Applied Mathematics Seminars Autumn 2012

All seminars (unless otherwise stated) will take place on Mondays at 3.00 pm in Room 505 which is located on the 5th floor of the Mathematics Department. See How to Find Us for further details. There will be tea afterwards in room 606. If you require any more information on the Applied seminars please contact Professor Yaroslav Kurylev e-mail: y.kurylev AT ucl.ac.uk or tel: 020-7679-7896.

01 October 2012

Dr Ben Cox - Department of Medical Physics, UCL

Modelling and Inverse Problems in Photoacoustic Tomography

Abstract:
There are two uncoupled inverse problems in photoacoustic tomography, an acoustic inversion and a diffuse optical inversion. This talk will discuss both numerical modelling of the direct problems as well as the inverse problems. The acoustic inversion is well posed and well understood, and some experimental images will be shown that were obtained using a time-reversal image reconstruction algorithm. The optical inversion is ill-posed and, while progress has been made, practically useful results have yet to be obtained. Promising recent work on this problem using optimisations based on the radiative transfer equation will be described.

08 October 2012

NO SEMINAR

15 October 2012

Dr Mark Dickinson - Enthought Ltd.

Python, NumPy and EPD as a platform for scientific computing and visualisation.

Abstract:
Over the last few years, increasing numbers of researchers in scientific and financial domains have adopted the programming language Python, together with the de-facto standard libraries NumPy and SciPy, as a platform for numeric computing and visualisation.  In this talk we'll briefly introduce Python and its libraries and demonstrate some of the features that have made Python so successful in this area.

We'll also introduce EPD, a pre-packaged Python distribution comprising Python together with several of the most commonly used libraries.

22 October 2012

Dr Monika Nitsche - Dept. of Maths, Univ. of New Mixico, USA

On regularizations of vortex sheet motion

Abstract:
The vortex sheet is a mathematical model for a shear layer, in which the layer is approximated by a surface. Vortex sheets generally develop singularities in finite time. To approximate the fluid past this time, the motion is regularized and the sheet defined as the limit of zero regularization. However, very little is known about this limit. For example, it is not known whether it is unique or whether it depends on the regularization. I will discuss several regularization mechanisms, including physical ones such as fluid viscosity, and purely numerical ones such as the vortex blob and the Euler-alpha regularization. I will show results for a model problem and discuss some of the unanswered questions of interest.

29 October 2012

Dr D J Smith - Dept. of Mathematics, University of Birmingham

The fluid mechanics of sperm motility

Abstract:
Motility, the ability of sperm to propel themselves through fluid through the beating of a flagellum, is essential to life and has since the 1950s been an inspirational topic for applied mathematicians. However, despite decades of work, we still do not understand what sperm have to do `mechanically' in order to fertilise. Central questions concern the transport of energy within the flagellum, the ability of sperm to penetrate the thin films of highly viscous non-Newtonian biological fluids lining the female reproductive tract, and regulation of the flagellar waveform associated with progression and directional changes. This seminar will describe recent multidisciplinary work with Birmingham Women's Hospital Fertility Centre, combining digital imaging with computational modelling of flagellum/fluid interaction. Key ideas include geometrically nonlinear elastohydrodynamics of the sperm flagellum, the interaction of sperm with surrounding boundaries, estimates of energy transport, and the effect of non-Newtonian rheology on motility.

05 November 2012

NO SEMINAR

12 November 2012

Dr. Mikhail Cherdantsev (School of Maths, Univ. of Cardiff)

Homogenisation of high-contrast periodic problems in non-linear elasticity

Abstract:
I will speak about periodic composites in non-linear elasticity setting consisting of two materials one of which is soft and another is stiff. I will describe the homogenisation limit of corresponding non-linear functional as the period tends to zero and the contrast (coupled with the period) tends to infinity. The method essentially employs a two-scale version of Gamma-convergence and multi-scale analysis of high-contrast problems.

19 November 2012

Prof. Alexander Tovbis - Department of Mathematics, University of Central Florida, Orlando

Universality for the focusing Nonlinear Schrödinger equation at the gradient catastrophe point: Rational breathers, poles of the tritronqué solution to Painlevé I and the large amplitude waves formation

Abstract:
We consider the point of gradient catastrophe for the dispersionless limiting system of the focusing Nonlinear Schrö̈dinger equation (NLS) with small dispersion. As shown by various analytical and numerical methods, this is the point where a slowly modulated high frequency plane wave solution to the small dispersion (semiclassical) focusing NLS suddenly burst into rapid amplitudial oscillations (spikes). We give complete description of the leading order solution to the semiclassical focusing NLS near the point of gradient catastrophe in terms of the tritronqué solution to the Painlevé I and rational breathers for the NLS. In particular, each spike corresponds to a pole of the tritronqué and has the universal shape of a scaled rational (Peregrine) breather. The height of each spike is three times the height of the background oscillations. Thus, we are able to demonstrate how certain slowly modulated one-humped plane waves develop into large amplitude (rogue) waves, radiating right and left in the 1D medium. Our calculations are rigorous and the corresponding error estimates (based on the nonlinear steepest descent (Deift-Zhou) method Riemann-Hilbert problems) are provided. The talk is based upon joint work [1] with Marco Bertola.

References:
1. M. Bertola and A. Tovbis, Universality for the focusing nonlinear Schroedinger
equation at the gradient catastrophe point: Rational breathers and poles of
the tritronqué solution to Painleve I, to appear in Comm. Pure and Appl.
Math..

26 November 2012

Prof. Robin Tucker - Dept. of Physics, University of Lancaster

Eigenvalue Distributions and Casimir Stresses in Media

Abstract:
A new mathematical and computational technique for calculating finite quantum expectation values of energy, momentum and angular momentum currents and densities associated with electromagnetic fields in both bounded and unbounded domains will be presented. This technique will be used to analyze the effects of quantum fluctuations of such fields in  continuous and piecewise continuous inhomogeneous dielectrics in an attempt to understand some aspects of  the Lifshitz theory.

03 December 2012

NO SEMINAR

10 December 2012 (Please note: This is a date change from 03 December 2012)

Dr W  Parnell - Dept. of Mathematics, University of Manchester

Elastodynamic cloaking: Transformation elasticity with prestressed hyperelastic solids

Abstract:
Interest in cloaking theory (i.e. rendering objects near-invisible to incident waves) and its practical realization has grown significantly since the early theoretical work in 2006 of Leonhardt and the Pendry group in optics and electromagnetism respectively. Methods have largely been based on the idea of coordinate transformations, which motivate the design of cloaking metamaterials. These materials are able to guide waves around a specific region of space. Research has subsequently focused on the possibility of cloaking in the contexts of acoustics, surface waves in fluids, heat transfer, fluid flow and linear elastodynamics.

It was shown by Milton and co-workers that elastodynamic cloaking is made difficult due to the lack of invariance of Navier's equations under general coordinate transformations which retain the symmetries of the elastic modulus tensor. Invariance of the governing equations can be achieved if assumptions are relaxed on the minor symmetries of the elastic modulus tensor but commonly occurring elastic materials do not possess this property.

In this talk after giving a brief introduction to cloaking theory and the difficulties that arise in elastodynamics, we shall show that it is theoretically possible to construct elastodynamic cloaks by pre-stressing hyperelastic (nonlinear) solids (e.g. rubber). We shall discuss an initial simple case (antiplane waves) as was studied in [1] before moving on to describe various generalizations including finite cloaks [2] and more general elastodynamic cloaking pre-stress [3]. Much of this work has been done in collaboration with Prof. Andy Norris (Rutgers, USA)

[1] Parnell, W.J. "Nonlinear pre-stress for cloaking from antiplane elastic waves", Proc Roy Soc A 468, 563-580.
[2] Parnell, W.J., Norris, A.N. and Shearer, T. "Employing pre-stress to generate finite cloaks for antiplane elastic waves." Appl. Phys. Letters 100, 171907.
[3] Norris, A.N. and Parnell, W.J. "Hyperelastic cloaking theory: Transformation elasticity with prestressed solids". Proc. Roy. Soc. A. In press