil rigs.

Here is the breakdown of their new method using simple analogies:

1. The Problem with the Old "Recipe"

The traditional method (Second-Order Theory) is like following a recipe that says: "Add a little bit of wave force, then add a little bit of ship movement, and mix them."

• The Flaw: It assumes the ship's movement is tiny compared to the wave. But for anchored ships, the ship can swing back and forth wildly (resonance) due to the slow, heavy push of the waves.
• The Result: The old math gets confused. It tries to calculate the force based on where the ship was a moment ago, rather than where it is right now. It's like trying to aim a cannon at a moving target by calculating where the target was 10 seconds ago. You miss.

2. The New "Motion-Explicit" Approach

The authors (Dermatis, Bredmose, et al.) developed a method they call Quadratic Motion-Explicit (QME).

The Analogy: The Dance Floor
Imagine the ocean is a dance floor and the ship is a dancer.

• Old Method: The choreographer (the math) tells the dancer, "Step left because the music (the wave) is loud." The dancer moves, but the choreographer doesn't update the instructions until the next song. If the dancer stumbles or spins wildly, the choreographer is still calculating based on the "gentle step" they expected.
• New Method (QME): The choreographer is watching the dancer in real-time. They see the dancer stumble, spin, and lean. They instantly recalculate the force needed to keep the dancer safe, accounting for the dancer's actual wild movement, not just the "expected" gentle movement.

3. How It Works (The Magic Trick)

The genius of this paper is that they didn't have to invent a whole new, incredibly slow computer program to do this. They found a way to use existing, fast tools but upgrade them.

• The "Frequency Domain" (The Blueprint): Engineers usually use a fast, standard tool to get a "blueprint" of how waves hit a ship. This tool is fast but only works for simple, linear waves.
• The "Time Domain" (The Live Action): The new method takes that fast blueprint and runs it through a "time machine." Instead of just looking at the average wave, it looks at the instantaneous wave and the instantaneous ship position.
• The Hybrid: They take the "blueprint" (from the fast tool) and apply it to the "live action" (the real-time, messy, nonlinear waves and ship movements).

The Metaphor:
Think of the old method as using a static map to navigate a car in a blizzard. You know the road is there, but you don't know the snowdrifts are shifting right now.
The new method is like having a GPS with live traffic and weather updates. It uses the same map (the standard engineering data), but it constantly updates the route based on the car's actual current speed and the actual current snowdrifts.

4. Why This Matters

• Safety: Anchored ships (like floating wind turbines) are vulnerable to "slow-drift" motions. If the math underestimates the force, the mooring lines (the ropes holding the ship) could snap, causing the structure to drift away or capsize.
• Accuracy: The authors tested this on a model of a massive container ship in a wave tank. They found that their new method predicted the ship's "surge" (moving forward and backward) much more accurately than the old method, especially in rough seas.
• Efficiency: Usually, calculating these complex, real-time forces requires supercomputers and takes days. This new method is fast enough to run in minutes because it cleverly reuses the standard "blueprint" data.

The Bottom Line

This paper gives engineers a way to stop guessing how floating structures will behave in a storm. Instead of assuming the ship is a calm, obedient passenger, the new math treats the ship as a wild, active participant in the ocean's chaos. It ensures that when we build floating cities or wind farms in the deep ocean, they are designed to survive the real ocean, not just the theoretical one.

Technical Summary

1. Problem Statement

The accurate prediction of hydrodynamic loads and motions for moored floating structures (e.g., floating wind turbines, container ships) is critical for safety and design. Current standard methods rely on second-order radiation-diffraction theory in the frequency domain, which is then transformed into the time domain. However, these methods suffer from two principal limitations when applied to resonant low-frequency motions (slow-drift):

1. Linearized Kinematics: Traditional approaches assume that second-order body motions are negligible compared to first-order motions. This assumption is violated in moored structures where resonant low-frequency motions can be significant, leading to inaccuracies in pressure integration.
2. Inertial Frame Limitations: Standard formulations often rely on Taylor series expansions around a mean body surface. When large second-order motions occur, the assumption that the instantaneous wetted surface is close to the mean surface breaks down, introducing errors.
3. Computational Cost: Fully nonlinear time-domain solvers (e.g., Boundary Element Methods or Finite Element Methods) are computationally expensive, making them impractical for long-term simulations or design optimization.

The paper aims to develop a method that incorporates fully nonlinear wave kinematics and explicit nonlinear body motions while retaining the computational efficiency of frequency-domain radiation-diffraction analysis.

2. Methodology: The Quadratic Motion-Explicit (QME) Approach

The authors propose a novel Quadratic Motion-Explicit (QME) approach. This method bridges the gap between frequency-domain efficiency and time-domain nonlinearity.

Core Concepts

• Hybrid Formulation: The method utilizes the output of a standard linear frequency-domain radiation-diffraction analysis (transfer functions) but applies them in the time domain using total nonlinear variables (wave elevation and body kinematics) rather than decomposing them into first- and second-order perturbation series.
• Force Decomposition: The total hydrodynamic force $\mathbf{F}(t)$ is derived by integrating pressure over the mean body surface $S_0$, expanded via Taylor series, resulting in four components:
  1. Hydrostatic Force ($\mathbf{F}_B$): Unchanged from standard theory.
  2. Restoring Force ($\mathbf{F}_H$): Proportional to the total nonlinear motion $\boldsymbol{\xi}(t)$, not just the first-order component.
  3. Potential Force ($\mathbf{F}_P$): Derived from the time derivative of the velocity potential. It is split into:
     • Diffraction/Wave Force ($\mathbf{F}_{PW}$): Calculated by applying linear transfer functions to the time-dependent modal amplitudes of a fully nonlinear incident wave field (e.g., from HOS-NWT).
     • Radiation Force ($\mathbf{F}_{PR}$): Calculated using the Cummins equation with infinite-frequency added mass and radiation convolution kernels, driven by the instantaneous total body velocity and acceleration.
     • Scattering Corrections ($\mathbf{F}_{PF}, \mathbf{F}_{PM}$): Terms accounting for free-surface forcing and body motion effects, treated via QTFs or time-domain kernels.
  4. Quadratic Force ($\mathbf{F}_Q$): Represents the quadratic interaction of wave and motion quantities. Crucially, this term is reformulated to use total nonlinear velocities and positions rather than first-order approximations. It includes:
     • Quadratic pressure terms ($\nabla \phi \cdot \nabla \phi$).
     • Interaction between body motion and potential time derivatives.
     • Waterline integrals.

Implementation

• Input: The method accepts fully nonlinear wave fields (e.g., from High-Order Spectral methods like HOS-NWT) and couples them with a time-domain motion solver.
• Efficiency: It avoids solving the full boundary value problem in the time domain. Instead, it reconstructs potentials using precomputed linear transfer functions (RAOs and QTFs) and impulse response functions derived from frequency-domain analysis.
• Coupling: The force calculation is explicitly coupled with the motion solver (DTUMotionSimulator). At every time step, the instantaneous body position and velocity are used to evaluate the forces, creating a two-way coupling that captures nonlinear feedback.

3. Key Contributions

1. Generalization of Pinkster Approximation: The derived potential force component is a generalization of the Pinkster approximation, valid for fully nonlinear incident waves rather than just linear waves.
2. Motion-Explicit Quadratic Forces: The quadratic force component is reformulated to account for total nonlinear body motion and velocity. This circumvents the traditional assumption that second-order motions are negligible, significantly improving accuracy for resonant low-frequency responses.
3. Decoupling of Radiation and Incident Waves: By treating the radiation potential in the time domain based on instantaneous kinematics, the method partially decouples the radiated wave field from the incident wave field. This allows for the explicit inclusion of other loads (e.g., aerodynamic loads on wind turbines) in a coupled analysis.
4. Computational Efficiency: The method achieves high-fidelity nonlinear modeling with a computational cost comparable to standard second-order frequency-domain approaches (approx. 30 minutes for a 20-minute simulation vs. 5 minutes for standard theory, both significantly faster than fully nonlinear BEM).

4. Results and Validation

The method was validated against experimental data from a 1/65 scale 6750-TEU container ship in the Ocean Engineering wave tank at École Centrale de Nantes.

• Test Cases:
  • Three irregular sea states (SS6, SS10, SS17), with SS17 representing an extreme 1000-year return period condition.
  • Design wave episodes (short-duration wave packets) designed to trigger extreme surge responses.
• Performance Metrics:
  • Surge Motion: The QME approach demonstrated significant improvements over standard second-order theory. It accurately captured the phase, amplitude, and low-frequency spectral content of the surge motion, particularly in the resonant region.
  • Pitch Motion: As pitch is predominantly a linear response, differences between QME and standard theory were less pronounced, though QME remained consistent with experiments.
  • Extreme Conditions (SS17): In the most severe sea state, the QME approach slightly overestimated the surge excursion magnitude. The authors attribute this to the violation of the "small motion" assumption in the Taylor expansion of the body surface under extreme conditions. However, QME remained more consistent with experiments than the standard second-order model.
• Sensitivity Analysis:
  • The study isolated the contributions of nonlinear waves vs. nonlinear body motions.
  • Finding: For the tested configuration, the use of instantaneous total body motions in the force calculation was the primary driver of improved accuracy. The inclusion of nonlinear wave kinematics (bound waves) had a secondary effect, though this may vary for other configurations.

5. Significance and Conclusion

This paper presents a robust, efficient, and practical framework for evaluating hydrodynamic loads on floating structures.

• Practicality: By leveraging existing frequency-domain radiation-diffraction outputs (standard in industry tools like Hydrostar or WAMIT), the method avoids the need for complex, computationally expensive fully nonlinear time-domain solvers.
• Accuracy: It resolves the inaccuracies associated with large resonant motions in moored structures by explicitly treating nonlinear kinematics in the force evaluation.
• Applicability: The approach is particularly well-suited for floating wind turbines and other multi-physics systems where the forcing is not solely wave-driven, as the explicit treatment of body kinematics allows for seamless integration with aerodynamic or other external load models.

In summary, the QME approach offers a "best of both worlds" solution: the computational speed of frequency-domain methods with the physical fidelity of time-domain nonlinear kinematics, significantly enhancing the reliability of load predictions for extreme sea states.