Unfitted Nitsche’s method for computing band structures of phononic crystals with periodic inclusions. (English) Zbl 1506.78012
Summary: In this paper, we propose an unfitted Nitsche’s method to compute the band structures of phononic crystal with periodic inclusions of general geometry. The proposed method does not require the background mesh to fit the interfaces of periodic inclusions, 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 periodic inclusions of general geometry. We demonstrate 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 periodic inclusions.
MSC:
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
| 35P99 | Spectral theory and eigenvalue problems for partial differential equations |
| 47A70 | (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces |
| 78A48 | Composite media; random media in optics and electromagnetic theory |
Software:
CutFEMReferences:
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