Towards Efficient Fully-Nonlinear Potential-Flow Solvers in Marine Hydrodynamics

Abstract

Solving potential-flow problems using the Boundary Element Method (BEM) is a strong tradition in marine hydrodynamics. An early example of the application of BEM is by Bai & Yeung [1]. The bottleneck of the conventional BEM in terms of CPU time and computer memory arises as the number of unknowns increases. Wu & Eatock Taylor [2] suggested that the Finite Element Method (FEM) field solver is much faster than the BEM based on their comparisons in a wave making problem. In this paper, we aim to find a highly efficient method to solve fully-nonlinear wave-body interaction problems based on potential-flow theory. We compare the efficiency and the accuracy of five different methods for the potential flows in two dimensions (2D), two of which are BEM-based while the other three are field solvers. The comparisons indicate that it is beneficial to use either an accelerated matrix-free BEM, e.g. Fast Multipole Method accelerated BEM (FMM-BEM), or any field solvers whose resulting matrix are sparse. Another highlight of this paper is that an efficient numerical potential-flow method named the harmonic polynomial cell (HPC) method is developed. The flow in each cell is described by a set of harmonic polynomials. The presented procedure has approximately 4th order accuracy, while its resulting matrix is sparse similarly as the other field solvers, e.g. Finite Element Method (FEM), Finite Difference Method (FDM) and Finite Volume Method (FVM). The method is verified by a linear wave making problem for which the steady-state analytical solution is available, and the forced oscillation of a semi-submerged circular cylinder for which the frequency-domain added mass and damping coefficients are compared. The fully-nonlinear wave making problem and nonlinear propagating waves over a submerged bar are also studied for validation purposes. Only 2D cases are studied in this paper.

Application of a 2D harmonic polynomial cell (HPC) method to singular flows and lifting problems

“…According to [22], the velocity potential at any point within the cell can be approximated by a linear combination of the first eight harmonic polynomials, which takes the following form: 22 1 2 3 4 5 3 2 2 3 4 2 2 4 6 7 8 , 1; , ; , ; , …”

“…These boundaries divide the fluid domain into a small inner region enclosing the sharp corner and an outer domain with only planar walls. In the original HPC method [22], the liquid domain is first discretized into a finite number of quadrilateral elements, as shown in Fig. 4.…”

“…The resulting equation system involves a sparse matrix, implying computational efficiency. Shao and Faltinsen [22] compared the HPC method with four other 2D Laplace equation solvers and found that the HPC method is highly efficient and the most accurate among the considered solvers. In addition, Faltinsen and Timokha [11] used harmonic polynomials to analytically study the 2D frequency-domain sloshing problem.…”

“…Although the HPC method has been used to solve a series of problems in marine hydrodynamics [22,23], it is not ideal. The HPC method must be modified to handle singular flows and discontinuous problems within the framework of potential flow.…”

“…Bingham and Zhang [5] used a higher-order finite difference method (FDM) to model nonlinear water waves using potential flow theory. Shao and Faltinsen [22] proposed a potential flow solver, the 2D harmonic polynomial cell (HPC) method, with an order greater than third order. When using this method, the liquid domain is divided into cells and harmonic polynomials, which satisfy the Laplace equation and are used to represent the velocity potential in each cell.…”