Fourier–Chebyshev approximation of Kochin functions in the Fourier–Kochin approach to panel methods

Abstract

The dominant computational task involved in the methods, widely called panel methods, currently used to compute 3D potential flows around large floating bodies such as ships and offshore structures (and other classes of dispersive waves such as flexural-gravity waves in very large floating structures or ice sheets) consists in evaluating N2 coefficients, called influence coefficients. These coefficients are associated with the flows created by distributions of singularities (sources, dipoles) over the N panels that approximate the surface of the body (ship, offshore structure) at every panel of the body surface. In contrast to the O(N2) computations required in existing panel methods, the dominant part of the computations of influence coefficients are independent of N in the Fourier–Kochin (FK) method if the Kochin functions in the FK flow representation are approximated by means of Fourier series and Chebyshev polynomials. The numerical analysis reported in this study shows that Fourier–Chebyshev approximations to the Kochin functions are feasible and indeed practical. In particular, the analysis shows that the number NF of basic Fourier integrals (the major and most difficult computational task) that must be evaluated within the approach to the numerical implementation of the FK method considered in the study is given by NF≈(600)2, i.e. is smaller than N2 if 600<N. Thus, one has NF≪N2 for typical panel numbers N=O(104), and the FK approach opens the way for a class of panel methods in which the dominant computations are independent of the number N of panels instead of O(N2) in existing panel methods. Another major advantageous feature of the FK method is that this approach circumvents the notorious difficulties involved in the numerical evaluation and the panel-surface-integration of the complicated Green functions associated with ship and offshore hydrodynamics.