In this paper, we consider the geometric inverse problem of recovering an obstacle ω immersed in a bounded fluid flow Ω governed by the time-dependent Brinkman model. We reformulate the inverse problem into an optimization problem using a least squares functional. We prove the existence of an optimal solution to the optimization problem. Then, we perform the asymptotic expansion of the cost function in a simple way using a penalty method. An important advantage of this method is that it avoids the truncation method used in the literature. To reconstruct the obstacle, we propose a fast algorithm based on the topological derivative. Finally, we present some numerical experiments in two- and three-dimensional cases showing the efficiency of the proposed method.

We consider the inverse problem of recovering piecewise scattering and piecewise absorption jump sets from boundary measurements and from a single measurement of the absorbed energy. We propose a reconstruction method based on a shape optimization approach combined with the level set techniques. Our main result includes the partial shape derivatives of two different shape functionals, using differentiability properties of the min-sup combined with a function space parameterization technique. In particular, it reveals the expression of the distributed partial shape derivative in tensor form. Based on the computed distributed partial shape derivatives, we introduce and implement a numerical approach based on the gradient approach and level set method. We present several numerical experiments to show the efficiency of the method both for exact reconstruction data and for realistic data with noise.