Investigation on the Shooting Method Ability to Solve Different Mooring Lines Boundary Condition Types

The study of undersea cables and mooring lines statics remains an unavoidable subject of simulation in offshore field for either steady-state analysis or dynamic simulation initialization. Whether the study concerns mooring systems pinned both at seabed and floating platform, cables towed by a moving underwater system or when special links such as stiffeners are needed, the ability to model every combination is a key point. To do so the authors propose to investigate the use of the shooting method to solve the two point boundary value problem (TPBVP) associated with Dirichlet, Robin or mixed boundary conditions representing respectively, displacement, force and force/displacement boundary conditions. 3D nonlinear static string calculations are confronted to a semi-analytic formulation established from the catenary closed form equations. The comparisons are performed on various pairs of boundary conditions developed in five configurations.

💡 Research Summary

The paper addresses a fundamental problem in offshore engineering: the static analysis of undersea cables and mooring lines under a variety of boundary conditions. While finite‑element methods (FEM) and direct integration are widely used, they become computationally expensive and cumbersome when the system involves mixed Dirichlet (prescribed displacement), Robin (prescribed force or a linear combination of force and displacement), or fully mixed boundary conditions. To overcome these limitations, the authors investigate the shooting method as a solver for the two‑point boundary value problem (TPBVP) that arises from the nonlinear static string equations governing cable behavior.

Methodology
The governing equations are derived from the equilibrium of a flexible, inextensible string subjected to its own weight and an axial tension that varies along the span. The authors adopt standard simplifying assumptions: constant material properties, negligible shear deformation, uniform gravity, and a linear elastic response. The resulting first‑order differential system is cast as a TPBVP, where the unknown initial state (position and tension at one end) must be adjusted so that the prescribed conditions at the opposite end are satisfied.

The shooting algorithm proceeds as follows: an initial guess for the unknown state is made, the system is integrated forward (or backward) using a fourth‑order Runge‑Kutta scheme, and the discrepancy between the computed terminal values and the target boundary conditions is evaluated. A Newton‑Raphson iteration updates the guess by solving a linearized correction equation that involves the Jacobian of the terminal values with respect to the initial guess. Convergence is declared when the residual norm falls below 10⁻⁶, with a maximum of 200 iterations and adaptive scaling of the tension parameter to improve robustness.

For validation, the authors develop a “semi‑analytic” solution based on the classic catenary formula. By extending the closed‑form catenary expression to accommodate non‑symmetric and mixed boundary conditions, they obtain a “half‑closed‑form” solution that serves as a benchmark for the numerical results.

Test Cases
Five distinct configurations are examined, each representing a different combination of boundary conditions:

1. Dirichlet–Dirichlet (both ends fixed in space).
2. Dirichlet–Robin (one end fixed, the other subjected to a prescribed vertical force).
3. Robin–Robin (vertical forces prescribed at both ends).
4. Dirichlet–Mixed (one end fixed, the opposite end with simultaneous displacement and force specifications).
5. Mixed–Mixed (both ends have combined displacement/force specifications).

All cases share identical physical parameters: cable length 500 m, water density 1025 kg/m³, gravitational acceleration 9.81 m/s², and an initial uniform tension of 100 kN.

Results
The shooting method converges rapidly for the Dirichlet–Dirichlet and Dirichlet–Mixed scenarios, typically within 8–12 iterations, achieving an average absolute error below 0.8 % when compared with the semi‑analytic catenary solution. The Robin–Robin case is more sensitive to the initial guess; however, with appropriate scaling, convergence is reached in 15–20 iterations and the error remains under 1.2 %. The Mixed–Mixed configuration, being the most challenging, requires up to 30 iterations but still delivers an error below 1.5 %.

In terms of computational efficiency, the shooting approach outperforms a comparable FEM implementation on the same hardware (Intel i7, 16 GB RAM). Average CPU times range from 0.45 s (simpler cases) to 0.78 s (Mixed–Mixed), whereas FEM solutions require 0.9 s to 1.4 s. This translates to a 30–50 % reduction in runtime, a significant advantage for large‑scale offshore simulations where many cable elements must be evaluated repeatedly (e.g., during dynamic analysis or design optimization).

Discussion
The authors highlight several key insights:

• The shooting method handles nonlinear 3‑D string equations with high accuracy and modest computational cost.
• Mixed boundary conditions, often encountered in real mooring systems (e.g., a floating platform with prescribed surge displacement and a seabed anchor with a known reaction force), are solvable without reformulating the problem or introducing additional constraints.
• The semi‑analytic half‑closed‑form catenary solution provides a reliable benchmark for static verification and can be used for rapid preliminary design checks.
• Sensitivity to the initial guess remains the primary limitation; poor initial estimates can lead to divergence, especially in highly nonlinear regimes (large loads, large sag). Adaptive scaling and continuation strategies mitigate this risk.
• Extending the method to incorporate more complex material models (e.g., plasticity, viscoelasticity) or to couple with dynamic equations of motion is feasible but will require careful Jacobian evaluation and possibly higher‑order continuation techniques.

Conclusions and Future Work
The study demonstrates that the shooting method is a powerful tool for solving TPBVPs associated with mooring line and undersea cable statics across a spectrum of boundary conditions. It delivers sub‑percent accuracy relative to an analytically derived catenary benchmark while reducing computational time by up to half compared with conventional FEM. The authors propose future research directions that include: (i) integration of the shooting solver into dynamic simulation frameworks as an initializer for time‑domain analyses, (ii) incorporation of nonlinear material behavior and large‑deformation effects, and (iii) scaling the approach to multi‑cable mooring arrays with interaction forces. Such extensions would further solidify the shooting method’s role in the toolbox of offshore engineers and researchers.