Computational Mechanics

, Volume 55, Issue 4, pp 659–672 | Cite as

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

  • Jifeng Zhao
  • Ying Li
  • Wing Kam LiuEmail author
Original Paper


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.


XFEM Phononic band gap Bloch wave analysis Mechanical metamaterials Parallel computing 



We are grateful to Dr. Hong Zhang from Argonne National Laboratory for helpful discussions on how to use PETSc and SLEPc. We express thanks to Jacob Smith from Northwestern University for revising English for this paper. Y.L. warmly expresses thanks for the financial support provided by Ryan Fellowship and Royal E. Cabell Terminal Year Fellowship, as well as a supercomputing grant on Quest from Northwestern University High Performance Computing Center. W.K.L. expresses thanks for the support from AFOSR Grant No. FA9550-14-1-0032.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Theoretical & Applied MechanicsNorthwestern UniversityEvanstonUSA
  2. 2.Department of Mechanical EngineeringNorthwestern UniversityEvanstonUSA
  3. 3.Distinguished Scientists Program CommitteeKing Abdulaziz University (KAU)JeddahSaudi Arabia

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