# Research¶

Broadly speaking, my research interests are in numerical methods and the application of high performance computing to mathematical and physical problems. In particular, I am interested in the numerical simulation of wave propagation problems with an emphasis on development and implementation of efficient and accurate methods for the truncation of unbounded domains. Additionally I am interested in investigating questions arising in mathematical biology, where my primary focus has been on micro-scale behavior of both passive objects and active swimmers in low Renyolds number flows.

## Publications & Preprints¶

### Complex dynamics of long, flexible fibers in shear.¶

#### LaGrone, J., Cortez, R. Yan, W., and Fauci, L.¶

#### Journal of Non-Newtonian Fluid Mechanics. Vol 269. July 2019.¶

### Abstract:¶

The macroscopic properties of polymeric fluids are inherited from the material properties of the fibers embedded in the solvent. The behavior of such passive fibers in flow has been of interest in a wide range of systems, including cellular mechanics, nutrient acquisition by diatom chains in the ocean, and industrial applications such as paper manufacturing. The rotational dynamics and shape evolution of fibers in shear depends upon the slenderness of the fiber and the non-dimensional “elasto-viscous” number that measures the ratio of the fluid’s viscous forces to the fiber’s elastic forces. For a small elasto-viscous number, the nearly-rigid fiber rotates in the shear, but when the elasto-viscous number reaches a threshold, buckling occurs. For even larger elasto-viscous numbers, there is a transition to a “snaking behavior” where the fiber remains aligned with the shear axis, but its ends curl in, in opposite directions. These experimentally-observed behaviors have recently been characterized computationally using slender-body theory and immersed boundary computations. However, classical experiments with nylon fibers and recent experiments with actin filaments have demonstrated that for even larger elasto-viscous numbers, multiple buckling sites and coiling can occur. Using a regularized Stokeslet framework coupled with a kernel independent fast multipole method, we present simulations that capture these complex fiber dynamics.

### Elastohydrodynamics of swimming helices: effects of flexibility and confinement.¶

#### LaGrone, J., Cortez, R., and Fauci, L.¶

#### Phys. Rev. Fluids 4, 033102. 2019.¶

### Abstract:¶

Motivated by bacterial transport through porous media, here we study the swimming of an actuated, flexible helical filament in both three-dimensional free space and within a cylindrical tube whose diameter is much smaller than the length of the helix. The filament, at rest, has a native helical shape modeled after the geometry of a typical bacterial flagellar bundle. The finite length filament is a free swimmer and is driven by an applied torque as well as a countertorque (of equal strength and opposite direction) that represents a virtual cell body. We use a regularized Stokeslet framework to examine the shape changes of the flexible filament in response to the actuation as well as the swimming performance as a function of the nondimensional Sperm number that characterizes the elastohydrodynamic system. We also show that a modified Sperm number may be defined to characterize the swimming progression within a tube. Finally, we demonstrate that a helical filament whose axis is not aligned with the tube axis can exhibit centering behavior in the narrowest tubes.

### Double Absorbing Boundaries for Finite-Difference Time-Domain Electromagnetics.¶

#### LaGrone, J. and Hagstrom, T.¶

#### Journal of Computational Physics, 326, pp.650-665. 2016.¶

### Abstract:¶

We describe the implementation of optimal local radiation boundary condition sequences for second order finite difference approximations to Maxwell’s equations and the scalar wave equation using the double absorbing boundary formulation. Numerical experiments are presented which demonstrate that the design accuracy of the boundary conditions is achieved and, for comparable effort, exceeds that of a convolution perfectly matched layer with reasonably chosen parameters. An advantage of the proposed approach is that parameters can be chosen using an accurate *a priori* error bound.

[Preprint]

### Radiation Boundary Condition Pack Library — rbcpack.org¶

Responsible for all code related to FDTD/Yee and second order finite difference interfaces along with the corresponding documentation and examples. This work was done in collaboration with HyPerComp, Inc.

## Selected Presentations¶

### Numerical methods for (a) Absorbing boundaries in wave propagation and (b) Elastohydrodynamics in flagellar swimming.¶

Sandia National Lab, Albuquerque, New Mexico

### Numerical Simulation of Viscoelastic Fibers.¶

Math Colloquium, April 2019, University of New Orleans.

### Helical Buckling of Elastic Fibers in Straining Flows.¶

GFS follow on: Mathematics of form in active and inactive media, March 2019, Isaac Newton Institute for Mathematical Sciences, Cambridge, United Kingdom.

### Numerical Simulation of Viscoelastic Fibers.¶

Oakridge National Lab, February 2019, Oak Ridge, Tennessee.

### Microdynamics in Regularized Brinkman Flow.¶

SIAM Texas-Louisiana Section Meeting, October 2018, Baton Rouge, Louisiana.

### Chemotaxis Modeling for Sperm Motility.¶

Society for Mathematical Biology Annual Meeting, July 2018, Sydney, Austrailia.

### Simulating Bacterial Motility in Confined Environments.¶

IUTAM Symposium on Motile Cells in Complex Environments, May 2018, Udine, Italy.

### Applications of Complete Radiation Boundary Conditions to Electromagnetic and Elastic Problems¶

Undergraduate Math Seminar, March 2017, Xavier University of New Orleans.

### High Order Radiation Boundary Conditions for Elastic Waves.¶

ICOSAHOM 2016, July 2016, Rio de Janeiro, Brazil.

### Applications of Complete Radiation Boundary Conditions.¶

RTG Seminar, January 2016, Rensselaer Polytechnic Institute.

### Double Absorbing Boundaries for Finite-Difference Time-Domain Electromagnetics.¶

Applied Math Seminar, November 2015, University of New Mexico.