Identifying Lamé parameters from time-dependent elastic wave measurements

Abstract

In many sectors of today’s industry it is of utmost importance to detect defects in elastic structures contained in technical devices to guarantee their failure-free operation. As currently used signal processing techniques have natural limits with respect to accuracy and significance, modern mathematical methods are crucial to improve current algorithms. We consider in this paper a parameter identification approach for isotropic and linear elastic structures described by their Lamé parameters and a material density. This approach can be employed for non-destructive defect detection, location and characterization from time-dependent measurements of one elastic wave. To this end, we show that the operator linking the static parameters with the wave measurements is Fréchet differentiable, which allows to set up Newton-like methods for the non-linear parameter identification problem. We indicate the performance of a specific inexact Newton-like regularization method by numerical examples for a testing problem of a thin plate from measurements of the normal component of the displacement field on the boundary. As an extension, we further augment this method with a total variation regularization and thereby improve reconstructed parameters that feature edges.