Error Control for Molecular Statics Problems

In this paper, we present an extension of goal-oriented error estimation and adaptation to the simulation of multiscale problems of molecular statics. Computable error estimates for the quasicontinuum method are developed with respect to specific quantities of interest, and an adaptive strategy based on these estimates is proposed for error control. The theoretical results are illustrated on a nanoindentation problem in which the quantity of interest is the force acting on the indenter. The promising capability of such error estimates and adaptive procedure for the solution of multiscale problems is demonstrated on numerical examples.

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

[2]  Michael Ortiz,et al.  Hierarchical models of plasticity: dislocation nucleation and interaction , 1999 .

[3]  M. Baskes,et al.  Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals , 1984 .

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

[5]  Foiles,et al.  Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys. , 1986, Physical review. B, Condensed matter.

[6]  J. Tinsley Oden,et al.  Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials: I. Error estimates and adaptive algorithms , 2000 .

[7]  J. Tinsley Oden,et al.  Estimation of modeling error in computational mechanics , 2002 .

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

[9]  J. Q. Broughton,et al.  Concurrent coupling of length scales: Methodology and application , 1999 .

[10]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

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

[12]  James B. Adams,et al.  Interatomic Potentials from First-Principles Calculations: The Force-Matching Method , 1993, cond-mat/9306054.

[13]  Florian Theil,et al.  Validity and Failure of the Cauchy-Born Hypothesis in a Two-Dimensional Mass-Spring Lattice , 2002, J. Nonlinear Sci..

[14]  J. T. Oden,et al.  Adaptive modeling of wave propagation in heterogeneous elastic solids , 2004 .

[15]  M. Baskes,et al.  Semiempirical, Quantum Mechanical Calculation of Hydrogen Embrittlement in Metals , 1983 .

[16]  Michael Ortiz,et al.  Nanoindentation and incipient plasticity , 1999 .

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

[18]  J. Tinsley Oden,et al.  MultiScale Modeling of Physical Phenomena: Adaptive Control of Models , 2006, SIAM J. Sci. Comput..

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