Science and Engineering

# Applied and computational mathematics

## Overview

### What is applied and computational mathematics?

Applied mathematics and computational mathematics are closely linked branches of mathematics that are concerned with developing and analysing mathematical models and solving mathematical problems using numerical methods and computer simulation.

Our applied and computational mathematics discipline is devoted to:

• teaching mathematics with an emphasis on mathematical modelling and numerical simulation
• high quality research to develop new fundamental knowledge in mathematics
• applying new models, algorithms and software to solve complex problems for our partners in academia, industry and government.

Our discipline’s research has contributed to new advancements and insights into a wide variety of practical applications, including:

• the electrical activity of the heart
• the growth of cancerous tumours
• wound healing
• battery technology
• coal seam gas production
• wood drying processes
• agrichemical spraying of plants
• sugar cane production
• management of coastal aquifers.

### More than numbers

Wondering what a job in maths might be like? You'd be surprised how many exciting careers require a maths skill set. Check out our partner institute, the Australian Mathematical Sciences Institute (AMSI), for career guidance resources and career path inspiration.

### Research themes

Mathematical modelling of physical processes with industrial applications
Heat and mass transfer
Transport theory
Fluid dynamics, bubbles and droplets
Hydrology and groundwater flow
Electrochemical systems and solar cells
Porous materials
Phase-change phenomena
Multi-scaled modelling.
Modelling processes in biology, ecology and medicine
Computational biology
Intra-cellular processes
Cardiac modelling
Collective cell motion
Tissue engineering
Wound healing processes
Tumour growth and invasion
Infectious disease modelling
Gene regulation
Pattern formation
Fracture healing
Regulation of musculoskeletal tissues.
Computational simulation of large, multi-scaled or complex systems
Scientific computing and visualisation
High performance computing
Parallel computing with GPU hardware
Stochastic simulation
Computational efficiency
Numerical methods
Numerical linear algebra
Quantifying uncertainty.
New mathematical techniques and analysis of mathematical models
Analysis of differential equations
Numerical analysis
Analysis of stochastic models
Asymptotic analysis and perturbation methods
Applied complex analysis
Geometric singular perturbation theory
Mathematical cryptology.

### Rankings

Our recent Excellence in Research for Australia (ERA) ratings were:

• 5 (well above world standard) for numerical and computational mathematics (FOR 0103)
• 4 (above world standard) for applied mathematics (FOR 0102).

ERA (Excellence in Research for Australia) evaluates the quality of research undertaken in Australian universities against national and international benchmarks.

### Centres and groups

Our researchers collaborate on projects in specialised research groups and facilities across disciplines and institutions, including within the ARC Centre of Excellence for Mathematical and Statistical Frontiers and in groups like the QUT Plant nutrients and nutrition group, based at the Institute for Future Environments.

### Teaching

We offer a major in applied and computational mathematics in our Bachelor of Mathematics, which provides high-quality learning for students who want to combine their studies in mathematics with real-world applications and computational simulations. Computer simulation and visualisation techniques can be used in many research projects, from bone fracture and wound healing to modelling saltwater intrusion in coastal systems.

Our major introduces students to a wide range of concepts in mathematical foundations, modelling and computational methods, and provides strong links between theory and application.

Students investigate underlying mathematical theory to see how it can be applied to real-world scenarios from many fields of study including the physical and chemical sciences, biology, engineering and the social sciences. They also develop computational solution and simulation methods to couple with modelling skills in order to investigate large-scale applied problems.

#### Industry engagement in learning

In 2016 students in our capstone unit worked with Australasian Groundwater and Environmental Consultants (AGE) to model and analyse groundwater flow in aquifer systems used for crop irrigation in locations across regional Queensland. Through a combination of mathematical modelling and computational algorithms, students simulated different groundwater pumping scenarios and devised strategies to ensure this natural resource was being used sustainably.

As well as bringing together their three years of learning in modelling and computation, this project gave our students valuable experience in teamwork, communication, and the nature and language of industry projects.

## Projects

The Category 1 funded research projects we are currently leading are:

### Mathematical and computational analysis of ship wakes

Project leaders
Professor Scott McCue, Associate Professor Timothy Moroney.
Dates
2018-2020
Project summary

This project aims to develop mathematical and computational tools to compute the energy in a given ship wake and to determine a range of properties of a ship by taking simple measurements of the water height as the ship travels past.

The expected outcomes should have direct implications for measuring damage to coastal zones by ship wakes and for surveillance of shipping channels.

### A unifying framework for generalised distributed-order fractional models

Project leaders
Dates
2018-2020
Project summary

This project aims to develop a unifying theoretical framework for generalised fractional models using measure theory and a new class of distributed-order nonlocal operators to simulate anomalous transport processes in heterogeneous and anisotropic porous media.

Expected outcomes of the project include an evaluation of dimensionality and/or complexity reduction of the governing equations in fractional transport models with a focus on groundwater applications.

### Mathematical models of cell migration in three-dimensional living tissues

Project leader
Professor Matthew Simpson
Dates
2017-2020
Project summary

This project aims to develop mathematical models of cell migration in crowded, living tissues. Existing models rely solely on stochastic simulations, and therefore provide no general mathematical insight into how properties of the crowding environment (obstacle shape, size, density) affect the migration of cells through that environment.

This project will produce mathematical analysis, mathematical calculations and exact analytical tools that quantify how the crowding environment in threedimensional living tissues affects the migration of cells within these tissues. Long term effects will be the translation of this new mathematical knowledge into decision support tools for researchers from the life sciences.

### Mathematical and computational models for agrichemical retention on plants

Project leader
Professor Scott McCue
Dates
2016-2019
Project summary

This project aims to build interactive software that simulates agrichemical spraying for multiple virtual plants reconstructed from scanned data. Mathematical modelling and computer simulation could offer an alternative to expensive experimental programs for agrichemical spraying of plants.

This project will use contemporary fluid mechanics to build practical mathematical models for droplet impaction, spreading and evaporation on leaf surfaces, and experimentally calibrate and validate the models.

### Image-based multiscale modelling of transport phenomena in porous media

Project leader
Professor Ian Turner
Dates
2015-2017
Project summary

This project aims to develop an innovative general framework to build multiphase porous media transport models directly from electron microscope images of the underlying microstructure. Leading edge experimental, computational and applied mathematical techniques are proposed to drive this novel approach of multiscale modelling, employable across numerous fields of science and engineering.

The central theme of developing an efficient and accurate multiscale model for simulating transport in heterogeneous porous media is expected to find application in the drying, timber and crop industries, and governmental agencies managing pollution in groundwater resources. This insight is likely to be invaluable for designing new industrial technologies and optimising current operations.

### Two-scale numerical modelling of coupled transport in heterogeneous media

Project leader
Dr Elliot Carr
Dates
2015-2017
Project summary

Groundwater constitutes a vital part of water resources in Australia, however, the quality of this water is susceptible to contamination. This project aims to develop an innovative two-scale mathematical model for contaminant transport that accounts for small-scale heterogeneities found in the unsaturated zone of an aquifer located between the ground surface and the underlying groundwater.

The work will develop valuable environmental insights, a simulation tool that will help in making decisions regarding the future management of Australian groundwater resources, and a general two-scale modelling and simulation framework for other important environmental and industrial problems involving coupled transport in heterogeneous media.

### New mathematical models for capturing heterogeneity of human brain tissue

Project leader
Dr Qianqian Yang
Dates
2015-2017
Project summary
This project aims to understand the impact of the heterogeneity of brain tissue on Magnetic Resonance Imaging (MRI) data in both healthy and diseased human brains, and to extract and quantify information on heterogeneity from the data.

A key goal is to develop novel mathematical and computational approaches to model the heterogeneity of the human brain. The work also aims to identify new biomarkers for classifying different brain diseases, based on the extent of heterogeneity across different brain tissue. Results will be validated against extensive MRI scanning data of patients. This project aims to advance state-of-the-art techniques in human brain MRI data analysis.

### New data-driven mathematical models of collective cell motion

Project leader
Professor Matthew Simpson
Dates
2013-2017
Project summary
Cancer and chronic wounds are a national, and indeed, international health problem set to worsen as our population ages. Predictive and interpretive tools are required to improve our understanding of collective cell migration in relation to cancer and chronic wounds. This project will produce new validated mathematical tools for predicting collective cell migration in a general framework that can deal with application-specific details, such as the role of cell shape and cell size.

Although cell shape and size are known to affect collective cell migration, standard mathematical models ignore these details. This project will produce new predictive mathematical modelling tools that are validated by new experimental data.

### Interdisciplinary and inter-institution projects

#### Novel multiscale model to investigate mechanical properties of cartilage

Leaders
Professor Yuantong Gu, Professor Kevin Burrage, Associate Professor Travis Klein.
Dates
2018-2020
Project summary

This project aims to develop a new multiscale model to investigate anisotropic and inhomogeneous mechanical properties of cartilage. It has been found that the mechanical properties of cartilage highly depend on its microstructures and components. The new model is proposed based on a new constitutive relation in the macroscale and a novel algorithm to obtain local stress distributions in the microscale as well as through rigorous experimental validations.

This model will be a powerful tool to understand cartilage mechanical properties. It will accelerate the design of mechanically viable artificial cartilage biomaterial, which will provide significant economic benefits and place Australia in the forefront of modelling and biomaterials.

## Student topics

Are you looking to further your career by pursuing study at a higher and more detailed level? We are currently looking for students to research with us. Contact our staff to find out more about research opportunities, or take a look at our student topics.

## Contact

#### School of Mathematical Sciences

• Level 6, O Block, Room O617
Gardens Point