Petrov–Galerkin method for the band structure computation of anisotropic and piezoelectric phononic crystals

Abstract In this paper, the Petrov-Galerkin finite element interface method is modified to the vectorial form and applied to compute the band structure of phononic crystals with complicated scatterer geometry. Value-periodic subspace and projective grid are employed in this method. Complete mathematical model together with the continuity and equilibrium conditions on the scatterer interface as well as the Bloch-periodic boundary conditions on the unit cell are presented for these systems. We then apply this method to compute the band structure of three kinds of special phononic crystals: phononic crystal with anisotropic inclusions, phononic crystal containing piezoelectric materials and phononic crystal containing nano-piezoelectric materials. Taking advantage of asymmetric basis functions and non-body-fitted meshes, our method is suitable for the calculation and analysis of phononic crystals with complicated scatterer geometries. With plenty of numerical experiments, the accuracy and convergency of the proposed method is demonstrated. The influences of the rotation angle of anisotropic inclusion, complicated scatterer shape, filling fraction of the scatterers, and the rotation angle of the scatterers to the band structure of these three kinds of phononic crystals are investigated. This will provide a new perspective for the design and manufacture of phononic crystals with specific band structures.