A mathematical framework of the bridging scale method

作者：Tang SQ^*; Hou TY; Liu WK

来源：International Journal for Numerical Methods in Engineering, 2006, 65(10): 1688-1713.

DOI：10.1002/nme.1514

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

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 Omega(D) and the complementary domain QC, 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 Omega(D). We further illustrate the framework and analysis with linear and non-linear lattices in one-dimensional space.

• 出版日期2006-3-5
• 单位北京大学