A variational level set method for the topology optimization of steady-state Navier–Stokes flow

• Published in: Journal of computational physics Vol. 227; no. 24; pp. 10178 - 10195
• Main Authors: Zhou, Shiwei, Li, Qing
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 20.12.2008; Elsevier
• Subjects: Computational techniques; Exact sciences and technology; Level set method; Mathematical methods in physics; Maximum permeability; Minimum energy dissipation; Navier–Stokes flow; Physics; Topology optimization; Variational method; Topology optimization; Variational method; Maximum permeability; Level set method; Minimum energy dissipation; Navier–Stokes flow; Hamilton-Jacobi equations; Lagrangian; Variational methods; Permeability; Topology; Optimization; Calculation methods; Navier-Stokes flow; Algorithms; Energy dissipation; First order; Cost function; Calculation; Stokes flow
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2008.08.022

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.