# Fluid flow topology optimization in PolyTop: stability and computational implementation

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## Abstract

We present a Matlab implementation of topology optimization for fluid flow problems in the educational computer code PolyTop (Talischi et al. 2012b). The underlying formulation is the well-established porosity approach of Borrvall and Petersson (2003), wherein a dissipative term is introduced to impede the flow in the solid (non-fluid) regions. Polygonal finite elements are used to obtain a stable low-order discretization of the governing Stokes equations for incompressible viscous flow. As a result, the same mesh represents the design field as well as the velocity and pressure fields that characterize its response. Owing to the modular structure of PolyTop, incorporating new physics, in this case modeling fluid flow, involves changes that are limited mainly to the analysis routine. We provide several numerical examples to illustrate the capabilities and use of the code. To illustrate the modularity of the present approach, we extend the implementation to accommodate alternative formulations and cost functions. These include topology optimization formulations where both viscosity and inverse permeability are functions of the design; and flow control where the velocity at a certain location in the domain is maximized in a prescribed direction.

## Keywords

Topology optimization Polygonal finite elements Matlab Stokes flow## Notes

### Acknowledgments

Glaucio H. Paulino acknowledges support from the US National Science Foundation through grants #1559594 (formerly #1335160) and #1624232 (formerly #1437535). Ivan F. M. Menezes and Anderson Pereira acknowledge the financial support provided by Tecgraf/PUC-Rio (Group of Technology in Computer Graphics), Rio de Janeiro, Brazil. The information presented in this publication is the sole opinion of the authors and does not necessarily reflect the views of the sponsors or sponsoring agencies.

## References

- Allaire G (2001) Shape optimization by the homogenization method. Springer, New YorkzbMATHGoogle Scholar
- Beirão Da Veiga L, Lipnikov K (2010) A mimetic discretization of the Stokes problem with selected edge bubbles. SIAM J Sci Comput 32(2):875–893. doi: 10.1137/090767029 MathSciNetCrossRefzbMATHGoogle Scholar
- Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer-Verlag, BerlinzbMATHGoogle Scholar
- Bochev P, Lehoucq RB (2001) On finite element solution of the pure Neumann problem. SIAM Rev 47:50–66. doi: 10.1137/S0036144503426074 MathSciNetCrossRefzbMATHGoogle Scholar
- Borrvall T, Petersson J (2003) Topology optimization of fluids in Stokes flow. Int J Numer Methods Fluids 41(1):77–107. doi: 10.1002/fld.426 MathSciNetCrossRefzbMATHGoogle Scholar
- Challis VJ, Guest JK (2009) Level set topology optimization of fluids in Stokes flow. Int J Numer Methods Eng 79(10):1284–1308. doi: 10.1002/nme.2616 MathSciNetCrossRefzbMATHGoogle Scholar
- Deng Y, Liu Z, Zhang P, Liu Y, Wu Y (2011) Topology optimization of unsteady incompressible Navier-Stokes flows. J Comput Phys 230(17):6688–6708. doi: 10.1016/j.jcp.2011.05.004 MathSciNetCrossRefzbMATHGoogle Scholar
- Donea J, Huerta A (2003) Finite element methods for flow problems. Wiley, West SussexCrossRefGoogle Scholar
- Evgrafov A (2005) The limits of porous materials in the topology optimization of Stokes flows. Appl Math Optim 52:263–277. doi: 10.1007/s00245-005-0828-z MathSciNetCrossRefzbMATHGoogle Scholar
- Gersborg-Hansen A, Sigmund O, Haber R (2005) Topology optimization of channel flow problems. Struct Multidiscip Optim 30(3):181–192. doi: 10.1007/s00158-004-0508-7 MathSciNetCrossRefzbMATHGoogle Scholar
- Guest JK, Prévost JH (2006) Topology optimization of creeping fluid flows using a Darcy-Stokes finite element. Int J Numer Methods Eng 66(3):461–484. doi: 10.1002/nme.1560 MathSciNetCrossRefzbMATHGoogle Scholar
- Kreissl S, Maute K (2012) Levelset based fluid topology optimization using the extended finite element method. Struct Multidiscip Optim 46(3):311–326. doi: 10.1007/s00158-012-0782-8
- Kreissl S, Pingen G, Maute K (2011) An explicit level set approach for generalized shape optimization of fluids with the Lattice Boltzmann method. Int J Numer Methods Fluids 65(5):496–519. doi: 10.1002/fld.2193 CrossRefzbMATHGoogle Scholar
- Kreissl S, Pingen G, Maute K (2011) Topology optimization for unsteady flow. Int J Numer Methods Eng 87(13):1229–1253. doi: 10.1002/nme.3151 MathSciNetzbMATHGoogle Scholar
- Pereira A, Menezes IFM, Talischi C, Paulino GH (2011) An efficient and compact Matlab implementation of topology optimization: application to compliant mechanisms. In: XXXII Iberian Latin-American Congress on Computational Methods in EngineeringGoogle Scholar
- Pingen G, Evgrafov A, Maute K (2007) Topology optimization of flow domains using the lattice Boltzmann method. Struct Multidisc Optim 34(6):507–524. doi: 10.1007/s00158-007-0105-7 10.1007/s00158-007-0105-7 MathSciNetCrossRefzbMATHGoogle Scholar
- Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373. doi: 10.1002/nme.1620240207 MathSciNetCrossRefzbMATHGoogle Scholar
- Talischi C, Paulino GH, Pereira A, Menezes IFM (2012) PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct Multidiscip Optim 45(3):309–328. doi: 10.1007/s00158-011-0706-z 10.1007/s00158-011-0706-z MathSciNetCrossRefzbMATHGoogle Scholar
- Talischi C, Paulino GH, Pereira A, Menezes IFM (2012) PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Struct Multidiscip Optim 45 (3):329–357. doi: 10.1007/s00158-011-0696-x MathSciNetCrossRefzbMATHGoogle Scholar
- Talischi C, Pereira A, Paulino GH, de Menezes IFM, da Silveira Carvalho M (2014) Polygonal finite elements for incompressible fluid flow. Int J Numer Methods Fluids 74(2):134– 151. doi: 10.1002/fld.3843 MathSciNetCrossRefGoogle Scholar
- Wiker N, Klarbring A, Borrvall T (2007) Topology optimization of regions of Darcy and Stokes flow. Int J Numer Methods Eng 69(7):1374–1404. doi: 10.1002/nme.1811 MathSciNetCrossRefzbMATHGoogle Scholar