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

  • Authors:

  • Shiwei Zhou;Qing Li

  • Affiliations:

  • School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia;School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2008

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Abstract

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.