The shooting methods to solve 3D nonlinear strings assemblies

Abstract: This article presents an alternative approach to finite elements for modeling and analyzing 3D static mooring lines using string theory and the shooting method (SM) to solve two-point boundary value problems (TPBVPs) for 3D nonlinear static string equations with various boundary condition (BC) types relevant to offshore slender system assemblies.The two-point boundary value problem for nonlinear extensible elastic strings was formulated by incorporating arbitrary 3D external distributed loads that are not restricted to gravity alone. The TPBVP was reformulated based on the formalism of the shooting method. A multi-body/multi-shooting approach is proposed to handle multi-material segments and line assemblies. A formulation of the boundary conditions that allows the modeling of Dirichlet, Robin, and mixed boundary conditions representing the displacement, force, and combined force/displacement constraints is presented.Four validation cases are presented, comparing the results to analytical solutions: (1) a single catenary segment with ball-prismatic joint boundary conditions under several imposed forces, (2) a review of all possible boundary conditions for strings, including spring-based BCs, (3) the Velaria problem with nonlinear radial distributed load, and (4) a three-segment hanging configuration with different material properties connected by a buoy.The results demonstrate an accuracy under 10-9 in terms of absolute errors for both positions and tensions along the entire length of the mooring lines. The proposed method also provides error control through adaptive step integration. It demonstrates high accuracy in modeling complex 3D kinematics and configurations for mooring lines, while limiting the iterative problem size.The proposed method provides an efficient alternative to discretization-based techniques for analyzing static configurations of string kinematics slender systems with various end constraints, such as mooring lines and hawsers assemblies in offshore engineering, while maintaining simplicity in approach and implementation.