Topology optimization for surface flows

Surface flows can represent the motions of the viscous and incompressible fluid at the solid/fluid interfaces. This paper presents a topology optimization approach for surface flows and extends the design space of topology optimization of fluidic structures onto the curved surfaces in the forms of 2-manifolds corresponding to the geometrical configurations of the solid/fluid interfaces. The presented approach is implemented by filling a porous medium onto the 2-manifolds. An artificial Darcy friction is correspondingly added to the area force term of the surface Navier-Stokes equations used to describe the surface flows and the physical area forces are penalized to eliminate their existence in the fluidic regions and to avoid the invalidity of the porous medium based topology optimization model. Topology optimization for the steady and unsteady surface flows has been executed by iteratively evolving the impermeability of the porous medium, where the impermeability is interpolated by the material density derived from a design variable. The related partial differential equations are solved by using the surface finite element method. Numerical examples have been provided to demonstrate this topology optimization approach for the surface flows, including the boundary velocity driven flows, the area force driven flows and the convection-diffusion flows.