Heterogeneous Multiscale Methods: A Review (Communications in Computational Physics, 2007)
This paper gives a systematic introduction to HMM, the heterogeneous multiscale methods, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for aparticular problem. This is illustrated by examples from several application areas, including complex fluids, micro-fluidics, solids, interface problems, stochastic problems,and statistically self-similar problems. Emphasis is given to the technical tools, suchas the various constrained molecular dynamics, that have been developed, in order toapply HMM to these problems. Examples of mathematical results on the error analysisof HMM are presented. The review ends with a discussion on some of the problemsthat have to be solved in order to make HMM a more powerful tool.
Keywords: Multi-scale modeling; heterogeneous multi-scale method; multi-physics models; constrained micro-scale solver; data estimation.