A mathematical framework of the bridging scale method

Shaoqiang Tang*, Thomas Y. Hou, Wing Kam Liu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

69 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1688-1713
Number of pages26
JournalInternational Journal for Numerical Methods in Engineering
Issue number10
StatePublished - Mar 5 2006


  • Bridging scale method
  • Coarse-fine decomposition
  • Molecular dynamics coupled with finite elements
  • Multiscale computation

ASJC Scopus subject areas

  • Numerical Analysis
  • General Engineering
  • Applied Mathematics


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