Unfitted Nitsche’s method for computing band structures of phononic crystals with periodic inclusions

Abstract 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 Babuska–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.