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A mathematical framework of the bridging scale method
Author(s) -
Tang Shaoqiang,
Hou Thomas Y.,
Liu Wing Kam
Publication year - 2005
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1514
Subject(s) - bridging (networking) , interpolation (computer graphics) , scale (ratio) , finite element method , mathematics , grid , computer science , mathematical optimization , algorithm , geometry , physics , animation , computer network , computer graphics (images) , quantum mechanics , thermodynamics
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 Ω D and the complementary domain Ω C , 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 Ω D . We further illustrate the framework and analysis with linear and non‐linear lattices in one‐dimensional space. Copyright © 2005 John Wiley & Sons, Ltd.