Topology optimization of fluid flows under vorticity local constraints

Abstract In the context of topology optimization of fluid flows, vorticity is treated as a global quantity. However, vorticity is inherently a local property of the flow field, analogous to mechanical stress in solid mechanics. In structural topology optimization, local constraints on stress are applied, but an equivalent local treatment for vorticity has not yet been explored in topology optimization of fluid flow. Therefore, this work proposes a topology optimization formulation for incompressible fluid flows in which vorticity is controlled through local constraints. The model considers the Navier–Stokes equations with a density-based material model and it is numerically discretized using the finite element method to solve the state equations. In this optimization problem, the design requirements on vorticity are enforced through local constraints, while the objective function (friction dissipation and regularization) is used to remove greyscale and promote well-defined designs. The vorticity local constraints are enforced using an Augmented Lagrangian (AL) algorithm. The sensitivities are calculated with pyadjoint libraries by performing the automatic differentiation and the optimization problem is solved using the L-BFGS-B algorithm adapted to the AL formulation. The implementation is evaluated through three examples: a preliminary case enforcing flow reversal, and two cases dealing upper and lower limits for vorticity local constraints.