Applied and Industrial Mathematics Research Group

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Areas of Research

Bushfire Modelling

(Rod Weber, Jason Sharples, Joanne Chapman, Karen King, Wendy Anderson, Phil Zylstra, Geoff Mercer )

• fireline growth
• spread rates
• meteorology in the high country and its effects on fire spread
• impacts of fire on vegetation
• spread and patterns in discontinuous fuels
• effects of prescribed burning

Modelling of many aspects of bushfires from the growth of the fireline, fuel dependent spread rates, impacts on live vegetation, determining damage criteria and effects of prescribed burning. Members of the group are actively involved with the Bushfire CRC.

Combustion Theory

(Harvi Sidhu, Geoff Mercer, Rod Weber, Vladimir Gubernov, Isaac Towers, Jason Sharples)

• combustion waves
  • stability
  • initiation
  • multistep reactions
  • Evans function analysis
• chemical reactors
• models of composting
• polymers

Investigation of combustion waves in both solid and gaseous fuels using a broad variety of both numerical and analytic approaches. This work ranges over new combustion wave solutions for chain branching models, stability analysis using Evans function methods, critical initial conditions for combustion to take place and the effects of ambient temperature and heat losses on combustion waves. Others aspects related to combustion studied include chemical reactors (batch and continual flow), models of industrial composting, landfill and waste water treatment.

Industrial Process Modelling

(Geoff Mercer, Harvi Sidhu, Steve Barry)

• wool scouring
• porous filtering
• wastewater treatment
• compost
• bioreactors
• bloom in chocolate
• submarine battery charging
• wine fermentation

Many different industrial processes are studied with in the research group. These research areas have often been initiated by members attendance at the annual Mathematics in Industry Study Group.

Modelling extreme temperature effects on living tissue

(Geoff Mercer, Harvi Sidhu, Steve Barry)

• skin burns
  • models of skin burns subject to fire exposure
  • automotive airbags
  • design of firefighter clothing to minimise skin burn
• the effect of cold water on a swimmers core temperature
• modelling hypothermia
• wart freezing

Human skin and subcutaneous tissue is a complex organ designed to efficiently cope with a wide range of temperatures and conditions. However, at extremely high temperatures, such as exposures during fires or explosions, the skin is unable to transport the heat away fast enough and burns occur. Similarly, at cold temperatures, the skin is unable to insulate properly causing long term heat loss and hypothermia. We look at various models of heat transfer through the skin and subcutanious tissue to better understand this behaviour.

Design of large area mode fibres

(Isaac Towers & Zlatko Jovanoski)

Single-moded optical fibres are preferred for optical communication and for high power applications. The small core of a conventional single-mode fibre leads to high power densities and gives rise to significant unwanted nonlinear optical effects. Nonlinear effects in a fibre can distort the pulses at high bit rate and can produce crosstalk among the closely spaced wavelengths. However, the use of large-core conventional fibres to overcome nonlinear effects is not advisable as the large number of modes in such a fibre reduces the data transfer rate in an optical communication system and affects the beam quality in fibre lasers. It is therefore preferred to use a fibre with a large core yet supporting a single guided mode. The goal of this project is to investigate designs for optical fibres which maximise mode area but allow only a single mode to propagate a meaningful distance.

Numerical schemes for parabolic partial differential equations

(Isaac Towers)

This research project is to develop and implement numerical schemes which solve multi-dimensional partial differential equations (PDE). By using an appropriate set of orthogonal basis functions the goal is to create a spectral method which allows for boundary conditions at infinity while maintaining the speed of an explicit time-marching approach. Operator splitting, integrating factors and the so-called explicit exponential methods are being investigated.

Numerical schemes for simulating optical beam propagation

(Isaac Towers)

To compliment theoretical investigations of nonlinear optical beams we develop numerical schemes to efficiently study beam propagation. A variety of techniques are being investigated to provide fast and accurate simulations.

Plant and Disease Spread

(Steve Barry, Geoff Mercer, Harvi Sidhu, Roslyn Hickson)

• models of weed spread
• intervention strategies
• seed banks
• flood events

There are several current models to simulate the spreading of weeds (or any population) in the environment, involving reaction-diffusion equations, integro-difference equations, cellular automata systems, SIR ordinary differential equation, stochastic differential equations, among others. We are currently developing PLANTSIM, a Matlab based package, which allows users to simulate and compare numerous population spread simulation models in either real or artificial situations. A key feature is the Graphical User Interface which allows users to manipulate the complex parameter data sets involved in real simulations. PLANTSIM allows the user to overlay, visualise and manipulate parameter data on satellite imagery, and then run a collection of different simulation routines, with output automatically visualised in a variety of formats with key results summarised and graphed in generated LaTeX and pdf files.

Additional work is being done on formulating new models for weed spread, particularly along river systems. This is being applied to the spread of Lippia in the Murray-Darling system.

Frailty Modelling

(Joanne Chapman with Prof. Robin Henderson)

Random errors and unmeasurable association within clusters cannot be assumed Normally distributed and ignored in survival analysis as in other areas. The frailty term is introduced to represent these unknowns. A typical assumption is that the frailty variable follows a gamma distribution. Models which extend existing models and allow for the possibility of negative correlation within clusters (as often seen in animal litters) are being developed.

Statistical Ecology

In order to ensure the survival of animal populations it is essential to be able to estimate mortality rates in the wild. Two ways of doing this are from the recovery of bands from animals found dead, and from mark-recapture experiments. Models involving survival, recapture and recovery probabilities as unknown parameters, are developed and fitted.

The following projects are currently being undertaken in this area.

•  Effect of banding on juvenile Little Penguins Eudyptula minor
    (Leesa Sidhu and Ted Catchpole)

This work is being conducted in collaboration with researchers from the Phillip Island Nature Park. For more than 50 years, researchers have been marking penguins with flipper bands. Studies of Adélie Penguins and King Penguins have shown that banded penguins have lower survival rates than unbanded birds, and that they use more energy than unbanded birds when swimming. Our earlier work in this area was the first to study the effect of banding on Little Penguins. We showed that banding had a detrimental effect on the survival of adult penguins, with banded birds having an annual survival probability 6% lower than unbanded birds. While the effect of banding on juvenile Little Penguins is currently unknown, it is likely that banding significantly reduces the survival of young birds, particularly in their first year of life, and that our existing estimates of first-year survival underestimate the true survival probability for unbanded birds. A study examining the effects of banding on juvenile Little Penguins is currently underway on Phillip Island. The main aim of this project is to analyse the data from the Phillip Island study, in order to obtain estimates for survival for banded and unbanded juvenile Little Penguins, and for band loss in penguin chicks. The Department of the Environment and Heritage will use the results of this project, together with those from our earlier work, to determine whether banding of Little Penguins will be allowed to continue in Australia.

•  Analysis of recovery/recapture data for Pacific Gulls Larus pacificus
    (Leesa Sidhu and Ted Catchpole)

This work is being conducted in collaboration with researchers from La Trobe University. The Pacific Gull is the only large gull occurring naturally in Australia. There is evidence that its population size is falling, its range contracting, and that it could become extinct. While earlier studies of Pacific Gulls have focused on their biology, there are no existing survival estimates for these birds. This study will be the first to produce age- and time-varying survival estimates for these birds, by analysing a long-term mark-recapture-recovery dataset. Such a study is crucial to improve our understanding of these birds, and to ensure the survival of this species.

•  Analysis of recovery/recapture data for Short-tailed Shearwaters Puffinus tenuirostris    (Leesa Sidhu and Ted Catchpole)

This work is being conducted in collaboration with researchers from La Trobe University. Short-tailed Shearwaters have been studied continuously on Fisher Island, Tasmania since 1946, making it one of the longest continuous studies of any wildlife population in the world. Although life-history data have been collected sporadically over this time period, a detailed mark-recapture-recovery analysis has not yet been conducted. This study will produce age- and time-varying survival estimates for Short-tailed Shearwaters, and examine the effect of individual covariates such as egg size on first year survival.

•  Comparing first year survival for Little Penguins Eudyptula minor in Phillip Island, Australia and Oamaru, New Zealand (Leesa Sidhu and Ted Catchpole)

This work is being conducted in collaboration with researchers from the Phillip Island Nature Park , the Department of Conservation, New Zealand and the Institute of Marine Research, Norway. While the Phillip Island study has been underway for almost 40 years, the New Zealand study consists of six years of data. Here we estimate and compare the survival of penguins in these two locations and determine to what extent the conclusions of covariate dependence of survival can be extended from Phillip Island to another penguin colony.

Modelling frost tolerance of Eucalyptus trees (Ted Catchpole)

Using computer algebra software to investigate parameter redundancy (Ted Catchpole)

Vision processes in humans and insects (Peter McIntyre and Colin Pask)

Stefan problems with two moving boundaries - modelling swelling processes

(Steve Barry, Geoff Mercer, Harvi Sidhu)

• cooking of whole grains
• swelling of polymer implants in drug delivery
• exact and asymptotic solutions of Stefan problems
• swelling of grease layers on wool fibres

During the cooking of whole grains, or the swelling of grease, water diffuses into the outside of the grain causing the outer boundary to swell. This creates a partially swollen region which expands inwards and outwards over time. This then leads to a diffusion problem where the two boundaries defining the diffusive region move (a Stefan problem). We consider modelling this process and finding exact, numerical and asymptotic solutions to the system.

Mathematics In Industry Study Group

(Geoff Mercer, Harvi Sidhu, Steve Barry, Rod Weber, Roslyn Hickson, Richard Pennifold)

Members of the AIMRG are actively involved in the annual Mathematics in Industry Study Group (MISG). They have, and continue to, moderate many problems at the MISG. This provides a rich source of new and exciting research areas of direct relevance to industry.

Dynamical Systems

(Harvi Sidhu, Geoff Mercer, Richard Pennifold, Zlatko Jovanoski)

• analysing chaotic behaviour
• dynamics in ecology
• spread of diseases
• estuary behaviour
• forced systems

We utilize methods and techniques from nonlinear dynamics theory such as bifurcation and singularity theories to investigate complex real-world phenomena such as population interactions in ecology, spread of diseases, dynamics of shallow estuaries under drought conditions and forced systems. We also investigate new methods to quantify chaotic behaviour in dynamical systems.

Defence Applications

(Geoff Mercer, Harvi Sidhu, Steve Barry)

• minimising detection by enemy radar
• safest routes through minefields
• dynamic route changes in real time to minimise risk
• optimal route for aerial surveillance of ships
• effects of classification range and turning radius on optimal routes
• travelling salesman problem applied to aerial surveillance

Different aspect of travel through a risk environment are studied. Determining the minimal detection route through an area under radar surveillance using differential equation based models. This has application to Unmanned Air Vehicle (UAV) deployment and design issues. Calculation of the safest route through a minefield when information on the location of the mines is only discovered once travel is underway. Solutions must be able to be calculated in real time. Many aspects of the aerial surveillance of maritime regions including use of a modified dynamic travelling salesman problem to determine the visiting order of detected but as yet unclassified ships and aspects of the airplanes capabilities such as turning radius. This is joint work with DSTO Air Operations Division.

Poroelasticity

(Steve Barry, Geoff Mercer)

• injections
• swelling
• drug delivery
• water extraction

Poroelasticity is the modelling of how fluid moves through elastic porous materials, such as consolidation of soils or compression of biological soft tissue. We investigate various methods for solving the nonlinear diffusion equations governing poroelasicity and how to appy this to various industrial and biological systems.

Spraying of Agrichemicals

(Geoff Mercer)

• spray drift management
• spray retention on plant leaves
• droplet impaction models
• uptake and diffusion through leaf cuticles
• minimising environmental impacts of agrichemical sprays

The study of many aspects of the spraying of agrichemicals on to plants with the aim of developing more efficent spray techniques and formulations so that reduced spray volumes can be used for the same efficacy. This has major environmental consequences leading to less pollution, lower residues in the plants and lower chemical usage. Models for the diffusion of agrichemicals through plant leaves are being developed that take into account the varying nature of different plant cuticle structures. Droplet impaction models that predict the droplet spread over the leaf surface and possible bounce of the droplets are under investigation. Spray drift mitigation by suitable shelterbelt design is also being studied. This is joint wok with the agrichemical research company Plant Protection Chemistry New Zealand.