Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems

This article concerns an extension of the topological sensitivity (TS) concept for 2D potential problems involving insulated cracks, whereby a misfit functional J is expanded in powers of the characteristic size a of a crack. Going beyond the standard TS, which evaluates (in the present context) the leading O(a²) approximation of J, the higher-order TS established here for a small crack of arbitrarily given location and shape embedded in a 2-D region of arbitrary shape and conductivity yields the O(a⁴) approximation of J. Simpler and more explicit versions of this formulation are obtained for a centrally symmetric crack and a straight crack. A simple approximate global procedure for crack identification, based on minimizing the O(a⁴) expansion of J over a dense search grid, is proposed and demonstrated on a synthetic numerical example. BIE formulations are prominently used in both the mathematical treatment leading to the O(a⁴) approximation of J and the subsequent numerical experiments.