628 research outputs found
Computing Nearly Singular Solutions Using Pseudo-Spectral Methods
In this paper, we investigate the performance of pseudo-spectral methods in
computing nearly singular solutions of fluid dynamics equations. We consider
two different ways of removing the aliasing errors in a pseudo-spectral method.
The first one is the traditional 2/3 dealiasing rule. The second one is a high
(36th) order Fourier smoothing which keeps a significant portion of the Fourier
modes beyond the 2/3 cut-off point in the Fourier spectrum for the 2/3
dealiasing method. Both the 1D Burgers equation and the 3D incompressible Euler
equations are considered. We demonstrate that the pseudo-spectral method with
the high order Fourier smoothing gives a much better performance than the
pseudo-spectral method with the 2/3 dealiasing rule. Moreover, we show that the
high order Fourier smoothing method captures about more effective
Fourier modes in each dimension than the 2/3 dealiasing method. For the 3D
Euler equations, the gain in the effective Fourier codes for the high order
Fourier smoothing method can be as large as 20% over the 2/3 dealiasing method.
Another interesting observation is that the error produced by the high order
Fourier smoothing method is highly localized near the region where the solution
is most singular, while the 2/3 dealiasing method tends to produce oscillations
in the entire domain. The high order Fourier smoothing method is also found be
very stable dynamically. No high frequency instability has been observed.Comment: 26 pages, 23 figure
Lattice Resistance and Peierls Stress in Finite-size Atomistic Dislocation Simulations
Atomistic computations of the Peierls stress in fcc metals are relatively
scarce. By way of contrast, there are many more atomistic computations for bcc
metals, as well as mixed discrete-continuum computations of the Peierls-Nabarro
type for fcc metals. One of the reasons for this is the low Peierls stresses in
fcc metals. Because atomistic computations of the Peierls stress take place in
finite simulation cells, image forces caused by boundaries must either be
relaxed or corrected for if system size independent results are to be obtained.
One of the approaches that has been developed for treating such boundary forces
is by computing them directly and subsequently subtracting their effects, as
developed by V. B. Shenoy and R. Phillips [Phil. Mag. A, 76 (1997) 367]. That
work was primarily analytic, and limited to screw dislocations and special
symmetric geometries. We extend that work to edge and mixed dislocations, and
to arbitrary two-dimensional geometries, through a numerical finite element
computation. We also describe a method for estimating the boundary forces
directly on the basis of atomistic calculations. We apply these methods to the
numerical measurement of the Peierls stress and lattice resistance curves for a
model aluminum (fcc) system using an embedded-atom potential.Comment: LaTeX 47 pages including 20 figure
Nonlocal Multiscale Hierarchical Decomposition on Graphs
International audienceThe decomposition of images into their meaningful components is one of the major tasks in computer vision. Tadmor, Nezzar and Vese [1] have proposed a general approach for multiscale hierarchical decomposition of images. On the basis of this work, we propose a multiscale hierarchical decomposition of functions on graphs. The decomposition is based on a discrete variational framework that makes it possible to process arbitrary discrete data sets with the natural introduction of nonlocal interactions. This leads to an approach that can be used for the decomposition of images, meshes, or arbitrary data sets by taking advantage of the graph structure. To have a fully automatic decomposition, the issue of parameter selection is fully addressed. We illustrate our approach with numerous decomposition results on images, meshes, and point clouds and show the benefits
Singular Cucker-Smale Dynamics
The existing state of the art for singular models of flocking is overviewed,
starting from microscopic model of Cucker and Smale with singular communication
weight, through its mesoscopic mean-filed limit, up to the corresponding
macroscopic regime. For the microscopic Cucker-Smale (CS) model, the
collision-avoidance phenomenon is discussed, also in the presence of bonding
forces and the decentralized control. For the kinetic mean-field model, the
existence of global-in-time measure-valued solutions, with a special emphasis
on a weak atomic uniqueness of solutions is sketched. Ultimately, for the
macroscopic singular model, the summary of the existence results for the
Euler-type alignment system is provided, including existence of strong
solutions on one-dimensional torus, and the extension of this result to higher
dimensions upon restriction on the smallness of initial data. Additionally, the
pressureless Navier-Stokes-type system corresponding to particular choice of
alignment kernel is presented, and compared - analytically and numerically - to
the porous medium equation
Renormalization group approach to multiscale modelling in materials science
Dendritic growth, and the formation of material microstructure in general,
necessarily involves a wide range of length scales from the atomic up to sample
dimensions. The phase field approach of Langer, enhanced by optimal asymptotic
methods and adaptive mesh refinement, copes with this range of scales, and
provides an effective way to move phase boundaries. However, it fails to
preserve memory of the underlying crystallographic anisotropy, and thus is
ill-suited for problems involving defects or elasticity. The phase field
crystal (PFC) equation-- a conserving analogue of the Hohenberg-Swift equation
--is a phase field equation with periodic solutions that represent the atomic
density. It can natively model elasticity, the formation of solid phases, and
accurately reproduces the nonequilibrium dynamics of phase transitions in real
materials. However, the PFC models matter at the atomic scale, rendering it
unsuitable for coping with the range of length scales in problems of serious
interest. Here, we show that a computationally-efficient multiscale approach to
the PFC can be developed systematically by using the renormalization group or
equivalent techniques to derive appropriate coarse-grained coupled phase and
amplitude equations, which are suitable for solution by adaptive mesh
refinement algorithms
Approximation of Parametric Derivatives by the Empirical Interpolation Method
We introduce a general a priori convergence result for the approximation of parametric derivatives of parametrized functions. We consider the best approximations to parametric derivatives in a sequence of approximation spaces generated by a general approximation scheme, and we show that these approximations are convergent provided that the best approximation to the function itself is convergent. We also provide estimates for the convergence rates. We present numerical results with spaces generated by a particular approximation scheme—the Empirical Interpolation Method—to confirm the validity of the general theory
Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier-Stokes Equations
Discontinuous Galerkin (DG) methods have a long history in computational
physics and engineering to approximate solutions of partial differential
equations due to their high-order accuracy and geometric flexibility. However,
DG is not perfect and there remain some issues. Concerning robustness, DG has
undergone an extensive transformation over the past seven years into its modern
form that provides statements on solution boundedness for linear and nonlinear
problems.
This chapter takes a constructive approach to introduce a modern incarnation
of the DG spectral element method for the compressible Navier-Stokes equations
in a three-dimensional curvilinear context. The groundwork of the numerical
scheme comes from classic principles of spectral methods including polynomial
approximations and Gauss-type quadratures. We identify aliasing as one
underlying cause of the robustness issues for classical DG spectral methods.
Removing said aliasing errors requires a particular differentiation matrix and
careful discretization of the advective flux terms in the governing equations.Comment: 85 pages, 2 figures, book chapte
Multidimensional Conservation Laws: Overview, Problems, and Perspective
Some of recent important developments are overviewed, several longstanding
open problems are discussed, and a perspective is presented for the
mathematical theory of multidimensional conservation laws. Some basic features
and phenomena of multidimensional hyperbolic conservation laws are revealed,
and some samples of multidimensional systems/models and related important
problems are presented and analyzed with emphasis on the prototypes that have
been solved or may be expected to be solved rigorously at least for some cases.
In particular, multidimensional steady supersonic problems and transonic
problems, shock reflection-diffraction problems, and related effective
nonlinear approaches are analyzed. A theory of divergence-measure vector fields
and related analytical frameworks for the analysis of entropy solutions are
discussed.Comment: 43 pages, 3 figure
The Simple View of Reading Made Complex by Morphological Decoding Fluency in Bilingual Fourth-Grade Readers of English
This is the author accepted manuscript. The final version is available from Wiley via the DOI in this recordThis study examined the complexity of the Simple View of Reading focusing on morphological
decoding fluency in fourth-grade readers of English in Singapore. The participants were three
groups of students who all learned to become bilingual and biliterate in the English language
(EL) and their respective ethnic language in school but differed in the home language they used.
The first group was ethnic Chinese students who used English as the dominant home language
(Chinese EL1); the other two groups were ethnic Chinese and Malay students whose dominant
home language was not English but Chinese (Chinese EL2) and Malay (Malay EL2),
respectively. The measures included pseudo word decoding (phonemic decoding), timed
decoding of derivational words (morphological decoding fluency), oral vocabulary, and passage
comprehension. Path analysis showed that oral vocabulary significantly predicted reading
comprehension across all three groups; yet a significant effect of morphological decoding
fluency surfaced in the Chinese EL1 and Malay EL2 groups but not the Chinese EL2 group.
Multi-group path analysis and commonality analysis further confirmed that morphological
decoding played a larger role in the in the Chinese EL1 and Malay EL2 groups. These findings
are discussed in light of the joint influence of target language experience and cross-linguistic
influence on second language or bilingual reading development.Office of Education Research, National Institute of Education, Nanyang Technological Universit
Effects of Crystalline Anisotropy and Indenter Size on Nanoindentation by Multiscale Simulation
Nanoindentation processes in single crystal Ag thin film under different crystallographic orientations and various indenter widths are simulated by the quasicontinuum method. The nanoindentation deformation processes under influences of crystalline anisotropy and indenter size are investigated about hardness, load distribution, critical load for first dislocation emission and strain energy under the indenter. The simulation results are compared with previous experimental results and Rice-Thomson (R-T) dislocation model solution. It is shown that entirely different dislocation activities are presented under the effect of crystalline anisotropy during nanoindentation. The sharp load drops in the load–displacement curves are caused by the different dislocation activities. Both crystalline anisotropy and indenter size are found to have distinct effect on hardness, contact stress distribution, critical load for first dislocation emission and strain energy under the indenter. The above quantities are decreased at the indenter into Ag thin film along the crystal orientation with more favorable slip directions that easy trigger slip systems; whereas those will increase at the indenter into Ag thin film along the crystal orientation with less or without favorable slip directions that hard trigger slip systems. The results are shown to be in good agreement with experimental results and R-T dislocation model solution
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