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

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.