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Zvi Artstein
Mathematics, Weizmann Institute, Rehovot, Israel

Title: The slow progress of a fast dynamics

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An approach to analyze the behavior of a fast dynamics drifted along with a slow movement will be reviewed. The mathematical model is a singularly perturbed equation and the limit sought after is when the small parameter tends to zero. A Young measure depicts then the limit motion. The slow movement may be of a slow state variable, or of an external observable. Comments on the modeling, the analysis and computations will be provided.

 

Armand J. Beaudoin
Mechanical Engineering, University of Illinois

Title: Applications of Field Dislocation Mechanics

Phenomenological Mesoscale Field Dislocation Mechanics (PMFDM) offers a framework for the study of plasticity wherein evolution of the mobile dislocation density, internal stress and length scale are inherent in the specification of the boundary value problem. This provides for access to many long-standing problems in the plastic deformation of materials. In this presentation, a review will be given of several studies using PMFDM. Examples demonstrating the evolution of mobile dislocation density in ice, metal whiskers, and in the presence of dislocation-solute interaction will be given. The interplay of internal stress with constitutive response will be highlighted. Validation of PMFDM will be shown through comparison of experiment and simulation. Finally, attention will turn to current challenges in the modeling of plastic deformation of materials.

 

Peter T. Cummings
Chemical and Biomolecular Engineering - Vanderbilt University and Center for Nanophase Materials Sciences - Oak Ridge National Laboratory

Title: Many-Scale and Multi-Scale Modeling in Computational Nanoscience and the Interplay Between Experiment and Theory

Theory, modeling and simulation (TMS) tools constitute key enabling technologies for making fundamental advances in nanoscience and for making nanotechnology a practical reality. Many of the problems encountered in this field are inherently multiscale. In this talk, we provide an overview of the role of TMS in nanoscience, as well as an overview of our many-scale and multi-scale TMS research in nanotribology and molecular electronics.

 

Kaushik Dayal
Civil & Environmental Engineering, Carnegie Mellon University

Title: Multiscale Atomistics for Defects in Ferroelectrics

Defects in multifunctional ferroelectrics play a central role in enabling their properties: for example, the electromechanics of ferroelectrics occurs by the nucleation and growth of domain wall defects, and hysteresis is believed to be dominated by the interaction of domain walls with point-defect oxygen vacancies. I will talk about our efforts to develop multiscale atomistic methods to understand the structure of defects in these materials. These materials have long-range electrostatic interactions between charges, as well as electric fields that exist over all space outside the specimen. I will describe a multiscale methodology aimed at accurately and efficiently modeling defects in such materials in complex geometries. Our approach is based on a combination of Dirichlet-to-Neumann maps to consistently transform the problem from all-space to a finite domain; the quasicontinuum method to deal with short-range atomic interactions, and rigorous thermodynamic limits of dipole lattices from the literature. We apply the method to understand the electromechanics of a ferroelectric under complex electrical loading.

 

Somnath Ghosh
Departments of Civil Engineering and Mechanical Engineering Johns Hopkins University

Title: Multi-time Scaling Methods in Image Based Crystal Plasticity FEM

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Cyclic finite elelent simulations are widely used to study fatigue behavior of metallic materials undergoing plasticity and damage. However, numerical integration of non-linear plasticity and damage laws allow only small time steps with conventional schemes and render cyclic FE simulations computationally intractable. This talk will discuss aspects of crystal plasticity finite element (CPFE) modeling of polycrystalline metals and alloys for predicting cyclic deformation leading to fatigue. Image-based crystal plasticity FEM models are developed, incorporating statistically equivalent distribution functions of grain morphology and crystallographic orientations from data obtained from orientation imaging microscopy. A grain-based crack nucleation model evolves from considerations of energy needed to open a free surface in a hard grain surrounded by dislocation pileup in neighboring soft grains.

To overcome the computational bottleneck associated with time integration, a wavelet transformation based multi-time scale (WATMUS) method will be presented for performing accelerated FE simulations of materials undergoing plasticity/damage for large number of cycles. Crystal plasticity based material model will be used to demonstrate the workability of WATMUS method to perform accelerated FE simulations of polycrystalline alloys till crack initiation. Accuracy of the proposed methodology is compared with cyclic single time scale CPFE simulations performed on statistically equivalent microstructure of Ti-6242. The resulting finite element scheme is able to allow simulations to continue for a large number of cycles to fatigue crack nucleation.

 

Turab Lookman
Physics of condensed matter and complex systems, Los Alamos National Laboratory

Title: Heterogeneities and Kinetics in Phase Transforming Systems

Recent results from flyer plate experiments on Zr under shock illustrate the need for in-situ time and spatially resolved measurements. I will describe plans for the Los Alamos signature concept facility MaRIE(Matter Radiation in Extreme). I will then present recent work comparing the use of linear and nonlinear strain compatibility relations in systems with significant rotations, as well as discrete "Ising Models" for obtaining microstructure.

 

Craig Maloney
Civil & Environmental Engineering, Carnegie Mellon University

Title: Large fluctuations and collective behavior in quasi-static shear flows: From metallic glasses to soft particle suspensions

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Bulk metallic glasses are amorphous alloys composed of metallic species that suppress crystallization. They have been the subject of intense interest to the materials science community in recent years owing to their unique mechanical properties. However, their underlying mechanisms of plasticity are poorly understood, and these uncertainties limit their structural applications. Colloidal suspensions, assemblies of discrete particles suspended in a continuous fluid phase, exhibit a strikingly similar phenomenology to conventional glasses. At high particle volume fraction, they form a so-called jammed state with solid-like properties. As with the metallic glasses, the mechanisms for plastic flow are poorly understood, and, ultimately, one would like to compute the mechanical response from first principles.

I will present an extensive set of computer simulations of both model metallic glasses and jammed suspensions of elastically deformable colloidal particles. The plastic deformation will be shown, in both cases, to exhibit similar complex spatio-temporal organization in the form of avalanches that can grow to the full size of the system. This organization makes a coarse-grained description particularly challenging and affects both the mechanical response and the statistics of tracer particle motion, giving rise to a novel shear-rate and system-size dependence of the diffusion constant. For the case of the jammed suspensions, I will also discuss how this picture changes near the jamming volume fraction at which the suspension loses rigidity.

 

Alan McGaughey
Mechanical Engineering, Carnegie Mellon University

Title: Combining atomistic calculations and the Boltzmann transport equation to predict nanostructure thermal conductivity

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I will present a technique based in lattice dynamics calculations, Monte Carlo sampling, and the Boltzmann transport equation for predicting the thermal conductivity of a semiconductor nanostructure with arbitrary geometry. Thermal transport in a semiconductor is dominated by phonons, quanta of energy associated with atomic vibrations. For silicon, the relevant length scales are the phonon wavelengths (< 5 nm), bulk phonon mean free paths (< 5 micron) and the nanostructure dimensions (10 nm - 10 microns). Size effects emerge with the nanostructure size is smaller than the bulk mean free paths.

Bulk phonon frequencies and mean free paths are first calculated from lattice dynamics calculations, an atomistic approach performed at the unit cell level. These phonon properties are then combined with a boundary scattering model to predict the thermal conductivity of a nanostructure (e.g., a thin film or nanowire). The effects of phonon-phonon and phonon-boundary scattering are integrated using a Monte Carlo-based approach. Both the initial position of the phonon and the phonon-phonon free path are drawn from their natural distributions. The results are compared to previous experimental measurements.

 

Assad A. Oberai
Scientific Computation Research Center, Mechanical Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY

Title: Subgrid Models in Fluid Mechanics

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click here for: Video One ¦ Video Two

Almost all interesting solutions to the equations of uid mechanics involve features that are challenging to represent numerically. These include sharp boundary layers in wall bounded ows, shocks in compressible ows and a hierarchy of cascading vortices in turbulent ows. In all these cases the computational cost of explicitly resolving such features in a numerical simulation is prohibitive. To overcome this, several numerical methods have been developed wherein such features are assumed to lie below the grid resolution and subgrid models are developed to account for their eect on the resolved components of the ow.

In this talk I will describe the role of the variational multiscale formulation in deriving subgrid models in uid mechanics. In this approach, within the variational statement of the underlying problem, the weighting functions and the trial solutions are spilt into resolved and unresolved scales. As a result two equations, one for the resolved scales and another for the unresolved scales, are obtained. The equation for the unresolved scales is then approximately solved and the result is substituted in the equation for the resolved scales thereby achieving closure. I will focus on the application of this approach in developing large eddy simulation models for turbulence in incompressible, compressible and magnetohydrodynamic ows. 1

 

Mark Robbins
Physics and Astronomy, Johns Hopkins University

Title: Concurrent Modeling of Fluid Flow and Solid Contact

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Many problems can be treated efficiently and accurately over most spatial regions using continuum models, but require atomistic resolution at interfaces or in regions of high stress. Hybrid atomistic/continuum approaches for such problems will be described, including techniques for incorporating flux of particles, heat, and charge between atomistic and continuum regions. Applications of the method to fluid flow will include corner flow, contact line motion and electroosmosis. Contact of solids with roughness on a wide range of scales will be described and the effect of near-surface plasticity quantified.

 

Don Saari
Mathematics, University of California, Irvine

Title: Unexpected problems coming from the usual "reductionist" approach

A standard practice in studying multiscale phenomena is to adopt the standard reductionist approach. This is where, to make a complex problem more tractable, it is divided into parts. Each part may be analyzed by using of multiscale approaches to find answers; the answers from these parts are then assembled. Examples of this approach range from physics and engineering to even how some solutions of partial differential equations are found. But, a common problem is the difficulties in piecing together answers from the parts. In explaining this phenomenon, current work to overcome these difficulties is introduced.

 

Marshall Slemrod
Department of Mathematics, University of Wisconsin, Madison and Faculty of Mathematics, Weizmann Institute of Sciences

Title: Recent Applications of Multi-Scale Averaging

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In a series of recent papers my co-authors A. Acharya, Z. Artstein, W. Gear, S-Y Ha, S. Jung, Y. Kevrekedis, E. Titi and I have found a variety of applications of the multi-scale averaging theory given in the paper of Artstein, Kevrekedis, Titi, and Slemrod ( Nonlinearity 20 (2007)). In this talk I plan to survey these applications as a way of illustrating the usefulness of the theory for future potential applications.

 

Pierre Suquet
UPR CNRS 7051. Laboratoire de Mécanique et d’Acoustique, 31, Chemin Joseph Aiguier, 13402 Marseille Cedex 20, France.

Title: Long memory effects arising from homogenization and internal variables
(Based on joint works with R. Brenner and N. Lahellec.)

This talk is devoted to the homogenization of viscoelastic composite materials or polycrystals. It is well known that short memory effects in the individual constituents give rise, after homogenization, to long memory effects in the composite (Sanchez-Hubert and Sanchez- Palencia, 1978; Suquet, 1986). In other words the constitutive equations, which are ordinary differential equations at the level of individual constituents, become integrodifferential equations after homogenization. It is also well-known that the main ingredient for establishing this result is the Laplace transform which does not apply any more when the constituents becomes nonlinear or simply aging (time dependent material properties).

The present talk will report on less well-known recent progress made on specific properties of the kernel characterizing the long memory effects and its relation with constitutive theories based on internal variables. First, four relations satisfied by this kernel at large and short times have been derived (Suquet, 2012). Second, in practice, the implementation of functional constitutive relations requires the storage of all the stress (or strain) history in the composite from the initial time to the present time t and is therefore of limited use. However, in certain circumstances this long memory effect can be expressed by means of a finite number of internal variables (Ricaud and Masson, 2009). The constitutive relations remain ordinary differential equations for a larger, but still finite, number of variables than for the original constituents. When this is not the case, an approximate (but often accurate) model involving two internal variables can be proposed, based on the abovementioned four relations.

With the question of nonlinear viscoelastic composites in mind, the end of the talk will discuss another approach of the problem, valid both for linear and nonlinear composites. Incremental variational principles can be used to introduce the approximation of effective internal variables (Lahellec and Suquet, 2007). The accuracy of two versions of this approximation will be discussed.

 

Robert H. Swendsen
Dept. of Physics, Carnegie Mellon University

Title: The inverse renormalization-group transformation

This talk will present work done in collaboration with Dorit Ron and Achi Brandt on the use of inverse renormalization-group transformations to calculate critical properties at second-order phase transitions. The method is related to both the usual Monte Carlo renormalization group (MCRG) and the Multigrid Monte Carlo (MGMC) methods, and it retains advantages of both. The approach leads to computer simulations that are not hampered by critical slowing down.

 

Luc Tartar,
Department of Mathematics, Carnegie Mellon University

Title: The goal of GTH, the general theory of homogenization

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The mathematical theory of homogenization (started in the late 1960s in Pisa, and in the early 1970s in Paris) is not about studying partial differential equations with periodically modulated coefficients (to which some add a stochastic grain of salt according to their taste) but rather to develop a (new) nonlinear microlocal tool which will permit to describe the EVOLUTION of mixtures in continuum mechanics or physics (which should then be useful for chemistry and biology). Of course, this is my point of view, and I think I have a good understanding of why I feel isolated within my community of mathematicians. My mathematical point of view necessarily involves serious learning of sensible facts about the real world, in mechanics or physics, and it is a grave concern of mine that the training of the young generation of students and researchers in mathematics for some time now has been sorely lacking in this respect.

The goal of GTH is then to improve the way one describes the laws of nature at the moment, with known defects which one prefers not to emphasize. It seems that these defects cannot be resolved with old mathematical tools like ordinary differential equations involving a fixed and small number of degrees of freedom, which is the 18th century point of view of mechanics, or partial differential equations without small scales, which is the 19th century point of view of mechanics. The new 20th century point of view of mechanics or physics is then to consider partial differential equations in a multiscale environment (since it is a little limited to imagine a world with only one small scale and then to attempt to transform information at that microscopic scale into laws valid at a unique mesoscopic scale) and understand what new mathematical problems arise, for which I coined the term "beyond partial differential equations".

I shall present a few (academic) examples where the limiting effective equation is known, but where simple "natural" expansions do not converge, and the similarity with questions about diagrams used by physicists suggests that a mathematical understanding of more general cases might clarify why some strange procedures give a reasonable result: in the same way that the theory of distributions (which Laurent Schwartz developed after the pioneering work of Sergei Sobolev and Jean Leray) clarified some strange rules proposed by Dirac (with bold uses of his "function"), it is hoped then that the future development of GTH will explain how to handle Feynman diagrams in a mathematical way, and teach how to extract meaningful mesoscopic and macroscopic information out of molecular dynamics simulation, or other techniques, whose "convergence" to a useful effective equation is not so clear at the moment.

 

Johannes Zimmer
Mathematics, University of Bath, UK

Title: Toward nonequilibrium: from particles to entropic gradient flows

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Nonequilibrium thermodynamics is a field where our mathematical understanding, in particular as far as averaging and scale-bridging is concerned, is at present very limited. A fundamental result by Otto and collaborators demonstrates that linear diffusion can be formulated as gradient flow of the entropy, in the Wasserstein metric. This result shows that diffusion is indeed driven by entropy, even far from equilibrium. But why the Wasserstein metric? And how can we derive this result directly from an underlying particle model? I will present a framweork in which these two questions can be answered. I will try to indicate which other results can be obtained with this approach, but also highlight current (or general?) stumbling blocks for the derivation of free energies