Phenomena in Engineering Materials

Workshop

April 2-4, 2012

Student Abstracts and Posters

**Amin Aghaei, Carnegie Mellon University**

**Co-Author:** Kaushik Dayal

**Title:** Phonon Analysis of Carbon Nanotubes Using Objective Structure Framework

*click here for larger view of poster*

We introduce a method to find the phonons of objective structures such
as nanotubes. Phonons are the normal modes of vibration of the atoms
in a solid. They are the solutions u (eigenvectors) of the equation of
motion Hu=ω^{2}u at equilibrium where H is the Hessian matrix of the
structure. Since the size of the Hessian matrix is very large (often
infinity), finding the eigensystem of it will be a time-consuming
procedure. In a periodic lattice the Hessian matrix is block-circulant
which ensures that it can get transformed to a block-diagonal form using
Fourier transform. This greatly reduces the amount of computations
needed to calculate the frequencies and normal modes of vibration. But
some of the objective structures are not periodic and in some others one
may find periodicity in a very large cell. Nanotubes are a good example
of objective structures. When the structure is not periodic Hessian matrix
is not block-circulant, and, as a result, Fourier transform cannot be used.
In our work, based on the symmetry of the objective structures, we first
introduce a transformation to transform the Hessian matrix to a blockcirculant
form. Using the symmetry of nanotubes, we then show that the
full phonon spectra of nanotubes can be obtained by considering only
two atoms in the unit cell.

**Yanjiun Chen, Cornell University **

**Title:** Surface energy anisotropy for FCC metals: Functional forms

**Co-Authors:** Mihir Khadilkar, James P. Sethna

The energy of a crystalline surface depends
on the angle of the surface normal with respect to the crystalline
axes (i.e. the Miller index). We show that the surface energy as a function
of angle is surprisingly easy to describe with a simplistic broken
bond model including only a few parameters. In particular, this model
is successful at capturing the cusps at high symmetry surfaces. We
will use our fitted functional form as a characteristic continuum description
of material surface energies for FCC metals. The anisotropy of
these surface energies can then be utilized in the study of many material
properties- equilibrium shapes (Wulff plots) of crystals and voids,
fracture mechanics, cleavage and faceting. We calculate the surface energies
using ab initio calculations and various interatomic potentials,
providing a measure for the fitness of the potentials in studying physical
systems with surfaces. Furthermore, we assemble systematic tables of
these results, available to the community through the Knowledgebase of
Interatomic Models (KIM, https://openkim.org/).

**Woosong Choi and Matthew Bierbaum, Cornell University **

**Title:** Continuum Dislocation Dynamics : Walls and Vacancies

**Co-Authors:** Yong Chen, Stefanos Papanikolaou, James Sethna

*click here for larger view of poster*

Many continuum theories of dislocation dynamics have been proposed to bridge
the gap in between discrete microscopic simulations and macroscale
phenomenology. As of yet, however, these theories had limited success in
explaining or predicting the physics of microstructure formation and evolution.
Recently, we have shown that a simple isotropic continuum model dynamically
forms walls and exhibits complicated microstructure formation and evolution
similar to experiments. Most other continuum theories have not seen such
structures emerging, and to what extent this theory explains the physics
remains to be answered. We compare our model with simple traditional slip
equations and modified versions of those, and discuss the relevant mechanisms
and their consequences in the dynamics of microstructures. We also investigate
vacancy assisted climb behaviors by coupling the dislocation density
to a vacancy field. We are able to study the effects of vacancies on
diffusion-limited dislocation motion. We use our model to explore applications of
vacancy assisted climb, including dislocation creep and absorption of dislocations
at grain boundaries.

**Lin Hu, Carnegie Mellon University**

**Title:** Gas diffusion in a carbon nanotube aerogel

**Co-Authors:** Scott N. Schiffres, Gabriella Coloyan, Jonathan A. Malen, and Alan J. H. McGaughey

*click here for larger view of poster*

We use molecular dynamics simulation, mesoscopic simulation, and kinetic theory modeling to understand gas diffusion in a carbon nanotube (CNT) aerogel. A CNT aerogel is a low-density (<10 kg/m3) network of single-walled CNTs held together by van del Waals forces. Experimental measurements show that the thermal conductivity of an evacuated CNT aerogel is 0.025±0.001 W/m-k, which is comparable to the thermal conductivity of gases. The fits of a kinetic theory-based model to aerogel thermal conductivity measurements made in the presence of different gases indicate that the gas-aerogel scattering length scale is 2-6 μm, which is much larger than the aerogel pore sizes (2-50 nm). To understand the large difference between these two length scales, a mesoscale 3-D network model of the aerogel, with each CNT represented by a set of beads, was built to estimate the average distance traveled by a diffusing gas molecule between collisions with the aerogel (i.e., its mean free path). The mesoscale model predicts the mean free path to be 250 nm, ten times less than the length scale predicted from kinetic theory. We hypothesize that this different is a result of imperfect energy transfer between the gas molecule and the CNT. To establish the link between the two length scales, we use molecular dynamics simulations to predict the accommodation coefficient for gas molecules interacting with CNTs. By combining the MD-predicted accommodation coefficients with the mesoscale-predicted mean free path, we are able to match the length scale extracted from the experimental measurements and kinetic theory modeling. This agreement provides important insight into the nature of the diffusion of gas molecules inside a CNT aerogel.

**Jason Larkin, Carnegie Mellon University**

**Title:** Predicting Phonon Properties and Thermal Conductivity in Nanostructures

**Co-Authors:** Ankit Jain, and Alan McGaughey

*click here for larger view of poster*

Controlling thermal transport is crucial for optimizing the performance, reliability, and lifetime of semiconductor devices for electronic, optoelectonic, and energy conversion devices. The component nanostructures (e.g., thin films, nanowires, superlattices) offer an intriguing opportunity to tune thermal conductivity, but multiscale models are necessary to make accurate predictions that can be compared to experimental measurements.

Thermal transport in a semiconductor is controlled by phonons, quanta of energy associated with atomic vibrations. Atomic vibrations take place at the level of Angstroms, phonons wavelengths are on the order of 1-10 um, and phonons mean free paths can extend up to 10 um, comparable to the size of many technologically relevant nanostructures. It is this overlap between the phonon mean free paths and the system size that provides the opportunity for controlling thermal conductivity.

We present a multiscale approach to predict the thermal conductivity of a nanostructure of arbitrary geometry. The required inputs are the nanostructure geometry and the bulk phonon properties. The bulk phonon properties are obtained using atomistic *ab-initio* (from first-principles) calculations, which include quantum effects. *Ab-initio* methods, while computationally more demanding, are much more accurate than classical methods. The bulk phonon properties and nanostructure geometry are used as inputs for a Monte Carlo sampling-based method that predicts the mean free paths in the nanostructure. These mean free paths are then used as input to the Boltzmann transport equation for predicting thermal conductivity. As a demonstration, we apply this technique to predict the thermal conductivity of thin films, nanowires, and nanoporous materials.

**Jianfeng Liu, Rensselaer Polytechnic Institute**

**Title:** The residual-based variational multiscale formulation for the large eddy simulation of compressible turbulent flows

**Co-Authors:** David Sondak, Assad Oberai

*click here for larger view of poster*

The residual-based variational multiscale (RBVM) formulation is applied to develop
a large eddy simulation (LES) model for compressible turbulent flows. The basic
idea of RBVM formulation is to split the solution and weighting function spaces
into coarse and fine scale partitions. Splitting the weighting functions in this way
yields two sets of coupled equations: one for the coarse scales and another for the
fine, or the unresolved, scales. The equations for the fine scales are observed to be
driven by the residual of the coarse scale solution projected onto the fine scale space.
These equations for the unresolved scales are solved approximately and the solution
is substituted in the equations for the coarse scales. In this way the effect of the fine
or the unresolved scales on the coarse scales is modeled. In addition, a mixed model,
in which a deviatoric Smagorinsky subgrid stress term is added to the RBVM terms
to better model the Reynolds stresses is proposed for compressible flows. These
models are tested in predicting the decay of compressible homogenous turbulence.
For most relevant measures, it is observed that the RBVM models are more accurate
than the dynamic Smagorinsky-type model.

**Chang-Tsan Lu, Carnegie Mellon University**

**Title:** Kinetic Relations Associated with Phase Transformation in A One-Dimensional Atomic Chain

**Co-Author:** Kaushik Dayal

*click here for larger view of poster*

A recent work done by Schwetlick and Zimmer concerning phase transformation in a one-dimensional atomic chain finds that the phase boundary moves in form of traveling waves, and it associates a one-parameter family of kinetic relations. The current analysis proposes that the kinetic relation parameter is actually related to temperature, and verifies that numerically. The expression of the driving force derived by Schwetilick and Zimmer from a view of purely mechanical theory is extended from the view of continuum thermoelasticity. The driving force calculated from different views are also compared. The current analysis therefore provides another view of interpreting the kinetic relation associated with the phase transformation taking place in the one-dimensional atomic chain.

**Jason Marshall, Carnegie Mellon University**

**Title:** Multiscale Mechanics with Long-Range Electrostatic Forces

**Co-Author:** Kaushik Dayal

*click here for larger view of poster*

A key challenge in modeling ferroelectrics and other electromechanically coupled materials is the longrange
nature of the electrostatic fields. The electrostatic fields, in addition to being long-ranged, are
not confined to the sample, but instead are present even in the surrounding medium. Typical
approaches to this problem involve assumptions of periodicity or other highly restrictive
approximations. We develop a real-space multiscale method to accurately and efficiently deal with
electric fields in complex geometries and with realistic boundary conditions. Our approach handles the
short-range atomic interactions with the quasicontinuum method. The long-range electrostatic
interactions are handled by rigorously extending (from literature) the thermodynamic limits of lattices
of dipoles to complex lattices of charges. Additionally, we derive rigorous error bounds via multipole
expansion of the long-range interactions to justify the use of a multiscale model. We apply the method
to understand the deformation of a ferroelectric with complex geometries and subject to various
mechanical and electrostatic loading conditions.

**Phani Motamarri, Univ. of Michigan**

**Title:** Higher-Order Adaptive Finite-Element Methods for Kohn-Sham Density Functional Theory

**Co-Author:** Vikram Gavini

*click here for larger view of poster*

Many macroscopic properties of solids are influenced by complex interplay of defects. A mathematical model which accurately describes such defects must include both the electronic structure of the defect core at the quantum mechanical length scale and also elastic and electrostatic interactions at the coarse (micrometer and beyond) scale. One of the most popular electronic structure theories derived from first principles is the widely used Kohn-Sham density-functional theory (KSDFT). Recently, there is an increasing thrust towards using real-space formulations using a finite-element basis to compute material properties using DFT. The ability to handle complex geometries, arbitrary boundary conditions and more importantly the coarse graining nature of the basis sets makes finite elements even more desirable especially in problems involving defects. However, the complexity of the DFT calculations still restricts the computation to a few hundred atoms. To extend the calculations to realistic sample sizes that can accurately capture the long ranged fields generated by defects, an efficient multi-scale approach that can seamlessly connect quantum mechanical and continuum scales is highly desirable. As a first step towards developing a multi-scale method with KSDFT solely as the input physics, it is highly desirable to have a robust and an efficient computational framework for the real space formulation of the KSDFT problem. One of the recent works in this direction is the development of a real space, non-periodic, finite element formulation for the KSDFT problem [2] by transforming the original variational problem into a saddle point problem. However, it is observed that there is an enormous computational cost associated with the use of linear finite elements and the solution strategies adopted in solving DFT problem using finite-element basis sets.

In this work, we present an efficient computational strategy to perform electronic structure calculations using adaptive higher-order finite-element discretization of the Kohn-Sham density functional theory. The performance and computational efficiency of higher order finite-elements is thereby investigated in this work. An apriori error estimate of the finite element approximation of the Kohn-Sham problem is first derived and the optimal coarse graining rates of the finite element triangulations are obtained using these estimates. We then investigate efficient computational strategies to solve the Kohn-Sham eigenvalue problem in a self-consistent way using the above optimal finite-element meshes. The computational advantage derived from the use of higher order elements is subsequently analyzed using the above optimal meshes in conjunction with efficient numerical strategies on sample benchmark systems. The study suggests that the reduction in CPU time associated with the use of higher order finite elements is about 10-100 fold in comparison to that of the linear finite elements. We finally extend our investigations to large atomic systems (of the order 2000 atoms). To this end, we compare the computational CPU time involved in finite-element simulations with plane-wave basis in the case of pseudopotential calculations and atomic-orbital basis sets in the case of all electron calculations. We consider this study as a step towards developing a robust and computationally efficient discretization of linear scaling electronic structure calculations using the finite-element basis and thereby the development of a multi-scale method with KSDT solely as the input physics.

**Ashivni Shekhawat, Cornell University **

**Title:** Fracture Statistics: Nucleation Versus Universality

*click here for larger view of poster*

We reexamine several common assumptions about fracture strength,
utilizing large-scale simulations of a fuse network model and applying
both renormalization-group and nucleation theory methods. Statistical
distributions of fracture strengths are believed to be universal and
material independent. The universal Weibull and Gumbel distributions
emerge as a consequence of the ``weakest-link hypothesis'' and have
been studied in the classical theory of extreme value statistics.
These distributions are also the fixed points of a renormalization
group (RG) flow. However, the engineering community often ignores the
Gumbel distribution and uses the Weibull form almost exclusively to
fit experimental data. Further, such fits are often extrapolated
beyond the available data to estimate the probability of rare events
in a variety of applications ranging from structural reliability to
insurance pricing. Our recent studies of the random fuse network model
raises doubts about most of these practices. We find that the emergent
distribution of fracture strengths is the Gumbel distribution.
However, the extremely slow convergence to the universal Gumbel form
renders it unusable at least in this case. On the other hand, we show
that a non-universal distribution derived by using a Griffiths type
nucleation theory (due to Duxbury et al.) converges rapidly even for
moderate system sizes. We find that while extrapolating the RG based
universal Gumbel distribution is perilous and gives wildly incorrect
predictions, the nucleation based non-universal results can be
extrapolated with confidence. It is entertaining that fracture
provides wonderful examples of the statistical mechanics tools
developed to study both continuous as well as abrupt phase
transitions.

**Cameron Talischi, Univ. of Illinois**

**Title:** On Optimization of Shape and Topology

**Co-Author:** Glaucio H. Paulino

*click here for larger view of poster*

This poster presentation deals with problems of shape and topology optimization where the
goal is to find the most efficient shape of a physical system. The behavior of this system is typically
captured by the solution to a set of differential equations that in turn depends on the given shape.
As such, optimal shape design can be viewed as a form of optimal control in which the control
is the shape or domain of the governing state equation. The resulting methodologies have found
applications in various scientific and engineering disciplines and have been used to successfully
design novel material systems (e.g., microstructure of materials with extreme properties), civil
and mechanical engineering structures (e.g, lateral bracing system of high-rise buildings) and
medical transplants (e.g. patient-tailored craniofacial reconstruction). However, the optimal shape
problems, more generally PDE-constrained inverse problems, pose several fundamental challenges.
Despite the ever-increasing computational resources available, the current methodologies in many
instances are not sufficiently robust to tackle the design of more complicated systems, in particular
those with nonlinear, transient, and heterogeneous (multi-physics, multi-scale) physical response.
The dependence of final design on the choice of algorithmic parameters and the heuristics employed
becomes pronounced as the scale of the problem is increased.

In this poster presentation, I will discuss the inherent mathematical ill-posedness of optimal
shape problems, in particular the lack of existence of solutions in the classical sense. The structure
of the problem favors non-convergent sequences of shapes that exhibit progressively finer features
and thus an appropriate regularization scheme must be introduced to exclude such behavior and
limit the complexity of the admissible shapes. One strategy investigated here is to prescribe the
degree of sparsity of of the unknown shape (e.g., sparsity of the spatial gradients) and recover
the optimal solution in a stable manner using a tailored sparse optimization framework. This
leads to the discussion of another challenging aspect of the optimal shape problem, which is the
resulting large-scale non-convex optimization system that contains many local minima and involves
expensive function evaluations and gradient calculations. I will discuss the development of an
operator splitting algorithm that allows for separate treatment of the cost functional (that depends
on the state equation) and the regularization term (that controls the geometric complexity). The
convex subproblems generated by the splitting algorithm in each iteration are significantly simpler
to solve since the dependence on the state equation is removed. The important consequence is
that one is able to accommodate nonsmooth regularization schemes. This optimization approach
is aligned with the renewed interest in accelerated first order methods for large-scale linear inverse
problems that arise in image processing.

**Likun Tan, Carnegie Mellon University **

**Title:** On Multi-scale Modeling of Autonomous ODE systems

**Co-Authors:** Amit Acharya, Kaushik Dayal

*click here for larger view of poster*

Given an autonomous system of ordinary differential equations, we consider developing practical models
for the deterministic, slow/coarse behavior of the fine system. The coarse variables are defined as finite
time averages of phase functions. Approaches to construct the coarse evolution equation are discussed
and implemented on a ‘Forced’ Lorenz system and a singularly perturbed system whose fast flow does
not necessarily converge to an equilibrium. We explore two strategies. In one, we compute (locally)
invariant manifolds of the fast dynamics, parametrized by the slow variables. In the other, the choice of
our coarse variables automatically guarantees them to be 'slow' in a precise sense. This allows their
evolution to be phrased in terms of averaging utilizing limit measures (probability distributions) of the
fast flow. Coarse evolution equations are constructed based on these approaches and coarse features of
the model problems are observed.