On the use of an advanced Kirchhoff rod model to study mooring lines

This paper presents and validates an advanced nonlinear Kirchhoff rod model enhanced with a penalty-based barrier function for simulating mooring lines, demonstrating its accuracy against established solutions and revealing key dynamic behaviors such as frequency-dependent regime transitions and axial-bending coupling under various loading conditions.

The Gist

Imagine you are trying to predict how a giant, heavy rope behaves when it's holding up a floating wind turbine in the middle of the ocean. This isn't just a simple rope; it's a complex system battling wind, waves, currents, and the friction of the ocean floor.

This paper introduces a new, high-tech "digital rope" called ARMoor (Advanced Rod Model for Mooring Lines) designed to simulate this behavior with incredible precision. Here is a breakdown of what the authors did, using simple analogies.

1. The Problem: Why do we need a new model?

Currently, engineers use different tools to design these ropes (mooring lines).

• The Old Way: Some tools treat the rope like a simple chain of beads (Lumped Parameter Models). It's fast and easy, like counting beads on a string, but it misses the subtle "bending" and "twisting" nuances of a real rope.
• The New Way: The authors built a model based on Kirchhoff rod theory. Think of this as treating the rope not as beads, but as a continuous, flexible, yet stiff piece of spaghetti. It accounts for how the rope bends, stretches, and interacts with the water in a mathematically perfect way.

2. The Secret Sauce: The "Seabed Bouncer"

One of the hardest things to simulate is when the rope hits the ocean floor.

• The Challenge: If you just tell a computer "the rope can't go below the floor," the math gets messy and crashes.
• The Solution: The authors added a "Barrier Function." Imagine the ocean floor isn't a hard wall, but a giant, invisible trampoline made of stiff springs.
  • When the rope touches the floor, the springs push back.
  • The harder the rope pushes down, the harder the springs push back.
  • This allows the computer to smoothly calculate the rope sliding along the bottom without getting stuck or crashing the simulation.

3. The Water Dance: Drag vs. Added Mass

When the rope moves through water, the water fights back in two main ways:

• Drag (The Honey Effect): If you move slowly, the water feels like thick honey. It resists your speed. This is Drag.
• Added Mass (The Inertia Effect): If you try to move the rope very quickly, you aren't just moving the rope; you are also dragging a chunk of water along with it. It feels heavier. This is Added Mass.

The Big Discovery: The authors found a fascinating switch happens based on speed (frequency):

• Slow movements: The rope is dominated by Drag (the honey).
• Fast movements: The rope is dominated by Added Mass (the heavy water chunk).
• Why it matters: If you design a system for slow waves but a storm hits with fast, jerky movements, your old model might fail because it didn't account for this switch.

4. The Push and Pull: Tangential vs. Normal Forces

The team tested what happens when you shake the top of the rope (the "fairlead") in different directions:

• Shaking it sideways (Normal): The rope swings like a pendulum. The up-and-down motion stays mostly separate from the side-to-side motion. It's predictable.
• Shaking it along the rope's length (Tangential): This is where it gets wild. Pulling and pushing the rope lengthwise causes it to bend and wiggle in unexpected ways. It's like pulling a slinky; the tension travels down and creates a wave that makes the whole thing dance. The authors found that pulling the rope creates a strong "coupling" where stretching it causes it to bend, which is a complex, non-linear dance that simpler models miss.

5. The Proof: Did it work?

To prove their "digital spaghetti" model works, they ran three tests:

1. The Hanging Rope: They compared their model to the classic math formula for a hanging chain (a catenary). Result: It matched almost perfectly.
2. The Seabed Slide: They simulated a rope lying on the ocean floor and compared it to established engineering software. Result: The "touchdown point" (where the rope leaves the floor) matched within 1-2%.
3. The Real Wind Turbine: They simulated the mooring lines for a massive 15MW floating wind turbine (the IEA 15MW) and compared their results to OpenFAST, the industry-standard software used by everyone.
   • Result: Their model was incredibly accurate, differing by less than 1% in tension and less than half a meter in position over an 850-meter rope.

The Bottom Line

This paper presents a super-accurate digital twin for floating wind turbine ropes. While it takes a bit more computer power to run than the simple "bead" models, it captures the complex physics of bending, seabed contact, and water resistance much better.

Why should you care?
As we move wind farms further out to sea where the waves are bigger and the water is deeper, we can't afford to guess. This new model helps engineers design safer, more efficient floating wind farms that won't snap in a storm, ensuring we get more clean energy from the ocean.

Technical Summary

1. Problem Statement

Floating Offshore Wind Turbines (FOWTs) rely on complex mooring systems to maintain station-keeping. The dynamic behavior of these systems is influenced by numerous factors, including hydrodynamic loads (drag, added mass), seabed interaction, and structural non-linearities (large rotations, bending).

Current industry-standard approaches face limitations:

• Static/Quasi-Static methods: Often neglect hydrodynamic and inertial forces, leading to load underestimation.
• Frequency-Domain methods: Limited to linear analysis and often decouple the mooring line from the floater dynamics.
• Lumped-Parameter Models (LPMs): Widely used (e.g., in OpenFAST/MoorDyn) but rely on discrete mass-spring-damper elements. They may struggle with capturing continuous curvature effects and complex non-linear couplings between axial and bending dynamics.
• Distributed-Parameter Models (DPMs): While more accurate, many existing formulations neglect torsion/shear or lack robust methods for simulating complex seabed contact and fluid-structure interaction (FSI) simultaneously.

The authors aim to develop and validate an Advanced Rod Model for Mooring Lines (ARMoor) based on a nonlinear Kirchhoff rod formulation to address these gaps, specifically focusing on static/dynamic behavior, seabed contact, and FSI.

2. Methodology

The paper proposes a modified nonlinear, shear- and torsion-free Kirchhoff rod model enhanced with specific features for offshore applications.

A. Theoretical Formulation

• Governing Equations: Derived from Hamilton's principle of least action. The rod is modeled as a transversely isotropic curve $\phi(s)$ with no shear deformation and no torsional energy contribution.
• Seabed Contact (Barrier Function): To simulate the unilateral constraint of the seabed, the authors introduce a penalty-based barrier function into the Lagrangian. This replaces inequality constraints with a penalizing term that forces the solution to remain within the feasible region (above the seabed). Several barrier types (logarithmic, reciprocal, quadratic, etc.) were evaluated, with the reciprocal barrier selected for its stability in dynamic relaxation.
• Fluid-Structure Interaction (FSI): A simplified fluid model accounts for non-conservative forces:
  • Added Mass: Inertial effects due to fluid acceleration.
  • Drag Forces: Both tangential and normal components (quadratic in velocity).
  • Buoyancy: Hydrostatic forces.
• Spatial Discretization: The model utilizes Isogeometric Analysis (IGA) with B-splines. This allows for higher-order continuity ($C^r$) and eliminates the need for a separate director field variable, improving accuracy in curvature representation.
• Time Integration: An implicit second-order scheme (hybrid of midpoint and trapezoidal rules) is used. This scheme is designed to preserve linear and angular momenta and approximately conserve energy.

B. Numerical Strategy

• Static Equilibrium: Since the barrier function introduces numerical instability in pure static solvers, the authors employ a dynamic relaxation approach. The system is simulated dynamically with fluid damping until it settles into a static equilibrium position.
• Validation Cases:
  1. Elastic Catenary: Comparison against analytical 3D elastic catenary solutions.
  2. Seabed Contact: Comparison against established catenary-based solutions for lines resting on the seabed (touchdown point and reaction forces).
  3. Pulsating Loads: Analysis of a mooring line under harmonic forces at the fairlead (tangential vs. normal) to study frequency-dependent regimes.
  4. Full System Simulation: Application to the IEA 15MW FOWT on the VolturnUS-S platform, comparing results against OpenFAST (using the MoorDyn module).

3. Key Contributions

1. Novel Formulation: The first application of the specific nonlinear Kirchhoff rod formulation (from [35]) to offshore mooring lines, explicitly incorporating seabed contact via barrier functions and simplified FSI within a single framework.
2. Barrier Function Analysis: A detailed parametric study on the selection of barrier functions (logarithmic vs. reciprocal) and penalty parameters ($\mu$) for simulating seabed contact, highlighting the trade-offs between constraint enforcement and numerical stability.
3. Dynamic Coupling Insights: Discovery of strong coupling between axial and bending dynamics under tangential forcing, contrasting with normal forcing where axial dynamics remain largely decoupled.
4. Regime Transition Identification: Clear demonstration of the transition from a drag-dominated regime at low frequencies to an added-mass-dominated regime at high frequencies.
5. High-Fidelity Validation: Comprehensive verification against both analytical solutions and industry-standard software (OpenFAST), demonstrating the model's capability to predict complex non-linear behaviors.

4. Key Results

A. Static and Contact Verification

• Elastic Catenary: ARMoor showed excellent geometric agreement with analytical catenary solutions. The model predicted slightly stiffer behavior due to the inclusion of bending stiffness (which is zero in pure catenary theory).
• Seabed Contact: The dynamic relaxation approach successfully found static equilibrium for lines resting on the seabed. The reciprocal barrier function with a penalty parameter $\mu = 25$ provided the best balance.
  • Results for touchdown points and fairlead tensions matched reference solutions (from [6]) with errors generally < 2%.
  • The study highlighted that small changes in fairlead position lead to significant shifts in the touchdown point, confirming the geometric stiffness of mooring lines.

B. Pulsating Force Analysis (ROD-D)

• Frequency Response:
  • Normal Forcing (ROD-DN): The system behaved like a damped mass-spring system. At low frequencies, normal drag dominated; at high frequencies (>3 Hz), added mass became the dominant force.
  • Tangential Forcing (ROD-DT): Revealed a strong coupling between axial and bending dynamics. The phase portraits indicated complex non-linear behavior, including period-2 motion and potential transitions toward chaos, which are not captured by linear models.
• Energy Balance: The numerical scheme successfully preserved energy balance ($\Delta E = W_{nc}$), confirming the absence of artificial numerical damping or energy generation.

C. Full System Simulation (VolturnUS-S)

• Comparison with OpenFAST: ARMoor was compared against OpenFAST (MoorDyn) for the IEA 15MW turbine under step-wind loading.
  • Static Configuration: The final equilibrium positions of the fairleads matched OpenFAST with a maximum difference of 0.73%.
  • Dynamic Response: Under step-wind loads, the mean absolute difference (MAD) in fairlead position was approximately 17 cm (X-axis) and 12 cm (Z-axis) over an 850m line length.
  • Tension: Reaction forces at the clamped end differed by less than 1%.
• Observation: ARMoor exhibited a "stiffer" response than OpenFAST. This is attributed to the continuous curvature constraints inherent in the Kirchhoff rod formulation, whereas OpenFAST uses a lumped-mass approach better suited for chains but less smooth in curvature transitions.

5. Significance and Limitations

Significance:

• The study validates that advanced geometric rod models can serve as high-fidelity alternatives to lumped-mass models for FOWT mooring analysis.
• It provides critical insights into the non-linear dynamics of mooring lines, particularly the coupling effects of tangential loads and the frequency-dependent dominance of hydrodynamic forces.
• The model is capable of integration with multibody dynamics for full system simulation, offering a path toward more accurate design and safety assessments for deep-water wind farms.

Limitations:

• Cross-Section: The current model assumes a circular cross-section and does not support arbitrary or composite cross-sections.
• Material: Only linear elastic material behavior is considered; viscoplasticity (creep) is not included.
• Computational Cost: Being a distributed-parameter model with higher-order continuity, ARMoor is computationally more expensive than lumped-parameter models like MoorDyn.

Conclusion:
The paper successfully demonstrates that the ARMoor framework is a robust, accurate, and novel tool for simulating mooring lines under complex environmental loads. It bridges the gap between simplified industry tools and high-fidelity theoretical models, offering new capabilities for analyzing non-linear dynamic phenomena in offshore wind energy.