Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T23:22:28.964Z Has data issue: false hasContentIssue false

Analysis of a quasicontinuum method in one dimension

Published online by Cambridge University Press:  12 January 2008

Christoph Ortner
Affiliation:
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK. [email protected]; [email protected]
Endre Süli
Affiliation:
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK. [email protected]; [email protected]
Get access

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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blanc, X., Le Bris, C. and Legoll, F., Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics: the convex case. Acta Math. Appl. Sinica English Series 23 (2007) 209216. CrossRef
Braides, A. and Gelli, M.S., Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids 7 (2002) 4166. CrossRef
Braides, A., Dal Maso, G. and Garroni, A., Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch. Ration. Mech. Anal. 146 (1999) 2358. CrossRef
Braides, A., Lew, A.J. and Ortiz, M., Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal. 180 (2006) 151182. CrossRef
Brezzi, F., Rappaz, J. and Raviart, P.-A., Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions. Numer. Math. 36 (1980) 125. CrossRef
Dobson, M. and Luskin, M., Analysis of a force-based quasicontinuum approximation. ESAIM: M2AN 42 (2008) 113139. CrossRef
Dolzmann, G., Optimal convergence for the finite element method in Campanato spaces. Math. Comp. 68 (1999) 13971427. CrossRef
E, W. and Engquist, B., The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87132. CrossRef
E, W. and Ming, P., Analysis of multiscale methods. J. Comput. Math. 22 (2004) 210219. 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. CAM 6, Higher Ed. Press, Beijing (2005) 18–32.
Higham, D.J., Trust region algorithms and timestep selection. SIAM J. Numer. Anal. 37 (1999) 194210. CrossRef
Jones, J.E., On the Determination of Molecular Fields. III. From Crystal Measurements and Kinetic Theory Data. Proc. Roy. Soc. London A. 106 (1924) 709718. CrossRef
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.
Lin, P., Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp. 72 (2003) 657675. CrossRef
Lin, P., Convergence analysis of a quasi-continuum approximation for a two-dimensional material without defects. SIAM J. Numer. Anal. 45 (2007) 313332 (electronic). CrossRef
Miller, R.E. and Tadmor, E.B., The quasicontinuum method: overview, applications and current directions. J. Computer-Aided Mater. Des. 9 (2003) 203239. CrossRef
Morse, P.M., Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev. 34 (1929) 5764. CrossRef
Ortiz, M., Phillips, R. and Tadmor, E.B., Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (1996) 15291563.
Ortner, C., Gradient flows as a selection procedure for equilibria of nonconvex energies. SIAM J. Math. Anal. 38 (2006) 12141234 (electronic). CrossRef
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).
Ortner, C. and Süli, E., Discontinuous Galerkin finite element approximation of nonlinear second-order elliptic and hyperbolic systems. SIAM J. Numer. Anal. 45 (2007) 13701397. CrossRef
Plum, M., Computer-assisted enclosure methods for elliptic differential equations. Linear Algebra Appl. 324 (2001) 147187. Special issue on linear algebra in self-validating methods. CrossRef
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.
Verfürth, R., A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comp. 62 (1994) 445475. CrossRef
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.