We present a 2D high‐order and easily accessible immersed‐boundary adaptive harmonic polynomial cell (IB‐AHPC) method to solve fully nonlinear wave‐structure interaction problems in marine hydrodynamics using potential‐flow theory. To reduce the total number of cells without losing accuracy, adaptive quad‐tree cell refinements are employed close to the free‐surface and structure boundaries. The present method is simpler to implement than the existing IB‐HPC alternatives, in that it uses standard square cells both in the fluid domain and at the boundaries, thus without having to use the more complex and expensive overlapping grids or irregular cells. The spurious force oscillations on moving structures, which is a known issue for immersed boundary methods (IBMs), are eliminated in this study by solving a separate boundary value problem (BVP) for a Lagrangian acceleration potential. We also demonstrate that solving a similar BVP for the corresponding Eulerian acceleration potential is far less satisfactory due to the involved second derivatives of the velocity potential in the body‐boundary condition, which are very difficult to calculate accurately in an IBM‐based approach. In addition, we present, perhaps for the first time since the HPC method was developed, a linear matrix‐based stability analysis for the time‐domain IB‐AHPC method. The stability analysis is also used in this study as a general guide to design robust and stable numerical algorithms, in particular related to the treatment of boundary conditions close to the intersection between a Dirichlet and a Neumann boundary, which is essential in time‐domain wave‐structure interaction analyses using IBMs. We confirm theoretically through the stability analysis that square cells have the best stability properties. The present method has been verified and validated satisfactorily by various cases in marine hydrodynamics, including a moving structure in an infinite fluid, fully nonlinear wave generation and propagation, and fully nonlinear diffraction and radiation of a ship cross section.

中文翻译：

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具有浸没边界的自适应谐波多项式单元法：精度，稳定性和应用

我们提出了一种二维高阶且易于访问的浸入边界自适应谐波多项式单元（IB-AHPC）方法，使用势流理论来解决海洋流体动力学中的完全非线性波结构相互作用问题。为了减少单元总数而不损失精度，在自由表面和结构边界附近采用了自适应四叉树单元改进。与现有的IB-HPC替代方案相比，本方法的实施更简单，因为它在流体域和边界都使用标准的方形单元，因此无需使用更复杂和昂贵的重叠网格或不规则单元。移动结构上的杂散力振荡，这是沉浸边界方法（IBM）的一个已知问题，通过解决拉格朗日加速度势的一个单独的边值问题（BVP），在本研究中消除了这些问题。我们还证明，由于在体边界条件下涉及速度势的二阶导数，因此很难用相应的欧拉加速度势求解相似的BVP，这在基于IBM的方法中很难精确计算。此外，也许是自HPC方法开发以来的第一次，是针对时域IB-AHPC方法的基于线性矩阵的稳定性分析。稳定性分析在本研究中还被用作设计鲁棒和稳定数值算法的一般指南，特别是与处理Dirichlet和Neumann边界之间的交点附近的边界条件有关，这对于使用IBM进行时域波结构相互作用分析至关重要。我们通过稳定性分析从理论上确认方形电池具有最佳的稳定性。在海洋流体动力学的各种情况下，包括在无限流体中的运动结构，完全非线性的波的产生和传播，以及船舶截面的完全非线性的衍射和辐射，本方法已得到令人满意的验证和验证。