A gradient method for solving the boundary value problem of the underwater large deformation cable

A nonlinear large deformation cable equation system based on arc length is employed to determine the configuration of the cable under current and buoyancy loads. Different from a traditional initial method and a shooting method, we have solved this nonlinear problem as a boundary value problem with nonlinear and global boundary conditions directly. A finite difference scheme is proposed to solve the large deformation cable equation, and the Newton–Raphson iteration is used to search for numerical approximate solution. We demonstrate that this system degenerates into the catenary equation in the case of vanishing bending stiffness for the first time. The solution of the catenary equation serves as the initial guess for the three-section cable problem. This method overcomes the disadvantages of the initial value method and step method, avoiding the need to adjust the boundary location. The spatial shapes of the large deformation cable and the effects of the length and position of the buoyancy section are discussed. The impact of ocean currents is also analyzed using Morison's formula.