  The gradient index phononic crystal (GRIN-PC) lens concept has been proven
very effective for focusing elastic waves at a desired location. Although
well-studied for planar structures, GRIN-PC lenses for elastic wave focusing in
curved structures are scarce and lack the theoretical framework for studying
the wave focusing mechanism. In this work, we develop conformal GRIN-PC theory
to analyze wave focusing in non-planar geometries and present a design
framework for conformal GRIN-PC lenses to be implemented over curved
structures. The proposed conformal GRIN-PC theory studies the wave propagation
in a curved GRIN-PC lens using ray trajectories that meet at the focal spot of
the lens. We apply the conformal GRIN-PC theory to accurately predict the focal
region of the GRIN-PC lens implemented over a steel pipe and validate the
results with numerical simulations. Further, the design framework is utilized
to design a 3D-printed conical GRIN-PC lens. The elastic wave focusing in the
conical lens is demonstrated using numerical simulations and is further
validated with experiments.
