The immersed interface method (IIM) for models of fluid flow and fluid-structure interaction imposes jump conditions that capture stress discontinuities generated by forces that are concentrated along immersed boundaries. Most prior work using the IIM for fluid dynamic applications has focused on smooth interfaces, but boundaries with sharp features such as corners and edges can appear in practical analyses, particularly on engineered structures. The present study builds on our work to integrate finite element-type representations of interface geometries with the IIM. Initial realizations of this approach used a continuous Galerkin (CG) finite element discretization for the boundary, but as we show herein, these approaches generate large errors near sharp geometrical features. To overcome this difficulty, this study introduces an IIM approach using discontinuous Galerkin (DG) representation of the jump conditions. Numerical examples explore the impacts of different interface representations on accuracy for both smooth and sharp boundaries, particularly flows interacting with fixed interface configurations. We demonstrate that using a DG approach provides accuracy that is comparable to the CG method for smooth cases. Further, we identify a time step size restriction for the CG representation that is directly related to the sharpness of the geometry. In contrast, time step size restrictions imposed by DG representations are demonstrated to be insensitive to the presence of sharp features.

翻译：浸入界面法（IIM）在流体流动及流固耦合模型中施加跳跃条件，以捕捉沿浸入边界集中的力所产生的应力不连续性。先前大多数将IIM应用于流体动力学的研究集中于光滑界面，但在实际分析中（尤其在工程结构上）可能出现具有尖锐特征（如角点和棱边）的边界。本研究基于我们前期将界面几何的有限元类表示与IIM相结合的工作展开。该方法的初始实现采用连续伽辽金有限元离散化表示边界，但如本文所示，这类方法在尖锐几何特征附近会产生较大误差。为克服此困难，本研究提出一种采用间断伽辽金表示跳跃条件的IIM方法。数值算例探讨了不同界面表示对光滑与尖锐边界（特别是与固定界面构型相互作用的流动）计算精度的影响。我们证明，对于光滑情形，采用DG方法可获得与CG方法相当的精度。此外，我们发现了CG表示存在与几何尖锐度直接相关的时间步长限制。相比之下，DG表示所要求的时间步长限制被证明对尖锐特征的存在不敏感。