Unfitted Nitsche's method for computing band structures in phononic crystals with impurities

Hailong Guo School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia Xu Yang Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA Yi Zhu Yau Mathematical Sciences Center and Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

Numerical Analysis and Scientific Computing mathscidoc:2204.25002

2020.7
In this paper, we propose an unfitted Nitsche's method to compute the band structures of phononic crystal with impurities of general geometry. The proposed method does not require the background mesh to fit the interfaces of impurities, and thus avoids the expensive cost of generating body-fitted meshes and simplifies the inclusion of interface conditions in the formulation. The quasi-periodic boundary conditions are handled by the Floquet-Bloch transform, which converts the computation of band structures into an eigenvalue problem with periodic boundary conditions. More importantly, we show the well-posedness of the proposed method using a delicate argument based on the trace inequality, and further prove the convergence by the Babuška-Osborn theory. We achieve the optimal convergence rate at the presence of the impurities of general geometry. We confirm the theoretical results by two numerical examples, and show the capability of the proposed methods for computing the band structures without fitting the interfaces of impurities.
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@inproceedings{hailong2020unfitted,
  title={Unfitted Nitsche's method for computing band structures in phononic crystals with impurities},
  author={Hailong Guo, Xu Yang, and Yi Zhu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220422135615210193093},
  year={2020},
}
Hailong Guo, Xu Yang, and Yi Zhu. Unfitted Nitsche's method for computing band structures in phononic crystals with impurities. 2020. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220422135615210193093.
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