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, that is, 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 HPM has been successfully applied to the accurate and stable modeling of highly nonlinear free‐surface potential flows with and without moving bodies, that is, sloshing, water entry, and plunging breaker.

中文翻译：

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一种求解带复杂边界的拉普拉斯方程的高阶谐波多项式方法及其在自由表面流中的应用。第一部分：二维案例

为了解决具有复杂边界的拉普拉斯方程，开发了一种高阶谐波多项式方法（HPM）。提出“不规则单元”是为了精确地离散Laplace方程，在此难以构造高质量的模具。还开发了一种先进的离散化方案，用于精确评估复杂边界上潜在函数的正态导数。由于不规则单元和潜在函数的正态导数的离散化方案，本方法可以避免模具变形的缺点，即可能的数值不精确/不稳定性。此外，它可以以精确和简单的方式涉及笛卡尔网格上的静止或运动物体。利用适当的自由表面跟踪方法，