Begell House Inc.
International Journal for Multiscale Computational Engineering
JMC
1543-1649
10
1
2012
MULTISCALE COMPUTATION AND MODELING OF DEFECTS IN MATERIALS
vii
Vikram
Gavini
University of Michigan
THERMAL EXPANSION BEHAVIOR OF Al AND Ta USING AFINITE-TEMPERATURE EXTENSION OF THE QUASICONTINUUM METHOD
1-11
Gabriela
Venturini
Caltech
Jaime
Marian
Chemistry, Materials, and Life Sciences Directorate, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA
Jaroslaw
Knap
Chemistry, Materials, and Life Sciences Directorate, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA
G.
Campbell
Lawrence Livermore National Laboratory
Michael
Ortiz
California Institute of Technology
Numerical methods that bridge the atomistic andcontinuum scales concurrently have been applied successfully to anumber of materials science problems involving both nonlinear andlong-range deformation fields. However, extension of thesemethods to finite temperature, nonequilibrium dynamics isdifficult due to the intrinsic incoherency between moleculardynamics and continuum thermodynamics, which possess differentcrystal vibrational spectra and therefore result in unphysicalwave reflections across domain boundaries. Here we review ourrecent finite temperature extension of the three-dimensional,non-local quasicontinuum (QC) method based on Langevin dynamicsand carry out an analysis of the systematic errors associated withthe entropic depletion that results from the QC reduction. Weapply the method to Al and Ta structured meshes ranging fromatomistic resolution to minimum-node representations using thethermal expansion coefficient as the standard metric. We findthat, while Al errors scale linearly with the number of meshnodes, Ta displays a very erratic behavior that degrades rapidlywith mesh coarsening.
COARSE GRAINING OF ATOMISTIC DESCRIPTION AT FINITE TEMPERATURE USING FORMAL ASYMPTOTICS
13-31
Yashashree
Kulkarni
University of Houston
In this paper, we propose a computational method for coarse graining the atomistic description at finite temperature using formal asymptotics. The method is based on the ansatz that there exists a separation of scales between the time scale of the atomic fluctuations and that of the thermodynamic processes, such as thermal expansion. We use the WKB method to propose an averaging scheme for treating the thermal degrees of freedom and deriving an effective Hamiltonian for the atomistic system. This energy functional is incorporated into the quasicontinuum framework to achieve a seamless coarse graining on the spatial scale. Numerical validation is performed by computing the thermal equilibrium properties of selected materials. The scope of the method based on the use of perturbation theory is discussed, and its capability is illustrated by way of simulating dislocation nucleation under a nanoindenter.
ANALYSIS OF THE QUASI-NONLOCAL QUASICONTINUUM APPROXIMATION OF THE EMBEDDED ATOM MODEL
33-49
Xingjie Helen
Li
School of Mathematics, University of Minnesota, Minneapolis, MN, 55455
Mitchell
Luskin
University of Minnesota
The quasi-nonlocal quasicontinuum method (QNL) is a consistent hybrid coupling method for atomistic and continuum models. Embedded atom models are empirical many-body potentials that are widely used for fcc metals such as copper and aluminum. In this paper, we consider the QNL method for EAM potentials and give a stability and error analysis for a chain with next-nearest-neighbor interactions. We identify conditions for the pair potential, electron density function, and embedding function so that the lattice stability of the atomistic and EAM-QNL models are asymptotically equal.
STRESS-BASED ATOMISTIC/CONTINUUM COUPLING: A NEW VARIANT OF THE QUASICONTINUUM APPROXIMATION
51-64
C.
Makridakis
Department of Applied Mathematics, University of Crete, 71409 Heraklion-Crete, Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion-Crete
Christoph
Ortner
University of Oxford
E.
Suli
Mathematical Institute, 24-29 S. Giles', Oxford OX1 3LB, United Kingdom
The force-based quasicontinuum (QCF) approximation isthe principle that lies behind the most commonly usedatomistic/continuum hybrid models for crystalline solids. Recentanalyses have shown some potential pitfalls of the QCF method, particularly the lack of positive definiteness of the linearized QCF operator and the lack of uniform stability as the number ofatoms tends to infinity. We derive a weak variational representation of the QCF operator and identify the origin ofthese difficulties as the lack of an interface condition on thestresses. This leads us to propose an improved variant of the QCF method that can be understood as a coupling mechanism based onstresses rather than forces.
RECENT DEVELOPMENT IN QUANTUM MECHANICS/MOLECULAR MECHANICS MODELING FOR MATERIALS
65-82
Xu
Zhang
California State University Northridge
Yi
Zhao
Department of Physics and Astronomy, California State University Northridge, Northridge, CA
Gang
Lu
Department of Physics and Astronomy, California State University Northridge, Northridge, CA
We have introduced two quantum mechanics/molecularmechanics approaches for materials modeling. One is based onquantum mechanical coupling and the other on mechanicalcoupling. The formalism of both approaches is described indetail. The validations of the methods are demonstrated in termsof atomic and electronic structure. Finally, the applications ofthe methods are surveyed, including applications in vacancyclusters, dislocations, nanoindentations, and fractures.
LINEAR SCALING SOLUTION OF THE ALL-ELECTRON COULOMB PROBLEM INSOLIDS
83-99
J. E.
Pask
Condensed Matter and Materials Division, Lawrence Livermore National Laboratory, Livermore,CA 94550
N.
Sukumar
University of California Davis
S. E.
Mousavi
Department of Civil and Environmental Engineering, University of California, Davis, CA 95616
We present a linear scaling formulation for the solution of the all-electron Coulomb problem in crystalline solids. The
resulting method is systematically improvable and well suited to large-scale quantum mechanical calculations in which
the Coulomb potential and energy of a continuous electronic density and singular nuclear density are required. Linear
scaling is achieved by introducing smooth, strictly local neutralizing densities to render nuclear interactions strictly
local, and solving the remaining neutral Poisson problem for the electrons in real space. Although the formulation
includes singular nuclear potentials without smearing approximations, the required Poisson solution is in Sobolev space
H1, as required for convergence in the energy norm. We employ enriched finite elements, with enrichments from isolated
atom solutions, for an efficient solution of the resulting Poisson problem in the interacting solid. We demonstrate the
accuracy and convergence of the approach by direct comparison to standard Ewald sums for a lattice of point charges and
demonstrate the accuracy in all-electron quantum mechanical calculations with an application to crystalline diamond.
SADDLE NODE SCALING ON APPROACH TO DISLOCATION NUCLEATION
101-108
Asad
Hasan
Carnegie Mellon University
Craig
Maloney
CMU
We study the process of dislocation nucleation in aperfect 2D hexagonal crystal under nano-indentation loading in anumerical model using energy minimization techniques and analysisof the energy eigenmodes. The nucleation event takes the form ofa saddle-node catastrophe and is governed by associated scalinglaws In particular, on approach to nucleation, a single energyeigenmode descends through the spectrum and its eigenvaluevanishes as the square root of the distance to the nucleationpoint. The velocity of the system shows the same scalingbehavior, and its normal-mode decomposition demonstrates that itis dominated by the critical mode responsible for nucleation.
DYNAMICS OF NANOSCALE VOID-FIBER ASSEMBLY FOR MATION INIRRADIATED AMORPHOUS MATERIALS
109-116
Kun-Dar
Li
Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109; Department of Nuclear Engineering & Radiological Science, University of Michigan, Ann Arbor, Michigan 48109
Qiangmin
Wei
Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109
Lumin
Wang
Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109; Department of Nuclear Engineering & Radiological Science, University of Michigan, Ann Arbor, Michigan 48109
Wei
Lu
University of Michigan
Ion beam experiments have revealed an intriguingobservation that a nanoscale porous structure containing uniformlysized nanofibers can form spontaneously in preamorphizedgermanium. Depending on the ion energy and ion dose, thenanoporous fiber assembly can either be exposed from the samplesurface or embedded under an intact surface cover. This paperproposes a phase field model that describes the dynamic processfor the formation of such a structure in an amorphous matrix. Inthis model, vacancies in an amorphous matrix are defined by localreduction of atomic density relative to the reference solid beforeirradiation, and the cavity is treated as a phase with the vacancyconcentration close to one. It is shown that interface migrationtogether with the competing actions between vacancy production andannihilation determines the morphology of irradiated material.The model suggests that with continuous supply of vacanciesthrough increasing irradiation doses, the morphology evolves intoa network structure composed of nearly uniform sizes ofnanofibers. The morphology and characteristic wavelengthpredicted by the model are consistent with experimentalobservations. The model also well predicts the effects of ionflux, temperature, and material properties on the uniquenanostructure evolution under irradiation.