### Abstract

Topology optimization has been applied to a multitude of physical
systems and is now a mature technology used in industrial practice,
see [1] for an overview. Borrvall and Petersson
[2] introduced topology optimization of Stokes flow
problems which initiated works on extending topology optimization to
different flow problems. However, this research has focused on 2D
fluid modelling, which limits the practical impact of the computed
designs. The explanation of the limitation is that the finite size
domain used in topology optimization problems ensures that the
velocity components couples, even for Stokes flow [3].
Furthermore, it is questionable if such a coupling can be captured
by a 2D model especially in non-trivial geometries as typically seen
in topology design. This statement is widely accepted in the fluid
mechanics community, i.e. that planar fluid models are useful for
academic test problems only.
The motivation for considering topology optimization in 3D Stokes
flow originates from micro fluidic systems. At small scales the
Stokes equations are a reasonable mathematical model to use for the
fluid behavior. Physically Stokes flow is an exotic inertia free
flow, which in practice complicates mixing by passive devices.
Passive mixing devices are relevant particular at micro scales since
they are manufacturable (without moving parts) and maintenance free.
In order to tackle such a challenging problem a robust method is
needed which we approach by this contribution. It contains
fundamental aspects of the topology optimization method applied to
the Stokes equations as described below.
This work consists of two parts. The main part elaborates on effects
caused by 3D fluid modelling on the design. Numerical examples are
shown where the design is planar - relevant to micro fluidic
fabrication techniques - and where the designs are 3D. The second
part focuses on post--processing by a comparison of shape
optimization and topology optimization in 2D Stokes flow. This
investigates if topology design is sufficient for creating optimal
designs. This question is addressed in the setting of standard
analysis software which enables a credible performance
check relevant before design manufacturing. Note that this requires
a proper interpretation of a computed design used to generate a body
fitted mesh. In addition issues related to the parallel solution of
the linear algebra problems are discussed which is the critical
bottleneck for the 3D problem. The implementation uses
semi--analytical sensitivities to drive a gradient based
optimization algorithm.
[1]
M. P. Bendsøe and O. Sigmund, Topology Optimization -
Theory, Methods and Applications, Springer Verlag, Berlin
Heidelberg, 2003.
[2]
T. Borrvall and J. Petersson, Topology optimization of fluids in Stokes flow, Int. J. Num. Meth. Fluids 41, 77-107, 2003. DOI:10.1002/fld.426
[3]
E. Lauga, A. D. Stroock, and H. A. Stone, Three--dimensional flows
in slowly varying planar geometries. Physics of fluids 16(8), 3051-3062, 2004. DOI: 10.1063/1.1760105

Original language | English |
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Publication date | 2006 |

Publication status | Published - 2006 |

Event | 3rd European Conference on Computational Mechanics: Solids, Structures and Coupled Problems in Engineering - Lissabon, Portugal Duration: 5 Jun 2006 → 8 Jun 2006 Conference number: 3 |

### Conference

Conference | 3rd European Conference on Computational Mechanics |
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Number | 3 |

Country | Portugal |

City | Lissabon |

Period | 05/06/2006 → 08/06/2006 |

## Cite this

Gersborg-Hansen, A. (2006).

*Topology optimization of 3D Stokes flow problems*. Abstract from 3rd European Conference on Computational Mechanics, Lissabon, Portugal.