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Amin Aghaei, Carnegie Mellon University
Co-Author: Kaushik Dayal

Title: Phonon Analysis of Carbon Nanotubes Using Objective Structure Framework

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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=ω²u 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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.