  In this paper gradient based topology optimization (TO) is used to discover
3-D phononic structures that exhibit ultra-wide normalized all-angle all-mode
band gaps. The challenging computational task of repeated 3-D phononic
band-structure evaluations is accomplished by a combination of a fast mixed
variational eigenvalue solver and distributed Graphic Processing Unit (GPU)
parallel computations. The TO algorithm utilizes the material
distribution-based approach and a gradient-based optimizer. The design
sensitivity for the mixed variational eigenvalue problem is derived using the
adjoint method and is implemented through highly efficient vectorization
techniques. We present optimized results for two-material simple cubic (SC),
body centered cubic (BCC), and face centered cubic (FCC) crystal structures and
show that in each of these cases different initial designs converge to single
inclusion network topologies within their corresponding primitive cells. The
optimized results show that large phononic stop bands for bulk wave propagation
can be achieved at lower than close packed spherical configurations leading to
lighter unit cells. For tungsten carbide - epoxy crystals we identify all angle
all mode normalized stop bands exceeding 100%, which is larger than what is
possible with only spherical inclusions.
