Topology optimization for unsteady flow

A computational methodology for optimizing the conceptual layout of unsteady flow problems at low Reynolds numbers is presented. The geometry of the design is described by the spatial distribution of a fictitious material with continuously varying porosity. The flow is predicted by a stabilized finite element formulation of the incompressible Navier-Stokes equations. A Brinkman penalization is used to enforce zero-velocities in solid material. The resulting parameter optimization problem is solved by a non-linear programming method. The paper studies the feasibility of the material interpolation approach for optimizing the topology of unsteady flow problems. The derivation of the governing equations and the adjoint sensitivity analysis are presented. A design-dependent stabilization scheme is introduced to mitigate numerical instabilities in porous material. The emergence of non-physical artifacts in the optimized material distribution is observed and linked to an insufficient resolution of the flow field and an improper representation of the pressure field within solid material by the Brinkman penalization. Two numerical examples demonstrate that the designs optimized for unsteady flow differ significantly from their steady-state counterparts. Copyright (C) 2011 John Wiley & Sons, Ltd.