A spectral element solution of the 2D linearized potential flow radiation problem

TL;DR

This paper introduces a scalable spectral element method for solving the 2D linearized potential flow radiation problem, demonstrating high accuracy, efficiency, and validation against benchmarks for floating offshore structures.

Contribution

The paper develops a spectral element Galerkin method with stable time integration for the potential flow problem, including validation and analysis of non-resolved energy effects.

Findings

Spectral convergence achieved for affine and curvilinear elements.

Computational effort scales approximately linearly with grid points.

Excellent agreement with benchmark results validates the solver.

Abstract

We present a scalable 2D Galerkin spectral element method solution to the linearized potential flow radiation problem for wave induced forcing of a floating offshore structure. The pseudo-impulsive formulation of the problem is solved in the time-domain using a Gaussian displacement signal tailored to the discrete resolution. The added mass and damping coefficients are then obtained via Fourier transformation. The spectral element method is used to discretize the spatial fluid domain, whereas the classical explicit 4-stage 4th order Runge-Kutta scheme is employed for the temporal integration. Spectral convergence of the proposed model is established for both affine and curvilinear elements, and the computational effort is shown to scale with $\mathcal{O}(N^p)$, with $N$ begin the total number of grid points and $p \approx 1$. Temporal stability properties, caused by the spatial resolution, are considered to ensure a stable model. The solver is used to compute the hydrodynamic coefficients for several floating bodies and compare against known public benchmark results. The results are showing excellent agreement, ultimately validating the solver and emphasizing the geometrical flexibility and high accuracy and efficiency of the proposed solver strategy. Lastly, an extensive investigation of non-resolved energy from the pseudo-impulse is carried out to characterise the induced spurious oscillations of the free surface quantities leading to a verification of a proposal on how to efficiently and accurately calculate added mass and damping coefficients in pseudo-impulsive solvers.