Numerical Methods for Multilattices

@article{Abdulle2012NumericalMF,
  title={Numerical Methods for Multilattices},
  author={Assyr Abdulle and Ping Lin and Alexander V. Shapeev},
  journal={Multiscale Model. Simul.},
  year={2012},
  volume={10},
  pages={696-726}
}
Among the efficient numerical methods based on atomistic models, the quasi-continuum (QC) method has attracted growing interest in recent years. The QC method was first developed for crystalline materials with Bravais lattice and was later extended to multilattices [Tadmor et al., Phys. Rev. B, 59 (1999), pp. 235--245]. Another existing numerical approach to modeling multilattices is homogenization. In the present paper we review the existing numerical methods for multilattices and propose… 
View PDF on arXiv
8 Citations
Highly Influential Citations
1
Background Citations
3
Methods Citations
2

8 Citations

Citation Type

Citation Type

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.
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.
Finite Element Analysis of Cauchy–Born Approximations to Atomistic Models
This paper is devoted to a new finite element consistency analysis of Cauchy–Born approximations to atomistic models of crystalline materials in two and three space dimensions. Through this approach…
Formulation and optimization of the energy-based blended quasicontinuum method
An Atomistic/Continuum Coupling Method Using Enriched Bases
TLDR
This paper demonstrates that the error at the interface can be considerably reduced when new basis functions are introduced in a Galerkin projection formalism, and as a consequence, the overall accuracy is improved.
An efficient multigrid strategy for large-scale molecular mechanics optimization
A temporal multiscale method and its analysis for a system of fractional differential equations
TLDR
This paper analyzes a multiscale method for the nonlinear system where the fast system has a periodic applied force and the slow equation contains fractional derivatives as a simplication of the atherosclerosis with a plaque growth and finds that the larger the time scale separation is, the more effective the multiscales method is.
A Priori and A Posteriori W1, INFINITY Error Analysis of a QC Method for Complex Lattices
TLDR
A priori and a posteriori error estimates are proved for a multiscale numerical method for computing equilibria of multilattices under an external force and they are derived in a $W^{1,\infty}$ norm in one space dimension.

References

SHOWING 1-10 OF 74 REFERENCES
Homogenization-based Analysis of Quasicontinuum Method for Complex Crystals
Among the efficient numerical methods based on atomistic models, the quasicontinuum (QC) method, introduced by Tadmor, Ortiz, and Phillips (1996), has attracted growing interest in recent years.…
On a formulation for a multiscale atomistic-continuum homogenization method
Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model
  • P. Lin
  • Mathematics
    Math. Comput.
  • 2003
TLDR
The approximate error between the solution of the simplified problem and that of the full-interaction problem is analyzed so as to answer the question mathematically for a one-dimensional model.
A class of general algorithms for multi-scale analyses of heterogeneous media
Convergence of a force-based hybrid method for atomistic and continuum models in three dimension
We study a force-based hybrid method that couples atomistic models with nonlinear Cauchy-Born elasticity models. We show that the proposed scheme converges quadratically to the solution of the…
The Quasicontinuum Method: Overview, applications and current directions
TLDR
A review of the Quasicontinuum method, from its original motivations and formulation to recent improvements and developments, and a summary of the important mechanics of materials results that have been obtained using the QC approach is presented.
Generalized mathematical homogenization of atomistic media at finite temperatures in three dimensions
Computational method for atomistic homogenization of nanopatterned point defect structures
The development of an approximation method that rigorously averages small‐scale atomistic physics and embeds them in large‐scale mechanics is the principal aim of this work. This paper presents a…
A Computational and Theoretical Investigation of the Accuracy of Quasicontinuum Methods
We give computational results to study the accuracy of several quasicontinuum methods for two benchmark problems – the stability of a Lomer dislocation pair under shear and the stability of a lattice…
Finite Element Analysis of Cauchy–Born Approximations to Atomistic Models
This paper is devoted to a new finite element consistency analysis of Cauchy–Born approximations to atomistic models of crystalline materials in two and three space dimensions. Through this approach…
...
...