# The immersed boundary method

@article{Peskin2002TheIB, title={The immersed boundary method}, author={Charles S. Peskin}, journal={Acta Numerica}, year={2002}, volume={11}, pages={479 - 517} }

This paper is concerned with the mathematical structure of the immersed boundary (IB) method, which is intended for the computer simulation of fluid–structure interaction, especially in biological fluid dynamics. The IB formulation of such problems, derived here from the principle of least action, involves both Eulerian and Lagrangian variables, linked by the Dirac delta function. Spatial discretization of the IB equations is based on a fixed Cartesian mesh for the Eulerian variables, and a…

## 3,640 Citations

Hybrid finite difference/finite element version of the immersed boundary method

- Computer Science
- 2012

An approach to discretizing the interaction equations that enables us to use Lagrangian meshes that are significantly coarser than the background Eulerian grid is introduced.

A simple immersed boundary technique for simulating complex flows with rigid boundary

- Physics
- 2004

A new immersed boundary technique for the simulation of flow interacting with solid boundary is presented. The method employs a mixture of Eulerian and Lagrangian variables, where the solid boundary…

Stability Results and Algorithmic Strategies for the Finite Element Approach to the Immersed Boundary Method

- Mathematics, Computer Science
- 2006

This work presents a stability analysis based on energy estimates for the variational formulation of the immersed boundary method and presents a computable CFL condition for the semi-implicit method where the fluid terms are treated implicitly while the structure is treated explicitly.

NUMERICAL STABILITY OF THE FINITE ELEMENT IMMERSED BOUNDARY METHOD

- Mathematics, Computer Science
- 2007

This work uses a linearization of the Navier–Stokes equations and a linear elasticity model to prove the unconditional stability of the fully implicit discretization, achieved with the use of a backward Euler method for both the fluid and the structure evolution.

An adaptive Newton continuation strategy for the fully implicit finite element immersed boundary method

- MathematicsJ. Comput. Phys.
- 2012

On the Volume Conservation of the Immersed Boundary Method

- Environmental Science
- 2012

The immersed boundary (IB) method is an approach to problems of fluid-struc- ture interaction in which an elastic structure is immersed in a viscous incompress- ible fluid. The IB formulation of such…

On the stability of the finite element immersed boundary method

- Mathematics, Computer Science
- 2008

## References

SHOWING 1-10 OF 79 REFERENCES

The Blob Projection Method for Immersed Boundary Problems

- Engineering
- 2000

A new finite difference numerical method for modeling the interaction between flexible elastic membranes and an incompressible fluid in a two-dimensional domain is presented. The method differs from…

An implicit numerical method for fluid dynamics problems with immersed elastic boundaries

- Mathematics
- 1993

Problems of biofluid-dynarnics often involve the interaction of a viscous incompressible fluid with an immersed elastic boundary. One method of computing solutions of such problems is by using a…

Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension

- Computer ScienceSIAM J. Sci. Comput.
- 1997

A second-order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented and an implicit quasi-Newton method is developed that allows reasonable time steps to be used.

An Adaptive Version of the Immersed Boundary Method

- Computer Science, Mathematics
- 1999

The main contributions of this work concern the formulation and the implementation of a multilevel self-adaptive version of the Immersed Boundary Method on locally refined meshes.

Analysis of Stiffness in the Immersed Boundary Method and Implications for Time-Stepping Schemes

- Computer Science
- 1999

Numerical results are presented that show how fully explicit Runge?Kutta schemes perform in comparison with the best of the semi-implicit schemes currently in use.

The initial-value problem for elastodynamics of incompressible bodies

- Mathematics
- 1986

AbstractWe prove short-time well-posedness of the Cauchy problem for incompressible strongly elliptic hyperelastic materials.Our method consists in:a)Reformulating the classical equations in order to…

The Dynamics of an Elastic Membrane Using the Impulse Method

- Physics, Engineering
- 1997

A validation study of the impulse method applied to the motion of a closed elastic membrane under tension separating two incompressible, inviscid fluids in two dimensions is presented. The approach…

Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface

- Physics
- 1965

A new technique is described for the numerical investigation of the time‐dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free. The full…