Analysis of the Residual-Type and the Gradient Recovery-Type a Posteriori Error Estimators for a Consistent Atomistic-to-Continuum Coupling Method in One-Dimension

Wang, Hao; Yang, Siyao

doi:10.1137/17m1118579

Cited by 6 publications

(4 citation statements)

References 59 publications

(161 reference statements)

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“…for QM/MM schemes may require more involved constructions and analysis, and will be investigated in our future work. See [32] for a recent advance in this direction.…”

Section: A Practical a Posteriori Error Indicatormentioning

confidence: 99%

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Adaptive QM/MM coupling for crystalline defects

Chen

Liao

Wang

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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QM (quantum mechenics) and MM (molecular mechenics) coupling methods are widely used in simulations of crystalline defects. In this paper, we construct a residual based a posteriori error indicator for QM/MM coupling approximations. We prove the reliability of the error indicator (upper bound of the true approximation error) and develop some sampling techniques for its efficient calculation. Based on the error indicator and Dörfler marking strategy, we design an adaptive QM/MM algorithm for crystalline defects and demonstrate the efficiency with some numerical experiments.keywords. qm/mm coupling, a posteriori error estimate, adaptive algorithm, crystal defects AMS subject classifications.

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“…for QM/MM schemes may require more involved constructions and analysis, and will be investigated in our future work. See [32] for a recent advance in this direction.…”

Section: A Practical a Posteriori Error Indicatormentioning

confidence: 99%

“…The design of reliable and efficient error indicator (with both upper and lower bound estimates) for QM/MM schemes may require more involved constructions and analysis, and will be investigated in our future work. See [32] for a recent advance in this direction.…”

Section: A Practical a Posteriori Error Indicatormentioning

confidence: 99%

Adaptive QM/MM coupling for crystalline defects

Chen

Liao

Wang

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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“…We expect higher order models can be used to alleviate this problem. Once the coupling model is developed, the study of adaptivity should be under way where [1,2,22,30,32,31] should provide good references.…”

Section: Numerical Experimentsmentioning

confidence: 99%

A Priori Analysis of a Higher Order Nonlinear Elasticity Model for an Atomistic Chain with Periodic Boundary Condition

Wang¹,

Wang²,

Zhang³

2019

Preprint

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Nonlinear elastic models are widely used to describe the elastic response of crystalline solids, for example, the well-known Cauchy-Born model. While the Cauchy-Born model only depends on the strain, effects of higher order strain gradients are significant and higher order continuum models are preferred, in various applications such as defect dynamics and modeling of carbon nanotubes. In this paper, we rigorously derive a higher order nonlinear elasticity model for crystals from its atomistic description in one dimension. We show that, compared to the second order accuracy of the Cauchy-Born model, the higher order continuum model in this paper is of fourth oder accuracy with respect to the interatomic spacing in the thermal dynamic limit. In addition, we discuss the key issues for the derivation of higher order continuum models in more general cases. The theoretical convergence results are demonstrated by numerical experiments.

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“…Using a weighted 2 -norm on the QM/MM force error leads to a simple and practical scheme, but makes it impossible to guarantee lower bounds on the error, which is important to guarantee the efficiency of the algorithm. There have also been investigations of related a posteriori error estimates for atomistic/continuum (A/C) coupling methods, which share many similarities [2,33,37,42,31,43,48,28,49].…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Error Estimates for Adaptive QM/MM Coupling Methods

Wang¹,

Chen²,

Liao³

et al. 2020

Preprint

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Hybrid quantum/molecular mechanics models (QM/MM methods) are widely used in material and molecular simulations when MM models do not provide sufficient accuracy but pure QM models are computationally prohibitive. Adaptive QM/MM coupling methods feature on-the-fly classification of atoms during the simulation, allowing the QM and MM subsystems to be updated as needed. In this work, we propose such an adaptive QM/MM method for material defect simulations based on a new residual based a posteriori error estimator, which provides both lower and upper bounds for the true error. We validate the analysis and illustrate the effectiveness of the new scheme on numerical simulations for material defects.

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Analysis of the Residual-Type and the Gradient Recovery-Type a Posteriori Error Estimators for a Consistent Atomistic-to-Continuum Coupling Method in One-Dimension

Cited by 6 publications

References 59 publications

Adaptive QM/MM coupling for crystalline defects

Adaptive QM/MM coupling for crystalline defects

A Priori Analysis of a Higher Order Nonlinear Elasticity Model for an Atomistic Chain with Periodic Boundary Condition

A Posteriori Error Estimates for Adaptive QM/MM Coupling Methods

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