SIAM Digital Library
 
 
 

Multiscale Modeling & Simulation

BROWSE VOLUMES

Year Range: 
Search Issue | RSS Feeds RSS
Previous Issue Next Issue

2005

Volume 4, Issue 2, pp. 359-708


Prediction of Dislocation Nucleation During Nanoindentation by the Orbital-Free Density Functional Theory Local Quasi-continuum Method

Robin L. Hayes, Matt Fago, Michael Ortiz, and Emily A. Carter

Multiscale Model. Simul. 4, pp. 359-389 (31 pages) | Cited 16 times

Online Publication Date: July 26, 2006

Full Text: | Download PDF

Show Abstract
We introduce the orbital-free density functional theory local quasi-continuum\linebreak (OFDFT-LQC) method: a first-principles-based multiscale material model that embeds OFDFT unit cells at the subgrid level of a finite element computation. Although this method cannot address intermediate length scales such as grain boundary evolution or microtexture, it is well suited to study material phenomena such as continuum level prediction of dislocation nucleation and the effects of varying alloy composition. The model is illustrated with the simulation of dislocation nucleation during indentation into the $(111)$ and $(\overline{1}10)$ surfaces of aluminum and compared against results obtained using an embedded atom method interatomic potential. None of the traditional dislocation nucleation criteria (Hertzian principal shear stress, actual principal shear stress, von Mises strain, or resolved shear stress) correlates with a previously proposed local elastic stability criterion, $\Lambda$. Discrepancies in dislocation nucleation predictions between OFDFT-LQC and other simulations highlight the need for accurate, atomistic constitutive models and the use of realistically sized indenters in the simulations.

Image Decomposition Using Total Variation and div(BMO)

Triet M. Le and Luminita A. Vese

Multiscale Model. Simul. 4, pp. 390-423 (34 pages) | Cited 13 times

Online Publication Date: July 26, 2006

Full Text: | Download PDF

Show Abstract
This paper is devoted to the decomposition of an image f into u + v, with u a piecewise-smooth or "cartoon" component, and v an oscillatory component (texture or noise), in a variational approach. Meyer [Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, Univ. Lecture Ser. 22, AMS, Providence, RI, 2001] proposed refinements of the total variation model (Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259--268]) that better represent the oscillatory part v: the spaces of generalized functions $G = {\rm div}(L^\infty)$ and $F = {\rm div}(BMO)$ (this last space arises in the study of Navier--Stokes equations; see Koch and Tataru [Adv. Math., 157 (2001), pp. 22--35]) have been proposed to model v, instead of the standard L2 space, while keeping u a function of bounded variation. Mumford and Gidas [Quart. Appl. Math., 59 (2001), pp. 85--111] also show that natural images can be seen as samples of scale-invariant probability distributions that are supported on distributions only and not on sets of functions. However, there is no simple solution to obtain in practice such decompositions f = u + v when working with G or F. In earlier works [L. Vese and S. Osher, J. Sci. Comput., 19 (2003), pp. 553--572], [L. A. Vese and S. J. Osher, J. Math. Imaging Vision, 20 (2004), pp. 7--18], [S. Osher, A. Solé, and L. Vese, Multiscale Model. Simul., 1 (2003), pp. 349--370], the authors have proposed approximations to the (BV,G) decomposition model, where the $L^\infty$ space has been substituted by Lp, $1 \leq p < \infty$. In the present paper, we introduce energy minimization models to compute (BV,F) decompositions, and as a by-product we also introduce a simple model to realize the (BV,G) decomposition. In particular, we investigate several methods for the computation of the BMO norm of a function in practice. Theoretical, experimental results and comparisons to validate the proposed new methods are presented.

From Atomic Scale Ordering to Mesoscale Spatial Patterns in Surface Reactions: A Heterogeneous Coupled Lattice-Gas Simulation Approach

Da-Jiang Liu and J. W. Evans

Multiscale Model. Simul. 4, pp. 424-446 (23 pages) | Cited 13 times

Online Publication Date: July 26, 2006

Full Text: | Download PDF

Show Abstract
A current challenge in the modeling of surface reaction-diffusion systems is to connect the length scales from a realistic atomistic treatment of local ordering of adsorbed reactants and of reaction kinetics to an "exact" description of mesoscale spatial pattern formation. We discuss a heterogeneous coupled lattice-gas approach which utilizes parallel kinetic Monte Carlo simulations of lattice-gas models to simultaneously determine the local reaction kinetics and diffusive transport properties at various macroscopic "points" distributed across the surface. These simulations are periodically coupled to reflect macroscopic mass transport via surface diffusion. We place particular emphasis on the key issue of correctly describing the associated chemical or collective diffusion flux. This method is general, but it is discussed here only in the context of simple models for the bistable CO oxidation reaction on surfaces, where macroscopic mass transport involves only diffusion of adsorbed CO. Detailed analysis is presented for a canonical model of CO oxidation which incorporates nearest-neighbor exclusion between adsorbed oxygen, leading to superlattice ordering of this reactant. In this model, interactions of adsorbed CO with other adspecies are ignored. This reduces CO surface transport from a many-particle to a single-particle problem, although one with the complexity of "ant-in-the-labyrinth"-type percolative diffusion (the labyrinth being formed by coadsorbed oxygen).

On A Priori Error Analysis of Fully Discrete Heterogeneous Multiscale FEM

Assyr Abdulle

Multiscale Model. Simul. 4, pp. 447-459 (13 pages) | Cited 14 times

Online Publication Date: July 26, 2006

Full Text: | Download PDF

Show Abstract
Heterogeneous multiscale methods have been introduced by E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87--132] as a methodology for the numerical computation of problems with multiple scales. Analyses of the methods for various homogenization problems have been done by several authors. These results were obtained under the assumption that the microscopic models (the cell problems in the homogenization context) are analytically given. For numerical computations, these microscopic models have to be solved numerically. Therefore, it is important to analyze the error transmitted on the macroscale by discretizing the fine scale. We give in this paper H1 and L2 a priori estimates of the fully discrete heterogeneous multiscale finite element method. Numerical experiments confirm that the obtained a priori estimates are sharp.

An Iterative Regularization Method for Total Variation-Based Image Restoration

Stanley Osher, Martin Burger, Donald Goldfarb, Jinjun Xu, and Wotao Yin

Multiscale Model. Simul. 4, pp. 460-489 (30 pages) | Cited 121 times

Online Publication Date: July 26, 2006

Full Text: | Download PDF

Show Abstract
We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total variation regularization. We obtain rigorous convergence results and effective stopping criteria for the general procedure. The numerical results for denoising appear to give significant improvement over standard models, and preliminary results for deblurring/denoising are very encouraging.

A Review of Image Denoising Algorithms, with a New One

A. Buades, B. Coll, and J. M. Morel

Multiscale Model. Simul. 4, pp. 490-530 (41 pages) | Cited 167 times

Online Publication Date: July 26, 2006

Full Text: | Download PDF

Show Abstract
The search for efficient image denoising methods is still a valid challenge at the crossing of functional analysis and statistics. In spite of the sophistication of the recently proposed methods, most algorithms have not yet attained a desirable level of applicability. All show an outstanding performance when the image model corresponds to the algorithm assumptions but fail in general and create artifacts or remove image fine structures. The main focus of this paper is, first, to define a general mathematical and experimental methodology to compare and classify classical image denoising algorithms and, second, to propose a nonlocal means (NL-means) algorithm addressing the preservation of structure in a digital image. The mathematical analysis is based on the analysis of the "method noise," defined as the difference between a digital image and its denoised version. The NL-means algorithm is proven to be asymptotically optimal under a generic statistical image model. The denoising performance of all considered methods are compared in four ways; mathematical: asymptotic order of magnitude of the method noise under regularity assumptions; perceptual-mathematical: the algorithms artifacts and their explanation as a violation of the image model; quantitative experimental: by tables of L2 distances of the denoised version to the original image. The most powerful evaluation method seems, however, to be the visualization of the method noise on natural images. The more this method noise looks like a real white noise, the better the method.

Derivation of Higher Order Gradient Continuum Models from Atomistic Models for Crystalline Solids

M. Arndt and M. Griebel

Multiscale Model. Simul. 4, pp. 531-562 (32 pages) | Cited 15 times

Online Publication Date: July 26, 2006

Full Text: | Download PDF

Show Abstract
We propose a new upscaling scheme for the passage from atomistic to continuum mechanical models for crystalline solids. It is based on a Taylor expansion of the deformation function and allows us to capture the microscopic properties and the discreteness effects of the underlying atomistic system up to an arbitrary order. The resulting continuum mechanical model involves higher order terms and gives a description of the specimen within the quasi-continuum regime. Furthermore, the convexity of theatomistic potential is retained, which leads to well-posed evolution equations. We numerically compare our technique to other common upscaling schemes for the example of an atomic chain. Then we apply our approach to a physically more realistic many-body potential of crystalline silicon. Here the above-mentioned advantages of our technique hold for the newly obtained macroscopic model as well.

A Projective Thermostatting Dynamics Technique

Zhidong Jia and Benedict J. Leimkuhler

Multiscale Model. Simul. 4, pp. 563-583 (21 pages) | Cited 4 times

Online Publication Date: July 26, 2006

Full Text: | Download PDF

Show Abstract
A dynamical framework is developed with several variations for modeling multiple timescale molecular dynamics at constant temperature. The described approach can be adapted to various applications, including mixtures of heavy and light particles and models with stiff potentials. Canonical sampling properties are proved under the ergodicity assumption. Implications for numerical method development are discussed, and the technique is validated in numerical experiments with model problems, including a simple model of a diatomic gas with anharmonic weak interaction.

Image Registration Based on Multiscale Energy Information

Stefan Henn and Kristian Witsch

Multiscale Model. Simul. 4, pp. 584-609 (26 pages) | Cited 3 times

Online Publication Date: July 26, 2006

Full Text: | Download PDF

Show Abstract
We propose a novel multiscale approach for the image registration problem, i.e., to find a deformation that maps one image onto another. The image registration problem is confirmed to be mathematical ill-posed due to the fact that determining the unknown components of the displacements merely from the images is an underdetermined problem. The approach presented here utilizes an auxiliary regularization term based on the energy of a plate with free edges, which incorporates smoothness constraints into the deformation field. One of the important aspects of this approach is that the energy does not penalize affine-linear functions. Consequently, the kernel of the Euler--Lagrange equation is spanned by all rigid motions. Hence, the presented approach is invariant under planar rotation and translation. In order to find an optimal deformation, we solve a sequence of subproblems with decreasing regularization parameter. In this framework the regularization parameter can be regarded as a scale parameter, which captures information at multiple spatial scales. We analyze the multiscale nature of a solution.

Imaging in Randomly Layered Media by Cross-Correlating Noisy Signals

Josselin Garnier

Multiscale Model. Simul. 4, pp. 610-640 (31 pages) | Cited 6 times

Online Publication Date: July 26, 2006

Full Text: | Download PDF

Show Abstract
We consider an active source embedded in a randomly layered medium. We study the cross-correlation functions of the signals recorded at a series of points located at the surface. We show that this information can be processed to locate and identify the source inside the medium. The analysis is based on a separation of scales technique and limit theorems for random differential equations. The statistical stability of the imaging method is proved. The analogy with the time reversal of waves is enlightened, but the main difference is also put forward: we propose a passive way of imaging an unknown medium without the use of any active device. Finally, we extend these ideas for the location of a scatterer illuminated by a controlled source located at the surface or by a set of unknown sources generating random noise.

Construction of Projection Operators for Nonlinear Evolutionary Dynamics

Andreas Degenhard and Javier Rodríguez-Laguna

Multiscale Model. Simul. 4, pp. 641-663 (23 pages) | Cited 1 time

Online Publication Date: July 26, 2006

Full Text: | Download PDF

Show Abstract
The concept of multiresolution analysis is applied to analyze the interference of scales in the reduction of the degrees of freedom for nonlinear partial differential equations. The selection of the relevant degrees of freedom is performed within a rigorous and systematic construction of projection operators in the context of renormalization group theory and homogenization. Using the representation of operators in a basis of compactly supported wavelets, we explicitly calculate the contributions from each scale for every order in the reduction of the degrees of freedom. Contrarily to representations of the projection operators in real space and Fourier space, we show that the selection of the relevant degrees of freedom follows a certain pattern in the off-diagonal elements in the wavelet representation. Finally, the wavelet representation allows us to identify each relevant degree of freedom for the Burgers nonlinearity within each scale explicitly.

A Two-Dimensional Nonlinear Nonlocal Feed-Forward Cochlear Model and Time Domain Computation of Multitone Interactions

Yongsam Kim and Jack Xin

Multiscale Model. Simul. 4, pp. 664-690 (27 pages) | Cited 5 times

Online Publication Date: July 26, 2006

Full Text: | Download PDF

Show Abstract
A two-dimensional cochlear model is presented, which couples the classical second order partial differential equations of the basilar membrane (BM) with a discrete feed-forward (FF) outer hair cell (OHC) model for enhanced sensitivity. The enhancement (gain) factor in the model depends on BM displacement in a nonlinear nonlocal manner in order to capture multifrequency sound interactions and compression effects in a time-dependent simulation. The FF mechanism is based on the longitudinal tilt of the OHCs in feeding the mechanical energy onto the BM. A numerical method of second order accuracy in space and time is formulated by reducing the unknown variables to the BM with the representation of eigenfunction expansions. Though the nonlinear coupling with OHCs created an implicit algebraic problem at each time step, the structure of the FF mass matrix is found to permit a decomposition into a sum of a time-independent symmetric positive definite part and the remaining time-dependent part. A fast iterative method is devised and shown to converge with only the inversion of the time-independent part of the mass matrix. The time-dependent computation is studied by comparing with steady state solutions (frequency domain solutions) in the linear regime and by a convergence study in the nonlinear regime. Results are shown on OHC amplification of BM responses, compressive output for large intensity input, and nonlinear multitone interactions such as tonal suppression, noise suppression, and distortion products. Qualitative agreement with experimental data is observed.

An Input/Output Model Reduction-Based Optimization Scheme for Large-Scale Systems

Eduardo Luna-Ortiz and Constantinos Theodoropoulos

Multiscale Model. Simul. 4, pp. 691-708 (18 pages) | Cited 6 times

Online Publication Date: July 26, 2006

Full Text: | Download PDF

Show Abstract
A reduced model based optimization strategy is presented for the cases where input/output codes are the process simulators of choice, and thus system Jacobians and even system equations are not explicitly available to the user. The former is the case when commercial software packages or legacy codes are used to simulate a large-scale system and the latter when microscopic or multiscale simulators are employed. When such black-box dynamic simulators are used, we perform optimization by combining the recursive projection method [G. M. Shroff and H. B. Keller, SIAM J. Numer. Anal., 30 (1993), pp. 1099--1120] which identifies the (typically) low-dimensional slow dynamics of the (dissipative) model with a second reduction to the low-dimensional subspace of the decision variables. This results in the solution of a low-order unconstrained optimization problem. Optimal conditions are then computed in an efficient way using only low-dimensional numerical approximations of gradients and Hessians. The tubular reactor is used as an illustrative example to demonstrate this model reduction-based optimization methodology.
Close

Your Recent History Recent History Help

Recently Viewed

  1. A Simple Linear Response Closure Approximation for Slow Dynamics of a Multiscale System with Linear ...
  2. Importance Sampling for Multiscale Diffusions
  3. Indefinite-Quadratic Estimation and Control
  4. The Art of Differentiating Computer Programs
  5. Numerical Methods for Special Functions
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close