Structural and Multidisciplinary Optimization

, Volume 42, Issue 4, pp 495–516 | Cite as

Topology optimization of flexible micro-fluidic devices

  • Sebastian Kreissl
  • Georg Pingen
  • Anton Evgrafov
  • Kurt MauteEmail author
Research Paper


A multi-objective topology optimization formulation for the design of dynamically tunable fluidic devices is presented. The flow is manipulated via external and internal mechanical actuation, leading to elastic deformations of flow channels. The design objectives characterize the performance in the undeformed and deformed configurations. The layout of fluid channels is determined by material topology optimization. In addition, the thickness distribution, the distribution of active material for internal actuation, and the support conditions are optimized. The coupled fluid-structure response is predicted by a non-linear finite element model and a hydrodynamic lattice Boltzmann method. Focusing on applications with low flow velocities and pressures, structural deformations due to fluid-forces are neglected. A mapping scheme is presented that couples the material distributions in the structural and fluid mesh. The governing and the adjoint equations of the resulting fluid-structure interaction problem are derived. The proposed method is illustrated with the design of tunable manifolds.


Fluid-structure interaction Hydrodynamic lattice Boltzmann method Non-linear elasticity Adjoint sensitivity analysis 



The authors acknowledge the support of the National Science Foundation under grant DMI-0348759. The opinions and conclusions presented in this chapter are those of the authors and do not necessarily reflect the views of the sponsoring organization.


  1. Aage N, Poulsen TH, Gersborg-Hansen A, Sigmund O (2008) Topology optimization of large scale Stokes flow problems. Struct Multidiscipl Optim 35(2):175–180. doi: 10.1007/s00158-007-0128-0 CrossRefMathSciNetGoogle Scholar
  2. Andreasen SC, Gersborg AR, Sigmund O (2009) Topology optimization of microfluidic mixers. Int J Numer Methods Fluids 61(5):498–513. doi: 10.1002/fld.1964 zbMATHCrossRefMathSciNetGoogle Scholar
  3. Babuška I (1973) The finite element method with penalty (variational principle with penalty for finite element solution of model Poisson equation with homogeneous Dirichlet boundary conditions, noting convergence). Math Comput 27:221–228zbMATHGoogle Scholar
  4. Bar-Cohen Y (2004) Electroactive polymer (EAP) actuators as artificial muscles—reality, potential, and challenges. Bellingham SPIE—The International Society for Optical EngineeringGoogle Scholar
  5. Belytschko T, Liu WK, Moran B (2005) Nonlinear finite elements for continua and structures. Wiley, New YorkGoogle Scholar
  6. Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, HeidelbergGoogle Scholar
  7. 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 zbMATHCrossRefMathSciNetGoogle Scholar
  8. Buhl T (2002) Simultaneous topology optimization of structure and supports. Struct Multidiscipl Optim 23(5):336–346. doi: 10.1007/s00158-002-0194-2 CrossRefGoogle Scholar
  9. Chen S, Doolen GD (1998) Lattice Boltzmann method for fluid flows. Annu Rev Fluid Mech 30:329–364CrossRefMathSciNetGoogle Scholar
  10. Dimitrov D, Schreve K, de Beer N (2006) Advances in three dimensional printing–state of the art and future perspectives. Rapid Prototyping J 12(3):136–147CrossRefGoogle Scholar
  11. Evgrafov A (2006) Topology optimization of slightly compressible fluids. ZAMM 86(1):46–62. doi: 10.1002/zamm.200410223 zbMATHCrossRefMathSciNetGoogle Scholar
  12. Evgrafov A, Pingen G, Maute K (2008) Topology optimization of fluid domains: kinetic theory approach. ZAMM 88(2):129–141. doi: 10.1002/zamm.200700122 zbMATHCrossRefMathSciNetGoogle Scholar
  13. Gersborg-Hansen A, Sigmund O, Haber RB (2005) Topology optimization of channel flow problems. Struct Multidiscipl Optim 30(3):181–192. doi: 10.1007/s00158-004-0508-7 CrossRefMathSciNetGoogle Scholar
  14. Guest JK, Prévost JH (2006) Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability. Int J Solids Struct 43(22–23):7028–7047. doi: 10.1016/j.ijsolstr.2006.03.001 zbMATHCrossRefGoogle Scholar
  15. Kim H, Lee HBR, Maeng WJ (2009) Applications of atomic layer deposition to nanofabrication and emerging nanodevices. Thin Solid Films 517(8):2563–2580CrossRefGoogle Scholar
  16. Klimetzek FR, Paterson J, Moos O (2006) Autoduct: topology optimization for fluid flow. In: Proceedings of Konferenz für angewandte Optimierung. Karlsruhe, GermanyGoogle Scholar
  17. Maute K, Allen M (2004) Conceptual design of aeroelastic structures by topology optimization. Struct Multidiscipl Optim 27:27–42CrossRefGoogle Scholar
  18. Maute K, Reich GW (2006) Integrated multidisciplinary topology optimization approach to adaptive wing design. AIAA J Aircr 43(1):253–263Google Scholar
  19. Moos O, Klimetzek FR, Rossmann R (2004) Bionic optimization of air-guiding systems. In: Proceedings of SAE 2004 world congress & exhibition. Detroit, MIGoogle Scholar
  20. Niklaus F, Stemme G, Lu J-Q, Gutmann RJ (2006) Adhesive wafer bonding. J Appl Phys 99(3):031101CrossRefGoogle Scholar
  21. Othmer C (2008) A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows. Int J Numer Methods Fluids 58(8):861–877. doi: 10.1002/fld.1770 zbMATHCrossRefMathSciNetGoogle Scholar
  22. Othmer C, Klimetzek T, Giering R (2006) Computation of topological sensitivities in fluid dynamics: cost function versatility. In: Proceedings of ECCOMAS CFD. Delft, NetherlandsGoogle Scholar
  23. Pingen G (2008) Optimal design for fluidic systems: topology and shape optimization with the lattice Boltzmann method. PhD thesis, University Of Colorado at BoulderGoogle Scholar
  24. Pingen G, Evgrafov A, Maute K (2006) Towards the topology optimization of fluid-structure interaction problems with immersed boundary techniques. In: NSF design, service, and manufacturing grantees and research conference, St. Louis, MissouriGoogle Scholar
  25. Pingen G, Evgrafov A, Maute K (2007a) Topology optimization of flow domains using the lattice Boltzmann method. Struct Multidiscipl Optim 36(6):507–524. doi: 10.1007/s00158-007-0105-7 CrossRefMathSciNetGoogle Scholar
  26. Pingen G, Waidmann M, Evgrafov A, Maute K (2007b) Application of a parametric level-set approach to topology optimization fluid with the Navier–Stokes and lattice Boltzmann equations. In: Proceedings of the 7th world congress of structural and multidisciplinary optimization, 21–25 May 2007, Seoul, Korea, ISSMOGoogle Scholar
  27. Pingen G, Evgrafov A, Maute K (2009a) Adjoint parameter sensitivity analysis for the hydrodynamic lattice Boltzmann method with applications to design optimization. Comput Fluids 38(4):910–923. doi: 10.1016/j.compfluid.2008.10.002 CrossRefGoogle Scholar
  28. Pingen G, Waidmann M, Evgrafov A, Maute K (2009b) A parametric level-set approach for topology optimization of flow domains. Struct Multidiscipl Optim. doi: 10.1007/s00158-009-0405-1 Google Scholar
  29. Ramm E, Maute K, Schwarz S (1998a) Adaptive topology and shape optimization. In: Proceedings of 4th world congress on computational mechanics, 29 June–2 July. Mendoza, Argentina, pp 19–38Google Scholar
  30. Ramm E, Maute K, Schwarz S (1998b) Conceptual design by structural optimization. In: Proceedings of EURO-C, 31 March–3 April. Badgastein, Austria, pp 879–896Google Scholar
  31. Spaid MAA, Phelan FR (1997) Lattice Boltzmann methods for modeling microscale flow in fibrous porous media. Phys Fluids 9(9):2468–2474zbMATHCrossRefMathSciNetGoogle Scholar
  32. Stadler W (1988) Multicriteria optimization in engineering and in the sciences. Springer, HeidelbergzbMATHGoogle Scholar
  33. Succi S (2001) The lattice Boltzmann equation: for fluid dynamics and beyond. Oxford University Press, OxfordzbMATHGoogle Scholar
  34. Svanberg K (1995) A globally convergent version of MMA without linesearch. In: Proceedings of the first world congress of structural and multidisciplinary optimization, 28 May–2 June 1995, pp 9–16, Goslar, GermanyGoogle Scholar
  35. Yoon GH (2009) Topology optimization for stationary fluid-structure interaction problems using a new monolithic formulation. In: Proceedings of 8th world congress on structural and multidisciplinary optimization, Lisbon, PortugalGoogle Scholar
  36. Yu D, Mei R, Luo LS, Shyy W (2003) Viscous flow computations with the method of lattice Boltzmann equation. Prog Aerosp Sci 39(5):329–367. doi: 10.1016/S0376-0421(03)00003-4 CrossRefGoogle Scholar
  37. Zhang XQ, Lowe C, Wissler M, Jahne B, Kovacs G (2005) Dielectric elastomers in actuator technology. Adv Eng Mater 7(5):361–367CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Sebastian Kreissl
    • 1
  • Georg Pingen
    • 2
  • Anton Evgrafov
    • 3
  • Kurt Maute
    • 1
    Email author
  1. 1.Center for Aerospace StructuresUniversity of Colorado at BoulderBoulderUSA
  2. 2.Department of Mechanical and Aerospace EngineeringUniversity of Colorado at Colorado SpringsColorado SpringsUSA
  3. 3.Department of MathematicsTechnical University of DenmarkLyngbyDenmark

Personalised recommendations