Computational analysis of modeling error for the coupling of particle and continuum models by the Arlequin method

Prudhomme, Serge; Dhia, Hachmi Ben; Bauman, Paul T.; Elkhodja, Nadia; Oden, J. Tinsley

doi:10.1016/j.cma.2008.03.014

Cited by 68 publications

(51 citation statements)

References 16 publications

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“…While our results are not be immediately applicable to these related schemes [41,2,1,3,14,22,35,37], we expect that many of the techniques we develop can be employed to develop such extensions.…”

Section: Introductionmentioning

confidence: 93%

“…Precisely, we will consider (i) the B-QCE scheme formulated in [38,26], which is closely related to methods proposed in [41,2,1]; and (ii) the B-QCF scheme formulated in [24,20,21], which is closely related to methods proposed in [1,2,3,14,22,35,37,41]. While our results are not be immediately applicable to these related schemes [41,2,1,3,14,22,35,37], we expect that many of the techniques we develop can be employed to develop such extensions.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of blended atomistic/continuum hybrid methods

et al. 2015

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Abstract. We present a comprehensive error analysis of two prototypical atomistic-tocontinuum coupling methods of blending type: the energy-based and the force-based quasicontinuum methods.Our results are valid in two and three dimensions, for finite range many-body interactions (e.g., EAM type), and in the presence of lattice defects (we consider point defects and dislocations). The two key ingredients in the analysis are (i) new force and energy consistency error estimates; and (ii) a new technique for proving energy norm stability of a/c couplings that requires only the assumption that the exact atomistic solution is a stable equilibrium.

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Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Analysis of blended atomistic/continuum hybrid methods

et al. 2015

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“…Many blended a/c coupling methods have been proposed in the literature; see, e.g., [1,2,3,4,17,22,33,34,40]. In the present work, we formulate a blended force-based quasicontinuum (B-QCF) method, similar to the method proposed in [23], which smoothly blends the forces of the atomistic and continuum model instead of the sharp transition in the QCF method.…”

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confidence: 99%

Positive Definiteness of the Blended Force-Based Quasicontinuum Method

Li¹,

Luskin²,

Ortner³

2012

Multiscale Model. Simul.

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Abstract. The development of consistent and stable quasicontinuum models for multidimensional crystalline solids remains a challenge. For example, proving the stability of the force-based quasicontinuum (QCF) model [M. Dobson and M. Luskin, M 2AN Math. Model. Numer. Anal., 42 (2008), pp. 113-139] remains an open problem. In one and two dimensions, we show that by blending atomistic and Cauchy-Born continuum forces (instead of a sharp transition as in the QCF method) one obtains positive-definite blended force-based quasicontinuum (B-QCF) models. We establish sharp conditions on the required blending width.Key words. quasicontinuum, atomistic-to-continuum, blending, stability AMS subject classifications. 65Z05, 70C20 DOI. 10.1137/1108592701. Introduction. Atomistic-to-continuum (a/c) coupling methods have been proposed to increase the computational efficiency of atomistic computations involving the interaction between local crystal defects with long-range elastic fields [6,7,16,19,21,24,25,37]. Energy-based methods in this class, such as the quasicontinuum model (denoted QCE [38]), exhibit spurious interfacial forces ("ghost forces") even under uniform strain [8,36]. The effect of the ghost force on the error in computing the deformation and the lattice stability by the QCE approximation has been analyzed in [8,9,11,26]. The development of more accurate energy-based a/c methods is an ongoing process [5,16,20,31,35,37].An alternative approach to a/c coupling is the force-based quasicontinuum (QCF) approximation [7,12,13,23,24], but the nonconservative and indefinite equilibrium equations make the iterative solution and the determination of lattice stability more challenging [13,14,15]. Indeed, it is an open problem whether the (sharp-interface) QCF method is stable in dimensions greater than one.Many blended a/c coupling methods have been proposed in the literature; see, e.g., [1,2,3,4,17,22,33,34,40]. In the present work, we formulate a blended force-based quasicontinuum (B-QCF) method, similar to the method proposed in [23], which smoothly blends the forces of the atomistic and continuum model instead of the sharp transition in the QCF method. In one and two dimensions, we establish sharp conditions under which a linearized B-QCF operator is positive definite.Our results have three advantages over the stability result proven in [23]. First, we establish H 1 -stability (instead of H 2 -stability), which opens up the possibility of including defects in the analysis, along the lines of [15,29]. Second, our conditions for

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“…The Bridging Domain method (BDM) uses the concepts of the Arlequin approach [65][66][67][68] which can intermix energies of several continuum mechanical models and constrain consistent displacements within an overlaping zone (also termed as the bridging domain). Xiao et al applied this strategy for coupling continuum models with molecular dynamics (MD) [49,69].…”

Section: State Of the Art Of Multiscale Methodsmentioning

confidence: 99%

MD/FE Multiscale Modeling of Contact

Ramisetti

Anciaux

Molinari

2014

Fundamentals of Friction and Wear on the Nanoscale

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Limitations of single scale approaches to study the complex physics involved in friction have motivated the development of multiscale models. We review the state-of-the-art multiscale models that have been developed up to date. These have been successfully applied to a variety of physical problems, but that were limited, in most cases, to zero Kelvin studies. We illustrate some of the technical challenges involved with simulating a frictional sliding problem, which by nature generates a large amount of heat. These challenges can be overcome by a proper usage of spatial filters, which we combine to a direct finite-temperature multiscale approach coupling molecular dynamics with finite elements. The basic building block relies on the proper definition of a scale transfer operator using the least square minimization and spatial filtering. Then, the restitution force from the generalized Langevin equation is modified to perform a two-way thermal coupling between the two models. Numerical examples are shown to illustrate the proposed coupling formulation.

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Computational analysis of modeling error for the coupling of particle and continuum models by the Arlequin method

Cited by 68 publications

References 16 publications

Analysis of blended atomistic/continuum hybrid methods

Analysis of blended atomistic/continuum hybrid methods

Positive Definiteness of the Blended Force-Based Quasicontinuum Method

MD/FE Multiscale Modeling of Contact

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