Application Of The Fast Multipole Boundary Element To Fully Nonlinear Water Wave Problems

In this paper, the fully nonlinear water wave problems in open sea and in numerical wave tank are investigated on the basis of three-dimensional Laplace equation by a time-domain method.Since much larger computational cost and computer storage are required in the fully nonlinear time-domain problem, it is difficult to satisfy the calculation demands by the tranditional numerical method. In this paper, an efficient boundary element method accelerated by the fast multipole expansion technique is developed. Numerical examninations on the first boundary value problem, the second boundary value problem and symmetry problem in frequency-domain are carried out and comparisons are made with the traditional boundary element methods. The investigation shows that the fast multipole boundary element method is more efficient on the computational cost and computer storage on the premise of enough calculational accuracy.In the fully nonlinear time-domain model, the method of semi-Mixed Eulerian-Lagrangian is employed to caputure the fluid particles on the instantaneous free surface at each time step. As for the time marching scheme, the fourth-order Runge-Kutta method is adopted to integrate and update the velocity potential and wave elevation at the instantaneous free surface with respect to time. In order to make the calculation program possess good versatility and adaption, two damping layers are emplaced in front of the incident and the output boundaries respectively. They are used to absorb the waves propagating outwards and reflected from the surface-piercing body as sufficiently as possible. The aim is to prevent reflection on the output boundary and re-reflection on the input boundary respectively. Therefore, we can effectively simulate problems as long as possible within a finite distance after such technology is employed in the numerical calculation.At first, the fast multipole desingularized boundary element method is applied to the open-sea water wave problems including forced surging motions of objects with linear free surface boundary conditions and accelerated advance of source-sink, objects with fully nonlinear free surface boundary conditions. Subsequently, the fast multipole higher order boundary element method is applied to the numerical wave tank, in which the efficiency of the damping layersemplaced in front of the incident and the output boundaries is checked, and the propagation of fully nonlinear waves, sloshing problems and interaction of fully nonlinear waves with vertical cylinder are also simulated. As indicated above, all of the numercal results in this paper agree very well with the frequency-domain results and published time-domain results.