Three-scale finite element analysis of heterogeneous media by asymptotic homogenization and mesh superposition methods

Naoki Takano, Yoshihiro Okuno

Research output: Contribution to journalArticlepeer-review

31 Citations (Scopus)


This paper studies a three-scale computational method that simultaneously considers the microstructure of heterogeneous materials, the macroscopic component, and the fracture origin such as interface or crack. The synergetic application of the asymptotic homogenization and mesh superposition methods to problems with strong scale mixing is emphasized. The scale gap between the microstructure and the component is very large, but the fracture origin is at the middle scale between them. The overall behavior is analyzed by means of the homogenization of the heterogeneity expressed by the unit cell model, while the fracture origin is modeled directly with the microscopic heterogeneity by another microscopic mesh. The microscopic mesh is superposed onto the macroscopic mesh. This mesh superposition method can analyze the non-periodic microscopic stress at the crack tip under a non-uniform macroscopic strain field with high gradient. Hence, the present three-scale method can accurately focus on the behaviors at arbitrary scale differently from the conventional hierarchical model. A demonstrative example of porous thin film on a substrate with an interface crack was solved and the microscopic stress was analyzed at the crack tip considering the random dispersion of pores and the high gradient of macroscopic strain field. To solve the large-scale 3D problem with approximately 80,000 solid elements, a renumbering technique and the out-of-core skyline solver was employed.

Original languageEnglish
Pages (from-to)4121-4135
Number of pages15
JournalInternational Journal of Solids and Structures
Issue number15
Publication statusPublished - 2004 Jul
Externally publishedYes


  • 3D analysis
  • Finite element method
  • Heterogeneity
  • Homogenization
  • Interface crack
  • Mesh superposition
  • Microstructure
  • Multi-scale analysis

ASJC Scopus subject areas

  • Modelling and Simulation
  • General Materials Science
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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