Past Industrial and Applied Mathematics Seminar

5 February 2015
Samuel Isaacson

Particle-based stochastic reaction diffusion methods have become a 
popular approach for studying the behavior of cellular processes in 
which both spatial transport and noise in the chemical reaction process 
can be important. While the corresponding deterministic, mean-field 
models given by reaction-diffusion PDEs are well-established, there are 
a plethora of different stochastic models that have been used to study 
biological systems, along with a wide variety of proposed numerical 
solution methods.

In this talk I will motivate our interest in such methods by first 
summarizing several applications we have studied, focusing on how the 
complicated ultrastructure within cells, as reconstructed from X-ray CT 
images, might influence the dynamics of cellular processes. I will then 
introduce our attempt to rectify the major drawback to one of the most 
popular particle-based stochastic reaction-diffusion models, the lattice 
reaction-diffusion master equation (RDME). We propose a modified version 
of the RDME that converges in the continuum limit that the lattice 
spacing approaches zero to an appropriate spatially-continuous model. 
Time-permitting, I will discuss several questions related to calibrating 
parameters in the underlying spatially-continuous model.

  • Industrial and Applied Mathematics Seminar
29 January 2015
Michael Dallaston, Jeevanjyoti Chakraborty, Roberta Minussi

In order:

1. Michael Dallaston, "Modelling channelization under ice shelves"

2. Jeevanjyoti Chakraborty, "Growth, elasticity, and diffusion in 
lithium-ion batteries"

3. Roberta Minussi, "Lattice Boltzmann modelling of the generation and 
propagation of action potential in neurons"

  • Industrial and Applied Mathematics Seminar
22 January 2015
Chris MacMinn
Coupling across scales is often particularly strong in porous rocks,
soils, and sediments, where small-scale physical mechanisms such as
capillarity, erosion, and reaction can play an important role in
phenomena at much larger scales. Here, I will present two striking
examples of this coupling: (1) carbon sequestration, where storage
security relies on the action of millimeter-scale trapping mechanisms
to immobilise kilometer-scale plumes of buoyant carbon dioxide in the
subsurface, and (2) fluid injection into a granular solid, where
macroscopic poromechanics drive grain-scale deformation and failure.
I will show how we derive physical insight into the behaviour of these
complex systems with an effective combination of theoretical models,
numerical simulations, and laboratory experiments.
  • Industrial and Applied Mathematics Seminar
4 December 2014
Andrew Hausrath
The folded structures of proteins display a remarkable variety of three-dimensional forms, and this structural diversity confers to proteins their equally remarkable functional diversity. The accelerating accumulation of experimental structures, and the declining numbers of novel folds among them suggests that a substantial fraction of the protein folds used in nature have already been observed. The physical forces stabilizing the folded structures of proteins are now understood in some detail, and much progress has been made on the classical problem of predicting the structure of a particular protein from its sequence. However, there is as yet no satisfactory theory describing the “morphology” of protein folds themselves. This talk will describe an approach to this problem based on the description of protein folds as geometric objects using the differential geometry of curves and surfaces. Applications of the theory toward modeling of diverse protein folds and assemblies which are refractory to high-resolution structure determination will be emphasized.
  • Industrial and Applied Mathematics Seminar
27 November 2014
Peter Hicks

Droplet impacts form an important part of many processes and a detailed
understanding of the impact dynamics is critical in determining any
subsequent splashing behaviour. Prior to touchdown a gas squeeze film is
set-up between the substrate and the approaching droplet. The pressure
build-up in this squeeze film deforms the droplet free-surface, trapping
a pocket of gas and delaying touchdown. In this talk I will discuss two
extensions of existing models of pre-impact gas-cushioned droplet
behaviour, to model droplet impacts with textured substrates and droplet
impacts with surfaces hot enough to induce pre-impact phase change.

In the first case the substrate will be modelled as a thin porous layer.
This produces additional pathways for some of the gas to escape and
results in less delayed touchdown compared to a flat plate. In the
second case ideas related to the evaporation of heated thin viscous
films will be used to model the phase change. The vapour produced from
the droplet is added to the gas film enhancing the existing cushioning
mechanism by generating larger trapped gas pockets, which may ultimately
prevent touchdown altogether once the temperature enters the Leidenfrost

  • Industrial and Applied Mathematics Seminar
13 November 2014
Larry Forbes (Tasmania)

It is well known that the Navier-Stokes equations of viscous fluid flow do not give good predictions of when a viscous flow is likely to become unstable.  When classical linearized theory is used to explore the stability of a viscous flow, the Navier-Stokes equations predict that instability will occur at fluid speeds (Reynolds numbers) far in excess of those actually measured in experiments.  In response to this discrepancy, theories have arisen that suggest the eigenvalues computed in classical stability analysis do not give a full account of the behaviour, while others have suggested that fluid instability is a fundamentally non-linear process which is not accessible to linearized stability analyses.

In this talk, an alternative account of fluid instability and turbulence will be explored.  It is suggested that the Navier-Stokes equations themselves might not be entirely appropriate to describe the transition to turbulent flow.  A slightly more general model allows the possibility that the classical viscous fluid flows predicted by Navier-Stokes theory may become unstable at Reynolds numbers much closer to those seen in experiments, and so might perhaps give an account of the physics underlying turbulent behaviour.

  • Industrial and Applied Mathematics Seminar
6 November 2014
Matthew Wright
Persistent homology is a tool for identifying topological features of (often high-dimensional) data. Typically, the data is indexed by a one-dimensional parameter space, and persistent homology is easily visualized via a persistence diagram or "barcode." Multi-dimensional persistent homology identifies topological features for data that is indexed by a multi-dimensional index space, and visualization is challenging for both practical and algebraic reasons. In this talk, I will give an introduction to persistent homology in both the single- and multi-dimensional settings. I will then describe an approach to visualizing multi-dimensional persistence, and the algebraic and computational challenges involved. Lastly, I will demonstrate an interactive visualization tool, the result of recent work to efficiently compute and visualize multi-dimensional persistent homology. This work is in collaboration with Michael Lesnick of the Institute for Mathematics and its Applications.
  • Industrial and Applied Mathematics Seminar
30 October 2014
Steven Dargaville
LiFePO4 is a commercially available battery material with good theoretical discharge capacity, excellent cycle life and increased safety compared with competing Li-ion chemistries. During discharge, LiFePO4 material can undergo phase separation, between a highly and lowly lithiated form. Discharge of LiFePO4 crystals has traditionally been modelled by one-phase Stefan problems, which assume that phase separation occurs. Recent work has been using phase-field models based on the Cahn-Hilliard equation, which only phase-separates when thermodynamically favourable. In the past year or two, this work has been having considerable impact in both theoretical and experimental electrochemistry. Unfortunately, these models are very difficult to solve numerically and involve large, coupled systems of nonlinear PDEs across several different size scales that include a range of different physics and cannot be homogenised effectively. This talk will give an overview of recent developments in modelling LiFePO4 and the sort of strategies used to solve these systems numerically.
  • Industrial and Applied Mathematics Seminar