Time domain analysis of second-order wave diffraction by an array of vertical cylinders

A time domain method is employed to analyze interactions of water waves and a group or an array of cylinders. The nonlinear free-surface boundary conditions are satisfied based on the perturbation method up to the second order. The first- and second-order velocity potential problems at each time step are solved through a finite element method (FEM). The mesh required is generated based on a 2-D unstructured grid on a horizontal plane and its extension in the vertical direction. The matrix equation of the FEM is solved through an iteration and the initial solution is obtained from the result at the previous time step. The radiation condition is imposed through a combination of the damping zone method and the Sommerfeld–Orlanski equation. Results for various configurations, including two cylinders, four cylinders, an array of 10 cylinders and two arrays of eight cylinders are provided to show the effect of the interaction and their behaviour near the trapped mode.