Finite element analysis of nonlinear wave resonance by multiple cylinders in vertical motions

The finite element method (FEM) is employed to analyse the resonant oscillations of the liquid confined within multiple or an array of floating bodies with fully nonlinear boundary conditions on the free surface and the body surface in two-dimensions. The velocity potentials at each time step are obtained through the FEM with 8-node quadratic shape functions. The finite element linear system is solved by the conjugate gradient (CG) method with a symmetric successive overelaxlation (SSOR) preconditioner. 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 an array of floating wedge-shaped cylinders and rectangular cylinders. Results are provided for heaving motions and wave elevations, wave profiles and hydrodynamic forces are obtained. It is found the wave amplitude and the hydrodynamic force on a cylinder increase with their locations closer to the middle of the array at the first order resonant frequency and their second order component obtained by Fourier analysis becomes larger at the second order resonant frequency. Comparisons are also made in several cases with the results obtained from the second order solution in the time domain. The difference between the fully nonlinear result and the second order solution is discussed at both the first- and second-order resonant frequencies.