Fully nonlinear simulation of resonant motion of liquid confined between floating structures

A finite element based numerical method is employed to analyze the resonant oscillations of the liquid confined within multiple floating bodies based on fully nonlinear wave theory. The velocity potentials at each time step are obtained through the finite element method (FEM) with quadratic shape functions. The matrix equation of the FEM is solved through an iteration. The waves at the open boundary are absorbed by the method of combination of the damping zone method and the Sommerfeld–Orlanski equation. Numerical examples are given by floating two rectangular cylinders, two wedge-shaped cylinders and two semi elliptic cylinders undergoing forced oscillations at the resonant frequencies. The numerical results are compared with the first and second order solutions by previous study, which showed that the first order resonance happens at the odd order natural frequencies ω2i-1(i=1,2,⋯)ω2i-1(i=1,2,⋯) and second order resonance at the half of even order ω2i/2(i=1,2,⋯)ω2i/2(i=1,2,⋯) for antisymmetric motions and happen at ω2i(i=1,2,⋯)ω2i(i=1,2,⋯) and ω2i/2(i=1,2,⋯)ω2i/2(i=1,2,⋯) for symmetric motions, and it is found that they are in good agreement in smaller amplitude motions within a sufficiently long period of time. However, difference begins to appear as time increases further. This happens even in smaller amplitude motions. In all the calculated cases, the results always become periodic in time eventually.