An introduction and tutorial on multiple-scale analysis in solids

Harold S. Park*, Wing Kam Liu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

161 Scopus citations


Concurrent multiple-scale methods can be defined as those which combine information available from distinct length and time scales into a single coherent, coupled simulation. These methods have recently become both popular and necessary for the following reasons. One is the recent discovery of new, nanoscale materials, and the corresponding boom in nanotechnology research. Another factor is that experiments have conclusively shown the connection between microscale physics and macroscale deformation. Finally, the concept of linking disparate length and time scales has become feasible recently due to the ongoing explosion in computational power. We present a detailed introduction to the available technologies in the field of multiple-scale analysis. In particular, our review centers on methods which aim to couple molecular-level simulations (such as molecular dynamics) to continuum level simulations (such as finite element and meshfree methods). Using this definition, we first review existing multiple-scale technology, and explain the pertinent issues in creating an efficient yet accurate multiple-scale method. Following the review, we highlight a new multiple-scale method, the bridging scale, and compare it to existing multiple-scale methods. Next, we show example problems in which the bridging scale is applied to fully non-linear problems. Concluding remarks address the research needs for multiple-scale methods in general, the bridging scale method in particular, and potential applications for the bridging scale.

Original languageEnglish (US)
Pages (from-to)1733-1772
Number of pages40
JournalComputer Methods in Applied Mechanics and Engineering
Issue number17-20
StatePublished - May 7 2004


  • Bridging scale
  • Coupling methods
  • Finite elements
  • Molecular dynamics
  • Multiple-scale simulations
  • Non-linear

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


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