Abstracts: IMA Workshop: Future Challenges in Multiscale Modeling and Simulation, November 18-20, 2004
University of Minnesota
University of Minnesota

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


IMA Workshop:

Future Challenges in Multiscale Modeling and Simulation

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
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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
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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)
Slides:   pdf
Journal:   pdf

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)
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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)
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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
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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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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.

Andrew Stuart (Mathematics Institute, Warwick University) stuart@maths.warwick.ac.uk

Conditional Path Sampling of SDEs and the Langevin MCMC Method
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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.

Yi Sun (Department of Applied and Computational Mathematics, Princeton University) yisun@Math.Princeton.EDU

Heterogeneous Multi-scale Methods for Interface Tracking of Combustion Fronts (poster)

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.

Li-Lian Wang (Department of Mathematics, Purdue University) lwang@math.purdue.edu http://www.math.purdue.edu/~lwang

Efficient Spectral Methods Using Generalized Jacobi Polynomials (poster)
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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.

Olaf Weckner (Department of Mechanical Engineering, MIT) olafbaji@MIT.EDU http://www.mit.edu/people/olafbaji/home.html

The Effect of Long-Range Forces on the Dynamics of a Bar (poster)
Poster:   pdf

Co-Author: Rohan Abeyaratne, Department of Mechanical Engineering, MIT

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

Material from Talks

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