A mathematical framework of the bridging scale method

In this paper, we present a mathematical framework of the bridging scale method (BSM), recently proposed by Liu et al. Under certain conditions, it had been designed for accurately and efficiently simulating complex dynamics with different spatial scales. From a clear and consistent derivation, we identify two error sources in this method. First, we use a linear finite element interpolation, and derive the coarse grid equations directly from Newton's second law. Numerical error in this length scale exists mainly due to inadequate approximation for the effects of the fine scale fluctuations. An modified linear element (MLE) scheme is developed to improve the accuracy. Secondly, we derive an exact multiscale interfacial condition to treat the interfaces between the molecular dynamics region ΩD and the complementary domain ΩC, using a time history kernel technique. The interfacial condition proposed in the original BSM may be regarded as a leading order approximation to the exact one (with respect to the coarsening ratio). This approximation is responsible for minor reflections across the interfaces, with a dependency on the choice of ΩD. We further illustrate the framework and analysis with linear and non‐linear lattices in one‐dimensional space. Copyright © 2005 John Wiley & Sons, Ltd.

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

[2]  Thomas Y. Hou,et al.  A pseudo-spectral multiscale method: Interfacial conditions and coarse grid equations , 2006, J. Comput. Phys..

[3]  Shaofan Li,et al.  Perfectly matched multiscale simulations , 2005 .

[4]  Eduard G. Karpov,et al.  A Green's function approach to deriving non‐reflecting boundary conditions in molecular dynamics simulations , 2005 .

[5]  Harold S. Park,et al.  The bridging scale for two-dimensional atomistic/continuum coupling , 2005 .

[6]  E. Weinan,et al.  Analysis of the heterogeneous multiscale method for elliptic homogenization problems , 2004 .

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

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

[9]  Eduard G. Karpov,et al.  Molecular dynamics boundary conditions for regular crystal lattices , 2004 .

[10]  Gregory J. Wagner,et al.  A multiscale projection method for the analysis of carbon nanotubes , 2004 .

[11]  William A. Curtin,et al.  Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics , 2004 .

[12]  Xiaobo Nie,et al.  A continuum and molecular dynamics hybrid method for micro- and nano-fluid flow , 2004, Journal of Fluid Mechanics.

[13]  Gregory J. Wagner,et al.  Coupling of atomistic and continuum simulations using a bridging scale decomposition , 2003 .

[14]  E. Weinan.  Analysis of the heterogeneous multiscale method for ordinary differential equations , 2003 .

[15]  M. Ortiz,et al.  Effect of indenter-radius size on Au(001) nanoindentation. , 2003, Physical review letters.

[16]  A. Chopra,et al.  Perfectly matched layers for time-harmonic elastodynamics of unbounded domains : Theory and finite-element implementation , 2003 .

[17]  L E Shilkrot,et al.  Coupled atomistic and discrete dislocation plasticity. , 2002, Physical review letters.

[18]  E. Weinan,et al.  A dynamic atomistic-continuum method for the simulation of crystalline materials , 2001, cond-mat/0112173.

[19]  W. Cai,et al.  Minimizing boundary reflections in coupled-domain simulations. , 2000, Physical review letters.

[20]  J. Q. Broughton,et al.  Concurrent Coupling of Length Scales in Solid State Systems , 2000 .

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

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

[23]  Noam Bernstein,et al.  Spanning the length scales in dynamic simulation , 1998 .

[24]  Marcus J. Grote,et al.  Nonreflecting Boundary Conditions for Time-Dependent Scattering , 1996 .

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

[26]  O'Connell,et al.  Molecular dynamics-continuum hybrid computations: A tool for studying complex fluid flows. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[27]  R. Higdon.  Radiation boundary conditions for dispersive waves , 1994 .

[28]  H. Fischmeister,et al.  Crack propagation in b.c.c. crystals studied with a combined finite-element and atomistic model , 1991 .

[29]  J. Keller,et al.  Non-reflecting boundary conditions for elastic waves , 1990 .

[30]  A. Majda,et al.  Radiation boundary conditions for acoustic and elastic wave calculations , 1979 .

[31]  J. D. Doll,et al.  Generalized Langevin equation approach for atom/solid‐surface scattering: Collinear atom/harmonic chain model , 1974 .

[32]  Christian Lubich,et al.  Fast Convolution for Nonreflecting Boundary Conditions , 2002, SIAM J. Sci. Comput..