Topology optimization of fluid domains: kinetic theory approach

Abstract

AbstractWe consider the problem of optimal design of flow domains for Navier–Stokes flows in order to minimize a given performance functional. We attack the problem using topology optimization techniques, or control in coefficients, which are widely known in structural optimization of elastic solids and structures for their flexibility, generality, yet ease of use, and integration with existing FEM software. We use a simple kinetic model to approximate the Navier–Stokes system. Arguably, we take a rather unconventional path in the kinetic theory, using it only to gain insight about the Navier–Stokes‐related system of hydrodynamical equations, which we take as our starting point. Thus all the modifications we make to the kinetic models are “hydrodynamically” inspired and we seek no particular physical explanation for them; the only requirement for us is the convergence (at least, formal) of the kinetic equations towards the correct hydrodynamical limit. We formally compute the hydrodynamical limit of the proposed kinetic system, and rigorously establish the existence of flow solutions and their continuous dependence on the design parameters. Optimal controls are shown to belong to a special class (0–1 solutions) for a popular power dissipation minimization problem for viscous fluids.