CS 267

The Immersed Boundary Method
	The Immersed Boundary Method was Developed by Charles Peskin in the late 70s. In the words of Peskin: "The Immersed Boundary Method is both a mathematical formulation and a computational method for the biofluid dynamic problem. In the immersed boundary formulation, the equations of fluid dynamics are used in an unconventional way, to describe not only the fluid but also the immersed elastic tissue with which it interacts." The basic idea of the method is to solve independently the Navier-Stokes equations that govern the behavior of the fluid over a regular grid, and the Equations governing the behavior of the elastic material in a Lagrangian reference frame. To understand the difference between the Eulerial reference frame under which the NS equations are solved, and the Lagrangian reference frame over which the equations governing the behavior of the elastic material are solved, consider the difference between studying a fluid flow by puting a sensor at a certain point in the water and measuring the velocity of the fluid at that point, the Eulerian point of view, and puting a marker in the fluid, and looking at how this marker drifts in the fluid, the Lagrangian point of view. The key to making the method work is to link the Lagrangian and Eulerian components of the fluid through a "smoothed version of the Dirac delta function." Through this functions, the elastic forces generated by the elastic material can be applied to the fluid, and the fluid velocities can be interpolated to determine how the material is going to move and deform.
	Applications of the Immersed Boundary Method
	The immersed boundary method has been used in a number of applications, from simmulations of the inner ear, to studies of swimming involving a wide range of organisms from fish to bacteria.One of the most famous uses of the immersed boundary method, it's motivating application, is the study of fluid flow in the heart.
	Implementations
	In 1993, Charles Pekin and Dave McQueen were able to simmulate a synthetic model of the heart using a 128x128x128 fluid grid, and over 100000 fiber points. The model ran on a Cray C90 at the Pittsburgh Supercomputing Center. At the moment, there is an ongoing effort at Berkeley to develop a scalable massively parallel implementation of the heart simmulation that will allow us to use much finer fluid meshes and more detailed heart models. The application is difficult to parallelize efficiently because the combination of the Eulerian and Lagrangian representations means that where as the fluid grid is static, the grid representing the elastic material is going to move through the domain. In a distributed memory machine, where both the fluid and the material grids are distributed, this means that at every interaction step, the fibers will potentially reside in a different processor from the piece of the fluid grid with which they have to interact, which makes the interaction phase very communication intensive. Another problem with scalability is the fact that current incarnations of the method use an FFT based solver for the NS equations, which has problems scaling to many hundreds of processors.
	Links and Refferences
	The C90 Implementation of the Heart
		General Specs from the implementation
		A document by McQueen and Peskin describing it
	The IB Method
		A paper by Peskin from Acta Numerica
	Titanium Implementation of the IB Method
		Ed Givelberg's Home Page
	Who is Armando Solar-Lezama?
	I am currently a first year graduate student in the CS department. My current interests include computational physics in general, and computational fluid dynamics and computational neutron transport in particular. I am also interested in programming languages and compilers, particularly as it relates to productivity and high performance computing.