Bao Yuanxun, Donev Aleksandar, Griffith Boyce E, McQueen David M, Peskin Charles S
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, USA.
Departments of Mathematics and Biomedical Engineering, Carolina Center for Interdisciplinary Applied Mathematics, and McAllister Heart Institute, University of North Carolina, Chapel Hill, NC, USA.
J Comput Phys. 2017 Oct 15;347:183-206. doi: 10.1016/j.jcp.2017.06.041. Epub 2017 Jun 28.
The Immersed Boundary (IB) method is a mathematical framework for constructing robust numerical methods to study fluid-structure interaction in problems involving an elastic structure immersed in a viscous fluid. The IB formulation uses an Eulerian representation of the fluid and a Lagrangian representation of the structure. The Lagrangian and Eulerian frames are coupled by integral transforms with delta function kernels. The discretized IB equations use approximations to these transforms with regularized delta function kernels to interpolate the fluid velocity to the structure, and to spread structural forces to the fluid. It is well-known that the conventional IB method can suffer from poor volume conservation since the interpolated Lagrangian velocity field is not generally divergence-free, and so this can cause spurious volume changes. In practice, the lack of volume conservation is especially pronounced for cases where there are large pressure differences across thin structural boundaries. The aim of this paper is to greatly reduce the volume error of the IB method by introducing velocity-interpolation and force-spreading schemes with the properties that the interpolated velocity field in which the structure moves is at least and satisfies a continuous divergence-free condition, and that the force-spreading operator is the adjoint of the velocity-interpolation operator. We confirm through numerical experiments in two and three spatial dimensions that this new IB method is able to achieve substantial improvement in volume conservation compared to other existing IB methods, at the expense of a modest increase in the computational cost. Further, the new method provides smoother Lagrangian forces (tractions) than traditional IB methods. The method presented here is restricted to periodic computational domains. Its generalization to non-periodic domains is important future work.
浸入边界（IB）方法是一种数学框架，用于构建强大的数值方法，以研究涉及浸入粘性流体中的弹性结构的流固相互作用问题。IB公式使用流体的欧拉表示和结构的拉格朗日表示。拉格朗日框架和欧拉框架通过具有狄拉克函数核的积分变换耦合。离散化的IB方程使用具有正则化狄拉克函数核的这些变换的近似来将流体速度插值到结构上，并将结构力扩散到流体中。众所周知，传统的IB方法可能存在体积守恒性差的问题，因为插值的拉格朗日速度场通常不是无散度的，因此这可能导致虚假的体积变化。在实际应用中，对于薄结构边界上存在较大压力差的情况，体积守恒性的缺乏尤为明显。本文的目的是通过引入速度插值和力扩散方案来大幅降低IB方法的体积误差，这些方案具有以下特性：结构在其中移动的插值速度场至少 且满足连续无散度条件，并且力扩散算子是速度插值算子的伴随算子。我们通过二维和三维空间的数值实验证实，与其他现有的IB方法相比，这种新的IB方法能够在体积守恒方面取得显著改进，但代价是计算成本略有增加。此外，新方法提供的拉格朗日力（牵引力）比传统的IB方法更平滑。这里提出的方法仅限于周期性计算域。将其推广到非周期性域是未来重要的工作。
