Zhou, Shiwei; Li, Qing A variational level set method for the topology optimization of steady-state Navier-Stokes flow. (English) Zbl 1218.76023 J. Comput. Phys. 227, No. 24, 10178-10195 (2008). Summary: The smoothness of topological interfaces often largely affects the fluid optimization and sometimes makes the density-based approaches, though well established in structural designs, inadequate. This paper presents a level-set method for topology optimization of steady-state Navier-Stokes flow subject to a specific fluid volume constraint. The solid-fluid interface is implicitly characterized by a zero-level contour of a higher-order scalar level set function and can be naturally transformed to other configurations as its host moves. A variational form of the cost function is constructed based upon the adjoint variable and Lagrangian multiplier techniques. To satisfy the volume constraint effectively, the Lagrangian multiplier derived from the first-order approximation of the cost function is amended by the bisection algorithm. The procedure allows evolving initial design to an optimal shape and/or topology by solving the Hamilton-Jacobi equation. Two classes of benchmarking examples are presented in this paper: (1) periodic microstructural material design for the maximum permeability; and (2) topology optimization of flow channels for minimizing energy dissipation. A number of 2D and 3D examples well demonstrated the feasibility and advantage of the level-set method in solving fluid-solid shape and topology optimization problems. Cited in 52 Documents MSC: 76D55 Flow control and optimization for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 76M25 Other numerical methods (fluid mechanics) (MSC2010) 76M30 Variational methods applied to problems in fluid mechanics 49Q10 Optimization of shapes other than minimal surfaces 65K10 Numerical optimization and variational techniques Keywords:topology optimization; level set method; variational method; Navier-Stokes flow; maximum permeability; minimum energy dissipation Software:top.m PDFBibTeX XMLCite \textit{S. Zhou} and \textit{Q. Li}, J. Comput. 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