Selective focusing of multiple traction-free elastic particles

Inverse elastic scattering of multiple well-separated particles has a wide range of applications but remains highly challenging due to the multiple scattering effect. In this paper, we applied the DORT (decomposition of the time reversal operator) method to this imaging problem, enabling selective focusing on each particle using far-field measurements. Through asymptotic analysis of the far-field operator and the decay properties of oscillatory integrals, we established a relationship between the eigenfunctions of the time reversal operator and the locations of the particles. Specifically, we show that under suitable conditions, each particle with traction-free boundary generates five significant eigenvalues, with the corresponding eigenfunctions producing incident waves that focus selectively on the associated particle. Application of this method to imaging elastic particles with multiple scattering effect is demonstrated through boundary integral formulations.