status of a mature and reliable model, which has been validated by cross-comparisons between different numerical codes (Nestegard 1994) or by comparison with experiments (Dommermuth et al 1988). In these conditions, it now offers a valuable power of insight into a wide variety of free surface problems, ranging from wave deformation and breaking (Dommermuth et al, 1988) to nonlinear wave body interactions (Cointe et al, 1990), coastal engineering (Grilli et al, 1993), long distance propagation of wave groups (Tulin et al, 1993), or computation of waves impacting on structures (Tanizawa & Yue 1992).
The implementation of the MEL method in three dimensions is much more difficult, and not only because of the larger number of unknowns. Of course, the typical size of the linear algebraic systems to be solved is one order of magnitude larger than in corresponding 2D applications, and the resulting memory and CPU requirements are a major issue, but other difficulties resulting from the three dimensionality of the domain must not be underestimated. This includes the accurate evaluation of the velocity components on moving surfaces, but also the implementation in three dimensions of numerical methods for the tracking of moving intersections lines between Dirichlet and Neumann boundaries, the possible regridding and/or smoothing techniques to be applied during the simulation and which are much more complicated than in two dimensions, where meshes are naturally structured and thus easy to manipulate.
These specific difficulties explain why, despite the continuous improvement of computer power and numerical methods, available results from three dimensional applications of the MEL method are still rare.
Some early publications directly tackled the problem of the diffraction of a nonlinear wave on a surface piercing body (Zhou & Gu 1990, Yang & Ertekin 1992, Chan & Calisal 1993). However, low order boundary element methods were used, with approximate treatment of free surface-body intersections and coarse discretizations, so that corresponding numerical results were merely qualitative.
At the same time, more refined numerical models were developed, based on higher order panel methods, and with special attention given to the specific difficulties of the three dimensional MEL scheme, such as the accurate computation of free surface velocities and gemetry, extrapolation techniques at the intersection lines, etc… In this category, first published results mainly concerned academic problems on simplified geometries. Kang & Gong (1990) computed waves produced by the forced motion of a submerged sphere, using a spline-based panel method with Adams-Bashforth-Moulton predictor-corrector scheme for the time stepping. Romate (1989), followed by Broeze (1993) developed a nonlinear simulation model for the propagation of waves explicitly given by a stream function theory. The interaction of these waves with bottom deformations was described in Broeze (1993). Their model is based on a second order panel method, with a fourth order Runge-Kutta scheme for the time stepping. The problem of the computation of overturning waves in three dimensions was successfully solved by Xü & Yue (1992). They used a boundary element method with bi-quadratic isoparametric curvilinear elements, under the assumption of double periodicity in horizontal directions, and without accounting for wave-body interactions, the overturning being induced by the imposition of non-zero pressure patches on parts of the free surface. More recently, Boo et al (1994) simulated the propagation of nonlinear irregular waves of moderate steepness, using an Eulerian scheme applied to a rectangular domain.
On the specific aspects of the three dimensional nonlinear radiation problem, recent advances have been reported by Beck et al (1993, 1994), using the so-called ‘desingularized' method. A variety of applications were described, including the computation of added mass and damping on a modified Wigley hull at forward speed, and showed satisfactory agreement with experimental results. However, no incoming waves were accounted for in the computations.
In this paper, we describe some of the recent advances of our own approach of the solution of fully nonlinear wave-body interaction problems in hree dimensions. This research started in Sirehna in 1989 and has since been mainly sponsored by the French Ministry of Defense (DRET), through successive research contracts. Unsteady linearized flows were first addressed, with the development and validation of a linearized numerical wave tank named ANSWAVE, including wave-body motion coupling and absorbing conditions, and based on a boundary element method (Ferrant 1991–1993). Nonlinear free