Heterogeneous multiscale methods: A review

Weinan E, Bjorn Engquist, Xiantao Li, Weiqing Ren, Eric Vanden-Eijnden

Research output: Contribution to journalReview article

543 Scopus citations

Abstract

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 a particular 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, such as the various constrained molecular dynamics, that have been developed, in order to apply HMM to these problems. Examples of mathematical results on the error analysis of HMM are presented. The review ends with a discussion on some of the problems that have to be solved in order to make HMM a more powerful tool.

Original languageEnglish (US)
Pages (from-to)367-450
Number of pages84
JournalCommunications in Computational Physics
Volume2
Issue number3
StatePublished - Jun 2007

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy (miscellaneous)

Keywords

  • Constrained micro-scale solver
  • Data estimation
  • Heterogeneous multi-scale method
  • Multi-physics models
  • Multi-scale modeling

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  • Cite this

    E, W., Engquist, B., Li, X., Ren, W., & Vanden-Eijnden, E. (2007). Heterogeneous multiscale methods: A review. Communications in Computational Physics, 2(3), 367-450.