  In this paper, we compute the band structure of one- and two-dimensional
phononic composites using the extended finite element method (X-FEM) on
structured higher-order (spectral) finite element meshes. On using
partition-of-unity enrichment in finite element analysis, the X-FEM permits use
of structured finite element meshes that do not conform to the geometry of
holes and inclusions. This eliminates the need for remeshing in phononic shape
optimization and topology optimization studies. In two dimensions, we adopt
rational B{\'e}zier representation of curved (circular) geometries, and
construct suitable material enrichment functions to model two-phase composites.
A Bloch-formulation of the elastodynamic phononic eigenproblem is adopted.
Efficient computation of weak form integrals with polynomial integrands is
realized via 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 on finite
elements that are cut by a hole. Band structure calculations on perforated
(circular holes, elliptical holes, and holes defined as a level set) 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. Several numerical examples demonstrate the
advantages of $p$-refinement made possible by the spectral extended finite
element method. In these examples, fourth-order spectral extended finite
elements deliver $\mathcal{O}(10^{-8})$ accuracy in frequency calculations with
more than thirty-fold fewer degrees-of-freedom when compared to quadratic
finite elements.
