### Abstract

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 hyper-elastic 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.

Original language | English (US) |
---|---|

Pages (from-to) | 2210-2231 |

Number of pages | 22 |

Journal | Computer Methods in Applied Mechanics and Engineering |

Volume | 197 |

Issue number | 25-28 |

DOIs | |

State | Published - Apr 15 2008 |

### Fingerprint

### Keywords

- Fluid-structure interaction
- Hyper-elasticity
- Immersed boundary method

### ASJC Scopus subject areas

- Computer Science Applications
- Computational Mechanics

### Cite this

*Computer Methods in Applied Mechanics and Engineering*,

*197*(25-28), 2210-2231. https://doi.org/10.1016/j.cma.2007.09.015

**On the hyper-elastic formulation of the immersed boundary method.** / Boffi, Daniele; Gastaldi, Lucia; Heltai, Luca; Peskin, Charles.

Research output: Contribution to journal › Article

*Computer Methods in Applied Mechanics and Engineering*, vol. 197, no. 25-28, pp. 2210-2231. https://doi.org/10.1016/j.cma.2007.09.015

}

TY - JOUR

T1 - On the hyper-elastic formulation of the immersed boundary method

AU - Boffi, Daniele

AU - Gastaldi, Lucia

AU - Heltai, Luca

AU - Peskin, Charles

PY - 2008/4/15

Y1 - 2008/4/15

N2 - 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 hyper-elastic 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.

AB - 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 hyper-elastic 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.

KW - Fluid-structure interaction

KW - Hyper-elasticity

KW - Immersed boundary method

UR - http://www.scopus.com/inward/record.url?scp=41849132500&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=41849132500&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2007.09.015

DO - 10.1016/j.cma.2007.09.015

M3 - Article

VL - 197

SP - 2210

EP - 2231

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0374-2830

IS - 25-28

ER -