ABSTRACTS
 

K.D. Cherednichenko (Mathematical Institute, Oxford)

Higher-order homogenised equations for nonlinear periodic media

  Motivated by the task of rigorous derivation of size effects in the
overall behaviour of heterogeneous media, we present higher-order
(elliptic) homogenised equations for uniformly elliptic periodic
rapidly oscillating problems. The well-posedness of the higher-order
homogenised equations is subject to certain non-degeneracy conditions on
the original heterogeneous material, which we believe hold for the
majority of periodic media.

This is joint work with V.P.Smyshlyaev.
 

S L Dudarev (EURATOM/UKAEA Fusion Association, Culham Science Centre)
From molecular dynamics to the evolution of microstructure of
fusion materials

Understanding the evolution of microstructure of metals under
high-energy neutron irradiation is important for developing
materials suitable for applications in fusion. Modelling
fusion materials requires knowing the structure and dynamical
properties of defects at the atomistic and the electronic time
and length scales and how to separate relevant and irrelevant
degrees of freedom of defects in constructing models based on
either kinetic Monte-Carlo algorithms or systems of kinetic
equations. The talk will focus on the problem of Brownian motion
of clusters of interstitial defects and on effects associated
with the unusual statistics of motion of these defects in a
crystalline environment. The somewhat unexpected conclusions
following from the analysis of Brownian motion are likely to have
significant implications for understanding the origin of anomalies
observed near grain boundaries in irradiated materials.
 
 

Alain Forclaz (Mathematical Institute, Oxford)
Metastability in martensite

This talk focusses on the mathematical analysis of biaxial
loading experiments in martensite, and more particularly on how hysteresis
relates to metastability. These experiments were carried out
by Chu and James and their mathematical treatment was initiated by
Ball, Chu and James. Experimentally it is observed that a homogeneous
deformation $y_1(x)= U_1x$ is the stable state for `small' loads while
$y_2(x)=U_2x$ is stable for `large' loads. A model was proposed by
Ball, Chu and James which, for a certain intermediate range of loads,
predicts crucially that $y_1(x)=U_1x$ remains metastable
(i.e., a local -- as opposed to global -- minimiser of the
energy). This result explains convincingly the hysteresis that is observed
experimentally. It is easy to get an upper bound for when
metastability finishes. However, it was also noticed that this bound
(the Schmid Law) may not be sharp, though this required some geometric
conditions on the sample. In this talk, we mention how one can justify
the Ball-Chu-James model by means of De Giorgi's $\Gamma$-convergence,
overview some properties of local minimisers of the (limiting)
energy and state the metastability result mentioned
above. The talk also addresses the question of which geometric conditions
are necessary and sufficient for the counter-example to the Schmid Law to
apply.
 
 
 
 
 
 
 

Ioannis Kevrekedis (Chemical Engineering, Princeton)
Equation-free multiscale computation

Textbook models of reaction and transport processes typically come in the form of
conservation  laws (mass, species, momentum, energy) closed through constitutive equations
(e.g. the representation of viscous stresses for Newtonian fluids, or mass-action chemical
kinetics  expressions). In contemporary engineering modeling we have entered an era --- ushered
through materials modeling as well as systems biology modeling --- where the time-honored
macroscopic  conservation equations are often not available any more.
Instead, microscopic evolution rules,  such as Molecular Dynamics, Monte Carlo or Kinetic
Schemes are available, at various  levels of coarse-graining.

In this talk we will explore computational approaches combining microscopic simulators
with computational superstructures inspired from continuum  numerical analysis, large scale
iterative  linear algebra and applied bifurcation theory.  These approaches promise to bypass the
derivation of explicit macroscopic equations, while still being able to deliver systems level
information operating directly on the microscopic evolution rules. An anthology of examples
will be presented, including kinetic models of multiphase flows and reaction-diffusion systems,
Monte-Carlo studies of surface reactions, as well as effective medium calculations for reaction
and transport in complex media.
 
 

 Thomas J. Pence (Michigan State University, visiting Glasgow University)
Multi-field models for displcive phase transformation in polycrystals

 This work is co-authored with Davide Bernardini of the University of Rome-La Sapienza

We present a continuum mechanical framework for modeling
displacive phase transformations in solids at the scale of polycrystals.
Scalar field variables for the mass fraction of various crystallographic
phases are central to the description as are tensor field variables
representing homogenized Bain strains associated with crystallographic
transformation. At polycrystalline scales, this tensor is allowed to vary
continuously with position and time, as opposed to the case in a single
crystal where such a tensor would be restricted to values associated with
particular crystallographic variants. Standard balance laws for stress,
energy and entropy are then augmented with additional balance principles
for these additional field variables. The constitutive theory involves
specification of a free energy function and an entropy production
functional. We present a treatment for: the basic structure of the
theoretical description, handling of the constraints associated with the
additional fields, objectivity requirements, formulation of free energy
functions that deliver physically motivated equilibrium configurations, and
formulation of entropy production functionals that deliver physically
motivated hysteretic response.
 

David Pettifor (Department of Materials, Oxford)
Bond-order potentials : bridging the electronic to atomistic modelling
hierarchies

Materials modelling is often contingent upon having reliable knowledge
about the key mechanisms operating at the atomistic level.  Unfortunately ,
however , in many materials systems of technological importance the results
of atomistic simulations are questionable due to the unsatisfactory nature
of the 'classical' interatomic potentials used.  This talk outlines the
derivation of a novel class of interatomic potentials , the so-called
bond-order potentials (BOPs) , which are obtained by coarse-graining the
quantum-mechanical electronic structure within the chemically-intuitive
tight-binding framework.  The challenges to develop these
'quantum-mechanical' interatomic potentials will be discussed with relation
to simulating the growth of thin films and to understanding the defect
behaviour in high-temperature intermetallics and fusion-reactor materials.
 
 
 

Mark Sansom (Molecular Biophysics, Oxford)
Molecular simulations of ion channels: looking beyond the nanosecond timescale

Ion channels play a key role in the electrical properties of the
membranes of excitable cells (e.g. neurones) [1]. Molecular
simulations enable us to build a bridge between static molecular
structures (determined by X-ray crystallography) and single
channel physiological measurements (obtained via patch clamp
recording). In this way we can explore physiological function at
atomic resolution. In particular, one can use multi-nanosecond
simulations to probe the dynamics of water molecules and of ions
within transbilayer pores [2]. However, the maximum practical
duration of such simulations (ca. 10-8 sec) is several orders of
magnitude shorter than the timescale of physiologically important
channel events (10-6 to 10-3 sec). Some current approaches to
such multiscale problems will be outlined [3].

1. Hille, B:  Ionic Channels of Excitable Membranes.
Sunderland, Mass.: Sinauer Associates Inc.; 2001.
2. Sansom, MSP, Shrivastava, IH, Ranatunga, KM, Smith, GR:
Simulations of ion channels - watching ions and water
move. Trends Biochem. Sci. 2000, 25: 368-374.
3. Tieleman, DP, Biggin, PC, Smith, GR, Sansom, MSP:
Simulation approaches to ion channel structure-function
relationships. Quart. Rev. Biophys. 2001, 34: 473-561.
 
 

Anja Schloemerkemper (Mathematical Institute, Oxford)
Continuum limit of a magnetic force
 

Starting from a discrete setting of magnetic dipoles we consider the force
between the dipoles in two parts of a bounded subset of $\mathbb{R}^3$.
For the passage to the continuum an appropriate force formula is derived
on a scaled lattice and the continuum limit is performed by letting the
lattice parameter tend to zero. This involves a regularization of the
occurring hypersingular kernel. It turns out that the limiting force
formula contains an additional term with respect to the usual formula
which is obtained in a macroscopic setting.
 
 

Constantinos I. Siettos (Chemical Engineering, Princeton)
Some control considerations in microscopic simulations
 

It has been established that `coarse timesteppers' provide a bridge
between traditional numerical analysis and microscopic simulation. The
analogy carries over to control and optimization computations: coarse
time steppers can also become a bridge between traditional control
design and microscopic modeling. Coarse fixed points and coarse
identified models (linear or nonlinear) can be used to design `coarse
controllers' through well-established control design techniques (pole
placement, optimal and/or nonlinear control methodologies). We are now
working on using such coarse controllers, along with observers also
designed based on coarse timestepping, to do coarse control of the
microscopic simulations.  We also address the development of an adaptive
scheme, which, when implemented as a shell around existing microscopic
simulators (or, for that matter, an experiment) enables the simulator
(or the experiment) to automatically trace a bifurcation diagram and to
converge on coarse bifurcation points.  This work is in collaboration
with Yannis Kevrekidis as well as Alexei Makeev, Ramiro Rico-Martinez

Andrew Stuart (Warwick)
Algorithms for Extracting Macroscopic Stochastic Dynamics

The purpose of this work is to shed light on the extraction of
  effective macroscopic models, from detailed microscopic simulations,
  based on a recently developed transfer operator approach due to
  Sch{\"u}tte et al. (\textit{J.~Comput.\ Phys.\ 151 (1999)
    146--168}).  The investigations involve the formulation, and
  subsequent numerical study, of a class of model problems. The model
  problems are ordinary differential equations constructed to have the
  property that, when projected onto a low dimensional subspace, the
  dynamics is approximately that of a stochastic differential equation
  exhibiting a finite state space Markov chain structure.  The
  numerical studies show that the transfer operator approach can
  accurately extract finite state Markov chain behavior embedded
  within high dimensional ordinary differential equations.  In so
  doing the studies lend considerable weight to existing applications
  of the algorithm to the complex systems arising in applications such
  as molecular dynamics. Further numerical studies probe the role of
  memory effects on an algorithm predicated on the assumption of
  Markovian input data. Although preliminary, these studies indicate
  interesting avenues for further development of the transfer operator
  methodology.
 

Joint work with Wilhelm Huisinga and Christof Schuette (FU, Berlin).
 
 
 

Endre Suli (Computing Laboratory, Oxford)
Analysis of a class of two-scale finite element methods

The classical Galerkin finite element method exhibits spurious numerical
oscillations when applied to a non-self-adjoint elliptic boundary value
problem with dominant hyperbolic behaviour and the subscales in the
problem (such as boundary layers) are not properly resolved.

The aim of this talk is to discuss the analysis of a class of two-scale
finite element methods based on the use of residual-free bubbles to
capture the subscales. In the case of general finite element spaces
consisting of continuous piecewise polynomials, the method is shown to
exhibit an optimal rate of convergence. The analysis relies on a delicate
function-space interpolation result due to Luc Tartar which states that
$(L_2(\Omega),H^1_0(\Omega)_{1/2,\infty}\supset
(L_2(\Omega),H^1(\Omega))_{1/2,1}$, with continuous embedding.

The theoretical part of the talk in based on joint work with Franco Brezzi
and Donatella Marini (University of Pavia); the algorithmic and
computational aspects rely on recent work by my D.Phil. student Andrea
Cangiani.