3D Topology optimization of Stokes flow problems

The present talk is concerned with the application of topology optimization to creeping flow problems in 3D. This research is driven by the fact that topology optimization has proven very successful as a tool in academic and industrial design problems. Success stories are reported from such diverse fields as solid mechanics and optics and is due to the method's flexibility in the (rough) parametrization of the design, see [1] and the reference therein for an overview. Borrvall and Petersson [2] is the seminal reference for topology optimization in fluid flow problems. They considered design of energy efficient devices for 2D Stokes flow. Creeping flow problems are described by the Stokes equations which model very viscous fluids at macro scales or ordinary fluids at very small scales. The latter gives the motivation for topology optimization problems based on the Stokes equations being a model for the fluid behavior in a micro fluidic device. Such a device has finite size and a large degree of freedom for the design of geometry. Physically Stokes flow is an exotic inertia free flow. This, however, complicates mixing by passive devices. Passive devices, that is, devices without moving parts, are often of practical interest since they are easily manufacturable and maintenance free. In order to tackle such a challenging problem a robust method is needed which we approach by this contribution. The finite size of a micro fluidic device calls for 3D modelling of the equations, in particular when the design geometry is non-trivial as typically seen in topology design. The presentation elaborates on effects caused by 3D fluid modelling on the design. Numerical examples relevant for optimal micro fluidic mixer design are shown where the design is planar - compliant with micro fabrication techniques - and where the designs are 3D. Moreover, preliminary results show that a formulation of an optimization problem that maximizes mixing with a constraint on the pressure drop across the device gives promising results. To measure the mixing a step temperature profile is convected through the design and the resulting temperature profile at the outlet boundary is measured. The closer the outlet temperature profile is to the average inlet temperature the better mixing occurs in the device. For this problem the P$\acute{\textrm{e}}$clet number $Pe\gg1$ such that mixing by pure diffusion is not an option. Instead, the optimizer suggests a design that stretches the hot-cold interface, which is encouraging since "stretching and folding" is known to be key ingredients in efficient mixing. The modelling is performed using a finite element based solver, with analytically derived sensitivities that drives a gradient based optimization algorithm. Secondly, this talk also has its focus on the parallel implementation of the solution procedures, using OpenMP [3] on medium to large SMP computers. The necessary setup to achieve a good performance is described in detail. Further, issues such as scalability and portability are discussed. [1] M.P. Bendsøe and O. Sigmund. Topology optimization - theory, methods and applications, 2nd Edition, Springer 2003. [2] T. Borrvall, J. Petersson. Topology optimization of fluids in Stokes flow. Int. J. Num. Meth. Fluids, Vol. 41, 77-107, 2003. DOI:10.1002/fld.426 [3] The OpenMP Application Program Interface, http://www.openmp.org/