We analyze a force-based quasicontinuum approximation to a one-dimensional system of atoms that interact by a classical atomistic potential. This force-based quasicontinuum approximation can be derived as the modification of an energy-based quasicontinuum approximation by the addition of nonconservative forces to correct nonphysical “ghost” forces that occur in the atomistic to continuum interface during constant strain. The algorithmic simplicity and consistency with the purely atomistic model at constant strain has made the force-based quasicontinuum approximation popular for large-scale quasicontinuum computations. We prove that the force-based quasicontinuum equations have a unique solution when the magnitude of the external forces satisfy explicit bounds. For Lennard-Jones next-nearest-neighbor interactions, we show that unique solutions exist for external forces that extend the system nearly to its tensile limit. We give an analysis of the convergence of the ghost force iteration method to solve the equilibrium equations for the force-based quasicontinuum approximation. We show that the ghost force iteration is a contraction and give an analysis for its convergence rate.

Classification: 65Z05, 70C20

Keywords: quasicontinuum, ghost force, atomistic to continuum

@article{M2AN_2008__42_1_113_0, author = {Dobson, Matthew and Luskin, Mitchell}, title = {Analysis of a force-based quasicontinuum approximation}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {42}, number = {1}, year = {2008}, pages = {113-139}, doi = {10.1051/m2an:2007058}, zbl = {1140.74006}, mrnumber = {2387424}, language = {en}, url = {http://www.numdam.org/item/M2AN_2008__42_1_113_0} }

Dobson, Matthew; Luskin, Mitchell. Analysis of a force-based quasicontinuum approximation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 1, pp. 113-139. doi : 10.1051/m2an:2007058. http://www.numdam.org/item/M2AN_2008__42_1_113_0/

[1] Nonlinear problems of elasticity, Applied Mathematical Sciences 107. Springer, New York, second edition (2005). | MR 2132247 | Zbl 1098.74001

,[2] Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: M2AN 39 (2005) 797-826. | Numdam | MR 2165680 | Zbl pre02213940

, and ,[3] Atomistic to continuum limits for computational materials science. ESAIM: M2AN 41 (2007) 391-426. | Numdam | MR 2339634 | Zbl 1144.82018

, and ,[4] A Topological Introduction to Nonlinear Analysis. Birkhäuser (2004). | MR 2020421 | Zbl 1061.47001

,[5] W. E and P. Ming, Analysis of multiscale methods. J. Comput. Math. 22 (2004) 210-219. | MR 2058933 | Zbl 1046.65108

[6] W. E and P. Ming, Analysis of the local quasicontinuum method, in Frontiers and Prospects of Contemporary Applied Mathematics, T. Li and P. Zhang Eds., Higher Education Press, World Scientific, Singapore (2005) 18-32. | MR 2249291 | Zbl pre05050158

[7] W. E and P. Ming, Cauchy-born rule and the stabilitiy of crystalline solids: Static problems. Arch. Ration. Mech. Anal. 183 (2007) 241-297. | MR 2278407 | Zbl 1106.74019

[8] W. E, J. Lu and J. Yang, Uniform accuracy of the quasicontinuum method. Phys. Rev. B 74 (2006) 214115.

[9] Functions of Several Variables. Springer-Verlag (1977). | MR 422527 | Zbl 0348.26002

,[10] An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49 (2001) 1899-1923. | Zbl 1002.74008

and ,[11] Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp. 72 (2003) 657-675 (electronic). | MR 1954960 | Zbl 1010.74003

,[12] Convergence analysis of a quasi-continuum approximation for a two-dimensional material. SIAM J. Numer. Anal. 45 (2007) 313-332. | MR 2285857 | Zbl pre05246529

,[13] Condensed Matter Physics. John Wiley & Sons (2000).

,[14] The quasicontinuum method: Overview, applications and current directions. J. Comput. Aided Mater. Des. 9 (2002) 203-239.

and ,[15] A coupled atomistic and discrete dislocation plasticity simulation of nano-indentation into single crystal thin films. Acta Mater. 52 (2003) 271-284.

, and ,[16] Multi-scale modeling of physical phenomena: Adaptive control of models. SIAM J. Sci. Comput. 28 (2006) 2359-2389. | MR 2272265 | Zbl 1126.74006

, , and ,[17] A posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension. Technical report, Oxford Numerical Analysis Group (2006).

and ,[18] A priori analysis of the quasicontinuum method in one dimension. Technical report, Oxford Numerical Analysis Group (2006).

and ,[19] Error control for molecular statics problems. Int. J. Multiscale Comput. Eng. 4 (2006) 647-662.

, and ,[20] Structure and strength of dislocation junctions: An atomic level analysis. Phys. Rev. Lett. 82 (1999) 1704-1707.

and ,[21] Matrices: Theory and applications, Graduate Texts in Mathematics 216. Springer-Verlag, New York (2002). Translated from the 2001 French original. | MR 1923507 | Zbl 1011.15001

,[22] An adaptive finite element approach to atomic-scale mechanics - the quasicontinuum method. J. Mech. Phys. Solids 47 (1999) 611-642. | MR 1675219 | Zbl 0982.74071

, , , , and ,[23] Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic regions. Phys. Rev. B 69 (2004) 214104.

, , and ,[24] Quasicontinuum analysis of defects in solids. Phil. Mag. A 73 (1996) 1529-1563.

, and ,