Research

The Immersed Boundary Method provides a powerful and robust method for simulating the interaction between a fluid and an immersed object (fluid-structure interactions). Often the immersed object is either highly rigid or nonextensible in one or more directions. Modeling these objects correctly requires extremely stiff equations, which in turn severely restrict the stable timesteps a simulation can take.

Because of this restriction, many important applications have remained out of reach of current hardware. My work has focused on developing new implicit methodologies that eliminate the stiffness with minimal computational cost, opening up the potential for the Immersed Boundary Method to be utilized in an even greater array of applications.

The new methodologies are demonstrated on a few toy problems in the gallery below. In each case the immersed object is incredibly stiff. Traditional methods would require months or years to complete the given simulation. The new implicit methods require only a few hours, even on a cheap laptop.

The following table shows the simulation time in hours for both explicit and implicit simulations of a rigid plate. &sigma is the stiffness constant of the plate, N is the discretization resolution of the fluid domain. Note that as &sigma increases the total CPU time of the explicit simulations increase. When &sigma = 1011 the speed up from using our new implicit solver is more than 7000 fold!

In my first paper I investigated the stiffness problem for 2D fluids. Utilizing novel approximations to the underlying fluid dynamics, I developed a highly efficient multigrid to solve an unconditionally stable discretization of the Immersed Boundary Method. The resulting methodology proved capable of handling a large class of applications.

For some cases, in particular when the force distribution on the immersed structure is nonlinear, the new methodology is ineffective. In the same paper I developed an alternative method, a splitting method, suited for the important case of highly rigid structures. In this new method the fluid and the structure can be effectively split and handled independently. In addition, because of the nature of the methodology, greater stiffness actually accelerates convergence of the iterative methods used, allowing for very fast convergence for rigid structures.

Extending the 2D methodologies to 3D proved nontrivial. I investigated new approaches to the 3D case in my second paper. Here I developed a treecode for handling a key multipole summation, making use of approximate far field expansions. The treecode was used to accelerate iterations of a Krylov subspace method, reducing costs from O(N3log(N)) to O(N2log(N)). Assymptotically (and in practice!), the new method effectively makes the implicit method no more expensive than the explicit method, while still eliminating the timestep restraints that high stiffness creates.

My current work is focused on exploring new applications of the Immersed Boundary Method previously too computationally expensive. Along with my advisor, Hector Ceniceros, I am investigating the properties of non-Newtonian fluids in a peristaltic pump. In addition, I am looking at further ways to increase the speed of the implicit methodologies, by coupling a Fast Multipole Method to a multigrid, as well as looking at ways to extend the range of the methodologies, especially for nonlinear cases.

Publications



Gallery



Induced flow from a deforming sphere.




Flow past a stiff plate.