Past Industrial and Applied Mathematics Seminar

19 May 2016

Permanent deformations of crystalline materials are known to be carried out by a large
number of atomistic line defects, i.e. dislocations. For specimens on micron scales or above, it
is more computationally tractable to investigate macroscopic material properties based on the
evolution of underlying dislocation densities. However, classical models of dislocation
continua struggle to resolve short-range elastic interactions of dislocations, which are believed
responsible for the formation of various heterogeneous dislocation substructures in crystals. In
this talk, we start with discussion on formulating the collective behaviour of a row of
dislocation dipoles, which would be considered equivalent to a dislocation-free state in
classical continuum models. It is shown that the underlying discrete dislocation dynamics can
be asymptotically captured by a set of evolution equations for dislocation densities along with
a set of equilibrium equations for variables characterising the self-sustained dislocation
substructures residing on a shorter length scale, and the strength of the dislocation
substructures is associated with the solvability conditions of their governing equilibrium
equations. Under the same strategy, a (continuum) flow stress formula for multi-slip systems
is also derived, and the formula resolves more details from the underlying dynamics than the
ubiquitously adopted Taylor-type formulae.

  • Industrial and Applied Mathematics Seminar
12 May 2016
Eddie Wilson

My main purpose in this talk is try and convey a sense of my enthusiasm for mathematical modelling generally and how I've come to use it in a range of transport applications. For concreteness, I am going to talk in particular about work I have been doing on EPSRC grant EP/K000438/1 (PI: Jillian Anable, Aberdeen) where we are using the UK government's Department for Transport MOT data to estimate mileage totals and study how they are broken down across the population in various different ways. Embedded inside this practical problem is a whole set of miniature mathematical puzzles and challenges which are quite particular to the problem area itself, and one wider question which is rather deeper and more general: whether it is possible (and how) to convert usage data that is low-resolution in time but high-resolution in individuals to knowledge that is high-resolution in time but only expressed at a population level.

  • Industrial and Applied Mathematics Seminar
5 May 2016
Ory Schnitzer

Surface plasmons are collective electron-density oscillations at a metal-dielectric interface. In particular, highly localised surface-plasmon modes of nanometallic structures with narrow nonmetallic gaps, which enable a tuneable resonance frequency and a giant near-field enhancement, are at the heart of numerous nanophotonics applications. In this work, we elucidate the singular near-contact asymptotics of the plasmonic eigenvalue problem governing the resonant frequencies and modes of such structures. In the classical regime, valid for gap widths > 1nm, we find a generic scaling describing the redshift of the resonance frequency as the gap width is reduced, and in several prototypical dimer configurations derive explicit expressions for the plasmonic eigenvalues and eigenmodes using matched asymptotic expansions; we also derive expressions describing the resonant excitation of such modes by light based on a weak-dissipation limit. In the subnanometric ``nonlocal’’ regime, we show intuitively and by systematic analysis of the hydrodynamic Drude model that nonlocality manifests itself as a potential discontinuity, and in the near-contact limit equivalently as a widening of the gap. We thereby find the near-contact asymptotics as a renormalisation of the local asymptotics, and in particular a lower bound on plasmon frequency, scaling with the 1/4 power of the Fermi wavelength. Joint work with Vincenzo Giannini, Richard V. Craster and Stefan A. Maier. 

  • Industrial and Applied Mathematics Seminar
28 April 2016
Bob Eisenberg

Life is different because it is inherited. All life comes from a blueprint (genes) that can only make proteins. Proteins are studied by more than one hundred thousand scientists and physicians every day because they are so important in health and disease. The function of proteins is on the macroscopic scale, but atomic details control that function, as is shown in a multitude of experiments. The structure of proteins is so important that governments spend billions studying them. Structures are known in exquisite detail determined by crystallographic measurement of more than 105 different proteins. But the forces that govern the movement and function of proteins are not visible in the structure. Mathematics is needed to compute both function and forces so comparison with experiment can be made. Experiments report numbers, typically sets of numbers in the form of graphs. Verbal models, however beautifully written in the biological tradition, do not provide numerical outputs, and so it is difficult to tell which verbal model better fits data.

The mathematics of molecular biology must be multiscale because atomic details control macroscopic function. The device approach of the engineering and English physiological tradition provides the dimensional reduction needed to solve the multiscale problem. Mathematical analysis of hundreds of experiments (reported in some fifty papers) has been successful in showing how some properties of an important class of proteins—ion channels— work. Ion channels are natural nanovalves as important to animals as Field Effect Transistors (FETs) are to computers. I will present the Fermi Poisson approach started by Jinn Liang Liu. The Fermi distribution is used to describe the saturation of space produced by crowded spherical ions. The Poisson equation (and continuity of current) is used to describe long range electrodynamics. Short range correlations are approximated by the Santangelo equation. A fully consistent mathematical description reproduces macroscopic properties of bulk solutions of sodium and calcium chloride solutions. It also describes several different channels (with quite different atomic detailed structures) quite well in a wide range of conditions using a handful of parameters never changed. It is not clear why the model works as well it does, nor is it clear how well the model will work on other channels, transporters or proteins.

  • Industrial and Applied Mathematics Seminar
17 March 2016
David Hu

Fluids and solids leave our bodies everyday.  How do animals do it, from mice to elephants?  In this talk, I will show how the shape of urinary and digestive organs enable them to function, regardless of the size of the animal.  Such ideas may teach us how to more efficiently transport materials.  I will show how the pee-pee pipe enables animals to urinate in constant time, how slippery mucus is critical for defecation, and how the motion of the gut is related to the density of its contents, and in turn to the gut’s natural frequency. 

More info is in the BBC news here:

  • Industrial and Applied Mathematics Seminar
10 March 2016
Richard Craster

The aim of this talk is to describe effective media for wave propagation through periodic, or nearly periodic, composites. Homogenisation methods are well-known and developed for quasi-static and low frequency regimes. The aim here is to move to situations of more practical interest where the frequencies are high, in some sense, and to compare the results of the theory with large scale simulations.

  • Industrial and Applied Mathematics Seminar
3 March 2016
Francesco dell'Isola
There are relatively few results in the literature of non-linear beam theory: we recall here the very first classical results by Euler–Bernoulli and the researches stemming from von Kármán for moderately large rotations but small strains. In this paper, we consider a discretized springs model for extensible beams and propose a heuristic homogenization technique of the kind first used by Piola. The homogenized energy obtained has some peculiar features which we start to describe by solving numerically some exemplary deformation problems. Furthermore we consider pantographic structures constituted by the introduced nonlinear beams and study some planar deformation problems. Numerical solutions for these 2D problems are obtained via minimization of energy and are compared via some experimental measurements, in which the importance of elongation phenomena are clearly pointed out. In the conclusions we indicate a list of some mathematical problems which seems worth of consideration. 
Indeed Already Piola in 1848 introduces for microscopically discrete systems to be described via a continuum model: i) the micro-macro kinematical map, ii) the identification of micro- macro work functional and iii) the consequent determination of macro-constitutive equations in terms of the micro properties of considered mechanical system.
Piola uses, following the standards of his time, a rigorous mathematical deduction process and considers separately one dimensional, two dimensional and three dimensional continua as continua whose reference configuration is a curve, a surface or a regular connected subset of Euclidean three dimensional space. This subdivision of the presented matter is also followed by Cosserat Brothers: how to detect the influence on their works exerted by Piola’s pioneering ones is a historical problem which deserves further in-depth studies.
In the present paper we follow the spirit of Piola while looking for Lagrange density functions for a class of non-linear one dimensional continua in planar motion: we focus on modeling phenomena in which both extensional and bending deformations are of relevance.
Usually in literature the simultaneous extension and bending deformation of a beam is not considered: however when considering two dimensional continua embedding families of fibers as a model of some specific microstructured mechanical systems (as fiber fabrics or pantographic sheets ) the assumption that the fibers cannot extend while bending is not phenomenologically well-grounded. Therefore, we are led in the second part of the present paper to present some two dimensional continua in which the second gradient of in plane displacement (involving so called geodesic bending) appears in the expression of deformation energy.
The modeling assumptions are, in both cases, based on a physically reasonable discrete microstructure of used beams: in engineering literature these microstructures, constituted by extensional and rotational springs and possibly rigid bars, were introduced in order to get discrete Lagrangian approximation of continuum models in linearized regimes.
A natural development, involving the study of spatial placements of one dimensional or two dimensional continua or the introduction of three dimensional continua embedding reinforcement fibers will be subject of further investigations.
The study of pantographic sheets by means of a micro model based on Cauchy first gradient continuum models involves the choice of relatively small length scale, implying the introduction of numerical models involving finite elements with several millions of degrees of freedom: the computational burden of such models makes their use, at least in the mid term horizon, absolutely inappropriate. The higher gradient reduced order model presented in this paper involves a rather more effective numerical modeling whose performances (as will be shown in a forthcoming paper Giorgio et al. in preparation) are however absolutely comparable.
However the problem of formulating intermediate meso modeling, involving a class of Generalised Beam Theories, will be necessarily to be confronted: for instance the deformation of beam sections involving warping, Poisson effects, elastic necking or large shear or twist deformation can definitively be studied via reduced order models not resorting to the most detailed micro Cauchy first gradient models.
One should also remark that higher gradient continuum models may require novel integration schemes, more suitable to their intrinsic structure: we expect that isogeometric methods may further increase the effectiveness of the reduced models we present here, especially when completely spatial models will be considered .
  • Industrial and Applied Mathematics Seminar
25 February 2016

Honey poured from a sufficient height onto toast undergoes the well-known `liquid rope coiling’ instability.

We have studied this instability using a combination of laboratory experiments, theory, and numerics, with the aim of determining phase diagrams and scaling laws for the different coiling modes. Finite-amplitude coiling has four distinct modes - viscous, gravitational, inertio-gravitational, and inertial - depending on how the viscous forces that resist deformation of the rope are balanced. The inertio-gravitational mode is particularly interesting as it involves resonance between the coiling portion of the rope and its long trailing `tail’. Further experiments using less viscous fluids reveal that the rope can exhibit five different morphologies, of which steady coiling is only one. We determine the detailed phase diagram of these morphologies, which includes a novel `liquid supercoiling’

state in which the coiled cylinder formed by the primary coiling instability undergoes in turn its own complex buckling instability.  We show that the onset of these different patterns is determined by a non-penetrability condition which takes different forms in the viscous, gravitational and inertial limits. To close, we will briefly evoke two additional related phenomena: spiral waves of bubbles generated by coiling, and the `fluid mechanical sewing machine’ in which the fluid falls onto a moving belt.

  • Industrial and Applied Mathematics Seminar
18 February 2016

While there have been recent advances for analyzing the complex deterministic
behavior of systems with discontinuous dynamics, there are many open questions about
understanding and predicting noise-driven and noise-sensitive phenomena in the
non-smooth context.  Stochastic effects can often change the picture dramatically,
particularly if multiple time scales are present.  We demonstrate novel approaches
for exploring and explaining surprising phenomena driven by the interplay of
nonlinearities, delays, randomness, in specific applications with piecewise smooth
dynamics - nonlinear models of balance,  relay control, and impacting dynamics.
Effective techniques typically depend on the combination of mathematical techniques,
multiple scales techniques, and phenomenological intuition from seemingly unrelated
canonical models of biophysics, mechanics, and chemical dynamics.  The appropriate
strategy is not always immediately obvious from the area of application or model
type. This gap may follow from the limited attention that stochastic models with
discontinuous dynamics have received in the past, or it may be the reason for this
limited attention.  Combining the geometrical perspective with asymptotic approaches
in physical and phase space appears to be a critical part of developing effective

  • Industrial and Applied Mathematics Seminar
11 February 2016
Anand Oza
Roughly a decade ago, Yves Couder and coworkers demonstrated that droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic quantum realm, including single-particle diffraction, tunneling, quantized orbits, and wave-like statistics in a corral. We here develop an integro-differential trajectory equation for these walking droplets with a view to gaining insight into their subtle dynamics. We then rationalize the emergence of orbital quantization in a rotating frame by assessing the stability of the orbital solutions. In the limit of large vibrational forcing, the chaotic walker dynamics gives rise to a coherent statistical behavior with wave-like features.
I will then describe recent efforts to model the dynamics of interacting flapping swimmers. Our study is motivated by recent experiments using a one-dimensional array of wings in a water tank, in which the system adopts “schooling modes” characterized by specific spatial phase relationships between swimmers. We develop a discrete dynamical system that models the swimmers as airfoils shedding point vortices, and study the existence and stability of steady solutions. We expect that our model may be used to understand how schooling behavior is influenced by hydrodynamics in more general contexts.
  • Industrial and Applied Mathematics Seminar