A Geometric Inverse Problem for the Detection of an Anomaly Using Topological Sensitivity Analysis Method
In this paper, we study a geometric inverse problem related to steady- state heat equation. This geometric inverse problem concerns the recovery of anomalies from the knowledge of overdetermined boundary data in a specimen of human tissue. We propose an alternative method based on the Khon-Vogelius formulation combined with the topological gradient approach. The inverse problem is formulated as a topology optimization one. An asymptotic expansion is derived for Khon-Vogelius design function with respect to the presence of an arbitrarily shaped hole (anomaly) with Neumann boundary condition inside the initial domain (human tissue). Then, we present a numerical approach for the reconstruction of the unknown anomaly using the level curve of the topological gradient. Some numerical simulations are presented to illustrate the efficiency and the accuracy of the proposed approach.