Mathematics of Materials and Fluids
Math. faculty - members of the group
(alphabetically):
Alexander Balk,
Andrej Cherkaev,
Elena Cherkaev,
David Eyre,
Paul Fife,
Tim Folias,
Kenneth Golden,
Graeme Milton,
Jungui Zhu.
We intersect with the Math Biology group (see the web site)
Seminar: Relevant talks are in
the applied math seminar.
We expect that our graduate students will take some of the following
offered graduate courses:
ODEs, Dynamical Systems and Chaos,
PDEs, Introduction to Applied Math.
Methods
of optimization,
Analysis of Numerical Methods, Numerical
PDE,
Linear Operators and Spectral Methods,
Continuum Mechanics, Asymptotic Methods,
Calculus
of Variations, Mathematical Modeling,
Structural
Optimization, Homogenization,
and the related courses in the departments in
the College of Engineering and College of Science.
The following tables outline
the mathematical tools we use and
the spectrum of physical problems encountered.
Methods
Areas of application
Dynamics of interfaces |
Mechanics of composites |
Effective properties |
Fracture |
Microstructures |
Structural optimization |
Structural dynamics |
Percolation |
Phase transition |
Waves |
Tomography |
Computational Geophysics |
Electromagnetics |
Properties of sea ice |
Combustion |
Polycrystals |
To the top of the page
Faculty and their interests
Alexander Balk:
My main research interest at the present is in the Rossby waves. These are
the waves that represent huge vortices in the ocean and atmosphere, due to
the the earth's rotation. They transport various important quantities (like
temperature or phytoplankton) and, to a large extent, determine climate on
our planet.
People encounter the same kind of waves (from the mathematical
viewpoint) in the
nuclear fusion, when they try to confine hot plasma by a strong magnetic
field.
I have found that the nonlinear dynamics of Rossby waves has an unexpected
extra invariant (in addition to the energy and momentum). It was proved
that
this invariant is unique. Recently, I have found that this invariant
implies
that the energy organizes itself, transfering from small scale eddies to
large scale zonal jets (along the earth's parallels). This is a widely
observed phenomenon on Earth and other planets (in particular, on
Jupiter and
Saturn).
My other long-standing research interest is in Turbulence.
In his famous Lectures on Physics, Richard Feynman described one
old problem, common for many sciences, the central problem that
people need to solve. This is not the problem of elementary
particles or unified field; this is the problem of turbulence.
Particularly, I am concerned with the Wave Turbulence. Examples are the
rough
sea (due to the water waves), the climate (due to the Rossby waves) or
optical turbulence (due to the electromagnetic waves). Recently I have
developed the statistical near identity transformation (SNIT) to derive the
averaged equations of wave turbulence. Confirming SNIT, numerical
evidence indicates that more problems can be treated perturbationally than
previously thought. Besides, SNIT is capable to describe anomalous
behavior,
when the time evolution is not autonomous.
To the top of the page
Andrej Cherkaev:
Homepage
I work in the theory of extremal problems. Specifically, I
study the mechanics and physics of structured materials,
structural
optimization and optimal
composites. Mathematically, most of these problems are addressed
by means of the calculus of variation and homogenization. An informal description
of the areas of my research,
the
vitae and the
list of publications
are displayed on my homepage.
Recently, I have started to work on two intriguing projects. They are:
Understanding the essence of optimality of "living structures" by observing
morthology of various bio-structures,
and (with Alexander Balk) the analysis of an unusual dynamic
behavior of nonlinear structures due to their inner instabilities.
I teach several related graduate courses: Methods
of optimization, Introduction to applied math, Homogenization,
Calculus
of variations, and Methods
for structural optimization. This year, Springer publishes my book
Variational Methods
for Structural Optimization, that covers many topics of these courses.
More information about my teaching and interesting math links can be found
here.
To the top of the page
Elena Cherkaev:
Homepage
My field is inverse problems, especially the recovery of properties
of a medium from measured responses. These problems arise in geophysics
and material sciences, in electromagnetics and engineering, in biomedical
and environmental applications. Mathematically, the problem can often be
formulated as a problem of the identification of the coefficients of the
corresponding differential equation. Its solution leads to a large computational
problem. Inverse problems are usually ill-posed and numerical solutions
are unstable. Various methods of regularization of ill-posed problems are
developed to deal with the instabilities. A group of the methods constrains
the class of possible solutions: One can a priori chose a class of smooth
or blocky functions, and obtain very different solutions! Therefore the
numerical schemes are very sensitive to the type of solution one intends
to construct.
Also, one has to decide what measurements are needed to recover the
properties of the medium. For example, in electrical tomography the voltages
are registered on the surface of the body, these voltages are generated
by the applied currents. The question is: What currents are to be injected
if one wishes to recover the conductivity of the body with a given accuracy?
This question leads to the source optimization problem. A different type
of inverse problem arises when the considered medium is very inhomogeneous,
say it is a fine mixture of two different materials. Then one is interested
in recovering some averaged description of this medium rather than the
exact configuration of the materials.
To the top of the page
David Eyre:
Homepage
I am an applied mathematics and make extensive use of simulations.
The problems I work on are in materials science and fluid dynamics
To the top of the page
Paul Fife:
Homepage
Paul Fife is currently engaged in the development and analysis of a
variety of continuum models for phase transitions. These lead to interesting
problems in nonlinear partial differential equations, which he studies
by rigorous or asymptotic methods. The emphasis is on transitions which
occur at material interfaces, such as grain boundaries in metallic alloys,
solidification fronts for pure materials or alloys, and interfaces in polymers.
He is also working with various abstract continuum models for phase changes.
In all of this, one object is to set a framework for the study of physical
properties of such interfaces, an example being the role that excess free
energy and entropy have on the mechanics of motion of the interface. This
is done by discovering mathematical properties which models exhibit and
which may correlate to physical information.
To the top of the page
Tim Folias:
homepage
Tim Folias' field is the theory of solid and fracture mechanics with
applications to metal and composite material structures, ceramics, and
organics.
To the top of the page
Kenneth Golden
In many composite materials, the effective behavior depends critically
on the connectedness, or percolation properties of a particular phase.
For example, sandstones are permeable to water only when there is a high
enough volume fraction of pores that they connect and form pathways. Likewise,
the electromagnetic properties of a polymer film with conducting particles
vary dramatically near the critical volume fraction where the particles
begin to form a connected matrix. This critical volume fraction is called
the percolation threshold, and near it the effective transport coefficients,
such as fluid permeability or electrical permittivity and conductivity,
typically display scaling behavior characterized by critical exponents,
similar to a phase transition in statistical mechanics. Much of my recent
mathematical work has focused on analysis of transport in lattice and continuum
percolation models, the connections to statistical mechanics, and the application
of these models to porous media, particulate composites used in smart devices
and conducting films, and electrorheological fluids.
A particularly interesting example is the case of sea ice, a composite
of pure ice with liquid brine and air inclusions. It is permeable to fluid
only when the brine volume fraction exceeds about 5%, which corresponds
to a critical temperature of around -5 degrees C, allowing transport of
brine, nutrients, biomass, and heat through the ice. In the Antarctic,
these processes play an important role in air-sea-ice interactions, in
the life cycles of sea ice algae, and in remote sensing of the pack. We
have been developing percolation models to understand fluid transport in
sea ice, and have also made measurements of percolation structures in Antarctic
sea ice. Motivated by sea ice remote sensing, we have also been developing
(with E. Cherkaev) inverse algorithms for recovering the physical properties
of sea ice or other composites via electromagnetic means. This leads to
other interesting mathematical problems in electromagnetic scattering,
which we have been studying.
In connection with the above areas of research, I have taught graduate
courses in the following areas: Mathematics of Materials, Percolation,
Statistical Mechanics, Applied Linear Operators and Spectral Methods, Applied
Complex Variables and Asymptotic Methods, and Several Complex Variables.
To the top of the page
Graeme Milton: homepage
Graeme Milton's areas are composite materials, structures with unusual
properties, bounds of effective properties.
To the top of the page
Jungui Zhu: homepage
Turbulent combustion is an old subject and remains challenging to most
modern mathematical approaches, including the available computational powers.
One particularly important concept is the turbulent burning velocity, that
is, the effective burning rate enhanced by the turbulence in the flow.
With the lack of good understanding of turbulence itself and the complicated
chemical reaction manifested in many different temporal and spatial scales,
a complete understanding of the turbulent burning velocity seems to be
remote at this stage. However, it is possible to gain important understanding
from some simplified systems by both asymptotic methods and large-scale
computing. The particular simplifications we focus on are use of passive
flows and restricting chemical reaction to some particular regimes. Both
asymptotic methods and numerical methods are integrated at different levels
to achieve the research goals.
To the top of the page