A coupled-mode technique for weakly nonlinear wave interaction with large floating structures lying over variable bathymetry regions

A coupled-mode method is developed and applied to hydroelastic analysis of large floating platforms of shallow draft over variable bathymetry regions, characterised by parallel bottom contours. We consider the scattering problem of surface waves, under the combined effects of variable bathymetry and a semi-infinite floating elastic plate, in the time domain. The present development is based on appropriate generalisation of the unconstrained variational principle of Luke [Luke JC. A variational principle for a fluid with a free surface. J Fluid Mech 1967;27:395–7], which models the evolution of nonlinear water waves in intermediate water depth over a general bathymetry. Assuming small plate deflections and neglecting the rotation of plate section, the large floating structure has been modelled as a thin elastic plate. The present approach is based on appropriate extensions of the nonlinear coupled-mode model developed by Athanassoulis and Belibassakis [A nonlinear coupled-mode model for water waves over a general bathymetry. In: Proc. 21st international conference on offshore mechanics and arctic engineering OMAE 2002. 2002] for waves propagating in variable bathymetry regions. In order to consistently treat the wave field beneath the elastic floating plate down to the sloping bottom boundary, a complete, local-mode series expansion of the wave field is used, enhanced by appropriate sloping-bottom and free-surface modes. The latter enable consistent satisfaction of the Neumann bottom-boundary condition on a general topography, as well as the kinematical conditions on the free surface and on the elastic plate surface. By introducing this expansion into the variational principle, an equivalent coupled-mode system of horizontal equations is derived, fully accounting for the effects of nonlinearity and dispersion. Boundary conditions are also provided by the variational principle, ensuring that the edges of the plate are free of moment and shear force. Numerical results concerning floating structures lying over sloping seabeds are presented, as obtained by simplifying the fully nonlinear coupled-mode system, keeping only up to second-order terms. The present method can be extended to treat large floating elastic bodies or structures characterised by variable thickness (draft), flexural rigidity and mass distributions.