A Robust Computational Algorithm for a Class of Unsteady-State Free-Surface Models in Ship Hydrodynamics

Abstract In this paper, we present an efficient stable set polynomial of the complete bipartite graph algorithm (SPBA) to address the time-dependent free-surface behavior resulting from unsteady ship movements. Analytical expressions for the horizontal and vertical derivatives are derived using the transient free-surface Green's function (TFSGF). The fundamental concept of the suggested approach is to convert fourth-order nonlinear differential equations into a system of algebraic equations through an operational matrix of derivatives, employing an optimized choice of collocation points in order to achieve precise analytical solutions. Results are compared against the shifted Legendre wavelet method (SLWM) and analytical solutions, illustrating that SPBA produces similar accuracy to SLWM with smoother error growth, reduced computational cost, and slightly improved long-term stability, making it more suitable for large-scale simulations. Eigenvalue spectrum analysis further confirms SPBA's superior long-term accuracy and enhanced stability over SLWM. Also, SPBA has been used to calculate nondimensional added mass and damping coefficients for the zero-velocity radiation problem of a floating hemisphere. Overall, SPBA proves to be a reliable, flexible, and computationally attractive approach.