Conformal gradient-index phononic crystal lenses: Design, theory, and application on non-planar structures

Gradient index phononic crystal (GRIN-PC) lenses have been widely recognized for their effectiveness in focusing or localizing elastic waves at specific target locations. This wave-focusing capability enhances the energy-harvesting performance of piezoelectric transducers and improves defect detection sensitivity in non-destructive evaluation (NDE) applications. While GRIN-PC lenses have been extensively studied for planar structures, their application to curved geometries remains limited, primarily due to the lack of a comprehensive theoretical framework for understanding wave behavior in non-planar phononic crystal structures. In this work, we develop a conformal GRIN-PC theory to analyze elastic wave focusing in curved structures and propose a systematic design framework for implementing GRIN-PC lenses on non-planar surfaces. The proposed theory models wave propagation within conformal GRIN-PC lenses using ray trajectory analysis, accurately predicting the focal region. We validate this framework through numerical simulations of a conformal GRIN-PC lens applied to a steel pipe and demonstrate its accuracy in predicting focal points. Furthermore, the design framework is applied to fabricate a 3D-printed conical GRIN-PC lens, with numerical simulations and experimental results confirming its wave-focusing performance. This work establishes a foundation for expanding GRIN-PC applications to non-planar structural components widely found in mechanical, aerospace, and civil engineering structures.