Identification of subwavelength microstructural information from macroscopic boundary measurements in elastodynamics

We present a method to obtain microstructural information from macroscopic boundary measurements exploiting scattering governed by the wave equation in a bounded linearly elastic domain in the long-wavelength regime. Applying a force to the outer boundary of the body on the macroscopic scale while measuring the resulting boundary displacement, we solve the inverse problem of identifying the geometry of the microstructure in the context of periodic homogenization minimizing a tracking-type objective functional as long as the geometry of the microstructure is parameterized by a finite set of parameters. Shape calculus is used to characterize the Gâteaux derivative of the objective function facilitating the use of gradient-based algorithms, and we present numerical experiments for a generic non-destructive testing problem for ellipsoidal microstructures showcasing the functioning of the identification method.