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Topology optimization of creeping fluid flows using a Darcy-Stokes finite element. (English) Zbl 1110.76310
Summary: A new methodology is proposed for the topology optimization of fluid in Stokes flow. The binary design variable and no-slip condition along the solid-fluid interface are regularized to allow for the use of continuous mathematical programming techniques. The regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by Darcy’s law. Fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the Stokes and Darcy equations. The mixed formulation of the new Darcy-Stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems. Convergence to the no-slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid-fluid interface are not needed. The optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature. The advantages of the Darcy-Stokes approach include that it uses existing stabilization techniques to solve the finite element problem, it produces 0-1 (void-solid) topologies (i.e. there are no regions of artificial material), and that it can potentially be used to optimize the layout of a microscopically porous material.

MSC:
76D55 Flow control and optimization for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
76S05 Flows in porous media; filtration; seepage
76M10 Finite element methods applied to problems in fluid mechanics
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