A Novel Density Based Approach for Topology Optimization of Stokes Flow
Johannes Haubner, Franziska Neumann, Michael Ulbrich · SIAM Journal on Scientific Computing · 2023
Abstract. A new method for performing density-based topology optimization for Stokes flow is presented. It differs from previous approaches in the way the underlying mixed integer problem is relaxed and in the choice of the space in which the density that describes the topology lives. Initial numerical experiments, using [Formula: see text] for the density and a discretization by continuous piecewise linear finite elements, showed unsatisfactory convergence properties of state-of-the-art optimization solvers. This motivated the work in this paper, which proposes solutions to these difficulties. We present a theoretically founded new problem formulation based on a space for the density that allows for jumps along hypersurfaces, such as BV or fractional order Sobolev spaces. We extend the existing theory for the generalized Stokes equations and investigate the arising optimization problems concerning existence of solutions, differentiability, and convergence of relaxed solutions toward solutions of the original problem. We motivate a localized fractional order Sobolev norm as an approximation of the [Formula: see text]-norm for [Formula: see text]-valued functions and discuss its discretization by piecewise constant finite elements. Building on these theoretical findings, we present some numerical realizations and show computational results.