# Immersed boundary method

In computational fluid dynamics, the immersed boundary method is an approach to simulate fluid-structure (fiber) interactions. Treating the coupling of the structure deformations and the fluid flow poses a number of challenging problems for numerical simulations (the elastic boundary changes the flow of the fluid and the fluid moves the elastic boundary simultaneously). In the immersed boundary method the fluid is represented on an Eulerian coordinate and the structure is represented on a Lagrangian coordinate. For Newtonian fluids governed by the incompressible Navier–Stokes equations, the fluid equations are

$\rho \left(\frac{\partial{u}({x},t)}{\partial{t}} + {u}\cdot\nabla{u}\right) = -\nabla p + \mu\, \Delta u(x,t) + f(x,t)$

and in case of incompressible fluids (assuming constant density) we have the incompressibility condition

$\nabla \cdot u = 0. \,$

The immersed structures are typically represented as a collection of one-dimensional fibers, denoted by $\Gamma$. Each fiber can be viewed as a parametric curve $X(s,t)$ where $s$ is the parameter and $t$ is time. Physics of the fiber is represented via the fiber force distribution $F(s,t)$. Spring forces, bending resistance or any other type of behavior can be built into this term. The force exerted by the structure on the fluid is then interpolated as a source term in the momentum equation using

$f(x,t) = \int_\Gamma F(s,t) \delta(x - X(s,t)) \, ds,$

where $\delta$ is the Dirac δ function. The forcing can be extended to multiple dimensions to model elastic surfaces or three-dimensional solids. Assuming a massless structure, the elastic fiber moves with the local fluid velocity and can be interpolated via the delta function

$\frac{\partial X(s,t)}{\partial t} = u(X,t) = \int_\Omega \delta(x - X(s,t)) u(x,t) \, dx,$

where $\Omega$ denotes the entire fluid domain. Discretization of these equations can be done by assuming an Eulerian grid on the fluid and a separate Lagrangian grid on the fiber. Approximations of the Delta distribution by smoother functions will allow us to interpolate between the two grids. Any existing fluid solver can be coupled to a solver for the fiber equations to solve the Immersed Boundary equations. Variants of this basic approach have been applied to simulate a wide variety of mechanical systems involving elastic structures which interact with fluid flows. See the references for more details.

## References

1. C. S. Peskin, The immersed boundary method, Acta Numerica, 11, pp. 1–39, 2002.
2. C.S. Peskin, Numerical analysis of blood ﬂow in the heart, J. Comput. Phys. 25 (1977) 220–252.
3. R. Mittal and G. Iaccarino, Immersed Boundary Methods, Annual Review of Fluid Mechanics, vol. 37, pp. 239–261, 2005.
4. Y. Mori and C. S. Peskin, Implicit Second Order Immersed Boundary Methods with Boundary Mass Computational Methods in Applied Mechanics and Engineering, 2007.
5. L. Zhua and C. S. Peskin, Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, Journal of Computational Physics, vol. 179, Issue 2, pp. 452–468, 2002.
6. P.J. Atzberger, Stochastic Eulerian Lagrangian Methods for Fluid Structure Interactions with Thermal Fluctuations, Journal of Computational Physics, 230, pp. 2821–2837, (2011)[DOI] .
7. P. J. Atzberger, P. R. Kramer, and C. S. Peskin, A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales, Journal of Computational Physics, vol. 224, Issue 2, 2007. [DOI] .
8. A. M. Roma, C. S. Peskin, and M. J. Berger, An adaptive version of the immersed boundary method, Journal of Computational Physics, vol. 153 n.2, pp. 509–534, 1999.
9. Jindal S. et al. “The Immersed Boundary CFD Approach for Complex Aerodynamics Flow Predictions” SAE Journal, Detroit, Michigan 2007-01-0109 (2007)
10. J.Kim, D.Kim, H.Choi, "An Immersed-Boundary Finite Volume Method for Simulations of Flow in Complex Geometries" Journal of Computational Physics, vol. 171, Issue 1, pp. 132–150, 2001.
11. A.P.S. Bhalla, R. Bale, B.E. Griffith, N.A. Patankar, "A unified mathematical framework and an adaptive numerical method for fluid–structure interaction with rigid, deforming, and elastic bodies", Journal of Computational Physics, vol. 250, pp. 446–476, 2013 [ DOI].
12. J. M. Stockie, Brian T.R. Wetton, "Stability analysis for the immersed fiber problem", SIAM Journal on Applied Mathematics, vol. 55, Issue 6, pp. 1577-1591 (1995)