Topological derivative for a fast identification of short, linear perfectly conducting cracks with inaccurate background information

In this study, we consider a topological derivative-based imaging technique for the fast identification of short, linear perfectly conducting cracks completely embedded in a two-dimensional homogeneous domain with smooth boundary. Unlike conventional approaches, we assume that the background permittivity and permeability are unknown due to their dependence on frequency and temperature, and we propose a normalized imaging function to localize cracks. Despite inaccuracies in background parameters, application of the proposed imaging function enables to recognize the existence of crack but it is still impossible to identify accurate crack locations. Furthermore, the shift in crack localization of imaging results is significantly influenced by the applied background parameters. In order to theoretically explain this phenomenon, we show that the imaging function can be expressed in terms of the zero-order Bessel function of the first kind, the crack lengths, and the applied inaccurate background wavenumber corresponding to the applied inaccurate background permittivity and permeability. Various numerical simulations results with synthetic data polluted by random noise validate the theoretical results.