# 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

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

 Calculus of variations Partial differential equations Dynamical systems Inverse problems Relaxation of ill posed problems Complex analysis Math. modeling Numerical PDE Optimization theory Algebraic methods Optimization and Homogenization Stability and bifurcation

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

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.

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.

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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.

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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

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.

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.

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.

Graeme Milton: homepage

Graeme Milton's areas are composite materials, structures with unusual properties, bounds of effective properties.
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.

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