Theory-based benchmarking of the blended force-based quasicontinuum method☆

@article{Li2014TheorybasedBO,
  title={Theory-based benchmarking of the blended force-based quasicontinuum method☆},
  author={Xingjie Helen Li and Mitchell Luskin and Christoph Ortner and Alexander V. Shapeev},
  journal={Computer Methods in Applied Mechanics and Engineering},
  year={2014},
  volume={268},
  pages={763-781}
}
  • X. Li, M. Luskin, A. Shapeev
  • Published 4 April 2013
  • Physics
  • Computer Methods in Applied Mechanics and Engineering
Analysis of blended atomistic/continuum hybrid methods
TLDR
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 is presented.
Force-based atomistic/continuum blending for multilattices
TLDR
Balancing the approximation parameters yields a convergent atomistic/continuum multiscale method for multilattices with point defects, including a rigorous convergence rate in terms of the computational cost.
Atomistic/Continuum Blending with Ghost Force Correction
TLDR
An energy-based atomistic/continuum coupling scheme which has, for a range of benchmark problems, the same convergence rates as optimal force-based coupling schemes.
Analysis of an optimization-based atomistic-to-continuum coupling method for point defects
We formulate and analyze an optimization-based Atomistic-to-Continuum (AtC) coupling method for problems with point defects. Application of a potential-based atomistic model near the defect core
Development of an Optimization-Based Atomistic-to-Continuum Coupling Method
TLDR
This note conjecture optimal error estimates for the multidimensional AtC, outline an implementation procedure, and provide numerical results to corroborate the conjecture for a 1D Lennard-Jones system with next-nearest neighbor interactions.
Blended Ghost Force Correction Method for 3D Crystalline Defects
TLDR
This paper presents the formulation, implementation and analysis of the BGFC method in three dimensions, and focuses on the difference and connection with other blending variants, such as energy based blended quasi-continuum method (BQCE) and force based blended semi-continuity method ( BQCF).
Atomistic-to-continuum coupling
TLDR
A rigorous numerical analysis approach that classifies and quantifies approximation errors in the construction of a/c coupling methods can give confidence in the simulation results, as well as enable optimization of the numerical methods for accuracy and computational cost.
Energy-based atomistic-to-continuum coupling without ghost forces
A Posteriori Error Control for Three Typical Force-Based Atomistic-to-Continuum Coupling Methods for an Atomistic Chain
  • Hao Wang
  • Computer Science
    Numerical Mathematics: Theory, Methods and Applications
  • 2019
TLDR
Combining the residual and the stability estimates, the computable a posteriori error estimators for the three typical force-based atomistic-to-continuum coupling methods are derived and adaptive algorithms using these estimators are formulated.
The Quasicontinuum Method: Theory and Applications
TLDR
The theoretical fundamentals are reviewed and the state of the art in QC theory, computational methods, and applications are given, particularly the fully nonlocal QC formulation which adaptively ties atomistic resolution to moving defects, and simulation results based on this framework are illustrated.
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References

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Analysis of blended atomistic/continuum hybrid methods
TLDR
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 is presented.
A Force-Based Blending Model forAtomistic-to-Continuum Coupling
A method for coupling atomistic and continuum models across a subdomain, or bridge region, is presented. Coupling is effected through a force-based blending model. The method properly accounts for
Consistent Energy-Based Atomistic/Continuum Coupling for Two-Body Potentials in Three Dimensions
TLDR
The main achievement of this work is in identifying and efficiently treating a modified Cauchy--Born continuum model which can be coupled to the exact atomistic model.
Analysis of an Energy-based Atomistic/Continuum Coupling Approximation of a Vacancy in the 2D Triangular Lattice
We present a comprehensive a priori error analysis of a practical energy based atomistic/continuum coupling method (Shapeev, arXiv:1010.0512) in two dimensions, for finite-range pair-potential
Analysis of an energy-based atomistic/continuum approximation of a vacancy in the 2D triangular lattice
TLDR
An a priori error analysis of a practical energy based atomistic/continuum coupling method in two dimensions, for finite-range pair-potential interactions, in the presence of vacancy defects, establishes first-order consistency and stability of the method.
Positive Definiteness of the Blended Force-Based Quasicontinuum Method
TLDR
In one and two dimensions, it is shown 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.
Analysis of Energy-Based Blended Quasi-Continuum Approximations
TLDR
An error analysis of the blended quasi-continuum energy (BQCE) for a periodic one-dimensional chain of atoms with next-nearest neighbor interactions is given, 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.
Iterative Methods for the Force-based Quasicontinuum Approximation
Atomistic-to-continuum coupling
TLDR
A rigorous numerical analysis approach that classifies and quantifies approximation errors in the construction of a/c coupling methods can give confidence in the simulation results, as well as enable optimization of the numerical methods for accuracy and computational cost.
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