450 research outputs found
Improved conditioning of the Floater--Hormann interpolants
The Floater--Hormann family of rational interpolants do not have spurious
poles or unattainable points, are efficient to calculate, and have arbitrarily
high approximation orders. One concern when using them is that the
amplification of rounding errors increases with approximation order, and can
make balancing the interpolation error and rounding error difficult. This
article proposes to modify the Floater--Hormann interpolants by including
additional local polynomial interpolants at the ends of the interval. This
appears to improve the conditioning of the interpolants and allow higher
approximation orders to be used in practice.Comment: 13 pages, 4 figures, 1 tabl
Basis functions on the grain boundary space: Theory
With the increasing availability of experimental and computational data
concerning the properties and distribution of grain boundaries in
polycrystalline materials, there is a corresponding need to efficiently and
systematically express functions on the grain boundary space. A grain boundary
can be described by the rotations applied to two grains on either side of a
fixed boundary plane, suggesting that the grain boundary space is related to
the space of rotations. This observation is used to construct an orthornormal
function basis, allowing effectively arbitrary functions on the grain boundary
space to be written as linear combinations of the basis functions. Moreover, a
procedure is developed to construct a smaller set of basis functions consistent
with the crystallographic point group symmetries, grain exchange symmetry, and
the null boundary singularity. Functions with the corresponding symmetries can
be efficiently expressed as linear combinations of the symmetrized basis
functions. An example is provided that shows the efficacy of the symmetrization
procedure
Classification of atomic environments via the Gromov-Wasserstein distance
Interpreting molecular dynamics simulations usually involves automated
classification of local atomic environments to identify regions of interest.
Existing approaches are generally limited to a small number of reference
structures and only include limited information about the local chemical
composition. This work proposes to use a variant of the Gromov-Wasserstein (GW)
distance to quantify the difference between a local atomic environment and a
set of arbitrary reference environments in a way that is sensitive to atomic
displacements, missing atoms, and differences in chemical composition. This
involves describing a local atomic environment as a finite metric measure
space, which has the additional advantages of not requiring the local
environment to be centered on an atom and of not making any assumptions about
the material class. Numerical examples illustrate the efficacy and versatility
of the algorithm
Continuous and Optimally Complete Description of Chemical Environments Using Spherical Bessel Descriptors
Recently, machine learning potentials have been advanced as candidates to
combine the high-accuracy of quantum mechanical simulations with the speed of
classical interatomic potentials. A crucial component of a machine learning
potential is the description of local atomic environments by some set of
descriptors. These should ideally be continuous throughout the specified local
atomic environment, twice-differentiable with respect to atomic positions and
complete in the sense of containing all possible information about the
neighborhood. An updated version of the recently proposed Spherical Bessel
descriptors satisfies all three of these properties, and moreover is optimally
complete in the sense of encoding all configurational information with the
smallest possible number of descriptors. The Smooth Overlap of Atomic Position
descriptors that are frequently visited in the literature and the Zernike
descriptors that are built upon a similar basis are included into the
discussion as being the natural counterparts of the Spherical Bessel
descriptors, and shown to be incapable of satisfying the full list of core
requirements for an accurate description. Aside being mathematically and
physically superior, the Spherical Bessel descriptors have also the advantage
of allowing machine learning potentials of comparable accuracy that require
roughly an order of magnitude less computation time per evaluation than the
Smooth Overlap of Atomic Position descriptors, which appear to be the common
choice of descriptors in recent studies.Comment: 15 pages, 5 figures, under review for Journal of Chemical Physic
A Novel Approach to Describe Chemical Environments in High Dimensional Neural Network Potentials
A central concern of molecular dynamics simulations are the potential energy
surfaces that govern atomic interactions. These hypersurfaces define the
potential energy of the system, and have generally been calculated using either
predefined analytical formulas (classical) or quantum mechanical simulations
(ab initio). The former can accurately reproduce only a selection of material
properties, whereas the latter is restricted to short simulation times and
small systems. Machine learning potentials have recently emerged as a third
approach to model atomic interactions, and are purported to offer the accuracy
of ab initio simulations with the speed of classical potentials. However, the
performance of machine learning potentials depends crucially on the description
of a local atomic environment. A set of invariant, orthogonal and
differentiable descriptors for an atomic environment is proposed, implemented
in a neural network potential for solid-state silicon, and tested in molecular
dynamics simulations. Neural networks using the proposed descriptors are found
to outperform ones using the Behler Parinello and SOAP descriptors currently in
the literature.Comment: 23 Pages, 5 figures, 2 tables, journal articl
Analysis of crystallographic texture information by the hyperspherical harmonic expansion
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Materials Science and Engineering, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 223-230).The field of texture analysis is fundamentally concerned with measuring and analyzing the distribution of crystalline orientations in a given polycrystalline material. Traditionally, the orientation distribution function describing crystallographic orientation information is written as a linear combination of the generalized spherical harmonics. Since the use of generalized spherical harmonics requires that orientations be described by sets of Euler angles, the field of texture analysis suffers from the inherent limitations of Euler angles. These include difficulty of presentation and interpretation, discontinuous changes in the description of a changing orientation, and singularities in many equations of Euler angles. An alternative expansion of the orientation distribution function as a linear combination of the hyperspherical harmonics is therefore proposed, with the advantage that this expansion allows rotations to be described by angles that directly relate to the axis and angle of a rotation. Apart from the straightforward and intuitive presentation of orientation statistics that this allows, the utility of the hyperspherical harmonic expansion rests on the fact that the orientation distribution function inherits the useful mathematical properties of the hyperspherical harmonics. The relationship of the hyperspherical harmonics to the three- and four-dimensional rotation groups is investigated, and expressions for the matrix elements of the irreducible representatives of these rotation groups as linear combinations of the hyperspherical harmonics are found.(cont.) These expressions allow an addition formula for the hyperspherical harmonics to be derived, and provide the means to write a simple conversion between the generalized spherical harmonic and hyperspherical harmonic expansions. This allows results derived via the hyperspherical harmonic expansion to be related to the texture analysis literature. Furthermore, a procedure for calculating the symmetrized hyperspherical harmonics consistent with crystal and sample symmetries is indicated, and used to perform the expansion of an orientation distribution function significantly more efficiently. The capability of the hyperspherical harmonic expansion to provide results not traditionally accessible is demonstrated by the generalization of the Mackenzie distribution to arbitrary textures. Finally, further areas where the application of the hyperspherical harmonic expansion is expected to advance the field of texture analysis are discussed.by Jeremy K. Mason.Ph.D
Statistical physics of dislocation nucleation by nanoindentation
Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Physics, 2005.Page 82 blank.Includes bibliographical references (page 79-81).Current understanding of the onset of plasticity during nanoindentation of crystalline materials involves homogenous dislocation nucleation in the crystal underneath the indenter. Through the use of cutting-edge nanoindentation techniques, this study examines the initiation of plastic deformation in single crystal oriented platinum samples. Variations in the temperature and loading rate during indentation reveal temporal and thermal dependencies, and support the stochastic and thermally-activated nature of the initial plastic event. These dependencies of dislocation nucleation are precisely quantified by developing analysis methods based on statistical thermodynamics, and are used to evaluate the probability of various atomistic mechanisms. The results of this procedure implicate a critical activation event occurring in a single atomic volume, with an activation enthalpy of a fraction of an electron volt. These findings strongly indicate that the initiation of plasticity begins with a heterogeneous dislocation nucleation event, in conflict with the current belief, and significantly advance understanding of the onset of plastic deformation during nanoindentation.by Jeremy K. Mason.S.B
Dependence of simulated radiation damage on crystal structure and atomic misfit in metals
This study investigates radiation damage in three metals in the low
temperature and high radiant flux regime using molecular dynamics and a Frenkel
pair accumulation method to simulate up to displacements per atom. The
metals considered include Fe, equiatomic CrCoNi, and a fictitious metal with
identical bulk properties to the CrCoNi composed of a single atom type referred
to as an A-atom. CrCoNi is found to sustain higher concentrations of
dislocations than either the Fe or A-atom systems and more stacking faults than
the A-atom system. The results suggest that the concentration of vacancies and
interstitials are substantially higher for the CrCoNi than the A-atom system,
perhaps reflecting that the recombination radius is smaller in CrCoNi due to
the roughened potential energy landscape. A model that partitions the major
contributions from defects to the stored energy is described, and serves to
highlight a general need for higher fidelity approaches to point defect
identification
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