Reciprocity identities for quasi-static piezoelectric transducer models: Application to cavity identification using iterated excitations and a topological sensitivity approach

The focus of this article is the transient wave-based detection and identification of defects embedded in isotropic elastic solids using piezoelectric transducers. This work addresses this problem within a comprehensive framework encompassing description of elastic wave propagation within the probed media as well as consideration of the coupling phenomena induced by the transducers. A fundamental reciprocity identity associated with a quasi-static piezoelectric model is derived to lay the foundations of ensuing developments and approach of this inverse scattering problem. Modeling of piezoelectric transducers is discussed and application of the proven reciprocity theorem enables the proposition of an iterative construction procedure of electric inputs generating waves expected to focus on the sought defects. The characteristic features of the inverse problem considered, which uses piezoelectric sensor-based measurements, are also discussed. Next, the identification problem is investigated by way of an adjoint field-based topological sensitivity approach that permits the construction of a defect indicator function based on the derived reciprocity identity. For simplicity of exposition, the studied configurations involve defects in the form of traction-free cavities. Finally, a set of 2D numerical examples based on the spectral finite-elements method is presented to assess the performances of the proposed approach in identifying embedded defects from electric measurements.