Highly Precise Approximation of Free Surface Green Function and Its High Order Derivatives Based on Refined Subdomains

Abstract

SummaryThe infinite depth free surface Green function (GF) and its high order derivatives for diffraction and radiation of water waves are considered. Especially second order derivatives are essential requirements in high-order panel method. In this paper, concerning the classical representation, composed of a semi-infinite integral involving a Bessel function and a Cauchy singularity, not only the GF and its first order derivatives but also second order derivatives are derived from four kinds of analytical series expansion and refined division of whole calculation domain. The approximations of special functions, particularly the hypergeometric function and the algorithmic applicability with different subdomains are implemented. As a result, the computation accuracy can reach 10 -9 in whole domain compared with conventional methods based on direct numerical integration. Furthermore, numerical efficiency is almost equivalent to that with the classical method.

Enhanced Endo's approach for evaluating free-surface Green's function with application to wave-structure interactions

“…Computational techniques related to free-surface Green's function in infinite water depth have been well established by many scholars. Important works have been undertaken in developing rigorous formulations, effective algorithms & computer codes (e.g., Newman 1985Newman , 1992Telste and Noblesse, 1986;Ponizy et al, 1994;Chen 1993Chen , 2004Peter and Meylan, 2004;Clement, 2013;Wu et al, 2017;Shan and Wu, 2018;etc. ).…”

“…The paper presents a newly developed enhanced algorithm based on Endo's approach, which employs a direct Gauss-Laguerre quadrature in the calculation of the principal value integrations. This methodology was firstly originated from Endo (1983), and had been subsequently improved by Liu et al (2008) and Shan et al (2018). Unfortunately, the latest version of Endo's approach still has two fatal problems with (1) incredibly large or small values at some "weird frequencies", and (2) nonsense values in the high-frequency region under a large depth.…”

“…The developed algorithm is thereafter tested against a variety of sea conditions, via a comparison with Newman's polynomial algorithm (Newman, 1985), certifying its good accuracy in various circumstances. By carrying out a benchmark test on the DeepCwind semisubmersible, through a comparison with Shan's algorithm (Shan et al, 2018) and the commericial software Hydrostar®, the cause of the occurrence of the "weird frequencies" is found, in association with other important findings. The computations demonstrate that the present enhanced algorithm is thoroughly free from the preceding problems and is sufficiently accurate to compute wave loads in practice.…”