Analysis of particular phononic structures using single integral equations

A fast method for determining phononic (and photonic) bandgaps in composite materials is developed. It is known that in the propagation of waves in a 3D medium containing N scatterers arranged periodically, there exist refractive indices for which such structures have bandgaps, i.e., frequencies for which no waves can propagate inside. Our task is to find the frequencies that generate these prohibited waves. This requires the solution of an eigenvalue problem for the Helmholtz operator. To solve this problem we choose an alternate route which uses boundary integral equations. We derive a single integral equation on each of the interfaces between the outer region and the scatterers, considering a general transmission boundary condition, by using a hybrid method using layer potentials and Green’s formula. This approach reduces the number of unknowns considerably in comparison to other methods, but requires the treatment of large dense matrices and many matrix vector multiplications. To remedy this, we use the Fast Multipole Method. For solving the eigenvalue problem we discuss two methods: the Newton method and a method based on the Cauchy formula. Details of the numerical implementation, and results will be presented for different cases: sound hard, sound soft, impedance and transmission boundary conditions. [Work partially supported by NSF Award 0219681 is gratefully acknowledged.]