Immersed boundary method

The immersed boundary method is an approach to model and simulate mechanical systems in which elastic structures (or membranes) interact with fluid flows. Treating the coupling of the structure deformations and the fluid flow poses a number of challenging problems for numerical simulations. In the immersed boundary method approach the fluid is represented in an Eulerian coordinate frame and the structures in a Lagrangian coordinate frame. For Newtonian fluids governed by the Navier–Stokes equations the immersed boundary method fluid equations are:$holeft(frac\{partial\{u\}(\{x\},t)\}\{partial\{t\; +\; \{u\}cdot\; abla\{u\}\; ight)=\; mu\; Delta\; u(x,t)\; -\; abla\; p\; +\; f(x,t)$with incompressibility condition

:$abla\; cdot\; u\; =\; 0.$The immersed structures are typically represented by a collection of interacting particles $Z\_j$ with a prescribed force law, where $F\_j$ is the force acting on the $j^\{th\}$ particle. The forces are accounted for in the fluid equations by the force density:$f(x,t)\; =\; sum\_\{j\; =\; 1\}^N\; delta\_a(x\; -\; Z\_j)F\_j$where $delta\_a$ is an approximation of the Dirac $delta$-function smoothed out over a length scale $a$. The immersed structures are then updated using the equation:$frac\{dZ\_j\}\{dt\}\; =\; int\; delta\_a(x\; -\; Z\_j)\; u(x,t)\; dx.$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.

See also

*Stokesian dynamics
*Charles S. Peskin

References

#C. S. Peskin, The immersed boundary method, Acta Numerica, 11, pp. 1– 39, 2002.
#R. Mittal and G. Iaccarino, Immersed Boundary Methods, Annual Review of Fluid Mechanics, vol. 37, pp. 239-261, 2005.
#Y. Mori and C. S. Peskin, Implicit Second Order Immersed Boundary Methods with Boundary Mass Computational Methods in Applied Mechanics and Engineering, 2007.
#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.
#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.
#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.

* [http://www.math.utah.edu/IBIS/ An implementation of the Immersed Boundary Method for Uniform Meshes in 2D (Numerical Codes).]
* [http://www.math.nyu.edu/~griffith/IBAMR/ An implementation of the Immersed Boundary Method for Adaptive Meshes in 3D (Numerical Codes).]
* [http://www.math.ucsb.edu/~atzberg/SIB_Codes/index.html An implementation of the Stochastic Immersed Boundary Method in 3D (Numerical Codes).]