Implementation of a continuous adjoint for topology optimization of ducted flows

Topology optimization of fluid dynamical systems is still in its infancy, with its first academic realizations dating back to just four years ago. In this paper, we present an approach to fluid dynamic topology optimization that is based on a continuous adjoint. We briefly introduce the theory underlying the computation of topological sensitivity maps, discuss our implementation of this methodology into the professional CFD solver OpenFOAM and present results obtained for the optimization of an airduct manifold wrt. dissipated power. I. Fluid dynamic topology optimization In structure mechanics, topology optimization is a well-established concept for design optimization with respect to tension or stiness. 1 Its transfer to computational fluid dynamics, however, began just four years ago with the pioneering work of Borrvall and Petersson. 2 Since then, this topic has received significant interest in both academia and industry 3 9 . The starting point for fluid dynamic topology optimization is a volume mesh of the entire installation space. Based on a computation of the flow solution inside this domain, a suitable local criterion is applied to decide whether a fluid cell is favourable or counterproductive for the flow in terms of the chosen cost function. In order to iteratively remove the identified counterproductive cells from the fluid domain, they are either punished via a momentum loss term, or holes are inserted into the flow domain, with their positions being determined from an evaluation of the so-called topological asymptotic. In the former case, the momentum loss term is usually realized via a finite cell porosity, i.e. the whole design domain is treated as a porous medium: Each cell is assigned an individual porosity i, which is modeled via Darcy’s law. The value of i determines if the cell is fluid-like (low porosity values) or has a rather solid character (high values of i). In other words, the porosity field controls the geometry, and the i are the actual design variables. Within this framework, an adjoint method can be applied to elegantly compute the sensitivities of the chosen cost function wrt. the porosity of each cell. The obtained sensitivities can then be fed into a gradientbased optimization algorithm ‐ possibly with some penalization of intermediate porosity values in order to enforce a “digital” porosity distribution, and after several iterations, the desired optimum topology is finally extracted as an iso-surface of the obtained porosity distribution or similar post-processing operations. In a recent study, Othmer et al. 8 were able to verify the applicability of this methodology to typical