FRG - Advanced Algorithms and Software for Problems in Computational Bio-Fluid Dynamics

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).

The numerical modeling of the dynamics of biological fluid-structure interactions is a rapidly expanding research area in mathematical biology. We will consider two successful strategies for computing fluid-structure interactions which are popular in the applied mathematics community: the Method of Regularized Stokeslets (appropriate in the Stokes regime), and the Immersed Boundary Method (when inertial forces cannot be ignored). Both of these methods are popular partially because they offer a reasonably straightforward option for modeling complex boundaries interacting with an incompressible fluid.

Unfortunately, in both cases there are substantial computational bottlenecks which severely limit the efficiency and/or accuracy of these methods for a wide class of problems. For example, in both methods, the stiffness of the structure can restrict the allowable explicit time-step of a computation by orders of magnitude. In addition, the desire to model fine scale features of biological structures requires spatially adaptive computations and parallel processing. Recent advances in multi-resolution temporal integration methods and integral based methods for fast summation and elliptic equations provide the technology to substantially increase the accuracy and efficiency of these methods, although the analysis and implementation of these algorithms has not been attempted. Our goal is to complete the mathematical and computational analysis necessary to implement efficient parallel, adaptive, multi-resolution time integration and fast summation methods for fluid-structure problems and to build a computational infrastructure that will enable researchers in the biological sciences to perform numerical simulations of biological systems on massively parallel computers.

The development of improved algorithms and a computational interface to facilitate massively parallel computation will enable scientists to obtain a much more detailed understanding of the fluid dynamics in a wide range of problems in the biological and medical fields such as the swimming of fish or flying of insects, the reduction of drag in plants and trees by changes in shape or texture, and the pumping of blood in the heart or the flow of materials in the digestive system, the kidney, and the lungs. In particular, our software will allow researchers to begin to study increasingly complex problems such as fluid flow through bristled appendages, or interactions or coordination between multiple structures or organisms. Beyond obtaining fundamental insight into biological design, this work could motivate innovation in the field of biomimetics, where engineers look to biology for design strategies. For example, improved understanding of fin deformations in fish or jet propulsion in jellyfish is of interest to both the biological community and engineers working on micro underwater vehicles. Our computational methods can be applied to model other systems of interest in biommetics including muscular hydrostats such as octopus arms, earthworm bodies, and elephant trunks.