An Immersed Boundary method with divergence-free velocity interpolation and force spreading
An Immersed Boundary method with divergence-free velocity interpolation and force spreading
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DOI：
10.1016/j.jcp.2017.06.041
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发表时间：
2017-10-15
影响因子：
4.1
通讯作者：
Peskin, Charles S.
中科院分区：
文献类型：
--
作者：
Bao, Yuanxun;Donev, Aleksandar;Peskin, Charles S.
浸没边界(IB)方法是一种构造稳健数值方法的数学框架，用于研究浸没在粘性流体中的弹性结构问题中的流固耦合问题。IB公式使用流体的欧拉表示和结构的拉格朗日表示。拉格朗日标架和欧拉标架通过带有Delta函数核的积分变换耦合。离散化的IB方程利用正则化的Delta函数核对这些变换进行近似，将流体速度插值到结构中，并将结构力传播到流体中。众所周知，由于插值后的拉格朗日速度场一般不是无散度的，所以传统的IB方法体积守恒性差，这会导致虚假的体积变化。在实践中，缺乏体积守恒在薄的构造边界上存在大的压力差的情况下尤其明显。本文的目的是通过引入速度插值法和力扩散法来大大减小IB法的体积误差，这些方法的性质是：结构运动的插值速度场至少为C-1且满足连续无散度条件，且力扩散算子是速度插值算子的伴随算子。通过在二维和三维空间上的数值实验，我们证实了这种新的IB方法能够在不增加计算代价的情况下，在体积守恒方面比现有的IB方法有显著的改进。此外，新方法提供了比传统IB方法更平滑的拉格朗日力(牵引力)。本文提出的方法仅限于周期计算域。将其推广到非周期区域是今后的重要工作。(C)2017 Elsevier Inc.保留所有权利。
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 C-1 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. (C) 2017 Elsevier Inc. All rights reserved.