Shape reconstruction for advection-diffusion problems by shape optimization techniques: The case of constant velocity

Abstract

We consider the inverse problem of identifying an unknown portion of the boundary of a two-dimensional body $ \Omega $ by using a pair of Cauchy data $ (f, g) $ on the accessible portion $ \Sigma $ associated with the solution of the advection-diffusion problem – with constant velocity and diffusivity coefficient – denoted as $ u $ in $ \Omega $. We propose to solve the considered inverse problem using shape optimization methods, which are well-suited to this type of problem. In this direction, we demonstrate that the state variable corresponding to the advection-diffusion problem is differentiable with respect to the shape, and we rigorously derive its material derivative. The problem is recast into two different shape optimization formulations, and the corresponding shape derivatives of the cost functions – in boundary integral forms – are obtained by introducing appropriate adjoint systems. The shape gradient information is then used in a gradient-based scheme to approximate a solution to the optimization problems. Numerical results are provided to illustrate the feasibility of the proposed numerical methods in two spatial dimensions.