Many scientific and engineering problems involve heterogeneous media, where material properties (e.g. permeability, conductivity, diffusivity) vary spatially. Practical applications include groundwater flow in aquifers, heat and mass transport during drying processes and diffusion of molecules in biological tissues, to name a few. For problems where the material properties vary at a small scale, solving the governing mathematical model is prohibitively expensive since a very fine mesh/grid is required to capture the heterogeneous detail. Broadly speaking, homogenisation refers to the process of upscaling the fine-scale description of the flow/transport to a coarser-scale, thus enabling computationally feasible simulations. This process essentially involves homogenising part of the domain (or the full-domain) by replacing the spatially-varying material properties with averaged or effective properties.
This course focuses on modelling and simulating continuum diffusion in complex two-dimensional heterogeneous media. We will see how homogenising a heterogeneous domain and computing its effective diffusivity involves solving a suitable partial differential equation over the domain with appropriate boundary conditions. Apart from some simple cases, analytical solutions to such problems are unavailable. Instead, numerical solutions will be developed using finite volume discretisation and implemented in MATLAB.
Students should have some prior experience in numerical solution of ordinary/partial differential equations. Proficiency in MATLAB (or similar) is also required.
Elliot Carr is a Senior Lecturer and former ARC DECRA fellow (2015-2018) within the School of Mathematical Sciences at QUT. He completed his Bachelors (2005-2007), Honours (2008) and PhD (2009-2012) degrees (all in applied and computational mathematics) at QUT. His PhD thesis was titled “Exponential integrators and a dual-scale model for wood drying”. Following his PhD, Elliot worked as a postdoctoral researcher, which included positions at École Centrale Paris (now CentraleSupélec) in France and Imperial College London in the UK. At the start of 2015, Elliot returned to QUT to take up a permanent lecturing/research position.
Currently, Elliot’s research focuses primarily on the development and application of mathematical and computational techniques to solve and analyse mathematical models, particularly those involving partial differential equations. His expertise includes finite volume methods, multiscale modelling, heterogenous diffusion and time scale analysis. His has published over 25 journal articles in internationally-prestigious journals including Journal of Computational Physics, International Journal of Heat and Mass Transfer, Applied Mathematical Modelling, Multiscale Modelling and Simulation and Physical Review E.
More information about Elliot is available on his personal website: https://elliotcarr.github.io.