Spectral extended finite element method for band structure calculations in phononic crystals

Abstract

This talk introduces the extended finite element method (X-FEM) on structured higher-order (spectral) finite element meshes for computing the band structure of one- and two-dimensional phononic composites. The X-FEM enables the modeling of holes and inclusions in geometry through the framework of partition-of-unity enrichment. This reduces the burden on mesh generation, since meshes do not need to conform to geometric features in the periodic domain. Further, with iterative design processes such as phononic shape optimization, the need for remeshing is eliminated. To obtain theoretical rates of convergence with higher-order extended finite elements, careful consideration of curved geometry representation and numerical integration is required. In two dimensions, we adopt rational Bezier representation of conic sections or optimized cubic Hermite functions to model implicit boundaries described by level sets. Efficient computation of weak form integrals is realized through the homogeneous numerical integration scheme—a method that uses Euler’s homogeneous function theorem and Stokes’s theorem to reduce integration to the boundary of the domain. Ghost penalty stabilization is used to improve matrix conditioning on finite elements that are cut by a hole. Band structure calculations on perforated materials as well as on two-phase phononic crystals are presented that affirm the sound accuracy and optimal convergence of the method on structured, higher-order spectral finite element meshes.