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

Project Summary

The numerical modeling of the dynamics of biological fluid-structure interactions is a rapidly expanding research area in mathematical biology. With recent computational advancements, researchers can now begin to study increasingly complex problems such as fluid flow through bristled appendages and interactions or coordination between multiple structures. Such biological problems can vary over a large range of scales and involve many different types of fluid-structure interactions such as pumping, swimming, flying, flapping, filtering, and reducing drag. This translates mathematically into a large range of Reynolds numbers over which problems are posed and a rich variety of fluid-structure geometries that require adaptive computations.

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. In both methods, the stiffness of the structure can restrict the allowable time-step of a computation by orders of magnitude. In the Regularized Stokeslet case, the relationship between the force and velocity results in a dense linear system (equivalent to an N-body problem) which must be evaluated or inverted. Similarly, in the Immersed Boundary Method, forces must be globally transferred to a grid which is usually done through a finite difference solution of an elliptic equation. Recent advances in multi-resolution temporal integration methods and integral based methods for fast summation and elliptic equations provide the technology to effectively overcome these bottlenecks. However, the solutions are unfortunately neither easy to adapt to the fluid-structure setting nor easily implemented. 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 these problems.