Topology optimization of creeping fluid flows using a Darcy-Stokes finite element

• Published in: International journal for numerical methods in engineering 2006-04, Vol.66 (3), p.461-484
• Main Authors: Guest, James K., Prévost, Jean H.
• Format: Article
• Language: English
• Subjects: Computational fluid dynamics; Computational techniques; coupled flow; Darcy's law; Exact sciences and technology; Finite element method; Flows through porous media; Fluid dynamics; Fluid flow; Fluids; Fundamental areas of phenomenology (including applications); General theory; Mathematical analysis; Mathematical methods in physics; Mathematical models; Nonhomogeneous flows; Physics; porous media; Solid mechanics; Solid phases; stabilized finite element methods; Static elasticity (thermoelasticity...); Stokes equations; Structural and continuum mechanics; Topology optimization
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.1560

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. Copyright © 2005 John Wiley & Sons, Ltd.