Non-radiating sources in elastodynamics and their applications in the exterior cloaking

Abstract

In this work, we develop a mathematical framework on constructed general non-radiating sources of elastic waves governed by the Navier equation via the approach of Helmholtz decomposition and potential theory in elastodynamics. Our study offers a rather comprehensive analysis. We first provide a rigorous justification of the general non-radiating sources. Based on the complete destructive interference of external elastic fields generated by specific radiating sources, a general non-radiating elastic source is derived and shown to possess a hidden interior wave field. For an incident wave, targets remain invisible within non-radiating source regions, and the geometry and boundary conditions of obstacles can be very general, which holds significant practical implications. Moreover, we introduce an effective novel method for designing such generalized non-radiating sources. To avoid the complex structure, we propose to use radiating source overlay construction on specific nodes at the boundary of non-radiating regions construction and derive sharp error estimates to evaluate the cloaking performance. The proposed scheme is capable of nearly cloaking arbitrary obstacles with a high accuracy. Numerical verifications validate the precision of our analytical findings.