Use of Clement’s ODEs for the Speedup of Computation of the Green Function and its Derivatives for Floating or Submerged Bodies in Deep Water

Abstract

A new acceleration technique for the computation of first order hydrodynamic coefficients for floating bodies in frequency domain and in deep water is proposed. It is based on the classical boundary element method (BEM) which requires solving a boundary integral equation for distributions of sources and/or dipoles and evaluating integrals of Kelvin’s Green function and its derivatives over panels. The Kelvin’s Green function includes two Rankine sources and a wave term. In present study, for the two Rankine sources, analytical integrations of strongly singular kernels are adopted for the linear density distributions. It is shown that these analytical integrations are more accurate and faster than numerical integrations. The wave term is obtained by solving Clément’s ordinary differential equations (ODEs) [1] and an adaptive numerical quadrature is performed for integrations over the panels. It is shown here that the computational time of the wave term by solving the ODEs is greatly reduced compared to the classical integration method [7].

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

“…In this section, the results of the evaluation of the Green function and its horizontal derivative by using the ODE are compared to the results obtained using the direct integration method as described in [23]. Therefore, the ODE-based method can predict the Green function as accurately (6D accuracy) as existing alternative methods.…”

“…Thus, equations (23) and (24) provide a way to evaluate the horizontal derivative of the Green function from the Green function itself without having to solve an additional ODE.…”

“…Thus, the Green function can be evaluated by integrating numerically this ODE, provided that initial conditions are available. Since then, similar ODEs for the gradient of the frequency domain Green function have been established (see [23] and [24]).…”

“…Note that it is not explained in [24] how the initial conditions are obtained for ω = 1. In [23], initial conditions for arbitrary values of ω were obtained using methods relying on the numerical approximations of the singular integral.…”

Boundary element and integral methods in potential flow theory: a review with a focus on wave energy applications

“…A complete overview of the different analytical expressions for the Green function used in the literature, and a comparison of different approximations between the methods of Newman, Telste-Noblesse, Delhommeau, and Wu et al has been reported by Xie et al (2018b), for infinite depth problems. A conclusion of this article is that the most accurate approach is Newman's algorithm with 6 decimal place accuracy (DA), but it is also the slowest (10 − 7 s average computation time (ACT)).…”