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Volume 2, Issue 3
Heterogeneous Multiscale Methods: A Review

W. E, B. Engquist, X. Li, W. Ren & E. Vanden-Eijnden

Commun. Comput. Phys., 2 (2007), pp. 367-450.

Published online: 2007-02

[An open-access article; the PDF is free to any online user.]

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  • 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.

  • Keywords

Multi-scale modeling, heterogeneous multi-scale method, multi-physics models, constrained micro-scale solver, data estimation.

  • AMS Subject Headings

65N30, 74Q05, 74Q20, 39A12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-2-367, author = {W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden}, title = {Heterogeneous Multiscale Methods: A Review}, journal = {Communications in Computational Physics}, year = {2007}, volume = {2}, number = {3}, pages = {367--450}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7911.html} }
TY - JOUR T1 - Heterogeneous Multiscale Methods: A Review AU - W. E, B. Engquist, X. Li, W. Ren & E. Vanden-Eijnden JO - Communications in Computational Physics VL - 3 SP - 367 EP - 450 PY - 2007 DA - 2007/02 SN - 2 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cicp/7911.html KW - Multi-scale modeling, heterogeneous multi-scale method, multi-physics models, constrained micro-scale solver, data estimation. AB -

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.

W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden. (2007). Heterogeneous Multiscale Methods: A Review. Communications in Computational Physics. 2 (3). 367-450. doi:
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