Analysis of a quasicontinuum method in one dimension

Christoph Ortner; Endre Süli

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 1, page 57-91
  • ISSN: 0764-583X

Abstract

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The quasicontinuum method is a coarse-graining technique for reducing the complexity of atomistic simulations in a static and quasistatic setting. In this paper we aim to give a detailed a priori and a posteriori error analysis for a quasicontinuum method in one dimension. We consider atomistic models with Lennard–Jones type long-range interactions and a QC formulation which incorporates several important aspects of practical QC methods. First, we prove the existence, the local uniqueness and the stability with respect to a discrete W1,∞-norm of elastic and fractured atomistic solutions. We use a fixed point argument to prove the existence of a quasicontinuum approximation which satisfies a quasi-optimal a priori error bound. We then reverse the role of exact and approximate solution and prove that, if a computed quasicontinuum solution is stable in a sense that we make precise and has a sufficiently small residual, there exists a `nearby' exact solution which it approximates, and we give an a posteriori error bound. We stress that, despite the fact that we use linearization techniques in the analysis, our results apply to genuinely nonlinear situations.

How to cite

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Ortner, Christoph, and Süli, Endre. "Analysis of a quasicontinuum method in one dimension." ESAIM: Mathematical Modelling and Numerical Analysis 42.1 (2008): 57-91. <http://eudml.org/doc/250345>.

@article{Ortner2008,
abstract = { The quasicontinuum method is a coarse-graining technique for reducing the complexity of atomistic simulations in a static and quasistatic setting. In this paper we aim to give a detailed a priori and a posteriori error analysis for a quasicontinuum method in one dimension. We consider atomistic models with Lennard–Jones type long-range interactions and a QC formulation which incorporates several important aspects of practical QC methods. First, we prove the existence, the local uniqueness and the stability with respect to a discrete W1,∞-norm of elastic and fractured atomistic solutions. We use a fixed point argument to prove the existence of a quasicontinuum approximation which satisfies a quasi-optimal a priori error bound. We then reverse the role of exact and approximate solution and prove that, if a computed quasicontinuum solution is stable in a sense that we make precise and has a sufficiently small residual, there exists a `nearby' exact solution which it approximates, and we give an a posteriori error bound. We stress that, despite the fact that we use linearization techniques in the analysis, our results apply to genuinely nonlinear situations. },
author = {Ortner, Christoph, Süli, Endre},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Atomistic material models; quasicontinuum method; error analysis; stability.; atomistic material models; error analysis; stability; linearization},
language = {eng},
month = {1},
number = {1},
pages = {57-91},
publisher = {EDP Sciences},
title = {Analysis of a quasicontinuum method in one dimension},
url = {http://eudml.org/doc/250345},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Ortner, Christoph
AU - Süli, Endre
TI - Analysis of a quasicontinuum method in one dimension
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/1//
PB - EDP Sciences
VL - 42
IS - 1
SP - 57
EP - 91
AB - The quasicontinuum method is a coarse-graining technique for reducing the complexity of atomistic simulations in a static and quasistatic setting. In this paper we aim to give a detailed a priori and a posteriori error analysis for a quasicontinuum method in one dimension. We consider atomistic models with Lennard–Jones type long-range interactions and a QC formulation which incorporates several important aspects of practical QC methods. First, we prove the existence, the local uniqueness and the stability with respect to a discrete W1,∞-norm of elastic and fractured atomistic solutions. We use a fixed point argument to prove the existence of a quasicontinuum approximation which satisfies a quasi-optimal a priori error bound. We then reverse the role of exact and approximate solution and prove that, if a computed quasicontinuum solution is stable in a sense that we make precise and has a sufficiently small residual, there exists a `nearby' exact solution which it approximates, and we give an a posteriori error bound. We stress that, despite the fact that we use linearization techniques in the analysis, our results apply to genuinely nonlinear situations.
LA - eng
KW - Atomistic material models; quasicontinuum method; error analysis; stability.; atomistic material models; error analysis; stability; linearization
UR - http://eudml.org/doc/250345
ER -

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Citations in EuDML Documents

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  1. Thomas Hudson, Christoph Ortner, On the stability of Bravais lattices and their Cauchy–Born approximations
  2. Thomas Hudson, Christoph Ortner, On the stability of Bravais lattices and their Cauchy–Born approximations
  3. Christoph Ortner, The role of the patch test in 2D atomistic-to-continuum coupling methods
  4. Christoph Ortner, The role of the patch test in 2D atomistic-to-continuum coupling methods
  5. Matthew Dobson, Mitchell Luskin, An analysis of the effect of ghost force oscillation on quasicontinuum error

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