Atomistic to Continuum limits for computational materials science

Blanc, Xavier; Bris, Claude Le; Lions, Pierre Louis

doi:10.1051/m2an:2007018

Cited by 60 publications

(52 citation statements)

References 108 publications

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“…It will be clear from the presentation that similar analysis carries over to the case of any finite range interaction (see [11] for details). We refer to [19,8,9,1,2,20,4,26] for related work on the analysis of the QC method.…”

Section: Error Estimates For the Geometrically Consistent Quasicontinuummentioning

confidence: 99%

Analysis of a One-Dimensional Nonlocal Quasi-Continuum Method

Ming¹,

Yang²

2009

Multiscale Model. Simul.

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Abstract. The accuracy of the quasicontinuum method is analyzed using a series of models with increasing complexity. It is demonstrated that the existence of the ghost force may lead to large errors. It is also shown that the ghost force removal strategy proposed by E, Lu and Yang leads to a version of the quasicontinuum method with uniform accuracy.Key words. Quasicontinuum method, ghost force, geometrically consistent scheme AMS subject classifications. 65N12, 65N06, 74G20, 74G151. Introduction. The quasicontinuum (QC) method [33] is among the most successful multiscale methods for modeling the mechanical deformation of crystalline solids. It is designed to deal with situations when the crystal is undergoing mostly elastic deformation except at isolated regions with defects. The QC method is usually formulated as an adaptive finite element method. But instead of relying on a continuum model, the QC method is based on an atomistic model. Its main ingredients are: adaptive selection of representative atoms (rep-atoms), with fewer atoms selected in regions with smooth deformation; division of the whole sample into local and nonlocal regions, with the defects covered by the nonlocal regions; and the application of the Cauchy-Born (CB) approximation in the local region as a device for reducing the complexity involved in computing the total energy of the system.The quasicontinuum method has several distinct advantages. First of all, it has a reasonably simple formulation. In fact, it can be considered as a natural extension of adaptive finite element methods in which one simply uses the atomistic model where the mesh is refined to the atomic scale. Secondly, in the QC method, the treatment in different regions is based on the same model, the atomistic model, with the additional Cauchy-Born approximation used in the local region. For this reason, it is also considered to be more seamless than methods that are based on an explicit coupling between continuum and atomistic models. We refer to the review articles [6,21] for a discussion of methods that are based on explicitly coupling atomistic and continuum models.However, this does not mean that the QC method is free of the problems that one encounters when formulating coupled atomistic-continuum methodologies. In some sense, one may also regard the QC method as an example of such a strategy, with the local region playing the role of the continuum region, and the Cauchy-Born nonlinear elasticity model playing the role of the continuum model. In particular, the issue of consistency between the continuum and atomistic models across the coupling interface is very much manifested in the accuracy at the local-nonlocal interface for the QC method. This is the issue we will focus on in this paper. In fact, even though the atomistic models are used in both the local and the nonlocal regions, the CauchyBorn approximation made in the local regions means that the effective model in this

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Section: Error Estimates For the Geometrically Consistent Quasicontinuummentioning

confidence: 99%

Analysis of a One-Dimensional Nonlocal Quasi-Continuum Method

Ming¹,

Yang²

2009

Multiscale Model. Simul.

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“…We call the energy Ψ cb the fully continuum Cauchy-Born energy. We refer the reader to [8] for a detailed discussion of the convergence of Φ(y ǫ ) to Ψ cb (Y ) as ǫ goes to zero. Now suppose that Y is in the Lagrange P 1 finite element space on [0, 1] with nodes at the reference position of each atom; so the nodes are ǫξ for ξ = 1, .…”

Section: Introductionmentioning

confidence: 99%

Analysis of Energy-Based Blended Quasi-Continuum Approximations

Koten¹,

Luskin²

2011

SIAM J. Numer. Anal.

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Abstract. The development of patch test consistent quasicontinuum energies for multi-dimensional crystalline solids modeled by many-body potentials remains a challenge. The original quasicontinuum energy (QCE) [28] has been implemented for many-body potentials in two and three space dimensions, but it is not patch test consistent. We propose that by blending the atomistic and corresponding Cauchy-Born continuum models of QCE in an interfacial region with thickness of a small number k of blended atoms, a general quasicontinuum energy (BQCE) can be developed with the potential to significantly improve the accuracy of QCE near lattice instabilities such as dislocation formation and motion.In this paper, we give an error analysis of the blended quasicontinuum energy (BQCE) for a periodic one-dimensional chain of atoms with next-nearest neighbor interactions. Our analysis includes the optimization of the blending function for an improved convergence rate. We show that the ℓ 2 strain error for the non-blended QCE energy (QCE), which has low order O(ε 1/2 ) where ε is the atomistic length scale [13,29], can be reduced by a factor of k 3/2 for an optimized blending function where k is the number of atoms in the blending region. The QCE energy has been further shown to suffer from a O(1) error in the critical strain at which the lattice loses stability [16]. We prove that the error in the critical strain of BQCE can be reduced by a factor of k 2 for an optimized blending function, thus demonstrating that the BQCE energy for an optimized blending function has the potential to give an accurate approximation of the deformation near lattice instabilities such as crack growth.

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“…In principle, the AtC transition could be done by taking the homogenization limit in the atomistic model, as lattice spacing tends to 0 (for a review of rigorous results on AtC limit, see, e.g., [17,18,20,24]). However, this problem is presently intractable because of the complex form of the atomistic energy (1); in particular, the constant ε in the Lennard-Jones potential can have strong variations even for neighboring atoms.…”

Section: Inextensible Constraint and Boundary Conditionsmentioning

confidence: 99%

Global Energy Matching Method for Atomistic-to-Continuum Modeling of Self-Assembling Biopolymer Aggregates

Zhang

Berlyand

Fedorov

et al. 2010

Multiscale Model. Simul.

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Abstract. This paper studies mathematical models of biopolymer supramolecular aggregates that are formed by the self-assembly of single monomers. We develop a new multiscale numerical approach to model the structural properties of such aggregates. This theoretical approach establishes micro-macro relations between the geometrical and mechanical properties of the monomers and supramolecular aggregates. Most atomistic-to-continuum methods are constrained by a crystalline order or a periodic setting and therefore cannot be directly applied to modeling of soft matter. By contrast, the energy matching method developed in this paper does not require crystalline order and, therefore, can be applied to general microstructures with strongly variable spatial correlations. In this paper we use this method to compute the shape and the bending stiffness of their supramolecular aggregates from known chiral and amphiphilic properties of the short chain peptide monomers. Numerical implementation of our approach demonstrates consistency with results obtained by molecular dynamics simulations.Key words. atomistic-to-continuum, self-assembly, biopolymer aggregates, multiscale AMS subject classifications. 75L15, 74Q05, 65Z05 DOI. 10.1137/090781619 1. Introduction. Fast development of bio-and nanosciences requires new methods for theoretical investigations of structural properties of supramolecular aggregates consisting of thousands to millions of atoms. However, the fully atomistic treatment of such big aggregates remains a challenge. These days, the most promising strategy for computer modeling of nano-sized supramolecular aggregates is the multiscale approach, e.g., [29,39]. The fundamental idea behind the multiscale approach is that detailed information from the atomistic scale is systematically coarsegrained and then used for continuous modeling of the aggregates in a self-consistent way.Using the multiscale approach, we have developed a new low-cost model for simulating the mechanical properties of supramolecular tapes formed by the self-assembling oligopeptides (a short chain of peptide) [1,2,4,3]. These oligopeptides can form regular tapelike and rodlike nanoaggregates in aqueous solutions [1,2,4,3]. The aggregates are important as promising "smart" biodegradable nanomaterials [52] because one could easily assemble and disassemble them by varying the solvent conditions (pH, temperature, etc.) [4,3]. These self-assembled systems are also important as model

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Atomistic to Continuum limits for computational materials science

Cited by 60 publications

References 108 publications

Analysis of a One-Dimensional Nonlocal Quasi-Continuum Method

Analysis of a One-Dimensional Nonlocal Quasi-Continuum Method

Analysis of Energy-Based Blended Quasi-Continuum Approximations

Global Energy Matching Method for Atomistic-to-Continuum Modeling of Self-Assembling Biopolymer Aggregates

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