On the hyper-elastic formulation of the immersed boundary method
On the hyper-elastic formulation of the immersed boundary method
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DOI：
10.1016/j.cma.2007.09.015
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发表时间：
2008-01-01
影响因子：
7.2
通讯作者：
Peskin, Charles S.
中科院分区：
文献类型：
--
作者：
Boffi, Daniele;Gastaldi, Lucia;Peskin, Charles S.
浸没边界(IB)方法是一种求解不可压缩粘弹性物体或边界与不可压缩流体相互作用的流固耦合问题的数学公式和数值方法。由于在固液界面法向应力非零跃变的情况下，浸没物体与周围流体之间缺乏适当的传输条件，所以IB方法以前的公式不能适当地处理由一般超弹性本构关系所模拟的有限、非零厚度的浸没材料。(当固体由平行于界面的纤维组成时，不会出现这种跳跃，但通常在其他情况下会出现，例如，当固体包含终止于固-液界面的弹性纤维时)。我们给出了IB方法的一种推导，它以适当的方式考虑了缺失项。本文的推导是从分离虚功原理中出现的应力开始的。将应力分为类流体应力和类固体应力，并在其自然框架下处理这两个项，即类流体应力的欧拉框架和类固体应力的拉格朗日框架。我们描述了IB方法如何与浸入不可压缩弹性材料的连续介质力学模型的标准公式相结合，并给出了一些说明性的数值实验。(C)2007 Elsevier B.V.保留所有权利。
The immersed boundary (IB) method is both a mathematical formulation and a numerical method for fluid-structure interaction problems, in which immersed incompressible visco-elastic bodies or boundaries interact with an incompressible fluid. Previous formulations of the IB method were not able to treat appropriately immersed materials of finite, nonzero thickness modeled by general hyperelastic constitutive laws because of the lack of appropriate transmission conditions between the immersed body and the surrounding fluid in the case of a nonzero jump in normal stress at the solid-fluid interface. (Such a jump does not arise when the solid is comprised of fibers that run parallel to the interface, but typically does arise in other cases, e.g., when the solid contains elastic fibers that terminate at the solid-fluid interface). We present a derivation of the IB method that takes into account in an appropriate way the missing term. The derivation presented in this paper starts from a separation of the stress that appears in the principle of virtual work. The stress is divided into its fluid-like and solid-like components, and each of these two terms is treated in its natural framework, i.e., the Eulerian framework for the fluid-like stress and the Lagrangian framework for the solid-like stress. We describe how the IB method can be used in conjunction with standard formulations of continuum mechanics models for immersed incompressible elastic materials and present some illustrative numerical experiments. (C) 2007 Elsevier B.V. All rights reserved.