A harmonic polynomial cell (HPC) method for 3D Laplace equation with application in marine hydrodynamics

Abstract

We propose a new efficient and accurate numerical method based on harmonic polynomials to solve boundary value problems governed by 3D Laplace equation. The computational domain is discretized by overlapping cells. Within each cell, the velocity potential is represented by the linear superposition of a complete set of harmonic polynomials, which are the elementary solutions of Laplace equation. By its definition, the method is named as Harmonic Polynomial Cell (HPC) method. The characteristics of the accuracy and efficiency of the HPC method are demonstrated by studying analytical cases. Comparisons will be made with some other existing boundary element based methods, e.g. Quadratic Boundary Element Method (QBEM) and the Fast Multipole Accelerated QBEM (FMA-QBEM) and a fourth order Finite Difference Method (FDM). To demonstrate the applications of the method, it is applied to some studies relevant for marine hydrodynamics. Sloshing in 3D rectangular tanks, a fully-nonlinear numerical wave tank, fully-nonlinear wave focusing on a semi-circular shoal, and the nonlinear wave diffraction of a bottom-mounted cylinder in regular waves are studied. The comparisons with the experimental results and other numerical results are all in satisfactory agreement, indicating that the present HPC method is a promising method in solving potential-flow problems. The underlying procedure of the HPC method could also be useful in other fields than marine hydrodynamics involved with solving Laplace equation.

Higher harmonic wave loads on a vertical cylinder in finite water depth

“…Adding the correction signal gives a clear improvement for the second-order parasitic waves, consistent with the experiments. The same oscillatory features are clearly seen in other studies where results from simulations of fully nonlinear wave tanks are compared with experimental data, see for instance Shao & Faltinsen (2014).…”

A stabilised nodal spectral element method for fully nonlinear water waves

“…This leaves BEM less efficient compared to volume-based discretisation methods such as FDM and FEM solvers as suggested in [72,57]. We note, that BEM is particularly attractive as a near-field solver for cases where waves interact with complex geometries [75] and may be combined with a far-field solver such as FEM [73].…”

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

“…Fredriksen et al [12] combined the HPC method and a Navier-Stokes equation solver employing the finite volume method (FVM) to investigate the piston-mode resonance in moonpools with a low forward speed by using domain decomposition and a 2D flow assumption. Recently, Shao and Faltinsen [23] conducted a comprehensive study of the 3D HPC method. In addition, fully nonlinear free-surface problems involving a numerical wave tank and wave interactions with a vertical free-surface piercing circular cylinder standing on the sea floor were investigated.…”

“…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.…”