# Informal Applied and Numerical Analysis Seminars:

## Spring 2012

• 3 Feb 2012
Why you should never employ an American builder
Matthew Crooks

The last few years have seen major earthquakes cause mass devastation across the world from both the actual tremors to resulting tsunamis. Although nothing can be done to stop earthquakes an understanding of the deformations involved can help minimise their destruction.

In this talk I shall give an overview of geophysicists' current theory of the mechanisms involved and then derive the general solution for the velocity field within the tectonic plate. I shall then give an introduction to the mechanics of granular materials and how this shall be used to model the fault material and its coupling to the tectonic plates.

• 10 Feb 2012
Numerical solution of stochastic differential equations (SDEs)
Phillip Taylor

A stochastic differential equation (SDE) is an equation of the form

dX(t) = f(X(t)) dt + g(X(t)) dW(t),

where W(t) is a Wiener process (aka Brownian motion). Numerical schemes for solving SDEs are similar to those for ODEs but the noise term g(X(t)) dW(t) adds extra complexity. In this talk I will be introducing some of the tools required to study SDEs and solve them numerically.

• 17 Feb 2012
Tomography Impedance Electrical to Introduction An (An Inverse Problem)
Russell Miller

Electrical Impedance Tomography (EIT) is a low resolution imaging modality with many practical applications. I will give an introduction to the mathematical problem and some reconstruction techniques that can be applied. I will also show some simple numerical examples to illustrate the ideas presented.

• 24 Feb 2012
How to make an invisible building - elastic waves, homogenization and cloaking
William Parnell

Firstly, no prior knowledge of elastic waves is required in order to understand this talk. The talk will be almost entirely of a `chalk and talk' nature. As I am sure you are all aware, elastic waves are important in a number of application areas, e.g. vibration control, seismic surveying and earthquakes. In particular as with lots of waves, it is frequently of interest to be able to guide and control the propagation of elastic waves. In this talk we'll first introduce what an elastic wave is and discuss some different types. We'll then move on to describe how we can use an elastic "cloak" in order to prevent waves entering specific regions of space and thus making them "invisible" to an outside observer. The creation of such conventional cloaks is a non-trivial matter and we discuss how homogenization theory can be used in order to help us design such cloaks.

Finally, we discuss an alternative to conventional cloaking that has arisen recently where we can use an elastic pre-stress in order to create an elastodynamic cloak. This is beneficial for a number of reasons which we'll describe.

• 2 Mar 2012
Some symmetries in (applied!) maths.
Stephen Clegg

I will be discussing the question: what sort of crystalline structure can a crystal have and how can we classify these structres? Along with related topics topics including wallpaper tilings, asymmetric tilings and asymmetric crystals....other aspects of symmetry in nature. At the end, I hope to pose a question and conduct an experminent involving fluids.

• 9 Mar 2012
Water run-off and surface tension in an aircraft fuel tank
Alice Thompson
Abstract (click to view)

When warm humid air enters an aircraft fuel tank, condensation can occur on cold surfaces within the tank, such as the tank walls and ceiling, and the fuel free-surface. Once formed, the condensed water then slowly drains downwards through the tank. I will describe some simple mathematical models and experiments to predict the influence of surface coatings on the water drainage. I will also discuss possible interactions between water drops and the fuel layer.

• 16 Mar 2012
Cellular Automata: A Max-plus Approach
Ebrahim Patel

Cellular automata (CA) are discrete dynamical systems consisting of an array of identical cells. They comprise simple local rules but exhibit complex global space-time patterns (sometimes fractal-like). CA have traditionally been studied in synchronous form (where each cell updates at the same time). However, such a system is a simplification. Thus, an asynchronous version adds realism, e.g. for modelling pattern creation that arises in natural objects such as snowflakes or animal skin patterns. In this talk, I will present a model of asynchrony that is based in max-plus algebra. This arises naturally from the CA requirement that a cell receives the state of each neighbour before updating. The significant result is the existence of a bijective mapping between this asynchronous CA and the synchronous CA. Moreover, this type of asynchrony is shown to always reach periodic behaviour, a feature not prevalent in many asynchronous CA models.

• 23 Mar 2012
Transient problems in periodic composite materials
Ellis Barnwell

The propagation of waves through periodic heterogeneous media is of particular interest in the engineering of composite materials. Time harmonic waves have been greatly covered through the method of asymptotic homogenisation. Transient problems, on the other hand, have not been extensively considered due to the inherent difficulties involved in the modelling of them. In my talk I will review the current theories that describe composite materials and discuss their limitations. I will then propose a new approach to modelling composites in which we may use the dispersion relation found using Bloch-Floquet theory. This allows us to investigate the effects of stop and pass bands on pulses.

• 20 Apr 2012
Plasticity Models in the mechanics of granular materials
David Harris

This talk is intended to give an introduction to the mechanics of granular materials. It is written from a personal viewpoint, that is to say, it introduces those aspects which are required to understand the models that I work with in my research: it does not pretend to be an objective introduction to the subject (if such a thing exists, which I doubt, there is no consensus).

It could be said that the state of the art in formulating predictive models for the behaviour of granular materials is akin to fluid mechanics prior to the advent of the Newtonian fluid or to solid mechanics prior to the advent of Hooke's law. There are no universally accepted models and no model has proven itself so superior to others as to cause researchers to gravitate towards it.

Granular materials may exhibit solid-like, liquid-like and gaseous-like behaviour (but the use of these words can be misleading, these "phases" are purely mechanical, there is no classical thermodynamics involved in this classification) and this talk is concerned with solid-like and what we may call liquid-like behaviour.

Despite the discrete nature of granular materials, I will try to convince you that continuum models may, possibly, be applicable in certain circumstances for answering certain questions. Having taken the decision to go for continuum models, I will explain the similarity and differences between the behaviour of granular materials and ordinary solid/fluid behaviour. In particular we will consider the concepts necessary to describe the deformation and flow, namely, stress, strain, velocity strain, velocity spin and intrinsic spin. The essential properties that granular materials exhibit (rather surprisingly, in common with metals) are yield, inelastic deformation, flow and history dependence. We will present a mathematical formulation for the equations, that is to say the balance equations (applicable to all continua), yield criteria (actually, an algebraic inequality rather than an equation) and flow rules. Geotechnical engineers use a flow rule called the "plastic potential model" but this is shunned by all other researchers. Unusual names for models abound, e.g. double-shearing; double-sliding free-rotating; double-slip and double-spin. Also unusual continua may be used, e.g. Cosserat continua (not so much in the UK where they have never been popular, but certainly on the continent, where, if you want to be taken seriously, you'd better incorporate some, possibly imaginary, Cosserat efects). One of the above names is the name I give to my class of models. If you are lucky I will present the equations for my models, on the other hand I may keep them a secret...

• 27 Apr 2012
The acoustic radiation force on spheres
Kate Saunders

The radiation force on a body in an acoustic field is a second-order (nonlinear) effect. Potential applications include particle manipulation, for example as a possible alternative to optical tweezers.

In this talk I will discuss a method for finding the radiation force on a submerged sphere subjected to a weakly nonlinear acoustic field. It is assumed that the wavelength of the incident wave is large compared to the radius of the sphere. In this long-wavelength limit, using the example of an incident plane wave, I will work through a matched asymptotic expansions method for finding the scattered field, and hence the acoustic radiation force. I will then discuss the force due to an incident Bessel beam. This problem proves to be more interesting, since placing the sphere away from the axis of the beam results in a radiation force acting in the transverse, as well as the axial, direction.

• 4 May 2012
Rayleigh-Benard convection generated by a diffusion flame
Philip Pearce

We investigate the Rayleigh-Benard convection problem within the context of a diffusion flame formed in a porous channel where the fuel and oxidiser concentrations are given. When formulated in the low Mach number approximation the problem depends on two control parameters, the Rayleigh number and the Damkohler number. For an infinite Damkohler number the top half of the channel is heated from below in a similar way to the standard Rayleigh-Benard problem, but the rigid wall boundary condition for velocity applies at the lower wall which is below the flame sheet. Once the system has become unstable convection rolls form, which interact with the diffusion flame to cause cellular flames. These have been studied in the literature in the context of a variable density model but this study is the first to include the effect of buoyancy to account for the full hydrodynamics. To give a background a quick summary of the main (well known) results on the instabilities of standard Rayleigh-Benard convection and the planar diffusion flame without hydrodynamics is given. We then go on to formulate the problem and numerically compute the critical Rayleigh number (and its dependence upon the aspect ratio of the numerical domain) in the case of very high Damkohler number.

• 11 May 2012
An introduction to seismic imaging
Francis Watson

Seismic imaging, or migration, is a method of estimating properties of the Earth's subsurface from reflected seismic waves. Waves are transmitted from a source such as air gun, and the reflected/scattered waves are recorded by several receivers, or geophones.

Data is recorded as a function of time and (x,y) position on the Earth's surface. We will derive one method for performing a linearised inversion of the data to form a 3D image of the subsurface. This linearisation can sometimes be unsatisfactory, leading to unphysical artefacts in the image and misplaced features. In light of this, we will move on to discuss a method of improvement (which the industry terms annihilators), forming a partially linearised inversion.