Penalization model for Navier-Stokes-Darcy equations with application to porosity-oriented topology optimization

Topology optimization for fluid flow aims at finding the location of a porous medium minimizing a cost functional under constraints given by the Navier-Stokes equations. The location of the porous media is usually taken into account by adding a penalization term alpha u, where alpha is a kinematic viscosity divided by a permeability and u is the velocity of the fluid. The fluid part is obtained when alpha = 0 while the porous (solid) part is defined for large enough a since this formally yields u = 0. The main drawback of this method is that only solid that does not let the fluid to enter, that is perfect solid, can be considered. In this paper, we propose to use the porosity of the media as optimization parameter hence to minimize some cost function by finding the location of a porous media. The latter is taken into account through a singular perturbation of the Navier-Stokes equations for which we prove that its weak-limit corresponds to an interface fluid-porous medium problem modeled by the Navier-Stokes-Darcy equations. This model is then used as constraint for a topology optimization problem. We give necessary condition for such problem to have at least an optimal solution and derive first, order necessary optimality condition. This paper ends with some numerical simulations, for Stokes flow, to show the interest of this approach.