Comparison of existing methods for the calculation of the infinite water depth free-surface Green function for the wave–structure interaction problem

Abstract

In this study, the mathematical expressions and numerical methods for the free-surface Green function of the linearized wave-structure problem in deep water and in the frequency domain are investigated. Twelve different expressions are reviewed and analyzed. All these expressions are exact mathematical solutions for the propagation of waves from a pulsating source located in the fluid domain. However, their numerical evaluation is challenging. Dedicated numerical methods have been developed. They include series expansions, polynomials, table interpolations, multipole expansions, approximations with elementary functions, etc. In this work, four methods were implemented: the Newman's method [1], the Delhommeau's method [2], the Telste-Noblesse's method [3] and the Wu et al.'s method [4]. Their CPU time and accuracy are compared. It is found that the average computational time for Newman's method is 5.745×10 −7. It is 5.782 × 10 −8 for the Delhommeau's method. For Telste-Noblesse's method and Wu et al.'s methods, they are 4.642 × 10 −8 and 1.491 × 10 −9 , respectively. The accuracy is respectively 6D(6 decimals), 5D and 3D for the Newman's method, the Telste-Noblesse's method and the Wu et al.'s method. For the Delhommeau's method, it is 3D except when the vertical coordinate is close to 0. The accuracy of the Delhommeau's method can be increased significantly by

A new ordinary differential equation for the evaluation of the frequency-domain Green function

“…The vertical derivative of the Green function can be obtained directly from the Green function itself [18], [6], [2]. In the following, we show that the same applies to the horizontal derivative.…”

“…However, the imaginary part can be expressed analytically [18], [6], [2]. Thus, in the following, we shall focus only on the evaluation of the real part of the Green function.…”

“…It does not required dividing the computational domain into multiple sub-domains. All those methods were recently reviewed in [18].…”

Capytaine: a Python-based linear potential flow solver

“…Thus, only the wet surface of the floating body needs to be discretized. The Green's function can be computed in several ways (Xie et al, 2018), which are implemented in several software, most of them commercial and closed-source.…”

“…The present work is nonetheless a step towards a fully modular code, in which the current Green's function evaluation routines could be used in another BEM solver (e.g. (Alouges & Aussal, 2018)), and conversely other methods for the computation of the Green's function (Xie et al, 2018) could be plugged in to the present user interface.…”