Distributed parameter identification for the Navier–Stokes equations for obstacle detection

We present a parameter identification problem for a scalar permeability field and the maximum velocity in an inflow, following a reference profile. We utilize a modified version of the Navier–Stokes equations, incorporating a permeability term described by the Brinkman’s Law into the momentum equation. This modification takes into account the presence of obstacles on some parts of the boundary. For the outflow, we implement a directional do-nothing condition as a means of stabilizing the backflow. This work extends our previous research published in (Aguayo et al 2021 Inverse Problems 37 025010), where we considered a similar inverse problem for a linear Oseen model with do-nothing boundary conditions on the outlet and numerical simulations in 2D. Here we consider the more realistic case of Navier–Stokes equations with a backflow correction on the outflow and 3D simulations of the identification of a more realistic tricuspid cardiac valve. From a reference velocity that could have some noise or be obtained in low resolution, we define a suitable quadratic cost functional with some regularization terms. Existence of minimizers and first and second order optimality conditions are derived through the differentiability of the solutions of the Navier–Stokes equations with respect to the permeability and maximum velocity in the inflow. Finally, we present some synthetic numerical test based of recovering a 2D and 3D shape of a cardiac valve from total and local velocity measurements, inspired from 2D and 3D MRI.