Topology optimization of 3D Stokes flow problems

Topology optimization has been applied to a multitude of physical systems and is now a mature technology used in industrial practice, see [1] for an overview. Borrvall and Petersson [2] introduced topology optimization of Stokes flow problems which initiated works on extending topology optimization to different flow problems. However, this research has focused on 2D fluid modelling, which limits the practical impact of the computed designs. The explanation of the limitation is that the finite size domain used in topology optimization problems ensures that the velocity components couples, even for Stokes flow [3]. Furthermore, it is questionable if such a coupling can be captured by a 2D model especially in non-trivial geometries as typically seen in topology design. This statement is widely accepted in the fluid mechanics community, i.e. that planar fluid models are useful for academic test problems only. The motivation for considering topology optimization in 3D Stokes flow originates from micro fluidic systems. At small scales the Stokes equations are a reasonable mathematical model to use for the fluid behavior. Physically Stokes flow is an exotic inertia free flow, which in practice complicates mixing by passive devices. Passive mixing devices are relevant particular at micro scales since they are manufacturable (without moving parts) and maintenance free. In order to tackle such a challenging problem a robust method is needed which we approach by this contribution. It contains fundamental aspects of the topology optimization method applied to the Stokes equations as described below. This work consists of two parts. The main part elaborates on effects caused by 3D fluid modelling on the design. Numerical examples are shown where the design is planar - relevant to micro fluidic fabrication techniques - and where the designs are 3D. The second part focuses on post--processing by a comparison of shape optimization and topology optimization in 2D Stokes flow. This investigates if topology design is sufficient for creating optimal designs. This question is addressed in the setting of standard analysis software which enables a credible performance check relevant before design manufacturing. Note that this requires a proper interpretation of a computed design used to generate a body fitted mesh. In addition issues related to the parallel solution of the linear algebra problems are discussed which is the critical bottleneck for the 3D problem. The implementation uses semi--analytical sensitivities to drive a gradient based optimization algorithm. [1] M. P. Bendsøe and O. Sigmund, Topology Optimization - Theory, Methods and Applications, Springer Verlag, Berlin Heidelberg, 2003. [2] T. Borrvall and J. Petersson, Topology optimization of fluids in Stokes flow, Int. J. Num. Meth. Fluids 41, 77-107, 2003. DOI:10.1002/fld.426 [3] E. Lauga, A. D. Stroock, and H. A. Stone, Three--dimensional flows in slowly varying planar geometries. Physics of fluids 16(8), 3051-3062, 2004. DOI: 10.1063/1.1760105