Goal-Oriented Adaptive Mesh Refinement for the Quasicontinuum Approximation of a Frenkel-Kontorova Model

Abstract The quasicontinuum approximation [E.B. Tadmor, M. Ortiz, R. Phillips, Quasicontinuum analysis of defects in solids, Philos. Mag. A 73(6) (1996) 1529–1563] is a method to reduce the atomistic degrees of freedom of a crystalline solid by piecewise linear interpolation from representative atoms that are nodes for a finite element triangulation. In regions of the crystal with a highly nonuniform deformation such as around defects, every atom must be a representative atom to obtain sufficient accuracy, but the mesh can be coarsened away from such regions to remove atomistic degrees of freedom while retaining sufficient accuracy. We present an error estimator and a related adaptive mesh refinement algorithm for the quasicontinuum approximation of a generalized Frenkel-Kontorova model that enables a quantity of interest to be efficiently computed to a predetermined accuracy.

[1]  Endre Süli,et al.  Adaptive finite element methods for differential equations , 2003, Lectures in mathematics.

[2]  Michael Ortiz,et al.  Quasicontinuum simulation of fracture at the atomic scale , 1998 .

[3]  PING LIN,et al.  Convergence Analysis of a Quasi-Continuum Approximation for a Two-Dimensional Material Without Defects , 2007, SIAM J. Numer. Anal..

[4]  Pavel B. Bochev,et al.  On Atomistic-to-Continuum Coupling by Blending , 2008, Multiscale Model. Simul..

[5]  Endre Süli,et al.  ANALYSIS OF A QUASICONTINUUM METHOD IN ONE DIMENSION , 2008 .

[6]  Mitchell Luskin,et al.  Goal-oriented Atomistic-Continuum Adaptivity for the Quasicontinuum Approximation , 2007 .

[7]  Ronald E. Miller,et al.  Atomistic/continuum coupling in computational materials science , 2003 .

[8]  F. Legoll,et al.  Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics , 2005 .

[9]  Ronald E. Miller,et al.  The Quasicontinuum Method: Overview, applications and current directions , 2002 .

[10]  M. Ortiz,et al.  An analysis of the quasicontinuum method , 2001, cond-mat/0103455.

[11]  Ping Lin,et al.  Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model , 2003, Math. Comput..

[12]  ON ATOMISTIC-TO-CONTINUUM ( ATC ) COUPLING BY BLENDING , 2007 .

[13]  M. Marder.  Condensed Matter Physics , 2015 .

[14]  J. Tinsley Oden,et al.  Error Control for Molecular Statics Problems , 2006 .

[15]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[16]  T. Belytschko,et al.  A bridging domain method for coupling continua with molecular dynamics , 2004 .

[17]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis: Oden/A Posteriori , 2000 .

[18]  E Weinan,et al.  Uniform Accuracy of the Quasicontinuum Method , 2006, MRS Online Proceedings Library.

[19]  Mitchell Luskin,et al.  Error Estimation and Atomistic-Continuum Adaptivity for the Quasicontinuum Approximation of a Frenkel-Kontorova Model , 2007, Multiscale Model. Simul..

[20]  M. Ortiz,et al.  An adaptive finite element approach to atomic-scale mechanics—the quasicontinuum method , 1997, cond-mat/9710027.

[21]  Mitchell Luskin,et al.  Analysis of a force-based quasicontinuum approximation , 2006 .

[22]  Paul T. Bauman,et al.  On the extension of goal-oriented error estimation and hierarchical modeling to discrete lattice models , 2005 .

[23]  Pavel B. Bochev,et al.  Connecting Atomistic-to-Continuum Coupling and Domain Decomposition , 2008, Multiscale Model. Simul..

[24]  Harold S. Park,et al.  An introduction to computational nanomechanics and materials , 2004 .

[25]  E. Süli,et al.  A-posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension , 2006 .

[26]  E Weinan,et al.  Analysis of multiscale methods , 2004 .

[27]  M. Ortiz,et al.  Quasicontinuum analysis of defects in solids , 1996 .