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SUMMARYA 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.

This study is concerned with a generalized shape optimization approach for finding the geometry of fluidic devices and obstacles immersed in flows. Our approach is based on a level set representation of the fluid-solid interface and a hydrodynamic lattice Boltzmann method to predict the flow field. We present an explicit level set method that does not involve the solution of the Hamilton-Jacobi equation and allows using standard nonlinear programming methods. In contrast to previous works, the boundary conditions along the fluid-structure interface are enforced by second-order accurate interpolation schemes, overcoming shortcomings of flow penalization methods and Brinkman formulations frequently used in topology optimization. To ensure smooth boundaries and mesh-independent results, we introduce a simple, computationally inexpensive filtering method to regularize the level set field. Furthermore, we define box constraints for the design variables that guarantee a continuous evolution of the boundaries. The features of the proposed method are studied by two numeric examples of two-dimensional steady-state flow problems.

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