MSc Projects in Numerical Analysis CONTACT: Dave Harris SEMESTER: Second Room: Alan Turing Building 2.218 Tel: 63683 email: david.harris"at"manchester.ac.uk
 Introduction: I am an applied mathematician whose research interests are based on the mechanics of granular materials, in particular in developing constitutive equations governing the deformation and flow of such materials (for example sand, clay, grain and rice). The constitutive equations together with various balance laws furnish mathematical models comprising systems of first (or sometimes second) order partial differential equations. In some models there are additional algebraic inequalities. The complexity of the models makes it very difficult to obtain analytic solutions. So, having constructed the equations I now wish to begin to solve them numerically! The projects listed below should really be regarded as the application of numerical analysis to continuum mechanics rather than as pure numerical analysis. Project 1: The simplest model corresponds to what might be called "post failure rigid plastic plane flows" - in some regions the material is rigid, while in other regions it has failed and is flowing freely. The key to a successful model is that the equations form a hyperbolic set of first order pde's. This project would involve carrying out an analysis of the characteristics, transforming the equations to characteristic coordinates, formulating some simple problems and writing a matlab/maple/mathematica program to solve these problems. Project 2: A considerably more complicated model corresponds to "pre-failure elasto-plastic planar deformations" of a soil or granular material. Many examples of an initial/boundary value problem for such materials arise in geotechnical engineering , an example being provided by the cone penetrometer test, where the strength of the soil is obtained by measuring the force required to push a cone-shaped tester into the soil. The finite element method is almost universally used for solving problems of this kind and the project would involve deciding on the appropriate way to write the equations for use with a finite element program, to research the literature to find an appropriate scheme and to develop a small-scale implementation of the chosen scheme using matlab. Project 3: Another interesting class of problems is provided by two phase mixtures of gas and grain. An example of such a problem is wind blown sand. The models used here are best described as "two fluid models" even though a set of discrete solid grains can hardly be described as a fluid! What is of primary interest here is the dynamic nature of the flow. The equations of motion are written in conservation form and various inhomogeneous terms are present which can (and usually do) wreak havoc with the chosen numerical scheme. It is crucial that the equations do not lose hyperbolicity and so viscosity is neglected (but care needs to be taken - models have a tendency to lose hyperbolicity even in the absence of viscosity, and it may be be necessary to take steps to avoid this). The project involves researching numerical schemes suitable for hyperbolic sets of equations and implementing them using matlab. Project 4: The neglect of viscosity in project 3 is physically unrealistic, so in this project we retain it. However, doing so radically alters the type of equations obtained, some of them becoming second order. A paradigm here is provided by the famous Burgers equation and this equation can be generalised in various ways. The project is to research into the numerical methods of solution suitable for Burgers-like equations (well, ok, finite difference) and to implement them on systems of Burgers-like equations using Matlab.