Unfitted Nitsche’s method for computing band structures of phononic crystals with periodic inclusions
Introduction
Phononic crystals are synthetic materials with periodic structure. Similar to photonic crystals, they exhibit band-gap structures related to topological properties, which prevent elastic waves propagating in certain frequencies. This leads to a series of important applications such as ultrasound imaging and wireless communications. In literature, Economou and Sigalas [1] experimentally observed the band-gap in phononic crystals. Ammari et al. [2] mathematically proved the existence of band-gap in the high-contrast phononic crystal using the asymptotic expansion and the generalized Rouché’s theorem. In general, phononic crystals with large band-gap is preferred due to the wide range of applications. One of the most influential accounts of band-gap optimization comes from Sigmund and Jensen who were one of the first to use topology optimization approach to design a phononic crystal with maximum relative band-gap size [3]. The main idea is to find the optimal arrangement of two different materials to achieve maximum band-gap. The geometric configuration of the two materials is continually updated during designing process. The main computational challenge is the numerical solution of heterogeneous eigenvalue problems with the moving material interface.
Another desire was brought about by the recent increasing interest in wave propagation in topological phononic materials. One of the key problems is to understand the band structure of bulk phononic crystals [4]. In generic cases, it is hard to obtain the explicit form of band structures with complete analytical techniques, and thus numerical computation plays an essential role. Early works on numerical approximations can be traced back to [5] where Kushwaha et al. used the plane-wave expansion to compute the band structure. The transfer matrix method was also adopted by Sigalas and Soukoulis [6] to simulate the propagation of elastic waves through disordered solid. To date, various methods have been developed to compute the band structure of phononic crystals including the multiple scattering method [7], the finite difference time domain method [8], the meshless method [9], the (multiscale) finite element method [10], [11], [12], [13], the homogenization method [14], [15], [16], and the singular boundary method [17].
Among the aforementioned methods, the numerical difficulties come from two different perspectives: one is the heterogeneous primitive cell of the phononic crystals and the other is how to efficiently impose the quasi-periodic boundary condition. Although extensive research has been carried out on the computing bandgap of phononic crystal, very few has addressed the complication brought by adjusting material interfaces for instance in the material design. Until recently, Wang et al. [18] proposed a Petrov–Galerkin immersed finite element method to compute the band structure of the phononic crystal and imposed the quasi-periodic boundary condition directly. However, the rigorous analysis of unfitted numerical methods is still lacking in the literature.
In this paper, we propose an unfitted Nitsche’s method to compute the band structures of phononic crystal with periodic inclusions of general geometry, and prove the convergence with rigorous mathematical analysis. The heterogeneous primitive cell of the phononic crystal is described by the interface condition which we can build into a variational framework with the help of the Floquet–Bloch theory. To handle the quasi-periodic boundary condition, the Floquet–Bloch transform is applied which reformulates the model equation with quasi-periodic boundary conditions into an equivalent model equation with periodic boundary conditions and Bloch-type interface condition. Then, the reformulated model equations can be numerically tackled by the unfitted Nitsche’s type method [19], [20], [21], [22], [23] using uniform meshes. The proposed unfitted finite element method is motivated by our previous work of computing edge modes in topological materials [24]. The first advantage is that it uses meshes independent of the location of the material interfaces. It reduces the computational cost of generating body-fitted meshes, especially in designing phononic crystals. The second advantage is that it is straightforward to impose the periodic boundary conditions since only uniform meshes are used. Remark that imposing periodic boundary conditions on general unstructured meshes is quite technically involved, and interesting readers are referred to [25], [26] and the references therein about the recent development of imposing periodic boundary condition on general unstructured meshes.
As mentioned in our previous work [24], the discrete Nitsche’s bilinear form involves the solution itself in addition to its gradient which cause the difficulties in the analysis. In this paper, we establish a solid theoretical analysis for the proposed unfitted finite element methods by conquering the above difficulties. Specifically, we show the discrete equation is well defined by using a delicate trace inequality on the cut element, the Poincaré inequality between the energy norm of the original model equation and the energy norm of the modified model equation, and the explicit relation between the strain tensor and stress tensor. By the aid of the Babus̀ka–Osborn spectral approximation theory [27], [28], the proposed unfitted finite element method is proven to have the optimal approximation property for the eigenvalues and eigenfunctions in the high-contrast heterogeneous primitive cell.
The paper is organized as follows. In Section 2, we introduce the model of plane-wave propagation in the phononic crystals. In Section 3, we propose the unfitted numerical method to compute the band structure of phononic crystal based on the Bloch–Floquet theory and prove the proposed method admits a unique solution. In Section 4, we carry out the optimal error analysis. In Section 5, we present some numerical examples in a realistic setting to verify and validate our theoretical discoveries. At the end, some conclusion is drawn in Section 6.
Section snippets
Model of phononic crystal
In this section, we first present a little digest to the two-dimensional phononic crystal. Then we consider the model of in-plane wave propagation.
Unfitted Nitsche’s method for computing band structure
In this section, we are going to propose an unfitted numerical method to efficiently compute band structures for generic phononic crystals. The numerical challenges brought by the eigenvalue problem (2.16) is twofold: one is quasi-periodic nature of the Bloch wave and the other one is the inhomogeneity of the material. These challenges shall be discussed in the following subsections.
Error estimates
In this section, we shall conduct the error analysis for the proposed unfitted Nitsche’s method (3.14) using the Babuška–Osborn theory. To this end, we introduce an extension operator () to extend an function defined on a subdomain to the fundamental cell . For a function , the extend function is defined to satisfy and for .
For , let be the Scott–Zhang interpolation operator [40] on . The interpolation operator on
Numerical examples
In this section, we shall use several benchmark numerical examples to validate our theoretical results and to illustrate the efficiency of the proposed unfitted numerical method in the computation of band structures of phononic crystals. In the following tests, we shall consider the aurum/epoxy phononic crystal and the aluminium/epoxy phononic crystal as in [18]. The aurum (Au) scatters or the aluminium (Al) scatters are embedded in the epoxy matrix. Their material constants are documented in
Conclusions
In this paper, a new finite element method for computing band structures of phononic crystals with general material interfaces is proposed. To handle the quasi-periodic boundary condition, we transform the equation into an equivalent interface eigenvalue problem with periodic boundary conditions by applying the Floquet–Bloch transform. The distinguishing feature of the proposed method is that it does not require the background mesh to fit the material interface which avoids the heavy burden of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
H.G. was partially supported by Andrew Sisson Fund of the University of Melbourne, Australia, X.Y. was partially supported by the National Science Foundation, USA under grant DMS-1818592, and Y.Z. was partially supported by National Natural Science Foundation of China under grant 11871299.
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