Analysis of an optimization-based atomistic-to-continuum coupling method for point defects

@article{Olson2014AnalysisOA,
  title={Analysis of an optimization-based atomistic-to-continuum coupling method for point defects},
  author={Derek Olson and Alexander V. Shapeev and Pavel B. Bochev and Mitchell Luskin},
  journal={Mathematical Modelling and Numerical Analysis},
  year={2014},
  volume={50},
  pages={1-41}
}
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 enables accurate simulation of the defect. Away from the core, where site energies become nearly independent of the lattice position, the method switches to a more efficient continuum model. The two models are merged by minimizing the mismatch of their states on an overlap region, subject to the… 

Figures from this paper

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.
Regularity and Locality of Point Defects in Multilattices
We formulate a model for a point defect embedded in a homogeneous multilattice crystal with an empirical interatomic potential interaction. Under a natural, phonon stability assumption we quantify
Formulation and Analysis of an Optimization-Based Atomistic-to-Continuum Coupling Algorithm
TLDR
This dissertation presents a meta-mathematical framework for estimating the number of steps in a discrete-time model and shows how the model derived in [Bouchut-Boyaval, M3AS (23) 2013] can be modified to approximate the rate of response of the immune system to natural disasters.
Algorithms for Propagating Uncertainty Across Heterogeneous Domains | SIAM Journal on Scientific Computing | Vol. 37, No. 6 | Society for Industrial and Applied Mathematics
TLDR
Two new algorithms are proposed that enforces the continuity of the conditional mean and variance of the solution across adjacent subdomains by using Schwarz iterations and are based on PDE-constrained multiobjective optimization, and it allows to set more general interface conditions.

References

SHOWING 1-10 OF 52 REFERENCES
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.
An Optimization-based Atomistic-to-Continuum Coupling Method
We present a new optimization-based method for atomistic-to-continuum (AtC) coupling. The main idea is to cast the latter as a constrained optimization problem with virtual Dirichlet controls on the
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.
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.
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.
Consistent Energy-Based Atomistic/Continuum Coupling for Two-Body Potentials in One and Two Dimensions
TLDR
A novel coupling that is free from "ghost forces" is proposed for a two-body interaction potential under the assumptions of either one spatial dimension, or two spatial dimensions and piecewise affine finite elements for describing the continuum deformation.
THE ROLE OF THE PATCH TEST IN 2D ATOMISTIC-TO-CONTINUUM COUPLING METHODS ∗
For a general class of atomistic-to-continuum coupling methods, coupling multi-body inter- atomic potentials with a P1-finite element discretisation of Cauchy-Born nonlinear elasticity, this paper
Construction and Sharp Consistency Estimates for Atomistic/Continuum Coupling Methods with General Interfaces: A Two-Dimensional Model Problem
TLDR
For many-body nearest-neighbor interactions on the two-dimensional triangular lattice, it is shown that patch test consistent a/c methods can be constructed for arbitrary interface geometries and proved that all methods within this class are first-order consistent at the atomistic/continuum interface and second- order consistent in the interior of the continuum region.
...
...