Computation of transient nonlinear ship waves using an adaptive algorithm

Abstract

Abstract An indirect boundary integral method is used to solve transient nonlinear ship wave problems. A resulting mixed boundary value problem is solved at each time-step using a mixed Eulerian– Lagrangian time integration technique. Two dynamic node allocation techniques, which basically distribute nodes on an ever changing body surface, are presented. Both two-sided hyperbolic tangent and variational grid generation algorithms are developed and compared on station curves. A ship hull form is generated in parametric space using a B-spline surface representation. Two-sided hyperbolic tangent and variational adaptive curve grid-generation methods are then applied on the hull station curves to generate effective node placement. The numerical algorithm, in the first method, used two stretching parameters. In the second method, a conservative form of the parametric variational Euler–Lagrange equations is used the perform an adaptive gridding on each station. The resulting unsymmetrical influence coefficient matrix is solved using both a restarted version of GMRES based on the modified Gram–Schmidt procedure and a line Jacobi method based on LU decomposition. The convergence rates of both matrix iteration techniques are improved with specially devised preconditioners. Numerical examples of node placements on typical hull cross-sections using both techniques are discussed and fully nonlinear ship wave patterns and wave resistance computations are presented.