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

TL;DR

This paper introduces an unfitted Nitsche's method for computing phononic crystal band structures with impurities, avoiding complex mesh fitting and ensuring optimal convergence through rigorous theoretical analysis and numerical validation.

Contribution

The paper presents a novel unfitted Nitsche's method that handles impurities of arbitrary geometry without mesh fitting, with proven well-posedness and optimal convergence.

Findings

Method achieves optimal convergence rates.

Successfully computes band structures with complex impurities.

Validated through numerical examples.

Abstract

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 Babu\v{s}ka-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.