## Abstract

This paper presents the topology optimization method for the steady and unsteady incompressible Navier–Stokes flows driven by body forces, which typically include the constant force (e.g. the gravity) and the centrifugal and Coriolis forces. In the topology optimization problem, the artificial friction force with design variable interpolated porosity is added into the Navier–Stokes equations as the conventional method, and the physical body forces in the Navier–Stokes equations are penalized using the power-law approach. The topology optimization problem is analyzed by the continuous adjoint method, and solved by the finite element method in conjunction with the gradient based approach. In the numerical examples, the topology optimization of the fluidic channel, mass distribution of the flow and local velocity control are presented for the flows driven by body forces. The numerical results demonstrate that the presented method achieves the topology optimization of the flows driven by body forces robustly.

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## Acknowledgements

This work was supported by NSFC (Nos. 50975272 and 11034007), the 863 Program (No. 2012AA040503) and Hundred Talent Project in CAS. The authors are grateful to Professor Krister Svanberg for supply of the MMA code. The authors are also grateful to the reviewers’ kind attention and valuable suggestions.

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Deng, Y., Liu, Z. & Wu, Y. Topology optimization of steady and unsteady incompressible Navier–Stokes flows driven by body forces.
*Struct Multidisc Optim* **47, **555–570 (2013). https://doi.org/10.1007/s00158-012-0847-8

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### Keywords

- Topology optimization
- Navier–Stokes flows
- Body force
- Artificial friction force
- Power-law approach
- Continuous adjoint method