Research Article

A mathematical framework of the bridging scale method

  1. Shaoqiang Tang1,2,*,
  2. Thomas Y. Hou2,
  3. Wing Kam Liu3

Article first published online: 6 OCT 2005

DOI: 10.1002/nme.1514

Issue

International Journal for Numerical Methods in Engineering

International Journal for Numerical Methods in Engineering

Volume 65, Issue 10, pages 1688–1713, 5 March 2006

Additional Information

How to Cite

Tang, S., Hou, T. Y. and Liu, W. K. (2006), A mathematical framework of the bridging scale method. International Journal for Numerical Methods in Engineering, 65: 1688–1713. doi: 10.1002/nme.1514

Author Information

  1. 1

    LTCS, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, People's Republic of China

  2. 2

    Applied and Computational Mathematics, Caltech, Pasadena, CA 91125, U.S.A.

  3. 3

    Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, U.S.A.

*Department of Mechanics and Engineering Science, Peking University, Beijing 100871, People's Republic of China

Publication History

  1. Issue published online: 25 JAN 2006
  2. Article first published online: 6 OCT 2005
  3. Manuscript Accepted: 17 AUG 2005
  4. Manuscript Revised: 11 AUG 2005
  5. Manuscript Received: 18 FEB 2005

Funded by

  • NSF. Grant Numbers: FRG DMS-0353838, ITR ACI-0204932, IGERT
  • Chinese Special Funds for Major State Basic Research Project. Grant Number: G2000077305
  • NSFC. Grant Numbers: 90407021, 10271003

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Keywords:

  • multiscale computation;
  • bridging scale method;
  • coarse–fine decomposition;
  • molecular dynamics coupled with finite elements

Abstract

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.

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