A high‐order harmonic polynomial method for solving the Laplace equation with complex boundaries and its application to free‐surface flows. Part I: Two‐dimensional cases

Abstract

A high-order Harmonic Polynomial Method (HPM) is developed for solving the Laplace equation with complex boundaries. The "irregular cell" is proposed for the accurate discretization of the Laplace equation, where it is difficult to construct a high-quality stencil. An advanced discretization scheme is also developed for the accurate evaluation of the normal derivative of potential functions on complex boundaries. Thanks to the irregular cell and the discretization scheme for the normal derivative of the potential functions, the present method can avoid the drawback of distorted stencils, i.e., the possible numerical inaccuracy/instability. Furthermore, it can involve stationary or moving bodies on the Cartesian grid in an accurate and simple way. With the proper free-surface tracking methods, the harmonic polynomial method has been successfully applied to the accurate and stable modeling of highly-nonlinear free-surface potential flows with and without moving bodies, i.e., sloshing, water entry and plunging breaker.

An adaptive harmonic polynomial cell method with immersed boundaries: Accuracy, stability, and applications

“…Due to the property of high accuracy and efficiency, more and more attention is being paid to this novel method by both the research and the engineering communities. In our present study, unlike the "irregular cells" strategy in Wang et al, 25 square cells are used in the whole computational domain. Since the HPC method is a field solver, the computational domain is, therefore, discretized into overlapping quadrilateral cells with local indices i = 1 ∼ 9, as shown in Figure 2.…”

“…In this case, the cell can be changed into irregular shapes containing the necessary number of nodes to reach a desired accuracy. 25,44 Establishing the adequate equations for all active nodes in the computational domain, the linear algebraic equation system is established as  = e, with  the coefficient matrix containing at most nine entries in each row, the unknown velocity-potential vector and e the boundary-condition vector.…”

“…Another interesting development in the 2D HPC method was made recently by Wang et al 25 and Shen et al, 55 where irregular cells close to the boundaries are constructed through a local approximation based on least-square fitting of harmonic polynomials. A similar idea has been discussed earlier by Shao and Faltinsen 44 in their formulation of the original 3D HPC method.…”

“…An advantage of using irregular cells is that arbitrary order of polynomials could be included in the local approximations. With a proper free-surface tracking method, Wang et al 25 have been able to deal with problems involving complex boundaries, that is, plunging breakers, sloshing and water entry. However, a local irregular cell has to be formed for each boundary point at each time step in fully nonlinear wave-structure interaction analysis, where the free surface and structure surfaces move or deform.…”

“…Since a number of stencil points, typically much larger than that of a standard cell, are needed to form a solvable WLS problem, it can potentially increase the complexity and the computational cost of the method. 55 In the present study, we propose a new immersed-boundary HPC method, which avoids having to use the overlapping-grid strategy, 47,51,52 or the irregular cells 25,55 to handle complex geometries. As will be shown in later sections, the method has been satisfactorily verified and validated by several cases including: a circular cylinder moving in an infinite fluid where an analytical solution exists, fully nonlinear wave generation and propagation, and fully nonlinear diffraction and radiation of a ship cross section.…”