Higher-order topological sensitivity for 2-D potential problems. Application to fast identification of inclusions

This article concerns an extension of the topological derivative concept for 2-D potential problems involving penetrable inclusions, whereby a cost function J is expanded in powers of the characteristic size ε of a small inclusion. The O(ε⁴) approximation of J is established for a small inclusion of given location, shape and conductivity embedded in a 2-D region of arbitrary shape and conductivity, and then generalized to several such inclusions. Simpler and more explicit versions of this result are obtained for a centrally-symmetric inclusion and a circular inclusion. Numerical tests are performed on a sample configuration, for (i) the O(ε⁴) expansion of potential energy, and (ii) the identification of a hidden inclusion. For the latter problem, a simple approximate global search procedure based on minimizing the O(ε⁴) approximation of J over a dense search grid is proposed and demonstrated.