Mathematics of Materials and Macromolecules: Multiple
Scales, Disorder, and Singularities, September 2004 - June 2005

## Abstracts:

## IMA Workshop:

**November 18-20, 2004 **

Material from Talks

**Grégoire
Allaire** (Centre de Mathématiques Appliquées, Ecole
Polytechnique) allaire@cmapx.polytechnique.fr
http://www.cmap.polytechnique.fr/~allaire/

**Examples of Multiscale Methods in Shape Optimization
**

Slides: pdf

We discuss two examples of multiscale methods in the context
of structural optimization. The first method, which is by now classical, is
the homogenization method based on the use of composite materials. Instead
of optimizing the position of macroscopic boundaries, the homogenization method
optimizes the layout of microscopic holes in a porous material. The two design
parameters are the local volume fraction of material and the local microstructure
or shape of the holes. The latter one is optimized at a mesh subscale level.
The second method is the more recent level set method which relies on the
classical Hadamard method of shape sensitivity. Although the level set method
is able to handle topology changes, it can not easily nucleate new holes.
Therefore it has been coupled with the topological asymptotic method which
decides when and where it is favorable to cut an infinitesimal new hole. In
these two examples a macroscopic shape optimization process is coupled with
a microscopic evaluation, either of the optimal hole microgeometry, or of
the potential gain in hole nucleation. In both cases their multiscale characters
improve the ability of the algorithms to escape from local minima. Numerical
examples in 2-d and 3-d will support this claim.

**Folkmar
Bornemann** (Munich University of Technology) bornemann@ma.tum.de

**Energy Level Crossings in Molecular Dynamics - Is there
a (Mathematical) Passage? **

We discuss the mathematical description of the quantum dynamics
of a molecular system that undergoes a conical intersection of energy levels.
At such intersections, because of nonlinear scale-interactions, leading order
transitions occur that are the reason for many important reaction mechanisms
studied in quantum chemistry. We will review recent work that could help to
develop mathematical well-founded versions (without any ad-hoc devices) of
the surface-hopping algorithms for the simulation of such systems. We will
focus on several challenging open problems.

**Achi Brandt**
(Department of Mathematics, UCLA) abrandt@math.ucla.edu

**From Fast Solvers to Systematic Upscaling **

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Most numerical methods for solving large-scale systems tend
to be extremely costly, for several general reasons, each of which can in
principle be removed by multiscale algorithms. Algorithms to be briefly surveyed:
fast multigrid solvers for discretized partial-differential equations (PDEs)
and for most other systems of local equations; fast summation of long-range
(e.g., electrostatic) interactions and fast solvers of integral and inverse
PDE problems; collective computation of many eigenfunctions; slowdown-free
Monte Carlo simulations; multilevel methods of global optimization; and general
procedures for "systematic upscaling."

SYSTEMATIC UPSCALING is amethodical approach for deriving, scale
after scale,collective variablesand governing numericalequations (or transition
probabilities rules) at increasingly larger scales, starting from a microscopic
scale where first-principle laws are known. Iterating back and forth between
all levels allows the computation at each scale to be short and confined to
small "windows."

The multiscale methods are key to removing computational bottlenecks
in many areas of science and engineering, such as: QCD (elementary particle)
computation; ab-initio quantum chemistry real-time path integrals; density-functional
calculation of electronic structures; molecular dynamics of fluids, materials
and macromolecules; turbulent flows; tomography (medical-imaging reconstruction);
image segmentation and picture recognition; and various large-scale graph
optimization, clustering and classification problems. Future directions will
be outlined.

**Russel
Caflisch** (Mathematics Department, UCLA) caflisch@math.ucla.edu

**Multiscale Modeling of Epitaxial Growth Processes**

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Epitaxy is the growth of a thin film by attachment to an existing
substrate in which the crystalline properties of the film are determined by
those of the substrate. No single model is able to address the wide range
of length and time scales involved in epitaxial growth, so that a wide range
of different models and simulation methods have been developed. This talk
will review several of these models - kinetic Monte Carlo (KMC), island dynamics
and continuum equations - in the context of layered semiconductors applied
to nanoscale devices. We describe a level set method for simulation of the
island dynamics model, validation of the model by comparison to KMC results,
and the inclusion of nucleation and strain. This model uses both atomistic
and continuum scaling, since it includes island boundaries that are of atomistic
height, but describes these boundaries as smooth curves.

**Emily A.
Carter** (Mechanical and Aerospace Engineering and Applied and
Computational Mathematics/Chemistry, Princeton University) eac@princeton.edu

**Challenges for Quantum-Mechanics-Based Multiscale Modeling**

Slides: pdf

In principle, the predictive power of multiscale modeling will
be greatly enhanced if information is provided by first principles methods
that do not rely on input from experiment. However, such methods, especially
for metallic systems, are extremely expensive to use. We have recently shown
(Fago et al., Phys. Rev. B, 2004) that it is possible to couple a linear scaling
density functional theory (DFT) method to the local quasicontinuum method,
thereby providing an on-the-fly two-scale method with feedback to both scales.
The current state of development of this orbital-free density functional theory
(OFDFT) method will be described, including achievements and limitations.
We will give an honest appraisal of what the challenges are and how we hope
to overcome them, such that predictive, on-the-fly multiscale modeling will
eventually be possible.

**Zhiming Chen**
(LSEC, Institute of Computational Mathematics, Academy of Mathematics and
Systems Science, Chinese Academy of Sciences) zmchen@lsec.cc.ac.cn
http://lsec.cc.ac.cn/~zmchen

**On the Upscaling of a Class of Nonlinear Parabolic Equations**
pdf

**Ravi
Chinnappan** (Ford Motor Company) rchinnap@ford.com

**First-Principles Calculation of Stable and Metastable
Precipitate Phase Solvus Boundaries in Al-Cu Alloys** (poster)

Precipitation strengthening via heat treatment is a common practice
to enhance the mechanical properties of many classes of industrial aluminum
alloys. In Al-Cu alloys, age hardening is controlled primarily by metastable
phases. Due to their metastable nature, their corresponding solvus boundaries
are difficult to determine experimentally. Knowledge of solid solution and
metastable phase boundaries is important for understanding the strengthening
contributions of various precipitate phases. We present here the first-principles
calculated metastable 'Z3' (Al3Cu-GP zones), theta'(Al2Cu) and equilibrium
theta (Al2Cu) phase solvus boundaries. The vibration contribution to alloy
free energy is obtained using the calculation of a full dynamical matrix based
on the frozen phonon approach. Comparison of the calculated results with measured
phase boundaries and their implications discussed.

**Ken A. Dill
**(Department of Pharmaceutical Chemistry, University of California
- San Francisco) dill@maxwell.compbio.ucsf.edu

**Protein Folding as a Global Optimization Problem**

A protein is a chain molecule having a large number of degrees
of freedom. In its biological state, it is folded into a single conformation,
out of a large conformational space. Finding its native state is a global
optimization problem that the protein can often solve in nanoseconds. We have
studied how the protein finds its global optimum so quickly, and are exploiting
the same strategies for use in computational protein structure prediction.

**Weinan
E** (Department of Mathematics & Applied Computational Mathematics,
Princeton University) weinan@Math.Princeton.EDU
http://www.math.princeton.edu/~weinan/

**Simple Concepts in Multiscale Modeling**

What is multiscale modeling about and why is there such a huge
interest right now? These questions are less trivial than one might initially
thought, given that almost every problem in nature is multiscaled, and there
has already been a long history of using multiscale ideas in scientific computing.
We will discuss these questions in the context of several canonical multiscale
problems and multiscale methods. This allows us to give a candid accessement
of the current status of multiscale modeling in several areas.

**Yalchin
Efendiev** (Department of Mathematics, Texas A&M University)
efendiev@math.tamu.edu

**Upscaling of Geocellular Models for Flow and Transport
Simulation in Heterogeneous Reservoirs**

It is difficult to fully resolve all of the scales that impact
flow and transport in oil reservoirs, so models for subgrid effects are often
required. In this talk, I will describe methods for the coarse scale modeling
of flow and transport (movement of injected fluid) in highly heterogeneous
systems. The representation of coarse scale permeability is accomplished with
a local technique. For the transport, I will describe non-local upscaling
methods as well as a generalized convection-diffusion model for subgrid effects.
The accuracy of these procedures will be illustrated for a variety of problems.
I will describe the use of coarse-scale models in inverse problems (history
matching). This is a joint work with Louis Durlofsky.

**Bjorn Engquist**
(Department of Mathematics, University of Texas at Austin) engquist@math.utexas.edu

**Heterogeneous Multiscale Methods**

The heterogeneous multiscale method (HMM) is a framework for
design and analysis of computational methods for problems with multiple scales.
If a macro-scale model is not fully known a more detailed micro-scale model
is used during the calculation to supply the missing data. We will present
how the HMM framework helps in understanding convergence properties and we
will discuss efficient ways of coupling the different scales during the simulation.
Examples of applications are stiff dynamical systems and molecular dynamics
coupled to continuum models.

**Matthias
K. Gobbert** (Department of Mathematics and Statistics, University
of Maryland, Baltimore County) gobbert@math.umbc.edu
http://www.math.umbc.edu/~gobbert

**Multiscale Models for Production Processes in Microelectronics
Manufacturing** (poster)

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Production steps in the manufacturing of microelectronic devices
involve gas flow at a wide range of pressures. We develop a kinetic transport
and reaction model (KTRM) based on a system of time-dependent linear Boltzmann
equations. This model is suitable for simulations across a range of pressures
and length scales. We will demonstrate this by results obtained for a large
range of the relevant dimensionless group in the model, the Knudsen number,
defined as the ratio of mean free path over length scale of interest.

**Thomas Yizhao
Hou** (Applied and Comp. Math, Caltech) hou@acm.caltech.edu
http://www.acm.caltech.edu/~hou

**Multiscale Modeling and Computation of Incompressible
Flow**

We perform a systematic multiscale analysis for incompressible
Euler equations with rapidly oscillating initial data. The initial condition
for velocity field is assumed to have a two-scale structure. One of the important
questions is how the two-scale velocity structure propagates in time and whether
nonlinear interaction will generate more scales dynamically. By making an
appropriate multiscale expansion for the velocity field, we show that the
two-scale structure is preserved dynamically. Further, we derive a well-posed
homogenized equation for the 2-D and 3-D incompressible Euler equations. Our
multiscale analysis also reveals an interesting structure of the Reynolds
stress, which provides useful guideline in designing systematic coarse grid
model for the incompressible flow.

**Richard D.
James** (Department of Aerospace Engineering and Mechanics,
University of Minnesota) james@aem.umn.edu
http://www.aem.umn.edu/people/faculty/bio/james.shtml

**Deformable Thin Films: from Macroscale to Microscale
and from Nanoscale to Microscale**

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1) A summary and brief discussion of the rigorous passage from
continuum to (small scale) continuum levels using \Gamma-convergence, and
its implications for phase transitions in films vs. bulk material. (Joint
work with Kaushik Bhattacharya)

2) An example at atomic level of how to choose variables for
a limiting continuum theory, based on the large body limit, where the choice
of these variables is not at all obvious. (Joint work with Gero
Freisecke)

3) A brief discussion of a new model for protein lattices and
its application to a fascinating contractile mechanism in the virus Bacteriophage
T-4. (Joint work with Wayne Falk)

**Yannis
G. Kevrekidis **(Department of Chemical Engineering, Princeton
University) yannis@Princeton.EDU

**Some Computational Examples of Equation-Free Modeling**

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I will discuss a number of modeling examples whose computation
is facilitated in an equation-free multiscale framework. The examples range
from MC computations of micelle formation to agent based simulation and studies
of coupled oscillators. I will also present some examples of equation-free
dynamic renormalization computations.

**Luigi
La Ragione** (Department of Theoretical and Applied Mechanics,
Cornell University) ll87@cornell.edu

**The Initial Incremental Response of a Random Aggregate
of Identical Spheres **(poster)

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We study the mechanical response of a random arrays of identical
spherical grains which interact through a non- central force. We focus on
the first increments in shear and compressive strain after the aggregate has
been isotropically compressed. The simplest approach to the problem is to
assume that the contact displacement is derived by an average strain. However,
this theory seems to predict a stiffer behavior of the material compared to
the results of numerical simulation. We give up the average strain assumption
and consider fluctuations in the kinematics of the particles and in the structure.
We provide an example of the features of this new theory in the context of
a frictionless aggregate.

**Frédéric
Legoll** (IMA Postdoc) legoll@ima.umn.edu
legoll@cermics.enpc.fr http://www.ima.umn.edu/~legoll/
http://cermics.enpc.fr/~legoll/home.html

**Analysis of a Prototypical Multiscale Method Coupling
Atomistic and Continuum Mechanics** (poster)

Slides: pdf

The description and computation of fine scale localized phenomena
arising in a material (during nanoindentation, for instance) is a challenging
problem that has given birth to many multiscale methods. In this work, we
propose a numerical analysis of a simple one-dimensional method that couples
two scales, the atomistic one and the continuum mechanics one. The method
includes an adaptative criterion in order to split the computational domain
into two subdomains, that are described at different scales. We will address
the questions of how to define the energy of the hybrid system and how to
split the computational domain.

**Xiantao Li**
(IMA Postdoc) xli@ima.umn.edu

**Heterogeneous Multiscale Method for the Modeling of
Dynamics of Solids at Finite Temperature** (poster)

We present a multiscale method for coupling atomistic and continuum
models of solids. Both models are formulated in the form of physical conservation
laws, and the coupling is achieved through balancing the fluxes. We shall
show some applications including phase transformation, and dynamic fracture
mechanics.

**Robert P.
Lipton** (Department of Mathematics, Louisiana State University)
lipton@math.lsu.edu

**Multi-Scale Stress Analysis for Composite Media**
(poster)

Mathematical objects are introduced that characterize the distribution
of stress inside heterogeneous media with fine scale structure. These quantities
are associated with the higher moments of corrector fields and are referred
to as macrostress modulation functions. They are used to determine the location
and extent of high stress zones near imperfections and reentrant corners in
composite materials. The stress assessment methodology is developed within
the mathematical context of G-convergence. Several examples illustrating the
method are provided in the contexts of periodic and random homogenization.

**Dionisios
Margetis** (Department of Mathematics, Massachusetts Institute
of Technology) dio@math.mit.edu

**Continuum Theory of Interacting Steps on Crystal Surfaces
in (2+1) Dimensions** (poster)

A continuum theory of crystal surface morphological evolution
in (2+1) dimensions below the roughening temperature is formulated on the
basis of motion of interacting atomic steps. The kinetic processes are isotropic
diffusion of adatoms across each terrace and attachment-detachment of atoms
at each bounding step, for a wide class of step-step interactions. The continuum
limit of the difference-differential equations for the step positions yields
an effective tensor diffusivity for the adatom current, along with a PDE for
the surface height profile. The tensor describes diffusion-induced adatom
flows parallel to steps as distinct from flows transverse to steps due to
asymmetry in the step geometry. The implications of this theory for recent
experiments of decaying surface corrugations are discussed.

**Daniel Onofrei**
(Department of Mathematics, Worcester Polytechnic Institute) onofrei@wpi.edu

**The Periodic Unfolding Method and Applications to the
Homogenization in Perforated Materials and Neumann Sieve** (poster)

In this poster we will present the applications of the > periodic
unfolding method developed by Cioranescu, Damlamian and Griso, to the homogenization
in perforated materials and Neumann sieve model.

The classical problem associated to the Laplace operator with
variable coefficients in a perforated domain will be studied, in the case
of n-dimensional perforations distributed in the volume, or on a hyperplane.
Also the Neumann sieve model with variable coefficients will be studied.

Many of the result are already known, but the method we are
using is new and give us the possibility to prove these results in a very
elegant manner.

Also we believe that this new approach give one the possibility
to improve the existent corrector results.

**Felix Otto**
(Institut fuer Angewandte Mathematik, Universitaet Bonn) otto@riemann.iam.uni-bonn.de
http://www-mathphys.iam.uni-bonn.de/~otto/

**The Onset of Switching in Thin Film Ferromagnetic Elements:
A Bifurcation Analysis by Gamma-Convergence**

Motivation for this joint work with Ruben Cantero-Alvarez is
the following experimental observation for thin film ferromagnetic elements.
Elements with elongated rectangular cross--section are saturated along the
longer axis by a strong external field. Then the external field is slowly
reduced. At a certain field strength, the uniform magnetization buckles into
a quasiperiodic domain pattern which resembles a concertina.

Starting point for the analysis is the micromagnetic model which
has three length scales. We identify the relevant parameter regime, which
has been overseen by the physics literature. In this parameter regime, we
identify a "normal form" for the bifurcation, which turns out to be supercritical.
The analysis amounts to the combination of an asymptotic limit with a bifurcation
argument. This is carried out by a suitable "blow-up'' of the energy landscape
in form of Gamma-convergence.

**Linda R.
Petzold** (Department of Mechanical and Environmental Engineering,
University of California, Santa Barbara) petzold@engineering.ucsb.edu

**Multiscale Stochastic Simulation Algorithm with Stochastic
Partial Equilibrium Assumption for Chemically Reacting Systems**

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In microscopic systems formed by living cells, small numbers
of reactant molecules can result in dynamical behavior that is discrete and
stochastic rather than continuous and deterministic. In simulating and analyzing
such behavior it is essential to employ methods that directly take into account
the underlying discrete stochastic nature of the molecular events. This leads
to an accurate description of the system that in many important cases is impossible
to obtain through deterministic continuous modeling (e.g. ODE's). Gillespie's
Stochastic Simulation Algorithm (SSA) has been widely used to treat these
problems. However as a procedure that simulates every reaction event, it is
prohibitively inefficient for most realistic problems. We report on our progress
in developing a multiscale computational framework for the numerical simulation
of chemically reacting systems, where each reaction will be treated at the
appropriate scale. We introduce a stochastic partial equilibrium approximation
which is valid even if the population of a fast chemical species is very small,
and present some preliminary numerical results from a multiscale numerical
simulation.

**Peter Philip**
(IMA Industrial Postdoc) philip@ima.umn.edu
http://www.ima.umn.edu/~philip/homepage/

**Simulation and Control of Sublimation Growth of SiC
Bulk Single Crystals** (poster)

Slides: pdf

A transient mathematical model for the sublimation growth of
silicon carbide single crystals (SiC) by physical vapor transport is presented.
Continuous mixture theory is used to obtain balance equations for energy,
mass, and momentum inside the gas phase. Heat transport by radiation is modeled
via the net radiation method for diffuse-gray radiation to allow for radiative
heat transfer between the surfaces of cavities. Induction heating is modeled
by an axisymmetric complex-valued magnetic scalar potential that is determined
as the solution of an elliptic problem. The resulting heat source distribution
is calculated from the magnetic potential. Ideas to model crystal growth and
source sublimation are presented. Results of numerical simulations, using
a finite volume method, are discussed.

**Olivier
Pironneau** (Laboratoire Jacques-Louis Lions, Université Paris
VI) pironneau@ann.jussieu.fr

**Nuclear Waste Safety of Repository Vaults: A Multi-Scale
Problem **

Slides: pdf

This analysis explores the possibilities of multiscale expansions
and domain decomposition to solve part of the Couplex 1 exercise posed by
the french agency for nucear waste, ANDRA. We concentrate on the hydrostatic
pressure and show that the slenderness of the domain and the large variations
of the Darcy constants allows an analytical approximation which our test reveals
to be true to relative errors smaller than 1/1000. The numerical tests are
done in 2D with freefem+ and in 3D with freefem3D, both in the public domain.
Some considerations will be also given for Iodine transport.

**James P.
Sethna** (Laboratory of Atomic and Solid State Physics (LASSP),
Cornell University) sethna@ccmr.cornell.edu
http://www.lassp.cornell.edu/sethna/sethna.html)

**Deriving Plasticity: Attempts at a Theory for Dislocation
Patterning and Work Hardening**

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Movie: avi

In collaboration with Surachate Limkumnerd,
Markus Rauscher, and Jean-Phillipe
Bouchaud.

When you abuse your fork in cutting a tough piece of meat, and
it bends irreversibly, plastic deformation has occured. Plastic deformation
in crystals arises because of the creation, motion, and tangling of myriads
of dislocation lines. These form complex patterns and cellular structures
whose evolution and properties pose perhaps the major unsolved problem in
the multiscale modeling of structural metals. We're developing a mesoscale
field theory for rate-independent plasticity, governing behavior on scales
large compared to the dislocations and explaining the emergence of these cellular
structures. Earlier, we attempted a scalar theory that seemed promising: it
exhibited a yield stress, work hardening, and cell boundary formation. Using
symmetry arguments, the Peach-Koehler force, and a closure approximation,
we're now developing dynamical laws for the dislocation--density tensor which
provides a clear, microscopic connection to the deformation field in the metal,
while exhibiting the same finite-time shock formation that led to irreversible
behavior in the scalar theory.

We describe a stochastic PDE based approach to sampling paths of SDEs, conditional
on observations. The SPDEs are derived by generalizing the Langevin MCMC method
to infinite dimensions. Various applications are described including sampling
paths subject to two end point conditions (bridges) and nonlinear filter/smoothers.

We present the heterogeneous multiscale methods (HMM) for interface tracking
and apply the technique to the simulation of combustion fronts. HMM overcomes
the numerical difficulties caused by different time scales between the transport
and reactive parts in the model. HMM relies on an efficient coupling between
the macroscale and microscale models. When the macroscale model is not fully
known explicitly or not valid in localized regions, HMM provides a procedure
for supplementing the missing data from a microscale model. A phase field or
front tracking method defines the interface on the macroscale. Numerical results
for Majda's model and multispecies reactive Euler equations show the efficiency
of HMM.

We introduce a family of generalized Jacobi polynomials (GJPs) with negative
integer indexes, which turns out to be the natural basis functions for spectral
approximations to PDEs in various situations. As examples of applications, we
analyze and implement generalized Jacobi spectral methods for general high-order
PDEs (such as KdV-type equations), and Helmholtz equation. We also apply it
to the time discretizations.

We show that spectral methods using GJPs lead to stable, well-conditioned
algorithms, and more precise error estimates.

The one-dimensional dynamic response of an infinite bar composed of a linear
"microelastic material" is examined. The principal physical characteristic of
this constitutive model is that it accounts for the effects of long-range forces.
The general theory that describes our setting, including the accompanying equation
of motion, was developed independently by (1), (2) and (3), and is called the
peridynamic theory. This theory is effectively an integral-type nonlocal model,
which, in contrast to other such models, only involves the displacement field,
not its gradient. This leads to a theory that formally appears to be a continuum
version of molecular dynamics. However this similarity is misleading since the
peridynamic theory is meant to apply at length-scales between those of classical
continuum mechanics and molecular dynamics. An attractive feature of peridynamic
theory is the computational advantage resulting from the absence of spatial
gradients, especially in settings that involve singularities (fracture mechanics,
phase transformations).

In this poster we present our results for the one-dimensional, linear bar involving
long-range forces as recently published in (4). The general initial-value problem
is solved and the motion is found to be dispersive as a consequence of the long-range
forces. The result converges, in the limit of short-range forces, to the classical
result for a linearly elastic medium. The most striking observations arise in
the Riemann-like problem corresponding to a constant initial displacement field
and a piecewise constant initial velocity field. Even though, initially, the
displacement field is continuous, it involves a jump discontinuity for all later
times, the Lagrangian location of which remains stationary. For some materials
the magnitude of the discontinuity-jump oscillates about an average value, while
for others it grows monotonically, presumably fracturing the material when it
exceeds some critical level. Solving the governing integrodifferential equation
numerically we could confirm the analytical results as predicted.

(1) Kunin, I.A., 1982. Elastic Media with Microstructure I. Springer, Berlin.

(2) Rogula, D., 1982. Nonlocal Theory of Material Media. Springer, Berlin.

(3) Silling, S.A., 2000. Reformulation of elasticity theory for discontinuities
and long-range forces. J. Mech. Phys. Solids 48, 175 - 209.

(4) Weckner, O. , Abeyaratne R, 2004. The Effect of Long-Range Forces on the
Dynamics of a Bar. J. Mech. Phys. Solids, accepted for publication