December 2-3, 2009
Unifying the Mechanics of Continua, Cracks Particles
Stewart Silling (Sandia National Labs)
Abstract
The standard theory of continuum mechanics, which is based on partial differential equations, is largely incompatible with some important phenomena in solids, such as the emergence and growth of cracks. Attempts to make a direct link between the standard continuum theory and molecular dynamics are equally problematic, because of the incompatibility between the PDEs and the mechanics of discrete particles.
An alternative formulation of mechanics, known as the peridynamic model, allows both continuous and discontinuous media to be represented within a mathematically consistent framework. This model reduces to the standard theory of solid mechanics as a special case. It predicts the spontaneous emergence and growth of cracks directly from the basic field equations, without any supplemental equations that govern fracture. The motion of a set of discrete particles interacting through a multibody potential can also be represented exactly within the peridynamic field equations.
This talk will review the purpose and basics of the peridynamic method, accomplishments and difficulties to date, and prospects for using it as a framework for multiscale mechanics.
Statistical Mechanical Foundations for Peridynamics
Richard Lehoucq (Sandia National Labs)
Abstract
My presentation derives the peridynamic momentum and energy balances conservation laws using the principles of statistical mechanics. In particular, I show that the peridynamic force density integral operator is the phase space expected value of internal force density given by a general multibody interatomic potential. The derivations generalize the seminal work of Irving-Kirkwood (1950), and build upon the elegant ideas due to Noll (1955) that generalized the former work. This is joint work with Mark Sears of Sandia National Labs.
Mesoscale Simulations with Microscale Tools: Peridynamics in a Molecular Dynamics Code
Michael Parks (Sandia National Labs)
Abstract
A particular discretization of peridynamics models a continuum body using discrete particles, where each particle’s motion is determined by summing interaction forces with neighboring particles. This discretization of the peridynamic model has the same computational structure as molecular dynamics, allowing for a natural implementation within a molecular dynamics framework. We discuss our implementation of the peridynamic model within the massively parallel molecular dynamics code LAMMPS, and demonstrate example problems.
Vector Nonlocal Calculus and Analysis of Peridynamic Models
Qiang Du (Pennsylvania State University)
Abstract
In this talk, a mathematical framework for a nonlocal calculus of vector-valued functions will presented first. It will then be used to analyze the nonlocal peridynamic models. The differences and connections between such nonlocal models and the classical PDE models of continuum mechanics will be illustrated from a functional analytical point of view. Implications of the analytical investigations on the accuracy of the finite dimensional numerical approximations will also be discussed. This is a joint work with Kun Zhou of Penn State Univ, Max Gunzburger of Florida State Univ and Rich Lehoucq of Sandia National Lab.
Multiscale Modeling for Heterogeneous Peridynamic Media
Robert Lipton (Louisiana State University)
Abstract
We introduce a method for simultaneously tracking both the global and local dynamics inside microstructured peridynamic media. This approach is capable of modeling the dynamics at the structural length scale while at the same time retaining the ability to resolve the dynamics on the length scale of the microstructure. The associated numerical scheme is able to extract this information at a cost that is anticipated to be far less than that obtained from direct numerical simulation. This is joint work with Bacim Alali.
Cloaking vs Shielding in Transformation Optics
Allan Greenleaf (University of Rochester)
Abstract
It is generally believed that transformation optics based cloaking, besides rendering the cloaked region invisible to detection by scattering of incident waves, also shields the region from those same waves. We exhibit a coupling between the cloaked and uncloaked regions, exposing a difference between cloaking for rays and waves. Interior resonances allow this coupling to be amplified, and careful choice of parameters leads to effective transformation optics cloaks with degraded shielding. As one application, this shows that transformation optics cloaking allows, as in the plasmon based approach of Al\’u and Engheta, for sensors to be hidden in the cloaked region, and yet be able to efficiently measure the waves incident on the exterior of the cloak. This is joint work with Slava Kurylev, Matti Lassas and Gunther Uhlmann.
Multiscale Simulation Methods for High-Contrast Flow Problems
Yalchin Efendiev (Texas A&M University)
Abstract
In this talk, I will discuss special coarse spaces for multiscale finite element and domain decomposition methods. These spaces allow a sparse representation of the solution of multiscale flow problem. The focus will be on problems that have high variations in the media properties. It is known that the number of iterations in domain decomposition and many iterative methods is adversely affected by the contrast in the media properties. One way to decrease the number of iterations is to choose coarse spaces appropriately. In the proposed methods, the coarse spaces are constructed based on a local eigenvalue problem. We show that if domain decomposition methods use these coarse spaces then the condition number of preconditioned system is independent of the contrast in media properties. The coarse space can have large dimension. In this talk, we discuss dimension reduction for the proposed coarse spaces and hierarchical computations of multiscale basis functions. The latter results to domain decomposition preconditioners that have the condition number which is independent of contrast. We will discuss the accuracy of coarse-scale solutions using these proposed coarse spaces. Numerical results will be presented. This is a joint work with Juan Galvis.
Efficient Sampling of the Protein Conformational Space
Yi Qin Gao (Texas A&M University)
Abstract
In this talk, I will first introduce an efficient sampling technique that was recently developed by our research group. This integrated tempering sampling (ITS) method not only allows quick and reliable search of the protein conformations, but also permits fast, accurate, and robust thermodynamics calculations. I will illustrate the usage of this method through applications to the folding of a series small proteins and show how these calculations lead to quantitative and atomic-detailed understanding of the folding mechanism and kinetics of these polypeptides. I will also discuss the roles of hydrophobic interactions, hydrogen bond formation, and turn structure formation in determining the structure and stability of proteins, to show that the protein native structure and main chain hydrogen bond formation are largely determined by the properties of the side chains. Finally, the effects of different solvent conditions on the protein structure formation will be discussed.
Viscometry of Bulk Materials and Atomic Structures
Richard D. James (University of Minnesota)
Abstract
The most important deformations in solid mechanics are those that represent the bending, twisting and extension of beams. The most important flows in fluid mechanics are viscometric flows. In both cases these are the motions that, when compared with the corresponding experiments, are used to measure the material constants. We give a universal molecular level interpretation of these motions. It is argued that all these motions are associated at molecular level with a time-dependent invariant manifold of the equations of molecular dynamics. The presence of this manifold can be used to simplify molecular-level computations, and deliver viscometric properties in the absence of a constitutive relation. Its presence also suggests a modification of the principle of material frame-indifference, a cornerstone of nonlinear continuum mechanics. Interesting links to theories of turbulence, to the kinetic theory of gases (i.e., the Boltzmann equation), to the dynamics of nanostructures, and to the Langevin equation will be briefly discussed. Joint work with Kaushik Dayal, Traian Dumitrica and Stefan Mueller.
Complete Characterization and Synthesis of the Response Function of Elastodynamic Networks
Graeme Milton (University of Utah)
Abstract
The response function of a network of springs and masses, an elastodynamic network, is the matrix valued function W(omega), depending on the frequency omega, mapping the displacements of some accessible or terminal nodes to the net forces at the terminals. We give necessary and sufficient conditions for a given function W(omega) to be the response function of an elastodynamic network, assuming there is no damping. In particular we construct an elastodynamic network that can mimic a suitable response in the frequency or time domain. Our characterization is valid for networks in three dimensions and also for planar networks, which are networks where all the elements, displacements and forces are in a plane. This is joint work with Fernando Guevara Vasquez, and Daniel Onofrei.