Topology optimization of fluid mechanics problems

While topology optimization for solid continuum structures have been studied for about 20 years and for the special case of trusses for many more years, topology optimization of fluid mechanics problems is more recent. Borrvall and Petersson [1] is the seminal reference for topology optimization using the material distribution technique with an underlying partial differential equation describing the fluid motion. The mathematical basis of departure is the incompressible Stokes equation with an extra absorption term which allows for material interpolation between Stokes flow and a model of Darcy flow (porous flow). This talk elaborates on the work of Borrvall and Petersson and has its focus on how to apply topology optimization using the material distribution technique to steady-state viscous incompressible flow problems. The target design applications are fluid devices that are optimized with respect to minimizing the energy loss, characteristic properties of the velocity field or mixing properties. To reduce the computational complexity of the topology optimization problems the primary focus is put on the Stokes equation in 2D and in 3D. However, the the talk also contains examples with the 2D Navier-Stokes equation as well as an example with convection dominated transport in 2D Stokes flow. Using Stokes flow limits the range of applications; nonetheless, the present work gives a proof-of-concept for the application of the method within fluid mechanics problems and it remains of interest for the design of microfluidic devices. Furthermore, the talk demonstrates how the software COMSOL can be used in topology optimization research to study different problem formulations related to topology optimization of fluid dynamics problems. Moreover, COMSOL has also been used as a post processing tool. Prior to design manufacturing this allows the engineer to quantify the performance of the computed topology design using standard, credible analysis tools with a body-fitted mesh. [1] Borrvall and Petersson (2003) "Topology optimization of fluids in Stokes flow", Int. J. Num. Meth. Fluids, 41, pp 77-107.