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Structural and Multidisciplinary Optimization

, Volume 59, Issue 4, pp 1105–1124 | Cite as

A “poor man’s” approach to topology optimization of natural convection problems

  • Janus Asmussen
  • Joe Alexandersen
  • Ole Sigmund
  • Casper Schousboe AndreasenEmail author
Research Paper

Abstract

Topology optimization of natural convection problems is computationally expensive, due to the large number of degrees of freedom (DOFs) in the model and its two-way coupled nature. Herein, a method is presented to reduce the computational effort by use of a reduced-order model governed by simplified physics. The proposed method models the fluid flow using a potential flow model, which introduces an additional fluid property. This material property currently requires tuning of the model by comparison to numerical Navier-Stokes-based solutions. Despite the significant simplifications, hereunder neglecting viscous boundary layers, topology optimization based on the reduced-order model is shown to provide qualitatively similar designs, as those obtained using a full Navier-Stokes-based model. The number of DOFs is reduced by 50% in two dimensions and the computational complexity is evaluated to be approximately 12.5% of the full model. We further compare to optimized designs obtained utilizing Newton’s convection law.

Keywords

Topology optimization Natural convection Reduced-order model Potential flow Heat sink design 

Notes

Acknowledgements

The authors would like to thank the TopOpt group for fruitful discussions.

Funding information

The work has been partly funded by the TopTEN project granted by Independent Research Fund Denmark.

References

  1. Alexandersen J (2011) Topology optimisation of convection problems. B.Eng. thesis, Technical University of Denmark.  https://doi.org/10.13140/RG.2.2.24635.72485
  2. Alexandersen J (2013) Topology optimization of coupled conveciton problems. Master’s thesis, Technical University of DenmarkGoogle Scholar
  3. Alexandersen J (2016) Efficient topology optimisation of multiscale and multiphysics problems. PhD thesis, Technical University of DenmarkGoogle Scholar
  4. Alexandersen J, Aage N, Andreasen C S, Sigmund O (2014) Topology optimisation for natural convection problems. Int J Numer Methods Fluids 76(10):699–721.  https://doi.org/10.1002/fld.3954 MathSciNetCrossRefGoogle Scholar
  5. Alexandersen J, Sigmund O, Aage N (2015) Topology optimisation of passive coolers for light-emitting diode lamps. In: 11th world congress on structural and multidisciplinary optimization.  https://doi.org/10.13140/RG.2.1.3906.5446
  6. Alexandersen J, Sigmund O, Aage N (2016) Large scale three-dimensional topology optimisation of heat sinks cooled by natural convection. Int J Heat Mass Transf 100:876–891.  https://doi.org/10.1016/j.ijheatmasstransfer.2016.05.013 CrossRefGoogle Scholar
  7. Alexandersen J, Sigmund O, Meyer K, Lazarov B S (2018) Design of passive coolers for light-emitting diode lamps using topology optimisation. Int J Heat Mass Transf 122:138–149.  https://doi.org/10.1016/j.ijheatmasstransfer.2018.01.103 CrossRefGoogle Scholar
  8. Andreasen C S, Gersborg A R, Sigmund O (2009) Topology optimization of microfluidic mixers. Int J Numer Methods Fluids 61(5):498–513.  https://doi.org/10.1002/fld.1964 MathSciNetCrossRefzbMATHGoogle Scholar
  9. Angot P, Bruneau C H, Fabrie P (1999) A penalization method to take into account obstacles in incompressible viscous flows. Numer Math 81(4):497–520.  https://doi.org/10.1007/s002110050401 MathSciNetCrossRefzbMATHGoogle Scholar
  10. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224.  https://doi.org/10.1016/0045-7825(88)90086-2 MathSciNetCrossRefzbMATHGoogle Scholar
  11. Bendsøe MP, Sigmund O (2003) Topology optimization—theory, methods, and applications. Springer, BerlinzbMATHGoogle Scholar
  12. Borrvall T, Petersson J (2003) Topology optimization of fluids in Stokes flow. Int J Numer Methods Fluids 41(1):77–107.  https://doi.org/10.1002/fld.426 MathSciNetCrossRefzbMATHGoogle Scholar
  13. Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50 (9):2143–2158.  https://doi.org/10.1002/nme.116 MathSciNetCrossRefzbMATHGoogle Scholar
  14. Brinkman H C (1947) A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl Sci Res Sect A-Mech Heat Chem Eng Math Methods 1(1):27–34zbMATHGoogle Scholar
  15. Brooks A N, Hughes T J (1982) Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 32 (1):199–259.  https://doi.org/10.1016/0045-7825(82)90071-8 MathSciNetCrossRefzbMATHGoogle Scholar
  16. Bruns T E (2007) Topology optimization of convection-dominated, steady-state heat transfer problems. Int J Heat Mass Transf 50 (15–16):2859–2873.  https://doi.org/10.1016/j.ijheatmasstransfer.2007.01.039 CrossRefzbMATHGoogle Scholar
  17. Bruns T E, Tortorelli D A (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26-27):3443–3459.  https://doi.org/10.1016/S0045-7825(00)00278-4 CrossRefzbMATHGoogle Scholar
  18. Coffin P, Maute K (2016a) A level-set method for steady-state and transient natural convection problems. Struct Multidiscip Optim 53(5):1047–1067.  https://doi.org/10.1007/s00158-015-1377-y MathSciNetCrossRefGoogle Scholar
  19. Coffin P, Maute K (2016b) Level set topology optimization of cooling and heating devices using a simplified convection model. Struct Multidiscip Optim 53(5):985–1003.  https://doi.org/10.1007/s00158-015-1343-8 CrossRefGoogle Scholar
  20. Deaton J D, Grandhi R V (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49 (1):1–38.  https://doi.org/10.1007/s00158-013-0956-z MathSciNetCrossRefGoogle Scholar
  21. Dede E (2009) Multiphysics topology optimization of heat transfer and fluid flow systems. In: Proceedings of the COMSOL Conference 2009, BostonGoogle Scholar
  22. Dilgen S B, Dilgen C B, Fuhrman D R, Sigmund O, Lazarov B S (2018) Density based topology optimization of turbulent flow heat transfer systems. Struct Multidiscip Optim 57(5):1905–1918.  https://doi.org/10.1007/s00158-018-1967-6 MathSciNetCrossRefGoogle Scholar
  23. Donea J, Huerta A (2003) Finite element methods for flow problems. Wiley, ChichesterCrossRefGoogle Scholar
  24. Donoso A, Sigmund O (2004) Topology optimization of multiple physics problems modelled by Poisson’s equation. Latin Am J Solids Struct 1(2):169–189Google Scholar
  25. Dugast F, Favennec Y, Josset C, Fan Y, Luo L (2018) Topology optimization of thermal fluid flows with an adjoint Lattice Boltzmann Method. J Comput Phys 365:376–404.  https://doi.org/10.1016/J.JCP.2018.03.040 MathSciNetCrossRefzbMATHGoogle Scholar
  26. Evgrafov A (2006) Topology optimization of slightly compressible fluids. ZAMM 86(1):46–62.  https://doi.org/10.1002/zamm.200410223 MathSciNetCrossRefzbMATHGoogle Scholar
  27. Fries T P, Matthies H G (2004) A review of Petrov-Galerkin stabilization approaches and an extension to meshfree methods. Tech. rep., Institute of Scientific Computing, Technical University Braunschweig, BraunschweigGoogle Scholar
  28. Gersborg-Hansen A, Sigmund O, Haber R (2005) Topology optimization of channel flow problems. Struct Multidiscip Optim 30(3):181–192.  https://doi.org/10.1007/s00158-004-0508-7 MathSciNetCrossRefzbMATHGoogle Scholar
  29. Gersborg-Hansen A, Bendsøe MP, Sigmund O (2006) Topology optimization of heat conduction problems using the finite volume method. Struct Multidiscip Optim 31(4):251–259.  https://doi.org/10.1007/s00158-005-0584-3. arXiv:1011.1669v3 MathSciNetCrossRefzbMATHGoogle Scholar
  30. Guest J K, Prévost J H (2006) Topology optimization of creeping fluid flows using a Darcy–Stokes finite element. Int J Numer Methods Eng 66(3):461–484.  https://doi.org/10.1002/nme.1560 MathSciNetCrossRefzbMATHGoogle Scholar
  31. Haertel J H, Nellis G F (2017) A fully developed flow thermofluid model for topology optimization of 3d-printed air-cooled heat exchangers. Appl Therm Eng 119:10–24.  https://doi.org/10.1016/j.applthermaleng.2017.03.030 CrossRefGoogle Scholar
  32. Haertel J H K, Engelbrecht K, Lazarov B S, Sigmund O (2015) Topology optimization of thermal heat sinks. In: Proceedings of COMSOL conference 2015Google Scholar
  33. Haertel J H, Engelbrecht K, Lazarov B S, Sigmund O (2018) Topology optimization of a pseudo 3D thermofluid heat sink model. Int J Heat Mass Transf 121:1073–1088.  https://doi.org/10.1016/j.ijheatmasstransfer.2018.01.078 CrossRefGoogle Scholar
  34. Iga A, Nishiwaki S, Izui K, Yoshimura M (2009) Topology optimization for thermal conductors considering design-dependent effects, including heat conduction and convection. Int J Heat Mass Transf 52(11–12):2721–2732.  https://doi.org/10.1016/J.IJHEATMASSTRANSFER.2008.12.013 CrossRefzbMATHGoogle Scholar
  35. Joo Y, Lee I, Kim S J (2017) Topology optimization of heat sinks in natural convection considering the effect of shape-dependent heat transfer coefficient. Int J Heat Mass Transf 109:123–133.  https://doi.org/10.1016/j.ijheatmasstransfer.2017.01.099 CrossRefGoogle Scholar
  36. Joo Y, Lee I, Kim S J (2018) Efficient three-dimensional topology optimization of heat sinks in natural convection using the shape-dependent convection model. Int J Heat Mass Transf 127:32–40.  https://doi.org/10.1016/J.IJHEATMASSTRANSFER.2018.08.009 CrossRefGoogle Scholar
  37. Koga A A, Lopes E C C, Nova H F V (2013) Development of heat sink device by using topology optimization. Int J Heat Mass Transf 64: 759–772.  https://doi.org/10.1016/j.ijheatmasstransfer.2013.05.007 CrossRefGoogle Scholar
  38. Laniewski-Wollk L, Rokicki J (2016) Adjoint lattice Boltzmann for topology optimization on multi-GPU architecture. Comput Math Appl 71(3):833–848.  https://doi.org/10.1016/j.camwa.2015.12.043 MathSciNetCrossRefzbMATHGoogle Scholar
  39. Lazarov B S, Alexandersen J, Sigmund O (2014) Topology optimized designs of steady state conduction heat transfer problems with convection boundary conditions. In: EngOpt 2014.  https://doi.org/10.13140/RG.2.2.29361.68966
  40. Lazarov B S, Sigmund O, Meyer K, Alexandersen J (2018) Experimental validation of additively manufactured optimized shapes for passive cooling. Appl Energy 226:330–339.  https://doi.org/10.1016/j.apenergy.2018.05.106 CrossRefGoogle Scholar
  41. Lei T, Alexandersen J, Lazarov B S, Wang F, Haertel J H, Angelis S D, Sanna S, Sigmund O, Engelbrecht K (2018) Investment casting and experimental testing of heat sinks designed by topology optimization. Int J Heat Mass Transf 127:396–412.  https://doi.org/10.1016/j.ijheatmasstransfer.2018.07.060 CrossRefGoogle Scholar
  42. Marck G, Nemer M, Harion J L (2013) Topology optimization of heat and mass transfer problems: Laminar flow. Numer Heat Transf Part B: Fundam 63(6):508–539.  https://doi.org/10.1080/10407790.2013.772001 CrossRefzbMATHGoogle Scholar
  43. Moon H, Kim C, Wang S (2004) Reliability-based topology optimization of thermal systems considering convection heat transfer. In: Proceedings of the 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization ConferenceGoogle Scholar
  44. Okkels F, Bruus H (2007) Scaling behavior of optimally structured catalytic microfluidic reactors. Phys Rev E 75(1):016,301.  https://doi.org/10.1103/PhysRevE.75.016301 CrossRefGoogle Scholar
  45. Olesen L H, Okkels F, Bruus H (2006) A high-level programming-language implementation of topology optimization applied to steady-state Navier-Stokes flow. Int J Numer Methods Eng 65(7):975–1001MathSciNetCrossRefzbMATHGoogle Scholar
  46. Rodrigues H, Fernandes P (1995) A material based model for topology optimization of thermoelastic structures. Int J Numer Methods Eng 38(12):1951–1965.  https://doi.org/10.1002/nme.1620381202 MathSciNetCrossRefzbMATHGoogle Scholar
  47. Saglietti C (2018) On optimization of natural convection flows. PhD thesis, KTH Royal Institute of Technology, iSBN: 978-91-7729-820-5Google Scholar
  48. Saglietti C, Wadbro E, Berggren M, Henningson DS (2018) Heat transfer maximization in a three-dimensional conductive differentially heated cavity by means of topology optimization. In: Proceedings of the Seventh European Conference on Computational Fluid Dynamics (ECCM-ECFD)Google Scholar
  49. Shakib F, Hughes T J, Johan Z (1991) A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations. Comput Methods Appl Mech Eng 89(1–3):141–219.  https://doi.org/10.1016/0045-7825(91)90041-4 MathSciNetCrossRefGoogle Scholar
  50. Sigmund O (2001) Design of multiphysics actuators using topology optimization—part I: one-material structures. Comput Methods Appl Mech Eng 190(49–50):6577–6604.  https://doi.org/10.1016/S0045-7825(01)00251-1 CrossRefzbMATHGoogle Scholar
  51. Subramaniam V, Dbouk T, Harion J L (2018) Topology optimization of conductive heat transfer devices: an experimental investigation. Appl Therm Eng 131:390–411.  https://doi.org/10.1016/J.APPLTHERMALENG.2017.12.026 CrossRefGoogle Scholar
  52. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373.  https://doi.org/10.1002/nme.1620240207 MathSciNetCrossRefzbMATHGoogle Scholar
  53. Thellner M (2005) Multi-parameter topology optimization in continuum mechanics. PhD thesis, Linköping University, The Institute of TechnologyGoogle Scholar
  54. 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.  https://doi.org/10.1002/nme.1811 MathSciNetCrossRefzbMATHGoogle Scholar
  55. Yaji K, Yamada T, Yoshino M, Matsumoto T, Izui K, Nishiwaki S (2016) Topology optimization in thermal-fluid flow using the lattice Boltzmann method. J Comput Phys 307:355–377.  https://doi.org/10.1016/j.jcp.2015.12.008 MathSciNetCrossRefzbMATHGoogle Scholar
  56. Yaji K, Ogino M, Chen C, Fujita K (2018) Large-scale topology optimization incorporating local-in-time adjoint-based method for unsteady thermal-fluid problem. Struct Multidiscip Optim 58(2):817–822.  https://doi.org/10.1007/s00158-018-1922-6 MathSciNetCrossRefGoogle Scholar
  57. Yamada T, Izui K, Nishiwaki S (2011) A level set-based topology optimization method for maximizing thermal diffusivity in problems including design-dependent effects. ASME J Mech Des 133(3):1–9.  https://doi.org/10.1115/1.4003684 CrossRefGoogle Scholar
  58. Yan S, Wang F, Sigmund O (2018) On the non-optimality of tree structures for heat conduction. Int J Heat Mass Transf 122:660–680.  https://doi.org/10.1016/j.ijheatmasstransfer.2018.01.114 CrossRefGoogle Scholar
  59. Yin L, Ananthasuresh G (2002) A novel topology design scheme for the multi-physics problems of electro-thermally actuated compliant micromechanisms. Sens Actuators, A 97–98:599–609.  https://doi.org/10.1016/S0924-4247(01)00853-6 CrossRefGoogle Scholar
  60. Yoon G H (2010) Topological design of heat dissipating structure with forced convective heat transfer. J Mech Sci Technol 24(6):1225–1233.  https://doi.org/10.1007/s12206-010-0328-1 CrossRefGoogle Scholar
  61. Zeng S, Kanargi B, Lee P S (2018) Experimental and numerical investigation of a mini channel forced air heat sink designed by topology optimization. Int J Heat Mass Transf 121:663–679.  https://doi.org/10.1016/j.ijheatmasstransfer.2018.01.039 CrossRefGoogle Scholar
  62. Zhao X, Zhou M, Sigmund O, Andreasen C S (2018) A “poor man’s approach” to topology optimization of cooling channels based on a Darcy flow model. Int J Heat Mass Transf 116:1108–1123.  https://doi.org/10.1016/j.ijheatmasstransfer.2017.09.090 CrossRefGoogle Scholar
  63. Zhou M, Alexandersen J, Sigmund O, W Pedersen C B (2016) Industrial application of topology optimization for combined conductive and convective heat transfer problems. Struct Multidiscip Optim 54 (4):1045–1060.  https://doi.org/10.1007/s00158-016-1433-2 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Section for Solid Mechanics, Department of Mechanical EngineeringTechnical University of DenmarkKongens LyngbyDenmark

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