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.
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