Comparative analysis of the methods to compute the radiation term in Cummins’ equation

Abstract

In the present paper, the methods provided in literature to compute the convolution integral in Cummins’ equation are compared. Direct computation of the convolution integral is revised to avoid truncation errors and to save computational cost. The three methods compared are the direct computation of the convolution integral, the approximation of the integral by a state space and the approximation of the impulse response function by Prony’s coefficients. These methods are used to simulate the movement of the water inside an oscillating water column (OWC) and a decay test in heave of a spar buoy. Cummins’ equation results in a system of ordinary differential equations with all the methods. All systems are computed using the same numerical scheme obtaining a fair comparison of the computational cost involved in each method. The results of the OWC are compared against CFD results and the results of the buoy against laboratory experiments. Results obtained by direct computation of the convolution integral show sensitivity to the time step used to precompute the impulse response function, while using state space or Prony’s approximations are dependent on the set of frequencies required for the identification of their coefficients. State space and Prony’s approximations evaluate the radiation force, including it in the matrix of the system, while direct integration computes it outside of the matrix. This modification in the matrix makes these approximations more sensitive to the data used to evaluate the radiation force.