Predicting band structure of 3D mechanical metamaterials with complex geometry via XFEM

Jifeng Zhao, Ying Li, Wing Kam Liu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

29 Scopus citations


Band structure characterizes the most important property of mechanical metamaterials. However, predicting the band structure of 3D metamaterials with complex microstructures through direct numerical simulation (DNS) is computationally inefficient due to the complexity of meshing. To overcome this issue, an extended finite element method (XFEM)-based method is developed to predict 3D metamaterial band structures. Since the microstructure and material interface are implicitly resolved by the level-set function embedded in the XFEM formulation, a non-conforming (such as uniform) mesh is used in the proposed method to avoid the difficulties in meshing complex geometries. The accuracy and mesh convergence of the proposed method have been validated and verified by studying the band structure of a spherical particle embedded in a cube and comparing the results with DNS. The band structures of 3D metamaterials with different microstructures have been studied using the proposed method with the same finite element mesh, indicating the flexibility of this method. This XFEM-based method opens new opportunities in design and optimization of mechanical metamaterials with target functions, e.g. location and width of the band gap, by eliminating the iterative procedure of re-building and re-meshing microstructures that is required by classical DNS type of methods.

Original languageEnglish (US)
Pages (from-to)659-672
Number of pages14
JournalComputational Mechanics
Issue number4
StatePublished - Apr 1 2015


  • Bloch wave analysis
  • Mechanical metamaterials
  • Parallel computing
  • Phononic band gap
  • XFEM

ASJC Scopus subject areas

  • Computational Mechanics
  • Ocean Engineering
  • Mechanical Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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