# Analysis of a quasicontinuum method in one dimension

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

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

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topOrtner, 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 -

## References

top- X. Blanc, C. Le Bris and F. Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics: the convex case. Acta Math. Appl. Sinica English Series23 (2007) 209–216.
- A. Braides and M.S. Gelli, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids7 (2002) 41–66.
- A. Braides, G. Dal Maso and A. Garroni, Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch. Ration. Mech. Anal.146 (1999) 23–58.
- A. Braides, A.J. Lew and M. Ortiz, Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal.180 (2006) 151–182.
- F. Brezzi, J. Rappaz and P.-A. Raviart, Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions. Numer. Math.36 (1980) 1–25.
- M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum approximation. ESAIM: M2AN42 (2008) 113–139.
- G. Dolzmann, Optimal convergence for the finite element method in Campanato spaces. Math. Comp.68 (1999) 1397–1427.
- W. E and B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci.1 (2003) 87–132.
- W. E and P. Ming, Analysis of multiscale methods. J. Comput. Math.22 (2004) 210–219. Special issue dedicated to the 70th birthday of Professor Zhong-Ci Shi.
- W. E and P. Ming, Analysis of the local quasicontinuum method, in Frontiers and prospects of contemporary applied mathematics, Ser. Contemp. Appl. Math. CAM6, Higher Ed. Press, Beijing (2005) 18–32.
- D.J. Higham, Trust region algorithms and timestep selection. SIAM J. Numer. Anal.37 (1999) 194–210.
- J.E. Jones, On the Determination of Molecular Fields. III. From Crystal Measurements and Kinetic Theory Data. Proc. Roy. Soc. London A.106 (1924) 709–718.
- B. Lemaire, The proximal algorithm, in New methods in optimization and their industrial uses (Pau/Paris, 1987), of Internat. Schriftenreihe Numer. Math.87, Birkhäuser, Basel (1989) 73–87.
- P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp.72 (2003) 657–675.
- P. Lin, Convergence analysis of a quasi-continuum approximation for a two-dimensional material without defects. SIAM J. Numer. Anal.45 (2007) 313–332 (electronic).
- R.E. Miller and E.B. Tadmor, The quasicontinuum method: overview, applications and current directions. J. Computer-Aided Mater. Des.9 (2003) 203–239.
- P.M. Morse, Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev.34 (1929) 57–64.
- M. Ortiz, R. Phillips and E.B. Tadmor, Quasicontinuum analysis of defects in solids. Philos. Mag. A73 (1996) 1529–1563.
- C. Ortner, Gradient flows as a selection procedure for equilibria of nonconvex energies. SIAM J. Math. Anal.38 (2006) 1214–1234 (electronic).
- C. Ortner and E. Süli, A posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension. Technical Report NA06/13, Oxford University Computing Laboratory (2006).
- C. Ortner and E. Süli, Discontinuous Galerkin finite element approximation of nonlinear second-order elliptic and hyperbolic systems. SIAM J. Numer. Anal.45 (2007) 1370–1397.
- M. Plum, Computer-assisted enclosure methods for elliptic differential equations. Linear Algebra Appl.324 (2001) 147–187. Special issue on linear algebra in self-validating methods.
- L. Truskinovsky, Fracture as a phase transformation, in Contemporary research in mechanics and mathematics of materials, R.C. Batra and M.F. Beatty Eds., CIMNE (1996) 322–332.
- R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comp.62 (1994) 445–475.
- E. Zeidler, Nonlinear functional analysis and its applications. I Fixed-point theorems. Springer-Verlag, New York (1986). Translated from the German by Peter R. Wadsack.

## Citations in EuDML Documents

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

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