Topology optimization of fluidic flows on two-dimensional manifolds
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This paper presents a topology optimization approach for the fluidic flows on two-dimensional manifolds, which can represent the viscous and incompressible material surfaces. The fluidic motion on such a material surface can be described by the surface Navier-Stokes equations, which are derived by using the elementary tangential calculus in terms of exterior differential operators expressed in a Cartesian coordinate 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 unsteady and steady surface flows is implemented by iteratively evolving the impermeability of the porous medium on two-dimensional manifolds, where the impermeability is interpolated by the material density derived from the design variable. The related partial differential equations are solved by using the surface finite element method. Numerical tests have been provided to demonstrated this topology optimization approach for fluidic flows on two-dimensional manifolds.
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