  This paper presents a topology optimization approach for surface flows, which
can represent the viscous and incompressible fluidic motions at the
solid/liquid and liquid/vapor interfaces. The fluidic motions on such material
interfaces can be described by the surface Navier-Stokes equations defined on
2-manifolds or two-dimensional manifolds, where the elementary tangential
calculus is implemented in terms of exterior differential operators expressed
in a Cartesian system. Based on the topology optimization model for fluidic
flows with porous medium filling the design domain, an artificial Darcy
friction is added to the area force term of the surface Navier-Stokes equations
and the physical area forces are penalized to eliminate their existence in the
fluidic regions and to avoid the invalidity of the porous medium model.
Topology optimization for steady and unsteady surface flows can be implemented
by iteratively evolving the impermeability of the porous medium on the
2-manifolds, 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 surface flows,
including the boundary velocity driven flows, area force driven flows and
convection-diffusion flows.
