A Conference in Memory of Thomas Bringley

to be held at the Courant Institute of Mathematical Sciences

Warren Weaver Hall, Room 1302

Saturday, March 28, 2009

The Courant Institute of Mathematical Sciences at New York University hosted a day conference in memory of Dr. Thomas Tyler Bringley. The speakers discussed problems related to Tom's mathematical interests. Professor Charles Peskin presented on Tom's dissertation work.

Invited Speakers:

Paul Atzberger (UCSB, Mathematics) | Yoichiro Mori (U Minnesota, Mathematics) |

Aaron Brown (AQR Capital Management) | Charles Peskin (NYU, Mathematics) |

Steve Childress (NYU, Mathematics) | Arjun Raj (MIT, Physics) |

Boyce Griffith (NYU, Medicine) | Jun Zhang (NYU, Physics) |

Tom Bringley passed away Sunday, June 22 [2008] after a lengthy battle with cancer. Tom came to Courant in the fall of 2003 after undergraduate work at Duke in math and physics. He was a superb student and colleague who continued to excel in his coursework and research despite the most adverse of circumstances. He was brilliant and kind and will truly be missed and long remembered as a colleague and friend. - Leslie Greengard, Director, Courant Institute.

A few photos from the event, presented in chronological order:

Schedule:

8:30-9:00am: Breakfast (Courant Lounge)

9:00-9:05am: Introductory remarks

9:05-9:35am: Boyce Griffith (NYU Medicine), *Simulating the fluid dynamics of the aortic heart valve*

9:40-10:10am: Steve Childress (NYU Mathematics), *Valveless pumping*

10:10-10:30am: Coffee break

10:30-11:30am: Charles Peskin (NYU Mathematics), *Tom Bringley's Ph.D. Thesis*

11:35-12:05pm: Yoichiro Mori (U Minn, Mathematics), *Convergence of the Immersed Boundary Method and the Even-Odd Condition*

12:05-1:30pm: Lunch (Courant Lounge)

1:30-2:00pm: Aaron Brown (AQR Capital Management), *Whiskey, Poker and the Global Derivatives Economy*

2:05-2:35pm: Jun Zhang (NYU Physics), *Reversed flapping flight and inverted hydrodynamical drafting*

2:35pm-3pm: Coffee break

3:00-3:30pm: Arjun Raj (MIT, Physics), *Nature, nurture or just dumb luck: Stochastic gene expression and cell fate*

3:35-4:05pm: Paul Atzberger (UCSB, Mathematics), *Stochastic Immersed Boundary Methods for simulation of microscopic fluid-structure systems with thermal fluctuations*

4:30-7:30pm: All are welcome to continue the celebration with dinner or drinks in the back room of Gonzalez y Gonzalez (625 Broadway, between Bleecker and Houston).

Abstracts:

Boyce Griffith: * Simulating the fluid dynamics of the aortic heart valve.*

The aortic heart valve is comprised of three elastic leaflets and connects the left ventricle of the heart to the aorta, the main artery through which blood is distributed to all the tissues of the body, including the heart itself. The primary functions of the aortic valve are to allow freely the flow of blood into the aorta when the left ventricle contracts, and to prevent the backflow of blood from the aorta into the left ventricle when the heart is relaxed. A healthy aortic valve opens with essentially no pressure drop across the valve, closes without leak, and, once closed, supports a significant pressure load.

Simulating the fluid dynamics of the aortic heart valve an approach in which the motion of the elastic valve leaflets is coupled to the motion of the blood in which the leaflets are immersed. The immersed boundary method is a mathematical formulation and a numerical scheme for such problems of fluid-structure interaction. In this talk, I shall discuss a version of the immersed boundary method which has enabled high-resolution three-dimensional simulations of the aortic heart valve. I shall highlight recent extensions to this methodology which allow detailed immersed boundary models to be coupled to reduced models of the circulation. This coupling is accomplished by imposing physical boundary conditions on the fluid in which the valve model is immersed. Simulation results obtained using this methodology will also be presented.

Charles Peskin: *Tom Bringley's Ph.D. Thesis*

(From Tom's dissertation:) The immersed boundary method is a numerical scheme for dynamical simulations of solid or elastic bodies immersed in a surrounding fluid. The method was originally introduced by Peskin to model the flow of blood in the human heart. It has since proven to be a general and robust method for diverse flow problems arising in biology and engineering. Recently, the immersed boundary method has been applied to biological problems at the micro-scale or smaller, including swimming microorganisms and biomolecular motors. At these scales, the Stokes equations govern the dynamics of the surrounding fluid, and it is particularly important to represent accurately the hydrodynamic interactions that govern the motion of immersed bodies. In this thesis, we construct a new version of the immersed boundary method for Stokes flow. Our analysis of this new method sheds light on several fundamental questions about the immersed boundary method. In the new method, the structures are immersed in an infinite three-dimensional fluid; no artificial boundaries are necessary. Although we start from a discrete representation of the fluid on an infinite grid, we are able to eliminate the fluid variables and reduce the dynamics to that of a finite collection of Lagrangian points. Asymptotic methods reduce the computational cost and reveal the magnitudes of the numerical errors, as well as the dependence of these errors on the approximate delta function used to regularize singular sources. We discuss the properties and the construction of these delta functions. We then study representations of simple rigid bodies: spheres and cylinders, by simplest-possible configurations of Lagrangian points, an isolated point for a sphere and a linear array of points for a cylinder. We use numerical experimentation to find how the physical parameters of these immersed bodies are related to the numerical parameters of the method. We find that the errors in our method are small if parameters are chosen appropriately, and we give recipes for parameter choices that should be helpful to future users.

Steve Chidress: *Valveless Pumping*

Valveless pumping is an important but poorly understood mechanism for moving a fluid through a simple tubular channel. Tom had a keen interest in the problem. He carried out experiments and developed a simple and elegant model of the process. In this talk we describe the phenomena, describe some of the analytical questions, and outline the elements of Tom's model.

Yoichiro Mori: *Convergence Proof of the Immersed Boundary Method and the Even-Odd Condition*

The immersed boundary method is a popular method for computations in fluid-structure interaction problems. It is charactrized by the use of an Eulerian grid for the fluid domain and a Lagrangian grid for the elastic structure, and the use of regularized dirac delta functions to establish communication between the two grids. In this talk, I will outline a convergence proof for a stationary Stokes flow immersed boundary problem. Computational results are presented to demonstrate that the error estimates obtained are close to optimal. The convergence analysis also suggests the existence of a family of discrete delta functions that may be of use in applications. I will end with a discussion of open problems.

Aaron Brown: *Whiskey, Poker and the Global Derivatives Economy*

In a light half-hour talk I will cover the economic history of the world three times (forward, backward, then forward again) to zero in on a key place and time at which whiskey and poker changed the world forever. Along the way we'll learn why we owe more to the 18th century Scottish professional gambler John Law than the 18th century Scottish moral philosopher Adam Smith, why there is no poker game named "Buffalo," why there were Whiskey Rebellions on both the US and Scotland, and why trade along river networks is so much more stimulative than land or ocean based trade routes. I will briefly sketch some similarities between the equations used to model network trade flows and fluid dynamics, but also why the analogy can be misleading.

Jun Zhang: *Reversed flapping flight and inverted hydrodynamical drafting*

We will discuss two recent laboratory experiments on interactions between moving boundaries and a surrounding fluid. Both experiments have shown surprising results. In the first experiment, we study the flapping flight of a rigid wing that is allowed to pitch (rotate) passively in response to fluid forces. We demonstrate a new state in which the flapping wing flies backward as the flapping frequency exceeds a certain threshold. This counter-intuitive transition occurs as the wing's leading edge in forward flight becomes the trailing edge in reversed motion. A dimensionless number -- the "passive pitching" -- demarcates the transition. In the second experiment, we investigate the schooling of fish or the flocking of birds by studying the group dynamics of interacting flapping flags in a moving fluid. We have found that, unlike the well-known hydrodynamic drafting of rigid objects placed in tandem, the leading flapping flag enjoys a reduced drag while the follower suffers a drag increase. If this result from passive, flexible objects can be applied to self-propelled locomoting bodies, it implies that it is easier to lead than to follow in a group.

Arjun Raj: *Nature, nurture or just dumb luck: stochastic gene expression and cell fate*

Gene expression, the fundamental process by which the program encoded in our DNA is executed, turns out to be a surprisingly stochastic, with large variability in the numbers of mRNAs and proteins in otherwise identical populations. This fact raises the question of how organisms are able to mitigate the effects of variability to produce reliable outcomes. To see if and how organisms do this, we studied the gene regulatory network responsible for gut formation during early embryonic development in C. elegans. We found that the normal gut development pathway is remarkably robust, but this robustness can be eliminated by mutations to a single gene that result in wildly varying embryonic fates. We have shown that these different fates result from the variable expression of a key gene in the rewired mutant gut network. These results suggest that redundancy in developmental networks can serve to mask otherwise hidden sources of gene expression variability.

Paul Atzberger: *Stochastic Immersed Boundary Methods for Simulation of Microscopic Fluid-Structure Systems with Thermal Fluctuations.*

The immersed boundary method is a numerical approach which has been applied to many macroscopic systems involving a fluid which interacts with flexible elastic structures. For microscopic systems of sufficiently small length-scale thermal fluctuations become significant and also must be taken into account. In this talk we shall discuss an extension of the immersed boundary method framework which incorporates thermal fluctuations through appropriate stochastic forcing terms in the fluid equations. This gives a system of stiff SPDE's for which standard numerical methods perform poorly. We shall discuss a few different approaches by which stochastic calculus can be used to obtain analytic results to help in handling the stiff features of the equations. We will further show how this can be used to formulate numerical methods for the fluid-structure equations both discretized on uniform and multilevel adaptive meshes. To demonstrate the approaches in practice we shall present simulation results the microscopic mechanics of polymers, polymer knots, membrane sheets, and vesicles.

Tom's thesis and papers may be downloaded here.