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

top
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

top

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 -

References

top
  1. 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.  Zbl1177.65091
  2. A. Braides and M.S. Gelli, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids7 (2002) 41–66.  Zbl1024.74004
  3. 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.  Zbl0945.74006
  4. A. Braides, A.J. Lew and M. Ortiz, Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal.180 (2006) 151–182.  Zbl1093.74013
  5. 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.  Zbl0488.65021
  6. M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum approximation. ESAIM: M2AN42 (2008) 113–139.  Zbl1140.74006
  7. G. Dolzmann, Optimal convergence for the finite element method in Campanato spaces. Math. Comp.68 (1999) 1397–1427.  Zbl0929.65096
  8. W. E and B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci.1 (2003) 87–132.  Zbl1093.35012
  9. 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.  Zbl1046.65108
  10. 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.  
  11. D.J. Higham, Trust region algorithms and timestep selection. SIAM J. Numer. Anal.37 (1999) 194–210.  Zbl0945.65068
  12. 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.  
  13. 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.  
  14. P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp.72 (2003) 657–675.  Zbl1010.74003
  15. 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).  Zbl1220.74010
  16. R.E. Miller and E.B. Tadmor, The quasicontinuum method: overview, applications and current directions. J. Computer-Aided Mater. Des.9 (2003) 203–239.  
  17. P.M. Morse, Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev.34 (1929) 57–64.  Zbl55.0539.02
  18. M. Ortiz, R. Phillips and E.B. Tadmor, Quasicontinuum analysis of defects in solids. Philos. Mag. A73 (1996) 1529–1563.  
  19. C. Ortner, Gradient flows as a selection procedure for equilibria of nonconvex energies. SIAM J. Math. Anal.38 (2006) 1214–1234 (electronic).  Zbl1117.35004
  20. 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).  
  21. 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.  Zbl1146.65070
  22. 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.  Zbl0973.65100
  23. 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.  
  24. R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comp.62 (1994) 445–475.  Zbl0799.65112
  25. 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.  Zbl0583.47050

Citations in EuDML Documents

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.