Development of an efficient method for tracking surface intersections with applications in time-domain ship hydrodynamics

Abstract

Accurate simulation of wave-body interaction is a fundamental requirement in ship hydrodynamics. The existing two-dimensional cBEM solver provides a time-domain framework for such simulations but is restricted to two dimensions. The extension to three dimensions introduces two geometric challenges that must be resolved at every time step: the construction of the waterline curve at the intersection of the hull and the free surface, and the estimation of missing free surface data in regions newly exposed by the moving hull. This thesis addresses both challenges as independent contributions towards the three-dimensional cBEM extension. The first is a six-stage geometry intersection pipeline for robust waterline computation, implemented using Open CASCADE Technology as the computational geometry kernel. The pipeline produces a closed, consistently oriented and globally parameterised waterline curve, together with the explicit separation of the hull into its submerged and above-waterline parts. Developed algorithmic contributions include a breadth-first search based shell separation, a parametric direction correction by endpoint matching to ensure consistent orientation of the intersecting waterline curve, and an arc-length parameterisation framework with a global-tolocal parameter lookup. The second contribution is a Z-Splines based extrapolation framework on bounded domains for filling missing free surface data. The framework derives boundarycorrected basis functions that restore polynomial reproduction near domain endpoints, which is essential for accurate derivative evaluation at the boundary. Two approaches are developed for computing the boundary derivatives required by the Taylor-step extrapolation: a numerical Taylor-system approach and an analytic approach based on explicit piecewise formulas for the Z-Spline kernel derivatives. A one-step extrapolation error bound of O(hm) is proved for Z-Spline order m in the grid spacing h, and a recursive error propagation analysis is developed that quantifies error accumulation across successive extrapolation steps. Both contributions are validated independently as concrete steps towards the full three-dimensional solver.